Small Scale Perturbations in a General MDM Cosmology

TL;DR

This work develops analytic small-scale perturbation theory for a mixed dark matter cosmology with massive neutrinos and baryons, extending to include Λ and spatial curvature. The authors derive a scale- and time-dependent growth framework, introducing growth functions D_{cb} and D_{cbν} that describe suppression and later neutrino infall, and a time-independent master transfer function T_{master} that captures the spatial shape. The resulting transfer functions T_{cb} and T_{cbν} factorize into growth factors and T_{master}, achieving 1–2% agreement with numerical codes in the small-scale limit and remaining valid across cosmologies via a cosmology-sensitive growth replacement D(y;Ω0,ΩΛ). The framework clarifies how baryons and neutrinos jointly shape structure growth, tightens neutrino-density constraints in low-density universes, and provides a practical approach for exploring MDM parameter space and interpreting high-redshift structure data.

Abstract

For a universe with massive neutrinos, cold dark matter, and baryons, we solve the linear perturbation equations analytically in the small-scale limit and find agreement with numerical codes at the 1-2% level. The inclusion of baryons, a cosmological constant, or spatial curvature reduces the small-scale power and tightens limits on the neutrino density from observations of high redshift objects. Using the asymptotic solution, we investigate neutrino infall into potential wells and show that it can be described on all scales by a growth function that depends on time, wavenumber, and cosmological parameters. The growth function may be used to scale the present-day transfer functions back in redshift. This allows us to construct the time-dependent transfer function for each species from a single master function that is independent of time, cosmological constant, and curvature.

Figures (4)

• Figure 1: Growth suppression from the neutrinos and baryons. The addition of neutrinos slows the growth of CDM $+$ baryon fluctuations after matter-radiation equality $y>1$ (upper panel, curve $b$ compared with curve $b$). Increasing the baryon fraction further suppresses fluctuations by a time-independent factor at $y>y_d$ due to growth suppression in the CDM fluctuations for $1<y<y_d$ and the lack of baryon fluctuations in the weighted density perturbations (upper panel, curve $c$). The analytic expressions (dashed lines) agree with numerical results (solid) for $\Omega_0=1$, $h=0.5$ and $q=160$ at the $1\%$ level except for $5\%$ discrepancies at $y \le 1$.
• Figure 2: Analytic vs. numerical descriptions of neutrino infall for $f_\nu=0.3$, $f_b=0.01$, $\Omega_0=1$, $h=0.5$. Upper panel: Growth functions compared to the fully free-streaming time-dependence of $y^{1-p_{cb}}$; numerical CDM $+$ baryon fluctuations compared with growth function $D_{cb}$ ( short-dashed lines) numerical CDM $+$ baryon $+$ neutrino density-weighted fluctuations compared with $D_{cb\nu}$ ( long-dashed lines). No infall is represented by a horizontal line of amplitude 1 ($cb$) and $f_{cb}=0.69$ ($cb\nu$) here. Lower panel: relative error.
• Figure 3: Demonstration of the existence of a time-independent master function. The transfer functions for the CDM $+$ baryon ($T_{cb}$) and CDM $+$ baryon $+$ neutrino ($T_{cb\nu}$) fluctuations at 3 different redshifts are divided by the growth factors $D_{cb}$ and $D_{cb\nu}$ respectively to obtain estimates of $T_{\rm master}$. That the 6 curves superimposed in the upper panel agree at the $1-2\%$ level (relative to $T_{cb}/D_{cb}$ at $z=0$, as shown in the lower panel) establishes the existence of the master function and verifies the accuracy of the growth functions. The analytic prediction for $T_{\rm master}$ (long-dashed line) converges to within $1\%$ of the numerical results at $q\gg 1$. The model here is $\Omega_0=1$, $h=0.5$, $f_\nu=0.4$, $f_b=0.2$.
• Figure 4: Demonstration of the invariance of the master function ($T_{\rm master}$) under changes in the cosmological constant $\Lambda$. The model has the same $\Omega_0 h^2$, $f_b$ and $f_\nu$ as Fig. \ref{['fig:transfer']} at $z=0$ but with $\Omega_0=0.25$. Division of $T_{cb}$ and $T_{cb\nu}$ by the growth functions returns the master function $T_{\rm master}$ over which the estimate of Fig. \ref{['fig:transfer']} and the small-scale analytic solution are plotted (top panel). The errors are plotted relative to $T_{\rm master}$ from the $T_{cb}$, $z=0$ estimate of Fig. \ref{['fig:transfer']} and show agreement at the $1-2\%$ level (bottom panel).