Neutrino Mass, Dark Energy, and the Linear Growth Factor

TL;DR

This work addresses how neutrino mass and dark energy shape cosmological structure, showing that the suppression of the matter power spectrum by massive neutrinos depends on more than the simple ratio $f_ν=Ω_ν/Ω_m$; it introduces a scale- and density-dependent analytical growth factor $f(z,k;f_ν,w,Ω_{DE})$, and provides a modified Eisenstein & Hu power-spectrum approximation with a public implementation. The authors present a compact form, $f(z,k;f_ν,w,Ω_{DE}) \approx μ(k,f_ν,Ω_{DE})\,Ω_m(z)^α$, where $μ=1 - A(k)Ω_{DE}f_ν + B(k)f_ν^2 - C(k)f_ν^3$ and $α$ encodes the influence of $w$, calibrated against CAMB and shown to be accurate for $f_ν\lesssim 0.15$; the approach extends across Einstein–de Sitter and Λ-dominated epochs. A central result is the identification of both indirect and direct $w$–$f_ν$ degeneracies in growth and power spectrum, including a degeneracy that can be lifted by measuring the absolute amplitude of fluctuations (e.g., via the CMB) and by exploiting probes like peculiar velocities and the ISW effect. Collectively, these contributions enhance modeling of massive neutrinos in cosmology and provide practical tools for breaking parameter degeneracies with current and upcoming data.

Abstract

We study the degeneracies between neutrino mass and dark energy as they manifest themselves in cosmological observations. In contradiction to a popular formula in the literature, the suppression of the matter power spectrum caused by massive neutrinos is not just a function of the ratio of neutrino to total mass densities f_nu=Omega_nu/Omega_m, but also each of the densities independently. We also present a fitting formula for the logarithmic growth factor of perturbations in a flat universe, f(z, k;f_nu,w,Omega_DE)= (1-A(k)*Omega_DE*f_nu+B(k)*f_nu^2-C(k)*f_nu^3)*Omega_m(z)^alpha, where alpha depends on the dark energy equation of state parameter w. We then discuss cosmological probes where the f factor directly appears: peculiar velocities, redshift distortion and the Intergrated Sachs-Wolfe effect. We also modify the approximation of Eisenstein & Hu (1999) for the power spectrum of fluctuations in the presence of massive neutrinos and provide a revised code (http://www.star.ucl.ac.uk/~lahav/nu_matter_power.f)

Figures (7)

• Figure 1: The quantity $\Delta P/ P$ at $z=0$ defined in equation (\ref{['eq:deltap1']}), derived using CAMB for fixed $w= -1, \Omega_{m}=0.25, \Omega_{b}=0.04, h=0.7$ (solid black line). The upper plot is for $f_{\nu}=0.04$ and the lower plot for $f_{\nu}=0.16$. For comparison we show the fitting formula from Eisenstein & Hu hu0 (dashed black line), and our modification to their formula (dotted black line). The horizontal blue line is the approximation $- 8 f_{\nu}$ from hu0, and the horizontal dashed red line is our numerical solution to equation (\ref{['eq:eq2']}).
• Figure 2: The dependence of $\Delta P/P$ at $z=0$ on $\Omega_{m}$ and $\Omega_{\nu}$, illustrating, via the scaling by $k_{fs}$, equation (\ref{['eq:kfs']}), that just the ratio $f_{\nu}=\frac{\Omega_{\nu}}{\Omega_{m}}$ is insufficient to fully parametrize $\Delta P /P$. From top to bottom: $f_{\nu}$=0.2 (dashed line), 2 models with $f_{\nu}$=0.1($\Omega_{\nu}$=0.03, $\Omega_{m}$=0.3 (dash-dotted line), $\Omega_{\nu}$=0.02, $\Omega_{m}$=0.2 (solid line)), $f_{\nu}$=0.03 (dotted line). We see that the ratio $\frac{\Delta P}{P}/ f_{\nu}$ tends to a constant only for $k>50k_{fs}$, but not to a universal constant.
• Figure 3: $\Delta P / P$ at $z=0$ for 1 massive, 2 massless neutrinos (top panel) and 3 massive neutrinos (bottom panel), using CAMB (full line), E&H (dotted line),and our modified fitting formula (dashed line).In all cases $f_{\nu}=0.04, \Omega_m =0.25, \Omega_b=0.04$.
• Figure 4: Density fluctuation $\delta (z)$ on comoving 8$h^{-1}\textrm{Mpc}$ scale, normalized to CMB (top panel) and normalized to the value at $z=0$ (middle panel). The models shown in the figure have $(w,f_\nu)$ equal to $(-1,0)$ (solid line), $(-1,0.04)$ (dotted line), $(-0.8,0)$ (dashed line), and $(-0.8,0.04)$ (dash-dotted line). In the case of $w=-0.8$, we included dark energy perturbations according to CMBFAST. We have assumed a spatially flat universe, adiabatic fluctuations, and fixed the matter density $\Omega_{m}=0.25$, baryon density $\Omega_{b}=0.04$, the Hubble constant $h=0.7$, scalar spectral index $n_{s}=1$, and the optical depth to reionization $\tau=0$. The lower plot shows the density fluctuation $\delta (z)$ as a function of $f_\nu$ at $z=0$ (solid line), $z=1$ (dotted line), and $z=2$(dashed line).
• Figure 5: Variation of the logarithmic growth factor $f=\frac{d\ln \delta}{d \ln a}$ with $f_{\nu}=\frac{\Omega_{\nu}}{\Omega_{m}}$ and with redshift at a fixed scale 8$h^{-1}\textrm{Mpc}$. The ratio $\frac{f(f_{\nu}> 0)}{f(f_{\nu}=0)}$ is shown as a function of $f_{\nu}$ for z=1 and z=2. he following examples are shown: $w=-0.8, z=1$(dashed line), $w=-1, z=1$(dotted line), $w=-0.8, z=2$(dash-dotted line) and $w=-1, z=2$(dash-triple dotted line).The solid line is $(1-f_{\nu})^{\alpha_0}$, from equation (\ref{['eq:fsimple']}), and the long-dashed line is the full polynomial found in equation (\ref{['eq:fgrow5']}), evaluated at $z=1$.Other parameters kept fixed $\Omega_{m}=0.3, \Omega_{b}=0.04, h=0.7$.