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

16 pages

/lp/ou_press/testing-the-consistency-of-three-point-halo-clustering-in-fourier-and-prbmsmnoQV

- Publisher
- The Royal Astronomical Society
- Copyright
- © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society
- ISSN
- 0035-8711
- eISSN
- 1365-2966
- D.O.I.
- 10.1093/mnras/sty187
- Publisher site
- See Article on Publisher Site

Abstract We compare reduced three-point correlations Q of matter, haloes (as proxies for galaxies) and their cross-correlations, measured in a total simulated volume of ∼100 (h−1 Gpc)3, to predictions from leading order perturbation theory on a large range of scales in configuration space. Predictions for haloes are based on the non-local bias model, employing linear (b1) and non-linear (c2, g2) bias parameters, which have been constrained previously from the bispectrum in Fourier space. We also study predictions from two other bias models, one local (g2 = 0) and one in which c2 and g2 are determined by b1 via approximately universal relations. Overall, measurements and predictions agree when Q is derived for triangles with (r1r2r3)1/3 ≳60 h−1 Mpc, where r1 − 3 are the sizes of the triangle legs. Predictions for Qmatter, based on the linear power spectrum, show significant deviations from the measurements at the BAO scale (given our small measurement errors), which strongly decrease when adding a damping term or using the non-linear power spectrum, as expected. Predictions for Qhalo agree best with measurements at large scales when considering non-local contributions. The universal bias model works well for haloes and might therefore be also useful for tightening constraints on b1 from Q in galaxy surveys. Such constraints are independent of the amplitude of matter density fluctuation (σ8) and hence break the degeneracy between b1 and σ8, present in galaxy two-point correlations. methods: analytical, methods: numerical, methods: statistical, large-scale structure of Universe 1 INTRODUCTION Higher-order correlations, induced by gravity into the distribution of large-scale matter density fluctuations, contain information that cannot be captured by second-order statistics. This information can be used to tighten constraints on cosmological models, as well as on models of galaxy formation. Key tools for obtaining such constraints are galaxy bias models (e.g. Desjacques, Jeong & Schmidt 2016). These models relate the density and the tidal field of the full matter content in a given region to the density of observable tracers, such as galaxies. They include a number of so-called bias parameters, which depend on the various processes that drive the tracer formation. Since these highly complex processes are only partly understood, the bias parameters cannot be predicted in a reliable way (e.g. Li et al. 2007; Müller, Hoffmann & Nuza 2011; Pujol et al. 2017; Springel et al. 2018) and hence need to be measured from the data. Such measurements can be obtained from the analysis of weak gravitational lensing signals, or redshift space distortions. However, these methods rely on good redshift estimations and imaging of the tracers (i.e. galaxies) as well as on various model assumptions. It is therefore interesting to obtain independent measurements of the bias parameters, which is possible with a joint analysis of second- and third-order statistics. This approach becomes increasingly interesting as errors on these statistics decrease with the increasing volumes of upcoming galaxy surveys. Going to third order in the statistical analysis of galaxy surveys does not only deliver bias measurements, but also measurements of the growth of matter fluctuations. The latter provide the aforementioned cosmological constraints, while the bias can be used to predict the number of galaxies per halo, which places constraints on galaxy formation models (e.g. Scoccimarro et al. 2001; Berlind & Weinberg 2002; Cooray & Sheth 2002). The most general third-order statistics is the three-point correlation function (hereafter referred to as 3PCF), which is defined in configurations space. Alternatively, one can study its Fourier space counterpart, the bispectrum. These two statistics contain, in principle, the same information. However, their analyses implicate different limitations and challenges, which can affect the physical interpretation of the results. A main advantage of the bispectrum is that an analysis in Fourier space allows for a clear exclusion of high-frequency modes in the density fluctuations, which are difficult to interpret theoretically due to their highly non-linear evolution. In configuration space, these high-frequency modes contribute to the 3PCF, in principle, at all scales. In practice, one therefore needs to restrict the analysis to large scales, where their contribution is negligible, lavishing a lot of valuable data. Another advantage of the bispectrum is that its covariance is diagonal for Gaussian density fluctuations. This approximation works well, even for evolved density fields, while deviations from Gaussianity can also be taken into account (Scoccimarro 2000; Sefusatti et al. 2006; Chan & Blot 2017). The covariance of the 3PCF, on the other hand, is not diagonal, even for Gaussian fluctuations, which makes the modelling more difficult (Srednicki 1993; Slepian & Eisenstein 2015; Byun et al. 2017; Gualdi et al. 2017). An additional difference in the analysis of the bispectrum and the 3PCF lies in the fact that the computation of the latter is more expensive. However, this aspect can be tackled by employing advanced algorithms and appropriate computational resources, as done in this work (see also, Barriga & Gaztañaga 2002; McBride et al. 2011a; Jarvis 2015; Slepian & Eisenstein 2015, and references therein). Besides its disadvantages, there are some arguments that speak for the 3PCF. One of them is the fact that the amplitude of the 3PCF (but not its errors) is not affected by shot-noise, whereas the latter affects the bispectrum amplitude at all scales and hence needs to be modelled for correcting the measurements. In addition, an analysis in configuration space has the advantage that complicated survey masks can be easily taken into account in the analysis of observational data, while in Fourier space such masks impose complicated effects on the measured bispectrum, which are difficult to model (e.g. Scoccimarro 2000). A more general consideration is that it is easier to interpret effects such as redshift space distortions or baryon acoustic oscillations (BAOs) on the statistics in configuration space, since that is where the physical processes that cause these effects happen. Studies of third-order correlations in the literature usually focus on either Fourier or configuration space (e.g. McBride et al. 2011b; Marín et al. 2013; Gil-Marín et al. 2015). However, it is worthwhile studying both statistics and cross-check the results, since their different advantages and disadvantages are quite complementary. In this work, we will conduct such a cross-check for the first time. Our main interest thereby is to verify if and when the bias parameters, obtained from the bispectrum, are consistent with those that affect the 3PCF in configuration space. Our approach is based on the analysis of Chan, Scoccimarro & Sheth (2012, hereafter CSS12). These authors measured the bias parameters of large-scale structure tracers in Fourier space from a set of N-body simulations, using a leading-order perturbative model of the bispectrum and restricting the analysis to large modes with wave numbers k ≤ 0.1 h Mpc−1. The tracers in their analysis are dark-matter haloes, while the same method for measuring the bias can be applied to any other type of tracers, such as galaxies or galaxy clusters. For our cross-check, we use the same perturbative model together with the bias parameters of CSS12 to predict the halo 3PCF in configuration space. We then measure the latter in the same set of simulations to test the predictions. This allows us to verify if and when the bias parameters measured from third-order statistics in Fourier space also describe the corresponding statistics in configuration space. Simultaneously, we test at which scales, redshifts and halo mass ranges the leading order perturbative modelling of the 3PCF is an appropriate approximation. 1.1 Bias models tested The bias model relates the density fluctuations and the tidal field of matter in a certain region to the density fluctuations of its tracers. These fluctuations are defined with respect to the mean density as $$\delta \equiv (\rho - \bar{\rho })/\bar{\rho }$$. Since the leading order perturbative expansion of third-order statistics, on which we focus in this analysis, is quadratic, we use the quadratic non-local bias model, \begin{equation} \delta _h= b_1 \biggl \lbrace \delta _m + \frac{1}{2}[ c_2(\delta _m^2 - \langle \delta _m^2 \rangle ) + g_2 \mathcal {G}_2] \biggr \rbrace . \end{equation} (1)The indices h and m refer to the halo and matter density fluctuations, respectively. The parameters b1 and c2 are hereafter referred to as local linear and quadratic bias (Fry & Gaztanaga 1993), while g2 will be referred to as quadratic non-local bias, since it scales with the tidal field term $$\mathcal {G}_2$$, which can be generated by masses outside of the volume in which δg is defined (see McDonald & Roy 2009; Baldauf et al. 2012, ; CSS12). The term for the smoothed tidal field is given by a second-order Gallileon \begin{equation} \mathcal { G}_2({\boldsymbol r})= \int \beta _{12}\theta _v({\boldsymbol k}_1) \theta _v({\boldsymbol k}_2) \ \hat{W}[k_{12}R]e^{i {\boldsymbol k}_{12}\cdot {\boldsymbol r} }d^3 {\boldsymbol k}_1 d^3 {\boldsymbol k}_2, \end{equation} (2)where $${\boldsymbol k}_i$$ and $${\boldsymbol k}_{12} \equiv {\boldsymbol k}_2 - {\boldsymbol k}_1$$ are wave vectors of density oscillations, $$\beta _{12} \equiv 1 - ({\hat{{\boldsymbol k}}}_1 \cdot {\hat{{\boldsymbol k}}}_2)^2$$ represents the mode coupling between density oscillations that describes tidal forces, θv ≡ ∇2Φv is the divergence of the normalized velocity field ($${\boldsymbol v}/\mathcal {H}/f$$), and $$\hat{W}[k_{12}R]$$ is the window function in Fourier space (CSS12). Note that the non-local bias has also been referred to as an additional local bias parameter, since the tidal field is a local observable, which depends on derivatives of the potential (see CSS12; Desjacques et al. 2016). However, in this work, we call it non-local, since it is non-local in the density. We use three sets of bias parametrizations for predicting the 3PCF. The first set consists of the bias parameters b1, c2 ≡ b2/b1 and g2 ≡ 2γ2/b1, obtained by CSS12 from fitting the non-local bias model predictions for the bispectrum at leading order to measurements in the same set of simulations as studied in this work. Here, b2 and γ2 are the quadratic local and the non-local bias parameters, respectively, in the notation of CSS12. The second set equals the first set, except for the non-local bias parameter g2, which is set to zero in order to verify the impact of the non-local contributions on the 3PCF predictions. In the third set, the linear bias b1 is the only input parameter. This is the same parameter, as in the two previous sets, but was obtained by CSS12 from fits of the linear bias model (δh = b1δm) to the power spectrum. The quadratic local bias parameter c2 in this third set is computed from the (approximately) universal relation $$c_2 \simeq 0.77 b_1^{-1} - 2.43 + b_1$$, given by Hoffmann, Bel & Gaztañaga (2017) (see also Hoffmann, Bel & Gaztañaga 2015b; Lazeyras et al. 2016, for similar relations). The non-local bias in the third set is obtained from the local Lagrangian model, g2 = −(4/7)(1 − 1/b1). The three sets of bias model parameters are summarized in Table 1 and will in the following be referred to as non-local, local, and $$b_{\delta ^2 {\rm {fix}}}$$ models, respectively. Table 1. Different bias models studied in this work. The bias parameters have been measured in Fourier space from the same set of simulations by CSS12. For the $$b_{\delta ^2 {\rm {fix}}}$$ model, we use a roughly universal relation for the quadratic local bias c2(b1) (Hoffmann et al. 2017) and the local Lagrangian model for the quadratic non-local g2(b1). Bias model Description Non-local b1, c2, g2 from bispectrum fits Local Same b1 and c2 as above, g2 = 0 $$b_{\delta ^2 {\rm {fix}}}$$ b1 from power spectrum fits, $$c_2 = 0.77 b_1^{-1} - 2.43 + b_1$$, g2 = −(4/7)(1 − 1/b1) Bias model Description Non-local b1, c2, g2 from bispectrum fits Local Same b1 and c2 as above, g2 = 0 $$b_{\delta ^2 {\rm {fix}}}$$ b1 from power spectrum fits, $$c_2 = 0.77 b_1^{-1} - 2.43 + b_1$$, g2 = −(4/7)(1 − 1/b1) View Large For a three-dimensional analysis of real spectroscopic surveys, one would further need to take into account redshift space distortions in the modelling. Redshift space distortions cancel out approximately at large scales in the reduced 3PCF (defined in Section 2.1, see e.g. Gaztañaga & Scoccimarro 2005), but there are non-linear contributions that could be as large as the non-local terms. There is some indication in simulations that non-local terms can cancel out with redshift space distortions (e.g. fig.17 of Hoffmann et al. 2015a), but this requires further study. Large volume photometric surveys, such as DES or LSST, will provide additional constraints from weak lensing of the projected 3PCF, both of galaxy and matter correlations, as well as galaxy–matter cross-correlations. Since those surveys measure redshifts from broad-band photometry, these probes will have little contamination by redshift space distortions. All this is beyond the scope of this paper, but should be a clear continuation of our study. 2 THREE-POINT CORRELATIONS Q 2.1 Definitions Our 3PCF analysis is applied on density fields $$\rho ^x(\boldsymbol r)$$, where x refers to the density of matter (x = m) or of its tracers, such as galaxies or, as in our case, dark matter haloes (x = h) at the position $$\boldsymbol r$$. The density fields are smoothed with a top-hat filter of scale R and described by the normalized density fluctuations $$\delta ^{x}({\boldsymbol r}_i) \equiv \delta ^{x}_i$$, introduced in Section 1.1. Note that, in contrast to the notation in equation (1), we now set x as upper index to avoid confusion between the position and the power indices in the following. The 3PCF can be defined as the average product of density fluctuations at three positions $$({\boldsymbol r}_{1}, {\boldsymbol r}_{2}, {\boldsymbol r}_{3})$$, which form a triangle. In the case of the halo-matter-matter cross-correlations, it is written as \begin{equation} \zeta ^{hmm} (r_{12}, r_{13}, r_{23}) \equiv \langle \delta ^h_1 \delta ^m_2 \delta ^m_3 \rangle (r_{12}, r_{13}, r_{23}), \end{equation} (3)where $$r_{ij} \equiv |{\boldsymbol r}_i - {\boldsymbol r}_j|$$ are the absolute values of the triangle legs and 〈…〉 denotes the average over all possible triangle orientations and translations. We proceed by defining the symmetric reduced three-point cross-correlation, \begin{equation} Q_{\times } \equiv \frac{1}{3} \frac{\zeta ^{hmm} + \zeta ^{mhm} + \zeta ^{mmh}}{\zeta ^{\times } _H} \end{equation} (4)and drop the expression reduced in the following. The hierarchical three-point cross-correlation in the denominator, defined as \begin{equation} \zeta ^{\times } _H \equiv \xi ^{hm}_{12} \xi ^{hm}_{13} + \xi ^{hm}_{12} \xi ^{hm}_{23} + \xi ^{hm}_{13} \xi ^{hm}_{23}, \end{equation} (5)is comprised of two-point cross-correlations $$\xi _{ij}^{xy} \equiv \langle \delta ^x_i \delta ^y_j \rangle (r_{ij})$$ between the density fields x and y (e.g. Peebles & Groth 1975; Fry 1984). The corresponding expressions for the three-point autocorrelations for matter and haloes (Qm and Qh, respectively) are defined analogously, i.e. $$Q_m \equiv \zeta ^{mmm} / \zeta _H^{mmm}$$ and $$Q_h \equiv \zeta ^{hhh} / \zeta _H^{hhh}$$. 2.2 Modelling Our predictions for Qh and Q× are based on the non-local quadratic bias model from equation (1), which yields at leading-order perturbative expansion in terms of δm \begin{equation} Q_h \simeq \frac{1}{b_1}[Q_m + c_2 + g_2 Q_{{\rm nloc}}], \end{equation} (6)where b1 and c2 are the local linear and quadratic bias parameters, respectively. The non-local contribution Qnloc scales with the non-local quadratic bias parameter g2 (see Baldauf et al. 2012; CSS12). The corresponding leading order expression for the halo-matter–matter cross-correlation is given by \begin{equation} Q_\times \simeq \frac{1}{b_1}[Q_m + \frac{1}{3}(c_2 + g_2 Q_{{\rm nloc}})]. \end{equation} (7)Equation (6) has an important application in the analysis of galaxy surveys, since it allows for bias measurements that are independent of the linear growth of matter fluctuations (e.g. Frieman & Gaztanaga 1994; Sefusatti et al. 2006; McBride et al. 2011b; Marín et al. 2013; Gil-Marín et al. 2015) and hence breaks the growth-bias degeneracy. However, cosmological constraints from such bias measurements are limited by the inaccuracies of the Qh modelling as explained in the following. The statistics of the full matter field, Qm and Qnloc, cannot be observed in galaxy surveys and hence need to be predicted for a given cosmology. Qm is therefore often predicted from N-body simulations. This approach has also been used by CSS12 for measuring the bias parameters in their simulations, as it captures the non-linear contributions to Qm. However, these authors employ an analytical expression for the quadratic non-local contribution Qnloc, which is in Fourier space simply related to the cosine of the angle between two wave vectors. Direct measurements of Qnloc would be more complicated (see Section 3.2). Another disadvantage of deriving the Qm and Qnloc from simulations is that a dense sampling of the cosmological parameter space for deriving constraints from observations would require enormous resources (albeit Qm and Qnloc are independent of the linear growth factor and hence only weakly depend on cosmology at large scales). In this analysis, we will therefore employ predictions from leading order perturbation theory for Qm and Qnloc (Jing & Boerner 1997; Gaztanaga & Bernardeau 1998; Barriga & Gaztañaga 2002; Bel, Hoffmann & Gaztañaga 2015). These leading order approximations, as well as those of Qh and Q× in equations (6) and (7), introduce inaccuracies in the modelling, in particular, at small triangle scales, which are strongly affected by high-frequency, non-linear modes (Scoccimarro et al. 1998; Pollack, Smith & Porciani 2012). Measurements of the different 3PCFs in N-body simulations allow us to validate these approximations. For comparing Qh and Q× with such measurements, we employ bias parameters measured from the power spectrum and the bispectrum in Fourier space by CSS12 in the same set of simulations as used in this analysis. We thereby do not only test the validity of the perturbation theory predictions for the 3PCFs, but also if the bias parameters in Fourier and configuration space are consistent with each other. Note that the Fourier space bias measurements are also based on leading order perturbation theory predictions for the cross-bispectrum Bhmm and the non-local contribution. However, non-linear contributions can be excluded in that case in a more reliable way than in configuration space by restricting the analysis to long wavelength modes. We therefore consider them to be robust. To summarize, the accuracy of the model of Qh in equation (6) depends on the accuracy of the leading order perturbative expansion of Qh, Qm, and Qnloc. The comparison with measurements in simulations will further depend on the accuracy of the bias measurements in Fourier space. In Section 3, we will test these different model ingredients using measurements of Qm, Qh, and Q×. 2.3 Measurements in simulations We verify the model predictions using the same set of Nsim = 49 cosmological N-body simulations, which was analysed by CSS12. Each simulation was run with 6403 dark matter particles, which reside in a cube with comoving side length of 1280 h−1Mpc, which results in a total simulated volume of ∼100 (h−1 Gpc)3. The cosmological parameters were set to Ωm = 1 − ΩΛ = 0.27, with Ωb = 0.046, h = 0.72, ns = 1, and σ8 = 0.9. Haloes were identified as friends-of-friends groups with a linking length of 0.2 of the mean particle separation. We split them into the same mass samples as CSS12, which are summarized in Table 2. Table 2. Halo mass samples with corresponding linear bias from the halo-matter cross-power spectrum from CSS12. The same samples are used in this work. z Halo sample Mass range (1013 M⊙ h−1) $$b^P_{hm}$$ 0.0 m0 4–7 1.43 0.0 m1 7–15 1.75 0.0 m2 >15 2.66 0.5 m0 3–5 1.88 0.5 m1 5–10 2.26 0.5 m2 >10 3.29 1.0 m0 2–3.1 2.43 1.0 m1 3.1–5.7 2.86 1.0 m2 >5.7 3.99 z Halo sample Mass range (1013 M⊙ h−1) $$b^P_{hm}$$ 0.0 m0 4–7 1.43 0.0 m1 7–15 1.75 0.0 m2 >15 2.66 0.5 m0 3–5 1.88 0.5 m1 5–10 2.26 0.5 m2 >10 3.29 1.0 m0 2–3.1 2.43 1.0 m1 3.1–5.7 2.86 1.0 m2 >5.7 3.99 View Large For measuring the 3PCFs in these simulations, we generate density maps of the simulated halo and matter distributions based on 8 h−1 Mpc cubical cells. The products of density contrasts δ1δ2δ3, over which we average to compute Q, are obtained from triplets of these cells, which we find using an algorithm described by Barriga & Gaztañaga (2002). This algorithm delivers measurements for triangle configurations, defined by the fixed leg sizes r1, r2 at different opening angles $$\alpha \equiv \arccos ( \hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$. The fixed triangle legs are defined with a tolerance ri ± δr, while we set δr to values between 1 and 4 h−1Mpc, depending on the triangle configuration. This tolerance is needed for finding a large number of triplets on the grid and thereby reduce the impact of shot-noise on the 3PCF measurements. We study the impact of this tolerance on the 3PCF, by computing the 3PCF predictions for the same set of triangles, which we find on the grid for a given (r1, r2) ± (δr1, δr2). We then bin the results for different opening angles α and compare them to predictions for exact (r1, r2) values, as we use them in our analysis (see Appendix A). This comparison shows that the effect of this tolerance and the binning on the 3PCF are small compared to inaccuracies of the 3PCF predictions. The 3PCFs are computed for 28 configurations (r1, r2), with 18 opening angles each, which leads to a total number of 504 triangles. 2.4 Error estimation To quantify the deviations between the mean 3PCF measurements from the 49 simulations, $$\bar{Q}_i$$, and the corresponding model predictions, $$Q_i^{{\rm mod}}$$, for a set of N∇ triangles (each defined by r1, r2 and α, with i ∈ {1, 2, …, N∇}), we want to compute \begin{equation} \chi ^2=\sum ^{N_{\nabla }}_{ij} \Delta _i {\hat{C}}_{ij}^{-1}\Delta _j, \end{equation} (8)where $$\Delta _i \equiv (Q^{{\rm mod}}_i-\bar{Q}_i)/\sigma _i$$. The standard deviation of $$\bar{Q}_i$$ is given by $$\sigma _i^2 = \langle (Q_i - \bar{Q}_i)^2 \rangle / N_{{\rm sim}}$$, while 〈…〉 denotes the mean over the Nsim measurements. The factor 1/Nsim accounts for the fact that we study the deviations of the mean measurements from the prediction, rather than deviations of measurements in individual realizations. The normalized covariance (or correlation) matrix is hence given by \begin{equation} \hat{C}_{ij} = \langle \Delta _{i}\Delta _{j} \rangle /N_{{\rm sim}}, \end{equation} (9)with $$\Delta _{i} \equiv (Q_i - \bar{Q}_i)/\sigma _i$$. We choose N∇ < Nsim to allow for the inversion of Cij, as pointed out by Hartlap, Simon & Schneider (2007), and set N∇ = 30 for the χ2 measurements shown in this paper. We tested that these measurements are consistent, but noisier (less noisy) when setting N∇ = 20 (40), which presumably results from the relatively low number of 49 realizations. To reduce this noise, we follow Gaztañaga & Scoccimarro (2005) by performing a Singular Value Decomposition of the covariance (hereafter referred to as SVD), i.e. \begin{equation} \hat{C}_{ij} = (U_{ik})^{\dagger } D_{kl} V_{lj}. \end{equation} (10)The diagonal matrix $$D_{kl} = \delta _{kl}\lambda _k^2$$ consists of the singular values λk (SVs), while the corresponding normalized modes $${\hat{\boldsymbol M}}_k$$ form the matrix Uik. \begin{equation} \chi ^2 \simeq \sum _k^{N_{{\rm mode}}} ({\bf \Delta} \cdot \hat {\boldsymbol M}_k)^2 / \lambda_k^2. \end{equation} (11)Note the elements of the vector Δ correspond to the quantity Δi, which appears in equation (8). Fig. C2 shows that $$\hat{C}_{ij}$$ is typically dominated only by a few modes. Assuming that the modes with the lowest SVs can be associated with measurement noise, we use only SVs with values larger than the sampling error estimate (i.e. $$\lambda ^2 \gtrsim \sqrt{2/N_{{\rm sim}}}$$) for our χ2 computation, as suggested by Gaztañaga & Scoccimarro (2005). The number of selected modes is hence the degree of freedom (d.o.f.) in our χ2 estimation, i.e. d.o.f. = Nmode < Nbin < Nsim. 3 ACCURACY OF Q PREDICTIONS The accuracy of bias measurements from the reduced three-point halo autocorrelation Qh in observations depends on how well it is approximated by the leading order perturbative model, given by equation (6). To verify this model, we test its different components separately with direct measurements in the simulations described in Section 2.3. We start with testing the modelling of Qm in Section 3.1 and proceed in Section 3.2 with tests of the quadratic component in equation (6). Our measurements of the latter are obtained by combining the three-point auto- and cross-correlations, Qh and Q×, respectively. Finally, we compare the complete predictions for Qh and Q×, given by equations (6) and (7), with the measurements in the simulations in Section 3.3. For modelling the quadratic components, we use bias parameters measured by CSS12 in the same set of simulations, using a leading order perturbative approximation of the 3PCF in Fourier space, i.e. the tree-level bispectrum. In addition, we employ simple relations between the linear and the quadratic bias parameters, i.e. b2(b1) and g2(b1). This leaves b1 as the only free input parameter in the bias model, which we adopt from the fits to the power spectrum, given by CSS12 (see Section 1.1). 3.1 Qm We start our verification of the Qm model from leading order (tree level) perturbation theory (hereafter referred to as $$Q_m^{PT}$$, see Section 2.2) by comparing its predictions to measurements in simulations. As examples we show in Fig. 1 results at redshift z = 0.5 for triangles with fixed legs (r1, r2) = (64, 32) and (80, 48) h−1Mpc versus the triangle opening angle $$\alpha \equiv \arccos ( \hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$. The Qm measurements are the mean results of the 49 simulations and are shown with 1σ errors bars. Here and throughout this paper, we display measurements at the mean opening angle in each bin. The predictions in Fig. 1 are computed from the non-linear power spectrum, which was measured in the simulations. Both measurements and predictions exhibit a u-shape, which is more strongly pronounced for the larger triangle configuration and originates from the filamentary structure of the cosmic web. The measurements clearly show the BAO feature for the (80, 48) h−1Mpc configuration at around 105°. Indications for similar BAO 3PCF features in real data have first been reported for luminous red galaxies in the SDSS DR7 sample by Gaztañaga et al. (2009). Slepian et al. (2015) later reported indications for the 3PCF BAO feature in the SDSS DR12 BOSS CMASS sample, which were confirmed by the 4.5σ detection in the same data set by Slepian et al. (2017). Figure 1. View largeDownload slide Reduced matter 3PCF Qm for triangles with fixed legs r1 and r2 (size is indicated in h−1Mpc) versus the mean triangle opening angle $$\alpha \equiv \arccos\, (\hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$ in each bin. Symbols show mean measurements from 49 simulations with 1σ errors at redshift z = 0.5. Lines show tree-level predictions from the measured (non-linear) power spectrum. Figure 1. View largeDownload slide Reduced matter 3PCF Qm for triangles with fixed legs r1 and r2 (size is indicated in h−1Mpc) versus the mean triangle opening angle $$\alpha \equiv \arccos\, (\hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$ in each bin. Symbols show mean measurements from 49 simulations with 1σ errors at redshift z = 0.5. Lines show tree-level predictions from the measured (non-linear) power spectrum. The significance of the deviations between measurements and predictions is shown for all redshifts in Fig. 2. In addition to the predictions from the non-linear power spectrum, we show in this figure also results based on the linear as well as the so-called de-wiggled power spectrum (hereafter also referred to as Plin and Pdw, respectively). The latter introduces non-linearities around the BAO scale in the 3PCF, coming from large-scale displacements (Crocce & Scoccimarro 2008; Carlson, Reid & White 2013; Baldauf et al. 2015; Senatore & Zaldarriaga 2015; Blas et al. 2016). It consists of the no-wiggle approximation of the power spectrum (Pnw) from Eisenstein & Hu (1998), Plin and a smearing function, i.e. Pdw ≡ Pnw + (Plin − Pnw) exp( − k2σv), where k is the wavenumber and σv is the variance of the displacement field (Eisenstein, Seo & White 2007).1 Figure 2. View largeDownload slide Significance of the deviations between the mean reduced matter 3PCF, measured in 49 simulations and different tree-level predictions for the same triangle configurations as shown in Fig. 1. Dashed and solid lines show predictions from the linear and the measured (non-linear) power spectrum, respectively. Predictions based on the de-wiggled power spectrum are shown as dash–dotted lines. The black dotted lines mark 3σ deviations from the measurements. Figure 2. View largeDownload slide Significance of the deviations between the mean reduced matter 3PCF, measured in 49 simulations and different tree-level predictions for the same triangle configurations as shown in Fig. 1. Dashed and solid lines show predictions from the linear and the measured (non-linear) power spectrum, respectively. Predictions based on the de-wiggled power spectrum are shown as dash–dotted lines. The black dotted lines mark 3σ deviations from the measurements. As a general trend we see in Fig. 2 that all predictions differ more significantly from the measurements for smaller triangles. This can be explained by the interplay of two effects. On one hand, terms in the perturbative expansion of Qm beyond leading order, which are neglected in our Qm model, contribute stronger at smaller scales. This explanation is consistent with the fact that the deviations are less significant at higher redshift and also when $$Q_m^{PT}$$ is computed from the non-linear, instead of the linear power spectrum. On the other hand, the signal-to-noise ratio is higher at small scales (see the bottom panel of Fig. B2). Note that the latter is specific to the joint volume of our 49 realizations of roughly ∼100 (h−1 Gpc)3. For the smaller volumes of current and near future galaxy surveys, we expect the model to deviate less significantly because of larger measurement errors. Predictions from the de-wiggled power spectrum are very similar to those from the linear power spectrum for smaller triangles (e.g. (r1, r2) = (64, 32) h−1Mpc, α ≲ 120°) and agree well with those from the non-linear power spectrum for large triangles (i.e. (r1, r2) = (80, 48) h−1Mpc, α ≳ 90°). The latter finding indicates that for the tree-level calculation of the 3PCF in configurations space, implementing resummations over large-scale displacements by using the de-wiggled power spectrum has almost the same effect as using the non-linear spectrum from the simulation. For both cases, the predictions are in 1σ agreement with the measurements at the BAO scale, while using the linear spectrum leads to 2–3σ deviations. For the remainder of our analysis, we will use predictions based on the non-linear power spectrum, as they show the best overall agreement with the measurements in Fig. 2. A convenient way to show results for all triangles in our analysis is to display them for a given opening angle α versus the triangle size, here defined as $$\sqrt{r_1r_2}$$. As an example, we show the measurements of Qm in Fig. 3. This figure demonstrates the strong increase of the u-shape of Qm(α) with the triangle scale. The minimum lies between 60° and 90°. Measurements for α ≳ 120° and $$\sqrt{r_1r_2} \gtrsim 50$$ are dominated by noise. Figure 3. View largeDownload slide Qm measurements at z = 0.5 versus the mean triangle opening angle per bin α and triangle scale (r1r2)1/2. Dots and solid contour lines show mean results from 49 simulations. Dashed contour lines show tree-level predictions based on the non-linear power spectrum. The colours indicate the amplitude of Qm. Figure 3. View largeDownload slide Qm measurements at z = 0.5 versus the mean triangle opening angle per bin α and triangle scale (r1r2)1/2. Dots and solid contour lines show mean results from 49 simulations. Dashed contour lines show tree-level predictions based on the non-linear power spectrum. The colours indicate the amplitude of Qm. The significance of the deviations between Qm model predictions and measurements are shown for redshift z = 0.5 versus α and $$\sqrt{r_1r_2}$$ in Fig. 4. We find that $$Q_m^{PT}$$ is below the measurements for opening angles between roughly 30 and 90° for triangles with $$30 \lesssim \sqrt{r_1r_2} \lesssim 50$$ h−1Mpc. For smaller and larger opening angles, the predictions tend to lie above the measurements. Similar results based on simulations with different cosmologies have been reported in the literature [see for instance Barriga & Gaztañaga (2002) or Hoffmann et al. (2015a), who use the same algorithms for the Qm predictions and measurements as employed in this study]. Scoccimarro et al. (1998) showed that such deviations can be explained by higher order contributions, as they reduce when the predictions are developed to next to leading order, including 1-loop terms (see also Sefusatti, Crocce & Desjacques 2010). As in Fig. 2 one can see in the top panel of Fig. 4 that using the linear power spectrum leads to strong deviations between predictions and measurements, in particular around the BAO peak, which are apparent as a red banana-shaped feature. This BAO feature follows roughly triangles with r3 ∼ 95 h−1Mpc, which are marked in the top panel as black lines. The deviations strongly reduce when the predictions are computed from the non-linear power spectrum for triangles scales $$\sqrt{r_1r_2} \gtrsim 50$$ h−1Mpc and α ≳ 30 to roughly 1σ. Figure 4. View largeDownload slide Significance of the deviations between Qm measurements and tree-level predictions versus the mean triangle opening angle per bin α and the triangle scale (r1r2)1/2 in h−1Mpc at redshift z = 0.5. The predictions are derived from the linear and measured (non-linear) power spectrum (top and bottom panels, respectively). Black lines in the top panel trace the BAO feature (r1r2)1/2(α) for r3 = 95 h−1Mpc. In the bottom panel, black lines indicate the triangle scale at which the model fails at 2σ, (r1r2r3)1/3 ≃ 60 h−1Mpc. In both cases, solid and dash–dotted lines correspond to triangle configurations of r2/r1 = 1.0 and 0.5, respectively. Figure 4. View largeDownload slide Significance of the deviations between Qm measurements and tree-level predictions versus the mean triangle opening angle per bin α and the triangle scale (r1r2)1/2 in h−1Mpc at redshift z = 0.5. The predictions are derived from the linear and measured (non-linear) power spectrum (top and bottom panels, respectively). Black lines in the top panel trace the BAO feature (r1r2)1/2(α) for r3 = 95 h−1Mpc. In the bottom panel, black lines indicate the triangle scale at which the model fails at 2σ, (r1r2r3)1/3 ≃ 60 h−1Mpc. In both cases, solid and dash–dotted lines correspond to triangle configurations of r2/r1 = 1.0 and 0.5, respectively. Defining the overall triangle size as (r1r2r3)1/3, we find that the deviations converge to 2σ at (r1r2r3)1/3 ≳ 60 (80) h−1Mpc, when using the non-linear (linear) power spectrum (Fig. B2). Triangles with (r1r2r3)1/3 = 60 h−1Mpc are therefore marked by black lines in the bottom panel of Fig. 4). The normalized Qm covariance matrix, shown in Fig. C1, reveals that the Qm measurements for different triangles are correlated with each other. Hence, we compute an SVD estimate of the χ2 in bins of (r1r2r3)1/3 to quantify the deviation between measurements and predictions, taking the covariance into account, as described in Section 2.4. Each (r1r2r3)1/3 bin includes measurements from 30 triangles, while we tested that our results change only weakly, when using 20 and 40 triangles per bin and do not affect our conclusions. In Fig. 5, we find χ2/d.o.f. values between 10 and 100 for (r1r2r3)1/3 ≲ 50 h−1Mpc at z = 1.0, where the d.o.f. is the number of singular values used for the χ2 estimation. At z = 0.0 the χ2/d.o.f. values are higher at small scales, indicating that Qm predictions agree better with measurements at higher redshifts. At (r1r2r3)1/3 ≳ 60 h−1Mpc the χ2/d.o.f. values are roughly constant, taking values between 0.6 and 4. An exception are the high values for the Qm model from the linear power spectrum at z = 0.0, whereas using the non-linear and linear power spectra lead to similar results at z = 1.0. These results indicated that non-linear contributions have a significant effect in Qm at small scales and low redshift and can partly be taken into account in the Qm predictions by using the non-linear power spectrum. Figure 5. View largeDownload slide χ2/d.o.f., quantifying the significance of the deviation between mean Qm, measured in the simulations and tree-level predictions versus the mean triangle size (r1r2r3)1/3 per bin. Lines and symbols show results for Qm predictions derived from the linear and non-linear power spectrum, respectively, at the redshifts z = 0.0 and z = 0.5. Figure 5. View largeDownload slide χ2/d.o.f., quantifying the significance of the deviation between mean Qm, measured in the simulations and tree-level predictions versus the mean triangle size (r1r2r3)1/3 per bin. Lines and symbols show results for Qm predictions derived from the linear and non-linear power spectrum, respectively, at the redshifts z = 0.0 and z = 0.5. Note that the differences between results at different redshifts do not only result from different model performance, but also from differences in the covariances and modes selected for the χ2 computation. Since these quantities are sensitive to noise, we will not enter a detailed discussion. 3.2 ΔQ In this subsection, we test how well the higher-order contributions to the halo 3PCF are described by the quadratic c2 + g2Qnloc term, which appears in equations (6) and (7). Following Bel et al. (2015), we obtain these higher-order contributions from the measurements by subtracting the halo-matter cross-correlation from the halo autocorrelation, \begin{equation} \Delta Q \equiv Q_h - Q_\times . \end{equation} (12)This subtraction leads to a cancellation of the linear Qm/b1 term in Qh and Q× and hence isolates the higher-order terms. The aforementioned quadratic term correspond to the leading order perturbative approximation of ΔQ, which follows from inserting the corresponding leading order approximations for Qh and Q× from equations (6) and (7) into equation (12), i.e. \begin{equation} \Delta Q \simeq \frac{2}{3 b_1}(c_2 + g_2 Q_{{\rm nloc}}). \end{equation} (13)The relation above allows us to test on one hand the accuracy of the quadratic model for the higher order terms in Qh and Q×, independently of inaccuracies in the Qm modelling, which we studied previously in Section 3.1. On the other hand, we test simultaneously if the bias parameters, which we adopt from the Fourier space measurements of CSS12, also describe the clustering statistics in configurations space. Regarding the latter case, we employ three sets of bias parameters to which we refer to as local, non-local, and $$b_{\delta ^2 {\rm {fix}}}$$ bias model, as described in Table 1 and in Section 1.1. The corresponding model predictions for ΔQ are compared to the measurements at different triangle opening angles in Fig. 6. For this comparison, we use the halo sample m2 at redshift z = 0.5 (defined in Table 2) and the same triangle configurations as for the Qm in Fig. 1. The ΔQ measurements in Fig. 6 show a clear dependence on the triangle opening angle α for the small (r1, r2) = (64, 32) h−1Mpc triangle configuration. This finding contrasts the local bias model prediction of a constant ΔQ = 2c2/3b1. However, at intermediate angles (60° ≲ α ≲ 120°), the local model predictions are in better agreement with the measurements than predictions from the non-local model. This result indicates that neglected higher order terms might compensate the quadratic non-local contribution. Figure 6. View largeDownload slide Top panels: ΔQ versus the mean triangle opening angle per bin $$\alpha \equiv \arccos\, ( \hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$, measured at z = 0.5 for the mass sample m2. Dashed and solid lines show tree-level predictions from the local and non-local bias models, respectively, using the non-linear power spectrum and bias parameters measured in Fourier space by CSS12 in equation (13). Bottom panel: the significance of the deviation between model predictions and measurements. Figure 6. View largeDownload slide Top panels: ΔQ versus the mean triangle opening angle per bin $$\alpha \equiv \arccos\, ( \hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$, measured at z = 0.5 for the mass sample m2. Dashed and solid lines show tree-level predictions from the local and non-local bias models, respectively, using the non-linear power spectrum and bias parameters measured in Fourier space by CSS12 in equation (13). Bottom panel: the significance of the deviation between model predictions and measurements. Similar trends are apparent for the larger (80,48) h−1Mpc triangle configuration, while here the large measurement errors lead to a similar significance of the different model deviations (see the bottom panel of Fig. 6). Note that for the presented results, we computed Qnloc in equation (13) from the non-linear power spectrum, which was measured in the simulation. This is motivated by the fact that the Qm model performs better in that case (see Section 3.1). However, using Qnloc predictions from the linear power spectrum delivers very similar result and does not affect the conclusions drawn above. Extending the comparison between models and measurements to all triangles in our analysis, we show in Fig. 7 the significance of the deviations between ΔQ measurements and model predictions versus the triangle opening angle and scale $$\sqrt{r_1r_2}$$ (analogously to Fig. 4). We use again the mass sample m2 at z = 0.5 (with b1 = 3.29) and show in addition also results for the sample m0 at z = 0.0 (with b1 = 1.43) to explore how differences in the bias effect the model performance. For the highly biased sample m2 at z = 0.5 (the bottom panel of Fig. 7), the results line up with those for the two single triangle configurations, shown Fig. 6. For small triangles with $$\sqrt{r_1r_2} \lesssim 40$$ h−1Mpc and triangle opening angles in the range of 60°–120°, the local bias model is in better agreement with the measurements than the non-local model. Overall both models tend to overpredict the measurements at small triangle scales. The results for the $$b_{\delta ^2 {\rm {fix}}}$$ model are very similar to those from the non-local model. This is also the case when the latter is based on the b2(b1) relation from Lazeyras et al. (2016). Figure 7. View largeDownload slide Significance of the deviations between predictions for ΔQ and measurements for the mass samples m0 at z = 0.0 and m2 at z = 0.5 (top and bottom panels, respectively). The results are shown versus the mean triangle opening angles per bin α and triangle scales (r1r2)1/2 in h−1Mpc. The predictions are derived from the local, the non-local, and the $$b_{\delta ^2 {\rm {fix}}}$$bias models (see Table 1), with bias parameters measured in Fourier space by CSS12. Solid and dashed black lines in the bottom right-hand panel show (r1r2)1/2(α) for (r1r2r3)1/3 = 60 h−1Mpc and triangle configurations of r1/r2 = 0.5 and 1.0, respectively. Figure 7. View largeDownload slide Significance of the deviations between predictions for ΔQ and measurements for the mass samples m0 at z = 0.0 and m2 at z = 0.5 (top and bottom panels, respectively). The results are shown versus the mean triangle opening angles per bin α and triangle scales (r1r2)1/2 in h−1Mpc. The predictions are derived from the local, the non-local, and the $$b_{\delta ^2 {\rm {fix}}}$$bias models (see Table 1), with bias parameters measured in Fourier space by CSS12. Solid and dashed black lines in the bottom right-hand panel show (r1r2)1/2(α) for (r1r2r3)1/3 = 60 h−1Mpc and triangle configurations of r1/r2 = 0.5 and 1.0, respectively. These findings differ from those of the low biased sample m0 at z = 0.0 (shown in the top panel of Fig. 7) in three aspects. The first aspect is that the local and non-local models tend to underpredict the measurements for $$\sqrt{r_1r_2} \lesssim 40$$ h−1Mpc. The second aspect is that for the low biased sample, the local and non-local models perform equally well. This can be expected, since the non-local bias, measured by CSS12, is close to zero in that case. The third aspect is that the $$b_{\delta ^2 {\rm {fix}}}$$ model differs from non-local model. In fact, it agrees better with the measurements than the other models. One interpretation of this result could be that the c2(b1) and g2(b1) relation is more accurate than the Fourier space measurements of the bias parameters from CSS12. Alternatively, one might conclude that inaccuracies of the $$b_{\delta ^2 {\rm {fix}}}$$ model compensate the neglected higher-order terms in the ΔQ model in equation (13), leading to a good agreement with the measurements by accident. To clarify this point, one could repeat the exercise, using a model for ΔQ that is developed beyond the second order. For a possible application of the c2(b1) and g2(b1) relations of the $$b_{\delta ^2 {\rm {fix}}}$$ model in observations, it would be interesting to test the dependence of our results on the cosmological parameters used. For bias measurements in observations, it is also interesting to note that deviations between measurements in our ∼100 (h−1 Gpc)3 volume and model predictions become insignificant for $$\sqrt{r_1r_2} \gtrsim 40$$ h−1Mpc as the measurement errors increase with scale. As for Qm we find an overall convergence of the deviation between measurements and predictions for triangles with (r1r2r3)1/3 ≳ 60 h−1Mpc in Fig. B3, which are marked in Fig. 7 with black lines. We quantify these deviations again by computing the χ2 via SVD, taking into account the covariance between measurements at different scales in (r1r2r3)1/3 bins with 30 triangles. Note that the ΔQ covariance is typically dominated by shot-noise, coming from the Qh contribution, which can be seen in Fig. C1. The results, shown in Fig. 8 are in line with our finding from Fig. B3 as results converge to χ2/d.o.f values around unity. The highly biased sample shows larger overall deviations between measurements and predictions, in particular for the non-local model at (r1r2r3)1/3 ≲ 60 h−1Mpc. Results for the $$b_{\delta ^2 {\rm {fix}}}$$ model are similar to those from the non-local model at large scales, while at small scales the former performs better as its χ2/d.o.f. values are lower. Figure 8. View largeDownload slide χ2/d.o.f., quantifying the significance of the deviation between the mean ΔQ, measured in the simulations and predictions based on Fourier space bias parameters. Results are shown versus the mean triangle size (r1r2r3)1/3 per bin. Dots are results based on the predictions from the non-local bias model. Results from the local and the $$b_{\delta ^2 \rm fix}$$ bias models (see Table 1) are shown as dashed and solid lines, respectively. Note that for the low biased mass sample m0 at redshift z = 0.0, the results for the local and non-local bias models are very similar, since the non-local bias contribution is very small. Figure 8. View largeDownload slide χ2/d.o.f., quantifying the significance of the deviation between the mean ΔQ, measured in the simulations and predictions based on Fourier space bias parameters. Results are shown versus the mean triangle size (r1r2r3)1/3 per bin. Dots are results based on the predictions from the non-local bias model. Results from the local and the $$b_{\delta ^2 \rm fix}$$ bias models (see Table 1) are shown as dashed and solid lines, respectively. Note that for the low biased mass sample m0 at redshift z = 0.0, the results for the local and non-local bias models are very similar, since the non-local bias contribution is very small. 3.3 Qh and Q× After validating the linear and quadratic components for the Qh and Q× models separately in Sections 3.1 and 3.2 we now compare the full models, given by equations (6) and (7) with the measurements in our simulations. As for ΔQ we focus on model predictions, which are based on the non-linear power spectrum and start the analysis by showing Qh and Q×, measured in the halo sample m2 at z = 0.5, for triangles with fixed legs of (r1, r2) = (64, 32) and (80, 48)h−1Mpc versus the triangle opening angle α in Fig. 9. We find that the models for both Qh and Q× tend to overpredict the measurements, which lines up with our corresponding results for ΔQ in Fig. 6. An exception of this trend are Qh results from the small triangle configuration with 60 ≲ α ≲ 90. This indicates that the neglected terms in the perturbative model beyond leading order affect Qh and Q× differently. Again, the model predictions based on the local bias model show the strongest deviations from the measurements, in particular for collapsed and relaxed triangles. This explains why neglecting the non-local term leads to an overestimation of the bias, when fitting Qh or Q× model predictions to measurements (see CSS12). For such a fit one would choose a higher b1, since this would flatten the curve and deliver the measured shape. The overall amplitude can then be adjusted by varying c2 (see equation 6). Such fits of the local model are in fact in very good agreement with the measurements. However, the linear bias is too high (e.g. Manera & Gaztañaga 2011; Bel et al. 2015). Note that the linear bias measurements based on the local bias model would be too low instead of too high when using the 3PCF or the bispectrum, as explained by CSS12. Figure 9. View largeDownload slide Left: top panels show the reduced halo 3PCF, Qh, for triangles with fixed legs r1 and r2 (size is indicated in h−1Mpc) versus the mean triangle opening angle per bin, $$\alpha \equiv \arccos ( \hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$. Symbols show mean measurements from 49 simulations with 1σ errors for the mass sample m2 at redshift z = 0.5. Lines show predictions from equation (6), using the non-linear power spectrum and the bias models from Table 1 with bias parameters measured by CSS12 in Fourier space. The bottom panel shows the significance of the deviations between model predictions and measurements. Right: analogous results for the reduced three-point halo-matter cross-correlations, while predictions are derived from equation (7). Figure 9. View largeDownload slide Left: top panels show the reduced halo 3PCF, Qh, for triangles with fixed legs r1 and r2 (size is indicated in h−1Mpc) versus the mean triangle opening angle per bin, $$\alpha \equiv \arccos ( \hat{\boldsymbol r}_1 \cdot \hat{\boldsymbol r}_2)$$. Symbols show mean measurements from 49 simulations with 1σ errors for the mass sample m2 at redshift z = 0.5. Lines show predictions from equation (6), using the non-linear power spectrum and the bias models from Table 1 with bias parameters measured by CSS12 in Fourier space. The bottom panel shows the significance of the deviations between model predictions and measurements. Right: analogous results for the reduced three-point halo-matter cross-correlations, while predictions are derived from equation (7). The best agreement between the Qh and Q× measurements and the corresponding models occurs at large opening angles (hence large triangles) when using the non-local bias model (1 − 2σ). This scale dependence can be expected since errors increase and higher order contributions decrease with the scale. Results based on the $$b_{\delta ^2 {\rm {fix}}}$$ model are again very similar to those from the non-local model. Interestingly, the deviations between the model predictions and measurements are less significant for Qh than for Q×, despite the fact that the neglected terms beyond leading order should have a higher contribution to Qh and therefore lead to stronger deviations from the model. However, the errors on Qh are more strongly affected by shot-noise than those for Q× ($$\sigma ^2_{Q_h} \sim n_h^3$$, $$\sigma ^2_{Q_\times } \sim n_h$$, where nh is the halo number density). This means that for observations with similar or larger errors than our measurements, a development of the Qh model beyond leading order might only lead to a marginal improvement of the model performance. In Fig. 10, we show the comparison between Q× and Qh models and measurements for all triangles, displaying them for different scales $$\sqrt{r_1r_2}$$ versus the triangle opening angle (as in Fig. 7). Results are shown for the low biased sample m0 at z = 0.0 and the highly biased sample m2 at z = 0.5. The latter confirm the trends from Fig. 9. In particular for small triangles ($$\sqrt{r_1r_2} \lesssim 40\,$$h−1Mpc), the Q× and Qh models overpredict the measurements for collapsed and relaxed triangles and underpredict them for triangles with 60 ≲ α ≲ 90. The Qh results for m2 at z = 0.5 are again an exception. In that case, the local model is in better agreement with the measurements than the non-local model, which is consistent with the ΔQ results for this sample and might be attributed to a compensation of quadratic non-local and neglected higher-order terms, as mentioned in the discussion of Fig. 6 in Section 3.2. Note that this compensation is shown here to occur for one particular halo sample, while this is not the case for other samples (not shown here). Figure 10. View largeDownload slide Significance of the deviations between predictions for Q× and Qh and measurements for the mass samples m0 at z = 0.0 and m2 at z = 0.5 (top and bottom panels, respectively). The results are shown versus the mean triangle opening angles per bin, α, and triangle scales (r1r2)1/2 in h−1Mpc. The predictions are derived from the bias models described in Table 1, with bias parameters measured in Fourier space by CSS12. Solid and dashed black lines in the bottom right-hand panel show (r1r2)1/2(α) for (r1r2r3)1/3 = 60 h−1Mpc and triangle configurations of r1/r2 = 0.5 and 1.0, respectively. Figure 10. View largeDownload slide Significance of the deviations between predictions for Q× and Qh and measurements for the mass samples m0 at z = 0.0 and m2 at z = 0.5 (top and bottom panels, respectively). The results are shown versus the mean triangle opening angles per bin, α, and triangle scales (r1r2)1/2 in h−1Mpc. The predictions are derived from the bias models described in Table 1, with bias parameters measured in Fourier space by CSS12. Solid and dashed black lines in the bottom right-hand panel show (r1r2)1/2(α) for (r1r2r3)1/3 = 60 h−1Mpc and triangle configurations of r1/r2 = 0.5 and 1.0, respectively. Overall the results from the non-local bias model are in better agreement with the measurements for the sample m2 at z = 0.5 than the local model at large triangles scales ($$\sqrt{r_1r_2} \gtrsim 40$$h−1Mpc or α ≳ 90) and are consistent with those from the $$b_{\delta ^2 {\rm {fix}}}$$ model. For the low biased sample m0 at z = 0.0 all bias models deliver similar results, since the non-local bias contribution is very weak. For some triangles, we find an increased significance of the deviations for that sample, compared to the m2 sample at z = 0.5, presumably because the shot-noise error contribution is decreased due to the higher halo density. In the case of Q×, where the shot-noise errors are the lowest, the deviations follow the BAO feature, which we saw already in the Qm model validation (Fig. 7). Using model predictions based on the linear power spectrum, we find a significant increase of the deviations in the case of Q× for both samples (not shown here). This indicates that neglected terms in the Qm model, in the bias model, or both affect the halo 3PCF, even at very large triangle scales. However, for the autocorrelation Qh at $$\sqrt{r_1r_2} \gtrsim 40$$h−1Mpc their contribution seems to be small compared to the measurement errors, as we find a similarly significant deviations for different power spectra and halo samples. This will, in particular, be also the case for the smaller volumes, covered by galaxy surveys, for which the measurement errors can be expected to be larger. The deviations between non-local bias model predictions and measurements converge to values of ≲ 2σ (r1r2r3)1/3 ≳ 60 h−1Mpc (marked in Fig. 10 as black lines) for both, Q× and Qh, as shown in Figs B4 and B5. This is consistent with our corresponding results for Qm and ΔQ. As in the case of Qm and ΔQ, measurements of Qh and Q× from different triangles are covariant (see Fig. C1). Quantifying the significance of deviations between model predictions and measurements, we show in Fig. 11 the χ2/d.o.f. in bins of (r1r2r3)1/3 for all three mass samples and redshifts. Each bin contains measurements from 30 triangles and the χ2 values have been computed via SVD (see Section 2.4) using only the dominant modes, as for the Qm and ΔQ analyses from Figs 5 and 8. We also tested that our results are not affected by the chosen number of triangles per bin. The results confirm the convergence of the deviations to 1 − 2σ for (r1r2r3)1/3 ≳ 60 h−1Mpc. However, they show strong variations for different scales, which might result from noise in our covariance estimation from only 49 realizations. Overall, the χ2/d.o.f. values for Q× are higher than those for Qh, presumably because of the higher signal-to-noise ratio of the measurements. Even at large scales above 60 h−1Mpc, we find χ2/d.o.f. ≃ 5 values. They might be explained by non-linearities around the BAO feature, which are not fully captured in our leading order perturbative model (see Fig. 10). For Q×, the χ2/d.o.f. values are lower at high redshift and higher mass samples. The latter result might be explained by larger shot-noise errors on the high-mass samples and agrees with the results from Figs B4 and B5. Smaller deviations at high redshifts might result from a smaller impact of next to leading order terms in the Q× model, which we neglect in our analysis. We do not see a clear dependence of the results on mass and redshift for Qh, possibly because of the low signal-to-noise ratio. It is interesting to note that the χ2/d.o.f. values for the $$b_{\delta ^2 {\rm {fix}}}$$ model are in very good agreement with those from the non-local model for highly biased sample (high halo mass and redshift). For samples with low bias (low mass, low redshift) the χ2/d.o.f. values for the $$b_{\delta ^2 {\rm {fix}}}$$ are even smaller than those for the non-local model. The latter finding is consistent with our model comparison for ΔQ. Figure 11. View largeDownload slide Top: χ2 per d.o.f., quantifying the difference between the Q× measurements for the mass samples m1, m2, m3 (defined in Table 2) and the corresponding predictions, based on the non-linear power spectrum at different redshifts z. Symbols show results for the non-local bias model, while lines show results using analytical relations between the linear and non-linear bias parameters ($$b_{\delta ^2 {\rm {fix}}}$$ model, see Table 1). The bias parameters were measured by CSS12 in Fourier space. Bottom: same as top panel, but for Qh. Figure 11. View largeDownload slide Top: χ2 per d.o.f., quantifying the difference between the Q× measurements for the mass samples m1, m2, m3 (defined in Table 2) and the corresponding predictions, based on the non-linear power spectrum at different redshifts z. Symbols show results for the non-local bias model, while lines show results using analytical relations between the linear and non-linear bias parameters ($$b_{\delta ^2 {\rm {fix}}}$$ model, see Table 1). The bias parameters were measured by CSS12 in Fourier space. Bottom: same as top panel, but for Qh. Our comparison between χ2/d.o.f. values for local and non-local model predictions in Fig. 12 demonstrates that setting the non-local term in the prediction to zero leads to higher deviations from Q× measurements for highly biased samples. The effect is also apparent for Qh, even for (r1r2r3)1/3 >60 h−1Mpc, while in that case the χ2/d.o.f. values are lower, presumably due to larger errors on the measurements. Again these results confirm those for ΔQ, shown in Fig. 8. Figure 12. View largeDownload slide The figure shows the same results for the non-local and $$b_{\delta ^2 {\rm {fix}}}$$ models as shown in Fig. 11 for the mass sample m0 at z = 0.0 and m2 at z = 0.5. In addition, we show here results for the local model as dashed lines. Figure 12. View largeDownload slide The figure shows the same results for the non-local and $$b_{\delta ^2 {\rm {fix}}}$$ models as shown in Fig. 11 for the mass sample m0 at z = 0.0 and m2 at z = 0.5. In addition, we show here results for the local model as dashed lines. 4 SUMMARY AND CONCLUSIONS The main result of this paper (summarized in Fig. 11) is an empirical determination of the scales at which three-point halo correlations in configuration space are consistent with the corresponding statistics in Fourier space, i.e. the bispectrum. To this end, we measured the reduced three-point autocorrelation function of matter and haloes, as well as the reduced halo-matter three-point cross-correlation (which are referred to as Qm, Qh, and Q× respectively) in a set of 49 cosmological simulations with a total volume of ∼100 (h−1 Gpc)3. The large volume provides small errors on the measurements. At the same time, we obtain rough estimates of the error covariances, which we analysed using singular value decomposition. The Qh and Q× measurements were compared to leading order perturbative models (equations 6 and 7), which relate these statistics to Qm via the linear, quadratic, and non-local bias parameters (referred to as b1, c2, and g2, respectively). For testing the consistency with results from Fourier space, we adopted bias parameters, which were measured in the same set of simulations by CSS12 using the same perturbative model of the halo-matter cross-bispectrum. We adopted the bias parameters in three different ways. The first way is to simply employ the set of Fourier space parameter from CSS12. The second set of parameters are identical to the first set, except for the non-local bias parameter g2, which is set to zero in order to study the contribution of the non-local terms to the Qh and Q× predictions. For the third set, we used the linear bias, measured by CSS12 from the halo-matter cross-power spectrum, while the quadratic bias is set by the (approximately) universal c2(b1) relation from Hoffmann et al. (2017) and the non-local bias is predicted using the g2(b1) relation from the local Lagrangian model, reducing the degrees of freedom in the bias model. These three sets of bias parameters are referred to as non-local, local, and $$b_{\delta ^2 {\rm {fix}}}$$ models, respectively, and are summarized in Table 1. Before predicting Qh and Q× using the bias parameters, we first had to obtain the matter contribution Qm and the non-local contribution Qnloc. To remain closer to an analysis of observational data, where these quantities cannot be directly measured, we modelled them from the linear, the linear de-wiggled, and the non-linear power spectrum. By comparing the Qh and Q× predictions to measurements, we therefore did not only test if the bias parameters in Fourier space describe the clustering in configurations space, but also simultaneously at which scales the perturbative model of the three-point correlation breaks down. We conducted this comparison in three steps. First, we studied in Section 3.1 how well Qm measurements are described by the leading order perturbative predictions from the different power spectra. Secondly, we investigate how well the higher-order contributions to Qh are described by the leading (quadratic) order perturbative models, based on the Fourier space bias parameters. These contributions are obtained from the measurements by the subtraction ΔQ ≡ Qh − Q×, as described in Section 3.2. Finally, we compare in Section 3.3 the full predictions for Qh and Q× with the corresponding measurements. Overall our results show that the deviations between the model predictions for Qm, Qh, Q×, and ΔQ and the corresponding measurements depends on the triangle scale as well as on the triangle shape (characterized by the triangle opening angle) for which these statistics are studied. The quantity (r1r2r3)1/3 turns out to be a convenient definition of the triangle scale, since it shows a tight correlation with the measurement errors. Furthermore, it separates well larger triangles for which the models perform well from smaller ones, for which the measurements are presumably strongly affected by higher order terms. We found that the deviation between the perturbative model predictions of the different three-point correlations from the measurements converge to the 1 − 2σ level for (r1r2r3)1/3 ≳ 60 h−1Mpc, while the noisy error estimation imposes some uncertainty on this value. Note here that the smallest ri value above zero in our analysis corresponds to the size of the 8 h−1Mpc grid cells into which we divided the simulations for computing the correlations. However, when the measurement errors are small (in particular their shot-noise contribution), which is the case for Qm and Q×, and when the predictions are computed from the linear instead of the non-linear power spectrum, we find deviations above 2σ for (r1r2r3)1/3 ≳ 60 h−1Mpc. We attribute this effect to non-linearities around the BAO peak from large-scale displacements and bias contributions not included in our treatment. The fact that this effect is much weaker when using the de-wiggled, or the non-linear power spectrum indicates that the latter can incorporate higher orders in the perturbative model for Q to some degree. Validating the model for the quadratic terms in Qh and Q× with the ΔQ measurements, we found the predictions based on the non-local bias model to show an overall better performance than those from the local model. An exception are measurements in highly biased halo samples from small triangles with intermediate opening angles, which are better described by the local than the non-local bias model. We interpret this effect as a compensation of the non-local and the higher order terms not included in our bias treatment, which occurs for these particular triangles and this particular halo sample. Interestingly, the deviations of the ΔQ predictions based on the $$b_{\delta ^2 {\rm {fix}}}$$ bias model from the measurements are similar, and for low biased samples even smaller than those based on the non-local model. For (r1r2r3)1/3 ≳ 60 h−1Mpc, the significance of the deviations between ΔQ predictions and measurement is similar for all bias models, presumably because of the low signal-to-noise ratio. From the Q× measurements, we conclude that the leading order perturbative model predictions in combination with the bias derived from the same statistics in Fourier space are a good approximation, with 1 − 2σ deviations (r1r2r3)1/3 ≳ 60 h−1Mpc. These deviation are slightly higher around the aforementioned BAO feature, but given the small errors on Q× this agreement is still good. The model performance for Qh at large scales is even better, despite the fact that terms beyond leading order, which are neglected in the model should affect Qh more strongly than Q×. This might be a result of the lower signal-to-noise ratio. However, the Qh predictions differ significantly from the measurements for (r1r2r3)1/3 ≲ 50 h−1Mpc. It is thereby important to note that these results are specific to our small measurement errors from the combined ∼100 (h−1 Gpc)3 of the 49 simulations studied in this work. In practice, the deviation between model predictions and measurements can be expected to be less significant, as the measurement errors are larger for the smaller volumes of current and upcoming galaxy surveys. As for ΔQ, the Qh and Q× predictions from the $$b_{\delta ^2 {\rm {fix}}}$$ model agree equally well with the measurements at large scales for highly biased samples (high masses, high redshift). For low biased samples (low mass, low redshift), this model describes the Qh and Q× measurements even better than the non-local model. Differences in the Qh predictions based on the linear and non-linear power spectra are negligible compared to the larger measurement errors. The good performance of predictions from the $$b_{\delta ^2 {\rm {fix}}}$$ model at large scales suggests that a roughly universal c2(b1) relation, together with the local Lagrangian g2(b1) relation, could tighten constraints on the linear bias, derived from third-order statistics in galaxy surveys. However, recent studies pointed out that assembly bias can lead to deviations from a universal c2(b1) relation (Modi, Castorina & Seljak 2017; Paranjape & Padmanabhan 2017). An application of the $$b_{\delta ^2 {\rm {fix}}}$$ model in the analysis of galaxy surveys therefore requires tests in mock catalogues (for instance from semi-analytic models of galaxy formation) to validate for which type of galaxy samples these bias relations are useful approximations. More generally, the results presented in this paper show a good overall agreement of the non-local quadratic bias models with simulations, using the same bias parameters for Fourier and configuration space, but the range of validity will depend strongly on the samples used (volume, redshift, and bias), so a detailed comparison with mock galaxy and corresponding dark matter catalogues with redshift space distortions will be needed. Acknowledgements We acknowledge support from the Spanish Ministerio de Ciencia e Innovacion (MICINN) projects AYA2012-39559 and AYA2015-71825, and research project 2014 SGR 1378 from the Generalitat de Catalunya. K.H. acknowledges the support by the International Postdoc Fellowship from the Chinese Ministry of Education and the State Administration of Foreign Experts Affairs. He also thanks the organizers and participants of the 2016 workshop Biased Tracers of Large-Scale Structure at the Lorentz Center as well as Kwan Chuen Chan for useful discussions. 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J., 2015, MNRAS , 454, 4142 https://doi.org/10.1093/mnras/stv2119 CrossRef Search ADS Slepian Z. et al. , 2015, preprint (arXiv:1512.02231) Slepian Z. et al. , 2017, MNRAS , 469, 1738 https://doi.org/10.1093/mnras/stx488 CrossRef Search ADS Springel V. et al. , 2018, MNRAS , 475, 676 Srednicki M., 1993, ApJ , 416, L1 https://doi.org/10.1086/187056 CrossRef Search ADS APPENDIX A: 3PCF BINNING When measuring the 3PCF on a grid of cubical cells, we need to allow for a tolerance of the triangle leg sizes (r12, r13) to obtain a sufficiently larger number of triangles in each bin of the triangle opening angle α (see Section 2.3). Here, we test for different triangle configurations how much the results are affected by this tolerance. We therefore compute the matter 3PCF prediction for all triangles on the grid, which fulfil the condition (r12, r13) ± (δr12, δr13). As an example we show the results for the configuration (64, 32) ± (2, 2) h−1Mpc in Fig. A1 as grey dots versus α. The average 3PCF predictions in bins of α are shown as black dots at the mean angle in each bin. These results are compared to predictions for exact values of (r12, r13) (i.e. (δr12, δr13) = (0, 0)). We find that the difference between the two types of predictions is small, compared to the difference between predictions and measurements (red symbols). We obtain the same results for different triangle configurations (not shown here) and conclude that the binning of the 3PCF measurements has no significant impact on the comparison with the unbinned theory predictions, which we use in our analysis. Figure A1. View largeDownload slide Testing the impact of the grid on the 3PCF. Grey dots show the predictions for the matter 3PCF, based on the measured power spectrum at z = 1.0, for all triangles with (r12, r13) ± (δr12, δr13) = (64, 32) ± (2, 2) h−1Mpc on the grid versus the opening angles α. The mean predictions in bins of α are shown as black dots. Predictions for (δr12, δr13) = (0, 0) (as we use them in our analysis) are shown as dashed blue line. Measurements of the 3PCF at z = 0.0 are shown with 1σ errors as red symbols. Figure A1. View largeDownload slide Testing the impact of the grid on the 3PCF. Grey dots show the predictions for the matter 3PCF, based on the measured power spectrum at z = 1.0, for all triangles with (r12, r13) ± (δr12, δr13) = (64, 32) ± (2, 2) h−1Mpc on the grid versus the opening angles α. The mean predictions in bins of α are shown as black dots. Predictions for (δr12, δr13) = (0, 0) (as we use them in our analysis) are shown as dashed blue line. Measurements of the 3PCF at z = 0.0 are shown with 1σ errors as red symbols. APPENDIX B: DEVIATIONS FOR INDIVIDUAL TRIANGLES The errors of the different Q measurements correlates strongly with the total triangle scale, defined by (r1r2r3)1/3 as shown in Fig. B1. We therefore study here the significance of the deviations between measurements and predictions versus this scale. Figure B1. View largeDownload slide Examples of 1σ errors of the different reduced 3PCFs studied in this paper versus the triangle scale. Results for different redshifts and mass samples are similar. Figure B1. View largeDownload slide Examples of 1σ errors of the different reduced 3PCFs studied in this paper versus the triangle scale. Results for different redshifts and mass samples are similar. Fig. B2 shows that the Qm predictions deviate from the measurements by less than 2σ for (r1r2r3)1/3 ≳ 80 when predictions are computed from the linear power spectrum and ≳ 60 h−1Mpc when using the non-linear power spectrum. Note that these results are specific for the joint ∼100 (Gpc/h)3 volume of the 49 simulations. For smaller volumes (as covered by current galaxy surveys), errors would be larger and the significance therefore smaller. Using alternative measures for the triangle scale, such as the triangle area or the sum of the triangle legs leads to a less clear separation between triangles with weak and strong significance of the deviations. Figure B2. View largeDownload slide Top panels: significance of the deviation between Qm measurements and tree-level predictions (based on the linear and non-linear power spectrum) versus the triangle scale. Dash–dotted lines denote 2σ deviations. Bottom panel: signal-to-noise ratio. Colours denote the triangle opening angle. Figure B2. View largeDownload slide Top panels: significance of the deviation between Qm measurements and tree-level predictions (based on the linear and non-linear power spectrum) versus the triangle scale. Dash–dotted lines denote 2σ deviations. Bottom panel: signal-to-noise ratio. Colours denote the triangle opening angle. The corresponding results for ΔQ are shown in Fig. B3 for the low biased sample (m0) at z = 0.0 and the highly biased sample (m2) at z = 0.5 (with b1 = 1.43 and b1 = 3.29, respectively). For the sample with the low linear bias, the model predictions are below the measurements at (r1r2r3)1/3 ≲ 60 h−1Mpc. Differences between local and non-local model predictions are not apparent, as expected from Fig. 7. For the sample with the higher linear bias, the predictions are above the measurements for (r1r2r3)1/3 ≳60 h−1Mpc and the non-local model performs slightly better than the local model at small scales. At large scales differences between model and predictions are not significant for both samples, due to the low signal-to-noise ratio, which is shown in the bottom panel of Fig. B3. Note that the predictions are based on the non-linear power spectrum, measured in the simulation, while the linear power spectrum leads to very similar results. Figure B3. View largeDownload slide Significance of the deviation between predictions for ΔQ and measurements versus triangle scale (r1r2r3)1/3 for the halo mass samples m0 at z = 0.0 and m1 at z = 0.5 (left and right panels, respectively). The top and central panels show results for predictions from the local and non-local models, respectively (Table 1), based on the non-linear power spectrum. The bottom panel shows signal-to-noise ratio of measurements. Figure B3. View largeDownload slide Significance of the deviation between predictions for ΔQ and measurements versus triangle scale (r1r2r3)1/3 for the halo mass samples m0 at z = 0.0 and m1 at z = 0.5 (left and right panels, respectively). The top and central panels show results for predictions from the local and non-local models, respectively (Table 1), based on the non-linear power spectrum. The bottom panel shows signal-to-noise ratio of measurements. The significance of the deviations between non-local bias model predictions for Qh and Q× and the corresponding measurements are displayed versus the triangle scale in Figs B4 and B5. Covering a larger range of bias values (1.43 ≲ b1 ≲ 3.99), we now show results for the mass samples m0 and m2, each at redshift z = 0.0 and 1.0. Also here the predictions are based on the non-linear power spectrum and we find very similar results when using the linear power spectrum. The results are consistent with those shown in Fig. 10, as the predictions are most significant for small triangles, where they show a strong dependence on the triangle opening angle for low biased samples, while samples with high bias (higher masses and redshifts) show a weaker dependence on the opening angle at small scales. Overall the deviations for both, Q× and Qh converge to values of ≲ 2σ for all samples for (r1r2r3)1/3 ≳ 60 h−1Mpc. An exception are results Q× for large opening angles, which can be attributed non-linearities around the BAO peak, as mentioned in the discussion of Fig. 10. Figure B4. View largeDownload slide Significance of the deviations between predictions for Q× and measurements versus triangle scale (r1r2r3)1/3. The predictions are based on the non-local bias model (Table 1) and the non-linear power spectrum. Results are shown for the halo mass samples m0 and m2 at z = 0.0 and z = 1.0 (left and right, top, bottom panels, respectively). The lower subpanels show the signal-to-noise ratios for the samples m2, which have higher shot-noise contributions than the m0 sample. Figure B4. View largeDownload slide Significance of the deviations between predictions for Q× and measurements versus triangle scale (r1r2r3)1/3. The predictions are based on the non-local bias model (Table 1) and the non-linear power spectrum. Results are shown for the halo mass samples m0 and m2 at z = 0.0 and z = 1.0 (left and right, top, bottom panels, respectively). The lower subpanels show the signal-to-noise ratios for the samples m2, which have higher shot-noise contributions than the m0 sample. Figure B5. View largeDownload slide Same as Fig. B4, but for Qh. Figure B5. View largeDownload slide Same as Fig. B4, but for Qh. APPENDIX C: COVARIANCES In Fig. C1, we show examples of the normalized covariance matrices at z = 0.5 for the different three-point statistics versus the triangle scale (r1r2r3)1/3. The covariances for Qm and Q× show strong off-diagonal elements, while those of Qh and ΔQ are dominated by the diagonal elements, which indicates high shot-noise contributions. Subsets of these covariances with 302 elements around the diagonal are used for the χ2 estimation, described in Section 2.4. Figure C1. View largeDownload slide Examples of normalized covariances between different reduced 3PCS from the 504 triangles used in this work. They were obtained from a set of 49 simulations. The low amplitude of the off-diagonal elements in the Qh covariances indicates a dominance of shot-noise errors. Results for Qh and Q× are almost identical, because the Q× errors are dominated by the Qh contribution. For computing the χ2 deviation from the model prediction, we select triangles in scale bins containing 30 triangles and perform a singular value decomposition, as described in Section 2.3. Figure C1. View largeDownload slide Examples of normalized covariances between different reduced 3PCS from the 504 triangles used in this work. They were obtained from a set of 49 simulations. The low amplitude of the off-diagonal elements in the Qh covariances indicates a dominance of shot-noise errors. Results for Qh and Q× are almost identical, because the Q× errors are dominated by the Qh contribution. For computing the χ2 deviation from the model prediction, we select triangles in scale bins containing 30 triangles and perform a singular value decomposition, as described in Section 2.3. In order to reduce the impact of noise on these estimations we perform a singular value decomposition of the covariances. The distribution of singular values is shown in Fig. C2 and reveals that a significant fraction of modes has only a minor contribution to the covariance. One can see how the singular values for the shot-noise dominated covariances of ΔQ and Qh show a slightly more pronounced drop, while those of the Qm and Q× covariances decay more slowly. We associate modes below $$\lambda ^2 \lesssim \sqrt{2/N_{{\rm sim}}}$$ with noise in the covariance measurement and neglect them in the χ2 computation. Figure C2. View largeDownload slide Singular values of the covariance matrices for the different three-point statistics studied in this work versus the mode number. The maximum mode number corresponds to the number of triangles in the (r1r2r3)1/3 bins. Results are shown for the mass sample m1 at z = 0.5. Modes with singular values of less than $$\lambda ^2 \lesssim \sqrt{2/N_{{\rm sim}}}$$ are associated with noise and therefore neglected in the χ2 computation. The total number of modes is 30, which corresponds to the number of triangles in each (r1r2r3)1/3 bin. Figure C2. View largeDownload slide Singular values of the covariance matrices for the different three-point statistics studied in this work versus the mode number. The maximum mode number corresponds to the number of triangles in the (r1r2r3)1/3 bins. Results are shown for the mass sample m1 at z = 0.5. Modes with singular values of less than $$\lambda ^2 \lesssim \sqrt{2/N_{{\rm sim}}}$$ are associated with noise and therefore neglected in the χ2 computation. The total number of modes is 30, which corresponds to the number of triangles in each (r1r2r3)1/3 bin. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

Monthly Notices of the Royal Astronomical Society – Oxford University Press

**Published: ** May 1, 2018

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