Power spectra in the eikonal approximation with adiabatic and non-adiabatic modes

TL;DR

The paper develops and applies the eikonal approximation to a multi-fluid cosmological perturbation theory, showing that large-scale adiabatic modes do not affect small-scale power spectra at any order in standard perturbation theory, while nonadiabatic isodensity modes can damp and anisotropically modulate small-scale power. By formulating a 2N-component perturbation framework, rotating to an eigenbasis, and performing a detailed 1-loop calculation, the authors identify a damping scale set by decaying isodensity modes and demonstrate that the full damping can be captured via the eikonal resummation, with explicit high-$k$ behavior showing a $k^2$-driven damping proportional to $oldsymbol{\sigma}^2_{ riangledown riangledown}$. The eikonal treatment yields equal-time multispectra that are robust to adiabatic long-wavelength modulations, while isodensity modes alter amplitudes and can regularize divergences, enabling a controlled description of small-scale CDM and baryon power spectra, including relative-velocity effects in environments such as the Local Group. A representative outcome is a damped small-scale spectrum of the form $P_{++}(k, au) o e^{2( au- au_0)}P^{ m in}_{++}(k)ig[1-(k/k_{ m damp})^2ig]$ with $k_{ m damp} o 380~h~{ m Mpc}^{-1}$ in LCDM, illustrating the practical implications for early structure formation and baryon-CDM dynamics.

Abstract

We use the so-called eikonal approximation, recently introduced in the context of cosmological perturbation theory, to compute power spectra for multi-component fluids. We demonstrate that, at any given order in standard perturbation theory, multipoint power spectra do not depend on the large-scale adiabatic modes. Moreover, we employ perturbation theories to decipher how nonadiabatic modes, such as a relative velocity between two different components, damp the small-scale matter power spectrum, a mechanism recently described in the literature. In particular, we do an explicit calculation at 1-loop order of this effect. While the 1-loop result eventually breaks down, we show how the damping effect can be fully captured by the help of the eikonal approximation. A relative velocity not only induces mode damping but also creates large-scale anisotropic modulations of the matter power spectrum amplitude. We illustrate this for the Local Group environment.

Figures (7)

• Figure 1: Diagrammatic representation of the series expansion of $\Psi_{\alpha}({\bf k})$ up to second order in the initial conditions at $\eta=\eta_0$. Time increases along each segment according to the arrow and each segment bears a factor $g_{\alpha}^{\ \beta}(\eta,\eta_0)$ if $\eta_{0}$ is the initial time and $\eta$ is the final time. At each initial point and each vertex point there is a sum over the component indices; a sum over the incoming wave modes is also implicit and, finally, the time coordinate of the vertex points is integrated from $\eta=\eta_0$ to the final time $\eta$ according to the time ordering of each diagram.
• Figure 2: Diagrammatic representation of the power spectra up to 1-loop order. The symbol $\otimes$ represents the power spectra at initial time. Note that the power spectra and cross spectra of all 4 modes should be taken into account.
• Figure 3: Power spectrum of the total matter fluctuations (solid blue and dashed red line) and of baryonic fluctuations (dot-dashed red line) computed at 1-loop order as a function of $k$ at $z=40$, in units of the linear total matter power spectrum $P_{++}^{(0)}$ (i.e. of the adiabatic mode). For the total matter fluctuations, the power spectrum is computed either including the contribution of the adiabatic modes only (blue line) and taking into account all modes including also the isodensity modes (dashed red line). In the two red lines one can see the damping of modes due to the isodensity modes.
• Figure 4: The same as in Fig. \ref{['P1loopz40']}, but at $z=10$.
• Figure 5: This plot illustrates the central property of diagram complementarity in the leading $k$ regime. The sum of these 3 diagrams vanishes when the dashed line corresponds to a given adiabatic mode and when the vertex it is connected to is computed in the soft limit $q/k \to 0$, Eqs. \ref{['vert_+']} and \ref{['vert_-']}, or in the eikonal limit, Eq. (\ref{['eikovertex']}). The only case for which these 3 contributions do not cancel is when the intervening mode is in the decaying isodensity mode.