Improved calculation of the primordial gravitational wave spectrum in the standard model

We show that the energy density spectrum of the primordial gravitational waves has characteristic features due to the successive changes in the relativistic degrees of freedom during the radiation era. These changes make the evolution of radiation energy density deviate from the conventional adiabatic evolution, ${\rho }_r\propto a^{-4}$, and thus cause the expansion rate of the universe to change suddenly at each transition which, in turn, modifies the spectrum of primordial gravitational waves. We take into account all the particles in the standard model of elementary particles. In addition, free-streaming of neutrinos damps the amplitude of gravitational waves, leaving characteristic features in the energy density spectrum. Our calculations are solely based on the standard model of cosmology and particle physics, and therefore these features must exist. Our calculations significantly improve the previous ones which ignored these effects and predicted a smooth, featureless spectrum.

Figure 1

The primordial gravitational wave spectrum at present, $\tau ={\tau }_0$, as a function of the comoving wavenumber, $k$ (or $kc$ in units of Hertz). The frequency of gravitational waves observed today is related to $k$ by $f_0=kc/2\pi$. The spectrum at large wavenumber is exactly scale-invariant as we have assumed de Sitter inflation. In this figure we have not taken into account the effects of the change in effective relativistic degrees of freedom or neutrino free-streaming.

Figure 2

Evolution of the effective number of relativistic degrees of freedom contributing to energy density, $g_{*}$, as a function of temperature. The solid and dashed lines represent $g_{*}$ in the SM and in the minimal extension of SM, respectively. At the energy scales above $\sim 1\mathrm{TeV}$, $g_{*}^{\mathrm{SM}}=106.75$ and $g_{*}^{\mathrm{MSSM}}\simeq 220$. At the energy scales below $\sim 0.1\mathrm{MeV}$, $g_{*}=3.3626$ and $g_{*s}=3.9091$; $g_{*}=g_{*s}$ otherwise.

Figure 3

Evolution of $a^{\prime }$ as a function of the conformal time. If $g_{*}$ and $g_{*s}$ were constant, $\rho \propto a^{-4}$ and $a^{\prime }$ would also be constant.

Figure 4

The primordial gravitational wave spectrum at present, ${\Omega }_h({\tau }_0,k)/{10}^{-10}$, as a function of the comoving wavenumber, $k$ (or $kc$ in units of Hertz). The frequency of gravitational waves observed today is related to $k$ by $f_0=kc/2\pi$. We have assumed a scale-invariant primordial spectrum and ${\Omega }_m=1-{\Omega }_r$, ${\Omega }_r=4.15\times {10}^{-5}{h}^{-2}$, $h=0.7$, and $E_{\inf }={10}^{16}\mathrm{GeV}$. We have included the effects of the effective number of relativistic degrees of freedom and neutrino free-streaming. The dashed line shows the envelope of the previous calculations which ignored the change in the number of relativistic degrees of freedom and neutrino free-streaming (Fig. 1).

Figure 5

A blow-up of Fig. 4. Note that density of vertical lines shows density of sampling points at which we evaluate ${\Omega }_h({\tau }_0,k)$. The dashed line shows the envelope of the spectrum in the SM of elementary particles.

Figure 6

Numerical solutions of tensor perturbations. The solid, dashed, and short-dashed lines show the high, medium, and low frequency modes, respectively. The higher $k$-modes enter the horizon earlier, and are damped more by the cosmological redshift. Vertical lines define the horizon-crossing time for each $k$-mode.

Figure 7

Comparison between numerical solutions and analytical solutions of tensor perturbations. The dashed and short-dashed lines show numerical solutions of the high and low frequency modes, respectively. The higher $k$-modes enter the horizon earlier, and thus the numerical solution is well approximated by the analytical solution during the radiation era, $\chi (k\tau )=j_0(k\tau )$ (solid line). On the other hand, the lower $k$-modes enter the horizon much later, and thus the numerical solution is close to the analytical solution during the matter era, $\chi (k\tau )=3j_1(k\tau )/k\tau$ (dotted line).

Figure 8

Comparison between numerical solutions and analytical solutions of tensor perturbations. The effect of neutrino free-streaming is included for numerical solutions, but not for analytical solutions. The dashed and short-dashed lines show numerical solutions of the high and low frequency modes, respectively. The higher $k$-modes enter the horizon during the radiation era after neutrino decoupling, and thus the numerical solution is damped by neutrino free-streaming compared to the analytical solution, $\chi (k\tau )=j_0(k\tau )$ (solid line). On the other hand, the lower $k$-modes enter the horizon much later, and thus the numerical solution is closer to the analytical solution during the matter era, $\chi (k\tau )=3j_1(k\tau )/k\tau$ (dotted line).

Figure 9

Derivatives of modes which entered the horizon before neutrino decoupling. The solid line shows an analytical solution, ${\chi }^{\prime }=-j_1(u)$, during the radiation era without neutrino decoupling. The dotted, short-dashed, and dashed lines show numerical solutions of ${\chi }^{\prime }(k\tau )$ for which neutrinos decoupled at ${\tau }_{\nu \mathrm{dec}}$ given by $k{\tau }_{\nu \mathrm{dec}}=1.25$, 2.5, 3.75, respectively. They are damped by giving energy to free-streaming neutrinos. Vertical lines indicate the neutrino decoupling time for each mode.

Figure 10

Derivative of a mode which entered the horizon before neutrino decoupling. The solid line shows an analytic solution,${\chi }^{\prime }=-j_1(u)$, during the radiation era without neutrino decoupling. The dashed line shows numerical solutions of ${\chi }^{\prime }(k\tau )$ for which neutrinos decoupled at ${\tau }_{\nu \mathrm{dec}}$ given by $k{\tau }_{\nu \mathrm{dec}}=5.0$. The wave is amplified by gaining energy from free-streaming neutrinos. The vertical line indicates the neutrino decoupling time.

Figure 11

The oscillatory factor $[C+D{]}^2$ with respect to $t\equiv k{\tau }_1$. The vertical line indicates $s\equiv k{\tau }_2=300\le t$ and $\Delta \tau \equiv {\tau }_1-{\tau }_2=0$. The solid, dashed, short-dashed, and dotted lines show $n=0$, 1, 2, and 5, respectively. The factor, $[C+D{]}^2$, takes on unity at $\Delta \tau \to 0$ regardless of $n$.