Inflation from cosmological constant and nonminimally coupled scalar

We consider inflation in a universe with a positive cosmological constant and a nonminimally coupled scalar field, in which the field couples both quadratically and quartically to the Ricci scalar. When considered in the Einstein frame and when the nonminimal couplings are negative, the field starts in slow roll and inflation ends with an asymptotic value of the principal slow-roll parameter, ${\epsilon }_E=4/3$. Graceful exit can be achieved by suitably (tightly) coupling the scalar field to matter, such that at late time the total energy density reaches the scaling of matter, ${\epsilon }_E={\epsilon }_m$. Quite generically the model produces a red spectrum of scalar cosmological perturbations and a small amount of gravitational radiation. With a suitable choice of the nonminimal couplings, the spectral slope can be as large as $n_s\simeq 0.955$, which is about one standard deviation away from the central value measured by the Planck satellite. The model can be ruled out by future measurements if any of the following is observed: (a) the spectral index of scalar perturbations is $n_s>0.960$; (b) the amplitude of tensor perturbations is above about $r\sim {10}^{-2}$; (c) the running of the spectral index of scalar perturbations is positive.

Figure 1

The effective potential (cosmological constant) $V_E$ in the Einstein frame as a function of the Einstein frame field ${\varphi }_E$. In this figure ${\xi }_4=-0.1$. Left panel: ${\xi }_2=-0.01$ (blue solid), ${\xi }_2=-0.1$ (red dashes) and ${\xi }_2=-1$ (long green dashes). Right panel: ${\xi }_2=0.01$ (blue solid), ${\xi }_2=0.1$ (red dashes) and ${\xi }_2=0.2$ (long green dashes). Note that when ${\xi }_2<0$, $V_E$ has a local maximum at ${\varphi }_E=0$ ($\varphi =0$), while for ${\xi }_2>0$, $V_E$ has a local minimum at ${\varphi }_E=0$ and two local maxima at some ${\varphi }_E=\pm {\varphi }_{E0}\ne 0$. The potential $V_E$ exhibits a $Z_2$ symmetry, i.e. it is symmetric under ${\varphi }_E\to -{\varphi }_E$.

Figure 2

Principal slow-roll parameter ${\epsilon }_E$ as a function of ${\varphi }_J$ for ${\xi }_4=-0.01$. Different curves show ${\xi }_2=-0.01$ (blue solid), ${\xi }_2=-0.02$ (short red dashes), ${\xi }_2=-0.1$ (green dashes) and ${\xi }_2=-0.5$ (long orange dashes). Note that, independently of the values of ${\xi }_2$ and ${\xi }_4$ (as long as they are both negative), ${\epsilon }_E\to 4/3$ (horizontal blue dashes) when $\varphi \to \infty$. Inflation ends when ${\epsilon }_E\to 1$ (short horizontal blue dashes).

Figure 3

Spectral index $n_s$ as a function of the number of $e$-folds $N$ (from bottom to top) for ${\xi }_2=-0.005$ (long orange dashes), ${\xi }_2=-0.001$ (blue solid), ${\xi }_2=-0.003$ (green dashes) and ${\xi }_2=-0.002$ (short red dashes). Left panel: ${\xi }_4=-0.1$. Right panel: ${\xi }_4=-0.01$. Numerical investigations show that the maximum value of $n_s$ is very weakly dependent on ${\xi }_4$, and peaks for ${\xi }_4\in (-{10}^{-2},-{10}^{-1})$. The dependence on ${\xi }_2$ is much more pronounced, and $n_s$ peaks around ${\xi }_2\simeq -0.002$ (short red dashes in both left and right panels).

Figure 4

Left panel: The spectral index $n_s$ as a function of ${\xi }_2$ for $N=50$ (short red dashes), $N=60$ (blue solid) and $N=65$ (long green dashed). The maximum value of $n_s$ is very weakly dependent on ${\xi }_4$, and peaks for ${\xi }_4\in (-{10}^{-2},-{10}^{-1})$. The dependence on ${\xi }_2$ is much more pronounced, and $n_s$ peaks around ${\xi }_2\simeq -0.002$: $n_s=0.955$ (the value which is about $1\sigma$ lower than the Planck satellite best-fit value) when $N\simeq 62$ at the pivotal scale $k_{*}=0.05\mathrm{Mpc}$. In this graph ${\xi }_4=-0.02$. Right panel: The ratio of the spectra of tensor and scalar cosmological perturbations $r$ as a function of ${\xi }_2$ for $N=50$ (short red dashes), $N=60$ (blue solid) and $N=65$ (long green dashes). The maximum value of $r$ is attained for ${\xi }_2\sim -0.005$, and it is approximately inversely proportional to $|{\xi }_4|$. This means that, to get values that are observable by the near-future experiments, $|{\xi }_4|$ needs to be sufficiently small. Roughly, we have (when $N=60$) $r\sim {10}^{-6}/|{\xi }_4|$, such that in order to get an observable $r$ one needs $|{\xi }_4|\le {10}^{-3}$. In this graph ${\xi }_4=-0.0002$.

Figure 5

The ratio of the spectra of tensor and scalar cosmological perturbations $r$ versus the spectral index $n_s$ for the number of $e$-folds $N=50$ (short red dashes), $N=60$ (blue solid) and $N=65$ (long green dashes). Left panel: ${\xi }_4=-0.02$. Right panel: ${\xi }_4=-0.0002$. From the figure we see that the maximum value of $n_s$ grows as $N$ increases, favoring thus models with a large number of $e$-folds, such as models with instant preheating and models which have a post-inflationary period of kination. We also see that $r$ decreases as $N$ increases and as ${\xi }_4$ increases. However, making $|{\xi }_4|$ smaller has as a consequence a slight reduction of $n_s$.

Figure 6

Left panel: The running of the spectral index $\alpha =d{n}_s/d\ln (k)$ as a function of $n_s$ for $N=50$ (short red dashes), $N=60$ (blue solid) and $N=65$ (long green dashed). Both the maximum value of $n_s$ and $\alpha$ are very weakly dependent on ${\xi }_4$. The typical value of the running is about $\alpha \sim -{10}^{-3}$, which is consistent with the current Planck data and it is about a factor of a few smaller in value than the central value favored by the Planck (and other) data, $\alpha \sim -0.003\pm 0.007$. In this graph ${\xi }_4=-0.02$. Right panel: The same as in the left panel but with ${\xi }_4=-0.2$. The value of the running $|\alpha |$ grows slowly as $|{\xi }_4|$ increases.

Figure 7

Left panel: The tensor-to-scalar ratio $r$ versus the running of the spectral index $\alpha =d{n}_s/d\ln (k)$ for $N=50$ (short red dashes), $N=60$ (blue solid) and $N=65$ (long green dashed). For these curves ${\xi }_4=-0.02$. Right panel: The same as in the left panel but with ${\xi }_4=-0.0002$. Note that as $N$ increases the values of $r$ and $\alpha$ decrease. A comparison of the left and right panels reveals that $r\propto 1/|{\xi }_4|$, which was already pointed out above. For wide ranges of values of ${\xi }_2$ and ${\xi }_4$ the values of $\alpha$ and $r$ are small enough to be consistent with the current observations.