Theory and numerics of gravitational waves from preheating after inflation

Preheating after inflation involves large, time-dependent field inhomogeneities, which act as a classical source of gravitational radiation. The resulting spectrum might be probed by direct detection experiments if inflation occurs at a low enough energy scale. In this paper, we develop a theory and algorithm to calculate, analytically and numerically, the spectrum of energy density in gravitational waves produced from an inhomogeneous background of stochastic scalar fields in an expanding universe. We derive some generic analytical results for the emission of gravity waves by stochastic media of random fields, which can test the validity/accuracy of numerical calculations. We contrast our method with other numerical methods in the literature, and then we apply it to preheating after chaotic inflation. In this case, we are able to check analytically our numerical results, which differ significantly from previous works. We discuss how the gravity-wave spectrum builds up with time and find that the amplitude and the frequency of its peak depend in a relatively simple way on the characteristic spatial scale amplified during preheating. We then estimate the peak frequency and amplitude of the spectrum produced in two models of preheating after hybrid inflation, which for some parameters may be relevant for gravity-wave interferometric experiments.

Figure 1

Would-be emission of a graviton $h_{ij}$ with momentum $\mathbf{k}$ from the annihilation of two scalar waves $\varphi (\mathbf{p})$ and $\varphi (\mathbf{k}-\mathbf{p})$ with momenta $\mathbf{p}$ and $\mathbf{p}-\mathbf{k}$. Helicity 2 of the emitted graviton cannot match the helicity zero of the incoming scalar waves.

Figure 2

Two-dimensional slice through a three-dimensional realization of the scalar field $\chi$ from a numerical simulation of preheating in the $\lambda {\varphi }^4+g^2{\varphi }^2{\chi }^2$ model of the next section. The horizontal axes correspond to spatial coordinates and the vertical axis corresponds to the field’s values.

Figure 3

Spectrum of energy density in gravity waves calculated along nine different directions in $\mathbf{k}$ space. The parameters for this run were the same as for Fig. 4; see Sec. 6b for details.

Figure 4

Spectrum of gravity-wave energy density in physical variables today (43), accumulated up to the time $x_f=240$, for the model (66) with $q=120$. The two spectra were obtained from simulations with different box sizes, and averaged over different directions in $\mathbf{k}$ space.

Figure 5

The thick curve shows the total energy density in gravity waves (42) accumulated up to the time $x_f$, as a function of $x_f$. The thin curve shows the evolution with time of the total particles number density, $n_{\mathrm{tot}}=n_{\chi }+n_{\varphi }$, rescaled to fit on the same figure.

Figure 6

Spectrum (71) of the gravity-wave energy density, accumulated up to different times $x_f$, as a function of the comoving momentum $k$ (in units of $\lambda {\varphi }_0$). The spectra are shown from $x_f=90$ to $x_f=240$ with spacing $\Delta x_f=10$.

Figure 7

Measure of the (unnormalized) total energy density in the two scalar fields per logarithmic momentum interval at different moments of time. The same times as in Fig. 6 are shown, the spectra moving towards UV from $x=90$ to $x=240$ with spacing $\Delta x=10$.

Figure 8

Evolution with time of the gravity-wave energy density spectrum (left panels) and of the scalar field spectrum (right panels) for different values of $q=g^2/\lambda$. From top to bottom, $q=1.2$, 120, 128, and 1130. The left panels show the spectra $({\Omega }_{\mathrm{gw}}{)}_p$ [see (71)] accumulated up to different times $x_f$, as a function of the comoving momentum $K=k/(\sqrt{\lambda }{\varphi }_0)$. The right panels show the total scalar field spectrum $K^2(|X_k|+|{\Phi }_k|)/{\varphi }_0$ at different times, as a function of $K$.

Figure 9

Typical behavior of the modes amplified by parametric resonance. The left figure shows the evolution with time of the real part of $X_k(x)$ [see (67)] for $k$ inside a resonance band, here for $q=5050$. The right figure shows the evolution of the corresponding occupation number, together with the inflaton zero mode. The two figures are taken from 36.

Figure 10

Spectrum of energy density in gravity waves at the time of their production, (71), as a function of their comoving wave number $k$ (in units of $\sqrt{\lambda }{\varphi }_0$). The two thin lines correspond to the spectrum averaged over different directions in $\mathbf{k}$ space, as obtained from the lattice calculation. The thick line corresponds to the spectrum obtained from the analytical formula (78). The spectra are calculated in the model (66) with $q=128$ during the linear stage of preheating [$x_f\simeq 50$ in (28)].