Testing one-loop galaxy bias: Cosmological constraints from the power spectrum

We investigate the impact of different assumptions in the modeling of one-loop galaxy bias on the recovery of cosmological parameters, as a follow-up of the analysis done in the first paper of the series at fixed cosmology. To carry out these tests we focus on the real-space galaxy-power spectrum from a set of three different synthetic galaxy samples whose clustering properties are meant to match the ones of the CMASS and LOWZ catalogs of BOSS and the SDSS Main Galaxy Sample. We investigate the relevance of allowing for either short range nonlocality or scale-dependent stochasticity by fitting the real-space galaxy autopower spectrum or the combination of galaxy-galaxy and galaxy-matter power spectrum. From a comparison among the goodness of fit (${\chi }^2$), unbiasedness of cosmological parameters (FoB), and figure of merit (FoM) of the model, we find that a simple four-parameter model (linear, quadratic, cubic nonlocal bias, and constant shot noise) with fixed quadratic tidal bias provides a robust modeling choice for the autopower spectrum of the three galaxy samples, up to $k_{\max }=0.3h{\mathrm{Mpc}}^{-1}$ and for an effective volume of $6h^{-3}{\mathrm{Gpc}}^3$. Instead, a joint analysis of the two observables fails at larger scales, and a model extension with either higher derivatives or scale-dependent shot noise is necessary to reach a similar $k_{\max }$, with the latter providing the most accurate and stable results. Throughout the majority of the paper, we fix the description of the nonlinear matter evolution using a hybrid perturbative-N-body approach, respresso, that was found in the first paper to be the closest performing to the measured matter spectrum. We also test the impact of different modeling assumptions based on perturbative approaches, such as galilean-invariant Renormalised Perturbation Theory (gRPT) and effective field theory (EFT). In all cases, we find the inclusion of scale-dependent shot noise to increase the range of validity of the model in terms of FoB and ${\chi }^2$. Interestingly, these model extensions with additional free parameters do not necessarily lead to an increase in the maximally achievable FoM for the cosmological parameters $(h,{\mathrm{\Omega }}_c{h}^2,A_s)$, which are generally consistent with those of the simpler model at smaller $k_{\max }$.

Figure 1

Ratio between the CMBFast and CAMB linear prediction at the reference cosmology of the lasdamas simulations. The two shaded gray bands mark the intrinsic 1-sigma error of the LOWZ and MGS galaxy-power spectrum $P_{\mathrm{gg}}$.

Figure 2

FoM, FoB and goodness of fit extracted from chains corresponding to the fits of the galaxy autopower spectrum $P_{\mathrm{gg}}$. Performance metrics refer to the combinations of $h$ and ${\mathrm{\Omega }}_{\mathrm{c}}{h}^2$, which are the only two free cosmological parameters of the model. The standard (STD), $k^2$-dependent shot noise (k2N), and higher derivatives (HD) models are color coded as shown in the legend. In all three cases, the nonlinear matter power spectrum is modeled with respresso, and the second-order tidal bias ${\gamma }_2$ is fixed to Eq. (40). The two shaded gray regions mark the 68th and 95th percentiles of the FoB and ${\chi }^2$ distribution.

Figure 3

Marginalized constraints on the linear bias $b_1{\sigma }_8$ (top) and the constant stochastic term $N_0$ (bottom) as a function of $k_{\max }$, for the fits of the galaxy autopower spectrum $P_{\mathrm{gg}}$. In all cases dark matter nonlinear evolution is modeled using respresso. Fiducial values of the linear bias and constant shot-noise contribution are reproduced with black dashed lines. Uncertainties on the fiducial values are marked with shaded gray bands (marking $1\text{−}\sigma$ and $2\text{−}\sigma$ confidence intervals). Fiducial measurements are listed in Table 1.

Figure 4

Same as Fig. 2 but for the combinations of the galaxy auto- and cross-power spectrum, $P_{\mathrm{gg}}$ and $P_{\mathrm{gm}}$. In this case, performance metrics (FoM and FoB) refer to the combinations of the three cosmological parameters $(h,{\mathrm{\Omega }}_{\mathrm{c}}{h}^2,A_{\mathrm{s}})$. The standard (STD), $k^2$-dependent shot noise (k2N), and higher derivatives (HD) models are color coded as shown in the legend. The solid-to-dashed transition marks the model-breaking scale $k^{†}$. In all three cases, the nonlinear matter power spectrum is modeled with respresso. The two shaded gray regions mark the 68th and 95th percentiles of the FoB and ${\chi }^2$ distribution.

Figure 5

Same as Fig. 3 but for the combined fits of the galaxy autopower spectrum $P_{\mathrm{gg}}$ and the galaxy-matter cross power spectrum $P_{\mathrm{gm}}$. In all cases dark matter nonlinear evolution is modeled using respresso. Fiducial values of the linear bias and constant shot-noise contribution are reproduced with black dashed lines. Uncertainties on the fiducial values are marked with shaded gray bands (marking 1- and $2\text{−}\sigma$ confidence interval).

Figure 6

Marginalized posterior of the $k^2$-dependent stochastic parameters $N_2$ and $N_2^{\times }$ (top), and the higher derivatives parameters ${\beta }_P$ and ${\beta }_P^{\times }$ (bottom), as a function of $k_{\max }$, for the combination of $P_{\mathrm{gg}}+P_{\mathrm{gm}}$. In all cases dark matter nonlinear evolution is modeled using respresso. For a consistency check we also show the $1\text{−}\sigma$ standard deviations of the parameters $N_2$ and ${\beta }_P$ obtained from the fit of $P_{\mathrm{gg}}$ only as a shaded gray area.

Figure 7

FoM, FoB and goodness of fit extracted from fits of $P_{\mathrm{gg}}+P_{\mathrm{gm}}$. Performance metrics are separated into each individual component, i.e. $h$, ${\mathrm{\Omega }}_{\mathrm{c}}{h}^2$ and $A_{\mathrm{s}}$. Different dark matter models and different one-loop modeling assumptions are color and style coded as shown in the legend. The two shaded gray regions mark the 68th and 95th percentiles of the FoB and ${\chi }^2$ distribution.

Figure 8

Comparison between the posterior parameter distribution (68% and 95% confidence intervals) obtained by fitting the galaxy autopower spectrum $P_{\mathrm{gg}}$ (blue) and the combination of galaxy-galaxy and galaxy-matter power spectrum $P_{\mathrm{gg}}+P_{\mathrm{gm}}$ (red). We show constraints obtained at $k_{\max }=0.3h{\mathrm{Mpc}}^{-1}$ for the cosmological parameters $h$, ${\mathrm{\Omega }}_{\mathrm{c}}{h}^2$, $A_{\mathrm{s}}$, and the linear bias $b_1{\sigma }_8$. In all cases we assume nonlinear dark-matter evolution to be described by respresso. Dashed solid lines correspond to the fiducial values of the corresponding parameter. Gray solid bands mark the $1\text{−}\sigma$ and $2\text{−}\sigma$ error on the fiducial linear bias parameter.