Vacuum and gravitons of relic gravitational waves and the regularization of the spectrum and energy-momentum tensor

The spectrum of a relic gravitational wave (RGW) contains high-frequency divergences, which should be removed. We present a systematic study of the issue, based on the exact RGW solution that covers the five stages, from inflation to the acceleration, each being a power law expansion. We show that the present RGW consists of vacuum dominating at $f>10^{11}\mathrm{Hz}$ and graviton dominating at $f<10^{11}\mathrm{Hz}$, respectively. The gravitons are produced by the four cosmic transitions, mostly by the inflation-reheating one. We perform adiabatic regularization to remove vacuum divergences in three schemes: at present, at the end of inflation, and at horizon exit, to the second adiabatic order for the spectrum, and the fourth order for energy density and pressure. In the first scheme, a cutoff is needed to remove graviton divergences. We find that all three schemes yield the spectra of a similar profile, and the primordial spectrum defined far outside horizon during inflation is practically unaffected. We also regularize the gauge-invariant perturbed inflaton and the scalar curvature perturbation by the last two schemes, and find that the scalar spectra, the tensor-scalar ratio, and the consistency relation remain unchanged.

Figure 1

The shape of ${\mathrm{\Delta }}_t^2(k,\tau )$ in Eq. (22) for $\beta =-2$. It is flat at $k\tau =0$, but has a steep slope $\propto k^1$ at $k|\tau |=1$ during inflation.

Figure 2

The spectral indices $n_t$ and ${\alpha }_t$ defined in (23) and (24) as functions of $k|\tau |$ during inflation. Note that $n_t={\alpha }_k=0$ at $k|\tau |=0$, but $n_t={\alpha }_k=1$ at $k|\tau |=1$.

Figure 3

The spectrum ${\mathrm{\Delta }}_t(f,\tau )$ at three different times: at the end of inflation, at $z\sim 1100$, and at present, respectively. The horizontal axis is the physical frequency $f=k/2\pi a({\tau }_H)$ at the present time ${\tau }_H$. The parameters $\beta =-2.0125$ and $r=0.12$ are taken for illustration.

Figure 4

The number density $|{\beta }_k{|}^2$ of gravitons produced in the $k$ mode is shown.

Figure 5

The present spectrum consists of the vacuum and graviton parts. At $f>10^{11}\mathrm{Hz}$, the vacuum is dominant and contains both quadratic and log divergences ${\mathrm{\Delta }}_t^2(k)\propto k^2$, $k^0$, and the gravitons gives only log divergence ${\mathrm{\Delta }}_t^2(k)\propto k^0$.

Figure 6

${\mathrm{\Omega }}_g(f)={\rho }_k({\tau }_H)/{\rho }_c$ at present consists of the vacuum and graviton parts, where ${\rho }_c$ is the critical density. $\beta =-2.0125$ and $r=0.12$ are taken.

Figure 7

Spectral pressure $p_k({\tau }_H)/{\rho }_c$ at present consists of the vacuum and graviton parts. $\beta =-2.0125$ and $r=0.12$.

Figure 8

Regularization at the present time ${\tau }_H$. The unregularized spectrum at the end of inflation is at the top, and the present spectrum is at the lower part.

Figure 9

The spectrum regularized at present is nonvanishing for $\beta =-2$, which would be zero if regularized during inflation [48].

Figure 10

Regularized spectral energy density ${\mathrm{\Omega }}_g(k{)}_{re}={\rho }_k({\tau }_H{)}_{re}/{\rho }_c$ at present.

Figure 11

Regularized spectral pressure $p_k({\tau }_H{)}_{re}/{\rho }_c$ at present.

Figure 12

The spectrum is regularized at the end of inflation, then evolves into the present spectrum, which turns out to be almost the same as that of Fig. 8 regularized at the present time.

Figure 13

The spectrum is regularized at the horizon exit, then evolves into the present spectrum, which is similar to that of Figs. 8 and 12, but with a lower, overall amplitude.