Matter-Era Distance Excess

• The paper identifies a surplus in the dimensionless distance (MEDI) from decoupling to z=2.33, exceeding ΛCDM predictions by about 1.7–2.6σ.
• It demonstrates that the excess is primarily sensitive to matter density, neutrino masses, and high-redshift physics, with minimal dark-energy influence.
• Robust acoustic-scale analyses combining BAO and CMB measurements yield a direct geometric probe challenging conventional post-decoupling expansion models.

Matter-era distance excess denotes an observed surplus in the dimensionless distance accumulated between a high-redshift anchor and a later epoch in the matter era, relative to a calibrated baseline prediction. In the usage developed in “High-redshift physics from the acoustic scale” (Weiner, 18 Mar 2026), the relevant quantity is the matter-era distance interval (MEDI), usually taken between photon–baryon decoupling and a late matter-era redshift such as $z_m=2.330$. In that formulation, the excess is the difference between the MEDI inferred from acoustic-scale data and the MEDI predicted by the CMB under a specified post-decoupling model; it can equivalently be read as a deficit in inferred matter density. Related literature uses similar language for broader late-time distance anomalies, including low-redshift cosmographic deviations and local expansion-rate excesses, but these are distinct constructions (Shafieloo et al., 2011, Dumin, 2018).

1. Definition and geometric content

In a flat universe, the comoving distance from scale factor $a$ to today is written as

$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$

with

$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$

The MEDI is the interval $\chi(a,a_m)$, where $a$ is typically the decoupling epoch and $a_m$ is chosen in the matter era. For curved universes, the transverse comoving distance is

$$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$

and the interval is $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$ (Weiner, 18 Mar 2026).

In a matter+radiation background with matter density $\omega_m$ and equality scale factor $a$0, the MEDI from photon–baryon decoupling at $a$1 to $a$2 is

$a$3

Including a cosmological constant gives the approximation

$a$4

accurate at $a$5 for $a$6. The operational choice in the acoustic-scale analysis is $a$7, the highest redshift at which DESI BAO currently provides good distance measurements. At that redshift, $a$8, producing only a $a$9 effect on $$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$0, so the interval is dominantly set by matter and radiation (Weiner, 18 Mar 2026).

The observable form used in acoustic analyses is the distance interval in units of the drag horizon $$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$1: $$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$2

Quantity	Definition	Role
$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$3	$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$4	Transverse acoustic scale
$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$5	$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$6	Radial acoustic scale
$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$7	$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$8	Isotropic BAO combination
$$\chi(a) = \int_{a}^{1} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})} = \chi(a,a_m) + \chi(a_m),$$9	$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$0	Measured MEDI

The matter-era distance excess is then

$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$1

When $$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$2, photons have travelled a longer dimensionless distance between decoupling and $$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$3 than expected (Weiner, 18 Mar 2026).

2. Acoustic-scale reconstruction and the present discrepancy

The acoustic-scale construction combines a CMB determination of the distance to decoupling with BAO determinations of the distance to a late matter-era epoch. The CMB gives

$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$4

while DESI DR2 Lyman-$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$5, marginalizing over $$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$6, gives

$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$7

and combining DR2 BAO with DR1 AP gives

$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$8

These imply a direct estimate of the MEDI through

$$\chi(a_1,a_2) \equiv \int_{a_1}^{a_2} \frac{d\ln \tilde{a}}{\tilde{a} H(\tilde{a})}.$$9

The paper reports a high-$\chi(a,a_m)$0 agnostic MEDI of $\chi(a,a_m)$1, while the ΛCDM CMB prediction is $\chi(a,a_m)$2 (Weiner, 18 Mar 2026).

This discrepancy is the specific sense in which the term “matter-era distance excess” is used in the current acoustic-scale literature. The same analysis states that in minimal ΛCDM with zero neutrino mass the measured MEDI exceeds the CMB prediction by about $\chi(a,a_m)$3, and that marginalizing over neutrino masses compatible with oscillations raises the discrepancy to $\chi(a,a_m)$4. The paper also gives an analytic estimate for the neutrino-mass dependence of the tension,

$\chi(a,a_m)$5

which yields about $\chi(a,a_m)$6 for the minimum normal hierarchy and $\chi(a,a_m)$7 for the minimum inverted hierarchy (Weiner, 18 Mar 2026).

A central claim of the analysis is that this excess is unlikely to be explained by modified dynamics at low redshift. The inferred $\chi(a,a_m)$8 changes very little when one moves among Λ, constant-$\chi(a,a_m)$9, and $a$0 parameterizations, and the uncertainty remains essentially unchanged. In this sense, the measured MEDI is described as a robust acoustic-scale summary statistic rather than a by-product of late-time dark-energy fitting freedom (Weiner, 18 Mar 2026).

3. Dependence on matter density, neutrino masses, curvature, and high-redshift physics

In the matter+radiation approximation, the MEDI is primarily a matter-density observable. The dominant logarithmic sensitivity is

$a$1

whereas the sensitivities to $a$2 and $a$3 are small: $a$4 for $a$5. The dark-energy sensitivity at $a$6 is also small: $a$7 corresponding to a $a$8 effect at that redshift (Weiner, 18 Mar 2026).

The response to a general perturbation in the background density is written as

$a$9

or, equivalently,

$a_m$0

The weighting grows roughly as $a_m$1, so the MEDI is most sensitive to late times within the matter era (Weiner, 18 Mar 2026).

Several classes of high-redshift or post-decoupling physics therefore imprint directly on the MEDI. Massive neutrinos shorten the interval once they become nonrelativistic; the analysis presents the MEDI as a direct geometric measurement of the late-time matter contribution from neutrino masses. Decaying dark matter decreases the matter density with time and increases the MEDI. Scalar-mediated dark forces can make matter dilute faster than $a_m$2, again increasing the interval. Spatial curvature affects both the Friedmann evolution and the $a_m$3 conversion; for the DESI $a_m$4, the dominant geometric effect is approximately

$a_m$5

Modified recombination changes the predicted MEDI through the dependence $a_m$6, while extra radiation is comparatively inefficient because $a_m$7 remains nearly invariant when $a_m$8 and $a_m$9 are fixed (Weiner, 18 Mar 2026).

This hierarchy of sensitivities explains why the MEDI is presented as a diagnostic of high-redshift physics. In the paper’s formulation, it probes nonstandard recombination, nonminimal dark matter dynamics, and spatial curvature more directly than it probes conventional low-$$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$0 dark-energy variations (Weiner, 18 Mar 2026).

4. Low-redshift reinterpretations, dynamical dark energy, and cosmographic degeneracy

Phenomenological dynamical dark-energy models can mediate the acoustic-scale matter-era distance excess, but the paper argues that they do so through unphysical extrapolation to high redshift. In the Chevallier–Polarski–Linder parameterization,

$$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$1

the best-fitting DESI+CMB solutions typically favor $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$2. In that limit, the high-redshift dark-energy density becomes negligible, erasing the already small Λ contribution to the MEDI. The paper therefore states that phenomenological models of dynamical dark energy mediate the excess in a manner reliant on their unphysical, extrapolated behavior at high redshift (Weiner, 18 Mar 2026).

The same analysis further states that invoking alternative explanations of the excess removes the CMB’s contribution to the evidence for these models. The residual preference of around $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$3 mostly derives from DESI’s two lowest-redshift measurements of the Alcock–Paczynski distortion, and without those measurements it drops to $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$4. In that framework, the matter-era distance excess is not merely another way of restating a generic $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$5 anomaly; it is the specific ingredient that gives CMB+DESI combinations leverage on $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$6 fits (Weiner, 18 Mar 2026).

A broader late-time perspective is provided by “Cosmographic Degeneracy” (Shafieloo et al., 2011). That paper shows that distances alone permit a large 2D region in the $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$7 plane, because $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$8, curvature $$D_M \equiv \chi \,\mathrm{sinc}\!\left(\frac{\chi}{R_k}\right),\quad R_k^2 = -1/(\omega_k H_{100}^2),$$9, and a nearly arbitrary $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$0 can reproduce the same $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$1 out to a chosen $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$2. In that framework, a several-percent distance excess at $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$3–3 does not uniquely identify new physics: it can be absorbed by lower $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$4, nonzero curvature, or a suitably chosen $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$5. The combination of distances with $D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$6, gravitational lensing, or other large-scale structure data is therefore described as essential to determining a robust cosmological model (Shafieloo et al., 2011).

5. Related but distinct formulations

The phrase can also refer to physically different late-time effects. In “Can the Dark-Matter Deficit in the High-Redshift Galaxies Explain the Persistent Discrepancy in Hubble Constants?” (Dumin, 2018), the relevant anomaly is a local excess in the inferred expansion rate rather than an acoustic-scale MEDI. The paper starts from the discrepancy between

$D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$7

or even

$D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$8

and the global CMB-inferred range

$D_M(a_1,a_2)\equiv D_M(a_1)-D_M(a_2)$9

Within its simplified local Friedmann-like treatment,

$\omega_m$0

and matching a $\omega_m$1 excess requires

$\omega_m$2

about twice the global $\omega_m$3. This is presented as a local matter-era expansion-rate excess generated by spatial variation in dark matter density, not as the high-redshift acoustic-scale interval emphasized in the MEDI literature (Dumin, 2018).

Related early-Universe studies use the matter-era language to discuss horizon and scale modifications rather than a DESI–CMB distance interval. “Spinning primordial black holes formed during a matter-dominated era” analyzes an early matter-dominated background with $\omega_m$4, $\omega_m$5, and a cosmological horizon at $\omega_m$6 identified by $\omega_m$7 (Jong et al., 2023). “Neutrino Portal to FIMP Dark Matter with an Early Matter Era” introduces a pre-BBN early matter-dominated era with $\omega_m$8 in the isentropic phase and $\omega_m$9 during entropy production, altering the age–temperature and horizon–temperature relations relative to radiation domination (Cosme et al., 2020). These are scale and causal-horizon effects associated with matter domination, but they are not the same observable as the acoustic-scale matter-era distance excess.

6. Independent measurements and observational tests

Independent geometric probes currently do not show a statistically significant matter-era distance excess. “Measurement of a Cosmographic Distance Ratio with Galaxy and CMB Lensing” defines

$a$00

a purely geometric ratio in which the halo mass profile cancels for thin lens slices. The measured combined value is

$a$01

to be compared with the Planck best-fit flat $a$02CDM prediction

$a$03

The paper characterizes the difference as entirely consistent with noise and explicitly states that there is no evidence for a significant excess or deficit in distances across the matter-dominated era (Miyatake et al., 2016).

An independent matter-era ruler is the equality horizon measured from the turnover of the matter power spectrum. “Measurement of the matter-radiation equality scale using the extended Baryon Oscillation Spectroscopic Survey Quasar Sample” reports

$a$04

corresponding to

$a$05

Combining that measurement with Pantheon gives

$a$06

and combining with eBOSS BAO gives

$a$07

The paper states that these results are entirely consistent with Planck and BAO within current errors and that there is no statistically significant matter-era distance excess, although the central values lean toward the locally high $a$08 (Bahr-Kalus et al., 2023).

Future tests are expected to sharpen the distinction between genuine matter-era anomalies and model degeneracy. The lensing-ratio paper projects $a$09 precision with next-generation galaxy and CMB surveys (Miyatake et al., 2016). The turnover-scale analysis forecasts $a$10 precision on the equality scale for DESI QSO, $a$11–$a$12 for MSE ELG and LBG samples, and $a$13 for MegaMapper, with corresponding $a$14 uncertainties falling to a few $a$15 (Bahr-Kalus et al., 2023). In parallel, the cosmographic-degeneracy analysis implies that distances alone will remain insufficient: robust interpretation requires joint constraints from distances, $a$16, weak lensing, redshift-space distortions, and other growth-sensitive observables (Shafieloo et al., 2011).

The term therefore names a specific geometric discrepancy only in its modern acoustic-scale usage. In that sense, it is a summary of the distance interval between decoupling and a late matter-era epoch, measured in units of $a$17, and compared to a CMB-calibrated prediction. Its importance lies in the fact that this interval is dominantly sensitive to matter density and post-decoupling high-redshift physics, only weakly sensitive to standard low-$a$18 dark energy, and closely tied to current discussions of the DESI–CMB inconsistency and the geometric implications of neutrino masses (Weiner, 18 Mar 2026).