What spectators do during inflation

Abstract

The inflaton equation of motion including one loop radiative corrections from spectator fields is obtained. We consider a massless scalar conformally coupled to gravity and a massless fermion Yukawa coupled to the inflaton as models for spectators that \emph{do not feature} gravitational particle production, their production during slow roll is solely a consequence of their coupling to the inflaton. The one-loop self energy and the fully renormalized equation of motion of the inflaton are obtained and solved explicitly for an inflaton potential $m^2\varphi^2/2$. The solution features Sudakov-type logarithmic secular terms, which are resumed via the dynamical renormalization group and compared to the solutions with a phenomenological friction term. During $N_e$ e-folds of slow roll inflation the inflaton evolves as $\varphi^{(0)}_{Isr}(t)\,e^{\frac{m^2Γ}{9H^3}\,N_e(t)}$ for the phenomenological friction term $Γ$ and $\varphi^{(0)}_{Isr}(t)\,e^{ΥN^2_e}$ with $Υ= -\frac{λ^2}{24π^2 H^2} ; \frac{y^2_R}{12π^2}$ for the radiative corrections from bosonic and fermionic spectators respectively where $\varphi^{(0)}_{Isr}(t)$ is the slow roll solution in absence of interactions, showing that a phenomenological friction term is not reliable. A generalization of the optical theorem to a finite time domain and cosmological expansion is introduced to obtain the distribution function $f(k,t)$ and total number of spectators produced \emph{during slow roll}. $f(k,t)$ is peaked at superhorizon scales and the total number of particles grows $\propto e^{3N_e}$. A non-perturbative mean field theory is introduced to describe the self-consistent evolution of the inflaton coupled to spectators, its linearized version reproduces the self-energy, the inflaton equation of motion and the results on particle production.

Figures (3)

• Figure 1: Interaction vertices with the composite operator $\mathcal{O}^\pm$ from the Lagrangian (\ref{['shiftedlag']}). The solid lines correspond to the fluctuations of the inflaton field $\delta^\pm$ the dashed lines represent both$:\chi^2_s$ and $:\overline{\psi} \psi:$ for the respective branches ($\pm$). The dark dot stands for the inflaton condensate $\mathcal{X}$ .
• Figure 2: Representation of the diagrams leading to the equation of motion (\ref{['1lupeom']}). The solid line is the propagator of the inflaton fluctuation $\langle \delta^+ \delta^+\rangle$, the gray square is the tree level term $[\mathcal{X}^{"}(\eta)+\frac{d}{d\mathcal{X}}\widetilde{V}(\mathcal{X};\eta)]$, the dark dot stands for $\mathcal{X}$, the dashed lines represent both $:\chi^2_s$ and $:\overline{\psi} \psi:$ for the respective branches ($\pm$). The loops represent the self energy $\Sigma$.
• Figure 3: The function $\Omega(q)$ (\ref{['omedis']}) for $\beta_-=-1$ vs. $q$.