Cosmological consequences of a scalar field with oscillating equation of state. III. Unifying inflation with dark energy and small tensor-to-scalar ratio

The present standard cosmological model is known as the Lambda Cold Dark Matter ($\Lambda$CDM) model, in which the cosmological constant $\Lambda$ is generally referred to as the simplest explanation of the late-time cosmic acceleration [1, 2]. However, simpleness is not an indicator of naturalness, and the latter seems to be more important for a physical theory ¹¹1Simpleness represents mathematical complexity and naturalness represents the probability of choosing by nature.. The $\Lambda$CDM model is unnatural due to an incredible coincidence: matter dominated the $\Lambda$ by many orders of magnitude through most of the cosmic history, but they are comparable at today [4]. Dynamical dark energy provides an alternative explanation to the cosmic acceleration, some of which can alleviate the coincidence problem with the tracker property [5, 6, 7, 8]. The core argument is that the tracker property allows the initial conditions of dark energy to vary within a wide range without affecting the late-time evolution. However, the problem is not solved as the relative dark energy density is still much less than $1$ through most of the history in the tracker models (see Fig. 6 in [7] for an example).

Just releasing the initial conditions of dark energy cannot solve the coincidence problem. Intuitively, what we need may be a multi-accelerating Universe (MAU), in which the normal matters and dark energy alternately dominated each other across the whole cosmic history. The present day belongs to one of the transitions, and thus the coincidence problem disappears. This scenario can be realized by oscillating dark energy, which was first studied in [9] with a sinusoidal-modulated exponential potential $V(\phi)\propto\exp(-\lambda\phi)[1+A\sin(B\phi)]$. Follow-up works mainly focus on exploring other forms of the oscillating dark energy, e.g., multiple scalar fields realization [10], everpresent $\Lambda$ [11], and parameterized equation of state (EOS) [12, 13, 14, 15, 16]. Note that the Lagrangian can be reconstructed from a given EOS [17, 18, 19, 20] and the result generally has a specified dependence on the Hubble constant $H_{0}$. This $H_{0}$-dependence is unattractive for a physical theory (see discussions in [1, 5, 6, 9]), and should not appear in the desired model. In [21], we proposed a new oscillating dark energy model with the potential

Meanwhile we showed that the parameter space of $\{\lambda_{1}+\lambda_{2}>4$, $0<\lambda_{2}<0.39$, $\alpha=\mathcal{O}(1)$ and $V_{0}$ is arbitrary$\}$ is able to explain the late-time cosmic acceleration and solve the coincidence problem. In this model, the theoretically preferred Universe evolves in an oscillating scaling manner in the radiation epoch and in a chaotic accelerating manner in the matter epoch [22]. Mathematically, the radiation-matter transition induces a process of period-doubling bifurcation to chaos.

Besides the late-time acceleration, inflation is the other widely accepted accelerating phase in the Universe [23]. These two accelerating phases may be driven by the same field. This idea emerged in the late 1980s [5, 6] and a concrete model, named quintessential inflation, is proposed in [24] after the observational confirmation of dark energy [25, 26]. The MAU can also provide such a unified picture as its name implies. On this topic, previous work, e.g., [12, 17, 20], has studied the cosmic background evolution based on the parameterized oscillating EOS or its reconstructed Lagrangian. However, in the era of precision cosmology [27], the performance of the MAU in inflation merits further discussion. In this paper, we study in detail the inflation model driven by the scalar field described by Eq. (1), and explore the general predictions of this kind of model.

This paper is organized as follows. In Sec. II, using the slow-roll approximation, we analyze the background evolution and the primordial inhomogeneities generated during inflation for our model. In particular, we prove an upper limit of the tensor-to-scalar ratio for a class of general MAU model. Section III analyzes the post-inflationary background evolution without considering the reheating process. Section IV shows that all matters of the familiar Universe can be from gravitational reheating in our model. Discussion is presented in Sec. V. Throughout this paper, we adopt the SI units and inherit the notations in [21], e.g., the slope $\lambda\equiv-V^{\prime}/V$, ${}^{\prime}\equiv{\rm d}/{\rm d}\phi$, $\phi$ is dimensionless and $[V]={\rm length}^{-2}$.

The early Universe is assumed to be dominated by the scalar field with the potential Eq. (1). As shown in Fig. 2(a), the potential changes periodically between flat ($\lambda\approx\lambda_{2}$) and steep ($\lambda\approx\lambda_{1}$). A natural idea is that the flat part can drive the slow-roll inflation [28, 29, 30] while the steep part can end it. To be consistent with the slow-roll conditions but without loss of generality, we assume $\phi\approx\alpha\pi$ ($\lambda\approx\lambda_{2}$) at the beginning of inflation. With this setting, inflation would end around $\phi\approx 2\alpha\pi$ ($\lambda\approx\lambda_{1}$). Here we present detailed analysis of this inflation process.

Generally, the single scalar field slow-roll inflation ends once the potential becomes steep, and then the scalar field oscillates near the minimum of its potential [57]. However, things will be different in our model as there is no local minimum of the potential (note that $0<\lambda_{2}\leqslant\lambda\leqslant\lambda_{1}$). A monotonic potential will drive the scalar field roll to infinity. Here we present a quantitative analysis of this process. In this section, reheating and its backreaction on the background dynamics [57] are not taken into account.

Here the slow-roll approximation is no longer applicable due to the non-negligible kinetic energy of the scalar field. Inspired by [22], what we need may be the dynamical system form of Eq. (2). Introducing the dimensionless variables $x_{1}\equiv\dot{\phi}/(\sqrt{6}H)$ and $\nu\equiv\sqrt{6}(\lambda^{2}-V^{\prime\prime}/V)$, Eq. (2) can be rewritten as

where $N\equiv\ln(a/a_{i})$. A constraint equation for $\lambda$ and $\nu$ is given by Eq. (9) in [21]. To characterize the evolution of this system, we use the EOS $w_{\phi}=2x_{1}^{2}-1$ [21] and the Hubble slow-roll parameters [42]

where the above derivations following the definitions use Eq. (2). To lowest order, we have $\epsilon_{H}=\epsilon_{V}$, $\eta_{H}=\eta_{V}-\epsilon_{V}$ and $n_{\rm s}=1+2\eta_{H}-4\epsilon_{H}$. The use of Hubble slow-roll parameters facilitates the analysis based on the dynamical system Eq. (18).

To get a first glance of the system’s property, based on Eq. (18), Fig. 2 plot the evolution of $w_{\phi}$, $\epsilon_{H}$ and $\eta_{H}$ for a representative set of model parameters. This figure shows that inflation ends naturally in our model as we expected. Especially, inflation will not happen again when the scalar field passes through the second flat part ($\phi\approx 3\alpha\pi$ in our settings, see the first paragraph in Sec. II). The reason is the scalar field accumulates sufficient kinetic energy as it passes through the first steep part ($\phi\approx 2\alpha\pi$) and then it will quickly pass through the following flat parts.

After the end of inflation, Fig. 2 shows the existence of oscillation. Here we investigate the oscillation frequency and the average EOS $\overline{w}_{\phi}$ during oscillation. Similar to [21, 22], the result of exponential potential may provide direct clues. For $V(\phi)\propto\exp(-\lambda\phi)$, where $\lambda$ is a positive constant, the system’s evolution equation is given by Eq. (18a), and the stable critical point is $x_{1}=\lambda/\sqrt{6}$ if $\lambda<\sqrt{6}$ and $x_{1}=1$ if $\lambda\geqslant\sqrt{6}$ [58]. For the dynamical system Eq. (18), considering $w_{\phi}=2x_{1}^{2}-1$ and the results obtained for the pure exponential potential, we may be able to find an approximate EOS

during oscillation. As $w_{\phi}\leqslant 1$ for all the quintessence models, we guess that the oscillation only occurs when $\overline{w}_{\phi}<1$, i.e., approximately $\lambda_{1}+\lambda_{2}<2\sqrt{6}$. Otherwise, if $\overline{w}_{\phi}=1$, there will be no upper room for $w_{\phi}$ to be oscillating. To find an approximation of the oscillation frequency within this parameter space, we can use the Fourier series method to solve Eq. (18). Details can be found in the appendix of [22], and the only difference is that the background values should be $x_{1}=(\lambda_{1}+\lambda_{2})/\sqrt{24}$, $\lambda=(\lambda_{1}+\lambda_{2})/2$ and $\nu=0$. Note that this setting is consistent with the $\lambda$-$\nu$ constraint given by Eq. (9) in [21] as the background means $\lambda_{1}=\lambda_{2}$. The result of the oscillation frequency is

Note that, in principle, we can only prove Eqs. (20) and (21) hold for $\lambda_{1}-\lambda_{2}\ll 1$, which is outside the model’s viable parameter space.

Fig. 3 plots the numerical results of $\overline{w}_{\phi}$ and the Fourier transform $\tilde{x}_{1}(f)$, together with Eqs. (20) and (21). When $\lambda_{1}\approx\lambda_{2}$ ($\lambda_{1}-\lambda_{2}\ll 1$), for both $\overline{w}_{\phi}$ and $f_{\rm os}$, the numerical and approximate results are in good agreement as we expected. The difference between the numerical results and the approximations grows as $\lambda_{1}$ increases. However, the difference miraculously becomes negligible when $\lambda_{1}\gtrsim 4$. Especially, the boundary $\lambda_{1}+\lambda_{2}=2\sqrt{6}\approx 4.899$ is verified. Therefore, for the viable parameter space ($\lambda_{1}+\lambda_{2}>4$), Eqs. (20) and (21) are good approximations of the exact results. Equation (20) together with $\lambda_{1}+\lambda_{2}>4$ give $\overline{w}_{\phi}>1/3$, i.e., $\overline{w}_{\phi}$ is larger than the EOS of radiation when the Universe is dominated by the oscillating scalar field. If radiation is generated during this period, then its relative energy density will increase with cosmic expansion and the Universe will gradually enter the radiation era. Fig. 3 also shows that there is no period-doubling bifurcation and chaos in the three-dimensional dynamical system Eq. (18). This is different with the properties of the four-dimensional dynamical system discussed in [22].

Terminology might be mentioned. In the literature, the decelerating post-inflationary period may be called “deflationary” [60] or “kination” [61]. In our model, the post-inflationary dynamics is dominated by the scalar field with an average EOS $\overline{w}_{\phi}>1/3$. In this period, both the kinetic and potential energy of the scalar field are not negligible. Therefore, we call this period “deflation” instead of “kination”.

At the end of the inflation, the Universe is very cold due to the fast cosmic expansion. A mechanism is needed to reheat the Universe [23]. In modern cosmology, the reheating is typically assumed to occur through the decay of inflaton, which is sourced by the nonminimal coupling of inflaton and other fields [62, 63, 64, 65, 66, 67]. A less commonly discussed case is gravitational reheating, i.e., particle creation in curved spacetime. This scenario was first investigated in [68] and applied to the quintessential inflation [24]. One important feature of the gravitational reheating is that it does not need to introduce any nonminimal coupling. Here we study whether the gravitational reheating can successfully create the hot Universe in our model. Especially, we concentrate on the radiation temperature $T_{\rm reh}$ at the end of deflation, i.e., the beginning of radiation era. The requirement is $T_{\rm reh}\gtrsim 10^{11}\,{\rm K}$ ($\sim 10\,{\rm MeV}$ in natural units, i.e., the temperature at which primordial nucleosynthesis starts [46]).

The Universe is assumed to undergo a transition from inflation (nearly de Sitter spacetime) to deflation (decelerating phase). For a massless scalar field, the created energy density at the end of inflation from the inflation-deflation transition is

where $R\sim 0.01$ for arbitrary power-law deflation [68, 69], and $\rho_{\rm inf}$ is given by Eq. (14c) in our model. This creation can be regarded as an instantaneous process. Based on the calculations presented in Sec. III, it is natural to assume that Eq. (22) applies to our model. There may be many scalar fields in the Universe and each one contributes $0.01$ to $R$. Following [24], hereafter we assume $0.01\lesssim R\lesssim 1$, which corresponds to $T_{\rm r}(a_{e})\sim 10^{23}\,{\rm K}$, where $T_{\rm r}(a_{e})$ is the radiation temperature at the end of inflation (after reheating). During deflation, the radiation energy density $\rho_{\rm r}=\rho_{r}(a_{e})\cdot(a_{e}/a)^{4}$ and the inflaton energy density $\rho_{\phi}\approx\rho_{\rm inf}\cdot(a_{e}/a)^{3(1+\overline{w}_{\phi})}$, where $1/3<\overline{w}_{\phi}\leqslant 1$. Thus, the ratio of $\rho_{\rm r}$ and $\rho_{\phi}$ is

Radiation era thus begins at the scale factor

Meanwhile the radiation temperature $T_{\rm reh}$ is

where $k_{\rm B}$ is the Boltzmann constant. Therefore, it is possible to realize $T_{\rm reh}\gtrsim 10^{11}\,{\rm K}$ with suitable parameters in our model. For example, parameters with $\overline{w}_{\phi}>0.825$ if $R=1$ and $\overline{w}_{\phi}>0.903$ if $R=0.01$. Considering Eq. (20) and $\lambda_{2}\ll 1$, we obtain $\lambda_{1}>4.68$ and $\lambda_{1}>4.78$ respectively.

Regarding the late-time acceleration as one of the accelerating phases in the MAU provides an attractive solution to the coincidence problem. Meanwhile, MAU provides a natural unification of inflation and dark energy. In this paper, we analyze the inflationary consequences of the MAU model described by Eq. (1). This model with $\lambda_{2}=\mathcal{O}(10^{-4})$ gives sufficient e-folding number, suitable $n_{\rm s}$ and extremely small $r$. The post-inflationary expansion is decelerating with an average EOS $\overline{w}_{\phi}>1/3$. In this framework, we show that gravitational particle creation at the end of inflation is sufficient to reheat the hot Universe.

Together with our previous works, here we summarize two observational probes for the general MAU scenario. One probe is the primordial gravitational waves: MAU generally predicts $r<0.01$. The physical origin of this upper limit is that the coincidence problem requires the increment of $\phi$ cannot be too large during one accelerating phase (including inflation) [21]. During inflation, in order to obtain sufficient e-folding number, the potential must be very flat, which results in small $\epsilon_{V}$ and $r$. The other probe is the Hubble expansion rate at cosmic dawn (denoted as $H_{\rm cd}$) measured by the 21 cm signals [22, 70]. In the MAU, the scalar field is comparable with the normal matters across the whole cosmic history, and thus $H_{\rm cd}$ should be much larger than that in the $\Lambda$CDM model. We expect future observations can prove or disprove these two predictions. But note that the MAU is not the unique model to obtain small $r$ or large $H_{\rm cd}$. For example, Starobinsky $R^{2}$ inflation also gives small $r$ [51, 27] and some interacting dark energy models may give large $H_{\rm cd}$ [71]. What we have done is to provide a physical motivation (i.e., the cosmological coincidence problem) for the small $r$ and large $H_{\rm cd}$.

There are several unnatural features of our model. In principle, arbitrary $V_{0}$ can be used to explain the late-time cosmic acceleration [21]. However, in order to obtain a suitable $A_{\rm s}$, we have to specify an extremely small value for $V_{0}$ relative to the Planck scale values. This is unnatural (a fine-tuning problem), as in the conventional inflation models [32, 34, 35]. In addition, the dimensionless parameter $\lambda_{2}=\mathcal{O}(10^{-4})$ indicates the other minor fine-tuning problem in our model. We believe that nature (cosmology) prefers models that only introduce Planck scale parameters and dimensionless parameters of order unity. Inspired by [24], those two fine-tuning problems may can be addressed in models with an extra inflation field $\chi$. In the new model, both $V_{0}$ and $\lambda_{2}$ are not constants but slowly varying functions of $\chi$. This possibility will be explored in the future.

This work was supported by the National Natural Science Foundation of China under Grants Nos. 11633001, 11920101003 and 12021003, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB23000000 and the Interdiscipline Research Funds of Beijing Normal University. S.X.T. was supported by the Initiative Postdocs Supporting Program under Grant No. BX20200065.