Sub-Planckian Scale and Limits for f(R) Models

It is generally believed that our Universe originated from Planck energies and evolved by expanding and cooling to its present state. The initial stage of quick expansion starting from the sub-Planckian energy density seems inevitable. We regard the sub-Planck scale as the highest energy scale in which classical behavior can dominate. The Planck scale is characterized by complete dominance of quantum fluctuations. The spontaneous creation of an inflationary universe is described in detail, for example, in Reference  Firouzjahi et al. (2004). At the same time, the effects associated with the quantization of gravity may be responsible for model parameter alternation if the energy scale is large enough. Additionally, the gravity quantization leads to a nonlinear geometric extension of the Einstein–Hilbert action. The first and most successful formulation of the inflationary model, the Starobinsky model Starobinsky (1980), considers nonlinear geometric terms belonging to the $f(R)$ class of theories. Gravity with higher derivatives is widely used in modern research Nojiri, Odintsov, and Oikonomou , despite the internal problems inherent in this approach Barrow and Cotsakis (1988); Woodard (2015). Attempts were made to avoid Ostrogradsky instabilities Paul (2017), and $f(R)$-gravity was one of the simplest extensions of Einstein–Hilbert gravity free from Ostrogradsky instability. A necessary element of such models is the fitting of the model parameters to reconstruct the Einstein–Hilbert gravity at low energies Fabris et al. (2020). For example, in Reference Odintsov et al. (2020), the authors reconstructed the form of the function $f(R)$ using the boundary conditions imposed on the scale factor so that it satisfied the observations in the early and late stages of the evolution of the universe. A  variety of ways to study the nonlinear multidimensional gravity was discussed in Reference Rubin et al. (2020).

A wide variety of functions $f(R)$ are presented in the literature. As explicit examples, it is worth citing a couple of functions that relate to a wide range of $f(R)$ functions. The specific model of $f(R)$ gravity

$$f(R)=R-2\Lambda\left(1-\text{e}^{-\frac{\beta R}{2\Lambda}}\right)\left[1-\frac{\gamma R}{2\Lambda}\log\frac{R}{4\Lambda}\right]$$

(1)

is considered in Reference Odintsov et al. (2018). This model unifies the early time inflationary era and the late time acceleration of the universe expansion. The  authors investigated the viability of the model and obtained corresponding constraints on free parameters.

The Tsujikawa model Tsujikawa (2008) is in agreement with the cosmological observations Cen et al. (2019) but is slightly different from the $\Lambda$CDM model predictions. The  function $f(R)$ chosen there has the following form:

$$f(R)=R-\lambda R_{c}\tanh{\frac{R}{R_{c}}}\,,\quad R_{c}\,,\lambda>0\,.$$

(2)

Other attempts were undertaken to describe the whole period of evolution of the universe with the multiparametric $f(R)$ function; see  References Starobinsky (2007); Cognola et al. (2008); Nojiri et al. (2020); Odintsov and Oikonomou (2020); Oikonomou (2020a, b); Lyakhova et al. (2018).

The uncertainty in the parameter values is one of the common questions for such models. To determine or at least to limit them, the authors used cosmological and astrophysical observational data, laboratory and solar system tests Tsujikawa (2008); Iorio (2019), binary pulsars, and GWobservations (see  References Nojiri et al. (2007); Cembranos (2009); Berti et al. (2015); Joyce et al. (2015); Koyama (2016); Freire and [et al.] (2012); Planck Collaboration et al. (2016); Yunes and Siemens (2013) and references therein).

In this article, we discuss the restrictions on the parameters of the following models based on the known behavior of the scale factor starting from the sub-Planck scale:

1. 1.

   $f(R)=R-2\Lambda$ ,

2. 2.

   $f(R)=a_{2}R^{2}+R+a_{0}$ ,

3. 3.

   $f(R)=a_{3}R^{3}+a_{2}R^{2}+R+a_{0}$ ,

4. 4.

   $f(R)=a_{4}R^{4}+a_{3}R^{3}+a_{2}R^{2}+R+a_{0}$,

In the last three cases, we neglect the cosmological constant compared to the energies we deal with and use $a_{0}=0$.

We assume that quantum fluctuations nucleate compact Planck-sized manifolds. Here, we rely on the quantum field theory, where a quantum transition is usually suppressed exponentially by a volume of nucleated systems. As the spatial part of the considered four-dimensional metric, we choose the metric of the three-dimensional sphere as the simplest representative:

$$ds^{2}=dt^{2}-\text{e}^{2\alpha(t)}\Bigl{(}dx^{2}+\sin^{2}{x}\,dy^{2}+\sin^{2}{x}\,\sin^{2}{y}\,dz^{2}\Bigr{)}$$

(3)

Other metrics are also nucleated on equal footing, and we plan to study some of them (compact hyperbolic and torus metrics) in the future.

Constraints on the parameters of the considered models of $f(R)$ gravity, under which exponential growth of the scale factor is possible, are investigated. It is also necessary to determine the conditions under which the exponential growth of the scale factor is replaced by the observed stage of slow expansion. The  parameters of the model are also limited by the condition that the current size of space must exceed the visible size of the universe.

During our study, we kept in mind the following issues:

1. –

   the requirement of model stability, i.e., $f^{\prime}(R)>0$ and $f^{\prime\prime}(R)>0$;

2. –

   the quick growth of the space size. It must exceed the size of the visible Universe, $\sim 10^{28}$ cm;

3. –

   extremely small space expansion at the present time.

These requirements are in addition to those usually imposed on the models by the observations at low energies, in particular, inside the solar system.

Consider the theory described by action:

$$S[g_{\mu\nu}]=\frac{m_{Pl}^{2}}{2}\int d^{4}x\sqrt{|g|}\,f(R)\,.$$

(4)

The corresponding extended field equations are as follows:

$$f_{R}R_{\mu\nu}-\frac{1}{2}\,fg_{\mu\nu}+\Bigl{[}\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box\Bigr{]}f_{R}=0\,,\quad\Box\equiv g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\,,\quad f_{R}=df/dR\,.$$

(5)

This system of equations coincides with Einstein’s field equations for $f(R)=R$. Throughout this paper, we use the conventions for the curvature tensor $R_{\mu\nu\alpha}^{\beta}=\partial_{\alpha}\Gamma_{\mu\nu}^{\beta}-\partial_{\nu}\Gamma_{\mu\alpha}^{\beta}+\Gamma_{\sigma\alpha}^{\beta}\Gamma_{\nu\mu}^{\sigma}-\Gamma_{\sigma\nu}^{\beta}\Gamma_{\mu\alpha}^{\sigma}$ and the Ricci tensor is defined as $R_{\mu\nu}=R^{\alpha}_{\mu\alpha\nu}\,$.

Let us suppose that the action and metric have the forms (4) and (3) consequently. In this case, the  nontrivial Equation (5) acquirse the following form:

	$\displaystyle 6\dot{\alpha}\dot{R}f_{RR}-6\Bigl{(}\ddot{\alpha}+\dot{\alpha}^{2}\Bigr{)}f_{R}+f(R)=0\,,$		(6)
	$\displaystyle 2\dot{R}^{2}f_{RRR}+2\Bigl{(}\ddot{R}+2\dot{\alpha}\dot{R}\Bigr{)}f_{RR}-\Bigl{(}2\ddot{\alpha}+6\dot{\alpha}^{2}+4\text{e}^{-2\alpha}\Bigr{)}f_{R}+f(R)=0\,,$		(7)

where Equation (6) correspond to the (tt)-component and Equation (7) corresponds to the coinciding components (xx) = (yy)=(zz) of system (5). The definition of the Ricci scalar for metric (3) is

$$R=12\dot{\alpha}^{2}+6\ddot{\alpha}+6\text{e}^{-2\alpha}.$$

(8)

Substituting $\ddot{\alpha}$ from (8) into the Equation (6), we obtain an equation that does not contain the second derivatives of the functions $\alpha$ and $R$:

$$6\dot{\alpha}\dot{R}f_{RR}+\Bigl{(}6\dot{\alpha}^{2}+6\text{e}^{-2\alpha}-R\Bigr{)}f_{R}+f(R)=0\,,$$

(9)

There are three Equations (6)–(8) with respect to the unknown functions $\alpha(t)$ and $R(t)$, but only two of them are independent. It is technically easier to solve Equations (7) and (8). Equation (9) plays the role of a restriction to the solutions of second-order differential Equations (7) and (8). This equation was used twofold. Firstly, this equation should be the identity when the solution of systems (7) and (8) are substituted. Secondly, applied at $t=0$, it was used to fix one of the initial variables.

We look for those solutions to this system of equations that have ‘‘correct" asymptotic behavior. The  latter are those that could describe our universe at present time. The space size should be not smaller than the size of the universe. Therefore, space should expand extremely quickly, at least in the beginning. The  asymptotic value of the Hubble parameter should not be bigger than the observable one. Due to its smallness, compared to the sub-Planckian energies, we use $H\xrightarrow{t\rightarrow\infty}0$. That means $\alpha(t)\xrightarrow{t\rightarrow\infty}\text{const}$ and the asymptotic value of the Ricci scalar $R(t)\xrightarrow{t\rightarrow\infty}0$. We also assume that $\alpha(t\rightarrow\infty)>140,$ where the value $\text{e}^{140}m^{-1}_{Pl}$ corresponds to the horizon scale $10^{28}$cm at present time.

Knowledge of the asymptotic behavior facilitates the analysis. We sought for the solutions with asymptote $\alpha(t)\xrightarrow{t\rightarrow\infty}Ht$. Therefore, $R(t)\xrightarrow{t\rightarrow\infty}12H^{2}+6\text{e}^{-2Ht}$ and $\dot{R}(t)\xrightarrow{t\rightarrow\infty}-12H\text{e}^{-2Ht}$. A t the end of the asymptotic regime, $R(t=\infty)=R_{c}=\text{const}$ and $\dot{R}(t=\infty)=0.$ In this case ($R=\text{const}$), the trace of system (5) leads to the algebraic equation

$$f_{R}(R_{c})R_{c}-2\,f(R_{c})=0\,.$$

(10)

Several solutions of this equation could take place for specific values of the physical parameters of function $f(R)$. The Ricci scalar averaged over large scale is negligibly small at present time. Therefore, our aim is the asymptotic solution $R_{c}=R_{Universe}\simeq 0$.

Let us fix the initial conditions for systems (7) and (8)

$$\alpha(0)=\alpha_{0}\,,\quad\dot{\alpha}(0)=\alpha_{1}\,,\quad\dot{R}(0)=R_{1}\,.$$

(11)

Restriction (9) is used to fix the initial value of the curvature $R(0)=R_{0}$.

We are interested in the dynamics of the maximally symmetric manifold starting from the sub-Planck scale. Therefore, the natural choice of the initial conditions is

$$\alpha_{0}\sim\ln{H_{\text{sub-Planck}}^{-1}}\,\,,\quad\alpha_{1}\,\sim H_{\text{sub-Planck}}\,\,,\quad H_{\text{sub-Planck}}\,\lesssim\,m_{Pl}\,.$$

(12)

Further, we work in the Planck units, $m_{Pl}=1$.

The sections below describe the rate of space growth for several forms of the $f(R)$ function depending on the initial data and physical parameters.

In this paper, we discuss new restrictions imposed on the parameters of some $f(R)$ models of the gravity. These restrictions are the result of studying the Universe evolution at high energies. We suppose that our Universe was nucleated with the size of the Planck scale order. It must expand rapidly to reach a size no smaller than that of our Universe at present time. We also choose the 3 dimensional spherical metric from the very beginning as the additional assumption.

These suppositions being quite natural lead to new restrictions compared to limits based on the observations in the Solar system. For example, the parameter range of $R^{3}$ gravity is severely tightened if we apply both our restriction and those in the paper of Cheong et al. (2020). In all models discussed here, the parameter ranges depend on the initial conditions that lead to their slight uncertainties. Nevertheless, these restrictions should be taken into account in the considered models based on gravity with higher derivatives. It is worth mentioning that the pure Einstein–Hilbert gravity with the $\Lambda$ term is not realized in the framework of our approach.

This research was funded by the Ministry of Science and Higher Education of the Russian Federation, Project ‘‘Fundamental properties of elementary particles and cosmology’’ N 0723-2020-0041. The work of A.P. and S.R. was supported by the Kazan Federal University Strategic Academic Leadership Program. The work of A.P. was partly funded by the Russian Foundation for Basic Research grant No. 19-02-00496. The work of A.P was also funded by the development program of the Regional Scientific and Educational Mathematical Center of the Volga Federal District, agreement N 075-02-2020.

The authors declare no conflict of interest.