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Constraints on a Vacuum Energy from Both SNIa and CMB Temperature Observations

Constraints on a Vacuum Energy from Both SNIa and CMB Temperature Observations Hindawi Publishing Corporation Advances in Astronomy Volume 2012, Article ID 528243, 6 pages doi:10.1155/2012/528243 Research Article Constraints on a Vacuum Energy from Both SNIa and CMB Temperature Observations Riou Nakamura, E. P. Berni Ann Thushari, Mikio Ikeda, and Masa-Aki Hashimoto Department of Physics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka City 812-8581, Japan Correspondence should be addressed to Riou Nakamura, riou@phys.kyushu-u.ac.jp Received 28 February 2012; Accepted 1 May 2012 Academic Editor: Rob Ivison Copyright © 2012 Riou Nakamura et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the cosmic thermal evolution with a vacuum energy which decays into photon at the low redshift. We assume that the vacuum energy is a function of the scale factor that increases toward the early universe. We put on the constraints using recent observations of both type Ia supernovae (SNIa) by Union-2 compilation and the cosmic microwave background (CMB) temper- ature at the range of the redshift 0.01 <z < 3. From SNIa, we find that the effects of a decaying vacuum energy on the cosmic expansion rate should be very small but could be possible for z< 1.5. On the other hand, we obtain the severe constraints for parameters from the CMB temperature observations. Although the temperature can be still lower than the case of the standard cosmological model, it should only affect the thermal evolution at the early epoch. 1. Introduction to a primordial light-element abundances [15] and the CMB intensity [16]. From the point of the thermodynamical evo- One of the biggest cosmological mysteries is the accelerating lution in the universe, a decaying vacuum energy modifies cosmic expansion which was discovered by the observations the temperature-redshift relation [17–19]. This modification of distant type Ia supernovae (SNIa) starting from more than affects the cosmic thermal evolution after a hydrogen recom- 10 years ago. For the origin of the accelerating expansion, bination [20], such as formation of molecules [21, 22]and the following possibilities have been proposed: the modified the first star [21]. In addition to this model, the angular gravity, such as f (R)gravity [1, 2], brane-world cosmology power spectrum of CMB could be also modified [23]. In the [3], inhomogeneous cosmology [4, 5], and existence of previous analysis, thermal history at the higher redshift has unknown energy called as dark energy which is equivalent been studied. However, as seen in [20, 21], the time variation to acosmicfluidwithanegative pressure. of temperature at low redshift differs from the model without It is strongly suggested that the dark energy amounts to a decaying vacuum. 70% of the total energy density of the universe from the In the present study, we update the observational consis- astronomical observations such as SNIa [6–8], the anisotropy tency of a vacuum energy (hereafter we denote as Λ)coupled of cosmic microwave background (CMB) [9–11], and the with CMB photon. Hereafter, we call this model DΛCDM baryon acoustic oscillation [12]. Although there are various (model). To examine the consistency with observations at theoretical models of dark energy (details are shown in a low redshift, we focus on the cosmic evolution comparing review [13, 14]), its physical nature is still unknown. Con- both the type Ia supernovae and the redshift dependence of sidering above-referenced observations, we can not exclude CMB temperatures. This is because the time dependence the constant Λ term which is the most simple model of dark of the CMB temperature for a DΛCDM differs from CDM energy. model with a constant Λ term of which hereafter we call The interacting dark energy with CMB photon has been SΛCDM (model). In Section 2, formulation of DΛCDM is discussed, whereas decaying vacuum into photon is related reviewed. In Section 3, the m − z relation is investigated for 2 Advances in Astronomy DΛCDM in a flat universe. Parameters inherent in this model where ρ is the critical density defined by the present Hubble cr are constrained from the CMB temperature observations in constant H .Wecan rewrite(8) as follows (for details also see Section 4. Concluding remarks are given in Section 5. [22, 23]): dΩ Ω γ γ dΩ (10) +4 =− . 2. Dynamics of the Decaying Λ Model da a da The Einstein’s field equation is written as follows (e.g., [24]): Models with time-dependent Λ-term have been studied as summarizedin[25, 26]. We adopt the energy of the (1) R − g R = 8πGT , μν μν μν vacuum which varies with the scale factor [20–23]: −m Ω (a) = Ω + Ω a , (11) where G is the gravitational constant and T is the energy Λ Λ1 Λ2 μν momentum tensor. If we assume the perfect fluid, T is μν where Ω , Ω ,and m are constants. Note that the present Λ1 Λ2 written as follows: value of Ω is expressed by Ω = Ω + Ω . Λ Λ0 Λ1 Λ2 T = diag −ρ, p, p, p . Integrating (10)with(11), we obtain the photon energy (2) μν density as a function of a [22]: Here ρ and p are the energy density and the pressure, res- ⎪ 4−m −4 pectively. Note that we choose the unit of c = 1. ⎪ Ω + α a − 1 a (m= 4), ⎨ γ0 The equation of motion is obtained with use of the Fried- Ω = (12) −4 ⎪ Ω +4Ω ln a a (m = 4), mann-Robertson-Walker metric of the homogeneous and γ0 Λ2 isotropic principle: −5 −2 where Ω = 2.471 × 10 h (T /2.725 K) is the present 2 γ0 γ0 dr 2 2 2 2 2 2 2 2 ds =−dt + a(t) + r dθ + r sin θdφ , photon energy density, h is the normalized Hubble constant 1 − kr (H = 100h km/sec/Mpc), and T is the CMB temperature 0 γ0 (3) at the present epoch. We define α as α = mΩ /(4 − m). Λ2 Since we assume the flat geometry (k = 0) in this work, where a(t) is the scale factor and k is the specific curvature we can write the condition of Ωs as follows: constant. This leads to the Friedmann equations: Ω + Ω + Ω = 1. (13) m0 Λ1 Λ2 a˙ 8πG k H ≡ = ρ − . (4) a 3 a From now on, we adopt the present cosmological parame- ters: Hubble constant h = 0.738 [27, 28], and the present From the conservation’s law of energy, we can obtain the density parameter of matter Ω = 0.2735 [11]. The present equation of the energy density: temperature is T = 2.725 K observed by COBE [29]. In our γ0 a˙ study, we search the parameter regions in the range of Ω ρ˙ =−3(1+ w) ρ, (5) Λ2 −7 −2 and m :10 < Ω < 10 and 0 <m< 4. The range of m Λ2 has already been obtained from the previous analysis [23]. where w is the coefficients of the equation of state, p = wρ, Here we note the range of Ω in the present analysis. Λ2 anddefinedby Kimura et al. [20] found that the radiation temperature (i.e., ⎪ the energy density of radiation) in DΛCDM model becomes for photon and neutrino, numerically negative at a point of a< 1 when m and/or Ω Λ2 w ≡ (6) 0 for baryon and cold dark matter, is too large. To avoid this unreasonable situation, that is, T < ⎪ γ −1 for vacuum energy. 0, Nakamura et al. [23] imposed the limit of Ω and m as Λ2 follows: We take the component of the energy density as α< Ω (m< 4), γ0 ρ = ρ + ρ + ρ + ρ , γ ν m Λ (7) (14) Ω < Ω /92 (m = 4). Λ2 γ0 where γ, ν, m,and Λ indicate photon, neutrino, nonrela- From the condition (14), we obtain the limit of Ω as 4.9 × tivistic matter (baryon plus cold dark matter), and the vac- Λ2 −7 10 for m = 4. We can consider that the effect of Ω is uum, respectively. Here we neglect the energy contribution Λ2 −7 negligible when Ω is smaller than 10 . of Λ on other components except for photon: the ρ and ρ Λ2 ν m −4 −3 evolves as ρ ∝ a and ρ ∝ a ,respectively. From (5), the ν m equation of the photon coupled with Λ is written as 3. SNIa Constraints ρ˙ +4Hρ =−ρ˙ . (8) γ γ Λ The cosmic distance measures depend sensitively on the spatial curvature and expansion dynamics of the models. Let us define the parameter of the energy density as follows: Therefore, the magnitude-redshift relation for distant stan- 8πG dard candles are proposed to constrain the cosmological Ω ≡ = ρ , i i (9) parameters. 3H cr 0 Advances in Astronomy 3 −1 Excluded region −2 −3 −4 −5 −6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −7 m = 7 m = 0 m = 3 Union2 99.73% C.L. Figure 1: Illustration of the magnitude-redshift relation in SΛCDM 95.4% C.L. −2 model (m = 0) and DΛCDM with Ω = 10 compared with SNIa 68.3% C.L. Λ2 observations (cross-shaped ones with error bars). Note that the Figure 2: Constraints on the m − Ω plane from the recent obser- Λ2 theoretical curve for m = 3 has no difference with that of m = 0. vations of CMB. The lines indicate upper bounds of 1, 2, and 3σ confidence level. The dashed and solid lines are the limits deduced from SNIa and CMB temperature observations. The shaded region shows the excluded region obtained from [23]. To calculate the effects of the comic expansion at low-z, we calculate the theoretical distance modules, μ = m − M = 5log d + 25, (15) th B L where μ and μ are the observed and the theoretical obs,i th,i values of the distance modules. σ is the dispersion in the where m , M,and d are the apparent and absolute magni- B L redshift due to the peculiar velocity v is written as: tudes, and the luminosity distance, respectively. Here d is related to the radial distance r in the metric as follows [24]: v dμ th d = (1+ z)r. (16) σ = . (21) c dz We can obtain r as follows: dt 1 dz −1 We adopt v = 300 km sec [40]. The total number of the r = = , (17) ( ) ( ) a t H E z united sample N is 557 for the present analysis. We confirm that Ω has done some contribution to where z is the redshift defined by a = 1/(1 + z). Now E(z)is Λ2 defined as change the μ−z relation. When Ω increases, the expansion Λ2 rate in the universe decreases, which is similar to the behavior H(z) 2 3 m of the matter dominant universe in the Friedmann model. E(z) = = Ω (1+ z) + Ω + Ω (1+ z) . m0 Λ1 Λ2 Figure 1 indicates the relation between the distance mod- (18) ules and the redshift against the SNIa observations in terms Then, (16) is rewritten as follows: of SΛCDM and DΛCDM with several values of m for a fixed −2 value of Ω = 10 . As the value of m increases, μ tends to (1+ z) dz d = . (19) decrease because of the increasing cosmic expansion rate. We H E(z) recognize that m is not seriously effective to change the μ − z As a consequence, the theoretical distance modules are cal- relation. Even if we chose the larger value of m> 4, the effect culated by (15)and (19). is still small. Recently, the Supernovae Cosmology Project (SCP) col- Figure 2 shows the allowed parameter region due to the laboration released their Union2 sample of 557 SNIa data [8]. χ fitting of (20)inDΛCDM from SNIa constraints. We 2 2 The Union2 compilation is the largest published and spec- obtain the minimum value of χ = 677 (the reduced χ is 2 2 troscopically confirmed SNIa sample. The observations are defined to be χ = χ /N  1.215, where N is the degree of in the redshift range of 0.01 <z < 2. For the SNIa data set, freedom). The best fit parameter region of Ω for m< 4is Λ2 −3 we evaluate χ value of the distance modules, investigated as Ω < 4.5×10 at 1σ confidence level (C.L.). Λ2 Since the parameter region is very loose compared with the N 2 μ − μ th,i obs,i 2 previous analysis [22, 23], this result implies that the decay- χ = , (20) σ + σ ing-Λ has minor effects on the expansion rate at z< 2. μ v,i i=1 obs,i Λ2 4 Advances in Astronomy 7 Table 1: Observational temperatures obtained from molecular excitation levels and S-Z effects. z T [K] Reference obs 1.776 7.4 ± 0.8CI[30] 1.9731 7.9 ± 1.0CI[31] 9 2.3371 6.0 <T [K] < 14.0CI[32] +1.7 8 3.025 12.1 C+ [33] −3.2 1.77654 7.2 ± 0.8CI[34] 2.4184 9.15 ± 0.72 CO [35] +0.8 2.6896 10.5 CO [36] 3 −0.6 4 +1.2 1.7293 7.5 CO [37] −1.6 3 +0.6 1.7738 7.8 −0.7 4 3 2 1.5 1 +1.0 2.0377 8.6 −1.1 1+ z 1+ z +0.101 0.203 3.377 S-Z [38] −0.102 SΛCDM Molaro (2002) +0.080 0.0231 2.789 −0.065 DΛCDM (m = 0.05) Battistelli (2003) 0.023 2.72 ± 0.10 S-Z [39] DΛCDM (m = 0.1) Luzzi (2008) DΛCDM (m = 0.15) Srianand (2008) 0.152 2.90 ± 0.17 Songaila (1994) Noterdaeme (2010) 0.183 2.95 ± 0.27 Ge (1997) Noterdaeme (2011) 0.20 2.74 ± 0.28 Srianand (2000) 0.202 3.36 ± 0.2 Figure 3: Illustration of the temperature evolutions compared with 0.216 3.85 ± 0.64 CMB observations. The left panel is the results at z> 0.6. The right 0.232 3.51 ± 0.25 is same, but at z< 0.6. 0.252 3.39 ± 0.26 0.282 3.22 ± 0.26 0.291 4.05 ± 0.66 4. Temperature Constraints 0.451 3.97 ± 0.19 It has been shown that the decaying-Λ term affects the 0.546 3.69 ± 0.37 photon temperature evolution at the early epoch [20]. In 0.55 4.59 ± 0.36 this section, we study about the consistency of DΛCDM with recent temperature observations. The temperature evolution is obtained from (10). Fol- larger value of m results in the lower-T .Whenweadopt lowing the Stefan-Boltzmann Law, ρ ∝ T , the relation γ m = 0.15, the theoretical curve of T should be inconsistent between the photon temperature and the cosmological red- with observations even for the temperature of the largest shift is obtained as follows [22, 23]: uncertainty [32]. We can conclude that the deviation from SΛCDM model might be small for m< 0.1. 1/4 γΛ Using the recent observations of the CMB temperature, T (z) = T (1+ z) 1+ , γ γ0 γ0 whose accuracy has been improved quantitatively, we can put severe constraints on the temperature evolution in DΛCDM (22) m−4 ⎨ 2 α (1+ z) − 1 (m= 4), model. We calculate χ -analysis as follows: Ω ≡ γΛ −4Ω ln(1+ z)(m = 4). Λ2 T (z) − T γ obs,i (23) χ = , From (22), the temperature-redshift relation at higher-z approaches that of SΛCDM model, T ∝ (1 + z). At lower-z, the temperature evolution deviates from the proportional where T (z) is theoretical temperature calculated by (22)and relation. T the observational data as shown in Table 1. σ is the obs,i i In the meanwhile, recently, the temperatures at higher-z uncertainty of the observation. N = 25 is the number of the are observed using the molecules such as the fine structure observational data. Note that the result of Srianand et al. [32] of excitation levels of the neutral or ionized carbon [30–34] does not have best-fit value. Therefore, we assume the best-fit and the rotational excitation of CO [35–37]and Sunyaev- value is the same as the mean value, T = 10.0 ± 4.0K. Srianand Zel’dovich (S-Z) effect [38, 39]. Furthermore, there are a lot In Figure 2, we show the limits in the m − Ω plane Λ2 of temperature observations for the lower-z owing to the S-Z calculated by (23). The shaded region indicates the param- effect as shown in Table 1. Therefore, we expect to constrain eter regions which should be excluded and are from (14). We more severely the allowable region on the m−Ω plane from Λ2 can obtain the allowed parameter range as follows: the new temperature observations. −4 −3 Figure 3 illustrates the temperature evolution in SΛCDM Ω < 6.1 × 10 at 1σ C.L.,1.7 × 10 at 2σ C.L. Λ2 −2 and DΛCDM model with Ω = 10 . 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Constraints on a Vacuum Energy from Both SNIa and CMB Temperature Observations

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Copyright © 2012 Riou Nakamura et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Astronomy Volume 2012, Article ID 528243, 6 pages doi:10.1155/2012/528243 Research Article Constraints on a Vacuum Energy from Both SNIa and CMB Temperature Observations Riou Nakamura, E. P. Berni Ann Thushari, Mikio Ikeda, and Masa-Aki Hashimoto Department of Physics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka City 812-8581, Japan Correspondence should be addressed to Riou Nakamura, riou@phys.kyushu-u.ac.jp Received 28 February 2012; Accepted 1 May 2012 Academic Editor: Rob Ivison Copyright © 2012 Riou Nakamura et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the cosmic thermal evolution with a vacuum energy which decays into photon at the low redshift. We assume that the vacuum energy is a function of the scale factor that increases toward the early universe. We put on the constraints using recent observations of both type Ia supernovae (SNIa) by Union-2 compilation and the cosmic microwave background (CMB) temper- ature at the range of the redshift 0.01 <z < 3. From SNIa, we find that the effects of a decaying vacuum energy on the cosmic expansion rate should be very small but could be possible for z< 1.5. On the other hand, we obtain the severe constraints for parameters from the CMB temperature observations. Although the temperature can be still lower than the case of the standard cosmological model, it should only affect the thermal evolution at the early epoch. 1. Introduction to a primordial light-element abundances [15] and the CMB intensity [16]. From the point of the thermodynamical evo- One of the biggest cosmological mysteries is the accelerating lution in the universe, a decaying vacuum energy modifies cosmic expansion which was discovered by the observations the temperature-redshift relation [17–19]. This modification of distant type Ia supernovae (SNIa) starting from more than affects the cosmic thermal evolution after a hydrogen recom- 10 years ago. For the origin of the accelerating expansion, bination [20], such as formation of molecules [21, 22]and the following possibilities have been proposed: the modified the first star [21]. In addition to this model, the angular gravity, such as f (R)gravity [1, 2], brane-world cosmology power spectrum of CMB could be also modified [23]. In the [3], inhomogeneous cosmology [4, 5], and existence of previous analysis, thermal history at the higher redshift has unknown energy called as dark energy which is equivalent been studied. However, as seen in [20, 21], the time variation to acosmicfluidwithanegative pressure. of temperature at low redshift differs from the model without It is strongly suggested that the dark energy amounts to a decaying vacuum. 70% of the total energy density of the universe from the In the present study, we update the observational consis- astronomical observations such as SNIa [6–8], the anisotropy tency of a vacuum energy (hereafter we denote as Λ)coupled of cosmic microwave background (CMB) [9–11], and the with CMB photon. Hereafter, we call this model DΛCDM baryon acoustic oscillation [12]. Although there are various (model). To examine the consistency with observations at theoretical models of dark energy (details are shown in a low redshift, we focus on the cosmic evolution comparing review [13, 14]), its physical nature is still unknown. Con- both the type Ia supernovae and the redshift dependence of sidering above-referenced observations, we can not exclude CMB temperatures. This is because the time dependence the constant Λ term which is the most simple model of dark of the CMB temperature for a DΛCDM differs from CDM energy. model with a constant Λ term of which hereafter we call The interacting dark energy with CMB photon has been SΛCDM (model). In Section 2, formulation of DΛCDM is discussed, whereas decaying vacuum into photon is related reviewed. In Section 3, the m − z relation is investigated for 2 Advances in Astronomy DΛCDM in a flat universe. Parameters inherent in this model where ρ is the critical density defined by the present Hubble cr are constrained from the CMB temperature observations in constant H .Wecan rewrite(8) as follows (for details also see Section 4. Concluding remarks are given in Section 5. [22, 23]): dΩ Ω γ γ dΩ (10) +4 =− . 2. Dynamics of the Decaying Λ Model da a da The Einstein’s field equation is written as follows (e.g., [24]): Models with time-dependent Λ-term have been studied as summarizedin[25, 26]. We adopt the energy of the (1) R − g R = 8πGT , μν μν μν vacuum which varies with the scale factor [20–23]: −m Ω (a) = Ω + Ω a , (11) where G is the gravitational constant and T is the energy Λ Λ1 Λ2 μν momentum tensor. If we assume the perfect fluid, T is μν where Ω , Ω ,and m are constants. Note that the present Λ1 Λ2 written as follows: value of Ω is expressed by Ω = Ω + Ω . Λ Λ0 Λ1 Λ2 T = diag −ρ, p, p, p . Integrating (10)with(11), we obtain the photon energy (2) μν density as a function of a [22]: Here ρ and p are the energy density and the pressure, res- ⎪ 4−m −4 pectively. Note that we choose the unit of c = 1. ⎪ Ω + α a − 1 a (m= 4), ⎨ γ0 The equation of motion is obtained with use of the Fried- Ω = (12) −4 ⎪ Ω +4Ω ln a a (m = 4), mann-Robertson-Walker metric of the homogeneous and γ0 Λ2 isotropic principle: −5 −2 where Ω = 2.471 × 10 h (T /2.725 K) is the present 2 γ0 γ0 dr 2 2 2 2 2 2 2 2 ds =−dt + a(t) + r dθ + r sin θdφ , photon energy density, h is the normalized Hubble constant 1 − kr (H = 100h km/sec/Mpc), and T is the CMB temperature 0 γ0 (3) at the present epoch. We define α as α = mΩ /(4 − m). Λ2 Since we assume the flat geometry (k = 0) in this work, where a(t) is the scale factor and k is the specific curvature we can write the condition of Ωs as follows: constant. This leads to the Friedmann equations: Ω + Ω + Ω = 1. (13) m0 Λ1 Λ2 a˙ 8πG k H ≡ = ρ − . (4) a 3 a From now on, we adopt the present cosmological parame- ters: Hubble constant h = 0.738 [27, 28], and the present From the conservation’s law of energy, we can obtain the density parameter of matter Ω = 0.2735 [11]. The present equation of the energy density: temperature is T = 2.725 K observed by COBE [29]. In our γ0 a˙ study, we search the parameter regions in the range of Ω ρ˙ =−3(1+ w) ρ, (5) Λ2 −7 −2 and m :10 < Ω < 10 and 0 <m< 4. The range of m Λ2 has already been obtained from the previous analysis [23]. where w is the coefficients of the equation of state, p = wρ, Here we note the range of Ω in the present analysis. Λ2 anddefinedby Kimura et al. [20] found that the radiation temperature (i.e., ⎪ the energy density of radiation) in DΛCDM model becomes for photon and neutrino, numerically negative at a point of a< 1 when m and/or Ω Λ2 w ≡ (6) 0 for baryon and cold dark matter, is too large. To avoid this unreasonable situation, that is, T < ⎪ γ −1 for vacuum energy. 0, Nakamura et al. [23] imposed the limit of Ω and m as Λ2 follows: We take the component of the energy density as α< Ω (m< 4), γ0 ρ = ρ + ρ + ρ + ρ , γ ν m Λ (7) (14) Ω < Ω /92 (m = 4). Λ2 γ0 where γ, ν, m,and Λ indicate photon, neutrino, nonrela- From the condition (14), we obtain the limit of Ω as 4.9 × tivistic matter (baryon plus cold dark matter), and the vac- Λ2 −7 10 for m = 4. We can consider that the effect of Ω is uum, respectively. Here we neglect the energy contribution Λ2 −7 negligible when Ω is smaller than 10 . of Λ on other components except for photon: the ρ and ρ Λ2 ν m −4 −3 evolves as ρ ∝ a and ρ ∝ a ,respectively. From (5), the ν m equation of the photon coupled with Λ is written as 3. SNIa Constraints ρ˙ +4Hρ =−ρ˙ . (8) γ γ Λ The cosmic distance measures depend sensitively on the spatial curvature and expansion dynamics of the models. Let us define the parameter of the energy density as follows: Therefore, the magnitude-redshift relation for distant stan- 8πG dard candles are proposed to constrain the cosmological Ω ≡ = ρ , i i (9) parameters. 3H cr 0 Advances in Astronomy 3 −1 Excluded region −2 −3 −4 −5 −6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −7 m = 7 m = 0 m = 3 Union2 99.73% C.L. Figure 1: Illustration of the magnitude-redshift relation in SΛCDM 95.4% C.L. −2 model (m = 0) and DΛCDM with Ω = 10 compared with SNIa 68.3% C.L. Λ2 observations (cross-shaped ones with error bars). Note that the Figure 2: Constraints on the m − Ω plane from the recent obser- Λ2 theoretical curve for m = 3 has no difference with that of m = 0. vations of CMB. The lines indicate upper bounds of 1, 2, and 3σ confidence level. The dashed and solid lines are the limits deduced from SNIa and CMB temperature observations. The shaded region shows the excluded region obtained from [23]. To calculate the effects of the comic expansion at low-z, we calculate the theoretical distance modules, μ = m − M = 5log d + 25, (15) th B L where μ and μ are the observed and the theoretical obs,i th,i values of the distance modules. σ is the dispersion in the where m , M,and d are the apparent and absolute magni- B L redshift due to the peculiar velocity v is written as: tudes, and the luminosity distance, respectively. Here d is related to the radial distance r in the metric as follows [24]: v dμ th d = (1+ z)r. (16) σ = . (21) c dz We can obtain r as follows: dt 1 dz −1 We adopt v = 300 km sec [40]. The total number of the r = = , (17) ( ) ( ) a t H E z united sample N is 557 for the present analysis. We confirm that Ω has done some contribution to where z is the redshift defined by a = 1/(1 + z). Now E(z)is Λ2 defined as change the μ−z relation. When Ω increases, the expansion Λ2 rate in the universe decreases, which is similar to the behavior H(z) 2 3 m of the matter dominant universe in the Friedmann model. E(z) = = Ω (1+ z) + Ω + Ω (1+ z) . m0 Λ1 Λ2 Figure 1 indicates the relation between the distance mod- (18) ules and the redshift against the SNIa observations in terms Then, (16) is rewritten as follows: of SΛCDM and DΛCDM with several values of m for a fixed −2 value of Ω = 10 . As the value of m increases, μ tends to (1+ z) dz d = . (19) decrease because of the increasing cosmic expansion rate. We H E(z) recognize that m is not seriously effective to change the μ − z As a consequence, the theoretical distance modules are cal- relation. Even if we chose the larger value of m> 4, the effect culated by (15)and (19). is still small. Recently, the Supernovae Cosmology Project (SCP) col- Figure 2 shows the allowed parameter region due to the laboration released their Union2 sample of 557 SNIa data [8]. χ fitting of (20)inDΛCDM from SNIa constraints. We 2 2 The Union2 compilation is the largest published and spec- obtain the minimum value of χ = 677 (the reduced χ is 2 2 troscopically confirmed SNIa sample. The observations are defined to be χ = χ /N  1.215, where N is the degree of in the redshift range of 0.01 <z < 2. For the SNIa data set, freedom). The best fit parameter region of Ω for m< 4is Λ2 −3 we evaluate χ value of the distance modules, investigated as Ω < 4.5×10 at 1σ confidence level (C.L.). Λ2 Since the parameter region is very loose compared with the N 2 μ − μ th,i obs,i 2 previous analysis [22, 23], this result implies that the decay- χ = , (20) σ + σ ing-Λ has minor effects on the expansion rate at z< 2. μ v,i i=1 obs,i Λ2 4 Advances in Astronomy 7 Table 1: Observational temperatures obtained from molecular excitation levels and S-Z effects. z T [K] Reference obs 1.776 7.4 ± 0.8CI[30] 1.9731 7.9 ± 1.0CI[31] 9 2.3371 6.0 <T [K] < 14.0CI[32] +1.7 8 3.025 12.1 C+ [33] −3.2 1.77654 7.2 ± 0.8CI[34] 2.4184 9.15 ± 0.72 CO [35] +0.8 2.6896 10.5 CO [36] 3 −0.6 4 +1.2 1.7293 7.5 CO [37] −1.6 3 +0.6 1.7738 7.8 −0.7 4 3 2 1.5 1 +1.0 2.0377 8.6 −1.1 1+ z 1+ z +0.101 0.203 3.377 S-Z [38] −0.102 SΛCDM Molaro (2002) +0.080 0.0231 2.789 −0.065 DΛCDM (m = 0.05) Battistelli (2003) 0.023 2.72 ± 0.10 S-Z [39] DΛCDM (m = 0.1) Luzzi (2008) DΛCDM (m = 0.15) Srianand (2008) 0.152 2.90 ± 0.17 Songaila (1994) Noterdaeme (2010) 0.183 2.95 ± 0.27 Ge (1997) Noterdaeme (2011) 0.20 2.74 ± 0.28 Srianand (2000) 0.202 3.36 ± 0.2 Figure 3: Illustration of the temperature evolutions compared with 0.216 3.85 ± 0.64 CMB observations. The left panel is the results at z> 0.6. The right 0.232 3.51 ± 0.25 is same, but at z< 0.6. 0.252 3.39 ± 0.26 0.282 3.22 ± 0.26 0.291 4.05 ± 0.66 4. Temperature Constraints 0.451 3.97 ± 0.19 It has been shown that the decaying-Λ term affects the 0.546 3.69 ± 0.37 photon temperature evolution at the early epoch [20]. In 0.55 4.59 ± 0.36 this section, we study about the consistency of DΛCDM with recent temperature observations. The temperature evolution is obtained from (10). Fol- larger value of m results in the lower-T .Whenweadopt lowing the Stefan-Boltzmann Law, ρ ∝ T , the relation γ m = 0.15, the theoretical curve of T should be inconsistent between the photon temperature and the cosmological red- with observations even for the temperature of the largest shift is obtained as follows [22, 23]: uncertainty [32]. We can conclude that the deviation from SΛCDM model might be small for m< 0.1. 1/4 γΛ Using the recent observations of the CMB temperature, T (z) = T (1+ z) 1+ , γ γ0 γ0 whose accuracy has been improved quantitatively, we can put severe constraints on the temperature evolution in DΛCDM (22) m−4 ⎨ 2 α (1+ z) − 1 (m= 4), model. We calculate χ -analysis as follows: Ω ≡ γΛ −4Ω ln(1+ z)(m = 4). Λ2 T (z) − T γ obs,i (23) χ = , From (22), the temperature-redshift relation at higher-z approaches that of SΛCDM model, T ∝ (1 + z). At lower-z, the temperature evolution deviates from the proportional where T (z) is theoretical temperature calculated by (22)and relation. T the observational data as shown in Table 1. σ is the obs,i i In the meanwhile, recently, the temperatures at higher-z uncertainty of the observation. N = 25 is the number of the are observed using the molecules such as the fine structure observational data. Note that the result of Srianand et al. [32] of excitation levels of the neutral or ionized carbon [30–34] does not have best-fit value. Therefore, we assume the best-fit and the rotational excitation of CO [35–37]and Sunyaev- value is the same as the mean value, T = 10.0 ± 4.0K. Srianand Zel’dovich (S-Z) effect [38, 39]. Furthermore, there are a lot In Figure 2, we show the limits in the m − Ω plane Λ2 of temperature observations for the lower-z owing to the S-Z calculated by (23). The shaded region indicates the param- effect as shown in Table 1. Therefore, we expect to constrain eter regions which should be excluded and are from (14). We more severely the allowable region on the m−Ω plane from Λ2 can obtain the allowed parameter range as follows: the new temperature observations. −4 −3 Figure 3 illustrates the temperature evolution in SΛCDM Ω < 6.1 × 10 at 1σ C.L.,1.7 × 10 at 2σ C.L. Λ2 −2 and DΛCDM model with Ω = 10 . 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