Full-shape cosmology analysis of the SDSS-III BOSS galaxy power spectrum using an emulator-based halo model: A 5% determination of ${\sigma }_8$

We present the results obtained from the full-shape cosmology analysis of the redshift-space power spectra for four galaxy samples of the SDSS-III BOSS DR12 galaxy catalog over $0.2<z<0.75$. For the theoretical template, we use an emulator that was built from an ensemble set of $N$-body simulations, which enables fast and accurate computation of the redshift-space power spectrum of “halos.” Combining with the halo occupation distribution to model the galaxy-halo connection, we can compute the redshift-space power spectrum of BOSS-like galaxies in less than a CPU second, for an input model under flat $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. In our cosmology inference, we use the monopole, quadrupole, and hexadecapole moments of the redshift-space power spectrum and include seven nuisance parameters, with broad priors, to model uncertainties in the galaxy-halo connection for each galaxy sample, but do not use any information on the abundance of galaxies. We demonstrate a validation of our analysis pipeline using the mock catalogs of BOSS-like galaxies, generated using different recipes of the galaxy-halo connection and including the assembly bias effect. Assuming weak priors on cosmological parameters, except for the BBN prior on ${\mathrm{\Omega }}_{\mathrm{b}}{h}^2$ and the CMB prior on $n_{\mathrm{s}}$, we show that our model well reproduces the BOSS power spectra. Including the power-spectrum information up to $k_{\max }=0.25h{\mathrm{Mpc}}^{-1}$, we find ${\mathrm{\Omega }}_{\mathrm{m}}=0.301_{-0.011}^{+0.012}$, $H_0=68.2\pm 1.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, and ${\sigma }_8=0.786_{-0.037}^{+0.036}$ for the mode and 68% credible interval, after marginalization over galaxy-halo connection parameters. We find little improvement in the cosmological parameters beyond a maximum wavelength $k_{\max }\simeq 0.2h{\mathrm{Mpc}}^{-1}$ due to the shot noise domination and marginalization of the galaxy-halo connection parameters. Our results are consistent with the Planck CMB results within $1\sigma$ statistical uncertainties.

Figure 1

Results of validation tests of our cosmology analysis pipeline: we apply the analysis pipeline to the mock signals for the four BOSS-like galaxy samples using the covariance matrix of the actual BOSS spectra. We include the monopole, quadrupole, and hexadecapole moments of the redshift-space power spectrum for each of the four galaxy samples over $0.005h{\mathrm{Mpc}}^{-1}\le k\le k_{\max }$, and we show the results for $k_{\max }=0.20h$, $0.25h$, or $0.30h{\mathrm{Mpc}}^{-1}$, respectively. The contours show the posterior distributions of ${\mathrm{\Omega }}_{\mathrm{m}}$, $H_0$, and ${\sigma }_8$ including marginalization over uncertainties in other parameters including the galaxy-halo connection parameters. The left plot shows the results for the galaxy mocks that are generated using a different recipe of the galaxy-halo connection from our fiducial HOD model: the mock galaxy catalogs used in the cosmology challenges in Ref. [37] (see text for details). Since we would like to blind the true values of the cosmological parameters, we show the results in terms of the parameter difference such as $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{m}}={\mathrm{\Omega }}_{\mathrm{m}}-{\mathrm{\Omega }}_{\mathrm{m},\mathrm{true}}$. The right plot shows the results for the mock catalogs generated using the same form of HOD model as in our HOD model. The dashed lines in the right panel are the true values of the cosmological parameters used in the mock catalogs—i.e., those for the Planck cosmology.

Figure 2

The posterior distributions of ${\mathrm{\Omega }}_{\mathrm{m}}$, $H_0$, and ${\sigma }_8$, obtained from the cosmology inference including the full shapes of the monopole, quadrupole, and hexadecapole moments of the BOSS DR12 galaxy power spectra up to $k_{\max }=0.25h{\mathrm{Mpc}}^{-1}$ for the flat $\mathrm{\Lambda }\mathrm{CDM}$ model. For the theoretical templates, we use the emulator-based halo model, and the posteriors include marginalization over uncertainties in other cosmological parameters and the nuisance parameters including the galaxy-halo connection parameters. For comparison, the red contours show the results from the Planck 2018 “TT, TE, $\mathrm{EE}+\mathrm{lowE}$” analysis.

Figure 3

The comparison of monopole (blue), quadrupole (red), and hexadecapole (green) moments between the data (symbols with error bars) and the model predictions (solid lines) at MAP of the MCMC chains in our cosmology analysis shown in Fig. 2. The error bars are computed from the diagonal elements of the covariance matrix, described in Sec. 2.

Figure 4

The median and 68% percentile interval of the HOD functions for each of the BOSS galaxy samples (low-$z$/high-$z$ NGC/SGC samples), obtained from the MCMC chains in Fig. 2. The solid and dashed lines are the medians of $\mathrm{central}+\mathrm{satellite}$ and central-only HOD functions, respectively.

Figure 5

The posterior distributions for cosmological parameters, obtained from the different galaxy subsamples. The left panel shows the results for four individual galaxy samples: low-$z$ NGC, low-$z$ SGC, high-$z$ NGC, and high-$z$ SGC. The right panel shows the results for each of the low-$z$ or high-$z$ $\mathrm{NGC}+\mathrm{SGC}$ samples. In the right panel, we overplot the gray contours to show the distribution for the full sample for comparison, which is the same as the blue contours in Fig. 2.

Figure 6

The posterior distributions of cosmological parameters in our fiducial cosmology analysis for the full galaxy sample, obtained both with and without including the hexadecapole moments of the redshift-space power spectrum in the parameter inference. The red contours are the same as those in Fig. 2.

Figure 7

Comparison of the posterior distributions obtained when including the redshift-space power-spectrum information up to a different $k_{\max }$ in the analysis. We show the cases of $k_{\max }=0.2$ (blue), 0.25 (red), and 0.3 (green) $h{\mathrm{Mpc}}^{-1}$, and also show the results of the Planck 2018 $\mathrm{\Lambda }\mathrm{CDM}$ “TT, TE, $\mathrm{EE}+\mathrm{lowE}$” (gray). The cases of $k_{\max }=0.25h{\mathrm{Mpc}}^{-1}$ and the Planck results are identical to those in Fig. 2.

Figure 8

The distribution of scatters in each cosmological parameter, obtained from the cosmology analysis of noisy mock power spectra for the low-$z$ and high-$z$ NGC samples, using different $k_{\max }$ cuts; the $x$ axis shows a shift in the mode value in the 1D posterior of each parameter at $k_{\max }=0.25h$ or $0.3h{\mathrm{Mpc}}^{-1}$, compared to that at $k_{\max }=0.2h{\mathrm{Mpc}}^{-1}$. We use 50 realizations of the noisy mock spectra, and the histogram in each panel displays the distribution of parameter shifts. The vertical red and green lines denote the shift found from the cosmology analyses of the real BOSS data, shown in Fig. 7. The dashed curves denote the Gaussian distributions specified by the mean and variance of the parameter shifts among the 50 realizations.

Figure 9

The posterior distributions of the cosmological parameters using the mock catalogs that are generated from the $N$-body simulations for a cosmological model at MAP in Table 2 (see text for details). The horizontal and vertical dashed lines show the input parameter values—i.e., the cosmology at MAP. This is a similar test to Fig. 1, and the mock catalogs are generated using the same recipe of galaxy-halo connection as that in the left panel of Fig. 1.

Figure 10

The posterior distributions of ${\mathrm{\Omega }}_{\mathrm{m}},H_0$, and ${\sigma }_8$, obtained when we apply our analysis pipeline to the mock power-spectrum signals that are generated from the mock galaxy catalogs including the assembly bias effect (see text for the details). The horizontal and vertical dashed lines show the input parameter values. The mock galaxy catalogs have the same HOD shape as that of the mock catalogs in the right panel of Fig. 1, but they were generated by populating mock galaxies preferentially into halos that have lower concentrations, in each halo mass bin.

Figure 11

The red contours show the posterior distributions of the galaxy-halo connection parameters for the low-$z$ NGC sample, obtained for the same analysis using the assembly bias mock catalog as in Fig. 10. For comparison, the blue contours show the posterior distributions obtained from the fiducial mock catalog, which are from the same analysis in the right panel of Fig. 1. The two mock catalogs are constructed using the same galaxy-halo connection model, and the horizontal and vertical dashed lines show the input parameter values.

Figure 12

Comparison of the multipole moments between the measurements from the mock catalog and the model predictions at MAP that are from the same analysis in Fig. 11. The gray shaded region is the $k$ range on which we use the data in the analysis—i.e., $k_{\max }=0.25h{\mathrm{Mpc}}^{-1}$.

Figure 13

The posterior distributions of the cosmological parameters, obtained from the cosmology analysis of the real BOSS data for either the low-$z$ or high-$z$ NGC sample, when fixing the seven galaxy-halo connection parameters to their values at MAP in Fig. 15. The dashed-line contours and 1D posterior distribution are the same as in Fig. 15.

Figure 14

The posterior distributions of the galaxy-halo connection parameters, for the low-$z$ NGC sample as an example, obtained from the combined analysis using the mock catalogs that are generated using the same HOD model as in our theoretical template (the right panel of Fig. 1). The dashed lines in each plot denote the input value of each parameter that is used when generating the mock galaxy catalogs.

Figure 15

The posterior distributions in each 2D subspace of the full parameters for each of the four galaxy samples, for the cosmology analysis in Fig. 2: the low-$z$ NGC, low-$z$ SGC, high-$z$ NGC, and high-$z$ SGC samples, respectively. The posterior distributions are from the joint parameter inference of these parameters (33 parameters in total, as given in Table 1): five cosmological parameters (while we impose tight Gaussian priors to ${\omega }_{\mathrm{b}}$ and $n_{\mathrm{s}}$), and each sample is characterized by seven nuisance parameters (five HOD parameters, the virial velocity parameter, and the residual shot noise parameter).

Figure 16

The comparison of the cosmological parameter inferences between the patchy mock power spectra with and without the fiber collisions. We show the cases of the low-$z$ (blue) and high-$z$ (red) NGCs and $k_{\max }=0.3h{\mathrm{Mpc}}^{-1}$. The empty dashed-line contours are the results for the patchy mocks where we include the fiber collision weights. The filled solid-line contours are those for the same mocks where we assume all of the fiber collision weights are unity—i.e., no fiber collisions. The gray horizontal and vertical dashed lines indicate the input cosmological parameter values used in the simulations from which the patchy mocks are created.

Figure 17

The comparison of the cosmological parameter inferences between the fiducial mock power spectrum with and without the fiber collision effect. For mock signals with the fiber collisions, we add the fiber collision effect estimated by Ref. [94] (see their Fig. 3) to the power-spectrum monopole and quadrupole of the fiducial mock signals.

Figure 18

The dependence of the cosmological parameter constraints on the minimum wave number $k_{\min }$ used in the analysis. We test the cases of $k_{\min }=0.005h$ (our fiducial setting, blue), $0.02h$ (red), and $0.05h{\mathrm{Mpc}}^{-1}$ (green), while fixing $k_{\max }=0.25h{\mathrm{Mpc}}^{-1}$.