## Author(s)

Garny, Mathias, Niedermann, Florian, Rubira, Henrique, Sloth, Martin S.## Abstract

We propose a simple model that can alleviate the $H_0$ tension while remaining consistent with big bang nucleosynthesis (BBN). It is based on a dark sector described by a standard Lagrangian featuring a $SU(N)$ gauge symmetry with $N\geq3$ and a massive scalar field with a quartic coupling. The scalar acts as dark Higgs leading to spontaneous symmetry breaking $SU(N)\to SU(N\!-\!1)$ via a first-order phase transition à la Coleman-Weinberg. This set-up naturally realizes previously proposed scenarios featuring strongly interacting dark radiation (SIDR) with a mass threshold within hot new early dark energy (NEDE). For a wide range of reasonable model parameters, the phase transition occurs between the BBN and recombination epochs and releases a sufficient amount of latent heat such that the model easily respects bounds on extra radiation during BBN while featuring a sufficient SIDR density around recombination for increasing the value of $H_0$ inferred from the cosmic microwave background. Our model can be summarized as a natural mechanism providing two successive increases in the effective number of relativistic degrees of freedom after BBN but before recombination $\Delta N_\mathrm{BBN} \to \Delta N_\mathrm{NEDE} \to \Delta N_\mathrm{IR}$ alleviating the Hubble tension. The first step is related to the phase transition and the second to the dark Higgs becoming non-relativistic. This set-up predicts further signatures, including a stochastic gravitational wave background and features in the matter power spectrum that can be searched for with future pulsar timing and Lyman-$\alpha$ forest measurements.

## Figures

Energy density of different dark sector components in units of $\rho_ {1,\nu} \equiv \frac{7}{4}\frac{\pi^2}{30} \left(\frac{4}{11}\right)^{4/3}T_{\rm vis}^4$. The blue line depicts the evolution of the total dark sector energy density, composed of the dark radiation plasma formed by the gauge (green line) and Higgs (red line) bosons as well as the latent heat $\rho_{\rm NEDE}$ (orange line and area), that rises above the radiation fluid before the supercooled phase transition. The purple line depicts $\Delta N_{\rm eff}$ before and after the phase transition at $a_*$, respectively, see \eqref{eq:rho_before} and \eqref{eq:rho_after}. The red shaded area illustrates BBN constraints, and the green area values of $\Delta N_{\rm eff}$ required for addressing the $H_0$ tension. For comparison grey dotted and dashed lines show the SIDR model and its stepped version named WZDR~\cite{Aloni:2021eaq} (see Sec.~\ref{introduction}).

Evolution of $\Delta N_{\rm eff}$, temperature ratio $\xi_d=T_d/T_{\rm vis}$, entropy ratio $s_d/s_{\rm vis}$ and equation of state $w=p_d/\rho_d$ for $SU(N)\to SU(N\!-\!1)$ with $N=3,4,5$. All are discontinuous at the phase transition at $a_*$ (first step), while only the entropy is conserved once the dark Higgs becomes non-relativistic at $a_t$ (second step). The second step disappears in the large-$N$ limit. Note that for a growing number of degrees of freedom $g_d$ (i.e. for $N=3,4,5$), the temperature $T_d$ decreases while $s_d\propto g_dT_d^3$ increases with $g_d$ for given $\Delta N_{\rm IR}\propto g_dT_d^4$.

Impact of BBN constraint on a simple SIDR model with constant $\Delta N_{\rm eff}$. Orange contours show $68\%$ C.L. and $95\%$ C.L. marginalized posteriors when including the BBN prior on $\Delta N_{\rm eff}$ and using the corresponding helium abundance predicted by BBN. The red contours show for comparison the case when ignoring BBN constraints and setting the helium fraction to the value found for $\Lambda$CDM ($Y_p = 0.2454$) by hand, as done in previous analyses. Filled contours correspond to the case without SH$0$ES, and open dotted contours illustrate the impact of including SH$0$ES. We observe that BBN disqualifies SIDR models without post-BBN heating mechanism for addressing the $H_0$ tension.

Comparison between the $\Lambda$CDM (blue), SIDR (orange), and Hot NEDE (green) $68\%$ C.L. and $95\%$ C.L. marginalized posteriors, with BBN information included in all cases, and without (filled) and with (open dotted) including SH$0$ES. Only Hot NEDE is marginally compatible with the SH$0$ES value of $h$ (gray vertical band).

Posteriors when including the redshift of the dark sector phase transition $z_\ast$ as a free parameter, compared to the fiducial case with fixed $z_\ast=10^6$. Including $z_\ast$ does not affect the posteriors of the other model parameters, and provides us with a lower bound $z_\ast > 4.4 \times 10^4 $ ($95\%$ C.L.).

Gravitational wave spectrum generated in the first-order phase transition using the envelope approximation. We show an estimated prediction for a benchmark scenario (black) and the impact when varying one of the parameters (other lines, see legend). We also include NANOGrav 15yr results from~\cite{NANOGrav:2023gor} and the expected sensitivity of SKA after 20 years of data.

CMB $C_{\ell}^{TT}$ (upper panel) and matter power spectra $P(k)$ (lower panel) for Hot NEDE models with various $z_\ast$ and $r_g$, normalized to $\Lambda$CDM. In all cases, we use the bestfit (base+BBN$+H_0$) parameters obtained for Hot NEDE with $z_\ast = 10^6,\, r_g = 0$ and match $\Delta N_{\rm IR} = 0.40$ for SIDR. Compatibility with Planck requires $z_\ast\gtrsim 10^5$, while for $z_\ast \lesssim 10^6$ Lyman-$\alpha$ scales overlap with the onset (yellow dashed) or even the first peaks (yellow dotted) of the dark sector acoustic oscillations. The grey area represents the region probed by Lyman-$\alpha$ forest data.

CMB $C_{\ell}^{TT}$ (upper panel) and matter power spectra $P(k)$ (lower panel) for Hot NEDE models with various $z_\ast$ and $r_g$, normalized to $\Lambda$CDM. In all cases, we use the bestfit (base+BBN$+H_0$) parameters obtained for Hot NEDE with $z_\ast = 10^6,\, r_g = 0$ and match $\Delta N_{\rm IR} = 0.40$ for SIDR. Compatibility with Planck requires $z_\ast\gtrsim 10^5$, while for $z_\ast \lesssim 10^6$ Lyman-$\alpha$ scales overlap with the onset (yellow dashed) or even the first peaks (yellow dotted) of the dark sector acoustic oscillations. The grey area represents the region probed by Lyman-$\alpha$ forest data.

Posteriors exploring the second step at redshift $z_t$ for which the dark Higgs becomes non-relativistic after the phase transition at $z_\ast$ in the Hot NEDE model. We compare the model considered in the main text with step size $r_g = 0$ (green contours) and another one where $r_g$ and $z_t$ are sampled (orange contours). While $r_g$ is bounded from above, $z_t$ is hardly constrained at all, in agreement with findings in the literature for stepped SIDR models~\cite{Allali:2023zbi}.

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