The Hubble Tension in Light of the Symmetry of Scale Invariance

TL;DR

This work investigates whether scale-invariant vacuum (SIV) cosmology can resolve the Hubble tension by linking early-universe thermodynamics to present-day expansion. The authors derive analytic H(z), t(z), and T(z) in SIV, and constrain the matter density to $\Omega_m\approx0.18$–$0.20$ from SN Ia data and age considerations, finding that $H_0$ can be anchored at $\approx74$ km s$^{-1}$ Mpc$^{-1}$ without tension when evaluated within SIV. A key result is that the same recombination state (temperature $T\approx3000$ K) corresponds to different present-day $H_0$ values in ΛCDM and SIV due to their distinct expansion histories, suggesting the tension may reflect neglected scale-invariance effects. The study also shows that the CMB acoustic scale and high-redshift observations can be made compatible within SIV for the appropriate $\Omega_m$, supporting a viable alternative to ΛCDM that preserves consistency across early- and late-Universe data. Overall, the paper argues that incorporating scale invariance provides a natural mechanism to reconcile local and CMB-derived measurements of the Hubble constant while remaining compatible with key cosmological observables.

Abstract

When the expansion rate of the Universe at recombination is used to infer the present expansion rate $H_0$, the value derived in the $Λ$CDM model, $H_0=67.4$ km/s/mpc, is about in 6$\, σ$ tension with the value measured locally, $H_0=74$ km/s/mpc. In this work, we consider instead the expansion history in the context of the symmetry of scale-invariant vacuum (SIV model). We first perform two major cosmological tests: the Hubble diagram for type-Ia supernovae and the fundamental relation between $H_0$, the age of the Universe, and the total density of matter, $Ω_m$. This allows us to constrain $Ω_m$ in SIV, with both tests giving the best agreement for $Ω_m \simeq 0.20$. We then study the physical connections of the dynamical and thermal states of the Universe at recombination with the present Hubble constant, $H_0$, and the present temperature, $T$, in the $Λ$CDM and SIV contexts. We find that, in SIV, the properties at recombination may be conveyed to the present ones ($T=2.726$ and $H_0$ at $z=0$) without any tension, indicating $H_0=74$ km/s/mpc in spite of the anchoring on the CMB. This is due to the slightly different expansion and temperature histories of the two cosmological models. Importantly, this happens to occur for $Ω_m \simeq 0.20$, as constrained in SIV with supernovae and cosmic age. This suggests that the Hubble tension currently found between $H_0$ values in the early and late Universe may simply be the result of $ΛCDM$ ignoring the small but still measurable effects of scale invariance.

Figures (5)

• Figure S1: Temperature variations in the CMB and of the matter density as a function of redshift $z$ for the $\Lambda$CDM and SIV models both with $\Omega_{\mathrm{m}}=0.30$. One notices that for SIV the recombination occurs a higher $z$. A mean value of $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ is adopted.
• Figure S2: Bayesian fit to the supernovae sample compiled by Pierel2024. We fit for $\Omega_{\mathrm{m}}$ and for an arbitrary constant, $K$, on the distance modulus without attempting to constrain $H_0$. We use a flat prior on $\Omega_{\mathrm{m}}$ such that $\Omega_{\mathrm{m}}$$\in [0, 0.5]$. The data points are exactly the same as in Pierel2024.
• Figure S3: Relations predicted in SIV between $H_0$ and $\Omega_{\mathrm{m}}$ for different values of the age of the Universe. The blue dashed line shows the curve for the standard age value of 13.92 $\times 10^9$ years, shown as the blue dashed curve Valcin25. The measured local value of $H_0$ and associated 1$\sigma$ error bar is shown in green (SH0ES) and the CMB value from Planck is shown in red, illustrating the tension between the late and early Universe values. For the SIV value of $\Omega_{\mathrm{m}}$$\sim$0.2 measured with type Ia supernovae, and given the estimated age of the Universe, $H_0 \sim 74$ km s$^{-1}$ Mpc$^{-1}$ is clearly favored.
• Figure S4: Expansion rate of the Universe as a function of redshift in $\Lambda$CDM (red) for $\Omega_{\mathrm{m}}$ = 0.315 and in SIV (blue) for $\Omega_{\mathrm{m}}$ = 0.2, as also found with supernovae and the $H_0$ - $\Omega_{\mathrm{m}}$ - age relation. The $\Lambda$CDM curve for $\Omega_{\mathrm{m}}$ = 0.315 connects $H_0= 67.4$ km s$^{-1}$ Mpc$^{-1}$ at $z=0$ to $H(z^*_{\mathrm{rec}}) = 1.381 \times 10^6$ km s$^{-1}$ Mpc$^{-1}$ at $z=1099.51$ where $T=3000$ K. The SIV curve for $\Omega_{\mathrm{m}}$ =0.20 connects $H_0= 74$ km s$^{-1}$ Mpc$^{-1}$ at $z=0$ to $H(z^*_{\mathrm{rec}}) = 1.382 \times 10^6$ km s$^{-1}$ Mpc$^{-1}$ at $z=1438.08$ where $T=3000$ K. Thus, identical thermal and dynamical states at recombination are connected to different $H_0$ values by $\Lambda$CDM and SIV models for their respective best observationally supported $\Omega_{\mathrm{m}}$ value. The correspondence of thermal and dynamical states is also present over a large range of high $z$ values as shown at $H(z) = 10^5$ and $10^4$ km s$^{-1}$ Mpc$^{-1}$; it progressively disappears at lower $z$. This correspondence only happens for $\Omega_{\mathrm{m}}$ = 0.20 in SIV. The points in green give $z^*_{\mathrm{rec}}$ and $H(z^*_{\mathrm{rec}})$ for other values of $\Omega_{\mathrm{m}}$ in SIV.
• Figure S5: Expansion rate of the Universe as a function of temperature, T, in $\Lambda$CDM (red) for$\Omega_{\mathrm{m}}$ = 0.315 and in SIV (blue) for $\Omega_{\mathrm{m}}$ = 0.20, as also found with supernovae and the $H_0$-$\Omega_{\mathrm{m}}$-age relation. From the same temperature $T$ and same $H(z^*_{\mathrm{rec}})$ at recombination, the $\Lambda$CDM and SIV models lead respectively to two different values $H_0 = 67.4$ and 74 km s$^{-1}$ Mpc$^{-1}$ at $z=0$ for $T=2.726$ K. As indicated, the redshifts at recombination are not the same. However, the figure shows that the models in $\Lambda$CDM and SIV that have the same dynamical and thermal states at recombination lead to slightly different $H_0$ values at the present epoch.