Characterising the Primordial Cosmic Perturbations

TL;DR

The paper addresses whether primordial perturbations were purely adiabatic or contained isocurvature components by formulating a five-mode framework described by a matrix-valued power spectrum $P_{ij}({\bf k})$ that captures auto- and cross-correlations among adiabatic and four isocurvature modes. It uses Fisher-matrix forecasts for MAP and PLANCK, incorporating temperature and optimistic polarization, and analyzes models with 0–3 isocurvature modes to quantify parameter degradation and the ability to detect isocurvature signals. The main contributions are (i) a systematic eigenmode analysis of the reduced Fisher matrix to map parameter sensitivities, and (ii) quantitative forecasts showing that MAP alone cannot tightly constrain isocurvature amplitudes, while PLANCK with polarization can limit them to below ~10% of the adiabatic amplitude and recover key cosmological parameters to a few percent or better. The results highlight the critical role of polarization in disentangling primordial perturbation components and establishing robust cosmological inferences, while acknowledging simplifications such as neglecting tensors, assuming sub-dominant isocurvature components, and fixing isocurvature spectral shapes.

Abstract

The most general homogeneous and isotropic statistical ensemble of linear scalar perturbations which are regular at early times, in a universe with only photons, baryons, neutrinos, and a cold dark matter (CDM) component, is described by a 5x5 symmetric matrix-valued generalization of the power spectrum. This description is complete if the perturbations are Gaussian, and even in the non-Gaussian case describes all observables quadratic in the small perturbations. The matrix valued power spectrum describes the auto- and cross-correlations of the adiabatic, baryon isocurvature, CDM isocurvature, neutrino density isocurvature, and neutrino velocity isocurvature modes. In this paper we examine the prospects for constraining or discovering isocurvature modes using forthcoming MAP and PLANCK measurements of the cosmic microwave background (CMB) anisotropy. We also consider the degradation in estimates of the cosmological parameters resulting from the inclusion of these modes. In the case of MAP measurements of the temperature alone, the degradation is drastic. When isocurvature modes are admitted, uncertainties in the amplitudes of the mode auto-- and cross--correlations, and in the cosmological parameters, become of order one. With the inclusion of polarisation (at an optimistic sensitivity) the situation improves for the cosmological parameters but the isocurvature modes are still only weakly constrained. Measurements with PLANCK's estimated errors are far more constraining, especially so with the inclusion of polarisation. If PLANCK operates as planned the amplitudes of isocurvature modes will be constrained to less than ten per cent of the adiabatic mode and simultaneously key cosmological parameters will be estimated to a few per cent or better.

Figures (7)

• Figure 1: CMB anisotropy power spectra $l(l+1)C_l$ are plotted versus $l$. The three plots show cross-correlation power spectra as solid lines: adiabatic-baryon isocurvature density (lower), adiabatic-neutrino isocurvature density (middle) and adiabatic-neutrino isocurvature velocity (upper). The relevant auto-correlation spectra are also shown on each plot as follows: adiabatic perturbations (dot-dashed, magenta), baryon isocurvature (dotted, cyan), neutrino density isocurvature (short dashed, yellow), and neutrino velocity isocurvature (long dashed, green). All are assumed to have scale invariant underlying power spectra.
• Figure 2: As in Figure 1, but for the baryon isocurvature -neutrino isocurvature velocity correlation (lower), baryon isocurvature-neutrino isocurvature density correlation (middle) and neutrino isocurvature density-neutrino isocurvature velocity correlation (upper). On each plot the cross-correlation power spectrum is shown as a solid black line, with the corresponding auto-correlation power spectra denoted as in Figure 1.
• Figure 3: Illustration of the degeneracy problem. Including isocurvature modes renders the determination of cosmological parameters from the MAP satellite hazardous. Deviations from the fiducial adiabatic model in certain directions in the space of cosmological parameters and density perturbation amplitudes produce nearly degenerate $C_l$ spectra. Here we show the deviations $\delta l(l+1) C_l$ for each parameter change multiplied by the appropriate component of the eigenvector of $\hat{F}_{ij}$ with smallest eigenvalue (given in Table 8-M-TP). The dotted lines show the individual contributions as a function of $l$, and the solid line the sum after the normalisation has been projected out. The heavy dashed line shows the sum multiplied by 50 to render it more visible. The latter deviation is not measurable with MAP, but is with PLANCK, presumably because the latter is more sensitive to the high $l$ structure.
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