Dark Photons in the Early Universe: From Thermal Production to Cosmological Constraints

TL;DR

This work addresses the thermal production of dark photons with kinetic mixing in the early universe across $m_{\tilde{\gamma}'}$ from $0.1\ \text{keV}$ to $100\ \text{MeV}$, deriving and validating analytical collision terms for inverse decay, annihilation, and semi-Compton processes and solving the Boltzmann evolution. A key result is the accurate freeze-in yield from resonant production for light dark photons and a universal resonance-to-off-resonance yield ratio of about $0.14$ for heavy dark photons, with medium effects properly incorporated through the photon self-energy and in-medium mixing. The paper also demonstrates how the dark-photon abundance leads to cosmological constraints from BBN and CMB observations, finding that cosmology provides the most stringent bounds in the $0.1-6\ \text{MeV}$ mass range for $\varepsilon$ in the $10^{-12}-10^{-10}$ range, complementing stellar cooling, supernova, and laboratory bounds. Overall, the combination of analytic insight, numerical Boltzmann solutions, and robust cosmological constraints offers a comprehensive framework for exploring dark photons in this mass window and guides future experimental and observational probes.

Abstract

Dark photons, a generic class of light gauge bosons that interact with the Standard Model (SM) exclusively through kinetic mixing, arise naturally in many gauge extensions of the SM. Motivated by these theoretical considerations, we present a comprehensive analysis of their thermal production in the early universe. Our calculation covers a broad range of dark photon masses from 0.1 keV to 100 MeV and include inverse decay, annihilation, and semi-Compton processes. Wherever possible, we present analytical estimates of the production rates and yields, and verify their accuracy numerically. For dark photons lighter than twice the electron masses (around 1 MeV), we find that our analytical estimate of the freeze-in yield based on resonant production is very accurate, implying that off-resonance contributions can be neglected in practice. For heavy dark photons, although this conclusion no longer holds, we derive an interesting ratio, $4πe/27\approx0.14$, with $e$ the coupling constant of QED, that can be used to estimate the relative importance of on- and off-resonance contributions. Finally, using the calculated abundance of dark photons in the early universe, we derive cosmological constraints on the dark photon mass and kinetic mixing. Compared with bounds from stellar cooling and supernovae, the cosmological constraints are most stringent in the mass range from 0.1 MeV to 6 MeV, within which kinetic mixing at the level of $10^{-12}\sim10^{-10}$ can be probed.

Figures (7)

• Figure 1: Thermal processes for dark photon production, including inverse decay (left), annihilation (middle), and semi-Compton (right). The hatched blobs indicate the effective coupling of $\gamma'$ to electrons in thermal plasma.
• Figure 2: Comparison of the analytical expressions for two-to-two collision terms with Monte-Carlo results.
• Figure 3: Schematic illustration of the slab approximation used to derive $\gamma$-$\gamma'$ oscillation.
• Figure 4: Upper panels: Resonances of the production rate; Lower panels: the evolution of the dark photon number density $n_{\gamma'}$ divided by the entropy density $s$. The dashed and dash-dotted lines represent our analytical estimates given by Eqs. \ref{['eq:-23']} and \ref{['eq:-42']}. The gap between them, $4\pi e/27\approx0.14$, is estimated by Eq. \ref{['eq:-59']}.
• Figure 5: Left: The maximum of $n_{\gamma'}/s$ during the evolution as a function of $m_{\gamma'}$, assuming $\varepsilon=10^{-10}$. Right: Contours on the $\varepsilon$-$m_{\gamma'}$ plane to indicate whether $\gamma'$ has ever reached equilibrium (black lines) and whether it is long-lived (purple lines).