Late-time vacuum phase transitions: Connecting sub-eV scale physics with cosmological structure formation

We show that a particular class of postrecombination phase transitions in the vacuum can lead to localized overdense regions on relatively small scales, roughly ${10}^6$ to ${10}^{10}{M}_{\odot }$, potentially interesting for the origin of large black hole seeds and for dwarf galaxy evolution. Our study suggests that this mechanism could operate over a range of conditions which are consistent with current cosmological and laboratory bounds. One byproduct of phase transition bubble-wall decay may be extra radiation energy density. This could provide an avenue for constraint, but it could also help reconcile the discordant values of the present Hubble parameter ($H_0$) and ${\sigma }_8$ obtained by cosmic microwave background (CMB) fits and direct observational estimates. We also suggest ways in which future probes, including CMB considerations (e.g., early dark energy limits), 21-cm observations, and gravitational radiation limits, could provide more stringent constraints on this mechanism and the sub-eV scale beyond-standard-model physics, perhaps in the neutrino sector, on which it could be based. Late phase transitions associated with sterile neutrino mass and mixing may provide a way to reconcile cosmological limits and laboratory data, should a future disagreement arise.

Figure 1

A comoving coordinate sphere expanding with the Hubble flow prior to the onset of the phase transition.

Figure 2

A bubble of the broken phase (shaded) nucleates inside a comoving spherical shell (dashed line) and expands relativistically. As it expands, it sweeps up most of the vacuum energy from the unbroken phase onto its wall (thick gray line).

Figure 3

The expanding bubble crosses the comoving sphere and collides with an adjacent bubble. We assume that the bubble walls disintegrate as they collide, radiating away the swept-up vacuum energy (i.e., the kinetic energy of the bubble walls).

Figure 4

Cartoon illustrating the energetics associated with fluctuation binding. For binding to be accomplished, the kinetic energy has to go to zero, which can only happen if the total energy $E_{\text{total}}$ (horizontal dashed line) intersects the potential energy curve ($E_{\text{grav}}$, solid line). Cases (ii) and (iii) represent configurations which allow binding [with (ii) being right at the threshold], whereas case (i) depicts a situation where binding cannot be accomplished in spite of a negative total energy.

Figure 5

Contours of halting time (solid) and time-to-nonlinearity (dashed), labeled in gigayears, across a parameter space spanned by critical temperature and early vacuum energy density. Plots (a) and (b) correspond to two different comoving shells, labeled with their respective coordinate radii in relation to the typical nucleation scale $\delta H_c^{-1}$. Shaded areas indicate regions of the parameter space where comoving fluctuations of the specified radii cannot become gravitationally bound, either within a $\mathcal{O}(1\mathrm{Gyr})$ time scale, or at all.

Figure 6

Halting times corresponding to different mass scales, for certain selected parameter values. The four curves (from left to right) correspond to the following parameter values: $T_c=0.03\mathrm{eV}$, ${\rho }_v=55.3\mathrm{GeV}/{\mathrm{cm}}^3$; $T_c=0.03\mathrm{eV}$, ${\rho }_v=38\mathrm{GeV}/{\mathrm{cm}}^3$; $T_c=0.05\mathrm{eV}$, ${\rho }_v=100.2\mathrm{GeV}/{\mathrm{cm}}^3$; and $T_c=0.05\mathrm{eV}$, ${\rho }_v=76\mathrm{GeV}/{\mathrm{cm}}^3$, respectively. Evidently, lower critical temperatures and higher early vacuum energy densities imply stronger binding.

Figure 7

Contours labeled with the early vacuum energy closure fraction at photon decoupling, plotted across a parameter space spanned by critical temperature and early vacuum energy density.

Figure 8

Contours labeled with the closure fraction, at the present epoch, of the leftover radiation from bubble collisions, plotted across a parameter space spanned by critical temperature and early vacuum energy density. Shaded areas represent regions of the parameter space that are disfavored by current observational data.

Figure 9

Contours labeled with the Hubble parameter, calculated at the present epoch in our model ($H_0$, in $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$), plotted across a parameter space spanned by critical temperature and early vacuum energy density. Shaded areas represent regions of the parameter space where the calculated $H_0$ values are not in good agreement with current observational data.

Figure 10

Contours labeled with the Hubble parameter, calculated at a redshift of $z=2.36$ in our model ($H_{2.36}$, in $\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$), plotted across a parameter space spanned by critical temperature and early vacuum energy density. The dashed line at $242\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ marks the 95% C.L. limit given by Ref. [53], while the shaded areas represent regions of the parameter space where the calculated $H_{2.36}$ values are significantly at odds ($3\sigma$ or more) with that result.

Figure 11

Contours labeled with the early vacuum energy closure fraction at the onset of the phase transition (i.e., at $T=T_c$), plotted across a parameter space spanned by critical temperature and early vacuum energy density.

Figure 12

Evolution of closure fractions contributed by the various components of energy density with cosmic temperature in the standard $\mathrm{\Lambda }\mathrm{CDM}$ cosmological model. The solid, dotted, and dashed lines represent nonrelativistic, relativistic, and vacuum energy densities, respectively. Neutrinos were approximated as being massless throughout the course of the evolution.

Figure 13

Evolution of closure fractions contributed by the various components of energy density with cosmic temperature in our modified cosmological model, for different parameter values. The solid, dotted, and dashed lines represent nonrelativistic, primordial relativistic, and vacuum energy densities, respectively, whereas the dot-dashed line represents the leftover radiation from the phase transition. Notice how the vacuum energy density peaks at $T=T_c$, then suddenly plummets to a nearly zero closure fraction (as most of it gets converted to relativistic particles), before eventually becoming significant again at late times.

Figure 14

Evolution of the closure fraction contributed by the total nonrelativistic energy density (${\mathrm{\Omega }}_{\mathrm{NR}}$) with time in our modified cosmological model, compared with the standard model picture. The solid line at the top represents the standard model, whereas the other two curves (from top to bottom) represent evolution in our cosmological model, with the following parameter values: $T_c=0.05\mathrm{eV}$, ${\rho }_v=76\mathrm{GeV}/{\mathrm{cm}}^3$; and $T_c=0.03\mathrm{eV}$, ${\rho }_v=38\mathrm{GeV}/{\mathrm{cm}}^3$, respectively.

Figure 15

Evolution of the scale factor with time in our modified cosmological model, compared with the standard model picture. The solid line on the far right represents the standard model, whereas the other two curves (from left to right) represent evolution in our cosmological model, with the following parameter values: $T_c=0.03\mathrm{eV}$, ${\rho }_v=38\mathrm{GeV}/{\mathrm{cm}}^3$; and $T_c=0.05\mathrm{eV}$, ${\rho }_v=76\mathrm{GeV}/{\mathrm{cm}}^3$, respectively.

Figure 16

Plot showing the variation of nonrelativistic energy densities in our model (both the local and the average) as a function of redshift. The solid, dashed, dot-dashed, and dotted curves correspond to ${\rho }_{\mathrm{NR}}$, ${\rho }_{\mathrm{NR}}^{\prime }$, $\delta \rho$, and $\delta \rho /{\rho }_{\mathrm{NR}}$, respectively. The parameters used for this calculation are $T_c=0.05\mathrm{eV}$ and ${\rho }_v=76\mathrm{GeV}/{\mathrm{cm}}^3$.

Figure 17

Plot showing the gravitational potentials $\varphi (z)$ (dotted curve) of individual density fluctuations generated in our model at different redshifts along the line of sight, along with the integrated Sachs-Wolfe effect $\mathrm{\Delta }\varphi (z)$ (solid curve) that a photon passing through these fluctuations would experience. The parameters used for this calculation are $T_c=0.05\mathrm{eV}$ and ${\rho }_v=76\mathrm{GeV}/{\mathrm{cm}}^3$.

Figure 18

Contours representing the angular scales in today’s sky (labeled in arc minutes) that would correspond to the nucleation scale. Plotted across a parameter space spanned by critical temperature in eV and early vacuum energy density in $\mathrm{GeV}/{\mathrm{cm}}^3$.