Emergent scale symmetry: Connecting inflation and dark energy

Quantum gravity computations suggest the existence of an ultraviolet and an infrared fixed point where quantum scale invariance emerges as an exact symmetry. We discuss a particular variable gravity model for the crossover between these fixed points which can naturally account for inflation and dark energy, using a single scalar field. In the Einstein-frame formulation, the potential can be expressed in terms of Lambert functions, interpolating between a power-law inflationary potential and a mixed-quintessence potential. For two natural heating scenarios, the transition between inflation and radiation domination proceeds through a “graceful reheating” stage. The radiation temperature significantly exceeds the temperature of big bang nucleosynthesis. For this type of model, the observable consequences of the heating process can be summarized in a single parameter, the heating efficiency. Our quantitative analysis of compatibility with cosmological observations reveals the existence of realistic models able to describe the whole history of the Universe using only a single metric and scalar field and involving just a small number of order 1 parameters.

Figure 1

The relation between $\chi$ and $\varphi$ given by Eq. (15). For this figure, we took $\sigma =4$, $\kappa =1$, and $m={10}^5\mu$ with $\mu$ given by Eq. (2). The black-dashed line corresponds to $\chi =M_P$. For reference, we indicate the values of the field $\varphi$ (in $M_P$ units) at $N=20$ $e$-folds before the end of inflation (${\varphi }_{N=20}$), at the inflationary exit (${\varphi }_{\text{end}}$), and at the onset of the kinetic domination regime (${\varphi }_{\text{kin}}$).

Figure 2

The cosmon potential for $\sigma =4$ and $\kappa =1$. The blue line corresponds to the exact expression (15), while the orange-dotted line and red-dashed lines are associated to the approximated expressions (26) and (24) respectively. For reference, we indicate the values of the cosmon field $\varphi$ (in $M_P$ units) at $N=20$ $e$-folds before the end of inflation (${\varphi }_{N=20}$), at the inflationary exit (${\varphi }_{\text{end}}$), and at the onset of the kinetic domination regime (${\varphi }_{\text{kin}}$).

Figure 3

Comparison between the inflationary predictions (30) and the latest Planck/BICEP2 data at 68% and 95% C.L. [32, 33]. The arrows go from the values obtained assuming $N=55$ $e$-folds to those for $N=65$ (cf. Sec. 7b for a more precise estimation of the number of $e$-folds). For $\sigma \gtrsim 4$, the predicted spectral tilt and tensor-to-scalar ratio lay inside the $2\sigma$ contour.

Figure 4

The cosmon equation of state in the potential (15) with $\sigma =4$ and $\kappa =1$ as a function of the number of $e$-folds $N$. The end of inflation after 60 $e$-folds is indicated with a black dot. The limit $w_{\varphi }=1$ corresponds to a kinetic dominated regime.

Figure 5

The effective coupling $\epsilon (\varphi )$ for ${\sigma }_h=2$, ${\epsilon }_{\infty }={10}^{-10}$, ${\epsilon }_1{M}_P^2={\varphi }_{\epsilon }^2$, and different values of ${\varphi }_{\epsilon }$. All cases share the same asymptotic behavior during inflation. The red-solid, blue-dashed, and orange-dotted lines correspond respectively to ${\varphi }_{\epsilon }=M_P,0.75M_P$, and ${10}^{-1}{M}_P$. In the inset, we plot $\epsilon (\varphi )$ logarithmically in order to better resolve the approach to zero and to facilitate the comparison with Fig. 2.

Figure 6

The spectra of produced $h$ particles for ${\sigma }_h=2$, ${\epsilon }_{\infty }={10}^{-10}$, ${\epsilon }_1={10}^{-1}{\varphi }_{\epsilon }^2/M_P^2$, and different choices of ${\varphi }_{\epsilon }$. All cases are evaluated at the onset of the kinetic domination regime. At that time, $H_{\text{kin}}\simeq 1.63\times {10}^{11}\mathrm{GeV}$, and ${\rho }_{\varphi }^{\text{kin}}\simeq (8.4\times {10}^{14}\mathrm{GeV}{)}^4$.

Figure 7

Evolution of the dark energy equation of state $w_{\varphi }=p_{\varphi }/{\rho }_{\varphi }$ from the inflationary era to the matter dominated era as a function of the number of $e$-folds. The end of inflation corresponds to $N=0$. For this plot, we chose $\sigma =4$, $\kappa =1$ and assumed a heating efficiency $\mathrm{\Theta }={10}^{-4}$.

Figure 8

Postinflationary evolution of the density parameters ${\mathrm{\Omega }}_{\varphi }$, ${\mathrm{\Omega }}_R$, and ${\mathrm{\Omega }}_M$ as a function of the number of $e$-folds. The end of inflation corresponds to $N=0$. Parameters are chosen as in Fig. 7.

Figure 9

Detailed view of the density parameters ${\mathrm{\Omega }}_R$, ${\mathrm{\Omega }}_M$, and ${\mathrm{\Omega }}_{\varphi }$ during matter and radiation domination as a function of the number of $e$-folds using the same parameters in Fig. 7. The expression (112) with $n=4$ and $n=3$ is depicted with black dashed lines. Nucleosynthesis corresponds to $N\simeq 41$.

Figure 10

The number of $e$-folds (104) as a function of the heating efficiency $\mathrm{\Theta }$. The blue region stands for variations of the tensor-to-scalar ratio $r$ within the range [0.01, 0.08]. The vertical lines indicate the typical values of the heating efficiency associated to gravitational and matter interactions. For these lines, we assume a number of production channels of the order of the number of degrees of freedom in the Standard Model, $\mathcal{O}({10}^2)$ [cf. Eq. (59) and Table 2]. Enhanced particle contents as those appearing in Standard Model extensions such as grand unification would translate into a larger heating efficiency and therefore into a smaller number of $e$-folds $N$.

Figure 11

The fraction of early dark energy (112) at BBN as a function of the model parameters $\kappa$ and $\sigma$. For this plot, we set $T_{\mathrm{BBN}}=1\mathrm{MeV}$ and $w_{\varphi }\simeq 1/3$.

Figure 12

Evolution of the cosmon field $\chi$ and the dimensionless cosmon potential (15) as a function of the number of $e$-folds. The end of inflation corresponds to $N=0$. For this plot, we chose $\sigma =4$, $\kappa =1$ and assumed a heating efficiency $\mathrm{\Theta }\simeq {10}^{-4}$.

Figure 13

Contour plot of the adiabaticity violation parameter ${\delta }_{\omega }$ according to Eq. (b7) in the plane of parameters ${\varphi }_{\epsilon }$ and ${\epsilon }_1$. For the other parameters, we employ ${\sigma }_h=2$, ${\epsilon }_{\infty }={10}^{-10}$, a typical field velocity $|\dot{\varphi }({\varphi }_a^{\max })|=6\times {10}^{-6}{M}_P^2$, and $k={10}^{-4}{M}_P$. The violation of adiabaticity increases toward smaller ${\varphi }_{\epsilon }$. The red solid line corresponds to ${\epsilon }_1{M}_P^2={\varphi }_{\epsilon }^2$.