Probing early-universe phase transitions with CMB spectral distortions

Global, symmetry-breaking phase transitions in the early universe can generate scaling seed networks which lead to metric perturbations. The acoustic waves in the photon-baryon plasma sourced by these metric perturbations, when Silk damped, generate spectral distortions of the cosmic microwave background (CMB). In this work, the chemical potential distortion ($\mu$) due to scaling seed networks is computed and the accompanying Compton $y$-type distortion is estimated. The specific model of choice is the $O(N)$ nonlinear $\sigma$-model for $N\gg 1$, but the results remain the same order of magnitude for other scaling seeds. If CMB anisotropy constraints to the $O(N)$ model are saturated, the resulting chemical potential distortion $\mu \lesssim 2\times 10^{-9}$.

Figure 1

After the phase transition the scalar fields reside on the vacuum manifold, picking out an (uncorrelated) “angle” in each Hubble patch. As the universe expands, the local gradients in the field vanish on subhorizon scales. This process leads to scaling network of the scalar field, that looks the same when compared to the contemporary Hubble horizon [54].

Figure 2

The dimensionless power spectrum of the scalar-field fluctuations at a given scale and time: $(N/v^2){\mathrm{\Delta }}_{\phi }^2(k,\eta )\equiv (N/v^2)k^3{P}_{\phi }(k,\eta )/(2{\pi }^2)$ in a radiation-dominated universe. This spectrum depends only on $k\eta$, and is thus self-similar. There exists an additional cutoff (not apparent in this plot) at $k\eta =\eta /{\eta }_t\gg 10^2$, set by the initial time of the phase transition ${\eta }_t$.

Figure 3

The dimensionless curvature potential spectrum due to the scalar field only in a radiation-dominated universe (under the scaling ansatz for the solutions). The spectrum on superhorizon scales is $\approx 0.94(k\eta {)}^{-1}$ whereas on subhorizon scales it is $\approx 4.2(k\eta {)}^{-4}\ln [0.56(k\eta )]$.

Figure 4

The dimensionless power spectrum of the difference between the Newtonian and curvature potentials due to the scalar field only (in a radiation-dominated universe, under the scaling ansatz for the solutions). The spectrum on superhorizon scales is $\approx 0.64(k\eta {)}^{-1}$ whereas on subhorizon scales it is $\approx 30.4(k\eta {)}^{-4}\ln [0.56(k\eta )]$.

Figure 5

The evolution of the radiation density perturbations $u_{\gamma }={\delta }_{\gamma }-4\mathrm{\Psi }$ in the presence of seeds, for $k=10^2{\mathrm{Mpc}}^{-1}$. The black line corresponds to the full evolution of the radiation density perturbation. The orange line follows the full evolution prior to the beginning of acoustic oscillations, and tracks the peak-to-peak envelope of acoustic oscillations once they begin. The dashed line on the left corresponds to horizon entry. Outside the horizon $k\eta \ll 1$, $|u_{\gamma }(k,\eta ){|}^2\propto {\eta }^3$. This also implies that the dimensionless power spectrum $\sim k^3|u_{\gamma }(k,\eta ){|}^2\propto (k\eta {)}^3$, corresponding to a scaling, white noise spectrum on superhorizon scales. For $k\eta \gtrsim 10$, we see the characteristic acoustic oscillations. For subhorizon scales, $\mathrm{\Psi }\ll {\delta }_{\gamma }$ even with seed potentials, hence $u_{\gamma }\approx {\delta }_{\gamma }$. Note the continued growth of the photon perturbation for roughly a decade inside the horizon. Finally when the diffusion wave number $k_D(\eta )<k$, diffusion takes away the acoustic energy of the mode. The dashed line on the right denotes $k_D(\eta )=k$.

Figure 6

The envelope of the dimensionless power spectrum ${\mathrm{\Delta }}_{{\overline{u}}_{\gamma }}^2(k,\eta )$ of the photon density perturbation evaluated at the beginning and end of the $\mu$ era. Explicitly the top curve corresponds to ${\mathrm{\Delta }}_{{\overline{u}}_{\gamma }}^2(k,{\eta }_f)$ and the bottom curve corresponds to ${\mathrm{\Delta }}_{{\overline{u}}_{\gamma }}^2(k,{\eta }_i)$ where ${\eta }_i$ and ${\eta }_f$ correspond to the conformal times at the beginning and end of the $\mu$ era. Black points come from the full evolution code, while the orange curve interpolates between these points. The spectral distortions may be estimated by integrating the difference of the two spectra (with a logarithmic measure). Note that in the text we carry out a more detailed calculation instead of using this integral estimate.