Nonlinear power spectrum in the presence of massive neutrinos: perturbation theory approach, galaxy bias and parameter forecasts

TL;DR

The work develops a perturbation-theory framework to model the weakly nonlinear galaxy power spectrum in a mixed dark matter universe with finite-mass neutrinos, incorporating self-consistent nonlinear galaxy bias. By treating neutrino perturbations linearly and capturing nonlinear CDM+$b$ growth with one-loop corrections that are efficiently approximated, the authors obtain accurate predictions for the total matter power spectrum up to $k oughly 1~h{ m Mpc}^{-1}$ for $f_ u oughly 0.05$. A Fisher-matrix analysis combining Planck CMB data with several galaxy surveys shows Stage-IV could reach $\sigma(m_{ u, m tot}) oughly 0.05$ eV, enabling a potential detection, while Stage-III / BOSS reach around $0.1$ eV. They also show significant degeneracies between neutrino mass and dark-energy parameters, with biases in $w_0$ arising if neutrino masses are ignored; including neutrino mass in the model is essential for unbiased dark-energy inferences. The framework provides a more robust, physically grounded route to extract neutrino and dark-energy information from upcoming large-scale structure data, and it outlines paths for refinement with simulations and higher-order perturbation methods.

Abstract

Future or ongoing galaxy redshift surveys can put stringent constraints on neutrinos masses via the high-precision measurements of galaxy power spectrum, when combined with cosmic microwave background (CMB) information. In this paper we develop a method to model galaxy power spectrum in the weakly nonlinear regime for a mixed dark matter (CDM plus finite-mass neutrinos) model, based on perturbation theory (PT) whose validity is well tested by simulations for a CDM model. In doing this we carefully study various aspects of the nonlinear clustering and then arrive at a useful approximation allowing for a quick computation of the nonlinear power spectrum as in the CDM case. The nonlinear galaxy bias is also included in a self-consistent manner within the PT framework. Thus the use of our PT model can give a more robust understanding of the measured galaxy power spectrum as well as allow for higher sensitivity to neutrino masses due to the gain of Fourier modes beyond the linear regime. Based on the Fisher matrix formalism, we find that BOSS or Stage-III type survey, when combined with Planck CMB information, gives a precision of total neutrino mass constraint, sigma(m_nu,tot) 0.1eV, while Stage-IV type survey may achieve sigma(m_nu,tot) 0.05eV, i.e. more than a 1-sigma detection of neutrino masses. We also discuss possible systematic errors on dark energy parameters caused by the neutrino mass uncertainty. The significant correlation between neutrino mass and dark energy parameters is found, if the information on power spectrum amplitude is included. More importantly, for Stage-IV type survey, a best-fit dark energy model may be biased and falsely away from the underlying true model by more than the 1-sigma statistical errors, if neutrino mass is ignored in the model fitting.

Figures (11)

• Figure 1: The 2nd- and 3rd-order growth functions at redshift $z=0$ are plotted as a function of two wavenumbers $k_1$ and $k_2$, for a MDM model with $f_\nu=0.05$ ($m_{\nu,{\rm tot}}\simeq 0.6~$eV). As representative examples, shown here is the growth functions divided by some powers of the linear growth rate: $A^{(2)}_\delta(k_1,k_2)/[D_{\rm cb}(k_1)D_{\rm cb}(k_2)(5/7)]$ ( left panel), $A^{(3)}_{\delta}(k_1,k_2,k_1)/[D_{\rm cb}(k_1)^2D_{\rm cb}(k_2)(5/18)]$ ( middle) and $A^{(3)}_{\delta}(k_1,k_1,k_2)/[D_{\rm cb}(k_1)^2D_{\rm cb}(k_2)(5/18)]$ ( right), respectively. Note that specific combinations of $k_i$-arguments in $A^{(3)}_\delta$ are chosen because the one-loop power spectrum $P_{\rm cb}^{(13)}$ (see Eq. [\ref{['eq:Pk13_1']}]) depends on the growth functions of specific configurations. The quantities shown become unity for the limit $f_{\nu}=0$, i.e. a model without massive neutrinos (see Eqs. [\ref{['eq:2ndNLgrowth']}] and [\ref{['eq:3rdNLgrowth']}]), which is shown by the plane in each plot. Therefore the deviations from unity reflect additional scale dependences arising from the mode-coupling. It is clear that scale-dependences of the higher-order growth functions are well-captured by combinations of the linear growth rate, and the approximations (\ref{['eq:2ndNLgrowth']}) and (\ref{['eq:3rdNLgrowth']}) hold valid with accuracy better than $\sim 5\%$ over a range of wavenumbers we have considered.
• Figure 2: Left panel: The dashed curves show the one-loop power spectra of CDM plus baryon perturbations, $P^{(22)}_{\rm cb}$ and $P^{(13)}_{\rm cb}$, which are obtained by numerical integrations of Eqs. (\ref{['eq:Pk22_1']}) and (\ref{['eq:Pk13_1']}), respectively, while the solid curves show the spectra computed using the approximations (\ref{['eq:P22approx']}) and (\ref{['eq:P13approx']}). Note that the $y$-axis is plotted in the linear scale, and we consider $f_\nu=0.05$ and $z=0$. For comparison, the rightmost solid curve labelled as "$P_{\rm cb}^{\rm L}$" shows the linear power spectrum. Right panel: The fractional difference of the total matter power spectrum including up to the one-loop corrections is shown in the left panel: $P^{\rm NL}_{\rm cb}(k)=P^{\rm L}_{\rm cb}(k)+P^{(13)}_{\rm cb}(k) +P^{(22)}_{\rm cb}(k)$. The approximation is found to be accurate to better than $1\%$ on scales $k\raisebox{-0.5ex}{$\:\stackrel{ <}{\sim}\:$} 1h$Mpc$^{-1}$.
• Figure 3: The nonlinear power spectra for a MDM model with $f_\nu=0.01$. The solid curves show the SPT predictions divided by the linear spectra for three redshifts $z=0, 1$ and $3$, while the dashed curves denote the halofit results.
• Figure 4: Left panel: The fractional difference between mass power spectra with and without massive neutrino contribution. The shaded boxes show the expected $1$-$\sigma$ errors on the power spectrum measurement for a Stage-III type survey of $z\sim 1$ slice that is characterized by the mean number density of galaxies and survey volume, $\bar{n}_{\rm g}=5\times 10^{-4}~h^3{\rm Mpc}^{-3}$ and $V_{\rm survey}=1.5~h^{-3}{\rm Gpc}^{3}$ (also see Table \ref{['table:survey']}). The two models of neutrino mass, $f_\nu=0.01$ and 0.03 ($m_{\nu, {\rm tot}}\simeq 0.12$ and $0.36~{\rm eV}$, respectively) are assumed, where other cosmological parameters are kept fixed. Right panel: It is shown how the neutrino suppression feature in the power spectrum amplitude varies with redshifts, comparing the results for the SPT, linear theory and halofit.
• Figure 5: Top panel: The perturbation theory predictions for nonlinear galaxy power spectrum at redshift $z=1$, which are computed from Eq. (\ref{['eq:Nonlinear_galaxy_Pk']}) assuming the three fiducial values of nonlinear bias parameter, $b_2=-0.25, 0.25$ and $1.2$, respectively. The results are divided by the nonlinear mass power spectrum multiplied by the same linear bias parameter $b_1^2$ such that the deviation from unity represents the nonlinear, scale-dependent bias effect. The positive and negative $b_2$ values, with $|b_2|<1$, cause enhanced and suppressed power spectrum amplitudes on smaller scales compared to the linearly biased power spectrum. The model with $b_2>1$ causes a complex scale-dependent bias (also see text for the details). The valid range of linear theory and SPT are indicated by the two arrows in the upper horizontal axis (see text for the definition). Middle panel: The neutrino suppression features for the nonlinear galaxy power spectra for different fiducial values of $b_2$. For comparison the two dashed curves are the results for mass power spectrum computed from the SPT and linear theory as in Fig. \ref{['fig:matter suppression']}. Bottom panel: The effect of residual shot noise contamination that arises from nonlinear clustering, which is modeled as $P_{\rm g}\rightarrow P_{\rm g}+N$. The three solid curves show the results for $N=0,1000$ and 2000, respectively, where $b_{1}=1.51$ and $b_{2}=0.25$ are kept fixed.