Timelike Vector Field Dynamics in the Early Universe

We study the dynamics of a timelike vector field which violates Lorentz invariance when the background spacetime is in an accelerating phase in the early universe. It is shown that a timelike vector field is difficult to realize an inflationary phase, so we investigate the evolution of a vector field within a scalar field driven inflation model. And we calculate the power spectrum of the vector field without considering the metric perturbations. While the time component of the vector field perturbations provides a scale invariant spectrum when $\xi = 0$, where $\xi$ is a nonminimal coupling parameter, both the longitudinal and transverse perturbations give a scale invariant spectrum when $\xi = 1/6$.

💡 Research Summary

The paper investigates the dynamics of a timelike vector field that breaks Lorentz invariance during an accelerating phase of the early universe. Starting from a generally covariant action, the authors introduce a vector field (A_{\mu}) with a standard kinetic term (-\frac14F_{\mu\nu}F^{\mu\nu}), a mass term (\frac12 m^{2}A_{\mu}A^{\mu}), and a non‑minimal coupling (\xi R A_{\mu}A^{\mu}) to the Ricci scalar (R). The coupling constant (\xi) controls the degree of Lorentz violation: for (\xi>0) the effective mass (m_{\rm eff}^{2}=m^{2}+\xi R) can become negative, leading to tachyonic instabilities; for (\xi<0) the field is overdamped and quickly decays. In a spatially flat Friedmann‑Robertson‑Walker (FRW) background the homogeneous solution is taken to be purely timelike, (A_{\mu}=(A_{0}(t),0,0,0)), preserving isotropy at the background level.

The authors first examine whether such a vector field alone can drive inflation. By solving the background equations they find that the required slow‑roll conditions cannot be satisfied simultaneously with stability: either the field grows without bound (tachyonic case) or it is damped away before a sufficient number of e‑folds can be generated. Consequently, a pure timelike vector field is deemed unsuitable as the sole inflaton.

To overcome this limitation the study embeds the vector field in a conventional scalar‑driven inflationary model. The scalar inflaton (\phi) evolves under a potential (V(\phi)) and satisfies the usual slow‑roll conditions (\epsilon,\eta\ll1). The vector field is then treated as a spectator that evolves on the scalar‑inflation background. Its background equation acquires a Hubble‑friction term and the non‑minimal coupling, but the dominant energy density remains that of (\phi), ensuring a quasi‑de Sitter expansion.

Perturbations of the vector field are analyzed next, while metric perturbations are deliberately omitted to isolate the intrinsic vector dynamics. The first‑order perturbation (\delta A_{\mu}) is Fourier‑decomposed into a temporal component (\delta A_{0}), a longitudinal spatial component (\delta A_{\parallel}) (parallel to the wave‑vector (\mathbf{k})), and two transverse components (\delta A_{\perp}^{(i)}). Each satisfies a second‑order differential equation of the form \