# Cosmology in modified *f*(*R*, *T*)-gravity

## Abstract

In the present paper we propose a further modification of *f*(*R*, *T*)-gravity (where *T* is trace of the energy-momentum tensor) by introducing higher derivatives matter fields. We discuss stability conditions in the proposed theory and find restrictions for the parameters to prevent appearance of main type of instabilities, such as ghost-like and tachyon-like instabilities. We derive cosmological equations for a few representations of the theory and discuss main differences with conventional *f*(*R*, *T*)-gravity without higher derivatives. It is demonstrated that in the theory presented inflationary scenarios appear quite naturally even in the dust-filled Universe without any additional matter sources. Finally, we construct an inflationary model in one of the simplest representation of the theory, calculate the main inflationary parameters and find that it may be in quite good agreement with observations.

## 1 Introduction

According to current knowledge, based on experimental data, there were (at least) two different epochs of dynamical evolution of our Universe when the key role was played dark energy (DE): an inflationary stage at the early times of evolution and a late time acceleration (l.t.a.) stage, which started recently (on cosmological scales) and continues till modern time. We know about the existence of modern DE (associated with l.t.a.) with high precision from the different experiments, the first of which relate to SNI data [1, 2], whereas about primordial DE (associated with inflation) we know only by indirect detection such as general isotropy and flatness of observable part of Universe and the non-flatness spectrum of primordial scalar perturbations [3, 4]. Nevertheless the true nature of both DEs is unknown yet and this fact stimulates researchers to find solutions of the DE problem outside of standard physics.

Modifications of the gravitational sector are well known from early times and still are very popular, because different corrections to the gravitational action follow for instance from string theory [5, 6] and one-loop quantum effects [7, 8, 9] (see also [10, 11] for cosmological applications). The number of different approaches in this way is actually huge and we only mention here examples such as *f*(*R*)-gravity [12, 13, 14], Horndeski theory [15], unimodular gravity [16], teleparallel gravity [17], theories with non-minimal kinetic coupling [18]; see also [19].

Nevertheless there is another possibility to solve DE problem: we can introduce some exotic matter or modify the right hand side (matter sector) of the equations. The activity in this direction is not so intensive, but we can mention such attempts as phenomenological higher derivative matter fields [20, 21], bulk viscosity and imperfect fluids [22, 23, 24], theories with non-minimally coupled Ricci scalar with matter lagrangian [25, 26] and one of the most popular subclasses of this model, *f*(*R*, *T*)-gravity [27], where *T* is the trace of the energy-momentum tensor (stress-energy tensor). Note that the dependence on *T* may be induced by exotic imperfect fluids or quantum effects (such as the conformal anomaly). Also we can study such kinds of models as some phenomenological models, which arise from some more general theories. Indeed it is well known that brane models can modify exactly the r.h.s. of the equations of motions on the brane [28, 29, 30, 31]. For these reasons in our paper we try to discuss a wider class of *f*(*R*, *T*)-gravity models and incorporate a function dependence by the derivatives of *T* (models containing \(\Box R\)-terms also are known as possible modifications of *f*(*R*)-gravity [32, 33]).

This paper is organized as follows: in Sect. 2 we derive general equations and discuss stability conditions; in Sect. 3 we study a few concrete examples of functions and find some cosmological solutions; in Sect. 4 we estimate inflationary parameters for one of the simplest shapes of the function; and in Sect. 5 we give some concluding remarks.

## 2 General equations and stability conditions

*R*is the Ricci scalar and

*T*is the trace of the energy-momentum tensor; \(\epsilon \) is equal to 1 or to 0. First of all let us ensure that this theory is ghost-free. For this task let us introduce Lagrange multipliers in the following way

^{1}:

^{2}which allows us to solve this problem: if we put

*T*we find the field equation

*F*(

*R*,

*T*)-gravity.

## 3 Some concrete examples for cosmological applications

### 3.1 The simplest case of functions: \(f(R,T)=R+2f(T)\), \(h(T)=\alpha T\)

*b*is some function of \(\rho \). Indeed, expressing \({{\dot{\rho }}}\) from (24) and substituting into (18) and taking into account (22) we find

#### 3.1.1 Comparison with the case \(h=0\) and analogy with scalar field inflation

First of all, note that Eqs. (20) and (21) are very similar to the equations that describe cosmology with a scalar field (we need to substitute (22) there). Indeed there is a kinetic term \(\frac{1}{2}\alpha {{\dot{\rho }}}^2\) and some kind of potential \(f-\frac{1}{2}\rho f'\). This fact is very well understandable from (28): if we can ignore by \(16\pi \rho \) with respect to \(2f -\rho f'\) (or put by hand \(\epsilon =0\)) we obtain \(w_{\mathrm{eff}}=-1\) in the slow-roll regime when \({{\dot{\rho }}}^2\ll 2f -\rho f'\). This means we have classical inflation on the scalar field. Thus our new term shows a behavior absolutely identical to the scalar field one.

#### 3.1.2 Unification of inflation and l.t.a.

*f*will play the role of a cosmological constant in the beginning (the case of large values of \(\rho \)) and in the end (the case of small values of \(\rho \)) of the evolution of the Universe, whereas the existence of a kinetic term \({{\dot{\rho }}}^2\) may provide a transition between these two limit regimes. So in realistic cosmological models, the constants must satisfy the following conditions:

### 3.2 Non-minimally coupling case: \(f(R,T)=R+2\gamma R T + 2f(T)\)

#### 3.2.1 Special solution

*H*

*x*)

*F*is the some function of parameters. Of course, such a kind of solution may not exist for arbitrary set of parameters, so let us ensure that the solution which we found satisfies all equations from the system (34)–(37). First of all note that all these equations will have a finite part, so for verifying we can neglect all terms \(\sim x\). Substituting our solution into (36) we find

*H*, respectively. Substituting Eq. (40) into (37) we find

*f*.

^{3}We have a future solution near which for zero energy density \(\rho \) we have non-zero Hubble parameter

*H*.

^{4}

#### 3.2.2 Special case \(\gamma =0,\,\,f(T)=0\)

#### 3.2.3 Future singularities in special case \(\gamma =0,\,f(T)=0\)

*H*near singularity, which occurs at the moment \(t_s\),

Type I: Let us suppose \(h\propto \rho ^n\) and \(\rho \propto (t_s-t)^{\alpha }\). Substituting these relations and (49) to (44) and taking into account (46), we find that such a kind of particular solution may be realized for \(\alpha =(1-\beta )/(n+1)\) and \(\beta >1\). Thus we can see that Type I of future singularity, which is also known as “Big Rip” may appear due to our new terms.

Type II: For this type of singularity we have \(\dot{H}\gg H^2=H_0^2\) near the \(t_s\) point. It means that terms from the r.h.s. of Eq. (45) also are much more higher than \(H^2\) and the only possibility to satisfy Eq. (44) is to put \(2h'^2=h''h\), which leads to a very specific shape of the function \(h=-1/(C_1\rho +C_2)\) and contradicts our basic requirements for stability. Thus we can see that this type of future singularity cannot be realized due to our new terms.

Type III: In this case the situation is very similar to the previous one: we have \(\dot{H}\gg H^2\gg 1\) and realization of this type of singularity is impossible.

Type IV: In this case we have \(H=H_0\), \(\dot{H}=0\), while higher derivatives of *H* diverge. From (45) we can see that the only possibility to satisfy this equation is to have \({{\dot{\rho }}}=\mathrm {const}\ne 0\) near the point \(t_s\). It implies the only possible solution \(\rho =\varLambda _0+\rho _0(t_s-t)\), but even in this case we need the additional condition \(8\pi \varLambda _0+h'\rho _0^2=0\) to have a consistent system (44) and (45). To satisfy this condition we need to put \(h'<0\), which contradicts the general stability condition, or to put \(\varLambda _0<0\), which breaks the null energy condition. Thus we can see that this type of singularity also cannot be realized due to our new terms.

Finally, we can see that only Type I future singularities may appear in our theory, but it is also quite clear that in the most general case of non-minimal coupling *f*(*R*, *T*) any types of future singularities may appear due to the non-trivial dependence of the function from *R*, as happens in the usual *f*(*R*)-gravity. We shall address this question in future investigations.

## 4 Basic inflationary model and its parameters

*r*and the spectral index of the primordial curvature perturbations \(n_s\) may be expressed by using the slow-roll parameters in the following way:

^{5}Since during the inflation stage we have a slow-roll approximation, we can put the next relations \(\ddot{\rho }\ll H{{\dot{\rho }}}\) and \({{\dot{\rho }}}^2\ll f\) and now Eqs. (18)–(22) take the form

*f*. Thus such a kind of regime may be realized for any \(f\propto \rho ^n\) with \(n>1\), because in this regime we have large values of \(\rho \). Finally, instead of (50) and (51) we have now

Now let us take for example \(n=m=2\). In this case we have, for \(N_e=50\), \(r=0.107\), \(n_s=0.9667\); and for \(N_e=60\), \(r=0.09\), \(n_s=0.9778\). We can see that the inflationary parameters lie near the boundary of viable region and taking more complicated functions may move them deeper into this region.

Finally, let us ensure that all variables have physical values. From (56) we can see that \(\rho _*\) has actually a large value. \({{\dot{\rho }}} <0\) and \(\dot{H} <0\); it means that these variables both decrease during inflation, as they must. \({\ddot{H}}\) may has a different sign, depending on the parameters, we see that for \(n=m=2\) it has negative values. Finally all derivatives \({{\dot{\rho }}}\), \(\dot{H}\), \({\ddot{H}}\) must be small in comparison with \(\rho \) and *H*; this fact puts some additional restrictions for the parameters *m* and *n* (otherwise our slow-roll approximation will broken). For instance in the case \(n=m=2\) we have according to our formulas \({{\dot{\rho }}}\propto \rho ^{-1}\), \(\dot{H}\propto \rho ^{-1}\) and \(\ddot{H}\propto \rho ^{-3}\) and since the energy density \(\rho \) has a large value all time derivatives actually are small.

## 5 Conclusions

In this paper we discuss the possibility of a further generalization of *f*(*R*, *T*)-gravity by incorporating higher derivative terms \(\Box T\) in the action. First of all we find that in the proposed theory inflationary scenarios appear quite naturally and may produce viable inflationary parameters. Moreover, higher derivative terms decrease more rapidly than the classic ones, but this may lead to future singularities of Type I. Another important thing: since new terms produce a contribution to the inflationary parameters it may resurrect such inflationary models as \(R^n\) with \(n>2\), which are already closed by modern observational data. We shall address this question in further investigations. It may be interesting also to generalize our theory by incorporating terms like \(\sum c_i\Box ^iT\), which may produce a ghost-free theory for some specific sets of coefficients \(c_i\). Thus we propose a theory which is free from standard pathologies and promising for cosmological applications.

## Footnotes

- 1.
In this section we exclude \(L_m\) from our discussion and concentrate our attention on the function

*F*. - 2.
One may note that the more general case of infinite rows like \(\sum c_i\Box ^iT\) may produce a ghost-free theory, as has happened in the pure

*f*(*R*)-gravity case [34]. - 3.
This situation may be changed if we use a more general function

*h*. - 4.
For this solution we have non-zero \(\dot{H}\) as well, thus it does not correspond to an exact \(\mathrm{d}S\)-solution.

- 5.
The simplest case discussed above cannot produce viable inflationary parameters.

## Notes

### Acknowledgements

This work was supported by the Russian Science Foundation (RSF) Grant 16-12-10401.

## References

- 1.A.G. Riess et al., Astron. J.
**116**, 1009 (1998)ADSGoogle Scholar - 2.S. Perlmutter, Astrophys. J.
**517**, 565 (1999)ADSGoogle Scholar - 3.Planck Collaboration, A&A
**594**, A13 (2016). arXiv:1502.01589 [astro-ph.CO] - 4.Planck Collaboration, Planck 2018 results. X. Constraints on inflation. arXiv:1807.06211 [astro-ph.CO]
- 5.F. Muller-Hoissen, Phys. Lett. B
**163**, 106 (1985)ADSMathSciNetCrossRefGoogle Scholar - 6.R.R. Metsaev, A.A. Tseytlin, Phys. Lett. B
**185**, 52 (1987)ADSMathSciNetCrossRefGoogle Scholar - 7.B.S. DeWitt, Phys. Rev.
**160**, 1113 (1967)ADSCrossRefGoogle Scholar - 8.B.S. DeWitt, Phys. Rev.
**162**, 1239 (1967)ADSCrossRefGoogle Scholar - 9.N.D. Birrell, P.C.W. Davies,
*Quantum Fields in Curved Space*(Cambridge University Press, Cambridge, 1982)zbMATHCrossRefGoogle Scholar - 10.V.T. Gurevich, A.A. Starobinskii, JETP
**50**(5), 844 (1979)ADSGoogle Scholar - 11.B. Whitt, Phys. Lett. B
**145**, 176 (1984)ADSMathSciNetCrossRefGoogle Scholar - 12.S. Nojiri, S.D. Odintsov, Int. J. Geom. Methods Mod. Phys.
**4**, 115 (2007)MathSciNetCrossRefGoogle Scholar - 13.S. Nojiri, S.D. Odintsov, Phys. Rep.
**505**, 59–144 (2011)ADSMathSciNetCrossRefGoogle Scholar - 14.K. Bamba, S. Capozziello, S. Nojiri, S.D. Odintsov, Astrophys. Space Sci.
**342**, 155 (2012)ADSCrossRefGoogle Scholar - 15.T. Kobayashi, M. Yamaguchi, J. Yokoyama, Prog. Theor. Phys.
**126**, 511 (2011)ADSCrossRefGoogle Scholar - 16.S. Nojiri, S.D. Odintsov, V.K. Oikonomou, JCAP
**1605**, 046 (2016)ADSCrossRefGoogle Scholar - 17.R. Aldrovandi, J.G. Pereira,
*Teleparallel Gravity: An Introduction*(Springer, Dordrecht, 2012)zbMATHGoogle Scholar - 18.S.V. Sushkov, Phys. Rev. D
**80**, 103505 (2009)ADSCrossRefGoogle Scholar - 19.T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rep.
**513**(1), 1–189 (2012)ADSMathSciNetCrossRefGoogle Scholar - 20.P. Pani, T.P. Sotiriou, D. Vernieri, Phys. Rev. D
**88**, 121502 (2013)ADSCrossRefGoogle Scholar - 21.P.V. Tretyakov, Eur. Phys. J. C
**76**, 497 (2016)ADSCrossRefGoogle Scholar - 22.S. Nojiri, S.D. Odintsov, Phys. Rev. D
**72**, 023003 (2005)ADSCrossRefGoogle Scholar - 23.K. Bamba, S.D. Odintsov, Eur. Phys. J. C
**76**, 18 (2016)ADSCrossRefGoogle Scholar - 24.I. Brevik, E. Elizalde, S.D. Odintsov, A.V. Timoshkin, Int. J. Geom. Methods Mod. Phys.
**14**, 1750185 (2017)MathSciNetCrossRefGoogle Scholar - 25.S. Nojiri, S.D. Odintsov, Phys. Lett. B
**599**, 137–142 (2004)ADSCrossRefGoogle Scholar - 26.N.J. Poplawski, A Lagrangian description of interacting dark energy. arXiv:gr-qc/0608031
- 27.T. Harko, F.S.N. Lobo, S. Nojiri, S.D. Odintsov, Phys. Rev. D
**84**, 024020 (2011)ADSCrossRefGoogle Scholar - 28.L. Randall, R. Sundrum, Phys. Rev. Lett.
**83**, 4690 (1999)ADSMathSciNetCrossRefGoogle Scholar - 29.L. Randall, R. Sundrum, Phys. Rev. Lett.
**83**, 3370 (1999)ADSMathSciNetCrossRefGoogle Scholar - 30.G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B
**485**, 208 (2000)ADSMathSciNetCrossRefGoogle Scholar - 31.G. Dvali, G. Gabadadze, Phys. Rev. D
**63**, 065007 (2001)ADSMathSciNetCrossRefGoogle Scholar - 32.A. Hindawi, B.A. Ovrut, D. Waldram, Phys. Rev. D
**53**, 5597 (1996)ADSMathSciNetCrossRefGoogle Scholar - 33.M. Skugoreva, A. Toporensky, P. Tretyakov, Gravit. Cosmol.
**17**, 110 (2011)ADSCrossRefGoogle Scholar - 34.T. Biswas, A. Mazumdar, W. Siegel, JCAP
**0603**, 009 (2006)ADSCrossRefGoogle Scholar - 35.T. Koivisto, Class. Quantum Gravity
**23**, 4289 (2006)ADSCrossRefGoogle Scholar - 36.S. Nojiri, S.D. Odintsov, S. Tsujikawa, Phys. Rev. D
**71**, 063004 (2005)ADSCrossRefGoogle Scholar - 37.K. Bamba, S. Nojiri, S.D. Odintsov, JCAP
**0810**, 045 (2008)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}