Dark photon dark matter from a rolling inflaton

Technical Summary: Dark Photon Dark Matter from a Rolling Inflaton

Problem and Motivation
The nature of dark matter remains unresolved, and the lack of detection for Weakly Interacting Massive Particles (WIMPs) has driven interest in non-thermal production mechanisms. Specifically, vector (or "dark photon") dark matter has emerged as a viable candidate, often produced via couplings between a dark abelian gauge boson and a separate sector, such as the inflaton. Previous work [5] proposed that a dark abelian gauge field coupled to a rolling inflaton via a $\phi F \tilde{F}$ coupling could be produced through a tachyonic instability, generating the observed relic abundance for masses in the range $\mu\text{eV} \lesssim m \lesssim 10\text{ TeV}$. However, a detailed examination of the cosmic evolution of the energy density spectrum, particularly the preservation of the spectral peak and the resulting spatial distribution (clumping) of the dark matter, was required. This paper addresses the specific case where dark photons are relativistic at the time their mass is generated, computing the late-time power spectrum and relic abundance.

Methodology
The authors analyze the production and evolution of a dark vector field $A_\mu$ coupled to a slow-rolling inflaton $\phi$ in a Friedmann-Robertson-Walker metric. The action includes a coupling term $\frac{\alpha}{4f} \phi F_{\mu\nu} \tilde{F}^{\mu\nu}$.

1. Production Mechanism: The coupling induces a time-dependent term in the equation of motion for the transverse modes, characterized by an instability parameter $\xi \equiv \frac{\alpha \dot{\phi}}{2Hf}$. When $\xi > 0$, one transverse polarization ($A_+$) experiences a tachyonic instability, leading to exponential growth. The authors derive equations of motion for both transverse and longitudinal modes.
2. Analytic and Numerical Approaches:
   • Analytic: Using WKB approximations and Whittaker functions, the authors derive approximate solutions for the mode functions during inflation, assuming a constant $\xi$. This provides estimates for the total energy density and parameter space constraints.
   • Numerical: Recognizing that $\xi$ varies significantly near the end of inflation (as the slow-roll parameter $\epsilon \to 1$), the authors numerically solve the coupled equations of motion for the inflaton and the vector field. This is crucial for accurately determining the shape and height of the power spectrum peak, which analytic approximations may misrepresent due to the exponential sensitivity to $\xi$.
3. Cosmic Evolution: The study tracks the redshift of the dark vector energy density from the end of inflation through reheating, radiation domination, and matter-radiation equality. They distinguish between regimes where the dark vector is relativistic ($q \gg m$) and non-relativistic ($q \ll m$), analyzing the transition where the mass term dominates.
4. Spectral Analysis: The authors compute the energy density spectrum ($d\rho/d\ln k$) and the density contrast power spectrum ($P_\delta(k)$) to determine the spatial distribution of the dark matter.

Key Contributions and Results

• Relic Abundance and Parameter Space: The paper derives the present-day relic abundance $\Omega_{DM}$ as a function of five parameters: the dark vector mass $m$, the Hubble scale during inflation $H$, the coupling strength at the end of inflation $\xi_{end}$, and the reheating efficiency parameters $\epsilon_R$ and $\epsilon_H$. They identify a viable parameter space where the transverse mode alone can account for dark matter, spanning $m \sim \mu\text{eV}$ to $\text{TeV}$ and $H \sim 100\text{ GeV}$ to $10^{14}\text{ GeV}$. Constraints are imposed to ensure efficient production, avoid overproduction, prevent thermalization with the visible sector, and neglect back-reaction effects.
• Preservation of the Spectral Peak: A central result is the demonstration that the peak in the power spectrum, generated at the end of inflation at scales corresponding to the Hubble scale ($k \sim a_{end}H_{end}$), is preserved throughout cosmic evolution. The authors explicitly show that while the electric and magnetic components evolve differently (with the magnetic component "catching up" to the electric component during super-horizon evolution), the total energy density spectrum retains the peak structure.
• Clumpy Nature of Dark Matter: The analysis reveals that the peak in the power spectrum corresponds to very small physical scales today, $\ell_{today} \sim \text{cm} - 100\text{ km}$. This implies that the dark photon dark matter produced via this mechanism is "clumpy" on these scales.
• Isocurvature Constraints: The authors compute the power spectrum of isocurvature perturbations. They find that at large scales relevant for the Cosmic Microwave Background (CMB), the power is highly suppressed (falling off as $\sim k^3$), ensuring that the mechanism evades current CMB constraints on isocurvature modes. This suppression arises because the tachyonic modes experience maximal growth only when their wavelength is of order the horizon, combined with the time dependence of $\xi$.
• Comparison with Longitudinal Modes: The paper contrasts the transverse mode (tachyonic production) with the longitudinal mode (produced via inflationary fluctuations). The longitudinal mode peaks at a scale $k_*$ determined by the mass $m$ and Hubble scale, potentially leading to much larger clumping scales ($10^{-1} \text{ km} - 10^{10} \text{ km}$). If both modes contribute significantly, a "doubly clumpy" structure with two distinct peaks in the power spectrum could emerge.

Significance and Outlook
The paper establishes that the tachyonic production mechanism at the end of inflation is a robust source of non-thermal dark photon dark matter. Its primary significance lies in the detailed characterization of the late-time energy density spectrum and the confirmation that the inflationary peak is preserved, leading to a distinct prediction of small-scale clumping ($\text{cm} - 100\text{ km}$). This clumping nature differentiates this scenario from other dark matter candidates and suggests specific phenomenological consequences for structure formation.

The authors note that the non-relativistic case (where the mass is generated before the dark vectors become non-relativistic) yields different results and is left for future work. They also highlight potential avenues for further study, including the production of dark charged particles via the Schwinger effect, gravitational wave signals from dark Higgs phase transitions, and the effects of kinetic mixing with the visible sector. The preservation of the spectral peak offers a potential link between the observed dark matter distribution and the properties of the inflaton potential, though this connection is reserved for future investigation.