Cosmological and Astrophysical Constraints on Late First-Order Phase Transitions

Abstract

First-order cosmological phase transitions (PT) can take place in a dark sector at relatively late times between the big-bang nucleosynthesis and recombination epochs. Because bubble nucleation is stochastic, the PT completes at different times in different regions of the Universe. This fluctuation sources a curvature perturbation whose (dimensionless) power spectrum ${\cal P}_ζ(k)$ features a universal infrared tail, independent of the microscopic details of the PT. Even in the absence of any non-gravitational interaction between the dark sector and the Standard Model, these additional curvature perturbations at small scales impact a variety of observables. We derive new constraints on dark sector phase transitions using {\it Planck}, baryon acoustic oscillation (BAO), Lyman-$α$ observations, spectral distortion limits from FIRAS, constraints on early reionization, and the existence of ultra-faint dwarf galaxies.

Figures (6)

• Figure 1: Schematic picture of dark first-order PT in two far apart patches, $A$ and $B$, initially in the false vacuum $F$. The transition from $F$ to dark fluid $X$ occurs in each patch at times $t^A_c$ and $t^B_c$, respectively. The PT generates an isocurvature perturbation between the two patches at $t^A_c$. This isocurvature perturbation sources a curvature (density) perturbation between $t^A_c$ and $t^B_c$, whenever $F$ and $X$ redshift differently.
• Figure 2: Plot of ${\cal P}_{\zeta, \rm PT}(k)$ for a choice of PT parameters showing modification of super-horizon slope for different redshift behaviors of the dark fluid $X$. We show the cases where $X$ redshifts as radiation ($d_X = 0$), as matter ($d_X = -1$) and as kination ($d_X = 2$). Compared to the $\sim k^3$ power law for radiation, matter exhibits an enhanced $\sim k$ power law in the IR slope while kination exhibits a diminished $\sim k^7$ power law. Slight modifications to the amplitude come from the ${(4+d_X)}/{4}$ prefactor (in Eq. \ref{['eq:fdx']}) due to different redshifts relative to the false vacuum during the $\delta t_c$-fluctuations of the PT. Note that the enhanced IR slope for matter is not expected to hold past matter-radiation equality $k_{\rm eq} \approx 0.01/$Mpc.
• Figure 3: Demonstration of how the curvature power spectrum ${\cal P}_{\zeta, \rm PT}(k)$, for $X={\rm DR}$, changes as the PT parameters $(z_{\rm PT},\ f_{\rm DR},\ \beta/H_{\rm PT})$ are varied. The sub-bubble region of each spectrum is left dotted to stress the uncertainty in ${\cal P}_{\zeta, \rm PT}(k)$ on scales $k \gtrsim a_{\rm PT}/d_b$. All the peaks exhibit the same shape with $k^3$ super-bubble slope. The variations are exaggerated by increasing each of the three PT parameters by an order of magnitude. Using the blue curve as a starting reference, observe that (i) increasing $z_{\rm PT}$ translates the peak position to higher $k$ (yellow), (ii) increasing $f_{\rm DR}$ increases the overall amplitude (green), and (iii) increasing $\beta/H_{\rm PT}$ both translates the peak to higher $k$ and decreases its amplitude (red).
• Figure 4: $2\sigma$ exclusion bounds on $f_{\rm DR}$ presented for $\beta/H_{\rm PT}= 10$ (top) and $\beta/H_{\rm PT}= 100$ (bottom). The CMB+Ext. bound (Ext.=BAO+KV450) was calculated using MCMC with the datasets in Refs. Planck:2019nipPlanck:2018lbuBeutler_2011Ross:2014qpaBOSS:2016wmcHildebrandt:2018yau (pale orange), where KV450 primarily serves to exclude higher temperature PTs that affect $k$-modes $\sim (0.1-0.3)h/{\rm Mpc}$. The Lyman-$\alpha$ bound was estimated using a reduced $2$-parameter likelihood study Bird:2023Fernandez:2024He:2025Bird_github:2026 (blue). We also include bounds calculated from early reionization Qin:2025ymc (brown), UFD galaxy heating Graham:2024 (red) and FIRAS 1994ApJ...420..439MFixsen:1996nj (bright orange). The corresponding bounds derived by direct translation are presented as dashed curves of the same color, using Ref. Planck_inf:2020 for CMB+BAO and Ref. 2011MNRAS.413.1717B for Lyman-$\alpha$. The existing $\Delta N_{\rm eff}$ constraint (light gray region) includes $\Delta N_{\rm eff} \leq 0.29$ (dashed, TTTEEE+lowE+lensing+BAO) and $\Delta N_{\rm eff} \leq 0.21$ (dotted, TTTEEE+lowE+lensing) from Planck2018 Planck:2018vyg.
• Figure 5: Results of MCMC analysis with CMB + BAO +KV450 data for $\beta/H_{\rm PT}= 10$ (left) and $\beta/H_{\rm PT}= 100$ (right), marginalized to $z_{\rm PT}$ and $f_{\rm DR}$ in log space (base $10$). 2D contour plots show the preferred regions up to 1, 2 and 3$\sigma$ boundaries. The MCMC scan was done with $\Delta N_{\rm eff}$ component varying with $f_{\rm DR}$, which contributes to the suppressed preference for large $f_{\rm DR}$ values.