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How to generate a significant effective temperature for cold dark matter, from first principles

Patrick McDonald

TL;DR

This work addresses a core deficiency of perturbation theory in cosmology: the neglect of velocity dispersion in modeling quasi-linear structure. By starting from the exact Vlasov equation and taking moments, the author derives evolution equations that couple density, bulk velocity, and dispersion, and shows that standard PT fails to remain controlled once dispersion corrections are included. Using renormalization-group ideas, the paper demonstrates how an initially tiny background dispersion can be amplified by small-scale fluctuations, with Jeans-like filtering providing a self-consistent saturation mechanism, especially for power-law initial spectra. For realistic \Lambda CDM initial conditions, the dispersion remains modest at linear order but is expected to grow through non-linear feedback, potentially becoming dynamically relevant at the non-linear scale, and the proposed framework offers a path to include these effects in higher-order PT and redshift-space distortions.

Abstract

I show how to reintroduce velocity dispersion into perturbation theory (PT) calculations of structure in the Universe, i.e., how to go beyond the pressureless fluid approximation, starting from first principles. This addresses a possible deficiency in uses of PT to compute clustering on the weakly non-linear scales that will be critical for probing dark energy. Specifically, I show how to derive a non-negligible value for the (initially tiny) velocity dispersion of dark matter particles, <δv^2>, where δv is the deviation of particle velocities from the local bulk flow. The calculation is essentially a renormalization of the homogeneous (zero order) dispersion by fluctuations 1st order in the initial power spectrum. For power law power spectra with n>-3, the small-scale fluctuations diverge and significant dispersion can be generated from an arbitrarily small starting value -- the dispersion level is set by an equilibrium between fluctuations generating more dispersion and dispersion suppressing fluctuations. For an n=-1.4 power law normalized to match the present non-linear scale, the dispersion would be ~100 km/s. This n corresponds roughly to the slope on the non-linear scale in the real \LambdaCDM Universe, but \LambdaCDM contains much less initial small-scale power -- not enough to bootstrap the small starting dispersion up to a significant value within linear theory (viewed very broadly, structure formation has actually taken place rather suddenly and recently, in spite of the usual "hierarchical" description). The next order PT calculation, which I carry out only at an order of magnitude level, should drive the dispersion up into balance with the growing structure, accounting for small dispersion effects seen recently in simulations.

How to generate a significant effective temperature for cold dark matter, from first principles

TL;DR

This work addresses a core deficiency of perturbation theory in cosmology: the neglect of velocity dispersion in modeling quasi-linear structure. By starting from the exact Vlasov equation and taking moments, the author derives evolution equations that couple density, bulk velocity, and dispersion, and shows that standard PT fails to remain controlled once dispersion corrections are included. Using renormalization-group ideas, the paper demonstrates how an initially tiny background dispersion can be amplified by small-scale fluctuations, with Jeans-like filtering providing a self-consistent saturation mechanism, especially for power-law initial spectra. For realistic \Lambda CDM initial conditions, the dispersion remains modest at linear order but is expected to grow through non-linear feedback, potentially becoming dynamically relevant at the non-linear scale, and the proposed framework offers a path to include these effects in higher-order PT and redshift-space distortions.

Abstract

I show how to reintroduce velocity dispersion into perturbation theory (PT) calculations of structure in the Universe, i.e., how to go beyond the pressureless fluid approximation, starting from first principles. This addresses a possible deficiency in uses of PT to compute clustering on the weakly non-linear scales that will be critical for probing dark energy. Specifically, I show how to derive a non-negligible value for the (initially tiny) velocity dispersion of dark matter particles, <δv^2>, where δv is the deviation of particle velocities from the local bulk flow. The calculation is essentially a renormalization of the homogeneous (zero order) dispersion by fluctuations 1st order in the initial power spectrum. For power law power spectra with n>-3, the small-scale fluctuations diverge and significant dispersion can be generated from an arbitrarily small starting value -- the dispersion level is set by an equilibrium between fluctuations generating more dispersion and dispersion suppressing fluctuations. For an n=-1.4 power law normalized to match the present non-linear scale, the dispersion would be ~100 km/s. This n corresponds roughly to the slope on the non-linear scale in the real \LambdaCDM Universe, but \LambdaCDM contains much less initial small-scale power -- not enough to bootstrap the small starting dispersion up to a significant value within linear theory (viewed very broadly, structure formation has actually taken place rather suddenly and recently, in spite of the usual "hierarchical" description). The next order PT calculation, which I carry out only at an order of magnitude level, should drive the dispersion up into balance with the growing structure, accounting for small dispersion effects seen recently in simulations.

Paper Structure

This paper contains 11 sections, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generation of velocity dispersion
  3. Evolution equations
  4. Standard perturbation theory
  5. Renormalization group approach
  6. RG method, 2nd iteration
  7. A more exact, general approach
  8. Power law power spectra
  9. $\Lambda$CDM
  10. Redshift-space distortions
  11. Discussion

Figures (1)

  • Figure 1: Ratio of linear power with Jeans-like suppression by velocity dispersion to without (i.e., to the ${\tilde{T}}\rightarrow 0$ limit). The left panel is for a power law power spectrum with $n=-1.4$, while the right is for $n=-2.75$. Lines are identified in the figure panel. $\sigma\equiv {\tilde{T}}^{1/2}$.