The Cardassian expansion revisited: constraints from updated Hubble parameter measurements and type Ia supernova data

The Cardassian expansion revisited: constraints from updated Hubble parameter measurements and... Abstract Motivated by an updated compilation of observational Hubble data (OHD) that consist of 51 points in the redshift range of 0.07 < z < 2.36, we study an interesting model known as Cardassian that drives the late cosmic acceleration without a dark energy component. Our compilation contains 31 data points measured with the differential age method by Jimenez & Loeb (2002), and 20 data points obtained from clustering of galaxies. We focus on two modified Friedmann equations: the original Cardassian (OC) expansion and the modified polytropic Cardassian (MPC). The dimensionless Hubble, E(z), and the deceleration parameter, q(z), are revisited in order to constrain the OC and MPC free parameters, first with the OHD and then contrasted with recent observations of type Ia supernova (SN Ia) using the compressed and full joint-light-analysis (JLA) samples (Betoule et al.). We also perform a joint analysis using the combination OHD plus compressed JLA. Our results show that the OC and MPC models are in agreement with the standard cosmology and naturally introduce a cosmological-constant-like extra term in the canonical Friedmann equation with the capability of accelerating the Universe without dark energy. cosmological parameters, dark energy, cosmology: observations 1 INTRODUCTION The cold dark matter (ΛCDM) with a cosmological constant (CC) model is the cornerstone of modern cosmology. It has shown an unprecedented success predicting and reproducing the dynamics and evolution of the Universe. ΛCDM is based on two important but unknown components, dark matter (DM) and dark energy (DE), which constitute ∼ 96 per cent of the total content of our Universe (Planck Collaboration XIII 2016). In this standard paradigm, the DE, responsible of the late cosmic acceleration, is supplied by a CC, which is associated with vacuum energy. Although several cosmological observations favour the CC, some theoretical problems arise when we try a renormalization of the quantum vacuum fluctuations using an appropriate cut-off at the Planck energy. However, the problem becomes insurmountable, giving a difference of ∼120 orders in magnitude between theory and observations (Weinberg 1989; Zeldovich 1968). In addition, the problem of coincidence, i.e. the similitude between the energy density of matter and DE at the present epoch, remains as an open question (Zeldovich 1968; Weinberg 1989). To overcome these problems, several alternatives to the CC are proposed, such as quintessence, phantom energy, Chaplygin gas, holographic DE, and Galileons (see Carroll 2001; Copeland, Sami & Tsujikawa 2006, for a complete review). Geometrical approaches are also used to explain the DE dynamics (i.e. brane theories) like Dvali, Gabadaze and Porrati (DGP; Deffayet, Dvali & Gabadadze 2002), Randall-Sundrum I and II (RSI, RSII; Randall & Sundrum 1999a,b), or f(R) theories (Buchdahl 1970; Starobinsky 1980; Cembranos 2009); each one having important pros and cons. An interesting alternative, closely related to geometrical models, is the Cardassian expansion model for which there is no DE and the late cosmic acceleration is driven by the modification of the Friedmann equation as H2 = f(ρ) (Xu 2012), where f(ρ) is a functional form of the energy density of the Universe. Freese & Lewis (2002) proposed f(ρ) ∝ ρ + ρn in order to obtain a late acceleration stage under certain conditions on the n parameter, naming the model as the Cardassian expansion1 [hereafter the original Cardassian (OC) model]. However, this expression can be naturally deduced from extra dimensional theories (DGP, RSI, RSII, etc.), which imprint the effects of a five-dimensional space–time (the bulk) in our four-dimensional (4D) space–time (the brane) at cosmological scales. In the case of the DGP model, a consequence of this kind of geometry is a density parameter that evolves as $$(\sqrt{\rho +\alpha }+\beta )^2$$, where α and β are constants related to the threshold between the brane and the bulk, allowing an accelerated epoch driven only by geometry. In the case of RS models, a quadratic term in the energy–momentum tensor modifies the right-hand side of the Friedmann equation as aρ + bρ2 (Shiromizu, Maeda & Sasaki 2000), with a correspondence to the Cardassian models when n = 2. Thus, the topological structure of the brane and the bulk can naturally produce the Cardassian Friedmann equation. Indeed, it is possible to obtain a n-energy–momentum tensor from a Gauss equation with a product of n-extrinsic curvatures, which leads to the ρn extra term in the Friedmann equation of the OC model. Therefore, the model motivation is based on extra dimensions arising from a fundamental theory (for an excellent review of extra dimensions models, see for instance Maartens 2004, or Maartens 2000, for a cosmological point of view). Another alternative interpretation is to consider a fluid (that may or may not be in an intrinsically 4D metric) with an extra contribution to the energy–momentum tensor (Gondolo & Freese 2003). Both interpretations are interesting and the standard cosmological dynamics can be mimicked without the need to postulate a DE component. In addition, we notice that it is possible to recover a CC when ρn → 1, without adding it by hand. An OC model generalization can be obtained by considering an additional exponent in the right-hand side of the Friedmann equation as f(ρ) ∝ ρ(1 + ρl(n−1))1/l that is called modified polytropic Cardassian (MPC) model by analogy with a fluid interpretation (Gondolo & Freese 2002). The Cardassian models are extensively studied in the literature. They have been tested with several cosmological observations (see e.g. Wang et al. 2003; Feng & Li 2010; Liang, Wu & Zhu 2011; Li, Wu & Yu 2012; Xu 2012; Wei, Ma & Wu 2015, and references therein). Wei et al. (2015) put constraints on the OC model parameters using a joint analysis of gamma-ray burst data and type Ia supernovae (SNe Ia) of the Union 2.1 sample (Suzuki et al. 2012). Recently, Magaña et al. (2015) used the strong lensing measurements in the galaxy cluster Abell 1689, baryon acoustic oscillations (BAOs), cosmic microwave background (CMB) data from 9 yr Wilkinson Microwave Anisotropy Probe (WMAP) observations (Hinshaw et al. 2013), and the SN Ia LOSS sample (Ganeshalingam, Li & Filippenko 2013) to constrain the MPC parameters. In this work, we revisit the Cardassian expansion models with an universe that contains baryons, DM, together with the radiation component. We explore two functional forms of the Friedmann equation: one with the OC parameter n (following Freese & Lewis 2002), and the other one considering also the l exponent (following Gondolo & Freese 2003). These Cardassian models are tested using an update sample of observational Hubble data (OHD) and the compressed joint-light-analysis (cJLA) SN Ia data by Betoule et al. (2014). As a final comment, while we were finalizing this paper, Zhai et al. (2017a) addressed a similar revision of the Cardassian models. While the main focus of Zhai et al. (2017a) is to match the Cardassian Friedmann equations to $$f(T, \mathcal {T})$$ theory through the action principle, our work focus on providing bounds to the Cardassian models using OHD (see also Zhai et al. 2017b). Nonetheless, the authors also provide constraints derived from SN Ia, BAO, and CMB data. The paper is organized as follows. In Section 2, the Cardassian cosmology is revisited, introducing two proposals for the Friedmann equation, which correspond to the OC and MPC models, and the deceleration parameter is calculated. In Section 3, we present the data and methodology in order to study the Cardassian models using OHD and SN Ia observations. In Section 4, we show the constraints for the free parameters presenting the novel contrast with the updated sample. Finally, Section 5 presents our conclusions and the possible outlooks into future studies. We will henceforth use units in which c = ℏ = 1 (unless explicitly written). 2 THE CARDASSIAN COSMOLOGY 2.1 OC model The OC model was introduced by Freese & Lewis (2002) to explain the accelerated expansion of the Universe without DE. Motivated by braneworld theory, this model modifies the Friedmann equation as   $$H^{2}=\frac{8\pi G \rho _{t}}{3} + B\rho _{t}^{n},$$ (1)where $$H=\dot{a}/a$$ is the Hubble parameter, a is the scale factor of the Universe, G is the Newtonian gravitational constant, B is a dimensional coupling constant that depends on the theory, and the total matter density is ρt = ρm + ρr. The recent Planck measurements (Planck Collaboration XVI 2014; Planck Collaboration XIII 2016) suggest a curvature energy density Ωk ≃ 0, thus we assume a flat geometry. The conservation equation is maintained in the traditional form:   $$\dot{\rho }+3H(\rho +p)=0.$$ (2)The matter density (DM and baryons), ρm = ρm0a−3, and the radiation density, ρr = ρr0a−4, evolution can be computed from equation (2). The second term in the right-hand side of equation (1), known as the Cardassian term, drives the universe to an accelerated phase if the exponent n satisfies n < 2/3. At early times, this corrective term is negligible and the dynamics of the universe is governed by the canonical term of the Friedmann equation. When the universe evolves, the traditional energy density and the one due to the Cardassian correction becomes equal at redshift $$z_{\text{Card}}\sim \mathcal {O}(1)$$. Later on, the Cardassian term begins to dominate the evolution of the universe and source the cosmic acceleration. The equation (1) reproduces the ΛCDM model for n = 0. As in the standard case, it is possible to define a new critical density for the OC model, ρNc, which satisfies the equation (1) and can be written as ρNc = ρcF(B, n), where ρc = 3H2/8πG is the standard critical density, and F(B, n) is a function that depends on the OC parameters and the components of the Universe. The Raychaudhuri equation can be written in the form:   $$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho _t+3p_t)-B\left[\left(\frac{3n}{2}-1\right)\rho _t^n+\frac{3}{2}n\rho _t^{n-1}p_t\right]\!,$$ (3)where equations (1) and (2) were used. From equation (1), it is possible to obtain the dimensionless Hubble parameter $$E^2(z)\equiv H^2(z)/H^2_0$$ as   $$E(z, {\mathbf {\boldsymbol\Theta }})^{2}= \Omega _{\mathrm{std}}+(1-\Omega _{m0}-\Omega _{r0})\left[\frac{\Omega _{\mathrm{std}}}{\Omega _{m0}+\Omega _{r0}}\right]^{n},$$ (4)where $${\mathbf {\boldsymbol\Theta} }=(\Omega _{m0},h,n)$$ is the free parameter vector to be constrained by the data, Ωr0 = ρr0/ρc is the current standard density parameter for the radiation component, Ωm0 = ρm0/ρc is the observed standard density parameter for matter (baryons and DM), and we define Ωstd ≡ Ωm0(1 + z)3 + Ωr0(1 + z)4. We compute Ωr0 = 2.469 × 10−5h−2(1 + 0.2271Neff) (Komatsu et al. 2011), where Neff = 3.04 is the standard number of relativistic species (Mangano et al. 2002). Notice that we have also imposed a flatness condition on the total content of the Universe (for further details on how to obtain equation (4), see Sen & Sen 2003a,b). The deceleration parameter, defined as $$q\equiv -\ddot{a}/aH^2$$, can be written as   \begin{eqnarray} q(z, \mathbf {\boldsymbol\Theta })&=&\frac{\Omega _{\mathrm{std}}-\frac{1}{2}\Omega _{m0}(1+z)^3}{E^2(z, \mathbf {\boldsymbol\Theta })}+\frac{1-\Omega _{m0}-\Omega _{r0}}{(\Omega _{m0}+\Omega _{r0})^n}\nonumber \\ &&\times\left[\left(\frac{3n}{2}-1\right)+\frac{n\Omega _{r0}}{2\Omega _{\text{std}}}(1+z)^4\right]\frac{\Omega _{\mathrm{std}}^n}{E^2(z, \mathbf {\boldsymbol\Theta })}. \end{eqnarray} (5) In order to investigate whether the OC model can drive the late cosmic acceleration, it is necessary to reconstruct the q(z) using the mean values for the $$\mathbf {\boldsymbol\Theta}$$ parameters. 2.2 MPC model Gondolo & Freese (2002, 2003) introduced a simple generalization of the Cardassian model, the MPC, by introducing an additional exponent l (see also Wang et al. 2003). The modified Friedmann equation with this generalization can be written as   $$H^{2} = \frac{8\pi G}{3} \rho _{t} \beta ^{1/l},$$ (6)where   $$\beta \equiv 1 + \left( \frac{\rho _{\text{Card}}}{\rho _{t}} \right)^{l(1-n)},$$ (7)and ρCard is the characteristic energy density, with n < 2/3 and l > 0. In concordance with the previous Friedmann equation (1) and following Planck Collaboration XVI (2014); Planck Collaboration XIII (2016), we also assume Ωk ≃ 0. The equation (6) reproduce the ΛCDM model for l = 1 and n = 0. Thus, the acceleration equation is   $$\frac{\ddot{a}}{a}= -\frac{4\pi G}{3}\rho _{t}\beta ^{1/l}+4\pi G(1-n)\rho _{t}\left(1-\frac{1}{\beta }\right)\beta ^{1/l}.$$ (8)The MPC model (equation 9) has been studied by several authors using different data with ρt = ρm (see e.g. Feng & Li 2010) and also with ρt = ρm + ρr together with a curvature term (Shi, Huang & Lu 2012). Here, we consider a flat MPC with matter and radiation components. After straightforward calculations, the dimensionless $$E^{2}(z, \boldsymbol {\boldsymbol\Theta })$$ parameter reads as   \begin{eqnarray} &&E^{2}(z, \boldsymbol {\boldsymbol\Theta })=\Omega _{r0}(1+z)^{4} + \Omega _{m0}(1+z)^{3}\beta (z, \boldsymbol {\boldsymbol\Theta })^{1/l}, \quad \end{eqnarray} (9)where   $$\beta (z, \boldsymbol {\boldsymbol\Theta })\equiv 1 + \left[ \left( \frac{1-\Omega _{r0}}{\Omega _{m0}}\right)^l - 1 \right](1+z)^{3l(n - 1)},$$ (10)being $$\boldsymbol {\boldsymbol\Theta }=(\Omega _{m0},h,l,n)$$, the free parameter vector to be fitted by the data. In addition, $$q(z, \boldsymbol {\boldsymbol\Theta })$$ can be written as   \begin{eqnarray} q(z, \mathbf {\boldsymbol\Theta })&=& \frac{\Omega _{m0}\beta (z, \mathbf {\boldsymbol\Theta })^{1/l}}{2E^2(z, \mathbf {\boldsymbol\Theta })}\left[1-3(1-n)\left(1-\frac{1}{\beta (z, \mathbf {\boldsymbol\Theta })}\right)\right] \nonumber \\ &&\times\,(1+z)^3+\frac{\Omega _{r0}}{E^2(z, \mathbf {\boldsymbol\Theta })}(1+z)^4. \end{eqnarray} (11)We use the $$\mathbf {\boldsymbol\Theta}$$ mean values in the last expression to reconstruct the deceleration parameter q(z) and investigate whether the MPC model is consistent with a late cosmic acceleration. 3 DATA AND METHODOLOGY The OC and MPC model parameters are constrained using an updated OHD sample, which contains 51 data points, and the compressed SN Ia data set from the JLA full sample by Betoule et al. (2014), which contains 31 data points. In the following, we briefly introduce these data sets. 3.1 Observational Hubble data The ‘differential age’ (DA) method proposed by Jimenez & Loeb (2002) allows us to measure the expansion rate of the Universe at redshift z, i.e. H(z). This technique compares the ages of early-type galaxies (i.e. without ongoing star formation) with similar metallicity and separated by a small redshift interval (for instance, Moresco et al. 2012, measure Δz ∼ 0.04 at z < 0.4 and Δz ∼ 0.3 at z > 0.4). Thus, a H(z) point can be estimated using   $$H(z)=-\frac{1}{1+z}\frac{{\rm{d}z}}{{\rm {d}}t},$$ (12)where dz/dt is measured using the 4000 Å break (D4000) feature as function of redshift. A strong D4000 break depends on the metallicity and the age of the stellar population of the early-type galaxy. Thus, the technique by Jimenez & Loeb (2002) offers to directly measure the Hubble parameter using spectroscopic dating of passively-evolving galaxy to compare their ages and metallicities, providing H(z) measurements that are model independent. These H(z) points are given by different authors as Zhang et al. (2014), Moresco et al. (2012), Moresco (2015), Moresco et al. (2016) and Stern et al. (2010), and constitute the majority of our sample (31 points). In addition, we use 20 points from BAO measurements, although some of them being correlated because they either belong to the same analysis or there is overlapping among the galaxy samples; throughout this work, we assume that they are independent measurements. Moreover, some data points are biased because they are estimated using a sound horizon, rd,2 at the drag epoch, zd, which depends on the cosmological model (Melia & López-Corredoira 2017). Points provided by different authors use different values for the rd in clustering measurements, for instance Anderson et al. (2014) take 153.19 Mpc while Gaztanaga, Cabre & Hui (2009) choose 153.3 Mpc, etc. Table 1 shows an updated compilation of OHD accumulating a total of 51 points (other recent compilations are provided by Yu & Wang 2016; Zhang & Xia 2016; Farooq et al. 2017). We have included all the points of the previous references, although priority has been given to the measurements that comes from the DA method and have also been measured with clustering at the same redshift. As reference to compare our results, we also give the data point by Riess et al. (2016) who measured a Hubble constant H0 with 2.4 per cent of uncertainty. Authors argue that this improvement is due to a better calibration (using Cepheids) of the distance to 11 SN Ia host galaxies, reducing the error by almost 1 per cent. We use this sample to constrain the free parameters of the OC and MPC models and look for an alternative solution to the accelerated expansion of the Universe. The figure-of-merit for the OHD is given by   $$\chi _{\mbox{OHD}}^2 = \sum _{i=1}^{N_{\text{OHD}}} \frac{ \left[ H(z_{i}) -H_{\text{obs}}(z_{i})\right]^2 }{ \sigma _{H_i}^{2} },$$ (13) where NOHD is the number of the observational Hubble parameter Hobs(zi) at zi, $$\sigma _{H_i}$$ is its error, and H(zi) is the theoretical value for a given model. Table 1. 52 Hubble parameter measurements H(z) (in km s−1Mpc−1) and their errors, σH, at redshift z. The first point is not included in the Markov chain Monte Carlo analysis, it was only considered as a Gaussian prior in some tests. The method column refers as to how to H(z) was obtained: DA stands for differential age method, and clustering comes from BAO measurements. z  H(z)  σH  Reference  Method    (km s− 1 Mpc− 1)  (km s−1 Mpc−1)      0  73.24  1.74  Riess et al. (2016)  SN Ia/Cepheid  0.07  69  19.6  Zhang et al. (2014)  DA  0.1  69  12  Stern et al. (2010)  DA  0.12  68.6  26.2  Zhang et al. (2014)  DA  0.17  83  8  Stern et al. (2010)  DA  0.1791  75  4  Moresco et al. (2012)  DA  0.1993  75  5  Moresco et al. (2012)  DA  0.2  72.9  29.6  Zhang et al. (2014)  DA  0.24  79.69  2.65  Gaztanaga et al. (2009)  Clustering  0.27  77  14  Stern et al. (2010)  DA  0.28  88.8  36.6  Zhang et al. (2014)  DA  0.3  81.7  6.22  Oka et al. (2014)  Clustering  0.31  78.17  4.74  Wang et al. (2017)  Clustering  0.35  82.7  8.4  Chuang & Wang (2013)  Clustering  0.3519  83  14  Moresco et al. (2012)  DA  0.36  79.93  3.39  Wang et al. (2017)  Clustering  0.38  81.5  1.9  Alam et al. (2017)  Clustering  0.3802  83  13.5  Moresco et al. (2016)  DA  0.4  95  17  Stern et al. (2010)  DA  0.4004  77  10.2  Moresco et al. (2016)  DA  0.4247  87.1  11.2  Moresco et al. (2016)  DA  0.43  86.45  3.68  Gaztanaga et al. (2009)  Clustering  0.44  82.6  7.8  Blake et al. (2012)  Clustering  0.4497  92.8  12.9  Moresco et al. (2016)  DA  0.47  89  34  Ratsimbazafy et al. (2017)  DA  0.4783  80.9  9  Moresco et al. (2016)  DA  0.48  97  62  Stern et al. (2010)  DA  0.51  90.4  1.9  Alam et al. (2017)  Clustering  0.52  94.35  2.65  Wang et al. (2017)  Clustering  0.56  93.33  2.32  Wang et al. (2017)  Clustering  0.57  92.9  7.8  Anderson et al. (2014)  Clustering  0.59  98.48  3.19  Wang et al. (2017)  Clustering  0.5929  104  13  Moresco et al. (2012)  DA  0.6  87.9  6.1  Blake et al. (2012)  Clustering  0.61  97.3  2.1  Alam et al. (2017)  Clustering  0.64  98.82  2.99  Wang et al. (2017)  Clustering  0.6797  92  8  Moresco et al. (2012)  DA  0.73  97.3  7  Blake et al. (2012)  Clustering  0.7812  105  12  Moresco et al. (2012)  DA  0.8754  125  17  Moresco et al. (2012)  DA  0.88  90  40  Stern et al. (2010)  DA  0.9  117  23  Stern et al. (2010)  DA  1.037  154  20  Moresco et al. (2012)  DA  1.3  168  17  Stern et al. (2010)  DA  1.363  160  33.6  Moresco (2015)  DA  1.43  177  18  Stern et al. (2010)  DA  1.53  140  14  Stern et al. (2010)  DA  1.75  202  40  Stern et al. (2010)  DA  1.965  186.5  50.4  Moresco (2015)  DA  2.33  224  8  Bautista et al. (2017)  Clustering  2.34  222  7  Delubac et al. (2015)  Clustering  2.36  226  8  Font-Ribera et al. (2014)  Clustering  z  H(z)  σH  Reference  Method    (km s− 1 Mpc− 1)  (km s−1 Mpc−1)      0  73.24  1.74  Riess et al. (2016)  SN Ia/Cepheid  0.07  69  19.6  Zhang et al. (2014)  DA  0.1  69  12  Stern et al. (2010)  DA  0.12  68.6  26.2  Zhang et al. (2014)  DA  0.17  83  8  Stern et al. (2010)  DA  0.1791  75  4  Moresco et al. (2012)  DA  0.1993  75  5  Moresco et al. (2012)  DA  0.2  72.9  29.6  Zhang et al. (2014)  DA  0.24  79.69  2.65  Gaztanaga et al. (2009)  Clustering  0.27  77  14  Stern et al. (2010)  DA  0.28  88.8  36.6  Zhang et al. (2014)  DA  0.3  81.7  6.22  Oka et al. (2014)  Clustering  0.31  78.17  4.74  Wang et al. (2017)  Clustering  0.35  82.7  8.4  Chuang & Wang (2013)  Clustering  0.3519  83  14  Moresco et al. (2012)  DA  0.36  79.93  3.39  Wang et al. (2017)  Clustering  0.38  81.5  1.9  Alam et al. (2017)  Clustering  0.3802  83  13.5  Moresco et al. (2016)  DA  0.4  95  17  Stern et al. (2010)  DA  0.4004  77  10.2  Moresco et al. (2016)  DA  0.4247  87.1  11.2  Moresco et al. (2016)  DA  0.43  86.45  3.68  Gaztanaga et al. (2009)  Clustering  0.44  82.6  7.8  Blake et al. (2012)  Clustering  0.4497  92.8  12.9  Moresco et al. (2016)  DA  0.47  89  34  Ratsimbazafy et al. (2017)  DA  0.4783  80.9  9  Moresco et al. (2016)  DA  0.48  97  62  Stern et al. (2010)  DA  0.51  90.4  1.9  Alam et al. (2017)  Clustering  0.52  94.35  2.65  Wang et al. (2017)  Clustering  0.56  93.33  2.32  Wang et al. (2017)  Clustering  0.57  92.9  7.8  Anderson et al. (2014)  Clustering  0.59  98.48  3.19  Wang et al. (2017)  Clustering  0.5929  104  13  Moresco et al. (2012)  DA  0.6  87.9  6.1  Blake et al. (2012)  Clustering  0.61  97.3  2.1  Alam et al. (2017)  Clustering  0.64  98.82  2.99  Wang et al. (2017)  Clustering  0.6797  92  8  Moresco et al. (2012)  DA  0.73  97.3  7  Blake et al. (2012)  Clustering  0.7812  105  12  Moresco et al. (2012)  DA  0.8754  125  17  Moresco et al. (2012)  DA  0.88  90  40  Stern et al. (2010)  DA  0.9  117  23  Stern et al. (2010)  DA  1.037  154  20  Moresco et al. (2012)  DA  1.3  168  17  Stern et al. (2010)  DA  1.363  160  33.6  Moresco (2015)  DA  1.43  177  18  Stern et al. (2010)  DA  1.53  140  14  Stern et al. (2010)  DA  1.75  202  40  Stern et al. (2010)  DA  1.965  186.5  50.4  Moresco (2015)  DA  2.33  224  8  Bautista et al. (2017)  Clustering  2.34  222  7  Delubac et al. (2015)  Clustering  2.36  226  8  Font-Ribera et al. (2014)  Clustering  View Large 3.1.1 An homogeneous OHD sample As mentioned above, the OHD from clustering (BAO features) are biased due to an underlying ΛCDM cosmology to estimate rd. Different authors used different values in the cosmological parameters and obtained different sound horizons at the drag epoch, which are used to break the degeneracy in Hrd. Furthermore, the determination of H(z) from BAO features is computed taking into account very conservative systematic errors (see the discussion by Leaf & Melia 2017; Melia & López-Corredoira 2017). As a first attempt to homogenize and achieve model independence for the OHD obtained from clustering, we take the value Hrd for each data point and assume a common value rd for the entire data set. We consider two rd estimations: rdpl = 147.33 ± 0.49 Mpc and rdw9 = 152.3 ± 1.3 Mpc from the most recent Planck (Planck Collaboration XIII 2016) and WMAP9 (Bennett et al. 2013) measurements, respectively. In addition, we also take into account three other sources of errors that could affect rd due to its contamination by a cosmological model. The first one comes from the error of each reported value. The second error considers the possible range of rd values provided by separate CMB measurements, i.e. the difference between the sound horizon given by WMAP9 and Planck. This error is the one producing the largest impact on the rd mean value (3.37  per cent and 3.26  per cent for the Planck and WMAP9 data point, respectively). The last error to take into account is the difference between rd used to obtain the OHD and the one that would be obtained if we assume another cosmological model instead of ΛCDM. Hereafter, we use the one obtained for a DE constant equation-of-state (w) CDM model, rdωcdm = 148.38 Mpc (the cosmological parameters for this model are provided by Neveu et al. 2017). Adding in quadrature the percentage for these three errors, we obtain rdpl = 147.33 ± 5.08 Mpc and rdw9 = 152.3 ± 6.42 Mpc. Finally, we propagate this new error to the quantity H(z) to secure a new homogenized and model-independent sample (Table 2). Table 2. Homogenized model-independent OHD from clustering (in km s−1Mpc−1) and its error, σH, at redshift z. The first and second columns were obtained using the sound horizon in the drag epoch from Planck and WMAP measurements, respectively. z  H(z) ± σH(rdpl)  H(z) ± σH(rdw9)    (km s− 1Mpc− 1)  (km s− 1Mpc− 1)  0.24  82.37 ± 3.94  79.69 ± 4.28  0.3  78.83 ± 6.58  76.26 ± 6.63  0.31  78.39 ± 5.46  75.83 ± 5.60  0.35  88.10 ± 9.45  85.23 ± 9.37  0.36  80.16 ± 4.37  77.54 ± 4.63  0.38  81.74 ± 3.40  79.08 ± 3.81  0.43  89.36 ± 4.89  86.44 ± 5.18  0.44  85.48 ± 8.59  82.69 ± 8.55  0.51  90.67 ± 3.66  87.71 ± 4.13  0.52  94.61 ± 4.20  91.52 ± 4.63  0.56  93.59 ± 3.96  90.54 ± 4.42  0.57  96.59 ± 8.76  93.44 ± 8.78  0.59  98.75 ± 4.66  95.53 ± 5.07  0.6  90.96 ± 7.04  87.99 ± 7.14  0.61  97.59 ± 3.97  94.41 ± 4.47  0.64  99.09 ± 4.53  95.86 ± 4.97  0.73  100.69 ± 8.03  97.40 ± 8.12  2.33  223.99 ± 11.12  216.69 ± 11.97  2.34  222.105 ± 10.38  214.85 ± 11.31  2.36  226.24 ± 11.18  218.86 ± 12.05  z  H(z) ± σH(rdpl)  H(z) ± σH(rdw9)    (km s− 1Mpc− 1)  (km s− 1Mpc− 1)  0.24  82.37 ± 3.94  79.69 ± 4.28  0.3  78.83 ± 6.58  76.26 ± 6.63  0.31  78.39 ± 5.46  75.83 ± 5.60  0.35  88.10 ± 9.45  85.23 ± 9.37  0.36  80.16 ± 4.37  77.54 ± 4.63  0.38  81.74 ± 3.40  79.08 ± 3.81  0.43  89.36 ± 4.89  86.44 ± 5.18  0.44  85.48 ± 8.59  82.69 ± 8.55  0.51  90.67 ± 3.66  87.71 ± 4.13  0.52  94.61 ± 4.20  91.52 ± 4.63  0.56  93.59 ± 3.96  90.54 ± 4.42  0.57  96.59 ± 8.76  93.44 ± 8.78  0.59  98.75 ± 4.66  95.53 ± 5.07  0.6  90.96 ± 7.04  87.99 ± 7.14  0.61  97.59 ± 3.97  94.41 ± 4.47  0.64  99.09 ± 4.53  95.86 ± 4.97  0.73  100.69 ± 8.03  97.40 ± 8.12  2.33  223.99 ± 11.12  216.69 ± 11.97  2.34  222.105 ± 10.38  214.85 ± 11.31  2.36  226.24 ± 11.18  218.86 ± 12.05  View Large 3.2 Type Ia supernovae The SN Ia observations supply the evidence of the accelerated expansion of the Universe. They have been considered a perfect standard candle to measure the geometry and dynamics of the Universe and have been widely used to constrain alternatives cosmological models to explain the late-time cosmic acceleration. Currently, there are several compiled SN Ia samples, for instance, the Union 2.1 compilation by Suzuki et al. (2012) that consists of 580 points in the redshift range of 0.015 < z < 1.41, and the Lick Observatory Supernova Search (LOSS) sample containing 586 SN Ia in the redshift range of 0.01 < z < 1.4 (Ganeshalingam et al. 2013). Recently, Betoule et al. (2014) presented the so-called full JLA (fJLA) sample that contains 740 points spanning a redshift range of 0.01 < z < 1.2. The same authors also provide the information of the fJLA data in a compressed set (cJLA) of 31 binned distance modulus μb spanning a redshift range of 0.01 < z < 1.3, which still remains accurate for some models where the isotropic luminosity distance evolves slightly with redshift. For instance, when the cJLA is used in combination with other cosmological data, the difference between fJLA and cJLA in the mean values for the wCDM model parameters is at most 0.018σ. Here, we use both, the fJLA and cJLA samples, to constrain the parameters of the OC and MPC models. 3.2.1 Full JLA sample As mentioned, the full JLA sample contains 740 confirmed SN Ia in the redshift interval 0.01 < z < 1.2, which is one of the most recent and reliable SN Ia samples. We use this sample to constrain the parameters of both Cardassian models. The function of merit for the fJLA sample is calculated as   $$\chi ^{2}_{fJLA}={\left(\hat{\mu } - \mu _{\text{Card}}\right)^{\dagger }\mathrm{C_{\eta }^{-1}}\left( \hat{\mu } - \mu _{\text{Card}} \right)},$$ (14)where μCard = 5log10(dL/10 pc), and Cη is the covariance matrix3 of $$\mathbf {\hat{\mu }}$$ provided by Betoule et al. (2014) and is constructed using   \begin{eqnarray} \mathbf {C_{\eta }}&=& \left( \mathbf {C}_{\text{cal}} + \mathbf {C}_{\text{model}} + \mathbf {C}_{\text{bias}} + \mathbf {C}_{\text{host}} + \mathbf {C}_{\text{dust}} \right) \nonumber \\ && +\, \left( \mathbf {C}_{\text{pecvel}} + \mathbf {C}_{\text{nonIa}} \right) + \mathbf {C}_{\text{stat}}, \end{eqnarray} (15)where $$\mathbf {C}_{\text{cal}}, \mathbf {C}_{\text{model}}, \mathbf {C}_{\text{bias}}, \mathbf {C}_{\text{host}}, \mathbf {C}_{\text{dust}}$$ are systematic uncertainty matrices associated with the calibration, the light-curve model, the bias correction, the mass step, and dust uncertainties, respectively. $$\mathbf {C}_{\text{pecvel}}$$ and $$\mathbf {C}_{\text{nonIa}}$$ corresponds to systematics uncertainties in the peculiar velocity corrections and the contamination of the Hubble diagram by non-Ia events, respectively, $$\mathbf {C}_{\text{stat}}$$ corresponds to an statistical uncertainty obtained from error propagation of the light-curve fit uncertainties. Finally, $$\hat{\mu}$$ is given by   $$\hat{\mu } = m_{b}^{\star } - \left( M_{B} - \alpha \times X_{1} + \beta \times C \right),$$ (16)where $$m_{b}^{\star}$$ corresponds to the observed peak magnitude, α, β, and MB are nuisance parameters in the distance estimates. The X1 and C variables describe the time stretching of the light curve and the Supernova colour at maximum brightness, respectively. The absolute magnitude MB is related to the host stellar mass (Mstellar) by the step function:   \begin{eqnarray} M_{B} = \left\lbrace \begin{array}{cc}M_{B}^{1} & \rm {if} \ M_{stellar} < 10^{10} M_{\odot } , \\ M_{B}^{1} + \Delta _{M} & \rm {otherwise.} \\ \end{array} \right. \end{eqnarray} (17)By replacing equations (4), (9), (15), and (16) in equation (14), we obtain the explicit figure-of-merit $$\chi ^{2}_{\mathrm{fJLA}}$$ for the Cardassian models. 3.2.2 Compressed form of the JLA sample Table A1 shows the 31 binned distance modulus at the binned redshift zb. The function of merit for the cJLA sample is calculated as   $$\chi ^{2}_{{cJLA}}=\boldsymbol {r}^{\dagger}\mathrm{\bf C}_{b}^{-1}\boldsymbol {r},$$ (18)where Cb is the covariance matrix4 provided by Betoule et al. (2014), and $$\boldsymbol {r}$$ is given by   $$\boldsymbol {r}=\boldsymbol {\mu }_{b}-M-\log _{10}d_{L}(\boldsymbol {z}_{b}, \boldsymbol {\boldsymbol\Theta }),$$ (19)where M is a nuisance parameter and dL is the luminosity distance given by   $$d_{L}=(1+z)\frac{c}{H_{0}}\int ^{0}_{z}\frac{\mathrm{dz}^{\prime }}{E(z^{\prime }, \boldsymbol {\boldsymbol\Theta })}.$$ (20)By replacing equations (4) and (9) in the last expression, we obtain the explicit figure of merit $$\chi ^{2}_{\mathrm{cJLA}}$$ for the OC and MPC models. 4 RESULTS A Markov chain Monte Carlo (MCMC) Bayesian statistical analysis was performed to estimate the (Ωm0, h, n) and the (Ωm0, h, n, l) parameters for the OC and MPC models, respectively. The constructed Gaussian likelihood function for each data set are given by $$\mathcal {L}_{\mathrm{OHD}}\propto \exp (-\chi _{\mathrm{OHD}}^{2}/2)$$, $$\mathcal {L}_{\mathrm{cJLA}}\propto \exp (-\chi _{\mathrm{cJLA}}^{2}/2)$$, $$\mathcal {L}_{\mathrm{fJLA}}\propto \exp (-\chi _{\mathrm{fJLA}}^{2}/2)$$, and $$\mathcal {L}_{\mathrm{joint}}\propto \exp (-\chi _{\mathrm{tot}}^{2}/2)$$, where χtot2 = χOHD2 + χcJLA2. We use the Affine Invariant MCMC Ensemble sampler from the emceepython module (Foreman-Mackey et al. 2013). In all our computations, we consider 3000 steps to stabilize the estimations (burn-in phase), 6000 MCMC steps and 1000 walkers that are initialized in a small ball around the expected points of maximum probability, is estimated with a differential evolution method. For both, OC and MPC models, we assume the following flat priors: Ωm0[0, 1], and n[−1, 2/3]. For the l MPC parameter, we consider the flat prior [0, 6]. For the h parameter three priors are considered: a flat prior [0, 1], and two Gaussian priors, one by Riess et al. (2016, the first point in Table 1), and the other one by Planck Collaboration XIII (2016) from Planck 2015 measurements (h = 0.678 ± 0.009). When the cJLA data are used, we also take a flat prior on the nuisance parameter M[−1, 1]. The following flat priors α[0, 2], β[0, 4.0], $$M^{1}_{B}[-20,-18]$$, and ΔM[−0.1, 0.1] are considered when the fJLA sample is employed. To judge the convergence of the sampler, we ask that the acceptance fraction is in the [0.2–0.5] range and check the autocorrelation time that is found to be $$\mathcal {O}(60\text{--}80)$$. We carry out four runs using different OHD sets: the full observational sample given in Table 1, the 31 data points obtained using the DA method (OHDDA), and two samples containing the DA points plus those homogenized points from clustering using a common rd estimated from Planck and WMAP measurements (Table 2). We also estimate the OC and MPC parameters using both the cJLA and fJLA samples. Moreover, we perform a joint analysis considering each OHD sample and the cJLA sample. Tables 3 and 4 provide the best fits for the OC and MPC parameters, respectively, using the different data sets and priors on h. Tables 5 and 6 give the constraints from the following joint analysis: OHD+cJLA (J1), OHDDA+cJLA (J2), OHDhpl+CJLA (J3), and OHDhw9+CJLA (J4). We also give the minimum chi-square, χmin, and the reduced χred = χmin/d.o.f, where the degree of freedom (d.o.f.) is the difference between the number of data points and the free parameters. Table 3. Mean values for the OC parameters (Ωm0, h, n) derived from OHD and SN Ia data of the cJLA and fJLA samples. OC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.37  15.22  21.25  22.52  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.52  0.54  0.44  0.46  1.22  0.93  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.30^{+0.06}_{-0.06}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.25^{+0.03}_{-0.03}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.65^{+0.03}_{-0.03}$$  $$0.69^{+0.06}_{-0.05}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.72^{+0.19}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.26^{+0.16}_{-0.15}$$  $$-0.19^{+0.51}_{-0.50}$$  $$0.23^{+0.20}_{-0.20}$$  $$0.16^{+0.22}_{-0.22}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.07^{+0.5}_{-0.66}$$  $$-18.96^{+0.49}_{-0.64}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  28.86  14.47  22.83  23.91  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.60  0.51  0.47  0.49  1.22  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.31^{+0.03}_{-0.04}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.01^{+0.08}_{-0.08}$$  $$-0.43^{+0.28}_{-0.30}$$  $$-0.02^{+0.09}_{-0.10}$$  $$-0.11^{+0.10}_{-0.11}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.24  14.53  20.79  22.04  32.95  –  $$\chi ^{2}_{\text{red}}$$  0.52  0.51  0.43  0.45  1.22  –  Ωm0  $$0.26^{+0.01}_{-0.01}$$  $$0.33^{+0.05}_{-0.07}$$  $$0.26^{+0.02}_{-0.02}$$  $$0.25^{+0.02}_{-0.02}$$  $$0.22^{+0.11}_{-0.12}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$0.15^{+0.06}_{-0.06}$$  $$-0.05^{+0.27}_{-0.32}$$  $$0.17^{+0.08}_{-0.08}$$  $$0.09^{+0.08}_{-0.09}$$  $$0.16^{+0.17}_{-0.26}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  OC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.37  15.22  21.25  22.52  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.52  0.54  0.44  0.46  1.22  0.93  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.30^{+0.06}_{-0.06}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.25^{+0.03}_{-0.03}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.65^{+0.03}_{-0.03}$$  $$0.69^{+0.06}_{-0.05}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.72^{+0.19}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.26^{+0.16}_{-0.15}$$  $$-0.19^{+0.51}_{-0.50}$$  $$0.23^{+0.20}_{-0.20}$$  $$0.16^{+0.22}_{-0.22}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.07^{+0.5}_{-0.66}$$  $$-18.96^{+0.49}_{-0.64}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  28.86  14.47  22.83  23.91  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.60  0.51  0.47  0.49  1.22  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.31^{+0.03}_{-0.04}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.01^{+0.08}_{-0.08}$$  $$-0.43^{+0.28}_{-0.30}$$  $$-0.02^{+0.09}_{-0.10}$$  $$-0.11^{+0.10}_{-0.11}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.24  14.53  20.79  22.04  32.95  –  $$\chi ^{2}_{\text{red}}$$  0.52  0.51  0.43  0.45  1.22  –  Ωm0  $$0.26^{+0.01}_{-0.01}$$  $$0.33^{+0.05}_{-0.07}$$  $$0.26^{+0.02}_{-0.02}$$  $$0.25^{+0.02}_{-0.02}$$  $$0.22^{+0.11}_{-0.12}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$0.15^{+0.06}_{-0.06}$$  $$-0.05^{+0.27}_{-0.32}$$  $$0.17^{+0.08}_{-0.08}$$  $$0.09^{+0.08}_{-0.09}$$  $$0.16^{+0.17}_{-0.26}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  View Large Table 4. Mean values for the MPC parameters (Ωm0, h, n, l) derived from OHD and SN Ia data of the cJLA and fJLA samples. MPC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.31  17.95  21.17  22.98  33.76  682.92  $$\chi ^{2}_{\text{red}}$$  0.53  0.66  0.45  0.48  1.29  0.93  Ωm0  $$0.25^{+0.04}_{-0.04}$$  $$0.32^{+0.06}_{-0.07}$$  $$0.25^{+0.04}_{-0.05}$$  $$0.25^{+0.04}_{-0.04}$$  $$0.22^{+0.12}_{-0.13}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.64^{+0.03}_{-0.02}$$  $$0.68^{+0.07}_{-0.05}$$  $$0.65^{+0.03}_{-0.03}$$  $$0.65^{+0.04}_{-0.03}$$  $$0.72^{+0.18}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.17^{+0.34}_{-0.68}$$  $$0.10^{+0.38}_{-0.60}$$  $$0.25^{+0.29}_{-0.68}$$  $$0.15^{+0.34}_{-0.65}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.33^{+0.09}_{-0.50}$$  l  $$0.77^{+1.45}_{-0.43}$$  $$2.13^{+2.34}_{-1.33}$$  $$0.95^{+1.90}_{-0.58}$$  $$0.92^{+1.66}_{-0.52}$$  $$2.61^{+2.27}_{-1.83}$$  $$2.09^{+2.51}_{-1.49}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.08^{+0.50}_{-0.67}$$  $$-18.97^{+0.49}_{-0.65}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  27.75  14.92  22.40  23.42  33.75  683.17  $$\chi ^{2}_{\text{red}}$$  0.59  0.55  0.47  0.49  1.29  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.32^{+0.03}_{-0.04}$$  $$0.24^{+0.02}_{-0.02}$$  $$0.23^{+0.02}_{-0.02}$$  $$0.22^{+0.12}_{-0.14}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.34^{+0.40}_{-0.42}$$  $$-0.03^{+0.24}_{-0.49}$$  $$-0.19^{+0.39}_{-0.50}$$  $$-0.28^{+0.39}_{-0.46}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.34^{+0.08}_{-0.47}$$  l  $$0.62^{+0.49}_{-0.20}$$  $$2.12^{+2.29}_{-1.20}$$  $$0.75^{+0.80}_{-0.30}$$  $$0.77^{+0.73}_{-0.28}$$  $$2.60^{+2.27}_{-1.81}$$  $$2.26^{+2.44}_{-1.63}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.03  14.70  20.84  21.96  33.79  –  $$\chi ^{2}_{\text{red}}$$  0.53  0.54  0.44  0.46  1.29  –  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.35^{+0.05}_{-0.08}$$  $$0.26^{+0.03}_{-0.03}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.22^{+0.12}_{-0.13}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$-0.01^{+0.35}_{-0.57}$$  $$0.24^{+0.19}_{-0.53}$$  $$0.16^{+0.26}_{-0.60}$$  $$0.03^{+0.31}_{-0.58}$$  $$0.36^{+0.07}_{-0.36}$$  –  l  $$0.71^{+0.95}_{-0.34}$$  $$2.08^{+2.44}_{-1.37}$$  $$0.96^{+1.55}_{-0.55}$$  $$0.87^{+1.22}_{-0.45}$$  $$2.59^{+2.27}_{-1.8}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  MPC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.31  17.95  21.17  22.98  33.76  682.92  $$\chi ^{2}_{\text{red}}$$  0.53  0.66  0.45  0.48  1.29  0.93  Ωm0  $$0.25^{+0.04}_{-0.04}$$  $$0.32^{+0.06}_{-0.07}$$  $$0.25^{+0.04}_{-0.05}$$  $$0.25^{+0.04}_{-0.04}$$  $$0.22^{+0.12}_{-0.13}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.64^{+0.03}_{-0.02}$$  $$0.68^{+0.07}_{-0.05}$$  $$0.65^{+0.03}_{-0.03}$$  $$0.65^{+0.04}_{-0.03}$$  $$0.72^{+0.18}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.17^{+0.34}_{-0.68}$$  $$0.10^{+0.38}_{-0.60}$$  $$0.25^{+0.29}_{-0.68}$$  $$0.15^{+0.34}_{-0.65}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.33^{+0.09}_{-0.50}$$  l  $$0.77^{+1.45}_{-0.43}$$  $$2.13^{+2.34}_{-1.33}$$  $$0.95^{+1.90}_{-0.58}$$  $$0.92^{+1.66}_{-0.52}$$  $$2.61^{+2.27}_{-1.83}$$  $$2.09^{+2.51}_{-1.49}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.08^{+0.50}_{-0.67}$$  $$-18.97^{+0.49}_{-0.65}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  27.75  14.92  22.40  23.42  33.75  683.17  $$\chi ^{2}_{\text{red}}$$  0.59  0.55  0.47  0.49  1.29  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.32^{+0.03}_{-0.04}$$  $$0.24^{+0.02}_{-0.02}$$  $$0.23^{+0.02}_{-0.02}$$  $$0.22^{+0.12}_{-0.14}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.34^{+0.40}_{-0.42}$$  $$-0.03^{+0.24}_{-0.49}$$  $$-0.19^{+0.39}_{-0.50}$$  $$-0.28^{+0.39}_{-0.46}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.34^{+0.08}_{-0.47}$$  l  $$0.62^{+0.49}_{-0.20}$$  $$2.12^{+2.29}_{-1.20}$$  $$0.75^{+0.80}_{-0.30}$$  $$0.77^{+0.73}_{-0.28}$$  $$2.60^{+2.27}_{-1.81}$$  $$2.26^{+2.44}_{-1.63}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.03  14.70  20.84  21.96  33.79  –  $$\chi ^{2}_{\text{red}}$$  0.53  0.54  0.44  0.46  1.29  –  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.35^{+0.05}_{-0.08}$$  $$0.26^{+0.03}_{-0.03}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.22^{+0.12}_{-0.13}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$-0.01^{+0.35}_{-0.57}$$  $$0.24^{+0.19}_{-0.53}$$  $$0.16^{+0.26}_{-0.60}$$  $$0.03^{+0.31}_{-0.58}$$  $$0.36^{+0.07}_{-0.36}$$  –  l  $$0.71^{+0.95}_{-0.34}$$  $$2.08^{+2.44}_{-1.37}$$  $$0.96^{+1.55}_{-0.55}$$  $$0.87^{+1.22}_{-0.45}$$  $$2.59^{+2.27}_{-1.8}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  View Large Table 5. Mean values for the OC parameters (Ωm0, h, n) derived from a joint analysis OHD+cJLA. OC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  M  Flat prior on h  J1  58.91  0.71  $$0.25^{+0.01}_{-0.01}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.12^{+0.06}_{-0.06}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.28  0.58  $$0.30^{+0.05}_{-0.05}$$  $$0.68^{+0.02}_{-0.02}$$  $$-0.001^{+0.15}_{-0.17}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.28  0.66  $$0.25^{+0.02}_{-0.02}$$  $$0.69^{+0.01}_{-0.01}$$  $$0.11^{+0.07}_{-0.07}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.17  0.67  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.10^{+0.07}_{-0.08}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  63.34  0.76  $$0.25^{+0.01}_{-0.01}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.05^{+0.05}_{-0.05}$$  $$0.001^{+0.02}_{-0.03}$$  J2  50.73  0.61  $$0.27^{+0.04}_{-0.05}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.001^{+0.14}_{-0.15}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.37  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.06^{+0.06}_{-0.07}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.96  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.04^{+0.07}_{-0.07}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  59.04  0.71  $$0.26^{+0.01}_{-0.01}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.13^{+0.05}_{-0.05}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.53  0.58  $$0.31^{+0.04}_{-0.05}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.001^{+0.15}_{-0.17}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.80  0.66  $$0.26^{+0.02}_{-0.02}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.13^{+0.06}_{-0.07}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.18  0.65  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.10^{+0.07}_{-0.07}$$  $$-0.06^{+0.02}_{-0.02}$$  OC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  M  Flat prior on h  J1  58.91  0.71  $$0.25^{+0.01}_{-0.01}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.12^{+0.06}_{-0.06}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.28  0.58  $$0.30^{+0.05}_{-0.05}$$  $$0.68^{+0.02}_{-0.02}$$  $$-0.001^{+0.15}_{-0.17}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.28  0.66  $$0.25^{+0.02}_{-0.02}$$  $$0.69^{+0.01}_{-0.01}$$  $$0.11^{+0.07}_{-0.07}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.17  0.67  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.10^{+0.07}_{-0.08}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  63.34  0.76  $$0.25^{+0.01}_{-0.01}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.05^{+0.05}_{-0.05}$$  $$0.001^{+0.02}_{-0.03}$$  J2  50.73  0.61  $$0.27^{+0.04}_{-0.05}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.001^{+0.14}_{-0.15}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.37  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.06^{+0.06}_{-0.07}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.96  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.04^{+0.07}_{-0.07}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  59.04  0.71  $$0.26^{+0.01}_{-0.01}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.13^{+0.05}_{-0.05}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.53  0.58  $$0.31^{+0.04}_{-0.05}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.001^{+0.15}_{-0.17}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.80  0.66  $$0.26^{+0.02}_{-0.02}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.13^{+0.06}_{-0.07}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.18  0.65  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.10^{+0.07}_{-0.07}$$  $$-0.06^{+0.02}_{-0.02}$$  View Large Table 6. Mean values for the MPC parameters (Ωm0, h, n, l) derived from a joint analysis OHD+cJLA. MPC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  l  M  Flat prior on h  J1  58.61  0.71  $$0.25^{+0.02}_{-0.02}$$  $$0.68^{+0.01}_{-0.01}$$  $$-0.03^{+0.34}_{-0.56}$$  $$0.74^{+0.90}_{-0.35}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.25  0.58  $$0.32^{+0.05}_{-0.07}$$  $$0.68^{+0.02}_{-0.02}$$  $$0.25^{+0.13}_{-0.51}$$  $$2.00^{+2.244}_{-1.33}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.23  0.66  $$0.25^{+0.03}_{-0.03}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.06^{+0.29}_{-0.58}$$  $$0.89^{+1.29}_{-0.47}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.14  0.67  $$0.25^{+0.03}_{-0.03}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.06^{+0.28}_{-0.58}$$  $$0.91^{+1.29}_{-0.49}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  62.91  0.75  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.14^{+0.36}_{-0.51}$$  $$0.70^{+0.72}_{-0.29}$$  $$-0.0006^{+0.03}_{-0.03}$$  J2  50.81  0.61  $$0.30^{+0.04}_{-0.06}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.22^{+0.14}_{-0.54}$$  $$1.77^{+2.44}_{-1.17}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.32  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.002^{+0.29}_{-0.55}$$  $$0.88^{+1.12}_{-0.44}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.91  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.007^{+0.29}_{-0.55}$$  $$0.89^{+1.15}_{-0.45}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  58.85  0.70  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$-0.02^{+0.33}_{-0.57}$$  $$0.74^{+0.92}_{-0.35}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.45  0.58  $$0.33^{+0.04}_{-0.06}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.26^{+0.12}_{-0.51}$$  $$2.10^{+2.42}_{-1.42}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.82  0.66  $$0.26^{+0.03}_{-0.03}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.09^{+0.28}_{-0.59}$$  $$0.91^{+1.35}_{-0.49}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.19  0.66  $$0.25^{+0.02}_{-0.03}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.06^{+0.27}_{-0.57}$$  $$0.91^{+1.25}_{-0.48}$$  $$-0.06^{+0.02}_{-0.032}$$  MPC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  l  M  Flat prior on h  J1  58.61  0.71  $$0.25^{+0.02}_{-0.02}$$  $$0.68^{+0.01}_{-0.01}$$  $$-0.03^{+0.34}_{-0.56}$$  $$0.74^{+0.90}_{-0.35}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.25  0.58  $$0.32^{+0.05}_{-0.07}$$  $$0.68^{+0.02}_{-0.02}$$  $$0.25^{+0.13}_{-0.51}$$  $$2.00^{+2.244}_{-1.33}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.23  0.66  $$0.25^{+0.03}_{-0.03}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.06^{+0.29}_{-0.58}$$  $$0.89^{+1.29}_{-0.47}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.14  0.67  $$0.25^{+0.03}_{-0.03}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.06^{+0.28}_{-0.58}$$  $$0.91^{+1.29}_{-0.49}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  62.91  0.75  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.14^{+0.36}_{-0.51}$$  $$0.70^{+0.72}_{-0.29}$$  $$-0.0006^{+0.03}_{-0.03}$$  J2  50.81  0.61  $$0.30^{+0.04}_{-0.06}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.22^{+0.14}_{-0.54}$$  $$1.77^{+2.44}_{-1.17}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.32  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.002^{+0.29}_{-0.55}$$  $$0.88^{+1.12}_{-0.44}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.91  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.007^{+0.29}_{-0.55}$$  $$0.89^{+1.15}_{-0.45}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  58.85  0.70  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$-0.02^{+0.33}_{-0.57}$$  $$0.74^{+0.92}_{-0.35}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.45  0.58  $$0.33^{+0.04}_{-0.06}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.26^{+0.12}_{-0.51}$$  $$2.10^{+2.42}_{-1.42}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.82  0.66  $$0.26^{+0.03}_{-0.03}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.09^{+0.28}_{-0.59}$$  $$0.91^{+1.35}_{-0.49}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.19  0.66  $$0.25^{+0.02}_{-0.03}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.06^{+0.27}_{-0.57}$$  $$0.91^{+1.25}_{-0.48}$$  $$-0.06^{+0.02}_{-0.032}$$  View Large 4.1 cJLA versus fJLA on the Cardassian parameter estimations The use of the fJLA sample to infer cosmological parameters has a high computational cost when several model are tested. To deal with this, we use the cJLA sample instead of the fJLA. Nevertheless, the former was computed under the standard cosmology. To assess how the Cardassian model constraints are biased when using each SN Ia sample, we perform the parameter estimation with different combinations of models, priors, and samples. The several constraints are presented in Tables 3 and 4. Notice that the mean values for the cosmological parameters in the OC model obtained from both SN Ia samples are the same. For the MPC model, the largest difference is observed on the l parameter (flat prior on h), ∼0.18σ. It is smaller for the n parameter when employing a Gaussian prior on h. Fig. 1 illustrates the comparison of the confidence contours for these parameters using the cJLA and fJLA samples (flat prior on h). Fig. 2 shows that there is no significant difference in the reconstruction of the q(z) parameter for the OC and MPC models using the constraints obtained from both SN Ia samples. Therefore, to optimize the computational time, in the following analysis we only use the compressed JLA sample. Figure 1. View largeDownload slide Comparison of the Ωm0−n (top panel) and n − l(bottom panel) confidence contours for the OC and MPC parameters within the 1σ and 3σ confidence levels, using the cJLA (dashed lines) and fJLA (filled and solid lines) samples, respectively. In the parameter estimation, a flat prior is considered. The cross and star mark the mean values for each data set. Figure 1. View largeDownload slide Comparison of the Ωm0−n (top panel) and n − l(bottom panel) confidence contours for the OC and MPC parameters within the 1σ and 3σ confidence levels, using the cJLA (dashed lines) and fJLA (filled and solid lines) samples, respectively. In the parameter estimation, a flat prior is considered. The cross and star mark the mean values for each data set. Figure 2. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from the cJLA and fJLA samples when a flat prior on h is considered. Notice that there is no significant differences in the q(z) behaviour using each SN Ia sample. Figure 2. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from the cJLA and fJLA samples when a flat prior on h is considered. Notice that there is no significant differences in the q(z) behaviour using each SN Ia sample. 4.2 The effects of the homogeneous OHD subsample in the parameter estimation In Section 3.1.1, an homogenized and model-independent OHD from clustering was constructed to avoid or reduce biased constraints due to the underlying cosmology or the underestimated systematic errors. Tables 3 and 4 provide the OC and MPC bounds estimated from the combination of the new computed unbiased OHD from clustering with those obtained from the DA method. The increase on the error of rd also increases the error on H(z), reducing the goodness of the fit (χred). In spite of this, the advantage of these new limits is that they could be considered unbiased by different cosmological models. Fig. 3 shows the contours of the Ωm0−n OC (top panel) and the n − l MPC (bottom panel) parameters, respectively, using the different OHD samples. Note that all the bounds are consistent within the 1σ and 3σ confidence levels (CLs). Fig. 4 illustrates the q(z) reconstructions using the different OHD data sets. Notice that for the OC model the homogenized OHD samples give slightly different q(0) values than the obtained from the sample in Table 1. For the MPC model, these differences are less significant. Figure 3. View largeDownload slide Confidence contours of the Ωm0−n (top panel) and n − l (bottom panel) constraints for the OC and MPC models within the 1σ and 3σ CLs, using the OHD sample in Table 1, the OHDDA data set, and two samples containing the DA points plus those homogenized OHD points from clustering using the rd values from WMAP and Planck measurements. A flat prior on h was considered in the parameter estimation. Figure 3. View largeDownload slide Confidence contours of the Ωm0−n (top panel) and n − l (bottom panel) constraints for the OC and MPC models within the 1σ and 3σ CLs, using the OHD sample in Table 1, the OHDDA data set, and two samples containing the DA points plus those homogenized OHD points from clustering using the rd values from WMAP and Planck measurements. A flat prior on h was considered in the parameter estimation. Figure 4. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from different OHD samples when a flat prior on h is considered. Figure 4. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from different OHD samples when a flat prior on h is considered. 4.3 The effects of a different Gaussian prior on h One of the most important problems in cosmology is the tension up to more than 3σ between the local measurements of the Hubble constant H0 and those obtained from the CMB anisotropies (Bernal, Verde & Riess 2016). The latest estimation by the Planck collaboration (Planck Collaboration XIII 2016), h = 0.678 ± 0.009, is in disagreement with the first value given in Table 1. Thus, using different Gaussian priors on h will lead to different constraints on the OC and MPC parameters. Therefore, we carried out all our computations with both priors. Fig. 5 illustrates how the confidence contours for the Ωm0−n and l − n parameters of the OC (top panel) and MPC (bottom panel) models obtained from OHDhpl are shifted using each Gaussian prior. Although they are consistent at 3σ, the tension in the constraints is important. In spite of these differences, Fig. 6 shows that both results drive the Universe to an accelerated phase but with slightly different transition redshifts (i.e. the redshift at which the Universe passes from a decelerated to an accelerated phase) and amplitude, q(0). In addition, the OC and MPC bounds are consistent with the standard cosmology even when different Gaussian priors are considered. Figure 5. View largeDownload slide Comparison of the Ωm 0−n (top panel) and l − n (bottom panel) confidence contours for the OC and MPC parameters, respectively, within the 1σ and 3σ CLs obtained from the OHDhpl analysis using two Gaussian priors on h: 0.732 ± 0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). The stars mark the mean values for each data set. Figure 5. View largeDownload slide Comparison of the Ωm 0−n (top panel) and l − n (bottom panel) confidence contours for the OC and MPC parameters, respectively, within the 1σ and 3σ CLs obtained from the OHDhpl analysis using two Gaussian priors on h: 0.732 ± 0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). The stars mark the mean values for each data set. Figure 6. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models, using the constraints from the OHDhpl sample and the joint analysis J3 when a different Gaussian prior on h is considered: 0.732 ±  0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). Figure 6. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models, using the constraints from the OHDhpl sample and the joint analysis J3 when a different Gaussian prior on h is considered: 0.732 ±  0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). 4.4 Cosmological implications of the OC and MPC constraints Fig. 7 shows the one-dimensional (1D) marginalized posterior distributions and the two-dimensional (2D) 68 per cent, 95 per cent, 99 per cent contours for the Ωm0, h, and n parameters of the OC model obtained from OHDhpl, cJLA, and J3 with flat (left-hand panel) and Gaussian (right-hand panel) priors on h. Assuming a flat prior on h, the Ωm0, h constraints obtained from the different data sets are consistent between them and are in agreement with Planck measurements for the standard model. For the n parameter, we found a tension in the constraints obtained from the different data sets. Nevertheless, the bounds have large uncertainties and are consistent among them within the 1σ CL. Our n constraints are consistent within the 1σ CL with those estimated by other authors, for instance, $$n=-0.04^{+0.07}_{-0.07}$$ (Xu 2012), $$n=0.16^{+0.30}_{-0.52}$$ (Wei et al. 2015), and $$n=-0.022^{+0.05}_{-0.05}$$ (Zhai et al. 2017a). It is worth to note that, when the cJLA data are used, Ωm drop at extremely low values (see the Ωm0−n contour), which is consistent with the results by Wei et al. (2015) who obtained a similar contour using the Union 2.1 data set. In addition, the $$\chi _{\text{red}}^{2}$$ values from the SN Ia data suggest that their errors (cJLA sample) are underestimated. Figure 7. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, and n parameters of the OC model assuming a flat and Gaussian (hRiess) prior on h. Figure 7. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, and n parameters of the OC model assuming a flat and Gaussian (hRiess) prior on h. On the other hand, when the Gaussian prior on h by Riess et al. (2016) is considered, the OHDhpl provides a better fitting for the OC parameters than those obtained when a flat prior is used (see the $$\chi _{\text{red}}^{2}$$ values). SN Ia data show no important statistical difference in the parameter estimation when flat or Gaussian priors are employed. Notice that we obtain stringent constraints from the joint analysis (see Fig. 7), which prefers values around n ∼ 0. Fig. 8 shows the fittings to the OHDhpl (top panel) and cJLA data (bottom panel), using the OHDhpl, cJLA, and J3 constraints for the OC model. A Monte Carlo approach was performed to propagate the error on the 1σ and 3σ CL. The comparison between these results and the ΛCDM fitting reveals that both models are in agreement with the data and there is no significant difference between them. In addition, when the J1, J2, and J4 constraints are used, we found consistent results within the 1σ CL. Therefore, the extra term in the equation (1) to the canonical Friedmann equation acts like a CC. However, in the OC models this term can be sourced by an extra dimension instead of the expected vacuum energy. Figure 8. View largeDownload slide Fitting to OHDhpl (top panel) and cJLA data (bottom panel), using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines), and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and OC model with a flat prior on h. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 8. View largeDownload slide Fitting to OHDhpl (top panel) and cJLA data (bottom panel), using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines), and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and OC model with a flat prior on h. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. To confirm that the OC model can drive to a late cosmic acceleration, we reconstructed the deceleration parameter using the mean values derived from the different data sets. Fig. 9 shows that the q(z) dynamics is similar for the ΛCDM and OC models when the OHDhpl, cJLA, and J3 constrains are used, i.e. the universe has a late phase of accelerated expansion. Notice that although the CLs in the q(z) reconstruction obtained from the SN Ia constraints are bigger that those from the OHDhpl, they are consistent. The difference could be explained by the extra free parameter (nuisance) in the SN Ia analysis. Figure 9. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC model and ΛCDM using the constraints from OHDhpl (top panel) and cJLA data (bottom panel) with a flat prior on h. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 9. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC model and ΛCDM using the constraints from OHDhpl (top panel) and cJLA data (bottom panel) with a flat prior on h. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. Fig. 10 shows the 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99 per cent contours for the Ωm0, h, n, and l parameters of the MPC model obtained from OHDhpl, cJLA, and J3 with flat (left-hand panel) and Gaussian (right-hand panel) priors on h. Considering a flat prior on h, the different data sets provide slightly different constraints on Ωm0 and h. For instance, the OHDDA estimates higher (lower) values on Ωm0 (h) and SN Ia lower (higher) values. However, the limits are consistent within the 1σ CL. For the n and l constraints, we also obtained a marginal tension using different data but they are consistent within the 1σ CL. Notice that our constraints include n = 0 and l = 1, which reproduces the ΛCDM dynamics. All our bounds are similar within the 1σ CL to those obtained by other authors, e.g. Li et al. (2012) combining SN Ia, BAO, and CMB data measure $$n=0.014^{+0.36}_{-0.94}$$, $$l=1.09^{+1.01}_{-0.46}$$, Magaña et al. (2015) using strong lensing features estimate n = 0.41 ± 0.25, l = 5.2 ± 2.25, Zhai et al. (2017a) provide $$n=0.16^{+0.08}_{0.09}$$, $$l=1.38^{+0.25}_{-0.22}$$ from the joint analysis of CMB, BAO plus SN Ia (JLA) data, and Zhai et al. (2017b) give $$n=0.02^{+0.26}_{-0.41}$$, $$l=1.1^{+0.8}_{-0.4}$$ from the joint analysis of CMB, BAO, SN Ia, fσ8 and the H0 value from Riess et al. (2016). In addition, the $$\chi _{\text{red}}^{2}$$ values point out that the OHDDA provides better (unbiased) MPC constraints and the values from SN Ia data suggest that their errors (cJLA sample) are underestimated. Considering the Gaussian prior on h by Riess et al. (2016), the OHD, OHDhpl, and OHDhw9 probes yield improvements in the MPC constraints (see the χred values). For the SN Ia (cJLA) test, there is no significant difference with the flat prior case. Notice that the stringent limits are estimated from the joint analysis (see also Fig. 10). Fig. 11 shows the fittings to the OHDhpl and cJLA data using the OHDhpl, cJLa, and J3 constraints of the MPC parameters and those of the ΛCDM model with a flat prior on h. To propagate the errors on OHD, μ(z), and q(z), we have used a Monte Carlo approach. For both, OHD and μ(z) fittings, there is no significant statistical difference between the MPC model and the standard one. In addition, a good agreement at 1σ is obtained employing the J1, J2, and J4 constraints. In addition, Fig. 12 shows the reconstruction of the q(z) parameter using the constraints from the OHD and SN Ia data. For the OHD constraints, the q(z) dynamics for the MPC is in agreement with that of the standard model. When the SN Ia estimations are used, the history of the cosmic acceleration for the MPC model is consistent with the ΛCDM within the 1σ and 3σ CL. Thus, the MPC scenario is viable to explain the late cosmic acceleration without a DE component and its cosmological dynamics is almost indistinguishable from the standard model. Figure 10. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, n, and l parameters of the MPC model assuming a flat and Gaussian (hRiess) prior on h. Figure 10. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, n, and l parameters of the MPC model assuming a flat and Gaussian (hRiess) prior on h. Figure 11. View largeDownload slide Fitting to OHDhpl (top panel) and SN Ia data (bottom panel) using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines) and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and MPC model when a flat prior on h is considered. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 11. View largeDownload slide Fitting to OHDhpl (top panel) and SN Ia data (bottom panel) using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines) and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and MPC model when a flat prior on h is considered. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 12. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the MPC model and ΛCDM using the constraints from OHDhpl (top panel) and SN Ia data (bottom panel) when a flat prior on h is considered. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 12. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the MPC model and ΛCDM using the constraints from OHDhpl (top panel) and SN Ia data (bottom panel) when a flat prior on h is considered. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. 5 CONCLUSIONS AND OUTLOOKS In this paper, we analyse two alternatives to explain the late cosmic acceleration without a DE component: theOC and MPC models that are also excellent laboratories to study deviations from GR. The Cardassian models establish the modification of the canonical Friedmann equation as a consequence of a braneworld dynamics that emerges from novel ideas of the space–time dimensions and is based on a generalized Einstein–Hilbert action. To constrain the exponents n and the n − l of the OC and MPC models, we used 51 OHD, 740 SNIa data points of the JLA sample (fJLA), and 31 binned distance modulus of the compressed JLA sample (cJLA). The OHD compilation contains 31 points measured using the differential age technique in early-type galaxies and 20 points from clustering. These last points are biased due to an underlying ΛCDM cosmology to estimate the sound horizon at the drag epoch, which is used to compute H(z). Moreover, these data points are estimated taking into account very conservative systematic errors. Therefore, we constructed two homogenized and model-independent samples for the clustering points using a common rd obtained from Planck and WMAP measurements. We found that the different OHD samples provide consistent constraints on the OC and MPC parameters. In addition, there is no significant differences on the constraints obtained from the cJLA and those estimated from fJLA. Furthermore, we obtained consistent constraints at 3σ CL when different Gaussian priors on h are employed. We performed a joint analysis with the combination of cJLA and one homogenized OHD sample. Our results shown that the OC and MPC free parameters are consistent with the traditional dynamics dictated by the Friedmann equation (see Tables 3–6) containing a CC. However, in the Cardassian models the extra terms in the canonical Friedmann equation mimic the CC but it comes from the n-term of the energy–momentum tensor, unlike in the traditional form where the CC is added by hand in the Friedmann equation. Of course, those problems affecting the CC will be transferred to the interpretation of n-dimensional geometry and, as a consequence, to the emerging of the n-term of the energy–momentum tensor. Therefore, the idea is to interpret and to know the global topology of our Universe to generate a solution for the DE problem and the current Universe acceleration. ACKNOWLEDGEMENTS We thank the anonymous referee for thoughtful remarks and suggestions. J.M. acknowledges support from CONICYT/FONDECYT 3160674. M.H.A. acknowledges support from CONACYT PhD fellow, Consejo Zacatecano de Ciencia, Tecnología e Innovación (COZCYT), and Centro de Astrofísica de Valparaíso (CAV). M.H.A. thanks the staff of the Instituto de Física y Astronomía of the Universidad de Valparaíso where part of this work was done. M.A.G.-A. acknowledges support from CONACYT research fellow, Sistema Nacional de Investigadores (SNI), and Instituto Avanzado de Cosmología (IAC) collaborations. Footnotes 1 The name Cardassian refers to a humanoid race in Star Trek series, whose goal is the accelerated expansion of their evil empire. 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Uspekhi , 11, 381 CrossRef Search ADS   Zhai X.-h., Lin R.-h., Feng C.-j., Li X.-z., 2017a, Phys. Rev. D , 95, 104030 https://doi.org/10.1103/PhysRevD.95.104030 CrossRef Search ADS   Zhai Z., Blanton M., Slosar A., Tinker J., 2017b, ApJ , 850, 183 https://doi.org/10.3847/1538-4357/aa9888 CrossRef Search ADS   Zhang C., Zhang H., Yuan S., Zhang T.-J., Sun Y.-C., 2014, RA&A , 14, 1221 https://doi.org/10.1088/1674-4527/14/10/002 Zhang M.-J., Xia J.-Q., 2016, JCAP , 1612, 005 https://doi.org/10.1088/1475-7516/2016/12/005 CrossRef Search ADS   APPENDIX A: COMPRESSED JLA SAMPLE Table A1. Compressed JLA sample that contains 31 binned distance modulus fitted to the full JLA sample by Betoule et al. (2014). The first column is the binned redshift and the second column is the binned distance modulus. zb  μb  0.010  32.953 886 976  0.012  33.879 003 4661  0.014  33.842 140 7403  0.016  34.118 567 0426  0.019  34.593 445 9829  0.023  34.939 026 5264  0.026  35.252 096 3261  0.031  35.748 501 6537  0.037  36.069 787 6073  0.043  36.434 570 4737  0.051  36.651 110 5942  0.060  37.158 014 1133  0.070  37.430 173 2516  0.082  37.956 616 3488  0.097  38.253 254 0406  0.114  38.612 869 3372  0.134  39.067 850 7056  0.158  39.341 401 9038  0.186  39.792 143 6157  0.218  40.156 534 6033  0.257  40.564 956 0582  0.302  40.905 287 7824  0.355  41.421 417 4356  0.418  41.790 923 4574  0.491  42.231 461 0669  0.578  42.617 047 0706  0.679  43.052 731 4851  0.799  43.504 150 8283  0.940  43.972 573 4093  1.105  44.514 087 5789  1.300  44.821 867 4621  zb  μb  0.010  32.953 886 976  0.012  33.879 003 4661  0.014  33.842 140 7403  0.016  34.118 567 0426  0.019  34.593 445 9829  0.023  34.939 026 5264  0.026  35.252 096 3261  0.031  35.748 501 6537  0.037  36.069 787 6073  0.043  36.434 570 4737  0.051  36.651 110 5942  0.060  37.158 014 1133  0.070  37.430 173 2516  0.082  37.956 616 3488  0.097  38.253 254 0406  0.114  38.612 869 3372  0.134  39.067 850 7056  0.158  39.341 401 9038  0.186  39.792 143 6157  0.218  40.156 534 6033  0.257  40.564 956 0582  0.302  40.905 287 7824  0.355  41.421 417 4356  0.418  41.790 923 4574  0.491  42.231 461 0669  0.578  42.617 047 0706  0.679  43.052 731 4851  0.799  43.504 150 8283  0.940  43.972 573 4093  1.105  44.514 087 5789  1.300  44.821 867 4621  View Large © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press

The Cardassian expansion revisited: constraints from updated Hubble parameter measurements and type Ia supernova data

Monthly Notices of the Royal Astronomical Society, Volume 476 (1) – May 1, 2018
14 pages

/lp/ou_press/the-cardassian-expansion-revisited-constraints-from-updated-hubble-xTgmC0ls2y
Publisher
Oxford University Press
ISSN
0035-8711
eISSN
1365-2966
D.O.I.
10.1093/mnras/sty260
Publisher site
See Article on Publisher Site

Abstract

Abstract Motivated by an updated compilation of observational Hubble data (OHD) that consist of 51 points in the redshift range of 0.07 < z < 2.36, we study an interesting model known as Cardassian that drives the late cosmic acceleration without a dark energy component. Our compilation contains 31 data points measured with the differential age method by Jimenez & Loeb (2002), and 20 data points obtained from clustering of galaxies. We focus on two modified Friedmann equations: the original Cardassian (OC) expansion and the modified polytropic Cardassian (MPC). The dimensionless Hubble, E(z), and the deceleration parameter, q(z), are revisited in order to constrain the OC and MPC free parameters, first with the OHD and then contrasted with recent observations of type Ia supernova (SN Ia) using the compressed and full joint-light-analysis (JLA) samples (Betoule et al.). We also perform a joint analysis using the combination OHD plus compressed JLA. Our results show that the OC and MPC models are in agreement with the standard cosmology and naturally introduce a cosmological-constant-like extra term in the canonical Friedmann equation with the capability of accelerating the Universe without dark energy. cosmological parameters, dark energy, cosmology: observations 1 INTRODUCTION The cold dark matter (ΛCDM) with a cosmological constant (CC) model is the cornerstone of modern cosmology. It has shown an unprecedented success predicting and reproducing the dynamics and evolution of the Universe. ΛCDM is based on two important but unknown components, dark matter (DM) and dark energy (DE), which constitute ∼ 96 per cent of the total content of our Universe (Planck Collaboration XIII 2016). In this standard paradigm, the DE, responsible of the late cosmic acceleration, is supplied by a CC, which is associated with vacuum energy. Although several cosmological observations favour the CC, some theoretical problems arise when we try a renormalization of the quantum vacuum fluctuations using an appropriate cut-off at the Planck energy. However, the problem becomes insurmountable, giving a difference of ∼120 orders in magnitude between theory and observations (Weinberg 1989; Zeldovich 1968). In addition, the problem of coincidence, i.e. the similitude between the energy density of matter and DE at the present epoch, remains as an open question (Zeldovich 1968; Weinberg 1989). To overcome these problems, several alternatives to the CC are proposed, such as quintessence, phantom energy, Chaplygin gas, holographic DE, and Galileons (see Carroll 2001; Copeland, Sami & Tsujikawa 2006, for a complete review). Geometrical approaches are also used to explain the DE dynamics (i.e. brane theories) like Dvali, Gabadaze and Porrati (DGP; Deffayet, Dvali & Gabadadze 2002), Randall-Sundrum I and II (RSI, RSII; Randall & Sundrum 1999a,b), or f(R) theories (Buchdahl 1970; Starobinsky 1980; Cembranos 2009); each one having important pros and cons. An interesting alternative, closely related to geometrical models, is the Cardassian expansion model for which there is no DE and the late cosmic acceleration is driven by the modification of the Friedmann equation as H2 = f(ρ) (Xu 2012), where f(ρ) is a functional form of the energy density of the Universe. Freese & Lewis (2002) proposed f(ρ) ∝ ρ + ρn in order to obtain a late acceleration stage under certain conditions on the n parameter, naming the model as the Cardassian expansion1 [hereafter the original Cardassian (OC) model]. However, this expression can be naturally deduced from extra dimensional theories (DGP, RSI, RSII, etc.), which imprint the effects of a five-dimensional space–time (the bulk) in our four-dimensional (4D) space–time (the brane) at cosmological scales. In the case of the DGP model, a consequence of this kind of geometry is a density parameter that evolves as $$(\sqrt{\rho +\alpha }+\beta )^2$$, where α and β are constants related to the threshold between the brane and the bulk, allowing an accelerated epoch driven only by geometry. In the case of RS models, a quadratic term in the energy–momentum tensor modifies the right-hand side of the Friedmann equation as aρ + bρ2 (Shiromizu, Maeda & Sasaki 2000), with a correspondence to the Cardassian models when n = 2. Thus, the topological structure of the brane and the bulk can naturally produce the Cardassian Friedmann equation. Indeed, it is possible to obtain a n-energy–momentum tensor from a Gauss equation with a product of n-extrinsic curvatures, which leads to the ρn extra term in the Friedmann equation of the OC model. Therefore, the model motivation is based on extra dimensions arising from a fundamental theory (for an excellent review of extra dimensions models, see for instance Maartens 2004, or Maartens 2000, for a cosmological point of view). Another alternative interpretation is to consider a fluid (that may or may not be in an intrinsically 4D metric) with an extra contribution to the energy–momentum tensor (Gondolo & Freese 2003). Both interpretations are interesting and the standard cosmological dynamics can be mimicked without the need to postulate a DE component. In addition, we notice that it is possible to recover a CC when ρn → 1, without adding it by hand. An OC model generalization can be obtained by considering an additional exponent in the right-hand side of the Friedmann equation as f(ρ) ∝ ρ(1 + ρl(n−1))1/l that is called modified polytropic Cardassian (MPC) model by analogy with a fluid interpretation (Gondolo & Freese 2002). The Cardassian models are extensively studied in the literature. They have been tested with several cosmological observations (see e.g. Wang et al. 2003; Feng & Li 2010; Liang, Wu & Zhu 2011; Li, Wu & Yu 2012; Xu 2012; Wei, Ma & Wu 2015, and references therein). Wei et al. (2015) put constraints on the OC model parameters using a joint analysis of gamma-ray burst data and type Ia supernovae (SNe Ia) of the Union 2.1 sample (Suzuki et al. 2012). Recently, Magaña et al. (2015) used the strong lensing measurements in the galaxy cluster Abell 1689, baryon acoustic oscillations (BAOs), cosmic microwave background (CMB) data from 9 yr Wilkinson Microwave Anisotropy Probe (WMAP) observations (Hinshaw et al. 2013), and the SN Ia LOSS sample (Ganeshalingam, Li & Filippenko 2013) to constrain the MPC parameters. In this work, we revisit the Cardassian expansion models with an universe that contains baryons, DM, together with the radiation component. We explore two functional forms of the Friedmann equation: one with the OC parameter n (following Freese & Lewis 2002), and the other one considering also the l exponent (following Gondolo & Freese 2003). These Cardassian models are tested using an update sample of observational Hubble data (OHD) and the compressed joint-light-analysis (cJLA) SN Ia data by Betoule et al. (2014). As a final comment, while we were finalizing this paper, Zhai et al. (2017a) addressed a similar revision of the Cardassian models. While the main focus of Zhai et al. (2017a) is to match the Cardassian Friedmann equations to $$f(T, \mathcal {T})$$ theory through the action principle, our work focus on providing bounds to the Cardassian models using OHD (see also Zhai et al. 2017b). Nonetheless, the authors also provide constraints derived from SN Ia, BAO, and CMB data. The paper is organized as follows. In Section 2, the Cardassian cosmology is revisited, introducing two proposals for the Friedmann equation, which correspond to the OC and MPC models, and the deceleration parameter is calculated. In Section 3, we present the data and methodology in order to study the Cardassian models using OHD and SN Ia observations. In Section 4, we show the constraints for the free parameters presenting the novel contrast with the updated sample. Finally, Section 5 presents our conclusions and the possible outlooks into future studies. We will henceforth use units in which c = ℏ = 1 (unless explicitly written). 2 THE CARDASSIAN COSMOLOGY 2.1 OC model The OC model was introduced by Freese & Lewis (2002) to explain the accelerated expansion of the Universe without DE. Motivated by braneworld theory, this model modifies the Friedmann equation as   $$H^{2}=\frac{8\pi G \rho _{t}}{3} + B\rho _{t}^{n},$$ (1)where $$H=\dot{a}/a$$ is the Hubble parameter, a is the scale factor of the Universe, G is the Newtonian gravitational constant, B is a dimensional coupling constant that depends on the theory, and the total matter density is ρt = ρm + ρr. The recent Planck measurements (Planck Collaboration XVI 2014; Planck Collaboration XIII 2016) suggest a curvature energy density Ωk ≃ 0, thus we assume a flat geometry. The conservation equation is maintained in the traditional form:   $$\dot{\rho }+3H(\rho +p)=0.$$ (2)The matter density (DM and baryons), ρm = ρm0a−3, and the radiation density, ρr = ρr0a−4, evolution can be computed from equation (2). The second term in the right-hand side of equation (1), known as the Cardassian term, drives the universe to an accelerated phase if the exponent n satisfies n < 2/3. At early times, this corrective term is negligible and the dynamics of the universe is governed by the canonical term of the Friedmann equation. When the universe evolves, the traditional energy density and the one due to the Cardassian correction becomes equal at redshift $$z_{\text{Card}}\sim \mathcal {O}(1)$$. Later on, the Cardassian term begins to dominate the evolution of the universe and source the cosmic acceleration. The equation (1) reproduces the ΛCDM model for n = 0. As in the standard case, it is possible to define a new critical density for the OC model, ρNc, which satisfies the equation (1) and can be written as ρNc = ρcF(B, n), where ρc = 3H2/8πG is the standard critical density, and F(B, n) is a function that depends on the OC parameters and the components of the Universe. The Raychaudhuri equation can be written in the form:   $$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho _t+3p_t)-B\left[\left(\frac{3n}{2}-1\right)\rho _t^n+\frac{3}{2}n\rho _t^{n-1}p_t\right]\!,$$ (3)where equations (1) and (2) were used. From equation (1), it is possible to obtain the dimensionless Hubble parameter $$E^2(z)\equiv H^2(z)/H^2_0$$ as   $$E(z, {\mathbf {\boldsymbol\Theta }})^{2}= \Omega _{\mathrm{std}}+(1-\Omega _{m0}-\Omega _{r0})\left[\frac{\Omega _{\mathrm{std}}}{\Omega _{m0}+\Omega _{r0}}\right]^{n},$$ (4)where $${\mathbf {\boldsymbol\Theta} }=(\Omega _{m0},h,n)$$ is the free parameter vector to be constrained by the data, Ωr0 = ρr0/ρc is the current standard density parameter for the radiation component, Ωm0 = ρm0/ρc is the observed standard density parameter for matter (baryons and DM), and we define Ωstd ≡ Ωm0(1 + z)3 + Ωr0(1 + z)4. We compute Ωr0 = 2.469 × 10−5h−2(1 + 0.2271Neff) (Komatsu et al. 2011), where Neff = 3.04 is the standard number of relativistic species (Mangano et al. 2002). Notice that we have also imposed a flatness condition on the total content of the Universe (for further details on how to obtain equation (4), see Sen & Sen 2003a,b). The deceleration parameter, defined as $$q\equiv -\ddot{a}/aH^2$$, can be written as   \begin{eqnarray} q(z, \mathbf {\boldsymbol\Theta })&=&\frac{\Omega _{\mathrm{std}}-\frac{1}{2}\Omega _{m0}(1+z)^3}{E^2(z, \mathbf {\boldsymbol\Theta })}+\frac{1-\Omega _{m0}-\Omega _{r0}}{(\Omega _{m0}+\Omega _{r0})^n}\nonumber \\ &&\times\left[\left(\frac{3n}{2}-1\right)+\frac{n\Omega _{r0}}{2\Omega _{\text{std}}}(1+z)^4\right]\frac{\Omega _{\mathrm{std}}^n}{E^2(z, \mathbf {\boldsymbol\Theta })}. \end{eqnarray} (5) In order to investigate whether the OC model can drive the late cosmic acceleration, it is necessary to reconstruct the q(z) using the mean values for the $$\mathbf {\boldsymbol\Theta}$$ parameters. 2.2 MPC model Gondolo & Freese (2002, 2003) introduced a simple generalization of the Cardassian model, the MPC, by introducing an additional exponent l (see also Wang et al. 2003). The modified Friedmann equation with this generalization can be written as   $$H^{2} = \frac{8\pi G}{3} \rho _{t} \beta ^{1/l},$$ (6)where   $$\beta \equiv 1 + \left( \frac{\rho _{\text{Card}}}{\rho _{t}} \right)^{l(1-n)},$$ (7)and ρCard is the characteristic energy density, with n < 2/3 and l > 0. In concordance with the previous Friedmann equation (1) and following Planck Collaboration XVI (2014); Planck Collaboration XIII (2016), we also assume Ωk ≃ 0. The equation (6) reproduce the ΛCDM model for l = 1 and n = 0. Thus, the acceleration equation is   $$\frac{\ddot{a}}{a}= -\frac{4\pi G}{3}\rho _{t}\beta ^{1/l}+4\pi G(1-n)\rho _{t}\left(1-\frac{1}{\beta }\right)\beta ^{1/l}.$$ (8)The MPC model (equation 9) has been studied by several authors using different data with ρt = ρm (see e.g. Feng & Li 2010) and also with ρt = ρm + ρr together with a curvature term (Shi, Huang & Lu 2012). Here, we consider a flat MPC with matter and radiation components. After straightforward calculations, the dimensionless $$E^{2}(z, \boldsymbol {\boldsymbol\Theta })$$ parameter reads as   \begin{eqnarray} &&E^{2}(z, \boldsymbol {\boldsymbol\Theta })=\Omega _{r0}(1+z)^{4} + \Omega _{m0}(1+z)^{3}\beta (z, \boldsymbol {\boldsymbol\Theta })^{1/l}, \quad \end{eqnarray} (9)where   $$\beta (z, \boldsymbol {\boldsymbol\Theta })\equiv 1 + \left[ \left( \frac{1-\Omega _{r0}}{\Omega _{m0}}\right)^l - 1 \right](1+z)^{3l(n - 1)},$$ (10)being $$\boldsymbol {\boldsymbol\Theta }=(\Omega _{m0},h,l,n)$$, the free parameter vector to be fitted by the data. In addition, $$q(z, \boldsymbol {\boldsymbol\Theta })$$ can be written as   \begin{eqnarray} q(z, \mathbf {\boldsymbol\Theta })&=& \frac{\Omega _{m0}\beta (z, \mathbf {\boldsymbol\Theta })^{1/l}}{2E^2(z, \mathbf {\boldsymbol\Theta })}\left[1-3(1-n)\left(1-\frac{1}{\beta (z, \mathbf {\boldsymbol\Theta })}\right)\right] \nonumber \\ &&\times\,(1+z)^3+\frac{\Omega _{r0}}{E^2(z, \mathbf {\boldsymbol\Theta })}(1+z)^4. \end{eqnarray} (11)We use the $$\mathbf {\boldsymbol\Theta}$$ mean values in the last expression to reconstruct the deceleration parameter q(z) and investigate whether the MPC model is consistent with a late cosmic acceleration. 3 DATA AND METHODOLOGY The OC and MPC model parameters are constrained using an updated OHD sample, which contains 51 data points, and the compressed SN Ia data set from the JLA full sample by Betoule et al. (2014), which contains 31 data points. In the following, we briefly introduce these data sets. 3.1 Observational Hubble data The ‘differential age’ (DA) method proposed by Jimenez & Loeb (2002) allows us to measure the expansion rate of the Universe at redshift z, i.e. H(z). This technique compares the ages of early-type galaxies (i.e. without ongoing star formation) with similar metallicity and separated by a small redshift interval (for instance, Moresco et al. 2012, measure Δz ∼ 0.04 at z < 0.4 and Δz ∼ 0.3 at z > 0.4). Thus, a H(z) point can be estimated using   $$H(z)=-\frac{1}{1+z}\frac{{\rm{d}z}}{{\rm {d}}t},$$ (12)where dz/dt is measured using the 4000 Å break (D4000) feature as function of redshift. A strong D4000 break depends on the metallicity and the age of the stellar population of the early-type galaxy. Thus, the technique by Jimenez & Loeb (2002) offers to directly measure the Hubble parameter using spectroscopic dating of passively-evolving galaxy to compare their ages and metallicities, providing H(z) measurements that are model independent. These H(z) points are given by different authors as Zhang et al. (2014), Moresco et al. (2012), Moresco (2015), Moresco et al. (2016) and Stern et al. (2010), and constitute the majority of our sample (31 points). In addition, we use 20 points from BAO measurements, although some of them being correlated because they either belong to the same analysis or there is overlapping among the galaxy samples; throughout this work, we assume that they are independent measurements. Moreover, some data points are biased because they are estimated using a sound horizon, rd,2 at the drag epoch, zd, which depends on the cosmological model (Melia & López-Corredoira 2017). Points provided by different authors use different values for the rd in clustering measurements, for instance Anderson et al. (2014) take 153.19 Mpc while Gaztanaga, Cabre & Hui (2009) choose 153.3 Mpc, etc. Table 1 shows an updated compilation of OHD accumulating a total of 51 points (other recent compilations are provided by Yu & Wang 2016; Zhang & Xia 2016; Farooq et al. 2017). We have included all the points of the previous references, although priority has been given to the measurements that comes from the DA method and have also been measured with clustering at the same redshift. As reference to compare our results, we also give the data point by Riess et al. (2016) who measured a Hubble constant H0 with 2.4 per cent of uncertainty. Authors argue that this improvement is due to a better calibration (using Cepheids) of the distance to 11 SN Ia host galaxies, reducing the error by almost 1 per cent. We use this sample to constrain the free parameters of the OC and MPC models and look for an alternative solution to the accelerated expansion of the Universe. The figure-of-merit for the OHD is given by   $$\chi _{\mbox{OHD}}^2 = \sum _{i=1}^{N_{\text{OHD}}} \frac{ \left[ H(z_{i}) -H_{\text{obs}}(z_{i})\right]^2 }{ \sigma _{H_i}^{2} },$$ (13) where NOHD is the number of the observational Hubble parameter Hobs(zi) at zi, $$\sigma _{H_i}$$ is its error, and H(zi) is the theoretical value for a given model. Table 1. 52 Hubble parameter measurements H(z) (in km s−1Mpc−1) and their errors, σH, at redshift z. The first point is not included in the Markov chain Monte Carlo analysis, it was only considered as a Gaussian prior in some tests. The method column refers as to how to H(z) was obtained: DA stands for differential age method, and clustering comes from BAO measurements. z  H(z)  σH  Reference  Method    (km s− 1 Mpc− 1)  (km s−1 Mpc−1)      0  73.24  1.74  Riess et al. (2016)  SN Ia/Cepheid  0.07  69  19.6  Zhang et al. (2014)  DA  0.1  69  12  Stern et al. (2010)  DA  0.12  68.6  26.2  Zhang et al. (2014)  DA  0.17  83  8  Stern et al. (2010)  DA  0.1791  75  4  Moresco et al. (2012)  DA  0.1993  75  5  Moresco et al. (2012)  DA  0.2  72.9  29.6  Zhang et al. (2014)  DA  0.24  79.69  2.65  Gaztanaga et al. (2009)  Clustering  0.27  77  14  Stern et al. (2010)  DA  0.28  88.8  36.6  Zhang et al. (2014)  DA  0.3  81.7  6.22  Oka et al. (2014)  Clustering  0.31  78.17  4.74  Wang et al. (2017)  Clustering  0.35  82.7  8.4  Chuang & Wang (2013)  Clustering  0.3519  83  14  Moresco et al. (2012)  DA  0.36  79.93  3.39  Wang et al. (2017)  Clustering  0.38  81.5  1.9  Alam et al. (2017)  Clustering  0.3802  83  13.5  Moresco et al. (2016)  DA  0.4  95  17  Stern et al. (2010)  DA  0.4004  77  10.2  Moresco et al. (2016)  DA  0.4247  87.1  11.2  Moresco et al. (2016)  DA  0.43  86.45  3.68  Gaztanaga et al. (2009)  Clustering  0.44  82.6  7.8  Blake et al. (2012)  Clustering  0.4497  92.8  12.9  Moresco et al. (2016)  DA  0.47  89  34  Ratsimbazafy et al. (2017)  DA  0.4783  80.9  9  Moresco et al. (2016)  DA  0.48  97  62  Stern et al. (2010)  DA  0.51  90.4  1.9  Alam et al. (2017)  Clustering  0.52  94.35  2.65  Wang et al. (2017)  Clustering  0.56  93.33  2.32  Wang et al. (2017)  Clustering  0.57  92.9  7.8  Anderson et al. (2014)  Clustering  0.59  98.48  3.19  Wang et al. (2017)  Clustering  0.5929  104  13  Moresco et al. (2012)  DA  0.6  87.9  6.1  Blake et al. (2012)  Clustering  0.61  97.3  2.1  Alam et al. (2017)  Clustering  0.64  98.82  2.99  Wang et al. (2017)  Clustering  0.6797  92  8  Moresco et al. (2012)  DA  0.73  97.3  7  Blake et al. (2012)  Clustering  0.7812  105  12  Moresco et al. (2012)  DA  0.8754  125  17  Moresco et al. (2012)  DA  0.88  90  40  Stern et al. (2010)  DA  0.9  117  23  Stern et al. (2010)  DA  1.037  154  20  Moresco et al. (2012)  DA  1.3  168  17  Stern et al. (2010)  DA  1.363  160  33.6  Moresco (2015)  DA  1.43  177  18  Stern et al. (2010)  DA  1.53  140  14  Stern et al. (2010)  DA  1.75  202  40  Stern et al. (2010)  DA  1.965  186.5  50.4  Moresco (2015)  DA  2.33  224  8  Bautista et al. (2017)  Clustering  2.34  222  7  Delubac et al. (2015)  Clustering  2.36  226  8  Font-Ribera et al. (2014)  Clustering  z  H(z)  σH  Reference  Method    (km s− 1 Mpc− 1)  (km s−1 Mpc−1)      0  73.24  1.74  Riess et al. (2016)  SN Ia/Cepheid  0.07  69  19.6  Zhang et al. (2014)  DA  0.1  69  12  Stern et al. (2010)  DA  0.12  68.6  26.2  Zhang et al. (2014)  DA  0.17  83  8  Stern et al. (2010)  DA  0.1791  75  4  Moresco et al. (2012)  DA  0.1993  75  5  Moresco et al. (2012)  DA  0.2  72.9  29.6  Zhang et al. (2014)  DA  0.24  79.69  2.65  Gaztanaga et al. (2009)  Clustering  0.27  77  14  Stern et al. (2010)  DA  0.28  88.8  36.6  Zhang et al. (2014)  DA  0.3  81.7  6.22  Oka et al. (2014)  Clustering  0.31  78.17  4.74  Wang et al. (2017)  Clustering  0.35  82.7  8.4  Chuang & Wang (2013)  Clustering  0.3519  83  14  Moresco et al. (2012)  DA  0.36  79.93  3.39  Wang et al. (2017)  Clustering  0.38  81.5  1.9  Alam et al. (2017)  Clustering  0.3802  83  13.5  Moresco et al. (2016)  DA  0.4  95  17  Stern et al. (2010)  DA  0.4004  77  10.2  Moresco et al. (2016)  DA  0.4247  87.1  11.2  Moresco et al. (2016)  DA  0.43  86.45  3.68  Gaztanaga et al. (2009)  Clustering  0.44  82.6  7.8  Blake et al. (2012)  Clustering  0.4497  92.8  12.9  Moresco et al. (2016)  DA  0.47  89  34  Ratsimbazafy et al. (2017)  DA  0.4783  80.9  9  Moresco et al. (2016)  DA  0.48  97  62  Stern et al. (2010)  DA  0.51  90.4  1.9  Alam et al. (2017)  Clustering  0.52  94.35  2.65  Wang et al. (2017)  Clustering  0.56  93.33  2.32  Wang et al. (2017)  Clustering  0.57  92.9  7.8  Anderson et al. (2014)  Clustering  0.59  98.48  3.19  Wang et al. (2017)  Clustering  0.5929  104  13  Moresco et al. (2012)  DA  0.6  87.9  6.1  Blake et al. (2012)  Clustering  0.61  97.3  2.1  Alam et al. (2017)  Clustering  0.64  98.82  2.99  Wang et al. (2017)  Clustering  0.6797  92  8  Moresco et al. (2012)  DA  0.73  97.3  7  Blake et al. (2012)  Clustering  0.7812  105  12  Moresco et al. (2012)  DA  0.8754  125  17  Moresco et al. (2012)  DA  0.88  90  40  Stern et al. (2010)  DA  0.9  117  23  Stern et al. (2010)  DA  1.037  154  20  Moresco et al. (2012)  DA  1.3  168  17  Stern et al. (2010)  DA  1.363  160  33.6  Moresco (2015)  DA  1.43  177  18  Stern et al. (2010)  DA  1.53  140  14  Stern et al. (2010)  DA  1.75  202  40  Stern et al. (2010)  DA  1.965  186.5  50.4  Moresco (2015)  DA  2.33  224  8  Bautista et al. (2017)  Clustering  2.34  222  7  Delubac et al. (2015)  Clustering  2.36  226  8  Font-Ribera et al. (2014)  Clustering  View Large 3.1.1 An homogeneous OHD sample As mentioned above, the OHD from clustering (BAO features) are biased due to an underlying ΛCDM cosmology to estimate rd. Different authors used different values in the cosmological parameters and obtained different sound horizons at the drag epoch, which are used to break the degeneracy in Hrd. Furthermore, the determination of H(z) from BAO features is computed taking into account very conservative systematic errors (see the discussion by Leaf & Melia 2017; Melia & López-Corredoira 2017). As a first attempt to homogenize and achieve model independence for the OHD obtained from clustering, we take the value Hrd for each data point and assume a common value rd for the entire data set. We consider two rd estimations: rdpl = 147.33 ± 0.49 Mpc and rdw9 = 152.3 ± 1.3 Mpc from the most recent Planck (Planck Collaboration XIII 2016) and WMAP9 (Bennett et al. 2013) measurements, respectively. In addition, we also take into account three other sources of errors that could affect rd due to its contamination by a cosmological model. The first one comes from the error of each reported value. The second error considers the possible range of rd values provided by separate CMB measurements, i.e. the difference between the sound horizon given by WMAP9 and Planck. This error is the one producing the largest impact on the rd mean value (3.37  per cent and 3.26  per cent for the Planck and WMAP9 data point, respectively). The last error to take into account is the difference between rd used to obtain the OHD and the one that would be obtained if we assume another cosmological model instead of ΛCDM. Hereafter, we use the one obtained for a DE constant equation-of-state (w) CDM model, rdωcdm = 148.38 Mpc (the cosmological parameters for this model are provided by Neveu et al. 2017). Adding in quadrature the percentage for these three errors, we obtain rdpl = 147.33 ± 5.08 Mpc and rdw9 = 152.3 ± 6.42 Mpc. Finally, we propagate this new error to the quantity H(z) to secure a new homogenized and model-independent sample (Table 2). Table 2. Homogenized model-independent OHD from clustering (in km s−1Mpc−1) and its error, σH, at redshift z. The first and second columns were obtained using the sound horizon in the drag epoch from Planck and WMAP measurements, respectively. z  H(z) ± σH(rdpl)  H(z) ± σH(rdw9)    (km s− 1Mpc− 1)  (km s− 1Mpc− 1)  0.24  82.37 ± 3.94  79.69 ± 4.28  0.3  78.83 ± 6.58  76.26 ± 6.63  0.31  78.39 ± 5.46  75.83 ± 5.60  0.35  88.10 ± 9.45  85.23 ± 9.37  0.36  80.16 ± 4.37  77.54 ± 4.63  0.38  81.74 ± 3.40  79.08 ± 3.81  0.43  89.36 ± 4.89  86.44 ± 5.18  0.44  85.48 ± 8.59  82.69 ± 8.55  0.51  90.67 ± 3.66  87.71 ± 4.13  0.52  94.61 ± 4.20  91.52 ± 4.63  0.56  93.59 ± 3.96  90.54 ± 4.42  0.57  96.59 ± 8.76  93.44 ± 8.78  0.59  98.75 ± 4.66  95.53 ± 5.07  0.6  90.96 ± 7.04  87.99 ± 7.14  0.61  97.59 ± 3.97  94.41 ± 4.47  0.64  99.09 ± 4.53  95.86 ± 4.97  0.73  100.69 ± 8.03  97.40 ± 8.12  2.33  223.99 ± 11.12  216.69 ± 11.97  2.34  222.105 ± 10.38  214.85 ± 11.31  2.36  226.24 ± 11.18  218.86 ± 12.05  z  H(z) ± σH(rdpl)  H(z) ± σH(rdw9)    (km s− 1Mpc− 1)  (km s− 1Mpc− 1)  0.24  82.37 ± 3.94  79.69 ± 4.28  0.3  78.83 ± 6.58  76.26 ± 6.63  0.31  78.39 ± 5.46  75.83 ± 5.60  0.35  88.10 ± 9.45  85.23 ± 9.37  0.36  80.16 ± 4.37  77.54 ± 4.63  0.38  81.74 ± 3.40  79.08 ± 3.81  0.43  89.36 ± 4.89  86.44 ± 5.18  0.44  85.48 ± 8.59  82.69 ± 8.55  0.51  90.67 ± 3.66  87.71 ± 4.13  0.52  94.61 ± 4.20  91.52 ± 4.63  0.56  93.59 ± 3.96  90.54 ± 4.42  0.57  96.59 ± 8.76  93.44 ± 8.78  0.59  98.75 ± 4.66  95.53 ± 5.07  0.6  90.96 ± 7.04  87.99 ± 7.14  0.61  97.59 ± 3.97  94.41 ± 4.47  0.64  99.09 ± 4.53  95.86 ± 4.97  0.73  100.69 ± 8.03  97.40 ± 8.12  2.33  223.99 ± 11.12  216.69 ± 11.97  2.34  222.105 ± 10.38  214.85 ± 11.31  2.36  226.24 ± 11.18  218.86 ± 12.05  View Large 3.2 Type Ia supernovae The SN Ia observations supply the evidence of the accelerated expansion of the Universe. They have been considered a perfect standard candle to measure the geometry and dynamics of the Universe and have been widely used to constrain alternatives cosmological models to explain the late-time cosmic acceleration. Currently, there are several compiled SN Ia samples, for instance, the Union 2.1 compilation by Suzuki et al. (2012) that consists of 580 points in the redshift range of 0.015 < z < 1.41, and the Lick Observatory Supernova Search (LOSS) sample containing 586 SN Ia in the redshift range of 0.01 < z < 1.4 (Ganeshalingam et al. 2013). Recently, Betoule et al. (2014) presented the so-called full JLA (fJLA) sample that contains 740 points spanning a redshift range of 0.01 < z < 1.2. The same authors also provide the information of the fJLA data in a compressed set (cJLA) of 31 binned distance modulus μb spanning a redshift range of 0.01 < z < 1.3, which still remains accurate for some models where the isotropic luminosity distance evolves slightly with redshift. For instance, when the cJLA is used in combination with other cosmological data, the difference between fJLA and cJLA in the mean values for the wCDM model parameters is at most 0.018σ. Here, we use both, the fJLA and cJLA samples, to constrain the parameters of the OC and MPC models. 3.2.1 Full JLA sample As mentioned, the full JLA sample contains 740 confirmed SN Ia in the redshift interval 0.01 < z < 1.2, which is one of the most recent and reliable SN Ia samples. We use this sample to constrain the parameters of both Cardassian models. The function of merit for the fJLA sample is calculated as   $$\chi ^{2}_{fJLA}={\left(\hat{\mu } - \mu _{\text{Card}}\right)^{\dagger }\mathrm{C_{\eta }^{-1}}\left( \hat{\mu } - \mu _{\text{Card}} \right)},$$ (14)where μCard = 5log10(dL/10 pc), and Cη is the covariance matrix3 of $$\mathbf {\hat{\mu }}$$ provided by Betoule et al. (2014) and is constructed using   \begin{eqnarray} \mathbf {C_{\eta }}&=& \left( \mathbf {C}_{\text{cal}} + \mathbf {C}_{\text{model}} + \mathbf {C}_{\text{bias}} + \mathbf {C}_{\text{host}} + \mathbf {C}_{\text{dust}} \right) \nonumber \\ && +\, \left( \mathbf {C}_{\text{pecvel}} + \mathbf {C}_{\text{nonIa}} \right) + \mathbf {C}_{\text{stat}}, \end{eqnarray} (15)where $$\mathbf {C}_{\text{cal}}, \mathbf {C}_{\text{model}}, \mathbf {C}_{\text{bias}}, \mathbf {C}_{\text{host}}, \mathbf {C}_{\text{dust}}$$ are systematic uncertainty matrices associated with the calibration, the light-curve model, the bias correction, the mass step, and dust uncertainties, respectively. $$\mathbf {C}_{\text{pecvel}}$$ and $$\mathbf {C}_{\text{nonIa}}$$ corresponds to systematics uncertainties in the peculiar velocity corrections and the contamination of the Hubble diagram by non-Ia events, respectively, $$\mathbf {C}_{\text{stat}}$$ corresponds to an statistical uncertainty obtained from error propagation of the light-curve fit uncertainties. Finally, $$\hat{\mu}$$ is given by   $$\hat{\mu } = m_{b}^{\star } - \left( M_{B} - \alpha \times X_{1} + \beta \times C \right),$$ (16)where $$m_{b}^{\star}$$ corresponds to the observed peak magnitude, α, β, and MB are nuisance parameters in the distance estimates. The X1 and C variables describe the time stretching of the light curve and the Supernova colour at maximum brightness, respectively. The absolute magnitude MB is related to the host stellar mass (Mstellar) by the step function:   \begin{eqnarray} M_{B} = \left\lbrace \begin{array}{cc}M_{B}^{1} & \rm {if} \ M_{stellar} < 10^{10} M_{\odot } , \\ M_{B}^{1} + \Delta _{M} & \rm {otherwise.} \\ \end{array} \right. \end{eqnarray} (17)By replacing equations (4), (9), (15), and (16) in equation (14), we obtain the explicit figure-of-merit $$\chi ^{2}_{\mathrm{fJLA}}$$ for the Cardassian models. 3.2.2 Compressed form of the JLA sample Table A1 shows the 31 binned distance modulus at the binned redshift zb. The function of merit for the cJLA sample is calculated as   $$\chi ^{2}_{{cJLA}}=\boldsymbol {r}^{\dagger}\mathrm{\bf C}_{b}^{-1}\boldsymbol {r},$$ (18)where Cb is the covariance matrix4 provided by Betoule et al. (2014), and $$\boldsymbol {r}$$ is given by   $$\boldsymbol {r}=\boldsymbol {\mu }_{b}-M-\log _{10}d_{L}(\boldsymbol {z}_{b}, \boldsymbol {\boldsymbol\Theta }),$$ (19)where M is a nuisance parameter and dL is the luminosity distance given by   $$d_{L}=(1+z)\frac{c}{H_{0}}\int ^{0}_{z}\frac{\mathrm{dz}^{\prime }}{E(z^{\prime }, \boldsymbol {\boldsymbol\Theta })}.$$ (20)By replacing equations (4) and (9) in the last expression, we obtain the explicit figure of merit $$\chi ^{2}_{\mathrm{cJLA}}$$ for the OC and MPC models. 4 RESULTS A Markov chain Monte Carlo (MCMC) Bayesian statistical analysis was performed to estimate the (Ωm0, h, n) and the (Ωm0, h, n, l) parameters for the OC and MPC models, respectively. The constructed Gaussian likelihood function for each data set are given by $$\mathcal {L}_{\mathrm{OHD}}\propto \exp (-\chi _{\mathrm{OHD}}^{2}/2)$$, $$\mathcal {L}_{\mathrm{cJLA}}\propto \exp (-\chi _{\mathrm{cJLA}}^{2}/2)$$, $$\mathcal {L}_{\mathrm{fJLA}}\propto \exp (-\chi _{\mathrm{fJLA}}^{2}/2)$$, and $$\mathcal {L}_{\mathrm{joint}}\propto \exp (-\chi _{\mathrm{tot}}^{2}/2)$$, where χtot2 = χOHD2 + χcJLA2. We use the Affine Invariant MCMC Ensemble sampler from the emceepython module (Foreman-Mackey et al. 2013). In all our computations, we consider 3000 steps to stabilize the estimations (burn-in phase), 6000 MCMC steps and 1000 walkers that are initialized in a small ball around the expected points of maximum probability, is estimated with a differential evolution method. For both, OC and MPC models, we assume the following flat priors: Ωm0[0, 1], and n[−1, 2/3]. For the l MPC parameter, we consider the flat prior [0, 6]. For the h parameter three priors are considered: a flat prior [0, 1], and two Gaussian priors, one by Riess et al. (2016, the first point in Table 1), and the other one by Planck Collaboration XIII (2016) from Planck 2015 measurements (h = 0.678 ± 0.009). When the cJLA data are used, we also take a flat prior on the nuisance parameter M[−1, 1]. The following flat priors α[0, 2], β[0, 4.0], $$M^{1}_{B}[-20,-18]$$, and ΔM[−0.1, 0.1] are considered when the fJLA sample is employed. To judge the convergence of the sampler, we ask that the acceptance fraction is in the [0.2–0.5] range and check the autocorrelation time that is found to be $$\mathcal {O}(60\text{--}80)$$. We carry out four runs using different OHD sets: the full observational sample given in Table 1, the 31 data points obtained using the DA method (OHDDA), and two samples containing the DA points plus those homogenized points from clustering using a common rd estimated from Planck and WMAP measurements (Table 2). We also estimate the OC and MPC parameters using both the cJLA and fJLA samples. Moreover, we perform a joint analysis considering each OHD sample and the cJLA sample. Tables 3 and 4 provide the best fits for the OC and MPC parameters, respectively, using the different data sets and priors on h. Tables 5 and 6 give the constraints from the following joint analysis: OHD+cJLA (J1), OHDDA+cJLA (J2), OHDhpl+CJLA (J3), and OHDhw9+CJLA (J4). We also give the minimum chi-square, χmin, and the reduced χred = χmin/d.o.f, where the degree of freedom (d.o.f.) is the difference between the number of data points and the free parameters. Table 3. Mean values for the OC parameters (Ωm0, h, n) derived from OHD and SN Ia data of the cJLA and fJLA samples. OC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.37  15.22  21.25  22.52  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.52  0.54  0.44  0.46  1.22  0.93  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.30^{+0.06}_{-0.06}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.25^{+0.03}_{-0.03}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.65^{+0.03}_{-0.03}$$  $$0.69^{+0.06}_{-0.05}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.72^{+0.19}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.26^{+0.16}_{-0.15}$$  $$-0.19^{+0.51}_{-0.50}$$  $$0.23^{+0.20}_{-0.20}$$  $$0.16^{+0.22}_{-0.22}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.07^{+0.5}_{-0.66}$$  $$-18.96^{+0.49}_{-0.64}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  28.86  14.47  22.83  23.91  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.60  0.51  0.47  0.49  1.22  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.31^{+0.03}_{-0.04}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.01^{+0.08}_{-0.08}$$  $$-0.43^{+0.28}_{-0.30}$$  $$-0.02^{+0.09}_{-0.10}$$  $$-0.11^{+0.10}_{-0.11}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.24  14.53  20.79  22.04  32.95  –  $$\chi ^{2}_{\text{red}}$$  0.52  0.51  0.43  0.45  1.22  –  Ωm0  $$0.26^{+0.01}_{-0.01}$$  $$0.33^{+0.05}_{-0.07}$$  $$0.26^{+0.02}_{-0.02}$$  $$0.25^{+0.02}_{-0.02}$$  $$0.22^{+0.11}_{-0.12}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$0.15^{+0.06}_{-0.06}$$  $$-0.05^{+0.27}_{-0.32}$$  $$0.17^{+0.08}_{-0.08}$$  $$0.09^{+0.08}_{-0.09}$$  $$0.16^{+0.17}_{-0.26}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  OC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.37  15.22  21.25  22.52  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.52  0.54  0.44  0.46  1.22  0.93  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.30^{+0.06}_{-0.06}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.25^{+0.03}_{-0.03}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.65^{+0.03}_{-0.03}$$  $$0.69^{+0.06}_{-0.05}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.66^{+0.04}_{-0.03}$$  $$0.72^{+0.19}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.26^{+0.16}_{-0.15}$$  $$-0.19^{+0.51}_{-0.50}$$  $$0.23^{+0.20}_{-0.20}$$  $$0.16^{+0.22}_{-0.22}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.07^{+0.5}_{-0.66}$$  $$-18.96^{+0.49}_{-0.64}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  28.86  14.47  22.83  23.91  32.95  682.28  $$\chi ^{2}_{\text{red}}$$  0.60  0.51  0.47  0.49  1.22  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.31^{+0.03}_{-0.04}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.24^{+0.01}_{-0.01}$$  $$0.22^{+0.11}_{-0.12}$$  $$0.22^{+0.11}_{-0.12}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.01^{+0.08}_{-0.08}$$  $$-0.43^{+0.28}_{-0.30}$$  $$-0.02^{+0.09}_{-0.10}$$  $$-0.11^{+0.10}_{-0.11}$$  $$0.16^{+0.17}_{-0.26}$$  $$0.16^{+0.18}_{-0.26}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.24  14.53  20.79  22.04  32.95  –  $$\chi ^{2}_{\text{red}}$$  0.52  0.51  0.43  0.45  1.22  –  Ωm0  $$0.26^{+0.01}_{-0.01}$$  $$0.33^{+0.05}_{-0.07}$$  $$0.26^{+0.02}_{-0.02}$$  $$0.25^{+0.02}_{-0.02}$$  $$0.22^{+0.11}_{-0.12}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$0.15^{+0.06}_{-0.06}$$  $$-0.05^{+0.27}_{-0.32}$$  $$0.17^{+0.08}_{-0.08}$$  $$0.09^{+0.08}_{-0.09}$$  $$0.16^{+0.17}_{-0.26}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  View Large Table 4. Mean values for the MPC parameters (Ωm0, h, n, l) derived from OHD and SN Ia data of the cJLA and fJLA samples. MPC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.31  17.95  21.17  22.98  33.76  682.92  $$\chi ^{2}_{\text{red}}$$  0.53  0.66  0.45  0.48  1.29  0.93  Ωm0  $$0.25^{+0.04}_{-0.04}$$  $$0.32^{+0.06}_{-0.07}$$  $$0.25^{+0.04}_{-0.05}$$  $$0.25^{+0.04}_{-0.04}$$  $$0.22^{+0.12}_{-0.13}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.64^{+0.03}_{-0.02}$$  $$0.68^{+0.07}_{-0.05}$$  $$0.65^{+0.03}_{-0.03}$$  $$0.65^{+0.04}_{-0.03}$$  $$0.72^{+0.18}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.17^{+0.34}_{-0.68}$$  $$0.10^{+0.38}_{-0.60}$$  $$0.25^{+0.29}_{-0.68}$$  $$0.15^{+0.34}_{-0.65}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.33^{+0.09}_{-0.50}$$  l  $$0.77^{+1.45}_{-0.43}$$  $$2.13^{+2.34}_{-1.33}$$  $$0.95^{+1.90}_{-0.58}$$  $$0.92^{+1.66}_{-0.52}$$  $$2.61^{+2.27}_{-1.83}$$  $$2.09^{+2.51}_{-1.49}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.08^{+0.50}_{-0.67}$$  $$-18.97^{+0.49}_{-0.65}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  27.75  14.92  22.40  23.42  33.75  683.17  $$\chi ^{2}_{\text{red}}$$  0.59  0.55  0.47  0.49  1.29  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.32^{+0.03}_{-0.04}$$  $$0.24^{+0.02}_{-0.02}$$  $$0.23^{+0.02}_{-0.02}$$  $$0.22^{+0.12}_{-0.14}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.34^{+0.40}_{-0.42}$$  $$-0.03^{+0.24}_{-0.49}$$  $$-0.19^{+0.39}_{-0.50}$$  $$-0.28^{+0.39}_{-0.46}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.34^{+0.08}_{-0.47}$$  l  $$0.62^{+0.49}_{-0.20}$$  $$2.12^{+2.29}_{-1.20}$$  $$0.75^{+0.80}_{-0.30}$$  $$0.77^{+0.73}_{-0.28}$$  $$2.60^{+2.27}_{-1.81}$$  $$2.26^{+2.44}_{-1.63}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.03  14.70  20.84  21.96  33.79  –  $$\chi ^{2}_{\text{red}}$$  0.53  0.54  0.44  0.46  1.29  –  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.35^{+0.05}_{-0.08}$$  $$0.26^{+0.03}_{-0.03}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.22^{+0.12}_{-0.13}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$-0.01^{+0.35}_{-0.57}$$  $$0.24^{+0.19}_{-0.53}$$  $$0.16^{+0.26}_{-0.60}$$  $$0.03^{+0.31}_{-0.58}$$  $$0.36^{+0.07}_{-0.36}$$  –  l  $$0.71^{+0.95}_{-0.34}$$  $$2.08^{+2.44}_{-1.37}$$  $$0.96^{+1.55}_{-0.55}$$  $$0.87^{+1.22}_{-0.45}$$  $$2.59^{+2.27}_{-1.8}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  MPC model  Parameter  OHD  OHDDA  OHDhpl  OHDhw9  cJLA  fJLA  Flat prior on h  $$\chi ^{2}_{\text{min}}$$  25.31  17.95  21.17  22.98  33.76  682.92  $$\chi ^{2}_{\text{red}}$$  0.53  0.66  0.45  0.48  1.29  0.93  Ωm0  $$0.25^{+0.04}_{-0.04}$$  $$0.32^{+0.06}_{-0.07}$$  $$0.25^{+0.04}_{-0.05}$$  $$0.25^{+0.04}_{-0.04}$$  $$0.22^{+0.12}_{-0.13}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.64^{+0.03}_{-0.02}$$  $$0.68^{+0.07}_{-0.05}$$  $$0.65^{+0.03}_{-0.03}$$  $$0.65^{+0.04}_{-0.03}$$  $$0.72^{+0.18}_{-0.19}$$  $$0.72^{+0.18}_{-0.18}$$  n  $$0.17^{+0.34}_{-0.68}$$  $$0.10^{+0.38}_{-0.60}$$  $$0.25^{+0.29}_{-0.68}$$  $$0.15^{+0.34}_{-0.65}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.33^{+0.09}_{-0.50}$$  l  $$0.77^{+1.45}_{-0.43}$$  $$2.13^{+2.34}_{-1.33}$$  $$0.95^{+1.90}_{-0.58}$$  $$0.92^{+1.66}_{-0.52}$$  $$2.61^{+2.27}_{-1.83}$$  $$2.09^{+2.51}_{-1.49}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.08^{+0.50}_{-0.67}$$  $$-18.97^{+0.49}_{-0.65}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.732 ± 0.017  $$\chi ^{2}_{\text{min}}$$  27.75  14.92  22.40  23.42  33.75  683.17  $$\chi ^{2}_{\text{red}}$$  0.59  0.55  0.47  0.49  1.29  0.93  Ωm0  $$0.24^{+0.01}_{-0.01}$$  $$0.32^{+0.03}_{-0.04}$$  $$0.24^{+0.02}_{-0.02}$$  $$0.23^{+0.02}_{-0.02}$$  $$0.22^{+0.12}_{-0.14}$$  $$0.22^{+0.12}_{-0.13}$$  h  $$0.71^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.72^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  $$0.73^{+0.01}_{-0.01}$$  n  $$-0.34^{+0.40}_{-0.42}$$  $$-0.03^{+0.24}_{-0.49}$$  $$-0.19^{+0.39}_{-0.50}$$  $$-0.28^{+0.39}_{-0.46}$$  $$0.36^{+0.07}_{-0.33}$$  $$0.34^{+0.08}_{-0.47}$$  l  $$0.62^{+0.49}_{-0.20}$$  $$2.12^{+2.29}_{-1.20}$$  $$0.75^{+0.80}_{-0.30}$$  $$0.77^{+0.73}_{-0.28}$$  $$2.60^{+2.27}_{-1.81}$$  $$2.26^{+2.44}_{-1.63}$$  $$M(M_{B}^{1})$$  –  –  –  –  $$0.10^{+0.05}_{-0.05}$$  $$-18.94^{+0.05}_{-0.05}$$  ΔM  –  –  –  –  –  $$-0.06^{+0.02}_{-0.01}$$  α  –  –  –  –  –  $$0.14^{+0.006}_{-0.006}$$  β  –  –  –  –  –  $$3.10^{+0.08}_{-0.07}$$  Gaussian prior on h = 0.678 ± 0.009  $$\chi ^{2}_{\text{min}}$$  25.03  14.70  20.84  21.96  33.79  –  $$\chi ^{2}_{\text{red}}$$  0.53  0.54  0.44  0.46  1.29  –  Ωm0  $$0.25^{+0.02}_{-0.02}$$  $$0.35^{+0.05}_{-0.08}$$  $$0.26^{+0.03}_{-0.03}$$  $$0.25^{+0.02}_{-0.03}$$  $$0.22^{+0.12}_{-0.13}$$  –  h  $$0.67^{+0.008}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.009}$$  $$0.67^{+0.009}_{-0.008}$$  $$0.67^{+0.009}_{-0.009}$$  –  n  $$-0.01^{+0.35}_{-0.57}$$  $$0.24^{+0.19}_{-0.53}$$  $$0.16^{+0.26}_{-0.60}$$  $$0.03^{+0.31}_{-0.58}$$  $$0.36^{+0.07}_{-0.36}$$  –  l  $$0.71^{+0.95}_{-0.34}$$  $$2.08^{+2.44}_{-1.37}$$  $$0.96^{+1.55}_{-0.55}$$  $$0.87^{+1.22}_{-0.45}$$  $$2.59^{+2.27}_{-1.8}$$  –  M  –  –  –  –  $$-0.06^{+0.03}_{-0.03}$$  –  View Large Table 5. Mean values for the OC parameters (Ωm0, h, n) derived from a joint analysis OHD+cJLA. OC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  M  Flat prior on h  J1  58.91  0.71  $$0.25^{+0.01}_{-0.01}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.12^{+0.06}_{-0.06}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.28  0.58  $$0.30^{+0.05}_{-0.05}$$  $$0.68^{+0.02}_{-0.02}$$  $$-0.001^{+0.15}_{-0.17}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.28  0.66  $$0.25^{+0.02}_{-0.02}$$  $$0.69^{+0.01}_{-0.01}$$  $$0.11^{+0.07}_{-0.07}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.17  0.67  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.10^{+0.07}_{-0.08}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  63.34  0.76  $$0.25^{+0.01}_{-0.01}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.05^{+0.05}_{-0.05}$$  $$0.001^{+0.02}_{-0.03}$$  J2  50.73  0.61  $$0.27^{+0.04}_{-0.05}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.001^{+0.14}_{-0.15}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.37  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.06^{+0.06}_{-0.07}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.96  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.04^{+0.07}_{-0.07}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  59.04  0.71  $$0.26^{+0.01}_{-0.01}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.13^{+0.05}_{-0.05}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.53  0.58  $$0.31^{+0.04}_{-0.05}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.001^{+0.15}_{-0.17}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.80  0.66  $$0.26^{+0.02}_{-0.02}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.13^{+0.06}_{-0.07}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.18  0.65  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.10^{+0.07}_{-0.07}$$  $$-0.06^{+0.02}_{-0.02}$$  OC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  M  Flat prior on h  J1  58.91  0.71  $$0.25^{+0.01}_{-0.01}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.12^{+0.06}_{-0.06}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.28  0.58  $$0.30^{+0.05}_{-0.05}$$  $$0.68^{+0.02}_{-0.02}$$  $$-0.001^{+0.15}_{-0.17}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.28  0.66  $$0.25^{+0.02}_{-0.02}$$  $$0.69^{+0.01}_{-0.01}$$  $$0.11^{+0.07}_{-0.07}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.17  0.67  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.10^{+0.07}_{-0.08}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  63.34  0.76  $$0.25^{+0.01}_{-0.01}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.05^{+0.05}_{-0.05}$$  $$0.001^{+0.02}_{-0.03}$$  J2  50.73  0.61  $$0.27^{+0.04}_{-0.05}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.001^{+0.14}_{-0.15}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.37  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.06^{+0.06}_{-0.07}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.96  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.04^{+0.07}_{-0.07}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  59.04  0.71  $$0.26^{+0.01}_{-0.01}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.13^{+0.05}_{-0.05}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.53  0.58  $$0.31^{+0.04}_{-0.05}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.001^{+0.15}_{-0.17}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.80  0.66  $$0.26^{+0.02}_{-0.02}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.13^{+0.06}_{-0.07}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.18  0.65  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.10^{+0.07}_{-0.07}$$  $$-0.06^{+0.02}_{-0.02}$$  View Large Table 6. Mean values for the MPC parameters (Ωm0, h, n, l) derived from a joint analysis OHD+cJLA. MPC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  l  M  Flat prior on h  J1  58.61  0.71  $$0.25^{+0.02}_{-0.02}$$  $$0.68^{+0.01}_{-0.01}$$  $$-0.03^{+0.34}_{-0.56}$$  $$0.74^{+0.90}_{-0.35}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.25  0.58  $$0.32^{+0.05}_{-0.07}$$  $$0.68^{+0.02}_{-0.02}$$  $$0.25^{+0.13}_{-0.51}$$  $$2.00^{+2.244}_{-1.33}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.23  0.66  $$0.25^{+0.03}_{-0.03}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.06^{+0.29}_{-0.58}$$  $$0.89^{+1.29}_{-0.47}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.14  0.67  $$0.25^{+0.03}_{-0.03}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.06^{+0.28}_{-0.58}$$  $$0.91^{+1.29}_{-0.49}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  62.91  0.75  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.14^{+0.36}_{-0.51}$$  $$0.70^{+0.72}_{-0.29}$$  $$-0.0006^{+0.03}_{-0.03}$$  J2  50.81  0.61  $$0.30^{+0.04}_{-0.06}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.22^{+0.14}_{-0.54}$$  $$1.77^{+2.44}_{-1.17}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.32  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.002^{+0.29}_{-0.55}$$  $$0.88^{+1.12}_{-0.44}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.91  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.007^{+0.29}_{-0.55}$$  $$0.89^{+1.15}_{-0.45}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  58.85  0.70  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$-0.02^{+0.33}_{-0.57}$$  $$0.74^{+0.92}_{-0.35}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.45  0.58  $$0.33^{+0.04}_{-0.06}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.26^{+0.12}_{-0.51}$$  $$2.10^{+2.42}_{-1.42}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.82  0.66  $$0.26^{+0.03}_{-0.03}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.09^{+0.28}_{-0.59}$$  $$0.91^{+1.35}_{-0.49}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.19  0.66  $$0.25^{+0.02}_{-0.03}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.06^{+0.27}_{-0.57}$$  $$0.91^{+1.25}_{-0.48}$$  $$-0.06^{+0.02}_{-0.032}$$  MPC model  Data set  $$\chi ^{2}_{\text{min}}$$  $$\chi ^{2}_{\text{red}}$$  Ωm0  h  n  l  M  Flat prior on h  J1  58.61  0.71  $$0.25^{+0.02}_{-0.02}$$  $$0.68^{+0.01}_{-0.01}$$  $$-0.03^{+0.34}_{-0.56}$$  $$0.74^{+0.90}_{-0.35}$$  $$-0.04^{+0.03}_{-0.03}$$  J2  48.25  0.58  $$0.32^{+0.05}_{-0.07}$$  $$0.68^{+0.02}_{-0.02}$$  $$0.25^{+0.13}_{-0.51}$$  $$2.00^{+2.244}_{-1.33}$$  $$-0.03^{+0.06}_{-0.06}$$  J3  54.23  0.66  $$0.25^{+0.03}_{-0.03}$$  $$0.68^{+0.01}_{-0.01}$$  $$0.06^{+0.29}_{-0.58}$$  $$0.89^{+1.29}_{-0.47}$$  $$-0.02^{+0.04}_{-0.04}$$  J4  55.14  0.67  $$0.25^{+0.03}_{-0.03}$$  $$0.67^{+0.01}_{-0.01}$$  $$0.06^{+0.28}_{-0.58}$$  $$0.91^{+1.29}_{-0.49}$$  $$-0.07^{+0.04}_{-0.04}$$  Gaussian prior on h = 0.732 ± 0.017  J1  62.91  0.75  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.14^{+0.36}_{-0.51}$$  $$0.70^{+0.72}_{-0.29}$$  $$-0.0006^{+0.03}_{-0.03}$$  J2  50.81  0.61  $$0.30^{+0.04}_{-0.06}$$  $$0.71^{+0.01}_{-0.01}$$  $$0.22^{+0.14}_{-0.54}$$  $$1.77^{+2.44}_{-1.17}$$  $$0.03^{+0.04}_{-0.04}$$  J3  57.32  0.69  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$0.002^{+0.29}_{-0.55}$$  $$0.88^{+1.12}_{-0.44}$$  $$0.01^{+0.03}_{-0.03}$$  J4  60.91  0.73  $$0.24^{+0.02}_{-0.02}$$  $$0.70^{+0.01}_{-0.01}$$  $$-0.007^{+0.29}_{-0.55}$$  $$0.89^{+1.15}_{-0.45}$$  $$-0.01^{+0.03}_{-0.03}$$  Gaussian prior on h = 0.678 ± 0.009  J1  58.85  0.70  $$0.25^{+0.02}_{-0.02}$$  $$0.67^{+0.007}_{-0.007}$$  $$-0.02^{+0.33}_{-0.57}$$  $$0.74^{+0.92}_{-0.35}$$  $$-0.05^{+0.02}_{-0.02}$$  J2  48.45  0.58  $$0.33^{+0.04}_{-0.06}$$  $$0.67^{+0.008}_{-0.008}$$  $$0.26^{+0.12}_{-0.51}$$  $$2.10^{+2.42}_{-1.42}$$  $$0.05^{+0.03}_{-0.03}$$  J3  54.82  0.66  $$0.26^{+0.03}_{-0.03}$$  $$0.68^{+0.007}_{-0.007}$$  $$0.09^{+0.28}_{-0.59}$$  $$0.91^{+1.35}_{-0.49}$$  $$-0.04^{+0.02}_{-0.02}$$  J4  55.19  0.66  $$0.25^{+0.02}_{-0.03}$$  $$0.67^{+0.007}_{-0.007}$$  $$0.06^{+0.27}_{-0.57}$$  $$0.91^{+1.25}_{-0.48}$$  $$-0.06^{+0.02}_{-0.032}$$  View Large 4.1 cJLA versus fJLA on the Cardassian parameter estimations The use of the fJLA sample to infer cosmological parameters has a high computational cost when several model are tested. To deal with this, we use the cJLA sample instead of the fJLA. Nevertheless, the former was computed under the standard cosmology. To assess how the Cardassian model constraints are biased when using each SN Ia sample, we perform the parameter estimation with different combinations of models, priors, and samples. The several constraints are presented in Tables 3 and 4. Notice that the mean values for the cosmological parameters in the OC model obtained from both SN Ia samples are the same. For the MPC model, the largest difference is observed on the l parameter (flat prior on h), ∼0.18σ. It is smaller for the n parameter when employing a Gaussian prior on h. Fig. 1 illustrates the comparison of the confidence contours for these parameters using the cJLA and fJLA samples (flat prior on h). Fig. 2 shows that there is no significant difference in the reconstruction of the q(z) parameter for the OC and MPC models using the constraints obtained from both SN Ia samples. Therefore, to optimize the computational time, in the following analysis we only use the compressed JLA sample. Figure 1. View largeDownload slide Comparison of the Ωm0−n (top panel) and n − l(bottom panel) confidence contours for the OC and MPC parameters within the 1σ and 3σ confidence levels, using the cJLA (dashed lines) and fJLA (filled and solid lines) samples, respectively. In the parameter estimation, a flat prior is considered. The cross and star mark the mean values for each data set. Figure 1. View largeDownload slide Comparison of the Ωm0−n (top panel) and n − l(bottom panel) confidence contours for the OC and MPC parameters within the 1σ and 3σ confidence levels, using the cJLA (dashed lines) and fJLA (filled and solid lines) samples, respectively. In the parameter estimation, a flat prior is considered. The cross and star mark the mean values for each data set. Figure 2. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from the cJLA and fJLA samples when a flat prior on h is considered. Notice that there is no significant differences in the q(z) behaviour using each SN Ia sample. Figure 2. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from the cJLA and fJLA samples when a flat prior on h is considered. Notice that there is no significant differences in the q(z) behaviour using each SN Ia sample. 4.2 The effects of the homogeneous OHD subsample in the parameter estimation In Section 3.1.1, an homogenized and model-independent OHD from clustering was constructed to avoid or reduce biased constraints due to the underlying cosmology or the underestimated systematic errors. Tables 3 and 4 provide the OC and MPC bounds estimated from the combination of the new computed unbiased OHD from clustering with those obtained from the DA method. The increase on the error of rd also increases the error on H(z), reducing the goodness of the fit (χred). In spite of this, the advantage of these new limits is that they could be considered unbiased by different cosmological models. Fig. 3 shows the contours of the Ωm0−n OC (top panel) and the n − l MPC (bottom panel) parameters, respectively, using the different OHD samples. Note that all the bounds are consistent within the 1σ and 3σ confidence levels (CLs). Fig. 4 illustrates the q(z) reconstructions using the different OHD data sets. Notice that for the OC model the homogenized OHD samples give slightly different q(0) values than the obtained from the sample in Table 1. For the MPC model, these differences are less significant. Figure 3. View largeDownload slide Confidence contours of the Ωm0−n (top panel) and n − l (bottom panel) constraints for the OC and MPC models within the 1σ and 3σ CLs, using the OHD sample in Table 1, the OHDDA data set, and two samples containing the DA points plus those homogenized OHD points from clustering using the rd values from WMAP and Planck measurements. A flat prior on h was considered in the parameter estimation. Figure 3. View largeDownload slide Confidence contours of the Ωm0−n (top panel) and n − l (bottom panel) constraints for the OC and MPC models within the 1σ and 3σ CLs, using the OHD sample in Table 1, the OHDDA data set, and two samples containing the DA points plus those homogenized OHD points from clustering using the rd values from WMAP and Planck measurements. A flat prior on h was considered in the parameter estimation. Figure 4. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from different OHD samples when a flat prior on h is considered. Figure 4. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models using the constraints from different OHD samples when a flat prior on h is considered. 4.3 The effects of a different Gaussian prior on h One of the most important problems in cosmology is the tension up to more than 3σ between the local measurements of the Hubble constant H0 and those obtained from the CMB anisotropies (Bernal, Verde & Riess 2016). The latest estimation by the Planck collaboration (Planck Collaboration XIII 2016), h = 0.678 ± 0.009, is in disagreement with the first value given in Table 1. Thus, using different Gaussian priors on h will lead to different constraints on the OC and MPC parameters. Therefore, we carried out all our computations with both priors. Fig. 5 illustrates how the confidence contours for the Ωm0−n and l − n parameters of the OC (top panel) and MPC (bottom panel) models obtained from OHDhpl are shifted using each Gaussian prior. Although they are consistent at 3σ, the tension in the constraints is important. In spite of these differences, Fig. 6 shows that both results drive the Universe to an accelerated phase but with slightly different transition redshifts (i.e. the redshift at which the Universe passes from a decelerated to an accelerated phase) and amplitude, q(0). In addition, the OC and MPC bounds are consistent with the standard cosmology even when different Gaussian priors are considered. Figure 5. View largeDownload slide Comparison of the Ωm 0−n (top panel) and l − n (bottom panel) confidence contours for the OC and MPC parameters, respectively, within the 1σ and 3σ CLs obtained from the OHDhpl analysis using two Gaussian priors on h: 0.732 ± 0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). The stars mark the mean values for each data set. Figure 5. View largeDownload slide Comparison of the Ωm 0−n (top panel) and l − n (bottom panel) confidence contours for the OC and MPC parameters, respectively, within the 1σ and 3σ CLs obtained from the OHDhpl analysis using two Gaussian priors on h: 0.732 ± 0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). The stars mark the mean values for each data set. Figure 6. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models, using the constraints from the OHDhpl sample and the joint analysis J3 when a different Gaussian prior on h is considered: 0.732 ±  0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). Figure 6. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC (top panel) and MPC (bottom panel) models, using the constraints from the OHDhpl sample and the joint analysis J3 when a different Gaussian prior on h is considered: 0.732 ±  0.017 (Riess et al. 2016) and 0.678 ± 0.009 (Planck Collaboration XIII 2016). 4.4 Cosmological implications of the OC and MPC constraints Fig. 7 shows the one-dimensional (1D) marginalized posterior distributions and the two-dimensional (2D) 68 per cent, 95 per cent, 99 per cent contours for the Ωm0, h, and n parameters of the OC model obtained from OHDhpl, cJLA, and J3 with flat (left-hand panel) and Gaussian (right-hand panel) priors on h. Assuming a flat prior on h, the Ωm0, h constraints obtained from the different data sets are consistent between them and are in agreement with Planck measurements for the standard model. For the n parameter, we found a tension in the constraints obtained from the different data sets. Nevertheless, the bounds have large uncertainties and are consistent among them within the 1σ CL. Our n constraints are consistent within the 1σ CL with those estimated by other authors, for instance, $$n=-0.04^{+0.07}_{-0.07}$$ (Xu 2012), $$n=0.16^{+0.30}_{-0.52}$$ (Wei et al. 2015), and $$n=-0.022^{+0.05}_{-0.05}$$ (Zhai et al. 2017a). It is worth to note that, when the cJLA data are used, Ωm drop at extremely low values (see the Ωm0−n contour), which is consistent with the results by Wei et al. (2015) who obtained a similar contour using the Union 2.1 data set. In addition, the $$\chi _{\text{red}}^{2}$$ values from the SN Ia data suggest that their errors (cJLA sample) are underestimated. Figure 7. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, and n parameters of the OC model assuming a flat and Gaussian (hRiess) prior on h. Figure 7. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, and n parameters of the OC model assuming a flat and Gaussian (hRiess) prior on h. On the other hand, when the Gaussian prior on h by Riess et al. (2016) is considered, the OHDhpl provides a better fitting for the OC parameters than those obtained when a flat prior is used (see the $$\chi _{\text{red}}^{2}$$ values). SN Ia data show no important statistical difference in the parameter estimation when flat or Gaussian priors are employed. Notice that we obtain stringent constraints from the joint analysis (see Fig. 7), which prefers values around n ∼ 0. Fig. 8 shows the fittings to the OHDhpl (top panel) and cJLA data (bottom panel), using the OHDhpl, cJLA, and J3 constraints for the OC model. A Monte Carlo approach was performed to propagate the error on the 1σ and 3σ CL. The comparison between these results and the ΛCDM fitting reveals that both models are in agreement with the data and there is no significant difference between them. In addition, when the J1, J2, and J4 constraints are used, we found consistent results within the 1σ CL. Therefore, the extra term in the equation (1) to the canonical Friedmann equation acts like a CC. However, in the OC models this term can be sourced by an extra dimension instead of the expected vacuum energy. Figure 8. View largeDownload slide Fitting to OHDhpl (top panel) and cJLA data (bottom panel), using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines), and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and OC model with a flat prior on h. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 8. View largeDownload slide Fitting to OHDhpl (top panel) and cJLA data (bottom panel), using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines), and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and OC model with a flat prior on h. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. To confirm that the OC model can drive to a late cosmic acceleration, we reconstructed the deceleration parameter using the mean values derived from the different data sets. Fig. 9 shows that the q(z) dynamics is similar for the ΛCDM and OC models when the OHDhpl, cJLA, and J3 constrains are used, i.e. the universe has a late phase of accelerated expansion. Notice that although the CLs in the q(z) reconstruction obtained from the SN Ia constraints are bigger that those from the OHDhpl, they are consistent. The difference could be explained by the extra free parameter (nuisance) in the SN Ia analysis. Figure 9. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC model and ΛCDM using the constraints from OHDhpl (top panel) and cJLA data (bottom panel) with a flat prior on h. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 9. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the OC model and ΛCDM using the constraints from OHDhpl (top panel) and cJLA data (bottom panel) with a flat prior on h. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed and the dotted lines represent the 68 per cent and 99.7 per cent CLs, respectively. Fig. 10 shows the 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99 per cent contours for the Ωm0, h, n, and l parameters of the MPC model obtained from OHDhpl, cJLA, and J3 with flat (left-hand panel) and Gaussian (right-hand panel) priors on h. Considering a flat prior on h, the different data sets provide slightly different constraints on Ωm0 and h. For instance, the OHDDA estimates higher (lower) values on Ωm0 (h) and SN Ia lower (higher) values. However, the limits are consistent within the 1σ CL. For the n and l constraints, we also obtained a marginal tension using different data but they are consistent within the 1σ CL. Notice that our constraints include n = 0 and l = 1, which reproduces the ΛCDM dynamics. All our bounds are similar within the 1σ CL to those obtained by other authors, e.g. Li et al. (2012) combining SN Ia, BAO, and CMB data measure $$n=0.014^{+0.36}_{-0.94}$$, $$l=1.09^{+1.01}_{-0.46}$$, Magaña et al. (2015) using strong lensing features estimate n = 0.41 ± 0.25, l = 5.2 ± 2.25, Zhai et al. (2017a) provide $$n=0.16^{+0.08}_{0.09}$$, $$l=1.38^{+0.25}_{-0.22}$$ from the joint analysis of CMB, BAO plus SN Ia (JLA) data, and Zhai et al. (2017b) give $$n=0.02^{+0.26}_{-0.41}$$, $$l=1.1^{+0.8}_{-0.4}$$ from the joint analysis of CMB, BAO, SN Ia, fσ8 and the H0 value from Riess et al. (2016). In addition, the $$\chi _{\text{red}}^{2}$$ values point out that the OHDDA provides better (unbiased) MPC constraints and the values from SN Ia data suggest that their errors (cJLA sample) are underestimated. Considering the Gaussian prior on h by Riess et al. (2016), the OHD, OHDhpl, and OHDhw9 probes yield improvements in the MPC constraints (see the χred values). For the SN Ia (cJLA) test, there is no significant difference with the flat prior case. Notice that the stringent limits are estimated from the joint analysis (see also Fig. 10). Fig. 11 shows the fittings to the OHDhpl and cJLA data using the OHDhpl, cJLa, and J3 constraints of the MPC parameters and those of the ΛCDM model with a flat prior on h. To propagate the errors on OHD, μ(z), and q(z), we have used a Monte Carlo approach. For both, OHD and μ(z) fittings, there is no significant statistical difference between the MPC model and the standard one. In addition, a good agreement at 1σ is obtained employing the J1, J2, and J4 constraints. In addition, Fig. 12 shows the reconstruction of the q(z) parameter using the constraints from the OHD and SN Ia data. For the OHD constraints, the q(z) dynamics for the MPC is in agreement with that of the standard model. When the SN Ia estimations are used, the history of the cosmic acceleration for the MPC model is consistent with the ΛCDM within the 1σ and 3σ CL. Thus, the MPC scenario is viable to explain the late cosmic acceleration without a DE component and its cosmological dynamics is almost indistinguishable from the standard model. Figure 10. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, n, and l parameters of the MPC model assuming a flat and Gaussian (hRiess) prior on h. Figure 10. View largeDownload slide 1D marginalized posterior distributions and the 2D 68 per cent, 95 per cent, 99.7 per cent CLs for the Ωm0, h, n, and l parameters of the MPC model assuming a flat and Gaussian (hRiess) prior on h. Figure 11. View largeDownload slide Fitting to OHDhpl (top panel) and SN Ia data (bottom panel) using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines) and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and MPC model when a flat prior on h is considered. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 11. View largeDownload slide Fitting to OHDhpl (top panel) and SN Ia data (bottom panel) using the mean values from the OHDhpl (red solid lines), cJLA (blue solid lines) and J3 (yellow star and triangle) analysis for ΛCDM model (black squares) and MPC model when a flat prior on h is considered. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 12. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the MPC model and ΛCDM using the constraints from OHDhpl (top panel) and SN Ia data (bottom panel) when a flat prior on h is considered. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. Figure 12. View largeDownload slide Reconstruction of the deceleration parameter q(z) for the MPC model and ΛCDM using the constraints from OHDhpl (top panel) and SN Ia data (bottom panel) when a flat prior on h is considered. The q(z) reconstruction from J3 constraints is shown in both panels. The dashed-lines and dotted-lines represent the 68 per cent and 99.7 per cent CLs, respectively. 5 CONCLUSIONS AND OUTLOOKS In this paper, we analyse two alternatives to explain the late cosmic acceleration without a DE component: theOC and MPC models that are also excellent laboratories to study deviations from GR. The Cardassian models establish the modification of the canonical Friedmann equation as a consequence of a braneworld dynamics that emerges from novel ideas of the space–time dimensions and is based on a generalized Einstein–Hilbert action. To constrain the exponents n and the n − l of the OC and MPC models, we used 51 OHD, 740 SNIa data points of the JLA sample (fJLA), and 31 binned distance modulus of the compressed JLA sample (cJLA). The OHD compilation contains 31 points measured using the differential age technique in early-type galaxies and 20 points from clustering. These last points are biased due to an underlying ΛCDM cosmology to estimate the sound horizon at the drag epoch, which is used to compute H(z). Moreover, these data points are estimated taking into account very conservative systematic errors. Therefore, we constructed two homogenized and model-independent samples for the clustering points using a common rd obtained from Planck and WMAP measurements. We found that the different OHD samples provide consistent constraints on the OC and MPC parameters. In addition, there is no significant differences on the constraints obtained from the cJLA and those estimated from fJLA. Furthermore, we obtained consistent constraints at 3σ CL when different Gaussian priors on h are employed. We performed a joint analysis with the combination of cJLA and one homogenized OHD sample. Our results shown that the OC and MPC free parameters are consistent with the traditional dynamics dictated by the Friedmann equation (see Tables 3–6) containing a CC. However, in the Cardassian models the extra terms in the canonical Friedmann equation mimic the CC but it comes from the n-term of the energy–momentum tensor, unlike in the traditional form where the CC is added by hand in the Friedmann equation. Of course, those problems affecting the CC will be transferred to the interpretation of n-dimensional geometry and, as a consequence, to the emerging of the n-term of the energy–momentum tensor. Therefore, the idea is to interpret and to know the global topology of our Universe to generate a solution for the DE problem and the current Universe acceleration. ACKNOWLEDGEMENTS We thank the anonymous referee for thoughtful remarks and suggestions. J.M. acknowledges support from CONICYT/FONDECYT 3160674. M.H.A. acknowledges support from CONACYT PhD fellow, Consejo Zacatecano de Ciencia, Tecnología e Innovación (COZCYT), and Centro de Astrofísica de Valparaíso (CAV). M.H.A. thanks the staff of the Instituto de Física y Astronomía of the Universidad de Valparaíso where part of this work was done. M.A.G.-A. acknowledges support from CONACYT research fellow, Sistema Nacional de Investigadores (SNI), and Instituto Avanzado de Cosmología (IAC) collaborations. Footnotes 1 The name Cardassian refers to a humanoid race in Star Trek series, whose goal is the accelerated expansion of their evil empire. 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The first column is the binned redshift and the second column is the binned distance modulus. zb  μb  0.010  32.953 886 976  0.012  33.879 003 4661  0.014  33.842 140 7403  0.016  34.118 567 0426  0.019  34.593 445 9829  0.023  34.939 026 5264  0.026  35.252 096 3261  0.031  35.748 501 6537  0.037  36.069 787 6073  0.043  36.434 570 4737  0.051  36.651 110 5942  0.060  37.158 014 1133  0.070  37.430 173 2516  0.082  37.956 616 3488  0.097  38.253 254 0406  0.114  38.612 869 3372  0.134  39.067 850 7056  0.158  39.341 401 9038  0.186  39.792 143 6157  0.218  40.156 534 6033  0.257  40.564 956 0582  0.302  40.905 287 7824  0.355  41.421 417 4356  0.418  41.790 923 4574  0.491  42.231 461 0669  0.578  42.617 047 0706  0.679  43.052 731 4851  0.799  43.504 150 8283  0.940  43.972 573 4093  1.105  44.514 087 5789  1.300  44.821 867 4621  zb  μb  0.010  32.953 886 976  0.012  33.879 003 4661  0.014  33.842 140 7403  0.016  34.118 567 0426  0.019  34.593 445 9829  0.023  34.939 026 5264  0.026  35.252 096 3261  0.031  35.748 501 6537  0.037  36.069 787 6073  0.043  36.434 570 4737  0.051  36.651 110 5942  0.060  37.158 014 1133  0.070  37.430 173 2516  0.082  37.956 616 3488  0.097  38.253 254 0406  0.114  38.612 869 3372  0.134  39.067 850 7056  0.158  39.341 401 9038  0.186  39.792 143 6157  0.218  40.156 534 6033  0.257  40.564 956 0582  0.302  40.905 287 7824  0.355  41.421 417 4356  0.418  41.790 923 4574  0.491  42.231 461 0669  0.578  42.617 047 0706  0.679  43.052 731 4851  0.799  43.504 150 8283  0.940  43.972 573 4093  1.105  44.514 087 5789  1.300  44.821 867 4621  View Large © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

Journal

Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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