1 Introduction

The unsolved problems of General Relativity such as dark energy [1, 2] and dark matter [3] force us to investigate different alternatives. One of the most popular alternatives is well-known f(R)-gravity [4, 5] in the different forms [6–8]. There are also a number of another possibilities to modify gravity, such as Palatini f(R)-gravity [9], teleparallel gravity [10], Horndeski theory [11], theories with non-minimal kinetic coupling [12], and so on [13].

A new type of modified gravity was recently proposed [14]. The higher-derivative matter fields are implied in such a kind of theory, which can be interpreted as non-dynamical auxiliary fields,

$$\begin{aligned} G_{\mu \nu }+\Lambda g_{\mu \nu }= T_{\mu \nu }+ S_{\mu \nu }(\mathbf {g},\mathbf {T}). \end{aligned}$$
(1)

The most general form of the term \(S_{\mu \nu }\) if we take into account terms up to fourth order in derivatives is

$$\begin{aligned} S_{\mu \nu }= & {} \alpha _1 g_{\mu \nu }T + \alpha _2 g_{\mu \nu } T^2 + \alpha _3 T T_{\mu \nu }+\alpha _4 g_{\mu \nu }T^{\alpha \beta }T_{\alpha \beta }\nonumber \\&+\,\alpha _5 T^{\alpha }_{\,\,\,\,\,\mu }T_{\alpha \nu }+ \beta _1\nabla _{\mu }\nabla _{\nu }T+ \beta _2g_{\mu \nu }\Box T+ \beta _3\Box T_{\mu \nu }\nonumber \\&+\,2\beta _4\nabla ^{\alpha }\nabla _{(\mu }T_{\nu )\alpha }, \end{aligned}$$
(2)

where since we impose the divergence free condition \(S^{\mu \nu }_{;\mu }=0\) Footnote 1 there are a number of relations between the coefficients. So finally we have only two independent parameters, \(\beta _1\) and \(\beta _4\), and Eq. (1) takes the form [14]

$$\begin{aligned} G_{\mu \nu }= & {} T_{\mu \nu } -\Lambda g_{\mu \nu } -\beta _1\Lambda g_{\mu \nu }T+ \frac{1}{2}\beta _4 g_{\mu \nu }T^{\alpha \beta }T_{\alpha \beta } \nonumber \\&+\,\frac{1}{4}(1-2\beta _1\Lambda )(\beta _1-\beta _4) g_{\mu \nu } T^2\nonumber \\&+\, [\beta _4(1-2\beta _1\Lambda )-\beta _1] T T_{\mu \nu } -2\beta _4 T^{\alpha }_{\,\,\,\,\,\mu }T_{\alpha \nu }\nonumber \\&+\, \beta _1\nabla _{\mu }\nabla _{\nu }T - \beta _1g_{\mu \nu }\Box T - \beta _4\Box T_{\mu \nu }\nonumber \\&+\,2\beta _4\nabla ^{\alpha }\nabla _{(\mu }T_{\nu )\alpha }. \end{aligned}$$
(3)

Some of the terms from Eq. (3) also appear in f(R, T)-theory [15]. In this sense there may a further generalizations of such a kind of theories, for instance, in the light of \(f(R,T,R_{\mu \nu }T^{\mu \nu })\)-theory [16] or maybe \(f(R,T,R_{\mu \nu }T^{\mu \nu },T_{\mu \nu }T^{\mu \nu })\)-theory. Note that for a special choice of parameters some of the well-known theories are contained in the representation (3) as a limit. For instance, the case \(\beta _1=0\), \(\beta _4=-\kappa /2\) corresponds to EiBI gravity in the small coupling limit, or \(\beta _4=0\) corresponds to generic Palatini f(R) gravity [14]. In this sense the theory of (1) also may be interpreted as some kind of phenomenological theory of modified gravity. Cosmology in such a kind of theories was investigated in [17, 18]. There was obtained an effective dark energy sector and one found accelerating, de Sitter, and non-accelerating solutions. Also it was demonstrated that in such a kind of theories the effective dark energy e.o.s. may be quintessence-like, cosmological constant-like or even phantom-like. Nevertheless we need to note that the problem of multiple de Sitter solutions (which was successfully solved in f(R)-gravity [19]) is unsolved yet, and it seems that it cannot be solved without some additional terms in the equation. Generalizations for brane theories were studied in [20–22]. Some interesting remarks about this theory may be found also in [23].

For a standard perfect fluid \(T_{\mu \nu }=(p+\rho )U_{\mu }U_{\nu }+ p g_{\mu \nu }\) and FLRW metric \(\mathrm{d}s^2=-\mathrm{d}t^2+a^2(t) \delta _{\mu \nu }\mathrm{d}x^{\mu }\mathrm{d}x^{\nu }\) the Friedman-like equation takes the form [17]

$$\begin{aligned} 3H^2= & {} \rho +\Lambda +3H[\beta _1(\dot{\rho }-3\dot{p})-\beta _4(\dot{\rho }+2\dot{p})] -\beta _4\ddot{\rho }\nonumber \\&+\,\frac{1}{4}\left[ 3\beta _1(1+2\beta _4\Lambda )(\rho ^2-3p^2)\right. \nonumber \\&+\, 3\beta _4(\rho ^2+p^2) -12\beta _1(\beta _1+\beta _4)\Lambda \rho p \nonumber \\&-\, 6(\beta _1-\beta _4)\rho p-4\beta _1\Lambda (\rho -3p) \nonumber \\&\left. +\,2\beta _1^2\Lambda (\rho ^2+9p^2) \right] , \end{aligned}$$
(4)

and also we have the usual energy conservation law \(T^{\mu \nu }_{;\mu }=0\), which reads now

$$\begin{aligned} \dot{\rho }+3H(p+\rho )=0, \end{aligned}$$
(5)

where \(H=\frac{\dot{a}}{a}\) is usual Hubble parameter.

2 Stability conditions

It is well known that incorporating of higher-derivative terms can affect the stability of different solutions including the simplest cosmologically relevant one. One of the most important is the Minkowski solution. Let us study the stability of the Minkowski solution in such a kind of gravity. To discuss the stability problem we need to transform the equations to the form of a dynamical system. First of all we will discuss the simplest (and for cosmological applications the most convenient) equation of state, \(p=w\rho \). Further we can express the Hubble parameter from Eq. (5) and insert it into (4). In this way we obtain a second order differential equation for the function \(\rho \), which is also dependent on the parameters \(\beta _1\), \(\beta _4\), \(\Lambda \), and w:

$$\begin{aligned} \beta _4\ddot{\rho }=\rho +\Lambda +\frac{1}{4}\rho ^2f_2 - \frac{f_1}{1+w}\frac{\dot{\rho }^2}{\rho } - \frac{1}{3(1+w)^2}\frac{\dot{\rho }^2}{\rho ^2}, \end{aligned}$$
(6)

with

$$\begin{aligned} f_1=\beta _1(1-3w)-\beta _4(1+2w), \end{aligned}$$
(7)

and

$$\begin{aligned} f_2= & {} 3\beta _1(1-3w^2)(1+2\beta _4\Lambda )+3\beta _4(1+w^2)\nonumber \\&-12\beta _1(\beta _1+\beta _4)\Lambda w -6w(\beta _1-\beta _4)\nonumber \\&-4\beta _1(1-3w)\Lambda +2\beta _1\Lambda (1+9w^2). \end{aligned}$$
(8)

We can see that \(f_1\) and \(f_2\) are non-dynamical functions and they only depend on the parameters of the theory.

For further investigation let us rewrite Eq. (6) as a dynamical system:

$$\begin{aligned} \left\{ \begin{array}{r} \dot{\rho }=\pi ,\\ \dot{\pi }= F, \end{array} \right. \end{aligned}$$
(9)

with the function F

$$\begin{aligned} F(\rho ,\pi )\equiv & {} \frac{1}{\beta _4}\left( \rho +\Lambda +\frac{1}{4}\rho ^2 f_2 - \frac{f_1}{1+w}\frac{\pi ^2}{\rho }\right. \nonumber \\&\left. - \frac{1}{3(1+w)^2}\frac{\pi ^2}{\rho ^2} \right) . \end{aligned}$$
(10)

It is clear that the stationary points of system (9) are governed by the next simple equation,

$$\begin{aligned} \frac{1}{4}f_2\rho ^2+\rho +\Lambda =0. \end{aligned}$$
(11)

Now let us discuss the physical meaning of its solutions. First of all, note that we are interested in solutions with a true vacuum \(\Lambda =0\), but we need to keep the parameter \(\Lambda \) to investigate perturbations because it is coupled with the other parameters of our theory, \(\beta _1\) and \(\beta _4\). So we can let it vanish it only at the end of our investigation. Therefore, we may interpret \(\Lambda \) as a parameter which allows us to avoid degeneration of the solutions and we can put \(\Lambda \rightarrow 0\) at the end. A similar approach was successfully practiced to a stability investigation of the Minkowski solution in different modified gravity theories in our previous papers [24, 25]. The first solution of Eq. (11) reads

$$\begin{aligned} \rho =\frac{-2-2\sqrt{1-f_2\Lambda }}{f_2}. \end{aligned}$$
(12)

We can see that in the case \(\Lambda \rightarrow 0\) this solution reads \(\rho =-\frac{4}{f_2}\ne 0\); moreover, (5) tells us that \(H=0\). So this solution corresponds to the static Einstein universe and it is not interesting for our further investigations; but if we want to keep it, we must put \(f_2<0\) to satisfy the weak energy condition (wec)Footnote 2 and some additional restrictions for the parameters will follow from this inequality.Footnote 3 The second solution reads

$$\begin{aligned} \rho =\frac{-2+2\sqrt{1-f_2\Lambda }}{f_2}, \end{aligned}$$
(13)

and it has the limit \(\rho \rightarrow -\Lambda \) for \(\Lambda \rightarrow 0\), so we need to put \(\Lambda \rightarrow -0\) to satisfy wec. Note that this solution exists for any value of H including the case \(H=0\). In the last case this is a Minkowski solution and its stability is very important for us. Let us study the Minkowski stability conditions. It is well known that in the first order, stability is governed by the equation

$$\begin{aligned} \left| \begin{array}{l} -\mu \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\ (F_{\rho })_0\,\,\,\,\,\,(F_{\pi })_0-\mu \end{array} \right| =0, \end{aligned}$$
(14)

orFootnote 4

$$\begin{aligned} \mu ^2-(F_{\pi })_0\mu - (F_{\rho })_0=0, \end{aligned}$$
(15)

and it easy to see that the stability conditions take the formFootnote 5

$$\begin{aligned} \left\{ \begin{array}{l} (F_{\pi })_0<0,\\ (F_{\rho })_0<0. \end{array} \right. \end{aligned}$$
(16)

Let us calculate the values of these functions explicitly. For the function \(F_{\pi }\) we have

$$\begin{aligned} F_{\pi }=\frac{-2\dot{\rho }}{(1+w)\rho }\frac{1}{\beta _4}\left[ f_1 +\frac{1}{3(1+w)\rho } \right] . \end{aligned}$$
(17)

First of all note that at the interesting point this function takes a singular value, but we need only its sign. Second, we can put \(1+w>0\) without loss of generality, because the case \(w=-1\) corresponds to the case of the cosmological term, which is already incorporated in our equations. Now we can see from (5) that \(\frac{-2\dot{\rho }}{(1+w)\rho }=6H>0\) in an expanding universe. The value of the function \(f_1\) is always finite, see (7), whereas the value of the expression \(\frac{1}{3(1+w)\rho }\) is infinite at the point we study (\(\rho =0\)), and positive if we pose the quite natural conditions \(1+w>0\) and \(\rho >0\). So we find that the total sign of Eq. (17) at the point (0, 0) is governed by the sign of the parameter \(\beta _4\) and for Minkowski stability we must have \(\beta _4<0\).

Now let us discuss the second condition from (16). We have

$$\begin{aligned} F_{\rho }=\frac{1}{\beta _4} + \frac{1}{2}\rho f_2 +\frac{f_1}{\beta _4(1+w)}\frac{\dot{\rho }^2}{\rho ^2} + \frac{2}{3\beta _4(1+w)^2}\frac{\dot{\rho }^2}{\rho ^3}.\nonumber \\ \end{aligned}$$
(18)

First of all note that near the interesting point (0, 0) the second term may be neglected because \(f_2\) always has a finite value, see Eq. (8), and \(\rho \rightarrow 0\), whereas the first term is equal to a non-zero constant. According to (5) \(\frac{\dot{\rho }^2}{\rho ^2}=9H^2(1+w)^2\rightarrow 0\) for the Minkowski solution, so the third term may also be neglected. And for the fourth term we have \(\frac{\dot{\rho }^2}{\rho ^3(1+w)^2}=\frac{9H^2}{\rho }=3\) near the point \(\rho =0\); see (4). So finally we have

$$\begin{aligned} (F_{\rho })_0=\frac{3}{\beta _4}, \end{aligned}$$
(19)

and we can see that for stability we need \(\beta _4<0\) as well.

Now let us discuss possible instabilities from another point of view. The trace of Eq. (3) reads

$$\begin{aligned} (\beta _4+3\beta _1)\Box T +2\Lambda \left( 2 +2\beta _1 T +\beta _1^2T^2 \right) -T-R =0.\nonumber \\ \end{aligned}$$
(20)

Let us study a small perturbation \(\delta T\) for some solution of this equation with \(T_0\) and \(R_0\). So we put \(T=T_0+\delta T\) and the equation for \(\delta T\) takes the form

$$\begin{aligned} (\beta _4+3\beta _1)\Box \delta T +4\Lambda \left( \beta _1 +2\beta _1^2T_0 \right) \delta T - \delta T =0; \end{aligned}$$
(21)

we try to find possible solutions of (21) as standard decomposition on the functions

$$\begin{aligned} u_k\sim e^{i\mathbf {k}\mathbf {x}-i\omega t}, \end{aligned}$$
(22)

where \(\omega \equiv (k^2+\mu ^2)^{1/2}\), \(k\equiv \mid \mathbf {k}\mid \), and \(\mu \) is the mass of the effective scalar field (scalaron). After substituting this representation into (21) we obtain the next equation for \(\mu ^2\):

$$\begin{aligned} \mu ^2=\frac{1-4\Lambda \left( \beta _1+2\beta _1^2T_0\right) }{\beta _4+3\beta _1}, \end{aligned}$$
(23)

and if we turn back to the Minkowski limit \(\Lambda \rightarrow 0\), this relation give us an additional restriction to avoid the tachyon-like instability (\(\mu ^2<0\)):

$$\begin{aligned} \beta _1>-\frac{1}{3}\beta _4. \end{aligned}$$
(24)

3 Conclusion

In this paper we discuss instabilities in higher-derivative matter field theories. We found the condition for the Minkowski stability and another one to avoid the tachyon-like instability. Of course, these are only necessary but not sufficient conditions for stability of the theory. For instance we study Minkowski stability only with respect to the simplest class of isotropic perturbations and taking into account more complicated perturbations may provide us with some additional restrictions on the parameters. Nevertheless the conditions found must be satisfied and may be very helpful for further construction of the theory.