Correlated Primordial Perturbations in Light of CMB and LSS Data

TL;DR

This work tests whether primordial perturbations include a correlated CDM isocurvature component in addition to the standard adiabatic mode, allowing for separate spectral indices and a possible correlation. Using an 11-parameter MCMC analysis against CMB (WMAP, CBI, ACBAR) and SDSS data, the authors find no evidence for isocurvature and place a stringent upper limit of $α<0.18$ (95% C.L.) at the pivot scale $k_0 = 0.01\, mathrm{Mpc}^{-1}$, with the standard parameters $ω_b$, $ω_c$, and $H_0$ shifting modestly mainly through correlations, while $ω_b$ stays near its adiabatic value. The study reveals strong sensitivities to the choice of pivot scale and parametrization, showing that unconstrained spectral indices can bias posterior inferences; it argues for a two-scale amplitude approach to stabilize future analyses. Overall, the results reinforce the adiabatic paradigm but underscore methodological caveats relevant for inflationary model testing.

Abstract

We use cosmic microwave background (CMB) and large-scale structure data to constrain cosmological models where the primordial perturbations have both an adiabatic and a cold dark matter (CDM) isocurvature component. We allow for a possible correlation between the adiabatic and isocurvature modes, and for different spectral indices for the power in each mode and for their correlation. We do a likelihood analysis with 11 independent parameters. We discuss the effect of choosing the pivot scale for the definition of amplitude parameters. The upper limit for the isocurvature fraction is 18% around a pivot scale k = 0.01 Mpc^{-1}. For smaller pivot wavenumbers the limit stays about the same. For larger pivot wavenumbers, very large values of the isocurvature spectral index are favored, which makes the analysis problematic, but larger isocurvature fractions seem to be allowed. For large isocurvature spectral indices n_iso > 2 a positive correlation between the adiabatic and isocurvature mode is favored, and for n_iso < 2 a negative correlation is favored. The upper limit to the nonadiabatic contribution to the CMB temperature variance is 7.5%. Of the standard cosmological parameters, determination of the CDM density $ω_c$ and the sound horizon angle $θ$ (or the Hubble constant $H_0$) are affected most by a possible presence of a correlated isocurvature contribution. The baryon density $ω_b$ nearly retains its ``adiabatic value''.

Figures (21)

• Figure 1: The unit-amplitude component angular power spectra $\hat{C}^{\mathrm{ad}}_{l}$ (red), $\hat{C}^{\mathrm{iso}}_{l}$ (blue), and $\hat{C}^{\mathrm{cor}}_{l}$ (green) of Eqs. (\ref{['eq:ClTT']}) and (\ref{['eq:totCl']}) for the case of spectral indices $n_{\rm{ad}} = n_{\rm{iso}} = 1$ and other cosmological parameters representing median values of their marginalized likelihoods from our 11-parameter model. These curves would represent the relative contributions to the total $C_l$ for the case $\alpha = 0.5$, $\gamma = 1$, i.e., "equal" weights for the adiabatic and isocurvature contributions and a maximal positive correlation between them.
• Figure 2: The same as Fig. (\ref{['fig:cltt']}), but for (a) $\hat{C}_l^{\mathrm{TE}}$ and (b) the matter power spectrum $\hat{P}(k)$. We also show (c) the $\hat{C}_l^{\mathrm{TT}}$ of Fig. \ref{['fig:cltt']} with a logarithmic scale, so that the effect of changing the spectral indices can be readily estimated from the figure. The pivot scale $k_0 = 0.01 \hbox{Mpc}^{-1}$ becomes $k_0/h = 0.01418$Mpc$^{-1}$ for the parameter values used ($h = 0.7053$) for this plot.
• Figure 3: Marginalized likelihood functions for the standard cosmological parameters (i.e. those that exist in the adiabatic model). The solid (black) line is the likelihood in our 11-parameter model, the dashed (red) line is for the adiabatic model. Other line types show the effects of additional priors discussed in the text: dotted (blue) for Gaussian $\Omega_\Lambda = 0.70\pm0.04$, and dot-dashed (green) for Gaussian $\gamma = 0.0\pm0.02$.
• Figure 4: Marginalized likelihoods for two derived parameters, $\Omega_\Lambda$ and $H_0$. The line styles have the same meaning as in Fig. \ref{['fig:adiparam']}.
• Figure 5: Marginalized likelihoods for the parameters related to the isocurvature mode and correlation. We show the $4$ remaining independent parameters, $\alpha$, $\gamma$, $n_{\rm{iso}}$, $n_{\rm{ad2}}$. The solid (black) line is the full likelihood, other line types show the effects of additional priors discussed in the text: dotted (red) for Gaussian $\Omega_\Lambda > 0.82$, dashed (blue) for Gaussian $\Omega_\Lambda = 0.70\pm0.04$, and dot-dashed (green) for Gaussian $\gamma = 0.0\pm0.02$.