Particle creation in a cosmological background in analogy to the Schwinger effect

TL;DR

The paper develops a gravitational analogue of the Schwinger effect by analyzing quantum particle production in a long gravitational pulse within an FLRW spacetime, using canonical quantization and Bogolyubov transformations. For electromagnetism, it derives exact Bogolyubov coefficients from a hypergeometric equation and reproduces Schwinger’s long-pulse result $n(\boldsymbol p)=\exp[-\pi(m^2+p_\perp^2)/(eE)]$; for gravity, the problem reduces to a Heun equation with connecting formulas that yield the in/out mode mixing. A key finding is a low-frequency threshold in the gravitational case and, at high frequencies, a blackbody-like spectrum with inverse temperature $\\beta = 2\\pi\left(a_2^2+a_1^2-\\sqrt{a_2^4-a_1^4}\right) a_2 / b$, related to an apparent horizon radius $R_H$ via $\\beta = 2\\pi R_H$. This work links time-dependent background switching to horizon thermodynamics and provides a structured framework to compare EM and gravitational particle production, with implications for cosmological particle creation and possible connections to Hawking radiation in dynamical spacetimes.

Abstract

We consider a gravitational analogue of the Schwinger effect in a cosmological context. While the Schwinger effect is usually attributed to a static electric background, its derivation is actually based on a switching on/off of the electric field in the infinite past/future. Motivated by this, and our previous work on particle production in a gravitational background, we consider a long pulse of the gravitational field in an FLRW-spacetime, thus simulating a static background. We rigorously derive particle production by a novel application of the Heun equation. In fact, the recently obtained connecting formulas between its local solutions can be used to determine the Bogolyubov coefficients, and subsequently the particle production probabilities, in the limit of an infinitely long pulse. The particle production in the FLRW-model is found to have a lower threshold on the outgoing frequencies, which can be related to the duration of the time interval of the switching on/off the background field. For large frequencies and in the spatially flat case, we find black-body radiation whose temperature is inversely proportional to the radius of the apparent horizon that appears in the FLRW-model during the phase of scale change. We compare our findings to Schwinger's result on particle production for a long pulse of an electromagnetic background field, for which we also include a detailed derivation.

Figures (5)

• Figure 1: Vector potential (in units of $E/b$) and electric field (in units of $E$) for an electric field pulse as in \ref{['eq:AE']} (for $b=10^{-1}$).
• Figure 2: Scale function $a(\tau)$ as in Eq. \ref{['eq:scale-FLRW']} ($T=50, a_1=1, a_2=5$).
• Figure 3: Scalar curvature $R(\tau)$ for FLRW-spacetime with a gravitational pulse as in Eq. \ref{['eq:scale-FLRW']} and \ref{['eq:ricci']}, displayed at same scaling. ($T=50, b=1, a_1=1, a_2=5$).
• Figure 4: Probability $P_1(\lambda)$ defined in \ref{['eq:prob-n']} in terms of \ref{['eq:betaalpha-exact']} for $(a_1=1,a_2=5)$ at different time intervals $T = 50,100$ in (a),(c) and for $(a_1=2,a_2=5)$ and $T =50, 100$ in (b),(d). The thick blue line is the approximation of $P_1(\lambda)$ in terms of large $\lambda$-behavior of $|\beta_\lambda|^2/|\alpha_\lambda|^2$ as obtained in \ref{['eq:betaalpha-approx']}.
• Figure 5: Probability $P_1(\lambda)$ for $(a_1=1,a_2=5)$ and $(a_1=2,a_2=5)$ .