High-redshift physics from the acoustic scale

Abstract

We present a simplified and general description of the high-redshift information in acoustic scale measurements from the cosmic microwave background and large-scale structure. The transverse distance interval between photon--baryon decoupling and a late epoch in the matter era provides an analytically tractable summary statistic thereof and a general diagnostic of the current tension between the Dark Energy Spectroscopic Instrument and the CMB. We show that this "matter-era distance excess" is unlikely to be explained by modified dynamics at low redshift. We then analytically derive the matter-era distance interval's sensitivity to new physics at high redshift, including nonstandard recombination, nonminimal dark matter dynamics, and spatial curvature; in particular, we explain how this observable represents a direct geometric measurement of (and underlies the current incompatibility with) neutrino masses. Finally, we demonstrate that phenomenological models of dynamical dark energy mediate the matter-era distance excess in a manner reliant on their unphysical, extrapolated behavior at high redshift. Invoking alternative explanations of the excess removes the CMB's contribution to the evidence for these models; the residual preference of around $1.7σ$ mostly derives from DESI's two lowest-redshift measurements of the Alcock--Paczynski distortion, without which it drops to $0.5 σ$.

Figures (11)

• Figure 1: Robustness of the jointly inferred distance to DESI DR2's highest-redshift BAO tracer (Lyman-$\alpha$ at $z_m = 2.330$DESI:2025zpo) to increasing dynamical freedom of dark energy (by panel as labelled). The Lyman-$\alpha$ likelihood (whose $1$ and $2 \sigma$ mass levels appear in black) combines with the predictions of each model as calibrated to all lower-redshift DESI data (red) to yield joint measurements (blue) of the transverse distance $D_M(a_m) / r_\mathrm{d}$ with more than four times the precision of the direct one. The Lyman-$\alpha$ sample's transverse measurement is in fact almost completely irrelevant: its most important role is rather to constrain the expansion rate [$D_H(a_m) / r_\mathrm{d}$] at high redshift and therefore how much distance photons accumulate from lower redshift (as measured by the observations at lower redshift).
• Figure 2: Measurements of the late-time matter density $\omega_m r_\mathrm{d}^2$ from the acoustic scale and the matter-era distance interval $D_M(a_\mathrm{d}, a_m) / r_\mathrm{d} \equiv D_M(a_\mathrm{d}) / r_\mathrm{d} - D_M(a_m) / r_\mathrm{d}$. The $1$ and $2 \sigma$ mass levels of the joint distribution over $\omega_m r_\mathrm{d}^2$ and $D_M(a_m) / r_\mathrm{d}$ deriving from DESI DR2 and from all acoustic scale data (i.e., DESI and the CMB's geometric information) appear in blue and red, respectively. Black and gold regions depict results computed analytically with a minimal prior as described in the main text, with the MEDI respectively drawn from a broad, uniform distribution and computed as the difference of the CMB's measured $1 / \theta_\perp(a_\mathrm{d})$ and DESI's measured $D_M(a_m) / r_\mathrm{d}$ (horizontal blue bands); the latter represents an acoustic scale measurement agnostic to dynamics at $a > a_m$. The right axis depicts the MEDI corresponding to $D_M(a_m) / r_\mathrm{d}$, taking the CMB's mean $1 / \theta_\perp(a_\mathrm{d})$ (neglecting its small uncertainty). Comparing the marginal measurements of the matter density demonstrates the critical role of the MEDI and the relative unimportance of DESI's matter density information from late times ($a > a_m$).
• Figure 3: Quantification of the matter-era distance excess that underlies the BAO--CMB tension, i.e., between CMB predictions and acoustic scale measurements of the matter-era distance interval. Blue posteriors depict $\Lambda$CDM predictions calibrated by CMB temperature and polarization data from Planck PR3, ACT DR6, and SPT-3G D1, either fixing the neutrino mass sum to zero, marginalizing it over nonnegative values, or marginalizing it over values compatible with neutrino oscillations; these results depend only on nongeometric information. Red/orange results indicate measurements from the acoustic scale, with the darkest red indicating the constraint in standard $\Lambda$CDM from DESI DR2 data and the CMB's geometric information [$1 / \theta_\perp(a_\mathrm{d})$] alone, in noticeable tension with the prediction from the CMB's nongeometric information. Lighter shades display more conservative results that assume nothing about the expansion history at $z > z_m$, instead taking $D_M(a_\mathrm{d}) / r_\mathrm{d}$ from the CMB's fully marginalized measurement of $1 / \theta_\perp(a_\mathrm{d})$. The low-redshift expansion history is then modeled under varying assumptions in order to combine all DESI data to measure the "distance to matter domination" $D_M(a_m) / r_\mathrm{d}$, taking a cosmological constant, a dark energy fluid with fixed equation of state, or one with equation of state evolving linearly with $a$. These results thus represent measurements of the matter-era distance interval that are independent of physics between $a_m$ and $a_\mathrm{d}$, which remain in tension with the CMB's predictions thereof, especially when massive neutrinos are properly accounted for.
• Figure 4: Concordance of CMB predictions and acoustic scale measurements in a selection of scenarios modifying physics before decoupling: varying the effective number of neutrino species $N_\mathrm{eff}$ (red), the early-time value of the fine-structure constant (gold), or both (blue). Middle and bottom panels display the joint posterior over each parameter and the "distance to matter domination" $D_M(a_m) / r_\mathrm{d}$ and the matter-era distance interval $D_M(a_\mathrm{d}, a_m) / r_\mathrm{d}$, respectively; up to the small effect of $\Delta N_\mathrm{eff}$ on the inferred $D_M(a_\mathrm{d}) / r_\mathrm{d}$, both rows contain the same information. Horizontal shaded regions depict the $1$ and $2 \sigma$ limits of acoustic scale measurements from fig:distance-to-md,fig:medi-tension. Current CMB data alone prefer $N_\mathrm{eff}$ below the Standard Model prediction, which exacerbates the matter-era distance tension, but prefer $\alpha_i$ larger than the present-day value by $\gtrsim 2 \sigma$, which alleviates the tension. Results fix the neutrino mass sum to zero to isolate early-time effects on extrapolated distances; since CMB predictions at best match acoustic scale measurements in each scenario, none can fully reconcile the matter-era distance excess for arbitrarily large $M_\nu$. Varying the helium yield to absorb the impact of $\Delta N_\mathrm{eff}$ on diffusion has no substantive impact on acoustic scale predictions.
• Figure 5: Posterior distribution over the matter-era distance interval and the density in baryons and CDM, the drag horizon, and the neutrino mass sum. Results that vary $M_\nu$ and exclude low-$\ell$ polarization data are displayed as scatter colored by the optical depth, with the $1$ and $2 \sigma$ mass levels thereof that include all CMB data appearing in black. The same mass levels of posteriors that fix $M_\nu = 0$ and either exclude and include low-$\ell$ polarization data appear in red and blue, respectively. Horizontal shaded regions depict the $1$ and $2 \sigma$ limits of the MEDI as measured by the acoustic scale (see fig:medi-tension).