$S_8-H_0$ tension in a SI-ULDM scenario

TL;DR

This work investigates whether a transient self-interaction phase in Ultra-Light Dark Matter (SI–ULDM) can alleviate the $H_0$–$S_8$ tensions by briefly modifying the expansion history. The authors develop a perturbative, model-independent framework that parameterizes the self-interaction energy as a localized bump $f_{ m SI}(a)$ in the cosmic energy budget and show that the sound horizon and late-time growth respond as weighted integrals with distinct kernels. They derive analytic relations linking the early-time reduction of the sound horizon to the late-time suppression of structure growth, predicting correlated shifts in $H_0$ and $S_8$ whose sign and magnitude depend on the episode timing, notably peaking near matter–radiation equality. The results offer a unified physical interpretation of how a single transient episode can connect pre- and post-recombination physics, providing testable signatures that can be explored with Boltzmann solvers and data. Extensions to perturbations of the scalar field and broader interactions are marked for future work.

Abstract

We study the cosmological impact of a transient self-interaction phase in Ultra-Light Dark Matter (ULDM), focusing on its simultaneous effects on the sound horizon and the late-time growth of structure. In the presence of a quartic self-interaction, the scalar field undergoes a short-lived radiation-like phase before evolving into matter-like behaviour, inducing a localized modification of the expansion history at early times. We develop a perturbative and model-independent framework in which the self-interaction energy density is parametrized as a localized contribution to the total energy budget. Within this approach, the responses of the sound horizon and the linear growth factor can be expressed as weighted integrals over cosmic time, with distinct kernels encoding the temporal sensitivity of each observable. This structure leads to a simple analytic relation linking the corresponding early- and late-time responses, and naturally predicts correlated shifts in $H_0$ and $S_8$ whose sign and magnitude depend on the timing of the self-interaction episode. Our results show that a single transient modification of the expansion history can interpolate between early-time effects on the sound horizon and late-time suppression of structure growth within a unified physical framework, providing an analytical understanding of their joint response.

Figures (3)

• Figure 1: Sensitivity kernels for the sound horizon and linear growth. The blue curve shows the sound–horizon sensitivity, which peaks before recombination and vanishes afterwards, while the orange curve illustrates the cumulative sensitivity of linear growth, becoming effective after matter–radiation equality. The dashed green curve represents a localized self–interaction fraction $f_{\text{SI}}(a)$ centered at $a_{\star}$. Vertical lines mark equality $a_{\text{eq}}$ and recombination $a_{\text{rec}}$. Shaded regions indicate the portions of the self–interaction phase that contribute to the reduction of the sound horizon (blue) and to the suppression of structure growth (orange).
• Figure 2: Impact of a transient self–interaction phase on early- and late-time cosmological observables as a function of its characteristic epoch. Left: fractional shift of the sound horizon at recombination, $\Delta r_s/r_s$, as a function of the bump location $a_\star$, for fixed peak amplitude $f_{\rm pk}=0.02$ and different widths $\Delta$. The effect is maximal when the self–interaction occurs close to matter–radiation equality and rapidly vanishes for transitions taking place well before or after recombination. Right: corresponding fractional suppression of the linear growth amplitude, $\Delta S_8/S_8\simeq\Delta D/D$, obtained by solving the linear growth equation. In contrast to the sound horizon, the impact on structure growth remains significant even when the self–interaction occurs after recombination, reflecting the cumulative nature of growth. Vertical lines indicate matter–radiation equality ($a_{\rm eq}$) and recombination ($a_{\rm rec}$).
• Figure 3: Correlated response of the Hubble constant and the amplitude of matter fluctuations induced by a transient self--interaction phase. The figure shows the parametric trajectories in the $(\Delta H_0/H_0,\Delta S_8/S_8)$ plane obtained by varying the location of the self--interaction bump, $a_\star$, for fixed peak amplitude $f_{\rm pk}=0.08$ and different widths $\Delta$ (solid colors). The color of the markers encodes the value of $\log_{10} a_\star$. Solid lines correspond to self--interaction episodes occurring before recombination, for which both the sound horizon and the growth of structure are affected. The transition to dashed lines marks the regime in which the bump moves past recombination: beyond this epoch, further changes in $a_\star$ no longer modify the sound horizon, leading to a saturation of $\Delta H_0$, while the suppression of structure growth continues to accumulate. The resulting "kink" in the trajectories reflects this change of temporal sensitivity and illustrates that the relation between $\Delta H_0$ and $\Delta S_8$ is not one--to--one, but depends on the epoch at which the transient self--interaction takes place.