# Anisotropic cosmological solutions in \(R + R^2\) gravity

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## Abstract

In this paper we investigate the past evolution of an anisotropic Bianchi I universe in \(R+R^2\) gravity. Using the dynamical system approach we show that there exists a new two-parameter set of solutions that includes both an isotropic “false radiation” solution and an anisotropic generalized Kasner solution, which is stable. We derive the analytic behavior of the shear from a specific property of *f*(*R*) gravity and the analytic asymptotic form of the Ricci scalar when approaching the initial singularity. Finally, we numerically check our results.

## 1 Introduction

Inflation is a generic intermediate attractor in the direction of expansion of the universe, and in the case of pure \(R^2\) gravity it is an exact attractor [1]. However, it is not an attractor in the opposite direction in time. Thus, if we are interested in the most generic behavior before inflation, more general anisotropic and inhomogeneous solutions should be considered. We know from general relativity (GR) that already anisotropic homogeneous solutions help us much in understanding the structure of a generic space-like curvature singularity. Thus, a natural question is to investigate anisotropic solutions in the \(R+R^2\) gravity, too.

In the light of the latest Cosmic Microwave Background constraints by PLANCK [2], the pioneer inflationary model based on the modified \(R+R^2\) gravity (with small one-loop corrections from quantum fields) [3] represents one of the most favorable models. It lies among the simplest ones from all viable inflationary models since it contains only one free adjustable parameter taken from observations. Also it provides a graceful exit from inflation and a natural mechanism for creation of known matter after its end, which is actually the same as that used to generate scalar and tensor perturbations during inflation. This theory can be read as a particular form of *f*(*R*)-theories of gravity which, in turn, is a limiting case of scalar-tensor gravity when the Brans–Dicke parameter \(\omega _{BH}\rightarrow 0\), and it contains an additional scalar degree of freedom (scalar particles, or quasi-particles, in quantum language) compared to GR which is purely geometrical. The existence of a scalar degree of freedom (an effective scalar field) is needed if we want to generate scalar (matter) inhomogeneities in the universe from “vacuum” fluctuations of some quantum field [4, 5]. Such generalizations of the familiar Einstein–Hilbert action have been also studied as an explanation for dark energy and late-time acceleration of the universe’s expansion [6, 7, 8, 9] and to include quantum behavior in the gravitational theory [10].

As is already very well known, through the Gauss–Bonnet term which in four dimensions is a surface term, the most general theory up to quadratic in curvature terms is of the type \(R+R^2+C_{abcd}C^{abcd}\), where \(C_{abcd}\) is the Weyl tensor. The investigations of this type of models began with [11, 12, 13, 14]. After them, many authors have analyzed the cosmological evolutions of such a model [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Particular attention is given to the asymptotic behavior in [30, 31, 32, 33, 34]. The addition of a Weyl square term creates a richer set of solutions, and in particular, in [1, 35, 36], it has been shown that the cosmic no-hair theorems no longer hold.

Quadratic theory, like \(R+R^2\) gravity, is a particular case of the more general quadratic type, and it has higher order time derivatives in the equations of motion and this leads to the appearance of solutions which have no analogs in GR. One of such solutions corresponds to the scale factor \(a \sim \sqrt{t}\) behavior, which coincides with a radiation dominated solution in GR. In quadratic gravity, this solution represents a vacuum solution which is also a stable past attractor for Bianchi I model and probably for all Bianchi models [36]. The other solution, being an analog of a GR solution with matter with equation of state, \(p=(\gamma -1)\rho \), is \(a \sim t^{4/3 \gamma }\) (instead of usual \(a \sim t^{2/3 \gamma }\) in GR), and it can be naively considered as a solution which would describe the last stages of a collapsing universe when quadratic terms dominate. However, this solution appears to be a saddle, so a collapsing universe for a general initial condition ends up with a vacuum “false radiation” regime, in principle not possible in GR.

The \(a(t)\propto \sqrt{t}\) behavior near singularity in the \(R+R^2\) model does not mean that the \(R^2\) term behaves as radiation generically. This behavior is specific only for the purely isotropic case, as it will be shown in the present paper (and even in the isotropic case the late-time behavior is different: \(a(t)\propto t^{2/3}\) modulated by small high-frequency oscillations). Neither does it behave as an ideal fluid in the anisotropic case.

When shear is taken into account the situation becomes more complicated. Vacuum solutions exist in GR also, and in the simplest case of a flat anisotropic metrics (which is the case analyzed in the present paper) this is the famous Kasner solution [37]. On the other hand, studies of cosmological evolution in a general quadratic gravity (which includes apart from \(R^2\) the Weyl tensor square term in the quadratic part of the action) indicate that the isotropic vacuum “false radiation” still exists and, moreover, it is an attractor [36]. Kasner solution is also a solution in quadratic gravity. Due to complicated nature of dynamics near the Kasner solution a generic trajectory could end up in Kasner or isotropic solution depending on initial conditions [38].

For these reasons, the stability and the full behavior of quadratic theories of gravity is still subject of investigation and one powerful tool to address these problems is the dynamical system approach [39] which allows one to find exact solutions of the theory through the determination of fixed points and gives a description of the evolution of the system, at least at qualitative level.

Despite the obvious fact that the general quadratic gravity at the level of the action includes \(R+R^2\) gravity (which can be obtained setting the coefficient before Weyl square term to zero) it is not so at the level of corresponding equations of motion for the universe model in question. The reason is that the number of degrees of freedom in a general quadratic gravity is bigger than in \(R+R^2\) theory (which, in its turn, is bigger than in GR). That is why we cannot simply put the corresponding constant to zero in cosmological equations of motion. This means that cosmological evolution of a flat anisotropic universe in \(R+R^2\) gravity needs a special investigation which is the matter of the present paper.

The paper is organized as follows: in Sect. 2, we present the basic equations of the model. In Sect. 3 we describe schematically the strategy adopted to obtain the correct degrees of freedom and then we analyze the dynamics of \(R^2\)-gravity both in the vacuum case and in the case with matter; we find exact solutions and determine their stability. In Sect. 4, we derive the analytic behavior of the shear using a general line element. Finally, Sect. 5 contains a summary of the results and conclusions.

## 2 System under consideration

*g*is the determinant of the metric,

*G*is the Newton constant, and \(\beta \) is a parameter.

This theory can be interpreted as a particular form of *f*(*R*) gravity. Observations tell us that the dimensionless coefficient \(\frac{\beta }{16\pi G}\) is very large, \(\approx 5 \times 10^8\). This follows from the fact that its expression in terms of observable quantities, in the leading order of the slow-roll approximation, is \(\frac{\beta }{16\pi G}= \frac{N^2}{288\,\pi ^2 P_{\zeta }(k)}\) where \(P_{\zeta }(k)\) is the power spectrum of primordial scalar (adiabatic) perturbations, *N* is both \( \ln {k_f/k}\) and the number of e-folds from the end of inflation, \(k_f\) being the wave vector corresponding to the comoving scale equal to the Hubble radius at the end of inflation (\(k_f/a(t_0)\) is slightly less than the CMB temperature now); see e.g. [40]. For the \(R+R^2\) inflationary model, \(P_{\zeta }\propto N^2\), so \(\beta \) is a constant indeed. Note also that \(\beta =\frac{1}{6M^2}\) where *M* is the scalaron mass after the end of inflaton (and in flat space-time too). On the other hand, the coefficient of the Weyl square term (which is present in a general quadratic model of gravity) in the Lagrangian density generated by one-loop quantum gravitational effects is not expected to be so large. Typically it is of the order of unity (or even significantly less due to small numerical factors) multiplied by the number of elementary quantum fields. Thus, there exists a large range of the Riemann and Ricci curvature where the \(R^2\) term dominates, while the contribution from the Weyl square term is still small. For this reason, anisotropic solutions preceding the inflationary stage may be studied using the same \(R+R^2\) model up to curvatures of the order of the Planck one.

*w*is the equation of state (EoS) parameter.

*H*, defining in this way the new dimensionless expansion-normalized variables (ENVs)

## 3 Generalized anisotropic solutions

^{1}We also remember that, in terms of the ENV, the generalized Kasner solution is given by \(Q_1=-3\), \(\Sigma _+^2+\Sigma _-^2=1\), and the isotropic vacuum solution by \(Q_1=-2\) and \(\Sigma _+=\Sigma _-=0\). Both of these solutions belong to the solution set given by (20).

*s*and

*u*, like

### 3.1 Stability analysis

In the dynamical system approach, the field equations are rewritten with respect to the ENV, such that the solutions are fixed points. In particular, the solution space described in the previous section constitutes an invariant set of fixed points. The linearization around the fixed points reveals the local stability of the theory. In fact, since all eigenvalues \(\lambda _i \ge 0\), this solution set is an attractor to the past, as all trajectories to the future deviate exponentially from this solution set. Stability with and without a matter source is going to be addressed, and the presence of matter is irrelevant for sufficiently big shear.

#### 3.1.1 Obtaining the dynamical system

*f*is a generic function of all the remaining ENV. From the \(E_{00}\) component of (2), we obtain a constraint equation,

Looking at the above set of equations, it can be noted that there is only one additional dynamical degree of freedom compared to General Relativity, which is the first equation for \(\dot{Q}_{1}\). This can be easily understood by remembering that, through a conformal transformation, this gravitational theory is equivalent to GR plus a scalar field [42].

As we have described above, the linearization of the dynamical system (31) around the solution (21) gives rise to the following eigenvalues, which will be discussed in the next subsections.

#### 3.1.2 Pure geometric modes

#### 3.1.3 Matter modes

We focused on solutions which are attractors to the past; however, there can be other solutions [36].

## 4 Analytic behavior

*f*(

*R*) gravity, obtained in [14], it is possible to determine the dynamical evolution of the shear as an exact, analytical result. Considering a Lagrangian like

*R*, and \('\) represents derivative w.r.t. to the proper time. \(T_{ab}\) is the energy-momentum tensor of some matter source. Since we are considering a spatially homogeneous space-time, with corresponding sources, the following combinations are zero: \(a)\,T_{22}-T_{33}=0\), \(b)\,T_{11}-T_{22}/2-T_{33}/2=0\). In the same way, taking the same combinations of the left hand side of the field equations (35), we have

*R*in the absence of sources:

*C*are constants.

*R*can be substituted into (39) for the particular theory analyzed hitherto, for which we have \(f^\prime =1\,+\,2\beta R\),

*C*since we know that, interchangeably when \(s<1\) or \(Q_1<-3\), the asymptotic solution set (22) must continue to be a past attractor; see Sect. 3.1. This attractor has constant well-defined values for \(Q_1\), \(\Sigma _\pm \) satisfying (19) and when \(Q_1<-3\) this will only occur if there is a particular cancellation in the denominator of (55) \(f^\prime \rightarrow 0\) giving a well-defined limit for \(\Sigma _\pm \) at the singularity

## 5 Conclusions

In the present paper we have considered the past attractor solution for the evolution of the flat anisotropic universe in \(R+R^2\) gravity. Our results, in combination with already known results, indicate that the properties of the evolution of the universe near a cosmological singularity change significantly taking into account anisotropy and/or modifications of gravity. Indeed, the evolution of an isotropic universe is determined solely by the matter equation of state. When anisotropy is taken into account, this isotropic solution becomes a future asymptotic solution, while the generalized vacuum Kasner solution becomes a past attractor (except for a stiff fluid with a Jacobs solution).

In general quadratic gravity without anisotropy new vacuum isotropic solution (“false radiation” solution) is stable to the past. The anisotropic case instead has two sub-cases, because general quadratic corrections to the gravitational action have two independent terms, which can be chosen as proportional to the squares of scalar curvature and the Weyl tensor. In a general situation, when these two terms are of the same order, the dynamical system describing the universe past evolution has both “false radiation” and a generalized Kasner solution as attractors (the latter is, more precisely, a saddle-node fixed point). So the nature of a cosmological singularity (isotropic or anisotropic) depends on the initial conditions imposed.

However, since the \(R+R^2\) inflationary model is observationally well motivated, and we have argued in Sect. II above that there exists a large range of the Riemann and Ricci curvatures where the anomalously large \(R^2\) term dominates the Einstein term *R*, while a “normal-size” Weyl squared term is still small, one can expect that a new solution appears. It has two parameters (so it is a two-dimensional set of solutions) and includes both isotropic “false radiation” and a generalized anisotropic Kasner solution (which is a one-dimensional set) as subsets. Moreover, in some sense, it interpolates between them, because it is possible to construct a line of solutions with one end being an isotropic solution and the other end being a point in the generalized Kasner set.

All these intermediate points disappear when the correction proportional to the Weyl square term is added to the action, leaving only isotropic and generalized Kasner solutions and this represents one of the main results of our paper. However, since an \(R^2\) inflation model is observationally well motivated we can neglect the coefficient in front of the Weyl term, and we can expect that the two-dimensional set of solutions discussed in this paper could be a good approximation for realistic models in quadratic gravity.

In the present paper we have restricted the analysis to flat metrics. However, a positive spatial curvature could, in principle, destroy this regime and generate a more complicated behavior similar to the Belinsky–Khalatnikov–Lifshitz (BKL) [43] singularity in General Relativity. We leave this problem for future analysis.

## Footnotes

## Notes

### Acknowledgements

We are delighted to thank Sigbjørn Hervik for illuminating discussions and comments. D. M. and A. T. thank the University of Stavanger for warm hospitality when this paper was started. D. Müller would like to thank the Brazilian agency FAPDF process no. 193.000.181/2016 for partial support. The work of A. S. and A. T. was supported by the RSF grant 16-12-10401. The computations performed in this paper have been partially done with Maple 16 and with the *Ricci.m* package for Mathematica. For numerical codes we used GNU/GSL ode package, explicit embedded Runge–Kutta Prince–Dormand on Linux.

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