Dark Energy Probes in Light of the CMB

TL;DR

The paper analyzes how CMB observables, particularly self-calibrated standards like the sound horizon $s_*$ and the amplitude of initial fluctuations, fix the high-redshift expansion history and calibrate dark energy probes. It examines standard parameterizations of the dark energy equation of state $w(a)$, the role of recombination physics, and the internal consistency tests (e.g., damping scale) that validate the CMB-based standards. With the high-redshift standards fixed, most deviations due to dark energy appear at low redshift, making a percent-level measurement of the Hubble constant $H_0$ the most powerful complement to CMB data for constraining $w(a)$, while degeneracies require additional percent-level intermediate-redshift distance or growth measurements. The paper then forecasts how combining CMB priors with optical survey probes—baryon features, galaxy–galaxy lensing, cosmic shear, and cluster counts—can constrain $w_0$ and $w_a$, emphasizing the need for accurate distance standard calibration, mass calibration, and internal cross-checks to realize the full potential of CMB-driven dark energy probes.

Abstract

CMB observables have largely fixed the expansion history of the universe in the deceleration regime and provided two self-calibrated absolute standards for dark energy studies: the sound horizon at recombination as a standard ruler and the amplitude of initial density fluctuations. We review these inferences and expose the testable assumptions about recombination and reionization that underly them. Fixing the deceleration regime with CMB observables, deviations in the distance and growth observables appear most strongly at z=0 implying that accurate calibration of local and CMB standards may be more important than redshift range or depth. The single most important complement to the CMB for measuring the dark energy equation of state at z~0.5 is a determination of the Hubble constant to better than a few percent. Counterintuitively, with fixed fractional distance errors and relative standards such as SNe, the Hubble constant measurement is best achieved in the high redshift deceleration regime. Degeneracies between the evolution and current value of the equation of state or between its value and spatial curvature can be broken if percent level measurement and calibration of distance standards can be made at intermediate redshifts or the growth function at any redshift in the acceleration regime. We compare several dark energy probes available to a wide and deep optical survey: baryon features in galaxy angular power spectra and the growth rate from galaxy-galaxy lensing, shear tomography and the cluster abundance.

Figures (5)

• Figure 1: Recombination and the accuracy of CMB calibrations. (a) The current state-of-the-art in recombination involves a calibrated fudge in rescaling $\alpha_{B}$ for hydrogen in a 2 level atom to a multilevel atom. Shown is precision to which $\alpha_{B}$ will need to be calibrated so as to not introduce systematic errors that are larger than the cosmic variance out to a given $\ell$. (b) The damping tail contains a consistency check for recombination, here illustrated through a 5% variation in the fine structure constant $\alpha$ or a $\sim 10\%$ variation in $z_*$. The first 3 peaks measured by WMAP (points) determines the photon-baryon ratio $R_{*}$ and radiation matter ratio $r_{*}$ at recombination, here held fixed. The damping tail breaks the degeneracy and measures $z_*$ independently, here accurately analytically modeled through a change in the damping scale (dashed lines).
• Figure 2: Generalizing the sound horizon dark energy constraint. Given CMB constraints on the acoustic scale $\ell_{A}$, $\Omega_{m}h^{2}$ and $\Omega_{b}h^{2}$, CMB constraints can be applied to any set of dark energy parameters by propagation of errors. Shown here are WMAP constraints (a) in a 3D space of $\Omega_{m}$, $\Omega_{\rm DE}$, and $w_{\rm DE}$ and (b) projected onto the usual space of flat cosmologies.
• Figure 3: Deviations in the dark energy observables holding CMB observables ${\cal D}_{*}$ and $G_{*}$ fixed by varying $\Omega_{\rm DE}$ to compensate a variation in (a) a constant $w_{\rm DE}$; (b) $w_{a}=-dw/da$ at fixed $w(a=1)=w_{0}$; compensating variations which leave $H_0$ fixed (c) $\Delta w_{a}/\Delta w_{0}\approx -10/3$ (d) $\Delta w_{\rm DE}/\Delta \Omega_{T} \approx -15$. With fixed high-$z$ observables, the main deviations due to the dark energy equation of state appear as variations in the Hubble constant which can be measured at low redshift by absolute standards through ${\cal D}$, $H$ or at high redshift through relative standards $H_{0}{\cal D}$. The local value of the growth function $G_0=G(a=1)$ is useful in breaking the degeneracy left by variations at fixed $H_{0}$.
• Figure 4: Acoustic or baryon features in the galaxy angular power spectrum. (a) angular power spectrum at $z=1$ with $\Delta z=0.1$ for galaxies in halos above $M_{\rm th} = 10^{12.5} h^{-1} M_\odot$; (b) constraints on $w_{\rm DE}$ and $\Omega_{\rm DE}$ from $50 \le \ell \le 300$ and 10 angular spectra out to $z=1$ with Planck CMB priors; (c) constraints in the $w_{0}-w_{a}$ plane marginalized over $\Omega_{\rm DE}$.
• Figure 5: Dark energy constraints from (a) galaxy and lensing spectra (b) cluster abundance. 10 galaxy bins of $\Delta z=0.1$ out to $z=1$ and 4 shear bins of $\Delta z=0.25$ out to $z=1$ plus a $z \ge 1$ bin with shear noise corresponding to $\bar{n}=10$ gal arcmin$^{-2}$ and $\gamma_{\rm rms}=0.16$. Galaxy-galaxy and galaxy-shear power spectra constraints from $50 \le \ell \le 1000$ are compared with shear-shear spectra from $50 \le \ell \le 3000$. The cluster abundance is divided into the same bins as the galaxies for $M> 10^{14.2} h^{-1} M_\odot$ with a comparison between perfect mass calibration, marginalization over a power law mass-observable relation, and self-calibration through employing the sample variance as a measure of the mass dependent clustering of clusters.