Stochastic quintessence

The behavior of the quintessence field is studied during inflation. In order to have a satisfactory model of dark energy, the quintessence field value today should be as insensible to the initial conditions as possible. Usually, only the dependence on the initial conditions specified at the end of inflation or, equivalently, at the beginning of the radiation dominated era, is considered. Provided the quintessence field is initially within a large but, crucially, finite interval, its present value becomes independent of the initial value it started from. The question as to whether inflation naturally drives the quintessence field to the above-mentioned interval is addressed. Since the quantum effects turn out to be important, the formalism of stochastic inflation is used in order to calculate the evolution of the quintessence field. Moreover, the quantum effects originating from the inflaton field are also taken into account and are proved to be subdominant in most cases. Finally, the requirement that the quintessence field is on tracks today is shown to imply quite tight constraints on the initial values of the quintessence and inflaton fields at the beginning of inflation. In particular, the initial value of the inflaton field cannot be too large which indicates that the quintessential scenario seems to be compatible with inflation only if the total number of e-folds is quite small.

Figure 1

Evolution of $⟨\delta {\phi }_1^2⟩/{\phi }_{\mathrm{cl}}^2$ versus the number of e-folds during inflation for three different chaotic potentials characterized by $n=2$, 4, 6. The initial values of the inflaton field correspond to an initial energy density of $V_{\mathrm{in}}=0.5m_{\mathrm{Pl}}^4$ and imply that the total number of e-folds during inflation is not the same for the three models under consideration. This is also the reason why the curves do not stop at the same place. For the $n=2$ model, $⟨\delta {\phi }_1^2⟩$ always increases and, at some point, the perturbative scheme used to calculate it breaks down. For $n=4$, $⟨\delta {\phi }_1^2⟩/{\phi }_{\mathrm{cl}}^2$ tends to a constant that is difficult to visualize because of the e-folds logarithmic scale. For $n\ge 6$, $⟨\delta {\phi }_1^2⟩$ possesses a maximum during inflation and then decreases.

Figure 2

Evolution of $⟨\delta {\phi }_2⟩/{\phi }_{\mathrm{cl}}$ with the number of e-folds during inflation for three different potentials. The initial values of the scalar field always correspond to an initial energy density of $V_{\mathrm{in}}=0.5m_{\mathrm{Pl}}^4$. As a consequence, the total number of e-folds during inflation is not the same in these three models and this explains why the three curves stop at different $N$. A value of $⟨\delta {\phi }_2⟩/{\phi }_{\mathrm{cl}}$ greater than 1 signals a breakdown of the perturbative approach. For instance, this is the case for the $n=2$ model at the end of inflation.

Figure 3

Classical evolution of the quintessence field for different initial conditions. A massive chaotic inflaton potential (i.e., $n=2$) has been assumed and the parameter $\alpha$ in the Ratra-Peebles potential has been taken to be equal to 6. The initial condition for the inflaton field is such that ${\phi }_{\mathrm{in}}=2.5\times {10}^6{m}_{\mathrm{Pl}}$ corresponding to an initial energy density of $V_{\mathrm{in}}=0.1m_{\mathrm{Pl}}^4$. The dotted-dashed line signals the limit of the slow-roll approximation. Below this line, the slow-roll approximation is no longer valid.

Figure 4

Constraints coming from the requirement that the perturbative approach is valid for the case $n=2$ and $\alpha =6$. The values of the initial conditions ${\phi }_{\mathrm{in}}$ and $Q_{\mathrm{in}}$ are chosen such that the corresponding energy density is never Planckian, i.e., $V({\phi }_{\mathrm{in}})<m_{\mathrm{Pl}}^4$ and $V(Q_{\mathrm{in}})<m_{\mathrm{Pl}}^4$. Each line corresponds to a condition specified in the figure and that we describe below. By convention, the allowed region is always the region above the corresponding line. The solid black line indicates the regime where $Q_{\mathrm{cl}}$ can be considered as constant, see Eq. (53). The dot-dashed line corresponds to the region where $⟨\delta Q_1^2⟩/Q_{\mathrm{cl}}^2<1$, originating from the quintessence noise, is small, see Eq. (68). The dotted line corresponds to the same condition except that the variance now comes from the inflaton noise. Finally, the dashed line corresponds to the condition $⟨\delta Q_2⟩/Q_{\mathrm{cl}}<1$ where the mean value originates from the quintessence noise. As we have seen before, see Eq. (64), there are four other contributions but one can easily show that they do not lead to new constraints and this is the reason why they are not represented in this figure. As discussed in the text, the most stringent constraint comes from the requirement $⟨\delta Q_1^2⟩/Q_{\mathrm{cl}}^2<1$.

Figure 5

Evolution of $⟨Q⟩$ (solid black line) calculated from Eq. (73) (but with the inflaton noise taken into account) and compared with the corresponding classical evolution (dotted-dashed line) $Q_{\mathrm{cl}}$. The dashed area represents the zone $\pm \sigma$ around the mean value where $\sigma$ has been obtained with the help of Eq. (81). The difference between $⟨Q⟩$ and $Q_{\mathrm{cl}}$ at the end of inflation can be many orders of magnitude (and is, in the present case, of about 8 orders of magnitude) demonstrating that it is crucial to take into account the quantum effects during inflation. The curves have been obtained for a massive chaotic model, i.e., $n=2$ and for the Ratra-Peebles potential with $\alpha =6$. The initial value of the inflaton is ${\phi }_{\mathrm{in}}=5.4\times {10}^6{m}_{\mathrm{Pl}}$ corresponding to an initial energy density of $V_{\mathrm{in}}=0.5\times m_{\mathrm{Pl}}^4$. The initial value of the quintessence field is $Q_{\mathrm{in}}={10}^{-4}{m}_{\mathrm{Pl}}$ corresponding to $W_{\mathrm{in}}=1.25\times {10}^{-99}{m}_{\mathrm{Pl}}^4$.

Figure 6

The quantity $⟨Q⟩$ at the end of inflation versus the initial value of the inflaton field calculated according to Eq. (83) for two chaotic models, $n=2$ and $n=4$. As noticed in the text, $⟨Q⟩$ at the end of inflation does not depend on the initial value of the quintessence field because there is an attractor solution given by Eq. (76). The fact that the final value does not depend on the parameter $\alpha$ is due to the description of the classical drift term by means of the reflecting wall. The two curves start and stop at different values of the initial inflaton field because the minimal value (defined to be the one which leads to at least 60 e-folds) and the maximal value (defined to be the one which corresponds to $V_{\mathrm{in}}=m_{\mathrm{Pl}}^4$) are not the same according to the model (i.e., $n$) under considerations (for instance, ${\phi }_{\mathrm{in}}=3.1m_{\mathrm{Pl}}$ for $n=2$ while ${\phi }_{\mathrm{in}}=4.4m_{\mathrm{Pl}}$ for $n=4$). The horizontal lines represent the maximal values of $Q$ (for different values of the index $\alpha$) at the end of inflation such that the attractor is joined today. These values are computed by means of Eq. (85).

Figure 7

Probability computed from Eq. (88) that the quintessence field is on track today for various models with $n=2$ and different power indices $\alpha$ (from top to bottom, $\alpha =2$, 6, 11). For all the models under consideration, the maximum allowed value of the initial inflaton is ${\phi }_{\mathrm{in}}=7.6\times {10}^5{m}_{\mathrm{Pl}}$ (coming from the fact that $V_{\mathrm{in}}<m_{\mathrm{Pl}}^4$). From the plots, one notices that values larger than $27m_{\mathrm{Pl}}$, $1.3\times {10}^3{m}_{\mathrm{Pl}}$, and $3.0\times {10}^3{m}_{\mathrm{Pl}}$, respectively, (corresponding to an initial energy density $V_{\mathrm{in}}$ of $1.3\times {10}^{-9}{m}_{\mathrm{Pl}}^4$, $2.9\times {10}^{-6}{m}_{\mathrm{Pl}}^4$, and $1.5\times {10}^{-5}{m}_{\mathrm{Pl}}^4$) are excluded at 99% CL. One also notices that, since the confidence region enlarges for increasing values of $\alpha$, large values of the power index $\alpha$ are statistically preferred.

Figure 8

Left panel: evolution of $⟨\delta Q_1^2⟩/Q_{\mathrm{cl}}^2$ calculated in the perturbative regime for two different set of initial conditions (i.e., two different values of ${\phi }_{\mathrm{in}}$ and $Q_{\mathrm{in}}$) and for a model with $n=2$ and $\alpha =6$. For each set of initial conditions specified explicitly in the figure, the contribution coming from the quintessence noise and the contribution originating from the inflaton noise are represented. The perturbative treatment is under control only in the case ${\phi }_{\mathrm{in}}=221m_{\mathrm{Pl}}$ and $Q_{\mathrm{in}}=0.06m_{\mathrm{Pl}}$ corresponding to initial energy densities given by $V_{\mathrm{in}}=3.7\times {10}^{-8}{m}_{\mathrm{Pl}}^4$ and $W_{\mathrm{in}}=2.8\times {10}^{-114}{m}_{\mathrm{Pl}}^4$ (since all the corresponding curves lie in the region $⟨\delta Q_1^2⟩/Q_{\mathrm{cl}}^2<1$). The plot demonstrates clearly that, in the perturbative regime, the inflaton noise is totally negligible. Right panel: evolution of $⟨\delta Q_2⟩/Q_{\mathrm{cl}}$ for the initial conditions ${\phi }_{\mathrm{in}}=221m_{\mathrm{Pl}}$ and $Q_{\mathrm{in}}=0.06m_{\mathrm{Pl}}$. The five curves correspond to the five contributions of Eq. (64). Again, it is obvious that the four contributions originating from the inflaton noise are unimportant.