Cosmology in modified $f(R,T)$-gravity

In present paper we propose further modification of $f(R,T)$-gravity (where $T$ is trace of energy-momentum tensor) by introducing higher derivatives matter fields. We discuss stability conditions in proposed theory and find restrictions for 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 theory and discuss main differences with convenient $f(R,T)$-gravity without higher derivatives. It is demonstrated that in presented theory inflationary scenarios appears quite naturally even in the dust-filled Universe without any additional matter sources. Finally we construct inflationary model in one of the simplest representation of the theory, calculate main inflationary parameters and find that it may be in quite agreement with observations.

where R is Ricci scalar and T is the trace of energy-momentum tensor and ǫ 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 by the next way 1 variations with respect to µ i give us and variation with respect to λ i : thus initial action (1) may be rewritten in the form We can see that field µ 2 is non-physical and as consequence decouple from equation. Now let us focus no the second term. It may be reorganized by introducing new fields λ 2 = χ 2 + ψ 2 and µ 3 = χ 2 − ψ 2 : where we used integration by the parts. Formula (6) tell us that independently on the type of function F (R, T, ✷T ) this theory contain three scalar fields (one additional will appear after µ 1 R decoupling) and at least one from it is ghost-like. But there is one special case 2 , which allow us to solve this problem: if we put F (R, T, ✷T ) = f (R, T ) + h(T )✷T.
Note that h(T ) = const because in this case contribution to the equations will trivial. In this case theory will contain only two physical scalar fields and both of them may be non-ghost depending on signs of h ′ and f R . Indeed, if we start from lagrangian (7) and introduce auxiliary fields as λ = R, µ = f λ we gain the next action where potential V = f (λ, T ) − µλ and last term from (7) was integrated by the parts. Further, producing conformal transformation of the metricḡ ik = e χ g ik , χ = ln µ we have the action in canonical form We can see that last kinetic term contain multiplier h ′ /f R thus we need f R > 0 and h ′ > 0 for ghost-free theory. Now varying lagrangian (7) with respect to T we find field equation and finally varying (7) with respect to metric we have We can see that if take into account field equation (10), Einstein-like equation (11) has essential simplification Let us consider solution of equation (10) in the form T = T 0 + δT and for the trivial solution R 0 = 0, T 0 = 0 (under the flat background) we find additional restriction f T T 0 for the absence of tachyon-like effective particles in the theory (case f T T = 0 can not be totally excluded). For more complicate cases of non-flat background this relation will has more complicate structure and will contain h ′′ as well, but it is clear that theory may be free from tachyon instability.
Let us take divergence of equation (11). Divergence of l.h.s. reads where we following to [21] used which is also may be simplified by using (10) as Note here very essential thing. Equations (13) and (16) are not contain true limits at h = 0. If we want to find this limits, we must use equations (11) and (15). The reason of such kind situation quite understandable: for the limit h = 0 field equation (10) just absent (trivial) and simplifications which was produced became impossible. Exactly by this reason the proposed theory has very significant difference with respect to usual F (R, T )-gravity.
III. SOME CONCRETE EXAMPLES FOR COSMOLOGICAL APPLICATIONS. Now let us discuss some particular cases of gravitational field equations. For cosmological application usually used with u i u i = 1 and u i ∇ k u i = 0. In this case expression for Θ ik takes very simple form A. The simplest case of functions: Let us discuss firstly the simplest case f (R, T ) = R + 2f (T ) for the universe with FLRW-metric filled by the dust matter (p = 0, T = ρ). In this case equation (11) gives us 2ä a +ȧ equation (10) reads and finally equation (15) gives us which may be transformed to It is easy to test our system: let us take time derivative from (20), add (21) multiplied by −3H and cut from result −9H 3 -term by using (20), as result we gain equation (23). Now we can see that in this simplest case it is possible to write Fridman-like equation in the form H 2 = b(ρ), where b -some function of ρ. Indeed, expressingρ from (24) and substituting to (20) and taking into account (22) we find which for instance for f = 2λρ, ǫ = 1 reads and for f = λρ 2 , ǫ = 1 Note that these expressions look like expressions, which arise from brane cosmological models. Also it is useful calculate value w ef f ≡ −1 − 2Ḣ/H 2 : Comparison with the case h = 0 and analogy with scalar field inflation.
First of all note, that equations (20) and (21) very similar to equations described cosmology with scalar field (we need to substitute (22) there). Indeed there is a kinetic term 1 2 αρ 2 and some kind of potential f − 1 2 ρf ′ . This fact very well understandable from (28): if we can neglect by 16πρ with respect to 2f − ρf ′ (or put by the hand ǫ = 0) we obtain w ef f = −1 in slow-roll regime whenρ 2 ≪ 2f − ρf ′ . It means the classical inflation on the scalar field. Thus our new term has behavior absolutely identical to the scalar field one. Now let us compare our theory with the limit case h = 0, which was studied in previous investigations. In this case field equation (10) is absent and we need to use (11) instead of (13). Equations (20) and (21) now reads which corresponds to eos and not equal to −1 even for ǫ = 0. Moreover there is no kinetic term here, which may provide exit from inflation. Thus we can see that difference is very significant.

Unification of inflation and l.t.a.
Note here also that it is possible to construct cosmological model which will unify inflation and late time acceleration by special shape of function f (ρ). Indeed let us put where we imply all constants are positive and n > m > 0. This function has the limits and we can see from equations (20) and (21) that term f will play the role of cosmological constant in the beginning (case of big values of ρ ) and in the end (case of small values of ρ) of Universe evolution, whereas existence of kinetic termρ 2 may provide transition between these two limit regimes. So in realistic cosmological models constants must satisfy next conditions: where Λ inf -value of cosmological constant in inflation epoch and Λ 0 -value of cosmological constant in present time.
In this case we have equation (10) reads and finally equation (16) gives us It is easy to verify our system like in previous case: taking time derivative from (34), adding (35) multiplied by −3H and cut from result −9H 3 -term by using (34), we gain equation (37). Note also that energy conservation law (37) takes very complicate form in this case. Now let us try to find some special solutions. We will find solution in the form which near the the critical point t = t c may be decomposed as and thereforρ Now let us suppose for the sakes of simplicity h = αρ, and f (0) = 0, which is quite natural if we don't want to introduce cosmological constant by the hands. Substituting these expressions to (34) and taking into account that f = f ′ (0)ρ we find near critical point t = t c algebraical equation for H where we denote x ≡ (t c − t). Equation (38) has the only positive solution, which is reads (after linearization with respect to x) as where F is the some function of parameters. Of cause, such kind of solution may not exist for arbitrary set of parameters, so let us ensure that solution which we found satisfy to all equations from the system (34)-(37). First all note that all these equations will have finite part, so for verifying we can neglect by the all terms ∼ x. Substituting our solution to (36) we find whereḢ 0 and H 0 denotes finite part ofḢ and H correspondingly. Substituting expression (40) to (37) we find and this expression provide us relation between ρ 0 and parameters of the theory, which quite natural because ρ 0 is integration constant and must be determined from equations (constraint equation). Finally combining (40) and (35) we have which may be transform by using (34) to which tell us that found solution exist only for some specific shape of function f 3 . As a dry remnant we have future solution near which for zero energy density ρ we have non-zero Hubble parameter H 4 .
In this special case equations (34)-(35) reads and equation (37) takes the form and effective eos We can see that in the case whenρ 2 ≪ ρ, which is quite natural for the power-law solutions H ∝ 1/t, w ef f = 0 and dust stage is realizing in the Universe. In some sense this is the limit case without potential, but only with kinetic term. Note also that this term may play the key role near Big Rip solution.
Let us discuss possibility of future singularities in our theory. Since we interested in future singularities which may appears due to new kinetic term, we discuss this question in the special case γ = 0, f (T ) = 0. According to conventional classification [22] there are four types of future singularities, which may be parameterized as follows (for more details see original paper). If present Hubble parameter H near singularity, which occur at the moment t s as values of parameter β 1 corresponds to the Type I of singularities, values −1 < β < 0 to the Type II, values 0 < β < 1 to the Type III and β < −1 to the Type IV [23]. Let us study possibility of realization for every type consistently.
Type I. Let us suppose h ∝ ρ n and ρ ∝ (t s − t) α . Substituting these relations and (49) to (44) and taking into account (46), we find that such kind of particular solution may be realized for α = (1 − β)/(n + 1) and β > 1. Thus we can see that Type I of future singularity, which is also known as "Big Rip" may appears due to our new terms.
Type II. For this type of singularity we haveḢ ≫ H 2 = H 2 0 near the t s point. It mean that terms from r.h.s. of equation (45) also much more than H 2 and the only possibility to satisfy equation (44) is to put 2h ′2 = h ′′ h, which lead to very specific shape of function h = −1/(C 1 ρ + C 2 ) and contradict to our basic requirements for stability. Thus we can see that this type of future singularity can not be realized due to our new terms.
Type III. In this case situation very similar to previous one: we haveḢ ≫ H 2 ≫ 1 and realization of this type of singularity is impossible.
Type IV. In this case we have H = H 0 ,Ḣ = 0, while higher derivatives of H diverge. From (45) we can see that the only possibility to satisfy this equation is to haveρ = const = 0 near the point t s . It imply the only possible solution ρ = Λ 0 + ρ 0 (t s − t), but even in this case we need in additional condition 8πΛ 0 + h ′ ρ 2 0 = 0 to have consistent system (44)-(45). To satisfy this condition we need to put h ′ < 0, which contradict to general stability condition, or to put Λ 0 < 0 which broke null energy condition. Thus we can see that this type of singularity also can not be realized due to our new terms.
Finally we can see that only Type I of future singularities may appears 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 non-trivial dependence of function from R, as it happens in usual f (R)-gravity. We address this question to future investigations.
In this section let us try to calculate parameters of inflation, which may be constructed by using models previously described. Tensor-scalar ratio r and spectral index of primordial curvature perturbations n s may be expressed by using slow-roll parameters by the next way where the slow-roll indices being defined in terms of the Hubble rate as follows As example of calculations let us take the model described in sec. III A with arbitrary function h(ρ). 5 Since during inflation stage we have slow-roll approximation, we can put the next relationsρ ≪ Hρ andρ 2 ≪ f and now equations (20)-(22) take the form It is clear that in slow-roll approximation Hubble parameter change slowly, so we haveḢ ≪ H 2 and from comparison of equations (50) and (51) we can see that it equal to each other only if we can neglect by the 8πρ term with respect to f . Thus such kind of regime may be realized for any f ∝ ρ n with n > 1, because in this regime we have big values of ρ. Finally instead of (50)-(51) we have now Expressions forḢ andḦ which is also needed for slow-roll parameters may be calculated by consistent differentiating of expression (53). Indeed we have for instance forḢ Now according to definition the number of e-folding before end of inflation N e = ln ae a * , where a e scale factor related to the end of inflation and a * scale factor related to N e number. By using the definition of Hubble rate this formula may be transformed as follows where we used (52), and finally we obtain where ρ * ≫ ρ e . Now let us focus on the case of power functions and put f = µρ m , h = νρ n . In this case expression (55) may be easily integrated and we have Expression (52) takes the formρ and for Hubble rate and its derivatives we have and finally collecting all terms we find for slow-roll parameters the next expressions where we take into account solution (56). 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 inflationary parameters lies near the boundary of viable region and taking more complicate functions may move it deeper in this region.
Finally let us ensure, that all variables has physical values. From (56) we can see that ρ * has actually big value. ρ < 0 andḢ < 0 -it means that both these variables are decrease during inflation as it must be.Ḧ may has different sign depending on parameters, we can see that for n = m = 2 it has negative values. And finally all derivativesρ, H,Ḧ must be small in comparison with ρ and H -this fact put some additional restrictions for parameters m and n (otherwise our slow-roll approximation will broken). For instance in the case n = m = 2 we have according to our formulasρ ∝ ρ −1 ,Ḣ ∝ ρ −1 andḦ ∝ ρ −3 and since energy density ρ has a big value all time derivatives actually small.

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