Vector dark matter production during inflation in the gradient-expansion formalism

TL;DR

This work studies the production of a massive vector field as a dark matter candidate during inflation, focusing on direct inflaton couplings via a kinetic function $I_1(uthphi)$, an axial term $I_2(uthphi)$, and an inflaton-dependent mass $m(uthphi)$. The authors extend the gradient-expansion formalism (GEF), previously used for massless gauge fields, to include the longitudinal polarization by formulating a coupled set of equations for bilinear functions of the vector field and incorporating horizon-crossing boundary terms to account for quantum pumping. In a benchmark model with a Ratra-type exponential kinetic coupling and inflaton-dependent mass, they show that pure mass coupling preferentially enhances longitudinal modes, while adding kinetic coupling can shift dominance between transverse and longitudinal polarizations and drive backreaction regimes, extending inflation by several $e$-folds in strong-coupling cases. The framework remains accurate against mode-by-mode solutions in nondestructive regimes and offers a computationally efficient path to explore vector dark matter production and backreaction, paving the way for future work on kinetic mixing with the Standard Model sector.

Abstract

A massive vector field is a highly promising candidate for dark matter in the universe. A salient property of dark matter is its negligible or null coupling to ordinary matter, with the exception of gravitational interaction. This poses a significant challenge in producing the requisite amount of dark particles through processes within the Standard Model. In this study, we examine the production of a vector field during inflation due to its direct interaction with the inflaton field through kinetic and axion-like couplings as well as the field-dependent mass. The gradient-expansion formalism, previously proposed for massless Abelian gauge fields, is extended to include the longitudinal polarization of a massive vector field. We derive a coupled system of equations of motion for a set of bilinear functions of the vector field. This enables us to address the nonlinear dynamics of inflationary vector field production, including backreaction on background evolution. To illustrate this point, we apply our general formalism to a low-mass vector field whose kinetic and mass terms are coupled to the inflaton via the Ratra-type exponential function. The present study investigates the production of its transverse and longitudinal polarization components in a benchmark inflationary model with a quadratic inflaton potential. It has been demonstrated that pure mass coupling is able to enhance only the longitudinal components. By turning on also the kinetic coupling, one can get different scenarios. As the coupling function decreases, the primary contribution to the energy density is derived from the transverse polarizations of the vector field. Conversely, for an increasing coupling function, the longitudinal component becomes increasingly significant and rapidly propels the system into the strong backreaction regime.

Figures (6)

• Figure 1: Upper panel: the dependence of the energy densities $\rho_{\mathcal{Q}}$ (green dashed-dotted line) and $\rho_{\mathcal{F}}$ (purple dotted line) on the number of $e$-folds counted from the end of inflation in the model of mass coupling \ref{['eq:mass-fn']} of the vector field to inflaton with $\beta=12$, obtained by using the GEF truncated at the order $2p_{\rm max}+1=69$. Lower panel: evolution of the relative error of the GEF solution with respect to the mode-by-mode solution in momentum space.
• Figure 2: The dependence of the energy densities $\rho_{\mathcal{Q}}$ (green dashed-dotted line) and $\rho_{\mathcal{F}}$ (purple dotted line) of the longitudinal modes of the vector field as well as the inflaton energy density $\rho_{\rm inf}$ (black dashed line) on the number of $e$-folds in the model of mass coupling \ref{['eq:mass-fn']} of the vector field to inflaton with $\beta=16$, obtained by using the GEF truncated at the order $2p_{\rm max}+1=63$. The point $N=0$ is chosen at the moment of time when inflation would have ended in the absence of gauge fields. The backreaction prolongs the accelerated expansion of the universe by a few $e$-folds.
• Figure 3: Upper panel: the dependence of the energy densities $\rho_{\mathcal{E}}$ (blue solid line), $\rho_{\mathcal{B}}$ (red dashed line), $\rho_{\mathcal{Q}}$ (green dashed-dotted line), and $\rho_{\mathcal{F}}$ (purple dotted line) on the number of $e$-folds counted from the end of inflation in the model of kinetic \ref{['eq:Ratra-fn']} and mass couplings \ref{['eq:mass-fn']} of the vector field to inflaton with $\beta=8$, obtained by using the GEF truncated at the orders $n_{\rm max}=53$ for the transverse modes and $2p_{\rm max}+1=53$ for the longitudinal modes. Lower panel: evolution of the relative error of the GEF solution with respect to the mode-by-mode solution in momentum space.
• Figure 4: The same quantities as shown in Fig. \ref{['fig:km-noBR-pos']} for the case of a negative coupling constant $\beta=-6$. The GEF system was truncated at the orders $n_{\rm max}=71$ for the transverse modes and $2p_{\rm max}+1=71$ for the longitudinal modes.
• Figure 5: The evolution of the different components of the energy density of the vector field: $\rho_{\mathcal{E}}$ (blue solid line), $\rho_{\mathcal{B}}$ (red dashed line), $\rho_{\mathcal{Q}}$ (green dashed-dotted line), $\rho_{\mathcal{F}}$ (purple dotted line), and the inflaton energy density $\rho_{\rm inf}$ (thin black dashed line) in the model of kinetic \ref{['eq:Ratra-fn']} and mass couplings \ref{['eq:mass-fn']} of the vector field to inflaton with $\beta=10.4$, obtained by using the GEF truncated at the orders $n_{\rm max}=87$ for the transverse modes and $2p_{\rm max}+1=87$ for the longitudinal modes. The point $N=0$ is chosen at the moment of time when inflation would have ended in the absence of gauge fields.