Space-time evolution and CMB anisotropies from quantum gravity

We propose an evolutional scenario of the universe which starts from quantum states with conformal invariance, passing through the inflationary era, and then makes a transition to the conventional Einstein space-time. The space-time dynamics is derived from the renormalizable higher-derivative quantum gravity on the basis of a conformal gravity in four dimensions. Based on the linear perturbation theory in the inflationary background, we simulate evolutions of gravitational scalar, vector, and tensor modes, and evaluate the spectra at the transition point located at the beginning of the big bang. The obtained spectra cover the range of the primordial spectra for explaining the anisotropies in the homogeneous cosmic microwave background.

Figure 1

The time evolution of $H$ and $\rho$ for $b_1=7$, 10, and 15. $H_{\mathrm{D}}$ is normalized to be unity, and thus ${\tau }_{\Lambda }=60$ and the inflationary solution for $H$ becomes independent of $b_1$. The dashed line denotes the Friedmann solution for $H$.

Figure 2

The time evolution of $a$ and $\rho$ for $b_1=7$, 10, and 15 in a logarithmic plot, normalized by their values at ${\tau }_{\Lambda }(=60)$ denoted by $a_{\Lambda }$ and ${\rho }_{\Lambda }$, respectively. The dashed line denotes the Friedmann solution for $\rho$.

Figure 3

The evolution of Bardeen potential $\Phi$ for $b_1=10$ and $m=0.05{\mathrm{Mpc}}^{-1}$.

Figure 4

The evolution of vector and tensor perturbations, ${{\rm Y}}_i$ and $h_{ij}^{\mathrm{TT}}$, for $b_1=10$ and $m=0.05{\mathrm{Mpc}}^{-1}$.

Figure 5

The scalar and tensor spectra at the transition point for $b_1=7$, 10, 15 with $m=0.05$ and $m=0.06$. The initial amplitudes of $b_1=10$ for each $m$ are set as $\sqrt{A_{\mathrm{S}}}={10}^{-1}$ and $\sqrt{A_{\mathrm{T}}}={10}^{-5}$. The others are arbitrarily chosen to be equal to the result of $b_1=10$ at $k=m$ for each $m$.

Figure 6

The logarithmic plot of the scalar and tensor spectra at ${\tau }_{\Lambda }$.