Gravitational waves from preheating in inflation with Weyl symmetry

Inflation with Weyl scaling symmetry provides a viable scenario that can generate both the nearly scaling invariant primordial density fluctuation and a dark matter candidate. Here we point out that, in addition to the primordial gravitational waves (GWs) from quantum fluctuations, the production of high-frequency GWs from preheating in such inflation models can provide another probe of the inflationary dynamics. We conduct both linear analytical analysis and nonlinear numerical lattice simulation in a typical model. We find that significant stochastic GWs can be produced, and the frequency band is located around ${10}^8\mathrm{Hz}–{10}^{10}\mathrm{Hz}$, which might be probed by future resonance-cavity experiments.

Figure 1

Left: several examples of nonconvex inflaton potential derived from Weyl $F(\widehat{R},\varphi )$ gravity. The parameters are set as follows: $c_1={10}^9$, $c_2=1$, $k_1={10}^9$, $k_2=1$, $\alpha ={10}^9$, $\beta ={10}^9$, $\zeta ={10}^9$. Right: approximate potential in Eq. (16) with $\xi =0.1$, 1, 10, and $\alpha$ is chosen as $\alpha =5\times {10}^9\sqrt{\xi }$ obeying the CMB constraint.

Figure 2

Observational constraints vs theoretical predictions in Weyl polynomial model Eq. (15). Upper: predictions of tensor-to-scalar ratio $r$ and spectral index $n_s$ with various parameters $\zeta$, $\beta$ and $e$-folding number $N\sim (50,60)$. Blue area is the observational constraints given by the BICEP/Keck Collaboration [68]. Lower: viable parameter space allowed by BICEP/Keck observations with $N=60$. Here we have redefined $\xi \equiv \frac{{\alpha }^2}{3\beta \zeta }$. The color gradient represents the tensor-to-scalar ratio $r$ obtained with the corresponding parameters.

Figure 3

Resonance strength $\mathrm{Re}({\mu }_k)$ for the potential $V(\mathrm{\Phi })$ in Eq. (16) as a function of momentum $k$ and initial field value ${\mathrm{\Phi }}_i$ with the colorbar denoting the real part of the Floquet exponent $\mathrm{Re}({\mu }_k)$. $\alpha$ is chosen the same as in Fig. 1. The corresponding momentum $k$ is normalization by $\tilde{k}=\frac{k}{\omega }$, $\omega$ for different $\xi$ given in Sec. 3b. The initial field value ${\mathrm{\Phi }}_i$ is normalized by ${\tilde{\mathrm{\Phi }}}_i=\frac{{\mathrm{\Phi }}_i}{m_{\mathrm{pl}}}$.

Figure 4

Evolution of absolute value of the average field value $\tilde{\mathrm{\Phi }}$, the root-mean-squared of field fluctuation $(⟨\delta {\tilde{\mathrm{\Phi }}}^2⟩{)}^{1/2}$, the energy density $\tilde{\rho }$ and the energy density of fluctuation ${(⟨\delta {\tilde{\rho }}^2⟩)}^{1/2}$. The energy density $\rho$ is divided by ${\omega }^2{m}_{\mathrm{pl}}^2$, $\tilde{\rho }=\frac{\rho }{{\omega }^2{m}_{\mathrm{pl}}^2}$. The blue line captures the case when $\xi =1$, and the yellow line captures the case when $\xi =10$.

Figure 5

The evolution of power spectrum of $\mathrm{\Phi }$ with each curve as a snapshot. The spectra are plotted at time $\tilde{\eta }=0,5,10,\dots$, and evolve from the red curves (early times) to the blue ones (late times).

Figure 6

The evolution of GW energy spectrum with each curve as a snapshot. The spectra are plotted at time $\tilde{\eta }=0,5,10,\dots$, and evolve from the red curves (early times) to the blue ones (late times).

Figure 7

GW energy spectrum today for preheating in Weyl gravity for $\xi =0.1$, $\xi =1$, $\xi =10$ and $\xi =100$. The sensitivity of resonance cavities is shown as the blue curve [79, 80].

Figure 8

Evolution of Hubble violation for $\xi =1$ and $\xi =10$.

Figure 9

GW energy spectra comparison for different infrared momentum (lattice space).

Figure 10

GW energy spectra comparison for different $N$.