Gravitational waves from the end of inflation: Computational strategies

Parametric resonance or preheating is a plausible mechanism for bringing about the transition between the inflationary phase and a hot, radiation dominated universe. This epoch results in the rapid production of heavy particles far from thermal equilibrium and could source a significant stochastic background of gravitational radiation. Here, we present a numerical algorithm for computing the contemporary power spectrum of gravity waves generated in this post-inflationary phase transition for a large class of scalar-field driven inflationary models. We explicitly calculate this spectrum for both quartic and quadratic models of chaotic inflation, and low-scale hybrid models. In particular, we consider hybrid models with an inverted potential. These models have a very short and intense period of resonance which is qualitatively different from previous examples studied in this context, but we find that they lead to a similar spectrum of gravitational radiation.

Figure 1

The Mathieu Stability/Instability chart. The imaginary part of the Mathieu critical exponent is plotted, with darker colors corresponding to a larger imaginary component. Outside the heavy black lines the exponent is real-valued, and the corresponding solutions are strictly oscillatory. The diagonal line corresponds to $A_q=2q$.

Figure 2

From top to bottom: The scale factor, the variances of $\varphi$ (solid line) and $\chi$ (dotted line), $S_{11}^{TT}$ for a mode corresponding to $|k|\approx 1.4\times {10}^8\mathrm{Hz}$ today, and the maximum height of the gravitational wave spectrum for this mode as a function of time for $\lambda {\varphi }^4$ inflation with $\lambda ={10}^{-14}$ and $g^2/\lambda =120$.

Figure 3

The upper plot shows the magnitude, in arbitrary units, versus rescaled time of the diagonal components, $h_{11}$ (solid line), $h_{12}$ (dotted line), and $h_{13}$ (dashed line) for a mode corresponding to $|k|=1.4\times {10}^8$ today for a $\lambda {\varphi }^4$ model. The lower plot shows $k_1{h}_{11}(k)+k_2{h}_{12}(k)+k_3{h}_{13}(k)$, which demonstrates that perturbation is transverse via (2).

Figure 4

The upper plot shows the magnitude, in arbitrary units, versus rescaled time of the diagonal components, $h_{11}$ (solid line), $h_{22}$ (dotted line), and $h_{33}$ (dashed line) for a mode corresponding to $|k|=1.4\times {10}^8$ today for a $\lambda {\varphi }^4$ model. The lower plot shows their sum.

Figure 5

The evolution of the gravitational wave spectrum for a quartic inflation model, where $\lambda ={10}^{-14}$ and $g^2/\lambda =120$.

Figure 6

The evolution of the power spectra of the fields for both $\varphi$ (above) and $\chi$ (below). This is done for the case where $\lambda ={10}^{-14}$ and $g^2/\lambda =120$ on a ${128}^3$ lattice.

Figure 7

From top to bottom: The scale factor, the variances of $\varphi$ (dotted line) and $\chi$ (solid line), $S_{11}^{TT}$ for a mode corresponding to $|k|=5.4\times {10}^8\mathrm{Hz}$ today, and the maximum height of the gravitational wave spectrum as a function of time. This is for the $m={10}^{-6}{m}_p$ and $g^2{m}_{\mathrm{pl}}^2/m^2=2.5\times {10}^5$ model.

Figure 8

From top to bottom: The scale factor (normalized to one at the beginning of the simulation), the variances of $\varphi$ (solid line) and $\chi$ (dotted line), and one component, $S_{11}^{TT}$ for a mode corresponding to $|k|\approx 5.4\times {10}^7\mathrm{Hz}$ today, and the maximum height of the gravitational wave spectrum as a function of time. This is for the $g={10}^{-10}$ case.

Figure 9

From top to bottom: The scale factor (normalized to one at the beginning of the simulation), the variances of $\varphi$ (solid line) and $\chi$ (dotted line), and one component, $S_{11}^{TT}$ for a mode corresponding to $|k|\approx 5.2\times {10}^7\mathrm{Hz}$ today, and the maximum height of the gravitational wave spectrum as a function of time. This is for the $g=-{10}^{-10}$ case.

Figure 10

The overall spectrum (top panel) and the individual spectra (bottom panel) for the simulation shown in Fig. 8. In the lower plot, the dotted line shows gravitational waves due to $\varphi$-particle production while the solid line denotes the contribution from the $\chi$ field.

Figure 11

The spectrum of gravitational waves generated during preheating with a negative coupling, with the same parameters as the simulation shown in Fig. 9.

Figure 12

This shows a test of the integration step size. The dotted lines show simulations, with varying box sizes, where the metric perturbations are evolved at every time step, the solid line shows the case where the time step for the metric perturbation is increased.

Figure 13

We show the results of simulation a quartic inflation with $g^2/\lambda =120$ using both our spectral code, and the Green’s function algorithm. This simulation matches Figure 4 of 18, and the Green’s function results are shown with a dotted line. We show results from the spectral code for three different lattice sizes—${128}^3$ (solid line), ${256}^3$ (dashed line), and ${512}^3$ (dot-dashed line). The last two lines overlap, so we are presumably approaching the continuum limit, and we also have excellent agreement with the results of 18, in which the gravitational wave spectrum was obtained using a very different numerical algorithm.