Neutrino mass constraint with the Sloan Digital Sky Survey power spectrum of luminous red galaxies and perturbation theory

TL;DR

This work develops and tests a perturbation-theory–based framework to model the nonlinear galaxy power spectrum in a mixed CDM+neutrino universe, including nonlinear clustering and scale-dependent bias. By validating the PT model against N-body halo simulations, the authors justify a conservative analysis up to $k_{\max}=0.1\,h\,\mathrm{Mpc}^{-1}$ and then apply it to SDSS LRG power spectra combined with WMAP5 data. The joint analysis yields a neutrino-mass upper bound of $\sum m_{\nu} \le 0.81\,\mathrm{eV}$ (95% C.L.), improves significantly over WMAP5 alone, and reveals degeneracies with the dark-energy equation of state parameter $w$ and the nonlinear bias parameter $b_2$, with a notable preference for a nonzero $b_2$. The results demonstrate the potential of a PT-based, bias-aware modeling approach for extracting neutrino masses from galaxy clustering, while highlighting areas for improvement via higher-order corrections and simulation-calibrated refinements.

Abstract

We compare the model power spectrum, computed based on perturbation theory (PT) with the power spectrum of luminous red galaxies (LRG) measured from the SDSSDR7 catalog, assuming a flat, CDM-dominated cosmology. The model includes the effects of massive neutrinos, nonlinear matter clustering and nonlinear, scale-dependent galaxy bias in a self-consistent manner. We first test the accuracy of PT-model by comparing the model predictions with the halo power spectrum in real- and redshift-space measured from simulations without massive neutrinos. We show that the PT-model with bias parameters being properly adjusted can fairly well reproduce the simulation results. As a result the best-fit parameters obtained from the hypothetical parameter fitting recover, within statistical uncertainties, the input cosmological parameters in simulations, including an upper bound on neutrino mass, if the power spectrum information up to k~0.15h/Mpc is used. However, for the redshift-space power spectrum, the best-fit cosmological parameters show a sizable bias from the input values if using the information up to k~0.2h/Mpc, probably due to nonlinear redshift distortion effect. Given these tests, we decided, as a conservative choice, to use the LRG power spectrum up to k=0.1h/Mpc in order to minimize possible unknown nonlinearity effects. In combination with the recent results from Wilkinson Microwave Background Anisotropy Probe (WMAP), we derive a robust upper-bound on the sum of neutrino masses, given as m_nu,tot < 0.81eV (95% C.L.), marginalized over other parameters including nonlinear bias parameters and dark energy equation of state parameter. The neutrino mass limit is improved by a factor of 1.85 compared to the limit from the WMAP5 alone, m_nu,tot < 1.5eV.

Figures (5)

• Figure 1: Upper panel: The filled circles at each $k$ bin show the mean halo power spectrum measured from 70 simulation realizations at $z=0$ (see text for details), while the error bar shows the statistical measurement uncertainty at the $k$ bin for a simulation volume of 1 $h^{-3}$Gpc$^3$, roughly comparable with the SDSS survey volume. For illustrative purpose the halo spectrum is divided by the no-wiggle, linear power spectrum, multiplied by the linear halo bias squared, $b_1^2 P^L_{{\rm m,nw}}(k)$ ($b_{1}=1.66$). For comparison, the thin-dotted and -solid curves show the linear-theory and PT predictions for "mass" power spectrum, respectively, for the cosmological model assumed in the simulations. The bold solid curve shows the best-fit PT model for halo power spectrum, computed from Eq. (\ref{['eq:nonlinearPkmatter']}), where the best-fit model parameters including bias parameters are obtained by fitting the model predictions to the simulation spectrum up to $k=0.1~h$Mpc$^{-1}$ (see Fig. \ref{['fig:HaloTest_1D']}). Lower panel: Similar to the upper panel, but for redshift-space power spectrum ($b_{1}=1.81$). The redshift-space power spectrum is modified by redshift distortion effect due to peculiar velocities of halos. For comparison, the circle points without error bars show the simulation halo spectrum in real space (the same as in the upper panel).
• Figure 2: Testing the perturbation theory (PT) based model with the halo power spectrum measured from N-body simulations (70 realizations used). The solid curve in each panel is the posterior distribution of parameter, estimated by comparing the PT model with the halo power spectrum up to the maximum wavenumber $k_{\rm max}=0.1~h{\rm Mpc}^{-1}$. The input values of $\Omega_{\rm b0}/\Omega_{\rm m0}$ and $\Omega_{\rm m0}$, denoted by the vertical lines, are properly recovered within the statistical errors for the volume $1~h^{-3 }{\rm Gpc}^3$, which is comparable with the SDSS volume. For neutrino masses, which are not included in the N-body simulations, an upper limit is derived. Nonzero values of bias parameters ($b_1$ and $b_2$) and shot noise parameter ($N$) are obtained, implying that the parameters are needed to describe the halo power spectrum. The dotted curves represent the posterior distribution obtained by fixing the neutrino mass to zero, showing that the input values of the parameters $\Omega_{\rm b0}/\Omega_{\rm m0}$ and $\Omega_{\rm m0}$ are correctly reproduced together with a tighter constraint on the linear bias parameter.
• Figure 3: The best-fit parameters and the marginalized errors obtained by fitting the PT model with the halo spectrum up to a given maximum wavenumber $k_{\rm max}$, denoted in the horizontal axis. For each $k_{\rm max}$ the left-side point with error bar shows the results for the real-space halo spectrum, while the right-side point shows the results for redshift-space halo spectrum. The horizontal dashed line denotes the input value of each parameter.
• Figure 4: The parameter constraints obtained by comparing the PT model with the SDSS LRG power spectrum up to $k_{\rm max}=0.1~h{\rm Mpc}^{-1}$, in combination with the WMAP5 constraint, where we include 12 parameters given by Eq. (\ref{['eq:paras']}). The upper panel shows the posterior distribution of neutrino masses, yielding the upper limit $\sum m_{\nu}\le 0.81~{\rm eV}$ (95% C.L.), a factor 1.85 improvement over the limit $\sum m_\nu\le 1.5~{\rm eV}$ from the WMAP5 alone. The lower two panels show how the neutrino mass is degenerate with the dark energy equation of state parameter $w$ and the nonlinear bias parameter $b_2$, respectively, with 68% C.L. (dark shaded) and 95% C.L. (light shaded) regions. A nonzero $b_2$ or equivalently a scale-dependent bias is favored at 68% C.L. Our results are compared with the results derived using the halo-model based method in R10 for the same maximum wavenumber cutoff $k_{\rm max}=0.1~h$Mpc$^{-1}$.
• Figure 5: Comparing the best-fit PT model with the SDSS LRG spectrum, where the best-fit model is obtained from the fitting to $k_{\rm max}=0.1~h{\rm Mpc}^{-1}$. For illustrative clarity the power spectra are divided by the linear matter power spectrum for the best-fit cosmological model. For comparison, we also show the PT model, where the neutrino mass is changed to $\sum m_{\nu}=0.81\,$eV, corresponding to the 95% C.L. upper bound in Fig. \ref{['fig:result']}, but other parameters are kept fixed to the best-fit values.