Vacuum polarization and photon mass in inflation

TL;DR

The paper elucidates a mechanism by which inflation boosts vacuum polarization from light charged scalars, endowing long-wavelength photons with an effective mass during inflation. Using both a full polarization-tensor calculation and a Hartree approximation, it shows the mass scales with the inflationary Hubble parameter $H_I$ up to logarithmic factors, and discusses how the mass vanishes after inflation while the associated zero-point energy can seed large-scale magnetic fields. This links conformal invariance breaking in a time-dependent spacetime to a potential origin for cosmological magnetic seeds that could be amplified by dynamos to the micro-Gauss fields observed in galaxies. The work highlights the role of scalar QED in the inflationary era and provides a concrete mechanism connecting early-universe QED effects to observable cosmic magnetism.

Abstract

We give a pedagogical review of a mechanism through which long wave length photons can become massive during inflation. Our account begins with a discussion of the period of exponentially rapid expansion known as inflation. We next describe how, when the universe is not expanding, quantum fluctuations in charged particle fields cause even empty space to behave as a polarizable medium. This is the routinely observed phenomenon of vacuum polarization. We show that the quantum fluctuations of low mass, scalar fields are enormously amplified during inflation. If one of these fields is charged, the vacuum polarization effect of flat space is strengthened to the point that long wave length photons acquire mass. Our result for this mass is shown to agree with a simple model in which the massive photon electrodynamics of Proca emerges from applying the Hartree approximation to scalar quantum electrodynamics during inflation. A huge photon mass is not measured today because the original phase of inflation ended when the universe was only a tiny fraction of a second old. However, the zero-point energy left over from the epoch of large photon mass may have persisted during the post-inflationary universe as very weak, but cosmological-scale, magnetic fields. It has been suggested that these small seed fields were amplified by a dynamo mechanism to produce the micro-Gauss magnetic fields observed in galaxies and galactic clusters.

Figures (3)

• Figure 1: A gas of polarized atoms. In the absence of an external electric field the dipoles orient randomly (FIG. \ref{['figure-I']}a). When an external field $\vec{E}$ is applied, the dipoles tend to line up with it (FIG. \ref{['figure-I']}b). This alignment produces a net polarization, $\vec{P} = \chi_e \vec{E}$, which weakens the electric force by $1/(1 + \chi_e)$. There is a similar effect in vacuum due to oppositely charged pairs of evanescent, virtual particles.
• Figure 2: A sketch of the Maxwell-Cavendish experiment. Two metal concentric spheres of radii $a$ and $b$ are first grounded, then the outer sphere is raised to a high potential. If Cou-lomb's law were violated, the voltmeter $V$ would show a non-zero voltage.
• Figure 3: Evolution of a quantum particle's physical wave length $\lambda_{\rm ph} = a(t) \lambda$ as the universe expands. Wave lengths which are now of cosmological size were originally minuscule. First horizon crossing occurs (at $a_x$) when $\lambda_{\rm ph}$ becomes comparable to the inflationary Hubble radius $\sim c/H_I$. At first crossing massless, minimally coupled scalars and gravitons of that wave length are ripped out of the vacuum by the expansion of spacetime. These particles ride the subsequent evolution of the universe relatively undisturbed until the second horizon crossing at $\lambda_{\rm ph} \sim c/H(t)$. Then the particles manifest themselves as cosmological-scale correlations which could not have formed causally after inflation.