New physics in light of the $H_0$ tension: an alternative view

TL;DR

The paper investigates whether new physics that fixes beyond-$\Lambda$CDM parameters, notably the dark energy equation of state $w$ and the effective number of relativistic species $N_{\rm eff}$, can resolve the $H_0$ tension without enlarging uncertainties. It demonstrates that fixing $w\approx -1.3$ or $N_{\rm eff}\approx 3.95$ would yield $H_0$ in perfect agreement with local measurements when using $H_0$ from $H_0^{\rm local}$ and high-$z$ data, but such fixes are strongly disfavoured by Bayesian evidence relative to $\Lambda$CDM$. Extended models (where $w$ and/or $N_{\rm eff}$ vary) can reduce the tension to about $1.4$–$1.9\sigma$ but at a sizable Bayesian penalty, and none fully resolves the tension. The work provides dimensionless multipliers linking $\Delta H_0$ to changes in $w$ and $N_{\rm eff}$ to aid future forecasting and model-building with improved data.

Abstract

The strong discrepancy between local and inverse distance ladder estimates of the Hubble constant $H_0$ could be pointing towards new physics beyond $Λ$CDM. Several attempts to address this tension through new physics rely on extended models, featuring extra free parameters beyond the 6 $Λ$CDM parameters. However, marginalizing over extra parameters has the effect of broadening the uncertainties on the inferred parameters, and it is often the case that within these models the $H_0$ tension is addressed due to larger uncertainties rather than a genuine shift in the central value of $H_0$. What happens if a physical theory is able to fix the extra parameters to a specific set of non-standard values? The degrees of freedom of the model are reduced with respect to the standard case where the extra parameters are free to vary. Focusing on the dark energy equation of state $w$ and the effective number of relativistic species $N_{\rm eff}$, I find that physical theories able to fix $w \approx -1.3$ or $N_{\rm eff} \approx 3.95$ would lead to an estimate of $H_0$ from CMB, BAO, and SNeIa data in perfect agreement with the local distance ladder estimate, without broadening its uncertainty. These two non-standard models are, from a model-selection perspective, strongly disfavoured with respect to $Λ$CDM. However, models that predict $N_{\rm eff} \approx 3.45$ would be able to bring the tension down to $1.5σ$ while only being weakly disfavored with respect to $Λ$CDM, whereas models that predict $w \approx -1.1$ would be able to bring the tension down to $2σ$ (at the cost of the preference for $Λ$CDM being definite). Finally, I estimate dimensionless multipliers relating variations in $H_0$ to variations in $w$ and $N_{\rm eff}$, which can be used to repeat the analysis of this paper in light of future more precise local distance ladder estimates of $H_0$.

Figures (10)

• Figure 1: Normalized posterior distributions of $H_0$ (in ${\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$) for different choices of $w$, where $w$ is the dark energy equation of state fixed to non-standard values within the $\overline{w}$CDM model (see Tab. \ref{['tab:models']}). The models considered have values of $w$ fixed to $-1$ (i.e.$\Lambda$CDM, black curve), $-1.05$ (red), $-1.1$ (dark blue), $-1.15$ (green), $-1.2$ (purple), $-1.25$ (light blue), and $-1.3$ (yellow). The green shaded region is the 1$\sigma$ credible region for $H_0$ determined by the local distance ladder measurement of HSTRiess:2016jrr, yielding $H_0 = (73.24 \pm 1.74)\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$. When fixing $w=-1.3$, the high-redshift estimate of $H_0$ is $H_0 = (73.2 \pm 0.7)\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$, basically in agreement with the local distance ladder measurement.
• Figure 2: Bayesian evidence in favour of $\Lambda$CDM and tension between the high-redshift and local distance latter estimates of $H_0$ as a function of $w$, when the latter is fixed to non-standard values in the phantom region ($w<-1$) within the $\overline{w}$CDM model (see Tab. \ref{['tab:models']} for further details). The blue dashed curve (scale on the left $y$-axis) shows $-\ln B_{ij}$ [see Eq. (\ref{['eq:bayesfactor']}))], with ${\cal M}_i = \overline{w}$CDM and ${\cal M}_j = \Lambda$CDM. Therefore, a value $-\ln B_{ij}>0$ indicates that $\Lambda$CDM is favoured over the alternative model from the Bayesian evidence point of view. The Jeffreys scale used to quantify the strength of the evidence for $\Lambda$CDM (see Tab. \ref{['tab:kassraftery']}) is reflected in the colored regions (orange: weak preference for the extended model; blue: weak preference for $\Lambda$CDM; pink: definite preference for $\Lambda$CDM; green: strong preference for $\Lambda$CDM; grey: very strong preference for $\Lambda$CDM). The red dot-dashed curve quantifies the statistical significance of the $H_0$ tension through $\#\sigma$ [see Eq. (\ref{['eq:sigma']})].
• Figure 3: Index of inconsistency [see Eq. (\ref{['eq:ioi']})] as a function of $w$, when the latter is fixed to non-standard values in the phantom region ($w<-1$) within the $\overline{w}$CDM model (see Tab. \ref{['tab:models']} for further details). The scale of Lin:2017ikq used to quantify the strength of the inconsistency is reflected in the colored regions (blue: no significant inconsistency; pink: weak inconsistency; green: moderate inconsistency; grey: strong inconsistency), see Tab. \ref{['tab:ioi']} for further details.
• Figure 4: Normalized posterior distributions of $H_0$ (in ${\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$) for different choices of $N_{\rm eff}$, where $N_{\rm eff}$ is the effective number of relativistic species fixed to non-standard values within the $\overline{N}\Lambda$CDM model (see Tab. \ref{['tab:models']}). The models considered have values of $N_{\rm eff}$ fixed to $3.046$ (i.e.$\Lambda$CDM, black curve), $3.15$ (red), $3.35$ (dark blue), $3.55$ (green), $3.75$ (purple), and $3.95$ (light blue). The green shaded region is the 1$\sigma$ region of determined by the local distance ladder measurement of HST Riess:2016jrr, yielding $H_0 = (73.24 \pm 1.74)\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$. When fixing $N_{\rm eff}=3.95$, the high-redshift estimate of $H_0$ is $H_0 = (73.1 \pm 0.6)\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$, in complete agreement with the local distance ladder measurement.
• Figure 5: As in Fig. \ref{['fig:H0_w_tension_and_evidence']} but for the $\overline{N}\Lambda$CDM model.