Dark Photon Dark Matter from Quantum Fluctuations during Starobinsky Inflation

Technical Summary: Dark Photon Dark Matter from Quantum Fluctuations during Starobinsky Inflation

Problem Statement
The identification of the particle nature of dark matter (DM) remains a central challenge in cosmology and particle physics. While wave-like DM candidates, such as the dark photon (a massive vector field kinetically mixing with the ordinary photon), are well-motivated, their mass is typically treated as a free parameter. A specific production mechanism posits that dark-photon DM originates from quantum fluctuations of the longitudinal mode generated during inflation. Previous analyses of this scenario (e.g., Ref. [4]) assumed simple single-field inflation where the dark-photon mass remains constant. However, many realistic inflationary models, including the leading Starobinsky model, require a Weyl transformation to transition from the Jordan frame to the Einstein frame. In such models, the kinetic function of the dark photon's longitudinal mode becomes dependent on the inflaton amplitude, varying throughout and after the inflationary period. The authors investigate how this time-dependent kinetic function alters the predicted relic abundance of dark photons compared to previous estimates.

Methodology
The authors develop a formalism to calculate the dark-photon abundance in inflationary models necessitating a Weyl transformation.

1. Frame Transformation: They begin with the Lagrangian in the Jordan frame and perform a Weyl transformation ($g_{\mu\nu} = \Omega^2 \bar{g}_{\mu\nu}$) to reach the Einstein frame. The Weyl factor $\Omega$ is a function of the inflaton field $\phi$.
2. Action Analysis: In the Einstein frame, the dark-photon action reveals that the longitudinal mode possesses an inflaton-dependent kinetic function, while the transverse modes remain approximately conformal and negligible for DM production.
3. Quantum Fluctuations: The power spectrum of the longitudinal mode is calculated during inflation. The authors derive that the power spectrum $P_{AL}$ is enhanced by a factor of $\Omega^2(t_k)$ relative to the standard scenario, where $t_k$ is the time of horizon exit for mode $k$.
4. Classical Evolution: Post-inflation, the evolution of the mode amplitude is governed by a classical equation of motion including Hubble friction and an effective oscillation frequency. The authors numerically solve the background evolution of the inflaton, Standard Model (SM) radiation, and dark radiation (produced via inflaton decay) to track the cosmological history from inflation through reheating to the present epoch.
5. Numerical Implementation: They employ a hybrid numerical approach, solving the full mode equation when the Hubble parameter $H$ is comparable to the oscillation frequency $\omega$, and switching to an adiabatic approximation (using an action variable $I$) when $H \ll \omega$ to ensure computational efficiency and accuracy.

Key Contributions

• Formalism for Time-Dependent Kinetic Functions: The paper provides a systematic procedure to evaluate the relic density of dark photons in models where the kinetic function varies due to Weyl transformations, correcting previous estimates that assumed a constant kinetic term.
• Application to Starobinsky Inflation: The authors apply this formalism to the Starobinsky inflation model, a leading candidate consistent with Planck data. They derive the specific relationship between the Weyl factor $\Omega$ and the inflaton field, showing that $\Omega \propto \sqrt{N_e}$ (where $N_e$ is the number of e-folds) during the relevant epoch.
• Quantitative Constraints: By imposing observational constraints on the scalar amplitude ($A_s$), the scalar spectral index ($n_s$), the dark matter density ($\Omega_{DM}h^2$), and the effective number of relativistic species ($\Delta N_{eff}$), the authors determine the viable parameter space for the dark-photon mass.

Results

• Enhancement of Abundance: The inclusion of the Weyl factor $\Omega$ significantly enhances the dark-photon power spectrum by a factor of $\Omega^2 \approx 40$ in the Starobinsky scenario. Consequently, the dark-photon mass required to reproduce the observed dark matter density is substantially lower than in models neglecting this effect.
• Predicted Mass Range: Under the framework of Starobinsky inflation and the constraints $|\Omega_{DM}h^2 - 0.1200| < 0.0012$, $|A_s \times 10^9 - 2.100| < 0.030$, and $\Delta N_{eff} < 0.30$, the authors find that the dark-photon mass must lie in the range:
  $$5.6 < m < 7.4 \, \mu\text{eV}$$
  This corresponds to an oscillation frequency of $1.4 < f < 1.8$ GHz.
• Dark Radiation Constraints: The study highlights that if the inflaton decays significantly into dark photons (relative to SM particles), the resulting dark radiation ($\Delta N_{eff}$) can exceed observational bounds. The viable parameter space requires the inflaton to decay predominantly into the SM sector.

Significance and Claims
The paper claims that the effect of the Weyl transformation is not a minor correction but a dominant factor that shifts the predicted dark-photon mass by orders of magnitude compared to previous literature. The authors emphasize that the derived mass range ($5.6 - 7.4 \, \mu\text{eV}$) falls within the target sensitivity of current and future haloscope searches using resonant cavities and other platforms.

The authors maintain a modest stance regarding the universality of their results. They acknowledge that while Starobinsky inflation fits Planck data well, there is a noted tension with recent ACT observations. They state that while their formalism is directly applicable to other inflationary models requiring Weyl transformations (citing Refs. [44, 45]), a detailed investigation of dark-photon abundance in non-standard scenarios suggested by ACT data is a subject for future work. The paper does not propose new experimental setups but rather provides a theoretical target for existing experimental strategies.