Cosmological-Parameter Determination with Microwave Background Maps

TL;DR

This paper develops a covariance-matrix framework to forecast how well a high-resolution CMB temperature map can determine key cosmological parameters under primordial adiabatic perturbations. By computing the CMB power spectrum $C_ℓ=C_ℓ^S+C_ℓ^T$ with a fast semi-analytic method and relating parameter uncertainties to the curvature of the likelihood, the authors show that geometry ($Ω$) and the cosmological constant ($Λ$) can be measured with percent-level precision, especially when priors are available, and that parameters like $Ω_b h^2$, $h$, $N_ν$, and $τ_{\rm reion}$ are also tightly constrained. They analyze the impact of experimental factors (sky coverage, beam, noise) and discuss the dependencies and degeneracies among parameters, including the inflationary observables $n_S$, $n_T$, and $r$, which become measurable with adequate priors and polarization data. The work highlights the transformative potential of all-sky CMB maps for testing inflation, determining the Universe’s geometry, and probing the early Universe, while outlining avenues for code development and future polarization analyses.

Abstract

The angular power spectrum of the cosmic microwave background (CMB) contains information on virtually all cosmological parameters of interest, including the geometry of the Universe ($Ω$), the baryon density, the Hubble constant ($h$), the cosmological constant ($Λ$), the number of light neutrinos, the ionization history, and the amplitudes and spectral indices of the primordial scalar and tensor perturbation spectra. We review the imprint of each parameter on the CMB. Assuming only that the primordial perturbations were adiabatic, we use a covariance-matrix approach to estimate the precision with which these parameters can be determined by a CMB temperature map as a function of the fraction of sky mapped, the level of pixel noise, and the angular resolution. For example, with no prior information about any of the cosmological parameters, a full-sky CMB map with $0.5^\circ$ angular resolution and a noise level of 15 $μ$K per pixel can determine $Ω$, $h$, and $Λ$ with standard errors of $\pm0.1$ or better, and provide determinations of other parameters which are inaccessible with traditional observations. Smaller beam sizes or prior information on some of the other parameters from other observations improves the sensitivity. The dependence on the the underlying cosmological model is discussed.

Figures (6)

• Figure 1: Predicted multipole moments for standard CDM and variants. The heavy curves in each graph are for a model with primordial adiabatic perturbations with $\Omega=1$, $\Lambda=0$, $n_S=1$, $\Omega_bh^2=0.01$, $h=0.5$, $\alpha=0$, and no tensor modes. The graphs show the effects of varying $\Omega$, $\Lambda$, $h$, $\tau_{\rm reion}=0$, and $\Omega_b h^2$ while holding all other parameters fixed. In the $\Omega$ panel, from left to right, the solid curves are for $\Omega=1$, $\Omega=0.5$, and $\Omega=0.3$. The curves in the $\Omega_b h^2$ panel are (from lower to upper) for $\Omega_bh^2=0.01$, $\Omega_bh^2=0.03$, and $\Omega_b h^2=0.05$. In the $h$ panel, the heavy curves is for $h=0.5$, while the other two curves are for $h=0.3$ (the upper light curve) and $h=0.7$ (the lower light curve). The curves in the $\Lambda$ panel are for (from lower to upper) $\Lambda=0$, $\Lambda=0.3$, and $\Lambda=0.7$.
• Figure 2: Simulated data that might be obtained with a CMB mapping experiment, for beam sizes of $0.3^\circ$ and $0.1^\circ$, and a noise level of $w^{-1}=2\times 10^{-15}$.
• Figure 3: The standard errors for $\Omega$, $\Lambda$, $\Omega_b h^2$, and $h$ that can be obtained with a full-sky mapping experiment as a function of the beam width $\theta_{\rm fwhm}$ for noise levels $w^{-1}=2\times10^{-15}$, $9\times10^{-15}$, and $4\times10^{-14}$ (from lower to upper curves). The underlying model is "standard CDM." The solid curves are the sensitivities attainable with no prior assumptions about the values of any of the other cosmological parameters. The dotted curves are the sensitivities that would be attainable assuming that all other cosmological parameters, except the normalization ($Q$), were fixed. The results for a mapping experiment which covers only a fraction $f_{\rm sky}$ of the sky can be obtained by scaling by $f_{\rm sky}^{-1/2}$ [c.f., Eq. (20)].
• Figure 4: Like Fig. 3, but for $\alpha$, $N_\nu$, $\tau_{\rm reion}$, and $n_S$.
• Figure 5: The standard errors on the inflationary observables, $n_S$, $n_T$, $r=Q_T^2/Q_S^2$, and $Q$, that can be obtained with a full-sky mapping experiment as a function of the beam width $\theta_{\rm fwhm}$ for noise levels $w^{-1}=2\times10^{-15}$, $9\times10^{-15}$, and $4\times10^{-14}$ (from lower to upper curves). The parameters of the underlying model are the "standard=CDM" values, except we have set $r=0.28$, $n_S=0.94$, and $n_T=-0.04$. The solid curves are the sensitivities attainable with no prior assumptions about the values of any of the other cosmological parameters. The dotted curves are the standard errors that would be attainable by fitting to only these four inflationary observables and assuming all other cosmological parameters are known. (Note that this differs from the dotted curves in Fig. 4.) The results for a mapping experiment which covers only a fraction $f_{\rm sky}$ of the sky can be obtained by scaling by $f_{\rm sky}^{-1/2}$ [c.f., Eq. (20)].