Determining the Hubble Constant without the Sound Horizon Scale: Measurements from CMB Lensing

TL;DR

The paper devises an $r_s$-independent method to infer the Hubble constant $H_0$ using the CMB lensing power spectrum, by exploiting its sensitivity to the horizon scale at matter-radiation equality through the projected quantity $L_{ m eq}=k_{ m eq}\chi_*$. It shows that, with conservative priors on $A_s$ and external measurements of $\Omega_m$, one can constrain $H_0$ without referencing the sound horizon scale, obtaining $H_0=73.5\pm5.3$ km s$^{-1}$ Mpc$^{-1}$ from Planck lensing and Pantheon-like data, consistent with both early- and late-time measurements within errors. Forecasts indicate that future CMB surveys could reach about $\sigma(H_0)\sim3$ km s$^{-1}$ Mpc$^{-1}$ when combined with external $\Omega_m$ and $A_s$ priors, though improvements from lensing alone are limited by cosmic variance; galaxy power spectra offer a potential path to tighter, still-$r_s$-independent constraints. The work also argues that equality- and sound-horizon scales respond differently to new physics, so measuring $k_{ m eq}$-based $H_0$ provides a complementary diagnostic of early-universe modifications.

Abstract

Measurements of the Hubble constant, $H_0$, from the cosmic distance ladder are currently in tension with the value inferred from Planck observations of the CMB and other high redshift datasets if a flat $Λ$CDM cosmological model is assumed. One of the few promising theoretical resolutions of this tension is to invoke new physics that changes the sound horizon scale in the early universe; this can bring CMB and BAO constraints on $H_0$ into better agreement with local measurements. In this paper, we discuss how a measurement of the Hubble constant can be made from the CMB without using information from the sound horizon scale, $r_s$. In particular, we show how measurements of the CMB lensing power spectrum can be used to place interesting constraints on $H_0$ when combined with measurements of either supernovae or galaxy weak lensing, which constrain the matter density parameter. The constraints arise from the sensitivity of the CMB lensing power spectrum to the horizon scale at matter-radiation equality (in projection); this scale could have a different dependence on new physics than the sound horizon. From an analysis of current CMB lensing data from Planck and Pantheon supernovae with conservative external priors, we derive an $r_s$-independent constraint of $H_0 = 73.5\pm 5.3$ km/s/Mpc. Forecasts for future CMB surveys indicate that improving constraints beyond an error of $σ(H_0) = 3$ km/s/Mpc will be difficult with CMB lensing, although applying similar methods to the galaxy power spectrum may allow for further improvements.

Figures (6)

• Figure 1: Illustration of the effect of changes to the Hubble rate at different redshifts on the sound horizon scale at CMB last scattering (black solid line at left) and the matter-radiation equality scale (blue solid line at right). Dashed and dotted lines indicate the redshifts of CMB last-scattering and matter-radiation equality. This plot indicates that changes to the energy density in the decade of redshift before recombination would generically have a significantly different impact on these two scales.
• Figure 2: Forecast parameter constraints from CMB-S4-like measurements of the CMB lensing power spectrum. Information about $H_0$ enters via $L_{\rm eq} \equiv k_{\rm eq}\chi_* \propto \Omega_m^{0.6} h$; the black dashed curve shows the expected $h \propto \Omega_m^{-0.6}$ degeneracy direction. Degeneracy between $L_{\rm eq}$ and $A_s$ limits the ability of the CMB lensing power spectrum to constrain $L_{\rm eq}$, and thus $H_0$. We have marginalized over $\Omega_{\nu}h^2$ with the priors from Table \ref{['tab:priors']} when generating this figure; both parameters are prior dominated.
• Figure 3: Forecast constraints for a CMB-S4 measurement of the CMB lensing power spectrum upon including external $A_s$ and $\Omega_m$ information. By imposing a conservative, CMB-motivated (but $r_s$ independent) prior on $A_s$ we break the $A_s$-$L_{\rm eq}$ degeneracy, leading to a constraint on $L_{\rm eq} \equiv k_{\rm eq} \chi_{*}$ (blue curves; compare to Fig. \ref{['fig:no_priors']}). Since $L_{\rm eq} \propto \Omega_m^{0.6} h$, the resultant constraints on $\Omega_m$ and $H_0$ are degenerate (the black dashed curve illustrates the expected degeneracy). By including an additional $\Omega_m$ constraint from supernovae we break the $\Omega_m$-$H_0$ degeneracy, and obtain a constraint on $H_0$ (red curve, with marginalized parameter uncertainties reported along the diagonal). As an alternative to a supernova constraint on $\Omega_m$, we also consider adding information from an LSST-like measurement of cosmic shear (orange curve); in combination with CMB lensing, galaxy lensing also provides $\Omega_m$ information which serves to break the $H_0$-$\Omega_m$ degeneracy. Because of large-scale information in the CMB lensing power spectrum, our constraints are not very sensitive to the $A_s$ prior. If this prior is completely removed, we still obtain a constraint on $H_0$ (grey dashed curve).
• Figure 4: The linear matter power spectrum computed with CAMB (red solid) and with the toy model of Eq. \ref{['eq:toy_pk']} (blue dashed). In this toy model, we adopt the BBKS transfer function BBKS, and model the impact of baryons on the matter power with a step-like suppression on scales smaller than $r_s$. We use this toy model to explore how the CMB lensing power spectrum constrains $H_0$, and to confirm that our constraints are not informed by information about the sound horizon scale, $r_s$.
• Figure 5: Constraints on $H_0$ from current Planck CMB lensing measurements Plancklensing2015 before (orange) and after (blue) imposing the supernova $\Omega_m$ constraint Scolnic:2018 and the conservative, CMB-motivated (but $r_s$ independent) $A_s$ prior. The resultant constraint of $H_0 = 73.5\pm5.3$ km/s/Mpc is independent of the sound horizon scale, $r_s$. The black dashed curve illustrates the expected degeneracy between $h$ and $\Omega_m$ when information about $H_0$ enters via $L_{\rm eq} \equiv k_{\rm eq} \chi_*$. The grey dashed curves indicate the result obtained with an alternate choice of priors, see discussion in text.