The General Primordial Cosmic Perturbation

TL;DR

This paper investigates the most general primordial cosmological perturbation in a multi-component early universe consisting of photons, baryons, neutrinos, and cold dark matter, without assuming inflationary initial conditions. Using synchronous gauge and Boltzmann hierarchy formalisms, it identifies five regular, non-decaying modes (including two novel neutrino isocurvature modes: density and velocity) and shows that all linear-order observables are determined by a real symmetric 5×5 covariance matrix of mode amplitudes as a function of wavenumber k. The framework allows correlations between modes to be characterized without assuming Gaussianity, and discusses possible generation mechanisms for the neutrino isocurvature modes, along with the Newtonian potentials and early-time behavior. It also outlines how current and future CMB data (in a companion paper) could constrain or detect the amplitudes of these general perturbations, highlighting the potential impact on our understanding of initial conditions for structure formation.

Abstract

We consider the most general primordial cosmological perturbation in a universe filled with photons, baryons, neutrinos, and a hypothetical cold dark matter (CDM) component within the framework of linearized perturbation theory. We give a careful discussion of the different allowed modes, distinguishing modes which are regular at early times, singular at early times, or pure gauge. As well as the familiar growing and decaying adiabatic modes and the baryonic and CDM isocurvature modes we identify two {\it neutrino isocurvature} modes which do not seem to have been discussed before. In the first, the ratio of neutrinos to photons varies spatially but the net density perturbation vanishes. In the second the photon-baryon plasma and the neutrino fluid have a spatially varying relative bulk velocity, balanced so that the net momentum density vanishes. Possible mechanisms which could generate the two neutrino isocurvature modes are discussed. If one allows the most general regular primordial perturbation, all quadratic correlators of observables such as the microwave background anisotropy and matter perturbations are completely determined by a $5\times 5,$ real, symmetric matrix-valued function of co-moving wavenumber. In a companion paper we examine prospects for detecting or constraining the amplitudes of the most general allowed regular perturbations using present and future CMB data.

Figures (2)

• Figure 1: CMB Anisotropy for the Neutrino Isocurvature Density Mode. We plot $\ell (\ell +1)c_\ell /2\pi$ for the neutrino isocurvature density mode (the dashed curves) for initial power spectra $P_{\delta _\nu }\sim k^\alpha$ where $\alpha =-3.0,\ldots ,-2.4$ increasing in increments of $0.1$ from bottom to top at large $\ell .$ The adiabatic growing mode (the solid curve) with a scale-invariant spectrum is included for comparison. All curves are normalized to COBE. For the lowest curve the variations in the photon-to-neutrino ratio obey a scale-invariant initial power spectrum.
• Figure 2: CMB Anisotropy for the Neutrino Isocurvature Velocity Mode. We plot $\ell (\ell +1)c_\ell /2\pi$ for the neutrino isocurvature velocity mode (the dashed curves) for initial power spectra $P_{v_\nu }\sim k^\alpha$ with $\alpha =-3.0,\ldots ,-2.0$ increasing in increments of $0.2$ from bottom to top at large $\ell .$$\alpha =-2.0$ corresponds to a white noise initial power spectrum in the divergence of the velocity field, possibly resulting from a large number of small explosions.