Arguments against using $h^{-1}{\rm Mpc}$ units in observational cosmology

TL;DR

The paper argues that $h^{-1}{\\rm Mpc}$ units complicate the interpretation of density fluctuations by tying amplitude to the fiducial Hubble parameter through $\\sigma_8$. It proposes using $\\sigma_{12}$, the rms variance on 12 Mpc scales, as a robust amplitude descriptor and advocates describing growth via $f\\sigma_{12}$ to remove $h$-dependencies. Through analyses of power-spectrum normalization, AP distortions, and redshift-space distortions, the authors show improved cross-dataset consistency (Planck, DES, BOSS) and reduced apparent tensions when reframed in terms of $\\sigma_{12}$ and $f\\sigma_{12}$. Adopting these conventions would clarify cosmological inferences across redshifts and prevent misleading conclusions about tensions in the standard model.

Abstract

It is common to express cosmological measurements in units of $h^{-1}{\rm Mpc}$. Here, we review some of the complications that originate from this practice. A crucial problem caused by these units is related to the normalization of the matter power spectrum, which is commonly characterized in terms of the linear-theory rms mass fluctuation in spheres of radius $8\,h^{-1}{\rm Mpc}$, $σ_8$. This parameter does not correctly capture the impact of $h$ on the amplitude of density fluctuations. We show that the use of $σ_8$ has caused critical misconceptions for both the so-called $σ_8$ tension regarding the consistency between low-redshift probes and cosmic microwave background data, and the way in which growth-rate estimates inferred from redshift-space distortions are commonly expressed. We propose to abandon the use of $h^{-1}{\rm Mpc}$ units in cosmology and to characterize the amplitude of the matter power spectrum in terms of $σ_{12}$, defined as the mass fluctuation in spheres of radius $12\,{\rm Mpc}$, whose value is similar to the standard $σ_8$ for $h\sim 0.67$.

Figures (4)

• Figure 1: Panel $a)$: Linear matter power spectra at $z=0$ of three $\Lambda$CDM models defined by identical values of $\omega_{\rm b}$, $\omega_{\rm c}$, $\omega_{\nu}$, $A_{\rm s}$ and $n_{\rm s}$, and varying $h$, expressed in $h^{-1}{\rm Mpc}$ units. Panel $b)$: The same power spectra of panel $a$) shown in ${\rm Mpc}$ units. Panel $c)$: The power spectra of the same models of panel $b$) but with their values of $A_{\rm s}$ adapted to produce the same value of $\sigma_{12}$. Panel $d)$: Nonlinear matter power spectra corresponding to the same models of panel $c$).
• Figure 2: Two-dimensional 68% and 95% constraints recovered from Planck (green), the $3\times 2$pt analysis of DES (blue), and BOSS (orange) under the assumption of a $\Lambda$CDM cosmology on the parameters $\Omega_{\rm m}$ -- $\sigma_8$ [panel $a$)], $\Omega_{\rm m}$ -- $S_8=\sigma_8\left(\Omega_{\rm m}/0.3\right)^{0.5}$ [panel $b$)], $\omega_{\rm m}$ -- $\sigma_{12}$ [panel $c$)], and $\omega_{\rm m}$ -- $S_{12}=\sigma_{12}\left(\omega_{\rm m}/0.14\right)^{0.4}$ [panel $d$)].
• Figure 3: Posterior distributions of the dimensionless Hubble parameter $h$ [panel $a$)] and the reference scale $(8/h)\,{\rm Mpc}$ where $\sigma_8$ is measured [panel $b$)] recovered from DES, BOSS and Planck under the assumption of a $\Lambda$CDM universe.
• Figure 4: Panel $a$): constraints on $b\sigma_8(z)$ and $f\sigma_8(z)$ derived from synthetic Legendre multipoles $P_{\ell=0,2,4}(k)$. The contours correspond to the cases in which $A_{\rm s}$ and $h$ are fixed ($h=0.67$ orange, $h=0.54$ pink and $h=0.8$ gray), when $A_{\rm s}$ is varied and $h$ is fixed (green), and when both are varied (blue). Panel $b$): same constraints as panel $a$) but expressed in terms of $b\sigma_{12}(z)$ and $f\sigma_{12}(z)$.