Eur. Phys. J. C (2016) 76:449 DOI 10.1140/epjc/s10052-016-4303-6 Regular Article - Theoretical Physics Quantum Cosmology of f (R, T ) gravity Min-Xing Xu1,2,a , Tiberiu Harko3,4,b , Shi-Dong Liang1,5,c 1 School of Physics, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China Yat Sen School, Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China 3 Department of Physics, Babes-Bolyai University, Kogalniceanu Street, 400084 Cluj-Napoca, Romania 4 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 5 State Key Laboratory of Optoelectronic Material and Technology, Guangdong Province Key Laboratory of Display Material and Technology, Guangzhou, People’s Republic of China 2 Received: 29 May 2016 / Accepted: 30 July 2016 / Published online: 11 August 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Modified gravity theories have the potential of explaining the recent acceleration of the Universe without resorting to the mysterious concept of dark energy. In particular, it has been pointed out that matter–geometry coupling may be responsible for the recent cosmological dynamics of the Universe, and matter itself may play a more fundamental role in the description of the gravitational processes that usually assumed. In the present paper we study the quantum cosmology of the f (R, T ) theory of gravity, in which the effective Lagrangian of the gravitational field is given by an arbitrary function of the Ricci scalar, and the trace of the matter energy-momentum tensor, respectively. For the background geometry we adopt the Friedmann–Robertson– Walker metric, and we assume that matter content of the Universe consists of a perfect fluid. In this framework we obtain the general form of the gravitational Hamiltonian, of the quantum potential, and of the canonical momenta, respectively. This allows us to formulate the full Wheeler–de Witt equation describing the quantum properties of this modified gravity model. As a specific application we consider in detail the quantum cosmology of the f (R, T ) = F 0 (R) + θ RT model, in which F 0 (R) is an arbitrary function of the Ricci scalar, and θ is a function of the scale factor only. The Hamiltonian form of the equations of motion, and the Wheeler–de Witt equations are obtained, and a time parameter for the corresponding dynamical system is identified, which allows one to formulate the Schrödinger–Wheeler–de Witt equation for the quantum-mechanical description of the model under consideration. A perturbative approach for the study of this equation is developed, and the energy levels of the Universe are obtained by using a twofold degenerate pera e-mail: 253659701@qq.com b e-mail: t.harko@ucl.ac.uk c e-mail: stslsd@mail.sysu.edu.cn turbation approach. A second quantization approach for the description of quantum time is also proposed and briefly discussed. 1 Introduction One of the cornerstones of theoretical physics, general relativity (GR), formulated mathematically in terms of the Einstein field equations, proved to be a very successful gravitational theory at the scale of the solar system. By using GR we can describe the gravitational dynamics of the Solar system with a high precision, and phenomena like the perihelion precession of Mercury, the bending of light while passing the Sun, and the gravitational redshift can be fully understood. In GR the gravitational field equations can be obtained by √ 1 R −gd4 x + varying the Einstein–Hilbert action S = 16π  √ L m −gd4 x, where R is the Ricci scalar, and L m is the matter Lagrangian, with respect to the metric g μν , and they read Rμν − 21 gμν R = 8π Tμν , where Tμν is the energymomentum tensor. The energy-momentum tensor Tμν identically satisfies the mathematical relation ∇ μ Tμν = 0, which can be interpreted from the physical point of view as the energy conservation. Essentially GR is a beautiful geometric theory that establishes a deep connection between the geometry of the spacetime, matter fields, and gravitational interaction. Considering larger scales of the Universe, using GR we can numerically simulate galaxies’ formations and collisions, and the results of these simulations can be verified by the increasingly trustworthy data obtained due to the rapidly improving observational techniques. The predictions of general relativity have also been confirmed in the strong gravity regime by the discovery of the gravitational wave emission in the binary pulsars system PSR 1913 + 16 [1], a discovery that has opened a new testing ground for GR and for its gen- 123 449 Page 2 of 19 eralizations. The detection by LIGO of GW150914 from the inspiral and merger of a pair of black holes [2] will start a new era in observational astronomy, based on the theoretical and mathematical formalism of GR. Thus, with a large number of astronomical observations, as well as terrestrial experiments testing and confirming it both at the weak and strong gravity regime, it would seem that GR gives a full description of the gravitational interaction at the non-quantum level. However, several recent astrophysical and cosmological observations have raised the intriguing possibility that GR may not be able to model and explain the gravitational dynamics at scales much larger than the one of the solar system. On a fundamental theoretical level the two most important challenges GR must face are the dark energy and the dark matter problems, respectively.Several cosmological observations, obtained initially from the study of the distant Type Ia supernovae, have provided the unexpected result that the expansion of the Universe did accelerate lately [3–7]. The paradigmatic and usual explanation of the late time acceleration requires the existence of a mysterious and dominant component of the Universe, called dark energy (DE). Dark energy is responsible for the late time dynamics of the Universe [8,9], and it can explain the observed features of the recent cosmological evolution. The second and equally mysterious component in the Universe, called Dark Matter, an assumed non-baryonic and non-relativistic “substance”, is necessary for the explanation of the flat rotation curves of galaxies, and for the virial mass discrepancy in clusters of galaxies [10,11]. The detection/observation of dark matter is restricted by the fact that it interacts only gravitationally. Its effects can be observed by observations of the motion of the massive hydrogen clouds around galaxies, or by the motion of the galaxies in clusters [10]. However, despite many decades of intensive observational and experimental efforts the particle nature of dark matter still remains essentially unknown. One interesting possibility for explaining dark energy is based on theoretical models that contain a mixture of cold dark matter and a slowly varying, spatially inhomogeneous component, called quintessence [12]. The idea of quintessence can be implemented theoretically by assuming that it is the energy associated with a scalar field Q, having a self-interaction potential V (Q), and a pressure p = Q̇ 2 /2 − V (Q) associated to the quintessence Q-field. Such a model also allows a possible theoretical interpretation in terms of particle physics results. If the potential energy density V (Q) of the quintessence field is much greater than the kinetic one, then it follows that the pressure p of the field is negative. Quintessential cosmological models have been extensively investigated in the physical literature (for a recent review of quintessence cosmologies see [13]). Alternatively, the recent acceleration of the Universe can also be explained by scalar fields φ that are minimally cou- 123 Eur. Phys. J. C (2016) 76:449 pled to gravity via a negative kinetic energy, known as phantom fields, which have been proposed in [14]. Interestingly enough, these cosmological models allow values of the parameter of the equation of state w with w < −1. Hence, real or complex scalar fields may play a fundamental role in the cosmological processes describing the evolution of our Universe, and they may provide a realistic description of the observed cosmic dynamics. However, in order to explain the observed gravitational dynamics of the Universe a different line of thought on dark energy was also considered. It is based on the fundamental idea that dark energy is not a particular physical field, but it can be understood as a gravitational phenomenon induced at cosmological scales by the intrinsic modifications of the gravitational interaction itself. Hence the fundamental assumption of this theoretical approach is that at large astrophysical and cosmological scales standard general relativity cannot describe correctly the dynamical evolution of the Universe. In this context many modified gravity models, all trying to extend and generalize GR, have been proposed. Historically, in going beyond the standard gravitational models, the first step was to generalize the geometric part of the Einstein–Hilbert action. One of the first models of this type is f (R) gravity, in which the gravitational action is an function of the Ricci scalar R,  arbitrary √ √ 1 f (R) −gd4 x + L m −gd4 x [15–19]. so that S = 16π However, this and many other modifications of the Einstein– Hilbert action concentrate only on the geometric part of the action, by implicitly assuming that the matter Lagrangian plays a subordinate and passive role only [20], which naturally follows from its minimal coupling to geometry. But a fundamental theoretical principle forbidding a general coupling between matter and geometry has not been formulated yet, and in fact it does not exist a priori. If such matter– geometry couplings are allowed, many theoretical gravitational models with extremely interesting properties can be constructed. The first of these kind of models was the f (R, L m ) modia gravitational action of the fied gravity theory  [21–24], with √ 1 f (R, L m ) −gd4 x. A similar geometryform S = 16π matter type coupling is assumed in the f (R, T ) [25,26] gravity theory, where T is the trace of the energy-momentum tensor. For a recent review of the generalized f (R, L m ) and f (R, T ) type gravitational theories with non-minimal curvature-matter coupling see [27]. Several other gravitational theories involving geometry couplings have also been proposed, and extensively investigated, like, for example, the Weyl–Cartan–Weitzenböck (WCW) gravity theory [28], hybrid metric–Palatini f (R, R) gravity, where R is the Ricci scalar formed independent of the metric   from a connection [29,30], f R, T, Rμν T μν type models, where Rμν is the Ricci tensor, and Tμν the matter energy-momentum tensor, respectively [31,32], or f (T̃ , T ) gravity [33], in which a Eur. Phys. J. C (2016) 76:449 coupling between the torsion scalar T̃ and the trace of the matter energy-momentum tensor is assumed. For a review of hybrid metric–Palatini gravity see [34]. Modified gravity models with geometry-matter coupling are important since they can provide from a fundamental theoretical point of view a complete theoretical explanations for the late time acceleration of the Universe, without postulating the existence of dark energy. They can also offer some alternative explanations for the nature of dark matter. Moreover, these models show that matter itself may play a more fundamental role in the description of the gravitational dynamics that usually assumed [20], and they can also represent a bridge connecting the classical and the quantum worlds. For example, the dependence of the gravitational action on the trace of the energy-momentum tensor T may be due to the presence of quantum effects (conformal anomaly), or of some exotic imperfect quantum fluids [27]. Besides the difficulties presented by the present day cosmological observations, a central theoretical problem in present they physics is the unification of quantum mechanics and gravitation. Gravitation dominates the dynamics of objects at large scales, while quantum mechanics describes the microscopic behaviors of the particles. The study the Universe as a whole from the quantum mechanical point of view is the subject of quantum cosmology [35,36], which is based on the idea that quantum physics must apply to anything in nature, including the whole Universe. An unification of the electromagnetic force, of the strong force and of the weak force, respectively, is achieved in the standard model of particle physics, leaving the gravitational force as an exception that cannot be yet unified with the other fundamental forces. This is related to the fact that when considering gravitation in the framework of general relativity, we must consider not only matter, but also space and time, as physical objects. Space and time obey dynamical laws, and they have excitation such as gravitational waves that interact with each other and with matter. These aspects make quantizing the Universe far from being straightforward. Since the formation of cosmic structures is strongly dependent on the spacetime interaction, quantum cosmology is therefore closely related to quantum gravity, representing the quantum theory of the gravitational force and of spacetime [37]. Even being of a speculative and controversial nature, having several difficult conceptual problems to overcome, quantum cosmology has a long history [38–40], and various popular competing attempts have been proposed to quantized the gravitational field, like, for example, string theory, canonical quantum gravity and loop quantum gravity [35,36]. However, the lack of related observations reduces our abilities to resolve conceptual issues to all practical purposes. Since at the beginning of the Universe the average radius of each point is infinitely small, while the geometric curvature is infinitely large, quantum gravitational effects will dominate Page 3 of 19 449 the dynamics of the Universe, and therefore they cannot be neglected in the study of the very early Universe. One of the main obstacles in the understanding of quantum cosmology is the so-called problem of time, which comes from the Hamiltonian constraint in the Arnowitz–Deser–Misner (ADM) formalism, leading to the Wheeler–de Witt equation [38–40], a fundamental equation in canonical quantization of cosmology. The quantum cosmology of f (R) gravity theory with Schutz’s fluid is discussed in [41,42], with new perspectives to the time problem in quantum gravity considered. It is the purpose of the present paper to study the quantum cosmology of the f (R, T ) theory of gravity. Classical aspects of this theory have been extensively investigated [43–55], but its quantum implications have not been considered yet. In order to construct the quantum cosmology of the f (R, T ) gravity we adopt for the classical background metric the Friedmann–Robertson–Walker form, and we assume that the matter content of the very early Universe consists of a perfect fluid, described by two thermodynamic parameters only, the energy density, and the thermodynamic pressure, respectively. In order to introduce the canonical quantization scheme for the f (R, T ) gravity theory, as a first step in our study we obtain the general form of the gravitational Hamiltonian, of the quantum potential, and of the canonical momenta, respectively. Once these quantities are explicitly found we write down the full Wheeler–de Witt equation of the f (R, T ) modified gravity theory, which describes the quantum properties of the very early Universe, when quantum effects had a dominant influence on the dynamic evolution of the system. We introduce and consider in detail the quantum cosmological properties of a particular model of the f (R, T ) theory, namely, the quantum cosmology of the f (R, T ) = F 0 (R) + θ RT model, in which F 0 (R) is an arbitrary function of the Ricci scalar, and θ is a function of the scale factor only. We obtain the Hamiltonian form of the classical equations of motion for this model, and then we write down the Wheeler–de Witt equation, describing the evolution of the wave function of the early Universe. Starting from the Wheeler–de Witt equation we introduce a time parameter for the corresponding quantum dynamical system, which allows us to formulate the Schrödinger–Wheeler–de Witt equation for the quantum-mechanical model under consideration. In order to obtain the properties of the early Universe we develop perturbative approach for the study of this cosmological equation, and the energy levels of the Universe are obtained by using a twofold degenerate perturbation approach. Finally, we address the problem of the quantum time by introducing a second quantization approach for the description of the time in the quantum cosmology of the f (R, T ) modified gravity theory. The present paper is organized as follows. The Hamiltonian formulation of the f (R, T ) theory of gravity is presented in Sect. 2, where the canonical momenta associated 123 449 Page 4 of 19 Eur. Phys. J. C (2016) 76:449 to the field variables as well as the potential are obtained. This allows us to write down the Wheeler–de Witt equation. The quantum cosmological formulation of the particular model f (R, T ) = F 0 (R) + θ RT is presented in Sect. 3. In particular, we identify the canonical momentum associated to the time variable, which allows us to transform the Wheeler–de Witt equation into an equivalent Schrödinger–Wheeler–de Witt equation. The solutions of this Schrödinger–Wheeler–de Witt equation are studied, by using a perturbative approach, in Sect. 4. We consider the problem of the second quantization of time in Sect. 5. In Sect. 6 we discuss and conclude our results. The derivation of the field equations of the f (R, T ) theory of gravity, of the energy balance equation, as well as the canonical momenta for the f (R, T ) = F 0 (R) + θ RT model, and the expressions of the Ricci tensor for the Friedmann–Robertson–Walker metric are presented in Appendices A–D, respectively. 2 The Wheeler–de Witt equation in f (R, T ) gravity theory 2.1 The f (R, T ) gravity theory The gravitational action for the f (R, T ) gravity model is [25]   √ √ (1) S= f (R, T ) −gd4 x + L m −gd4 x. We define the energy-momentum tensor of the matter as Tμν (2) By assuming that the matter Lagrangian L m does not depend on the derivative of g μν , we obtain Tμν = gμν L m − 2 ∂ Lm . ∂g μν (3) The field equations of our model can then be obtained (see Appendix A for details): 123 = 1 f (R, T )gμν + (gμν  − ∇μ ∇ν ) f R R, T 2 1 Tμν − f T (R, T )Tμν − f T (R, T )μν , 2 (4) where μν ≡ g αβ δTαβ /δg μν . Contracting the above field equations we find f R (R, T )R − 3 f R (R, T ) − 2 f (R, T ) 1 = T − f T (R, T )T − f T (R, T ). 2 (5) For a perfect fluid, in the comoving frame, the energymomentum tensor takes the form Tμν = (ǫ + p)Uμ Uν + pgμν , (6) where ǫ and p are the matter energy density and thermodynamic pressure, respectively, and U μ is the four-velocity, satisfying the normalization condition g μν Uμ Uν = −1. For the perfect fluid we can fix the matter Lagrangian L m as L m = p, which gives μν = −2Tμν + pgμν . We begin our study of the quantum cosmological aspects of the f (R, T ) gravity theory by briefly presenting the classical field and conservation equations. Then we obtain the Hamiltonian formulation of the theory, which allows us to write down the Wheeler–de Witt equation in the f (R, T ) theory, which describes the quantum evolution of the very early Universe in the presence of geometry-matter coupling. In the present paper we use the natural system of units with c = h̄ = 16π G = 1. √ 2 δ( −gL m ) . = −√ −g δg μν f R (R, T )Rμν − (7) The f (R, T ) gravity theory is a non-conservative theory, and for the energy balance equation we obtain (see Appendix B for the details of the calculations) Uμ ∇ μ ǫ + (ǫ + p)∇ μ Uμ   1 fT × (ǫ + p)Uμ ∇ μ ln f T + Uμ ∇ μ (ǫ − p) . =−1 2 2 + fT (8) 2.2 The effective cosmological Lagrangian and the potential in f (R, T ) gravity theory We assume that the geometry of spacetime is described by the Friedmann–Robertson–Walker (FRW) metric, which in spherical coordinates is given by   dr 2 2 2 2 2 + r (dθ + sin θ dϕ ) , ds 2 = −N 2 (t)dt 2 + a 2 (t) 1 − kr 2 (9) where N (t) is the lapse function, a(t) is the cosmological scale factor, and k = 1, 0, −1, correspond to the closed, flat and open Universe models, respectively. To proceed, as a first step we obtain the effective Lagrangian for the f (R, T ) theory, whose variation with respect to its dynamical variables yields the appropriate equations of motion. The trace of the energy-momentum tensor is T = −ǫ + 3 p. In the comoving reference frame the components of the four-velocity are Uμ = (N (t), 0, 0, 0) and U μ = (−1/N (t), 0, 0, 0), respectively. Therefore the trace of the field equation (5) of the f (R, T ) gravity model becomes Page 5 of 19 449 Eur. Phys. J. C (2016) 76:449 f R (R, T )R − 2 f (R, T ) + 3 f R (R, T ) 1 = T + f T (R, T )T − 4 p f T (R, T ). (10) 2 With the use of the above identity, and by taking into account the explicit form of the components of the Ricci tensor, given in Appendix C, we obtain the cosmological action for the f (R, T ) theory of gravity as   Sgrav = dt N a 3 f (R, T )  ȧ 2 k N 2 N˙ȧ 6 ä −λ R − + + 2 − N a a a Na  1 − μ T + f T (R, T )T − f R (R, T )R 2  − 3 f R (R, T ) + 2 f (R, T ) − 4 p f T (R, T ) . (11) Now  λ can be expressed using the new variables as  λ= A− B(D M − C R + A − 3F) . 1/2 + 3B + E M − D R − 3G (17) In the following we further denote 1 + 3B + E M − D R − 3G, 2 Z = D M − C R + A − 3F. A= (18) (19) Thus we have BZ  λ= A− . A After taking the time derivative, we find (20) B Ż + ḂZ BZ Ȧ  λ˙ = Ȧ − + . A A2 (21) 2.3 The cosmological Hamiltonian in f (R, T ) gravity theory In Eq. (11) λ and μ are Lagrange multipliers. The term with the second Lagrange multiplier is chosen as the contracted field equation, because it is derived directly from the action, and no further assumptions need to be introduced. Moreover, if we use other formulations for the action, like adopting for the factor multiplying the second Lagrange multiplier the form T + ǫ − 3 p, we will lose important insights from the geometry of the modified gravity, and we will have to face conceptual problems when ǫ and p are related by the radiation equation of state p = ǫ/3. After taking the variation of Eq. (11) with respect to R and T , we obtain the following expressions of λ and μ: The canonical momentum associated to the coordinate q is given by Pq = ∂∂L q̇ . Hence the cosmological Hamiltonian of the f (R, T ) gravity is given by fT μ = 3 Na 1/2 + 3 f T + f T T T − f RT R − 3 f RT − 4 p f T T ≡ μ, (12) μ( f RT T − f R R R + f R − 3 f R R − 4 p f RT ) λ = fR − N a3 N a3  ≡ λ. (13) Hgrav = − Hgrav = ȧ Pa + Ȧ PA + Ḃ PB + Ċ PC + Ḋ PD + Ė PE + Ḟ PF + Ġ PG + Ṙ PR + Ṁ PM − Lgrav . After some simple calculations we obtain the explicit forms of the canonical momenta in the cosmological f (R, T ), which are presented in Appendix D. With these canonical momenta, we obtain the cosmological Hamiltonian of the f (R, T ) theory of gravity, 6 2 a ȧ  λ − 6k N a λ + N a3 V N  B Ȧ Ḃ Z 3B Ḃ Z B R Ċ 6 2 − + + − a ȧ Ȧ − N A A A2 A B Ḋ M B Z R Ḋ B Z Ė M 3B Ḟ 3B Z Ġ − + + − 2 2 A A A A A2  BC Ṙ B Z D Ṙ B D Ṁ B Z E Ṁ + − − + . (23) A A2 A A2 − Hence we obtain the gravitational part of the Lagrangian as 6 6 2 λ˙ + 6k N a λ − N a 3 V, a ȧ  λ − a 2 ȧ (14) N N where the potential V reads  1 V = − f (R, T ) +  λR +  μ T + f T (R, T )T − f R (R, T )R 2  −3 f R (R, T ) + 2 f (R, T ) − 4 p f T (R, T ) . (15) Lgrav = − In order to simplify the notation we will represent the Hamiltonian as Hgrav = (· · · ) − N6 a 2 ȧ[· · · ]. Now we can easily find the relation f R = A, f T = B, f R R = C, f RT = D, f T T = E, Pa PF = 2 6 N + Since (16) 6 N Pa PA = In order to make the presentation simpler, we introduce the following notations:  f R R = F,  f RT = G, T − 4 p = M. (22) 6 N λ 1− a 3 ȧ 2 2 a 4 ȧ − 2 λ 3 2a 3 ȧ 2 B A 2 6 N + a 4 ȧ[· · · ] B ˙  λ. A B A + (24) 6 N 2 3B λ˙ a 4 ȧ A , (25) 123 449 Page 6 of 19 Eur. Phys. J. C (2016) 76:449 by combining the above two equations, we obtain − 6 2 N a ȧ[· · · ] = − 2 N 6a Pa PA + 1 Pa PF 3 + 2·6 2 a ȧ  λ. N (26) Therefore we obtain the gravitational part of the cosmological Hamiltonian of the f (R, T ) theory of gravity as Hgrav = Therefore for the gravitational Hamiltonian we find 6 2 1 N BZ − 2 Pa PA + Pa PF a ȧ A − N A 6a 3 3  +N a V − 6k N a λ. (27)   B , after taking the Since we have PA = − N6 a 2 ȧ 1 − A square of it, we have  PA2 6 2 2B B2 6 a ȧ A =  2 − 2+ · a ȧ 2 A 6 N A A N a 4 ȧ 2 Hgrav = = N N PA2 6a 3 A+ 6 2 2B −B 2 a ȧ A + 2 N A A . (28) Thus we arrive at the following form of the Hamiltonian: Hgrav = Since N N 2 P A− 2 6a 3 A 6a Pa PA + 6 2 a ȧ N 2 B 2 RC B2Z R D − A2 A3 BM 6 2 2 a ȧ − − N A BD BZ E × − , + A A2 2 2 B ZM 6 1 PE PF = − a 2 ȧ , 3 N A3 2 9B 2 Z 2 6 PG2 = − a 2 ȧ , N A4 2 6 PF PC PA + = − a 2 ȧ 3 N PD PM = − + N A 6a 3 B + N 2PC 6a 3 PF 3 (A PA + F PF − a Pa ) PC PR + PD PM + PA + − 6k N a A − P2 PE PF E + G RE 3 9 1 PF C + N a 3 V 3 BZ A (36) , where the quantum potential V is given by V = − f + λR +  μ 1 T + f T M − f R R − 3 f R + 2 f . 2 (37) For the matter part of the Hamiltonian we have [56] (38) and thus the total Hamiltonian of the system is (29) , (30) BZ R A2 H = Hgrav + Hmatt . (39) The gravitational Hamiltonian constructed above consists of all canonical momenta associated to all variable of the f (R, T ) gravity theory, which can lead to the existence of a complex dynamics of these field variables, and of their associated canonical momenta. 2.4 The Wheeler–de Witt equation in f (R, T ) gravity (31) (32) (33) BR A , (34) 6a ȧ 2 A BZ E N A − 3 (PC PR + PD PM ) + 6a B N B A2 2 2BC R BZ R 6a ȧ BM − − . − × − 2 A A N A From the Hamiltonian given by Eq. (39) we immediately obtain the Wheeler–de Witt equation in the framework of the f (R, T ) gravity theory as H  = (Hgrav + Hmatt ) = N H = 0. (40) Here the Hamiltonian operator H takes the form H= we have 6 B(Z − 2 A) − a ȧ 2 N A 6 2 B (D M − C R + A − 3F − 2 A) = − a ȧ N 1/2 + 3B + E M − D R − 3G PF PF N = 3 PA + (A + 3F) 6a 3 3 123 PA + Hmatt = −Lmatt = −N a 3 p, 1 Pa PF 3 6 AB 2 B(Z − 2 A) 6 − a ȧ 2 2 − a ȧ 2 N A N A 3  + N a V − 6k N a λ. PC PR = − N 6a 3 F a 1 A P 2 A + PA PF F − PA Pa a + PF PA + PF2 − PF Pa 6a 3 A 3 3 3 − P2 1 A PE PF PC PR + PD PM + E + G RE 3 6a B 3 9 + 2 PC 6a 3 PA + PF 3 C + a 3 V − 6ka A − BZ A − a 3 p. (41) (35) In order to quantize the model we perform first parameter ordering. There are various ways to do it [57], but in the following we choose a procedure that keeps the Hamiltonian Hermitian [41]. Hence we assume the following relations (we let Pq = −i(∂/∂q)), Page 7 of 19 449 Eur. Phys. J. C (2016) 76:449 q Pq2 = 1 u q Pq q v Pq q w + q w Pq q v Pq q u 2 = −q ∂2 1 + uw , 2 ∂q q tensor can give a hint of the implications on the quantum cosmological evolution of the existence of such a coupling. (42) where the parameters u, v, w satisfy u + v + w = 1, and denote the ambiguity in the ordering of the factors q and Pq , q Pq =  ∂ 1 r q Pq q s + q s Pq q r = −i q +1 , 2 ∂q (43) where the parameters r, s satisfy r + s = 1, and denote the ambiguity in the ordering of factors q and Pq . Similarly we find q −2 2 1 ∂ Pq = −i − 3 + 2 q q ∂q . (44) Thus we obtain the quantized cosmological Hamiltonian in f (R, T ) gravity theory as 1 6a 3  ∂ ∂ ∂ ∂ +F + 2C −a +5 ∂A ∂F ∂C ∂a  1 1 ∂ 1 ∂ − u 1 w1 − u 2 w2 × + ∂A 3 ∂F A 3F  ∂ ∂ E ∂ ∂ R E ∂2 1 A ∂ ∂ + 3 + + − ∂D ∂M 3 ∂E ∂F 9 ∂G 2 6a B ∂C ∂ R H=− A R ∂ E ∂ A + EM ∂ D ∂ − + + B ∂C B ∂M B ∂D 3B ∂F  BZ RE − a 3 p. u 3 w3 + a 3 V − 6ka A − + 3BG A + − (45) Here u 1 , w1 , u 2 , w2 , and u 3 , w3 denote the ambiguity in the ordering of factors A, PA , F, PF , and G, PG , respectively. In the next section we will investigate some particular quantum cosmological models in the f (R, T ) theory of gravity. 3 The quantum cosmology of the f (R, T ) = F 0 (R) + θ RT gravity model After considering the general case of the Wheeler–de Witt equation in the previous section, we can see that an analytic general solution of this equation for arbitrary f (R, T ) would be difficult to obtain. Instead, in the present section we consider a specific case, in which the gravitational action is of the simple form f (R, T ) = F 0 (R) + θ RT, (46) where F 0 (R) is an arbitrary function of the Ricci scalar, and θ is an arbitrary function depending on the scale factor a(t) only. In this toy model, the coupling of the curvature of spacetime and the trace of the matter energy-momentum 3.1 The Hamiltonian and the Wheeler–de Witt equation In the newly born quantum Universe, the spacetime has a very high curvature, so that R → ∞. Accordingly, in the present model we have A= 1 + 2θ R, 2 B = θ R, B θR = . A 1/2 + 2θ R (47) Similarly, the other variables become A = FR0 + θ T, C = FR0 R , D = θ, E = 0, G = θ. (48) When R → ∞, B/A = 1/2. From the definition of PA and PF , we obtain PF = [3B/ (A − B)] PA ≈ 3PA . By assuming that in the new born quantum Universe f R ≫  f R R → A ≫ F, we obtain the gravitational Hamiltonian of the f (R, T ) = F 0 (R) + θ RT gravity as Hgrav = A N A N 2 − 2 Pa PA PA A 3 6a A− B 6a A− B N A 2N A PA PC C − 3 (PC PR + PD PM ) + 3 6a B 6a A − B BZ N + N a 3 V − 6k N a A − = −2 2 Pa PA A 6a N N 2 + 2 3 PA A − 2 3 (PC PR + PD PM ) 6a 6a 4N BZ . + 3 PA PC C + N a 3 V − 6k N a A − 6a A (49) In the above equation the quantum potential is defined as   2Rθ V = − 1 F 0 − FR0 R 1/2 + 2Rθ   1 Rθ 2 T + C R + 3F R − 3 f R . (50) + 1/2 + 2Rθ 2 In the limit R → ∞, and by assuming that θ = 0, the potential becomes   1 1 2 (51) T + C R + 3F R − 3 f R . V = 2 2 In this approximation the total Hamiltonian of the system becomes H = Hgrav + Hmatt   2N 2N = − 2 Pa PA + 3 PA2 A − 3k N a A 6a 6a   N a3 2N 2 C R − 3k N aC R + − 3 PC PR + 6a 2 123 449 Page 8 of 19  Eur. Phys. J. C (2016) 76:449 N a3 2N M + 3k N a D M PD PM + 3 6a 4   4N 3N a 3 F R −  fR . + 3 PA PC C + 6a 2 + −  (52) If in the Lagrangian/Hamiltonian we have some terms that can be omitted in the action, they can also be omitted in L and H , without causing any physical differences in the dynamics of the system. In Eq. (52), due to Gauss theorem, we have   √ 3N a 3 √  fR = dt −g d4 x −g f R (53) 2 M ;μ √ = (54) f R −gdσμ3 . ∂M Then the variational derivative of this term vanishes,  √ δ 2 −g f R d4 x = 0. √ μν −g δg (55) Therefore this term can be omitted in the Hamiltonian function. Thus we arrive at the Wheeler–de Witt equation for the f (R, T ) = F 0 (R) + θ RT gravitational model, which has the form  2 ∂2 ∂ ∂ 4 8 ∂ C H = − − 3 6a 2 ∂a∂ A 6a 3 ∂ A ∂C 6a ∂ A − 2 1 ∂2 2 A + u 1 w1 3 − 3ka A 3 2 6a ∂ A 6 Aa a3 2 ∂ ∂ + C R 2 − 3kaC R 6a 3 ∂C ∂ R 2   a3 ∂2 2 + M + 3ka D M + 6a 3 ∂ D∂ M 4  3 3a F R  = 0. + 2 + (56) 3.2 The Hamiltonian form of the field equations In classical mechanics the total time derivative of any function can be obtained with the use of the Poisson bracket {, } as ∂f d f = + { f, H }. (57) dt ∂t If the physical variables do not depend explicitly on the time t, we obtain d f = { f, H }. (58) dt Therefore we can formulate the classical equations of motion of the f (R, T ) gravity theory as N PA , 6a 3  2 1 − 3 PA Pa + 4 PA2 A + 3k A Ṗa = {Pa , H } = N 3a a ȧ = {a, H } = −2 123 3a 2 1 PC PR − C R 2 − 3kC R 4 a 2    1 3a 2 12 + − 4 PD PM − M + 3k D M + 4 PA PC C , a 4 6a (60)   2 4 1 Ȧ = {A, H } = N − 2 Pa + 3 A PA + 3 PC C , 3a 3a 6a (61)   1 P˙A = {PA , H } = N − 3 PA2 + 3ka , (62) 3a   N Ċ = {C, H } = − 3 PR − 2PA C , (63) 3a   4 a3 2 R + 3ka R − 3 PA PC , (64) P˙C = {PC , H } = N − 2 6a N Ṙ = {R, H } = − 3 PC , (65) 3a   3a 3 P˙R = {PR , H } = N − a 3 C R + 3kaC − F , (66) 2 N Ḋ = {D, H } = −2 3 PM , (67) 6a P˙D = −3k N a M, (68) N Ṁ = −2 3 PD , (69) 6a a3 P˙M = −N + 3ka D . (70) 4 + (59) − Now we define a new time variable τ , which has the following relation with the original time variable t:  τ = N (t)dt, (71) or, equivalently, dτ/dt = N (t). From the definition of PM , we have PM = 3a 2 ȧ D = 3a 3 h D, N (72) where h ≡ N1 ȧa is the Hubble function. Then the Hamilton equations of motion for a cosmological fluid become ′ D = −hθ = − M ′ = −2 a′ D, a 1 PD , 6a 3 ′ PD = −3ka M, ′ PM = a3 + 3ka D , 4 (73) (74) where a prime represents the derivative with respect to τ . The first equation of the above system gives us the coupling constant θ = D as δ , δ = constant. (75) D= a(τ ) This result tells us that the coupling between the gravitational field and the matter field decreases as the scale of the Universe increases. This result may be the reason why the coupling Page 9 of 19 449 Eur. Phys. J. C (2016) 76:449 between matter and gravity becomes so weak in the limit of large cosmological times, and the f (R, T ) gravity behaves as the standard gravity nowadays. 3.3 The time problem in the quantum cosmology of the f (R, T ) gravity The absence of the time evolution of the wave function of the Universe in the Wheeler–de Witt equation hinders our efforts of understanding quantum gravity in a way similar to standard quantum mechanics, or quantum field theory. Below we propose a way to turn the Wheeler–de Witt equation into a Schrödinger type equation. For the product PD PT we obtain immediately the result − 2 PD PM = −h D PD . 6a 3 (76) In the following we make the fundamental assumption that the above term can be interpreted as Pτ , which is the canonical momentum for time. There is a convincing reason for this ∂ , we assumption: after performing quantization Pq = −i ∂q obtain Pτ = −i ∂ da(τ ) d −i = dτ dτ ∂a =− a′ δ a a − a2 δ −i ∂ ∂a (77) = −h D PD . (78) This result holds if we assume that a(t) is the only time dependent variable in the model. The relation proved above allows us to perform the transformation − 2 PD PM → Pτ . 6a 3 (79) The validity of this transformation shows us that the coupling between the gravitational field and the matter field may play an important role in the evolution of the very early Universe. In the following we will make a further simplification of the gravitational action, by assuming it to be f (R, T ) = R + θ RT. (80) Then C = F = PR = 0, and for the cosmological Hamiltonian of the system we obtain 2N 2N Pa PA + 3 PA2 A 2 6a 6a N N a3 − 3k N a A − 3 PD PM + M + 3k N a M. (81) 6a 4 H = Hgrav + Hmatt = − This Hamiltonian is very similar to the Hamiltonian obtained in the f (R) model presented in [41], except for the 3 appearance of a new term −2 6aN3 PD PT + N ( a4 + 3ka)M, which shows us the effect of the coupling between spacetime and matter. The Wheeler–de Witt equation H = 0 for this f (R, T ) gravity model reads H =  2 2 2 Pa PA + 3 PA2 A − 3ka A − 3 PD PM 6a 2 6a 6a  a3 M + 3ka D M  = 0. (82) + 4 − After the use of the transformation introduced in Eq. (79), we obtain  2 2 Heff  = − 2 Pa PA + 3 PA2 A − 3ka A 6a 6a  3 a + M + 3ka D M  = −Pτ . (83) 4 ∂ to quantize the model, After substituting all Pq = −i ∂q we obtain the corresponding Schrödinger–Wheeler–de Witt (SWDW) equation describing the quantum evolution of the Universe as  2 ∂2 ∂2 4 ∂ 2 A − − Heff  = 6a 2 ∂a∂ A 6a 3 ∂ A 6a 3 ∂ A2  2 1 a3 M + 3ka D M  + u 1 w1 3 − 3ka A + 6 Aa 4 ∂ , (84) =i ∂τ which is just of the form of the standard Schrödinger equation, Heff  = i ∂ . ∂τ (85) Therefore in the f (R, T ) theory of gravity, we can generate a Schrödinger type equation from the Wheeler–de Witt equation, which can solve the time problem in quantum gravity. When dτ dt = N (t) = 1, the WDW equation will take the form of the Schrödinger equation we are familiar with, ∂ . (86) ∂t Let us take now a deeper look into the time problem of quantum gravity, and analyze the physical meaning of the time τ , and of the effective Hamiltonian Heff we have introduced here. In the Wheeler–de Witt equation Hψ = 0 there seems to be no dynamics of the system. Therefore it turns out that the wave function of the Universe (more precisely, the physical states) do not describe states of quantum gravity at a particular time, as in the standard quantum theory. Rather, they describe states for all times, or, more precisely, just that information as regards the state of the Universe that is invariant under all spacetime diffeomorphisms [58] In the modified gravity model f (R, T ) = R + θ RT , time may be introduced locally by the coupling of the gravitational field and the matter field, because the interaction between the gravitational field and the matter field is also local. The profound connections between thermodynamics and gravity tell us that the arrow of time may come from the second law H = i 123 449 Page 10 of 19 Eur. Phys. J. C (2016) 76:449 of thermodynamics, since both processes reveal irreversible dynamics. If we only consider a small pitch of the Universe, and think of it as an adiabatic system, with H = 0 still valid in it, the coupling between curvature and matter will generate an arrow of time to measure the increase of its entropy, contributed by matter. The other components of the Hamiltonian, included in Heff , show us the dynamics of the gravitational field, and of some matter components, so that the effective Hamiltonian takes the form Heff ∼ b1 pi p j + b2 xi x j , which is similar to the Hamiltonian we usually meet in quantum mechanics. Therefore one can suppose that the Schrödinger equation in ordinary quantum mechanics might describe just a locally effective theory of the Wheeler–de Witt equation. In other words, we may conjecture that the Wheeler–de Witt equation provides the global quantum description for the Universe, while the Schrödinger equation is just the local description for the present day microscopic regions of the Universe. 4 A perturbative approach to the cosmological SWDW equation in f (R, T ) gravity In order to solve the SWDW equation Eq. (84) for the wave function of the Universe, we look for stationary solutions, and we separate the variables as (a, A, D, M, τ ) = e−i Eτ ψ(a, A, D, M). (87) Here E = constant. Thus we obtain the following differential equation describing the time evolution of the wave function of the Universe: Hψ =  + 4 ∂ 2 ∂2 2 ∂2 − − A 6a 2 ∂a∂ A 6a 3 ∂ A 6a 3 ∂ A2 4.1 Time evolution as a non-constant energy perturbation In order to obtain a solution of the SWDW equation obtained above, we can estimate different terms’ order of magnitude. 5 We know that x = a A1/2 ∼ 1/a 2 , y = A ∼ 1/a 7 , and hence we can obtain 5 x 4 ∼ 1/a 10 , 1 y 2 /x 3 ∼ 1/a 8 , x 3 /y 2 ∼ 1/a 4 . (92) 1 2 Since a is very small, the term x 3 /y ∼ 1/a 4 can be thought of as a small perturbation. To obtain some analytic solutions, we consider the case when k = 0 (flat Universe). This choice can help us to get rid of the large coupling term 5 y 2 /x 3 ∼ 1/a 8 . Due to the discussion by Hawking and Page [59], we also assume that u 1 w1 , the ordering parameter, can be neglected. In our case this can be achieved by putting it into the energy term, or neglecting it directly, since it is small as compared to the variables related to a(τ ). After making the above assumptions, we obtain the Schrödinger–Wheeler–de Witt equation as 1   2 ∂ y 2 1 2 ∂2 1 ∂ 2 ∂ − 3w y  = i ∂ , x x − 2y − y − 4 ∂x ∂y 4 ∂τ 3x 3 4 ∂ x 2 ∂ y2 (93)  1 2 a3 u 1 w1 3 − 3ka A + M + 3ka D M − E ψ = 0. 6 Aa 4 (88) 1 By introducing the new variables x = a A 2 and y = A, we obtain the differential equation for ψ  2 ∂ 1 ∂ 1 2 ∂2 2 ∂ x x − 2y − y − + u 1 w1 − 9kx 4 4 ∂x2 4 ∂x ∂y ∂ y2  1 a6 + 3ka 4 D y − 3x 3 y − 2 E ψ = 0. (89) + 3M 4 In the following we approximate the equation of state of the early Universe by the stiff equation of state p = ǫ, since in the very high density Universe one expects the speed of sound cs to be of the same order of magnitude as the speed √ of light, cs = ∂ǫ/∂ p = 1. When p = ǫ, and since Ṙ = − 3aN3 PC = 2 ȧa 21 R = ȧa R, from Eq. (8) it follows that in this case the energy is conserved, ǫ ′ + 3(ǫ + p)h = 0. Hence we have 123 w , (90) a6 where w > 0 is a positive constant. By substituting this result into Eq. (89), and by letting D = δ/a(τ ), we find  2 1 2 ∂2 ∂ 1 ∂ 2 ∂ x x − 2y − y − + u 1 w1 − 9kx 4 4 ∂x2 4 ∂x ∂y ∂ y2  5 1 3w y2 (91) − y − 9wδk 3 − 3x 3 y − 2 E ψ = 0. 4 x M = T − 4p = − where  = (x, y, τ ). In the stationary situation, we can decompose the variables as (x, y, τ ) = e−i Eτ ψ(x, y), and thus we get 1   ∂ 1 ∂ 3w ∂2 y 2 1 2 ∂2 2 x − 2y −y y ψ = Eψ. − x − 4 ∂x ∂y 4 3x 3 4 ∂ x 2 ∂ y2 (94) Now we change the above equation into another form, and 3 multiply by 3x1 both sides. We thus have the equation  y  ∂ 3E x 3 ∂2 1 2 ∂2 1 ∂ 3w 2 x − 2y −y y− ψ = 0. − x − 1 4 ∂x2 4 ∂x ∂y 4 ∂ y2 y2 2 (95) In the first formulation of the SWDW equation, given by Eq. (94), we can think of the time evolution of the wave function as resulting in the addition of a constant E as a perturbation. But what we know from the perturbation theory of quantum mechanics tells us that a constant perturbation gives Page 11 of 19 449 Eur. Phys. J. C (2016) 76:449 us no changes in the energy level and in the wave function, which is to say that it does not affect the physical observables. After changing the unperturbed Hamiltonian in the following way:  1  2 ∂ 1 ∂ 3w y 2 1 2 ∂2 2 ∂ x x − 2y − y y − − 3x 3 4 ∂ x 2 4 ∂x ∂y ∂ y2 4   2 2 1 2 ∂ 1 ∂ 3w ∂ 2 ∂ → − x − x − 2y −y y , 4 ∂x2 4 ∂x ∂y ∂ y2 4 we obtain the new version of the SWDW equation, where we do not consider the constant E as induced by a time evolution effect anymore. Instead, we consider it as a non-constant perturbation Vpert = − 3E x 3 1 that the system is twofold degenerate, at the beginning of time τ = 0, the wave function can be written as  = c1 ψ1 + c2 ψ2 (100) where c1 , c2 are constants satisfying the relation |c1 |2 + |c2 |2 = 1, and   1 3wy , (101) ψ1 = x −1−v1 √ Jv1 y   1 ψ2 = x −1−v2 √ Jv2 3wy , (102) y where v1 and v2 are positive constants. For the perturbed 3 system with perturbation Vpert = −3E x 1 , we write y2 (96) , y2 of the system, such that the total Hamiltonian is H = H0 + Vpert . Hence we can say that the time evolution effect on the wave function in a non-perturbative system, like the f (R, T ) gravity theory, is equivalent to the splitting of the degenerate energy levels (loss of symmetries) in a perturbative, static system. This result shows that the time evolution in the quantum cosmology of f (R, T ) gravity leads to the splitting of the degenerate energy levels, which reveals the deep connections between energy and time. In the new form of the Hamiltonian the unperturbed component H0 satisfies the equation H0 ψ = 0, whose eigenvalue of energy is zero. In the unperturbed Hamiltonian we can separate the variables as ψ(x, y) = X (x)Y (y), and obtain   ∂2 ∂ + (1 − v 2 ) X (x) = 0, (97) x2 2 − x ∂x ∂x   2 3w v2 − 1 ∂ 2 ∂ Y (y) = 0, (98) + y− y + 2y ∂ y2 ∂y 4 4 2 where the separation constant is denoted as v 4−1 . Then we obtain the expression of the wave function for the unperturbed Hamiltonian H0 as   1 (x, y, τ ) = e−i Eτ A1 x 1−v + A2 x −1−v √ y     × B1 Jv ( 3wy) + B2 J−v ( 3wy) , (99) where A1 , A2 , B1 , B2 are integration constants. Jv (x) is the Bessel function, and Jv (x) and J−v (x) are linearly independent functions. Since in the early Universe, we have x → ∞, in order to have an analytic solution in the whole plane, we let A1 = B2 = 0, and we assume v ≥ 0. 4.2 The twofold degenerate case In order to investigate the energy level split, we consider the simplest case, the twofold degenerate case [60]. Assuming Vi j =  ψi∗ Vpert ψ j dxdy. (103) With the use of the unperturbed wave functions, we obtain  V11 = ψ1∗ Vpert ψ1 dxdy  2 1   = (−3E)x 1−2v1 3 Jv1 dxdy, (104) 3wy y2  V22 = ψ2∗ Vpert ψ2 dxdy  2 1   = (−3E)x 1−2v2 3 Jv2 dxdy, (105) 3wy y2   ∗ = ψ1∗ Vpert ψ2 dxdy = (−3E)x 1−v1 −v2 V12 = V21 × 1  y 3 2 Jv1     3wy Jv2 3wy dxdy. (106) Since the curvature scalar and the scale factor are always positive, we cannot define wave functions that are analytic on whole space, thus the orthogonality and normalization are not satisfied. In the following we define the quantities    1 S11 = ψ1∗ ψ1 = x −2−2v1 [Jv1 ( 3wy)]2 dxdy, (107) y    1 S22 = ψ2∗ ψ2 = x −2−2v2 [Jv2 ( 3wy)]2 dxdy, (108) y  ∗ S12 = S21 = ψ1∗ ψ2    1 = x −2−v1 −v2 Jv1 ( 3wy)Jv2 ( 3wy)dxdy. (109) y In the special function theory, we have the Schafheitlin integral [61], which reads  ∞ Jμ (at)Jν (bt) dt tλ 0 = Ŵ(λ)Ŵ( μ+ν−λ+1 )( a2 )λ−1 2 )Ŵ( ν−μ+λ+1 )Ŵ( μ+ν+λ+1 ) 2Ŵ( μ−ν+λ+1 2 2 2 , (110) 123 449 Page 12 of 19 Eur. Phys. J. C (2016) 76:449 where a and b are positive constants, and satisfy the relations below to make the integral convergent, Re(μ + ν + 1) > Re(λ) > 0. (111) For z ∈ ℜ the Gamma function Ŵ(z) satisfies the identities √ Ŵ(z + 1) = zŴ(z), Ŵ(1) = 1, Ŵ( 21 ) = π . In the following we will restrict our analysis to the simplest case v1 = 23 , v2 = 5 2 . With the use of these values we obtain √ 2  1 [J 23 ( 3wy)] dxdy V11 = −3E 3 x2 y2 2  ∞  ∞ 1 [J 23 (z)] =η dx dz x2 z2 0 l ∞ η dx 1 =η = , (112) 2 2π x 2πl l √ 2  1 [J 25 ( 3wy)] V22 = −3E dxdy 3 x4 y2 2  ∞  ∞ 1 [J 25 (z)] =η dz dx x4 z2 l 0  ∞ η dx 1 = =η , (113) 4 x 6π 18πl 3 l √ √  1 J 23 ( 3wy)J 25 ( 3wy) V12 = V21 = −3E dxdy 3 x3 y2  ∞  ∞ 1 J 32 (z)J 25 (z) dz =η dx x3 z2 l 0  ∞ η dx 1 = =η , (114) 3 x 15 30l 2 l √ 2  1 [J 23 ( 3wy)] S11 = dxdy y x5 2  ∞  ∞ 2 [J 23 (z)] = dz dx z x5 l 0  ∞ 1 2dx 1 = = 4, (115) 5 6l x 3 l √ 2  1 [J 25 ( 3wy)] dxdy S22 = x7 y 2  ∞  ∞ 2 [J 25 (z)] dz = dx x7 z 0 l ∞ 1 2dx 1 = = , (116) 7 5 6 x 15l l √ √  1 J 23 ( 3wy)J 25 ( 3wy) S12 = S21 = dxdy x6 y  ∞  ∞ 2 J 23 (z)J 25 (z) dz dx = x6 z l 0  ∞ 2dx 1 1 = = , (117) 6 x 2π 5πl 5 l 123 √ √ where we have denoted z = 3wy → dz = 2√3wy dy, and √ η = −6E 3w, respectively, and we have also assumed that x and y are independent variables. The upper limits in the integrals are both ∞ when a(τ ) → 0. As for the lower limits of integration we assumed them to be very small positive numbers l and l1 , which correspond to the transition from the quantum regime to the classical regime, and thus they describe the limit of applicability of the present quantum model of the Universe. From the numerical evaluation of the integrals it follows that when l1 < 0.5 the integral values will not vary much, and therefore we let l1 = 0. 4.3 The energy levels of the quantum Universe We use the degenerate perturbation theory to find the energy levels in the quantum cosmology of f (R, T ) gravity. The wave function is given by  = c1 ψ + c2 ψ2 , and we already know the eigenvalues and the eigenfunctions of the unperturbed Hamiltonian H0 , which are given as solutions of the equation H0 ψi = E (0) ψi , i = 1, 2. (118) By substituting the above equation in the Schrödinger equation, H  = (H0 + V ) = E, we obtain (H0 + V )(c1 ψ1 + c2 ψ2 ) = E(c1 ψ1 + c2 ψ2 ). (119) After multiplying Eq. (119) with ψ1∗ and ψ2∗ , and integrating over a volume V , we obtain the equations     c1 E 1(0) S11 + V11 − E S11 + c2 E 2(0) S12 + V12 − E S12 = 0, c1  (0) E 1 S21  + V21 − E S21 + c2  (120) (0) E 2 S22  + V22 − E S22 = 0. (121) In the degenerate situation, we let E = E (0) + E (1) , and (0) cn = cn , that is, we take for these coefficients the zeroth order (unperturbed) approximation. Then we have     (122) c1 V11 − E (1) S11 + c2 V12 − E (1) S12 = 0,     c1 V21 − E (1) S21 + c2 V22 − E (1) S22 = 0. (123) From these two equations we can find the coefficients c1 , c2 , after the perturbed energy E (1) is obtained. The secular equation is    V11 − E (1) S11 V12 − E (1) S12    (124)  V21 − E (1) S21 V22 − E (1) S22  = 0. Then we obtain the first-order modifications of the energy as √ −B ± B 2 − 4 AC E (1) = , (125) 2A Page 13 of 19 449 Eur. Phys. J. C (2016) 76:449 where 4.4 The transition probability in the quantum Universe A = S11 S22 − S12 S21 , (126) C = V11 V22 − V12 V21 . (128) B = S12 V21 + S21 V12 − S11 V22 − S22 V11 , In the case analyzed earlier, where v1 = obtain 1 , l 10 A= 1 1 − 90 25π 2 C= 1 1 − 2 36π 900 B=− (129) In the previous section we have considered the early Universe as a quantum system that has twofold degenerate energy levels corresponding to the wave functions ψ1 , ψ2 at the time τ = 0. In the following we will consider the probability of a transition of the Universe, from the ψ1 state at τ = 0, to the ψ2 state at time τ , the transition taking place due to a perturbation of the initial state. In the zeroth order approximation the wave functions are (130) ψ = c1 ψ1 + c2 ψ2 , ψ ′ = c1′ ψ1 + c2′ ψ2 , (127) 3 2 , v2 = 79 η , 2700π l 7 η2 . l4 5 2, we where c1 , c2 and c1′ , c2′ are the two pair of coefficients obtained previously. Here ψ and ψ ′ are the wave functions (1) (1) corresponding to two energy states E 0 + E + and E 0 + E − , (1) (1) respectively, where E 0 = 0 in our case, and E + and E − are the modifications of the energy due to the effect of the perturbation. From the above equation we obtain The two modified energy levels are (1) E± = √ −B ± 79 2700π = ± B 2 − 4 AC 2A   79 2 2700π 2 −4  1 90  1 90 − 1 25π 2 − = β± ηl 3 ≈ (0.660 ± 0.440)ηl 3 1 25π 2   1 36π 2 − 1 900  ηl 3 (131) where β± are two constants. From the above equation we know that the modified energies have the same sign, and they are proportional to E, as well as proportional to l 3 , which will tell us the lower limit of the size of the energy gap, if we know the numerical value of l. From Eq. (122) the wave function’s coefficients are obtained:  2   (1) 1  η − E  2 5 ± 30l 5πl c1 =   2   (1) 1 (1) 1 2 η + 30lη 2 − E ± 5πl 5 2πl − E ± 6l 4     β(±) 2  1 −  30 5π (132) = 2    β β± 2 1 1 ± 2 l + 30 − 5π 2π − 6 and     c2 = ±  1 2π −  1 2π β± 6 − 2  β± 2 2 l 6 l2 +  1 30 − β± 5π 2 , (134) (133) respectively. The above expressions show the dependence of the coefficients c1 and c2 on l. When l is large, the Universe will have a higher probability to be in the state ψ2 , and it will have a higher probability to be in the state ψ1 when l is small. Note that the coefficients c − 1 and c2 do not depend on η, which implies that the parameter E will not affect the state of the wave function. ψ1 = c2′ ψ − c2 ψ ′ . c1 c2′ − c1′ c2 (135) After reintroducing the time factor, we obtain the time dependent wave functions as i   (1) (1) e− h̄ E 0 τ − h̄i E + τ ′ − h̄i E − τ ′ c − c ψ e ψe 1 = 2 c1 c2′ − c1′ c2 2   (1) (1) 1 − h̄i E + τ ′ − h̄i E − τ ′ . c = − c ψ e ψe 2 c1 c2′ − c1′ c2 2 (136) Note that 1 = ψ1 at τ = 0. Then we use ψ1 , ψ2 to represent ψ, ψ ′ , and hence 1 becomes the linear combination of the wave functions ψ1 , ψ2 , with the combination coefficients time dependent. The absolute value of the coefficient multiplying ψ2 and integrating over the volume V is (after squaring) the transition probability w21 . Therefore we have w21 =  (1) (1) E E− 1 −i + ′ h̄ τ + c ′ c S e −i h̄ τ c c S e 1 12 2 12 2 1 c1 c2′ − c1′ c2  (1) (1) E+ E− + c2 c2′ S22 e−i h̄ τ − e−i h̄ τ   Q1 =  Q3 =2 e −i (1) E+ h̄ τ −e Q 21 Q1 Q2 − 2 Q3 Q3 −i (1) E− h̄ τ  1 − cos + Q2e (1) −i (1) E− h̄ τ 2    (1) E+ − E− τ h̄ + Q 22 ,  (137) where we have denoted Q 1 = c1 c2′ S12 + c2 c2′ S22 , Q 2 = S12 , Q 3 = c1 c2′ − c1′ c2 . (138) 123 449 Page 14 of 19 Eur. Phys. J. C (2016) 76:449 With the use of the previous results, the transition probability in ordinary units is w21 0.880802 3 0.0106435 0.00659062 − cos = ηl τ . l 10 l 10 h̄ (139) From the above equation it follows that the transition probability is a cosine function of the time τ . When τ is very small we have w21 proportional to τ 2 . Since we know that the probability is smaller than one, we have the restriction 0.0106/l 10 − 0.0066/l 10 ≤ 1, a condition which gives for the upper limit of l the numerical value l ≥ 0.666 → a(τ ) ≤ 1.177. (140) Therefore we get the upper limit of the size of the Universe for which our quantum model is applicable. The lower limit of y is 0.320 < 0.5, and therefore we can approximate the lower limit in the integral as l1 = 0. 5 The second quantization of time In standard quantum mechanics, time is not an operator. The energy of the system is the eigenvalue of the Hamiltonian, which in the case of the harmonic oscillator can be written in the form Ĥ = h̄ω(a + a + 21 ), where a and a + are the creation and annihilation operators. On the other hand the deep connections existing between energy and time suggest us to find a way to define the creation and annihilation operators of ’time’. After these operators are found, we can get rid of the concept of a time singularity at the beginning of the Universe, and we can properly define the distance between each pair of time slices. Consequently, we can obtain the quantum frequency of the 3-space evolution and therefore investigate from a quantum-mechanical point of view the birth of the Universe. In the previous section, by using the mathematical formalism of the f (R, T ) gravity theory we have defined a ’time’ variable in the WDW equation, and thus we have transformed it into a Schrödinger–Wheeler–de Witt equation. The time τ we have defined earlier is based on the relation Pτ = − 6a2 3 PD PM , and therefore in our analysis we have assumed that the idea of time in the f (R, T ) theory of gravity is related to the field variables D and M. In the following we want to define the creation/annihilation operators based on the term 3ka D M − 6a2 3 PD PM in the WDW Eq. (84). The procedure goes as follows. We assume the existence of the classical and quantum analogy for the f (R, T ) gravity model, which allows us to turn the classical Poisson brackets into quantum commutators, {. . .} → [. . .]. Thus we postulate the following commutation relations: 123     D̂, PˆD = M̂, PˆM = i h̄. (141) Since    D̂ + i M̂ PˆD − i PˆM = D̂ PˆD − i D̂ PˆM + i M̂ PˆD  PˆD − i PˆM   + M̂ PˆM , (142) + i PˆD M̂ + PˆM M̂, (143) D̂ + i M̂ = PˆD D̂ − i PˆM D̂ we have       D̂ + i M̂ PˆD − i PˆM − PˆD − i PˆM D̂ + i M̂ = 2i h̄. (144) In the following we denote v̂ = D̂ + i M̂, P̂v = PˆD − i PˆM . (145) Hence, by using the mathematical identities [ PˆD , M̂] = ∗ [ PˆM , D̂] = 0, D̂ = D̂ ∗ , M̂ = M̂ ∗ , PˆD = PˆD , PˆM = ∗ PˆM , where * denotes the complex conjugate, we obtain the commutation relations [v̂, P̂v ] = 2i h̄, (146)    ∗ ∗ P̂v P̂v − P̂v P̂v = PˆD + i PˆM PˆD + i PˆM    − PˆD − i PˆM PˆD − i PˆM = 4i PˆD PˆM ,      v̂ ∗ v̂ ∗ − v̂ v̂ = D̂ − i M̂ D̂ − i M̂ − D̂ + i M̂   × D̂ + i M̂ = −4i M̂ D̂, (147) (148) and ∗ ∗ v̂ v̂ − v̂ ∗ v̂ ∗ + P̂v P̂v − P̂v P̂v       ∗ ∗ v̂ ∗ − i P̂v , = v̂ − i P̂v v̂ + i P̂v − v̂ ∗ + i P̂v (149) respectively. Also   ∗ v̂ ∗ , P̂v = (2i h̄)∗ . (150) τˆ1 = v̂ − i P̂v , τˆ2 = v̂ + i P̂v . (151) Let us assume now that there are two directions of time, defined as Then we have       τˆ1 , τˆ2 = v̂ − i P̂v , v̂ + i P̂v = −4h̄, τˆ1 , τˆ1   = τˆ2 , τˆ2 = 0. (152) As a next step in our analysis we interpret τˆ1 as a creation operator τ̂ + and τˆ2 as an annihilate operator τ̂ . Thus we Page 15 of 19 449 Eur. Phys. J. C (2016) 76:449 further obtain v̂ v̂ − vˆ∗ vˆ∗ + P̂v P̂v − Pˆv∗ Pˆv∗       ∗ vˆ∗ − i Pˆv∗ = v̂ − i P̂v v̂ + i P̂v − v̂ ∗ + i P̂v  ∗ = τ̂ + τ̂ − τ̂ + τ̂ = Nˆτ − Nˆτ∗ = 2iIm Nˆτ = 2i N̂obs . (153) Here we define the complex time “number” operator Nˆτ ≡ τ̂ + τ̂ . (154) The above relation shows that the time “number” operator, defined as N̂obs ≡ Im Nˆτ , (155) is an observable in quantum mechanics. Although the complex time has two directions, there is only one real time observable that can be measured in the experiments. Now let us consider the case of our WDW equation, where  2  2 − 3 PD PM + 3ka D M = 3 − PˆD PˆM + 9ka 4 D̂ M̂ . 6a 6a (156) In the following we discuss again the specific cosmological model f (R, T ) = R + θ RT . We rescale v̂ as  (157) v̂ = 3a 2 k v̂. Thus we obtain the corresponding representation of the timerelated terms in the WDW equation as 2 ˆ ˆ PD PM + 3ka D̂ M̂ 6a 3  −i  ˆ∗ vˆ∗ + P̂v P̂v − Pˆv∗ Pˆv∗ = v̂ v̂ − v 12a 3    + ∗ −1 −i + = 3 Im N̂obs . τ̂ τ̂ − τ̂ τ̂ = 3 12a 6a  − 2a Pa PA + 2PA2 A − 18ka 4 A −  3a 6 M N = N N . 2 (161) Equation (161) may give us a clear picture of the evolution of the Universe when we use the Arnowitz–Deser–Misner (ADM) formalism, since the dynamics of system will change with the variation of the time quanta N . Therefore when we are at different time moments, we shall have different Schrödinger equations to describe the local dynamics of the Universe. Of course, we can also define the observable time vacuum (the beginning of the Universe) as τ |0τ  = τ ∗ |0τ ∗  = 0, |0obs  = |0τ |0τ ∗ . (162) In the present section we have tried to introduce the Fock space of the quantum time variable τ . If we define the time creation/annihilation operators in the Fock space, then we can get rid of the problem of time singularity at the beginning of the Universe, and we may have a deeper understanding of the discrete nature of time. But this will also require us to transform our wave function into the quantum occupation number picture. However, we must note that this technique works only when the canonical momentum and its corresponding canonical position are not coupling with each other. One of the difficulties of the canonical quantization of gravity is that it cannot define the Hilbert space. The second quantization of time procedure may provide a prospective way to think about the problems of quantum gravity in the Fock space, instead of in the Hilbert space. − 6 Discussions and final remarks (158) From this relation it follows that the coupling between the gravitational field and the matter field gives us a way to measure the quantum time number of a given Universe. With the use of the above relations the WDW equation becomes   3a 6 M  = N̂obs . − 2a Pa PA + 2PA2 A − 18ka 4 A − 2 (159) Equation (159) gives us the possibility of further constructing a wave function with N quanta of time, which has the property N̂obs  N = N  N . Therefore the WDW equation becomes (160) Quantum cosmology offers a large number of challenges, but also interesting insights into the fundamental nature of the spacetime. The physical problems related to the birth and very early evolution of the Universe might be better understood by using the mathematical formalism of quantum theory, including symmetries, discrete structures, or semiclassical features extracted from a generally covariant, and highly interacting quantum theory. In this paper we have investigated the quantum cosmology of the f (R, T ) gravity theory, a modified gravity theory in which the gravitational action is an arbitrary function of the Ricci scalar and of the trace of the energy-momentum tensor. In the present paper we have considered that the classical evolution of the Universe takes place in the background Friedmann–Robertson–Walker geometry, which we are using systematically to investigate the quantum properties of the early Universe. As a starting point in our analysis we have introduced the Hamiltonian formulation of the 123 449 Page 16 of 19 Eur. Phys. J. C (2016) 76:449 theory, which is constructed systematically from the action given by Eq. (11). In the action we have introduced two Lagrange multipliers λ and μ, with the first imposing the (purely geometric) definition of the Ricci scalar, while the second one imposes the trace constraint of the f (R, T ) theory of gravity. This constraint goes beyond a simple definition of the trace of the energy-momentum tensor, since it allows the investigation of the deep connection between matter and geometry at a more general level than the one that follows from the simple thermodynamic definition of T . From the cosmological gravitational action one can obtain the gravitational cosmological Hamiltonian, which, by canonical quantization, leads immediately to the general form of the Wheeler–de Witt equation, describing the evolution of the wave function of the quantum Universe. In order to obtain some physical insights in the quantum properties of the Universe we consider a simple extension of the standard general relativity, in which the gravitational Lagrangian is a “deformation” of the form θ RT of the general relativistic Lagrange function R. We have investigated in detail the properties of this quantum cosmological model. Its most interesting feature is the possibility of the definition of a quantum time, and of an associated canonical momentum operator. This leads to the reformulation of the Wheeler– de Witt equation as a Schrödinger type equation. We have studied in detail the mathematical properties of this equation, by using a perturbative approach, in which the small perturbation is proportional to the energy of the system, in the framework of a twofold degenerate quantum system. The probability of a transition between states is also obtained. As a theoretical possibility we have also discussed very briefly the second quantization of time, which leads to the interesting possibility of the extension of the Hilbert space of the canonical quantization method to the Fock space description of quantum phenomena in the very early Universe. The initial state of the Universe is essentially unknown. That is why the possibility that the initial geometry of the Universe was not an isotropic and homogeneous, Friedmann– Robertson–Walker type one, cannot be rejected a priori. This raises the interesting question of the applicability of the formalism developed in the present to describe the quantum cosmology of f (R, T ) gravity to more general geometries. In particular, in the following we briefly consider the quantum cosmology of f (R, T ) gravity in the anisotropic Bianchi type I geometry, with the metric given by ds 2 = −N 2 (t)dt 2 + a12 (t)dx 2 + a22 (t)dy 2 + a32 (t)dz 2 , (163) where ai , i = 1, 2, 3, are the directional scale factors. For the Bianchi type I geometry the scalar curvature is obtained: 123 2 N2 R=  2 Ṅ − N a¨1 a¨2 a¨3 + + a1 a2 a3 + a˙1 a˙2 a˙3 + + a1 a2 a3  . a˙1 a˙2 a˙1 a˙3 a˙2 a˙3 + + a1 a2 a1 a3 a2 a3 (164) For the Bianchi type I geometry the gravitational action reads     2 Sg = dt N a1 a2 a3 f (R, T ) − λ R − 2 (· · · ) N   1 M − ··· , (165) −μ 2 while the gravitational Lagrangian can also be obtained: 2 λ (a˙1 a˙2 a3 + a˙1 a˙3 a2 + a˙2 a˙3 a1 ) N a1 a3 2 ˙ d a1 a2 a1 a2 a3 + λ −  + N dt a2 a3 Lg = − − N a1 a2 a3 V. (166) Note that all the definitions of  λ,  μ, V remain unchanged. In 1 the following we introduce a new variable W = (a1 a2 a3 ) 3 . d Then we easily obtain Ẇ = dt (a1 a2 a3 )/3V 2 and 3W Ẇ = d dt (a1 a2 a3 ), respectively, as well as the relation W Ẇ 2 = 3 W 3  a˙i 9 ai i=1 2 2 + (a˙1 a˙2 a3 + a˙1 a˙3 a2 + a˙2 a˙3 a1 ) . 9 (167) Hence the gravitational Lagrangian of the f (R, T ) theory of gravity in a Bianchi type I geometry can be represented as Lg = − 3   λ a˙i 9 λ W Ẇ 2 + W 3 N N ai 2 i=1 6 ˙ 2 λW Ẇ − N W 3 V. (168) −  N The Hamiltonian corresponding to the Lagrangian (168) now reads Hg = Ẇ PW + a˙1 Pa1 + a˙2 Pa2 + a˙3 Pa3 + · · · − Lg , (169) and we obtain the canonical momenta corresponding to the variables (W, a1 , a2 , a3 ) as  λ 6 ˙ 2 18 λ a˙1 λW , Pa1 = 2 W 3 2 , W Ẇ −  N N N a1   λ λ a˙2 a˙3 Pa2 = 2 W 3 2 , Pa3 = 2 W 3 2 . (170) N N a2 a3 PW = − For the other canonical momenta we obtain the following correspondence between the isotropic and anisotropic case: −6 2 2 d a ȧ(· · · ) → Pq = − (a1 a2 a3 ) N N dt 6 = − W 2 Ẇ (· · · ). N Pq = (171) Page 17 of 19 449 Eur. Phys. J. C (2016) 76:449 Hence the gravitational Hamiltonian becomes Hg = − 3 2 λ 3  a˙i 9 λ W Ẇ 2 + W N N ai 2 i=1 6 + N W 3 V − W 2 Ẇ [· · · ], N where [· · · ] is represented by − N 6 2 PW W Ẇ [· · · ] = − N 6W 2 PA + (172) PF 3 + Since N ai2 2 2 λ 3 a˙i 2 P , i = 1, 2, 3, W 2 = N λ ai 2W 3 ai 18  λW Ẇ 2 . N (173) (174) for the Hamiltonian we obtain PF N 9 λ W Ẇ 2 − Hg = PW PA + 2 N 6W 3 + 3 N  2 2 a Pa + N a1 a2 a3 V. λ i=1 i i 2W 3 (175) The anisotropic cosmological Hamiltonian (175) of the f (R, T ) gravity theory is very similar to the one obtained in the isotropic case. When considering the f (R, T ) = F 0 (R) + θ RT model in the case of R → ∞, we obtain Ḋ = −2 N PM , 4W 3 PM = 1 6 2 W Ẇ D, 2N (176)   and hence we immediately arrive at Ḋ = −(3/2) Ẇ /W D, and D = δ0 /W 3/2 , respectively, where δ0 is a constant. Since dW ∂ 2W 5/2 ∂ ∂ = −i =i , PD = −i ∂D d D ∂W 3δ0 ∂ W 6Ẇ ∂ 2 PD D = −i Ẇ . PD PM = − 4W 3 4W ∂W By taking into account that we obtain the time canonical momentum ∂ Pτ = −i = −i ∂τ 3  dai ∂ dW ∂ + dτ ∂ W dτ ∂ai i=1 3  2 1 =− a 2 Pa2 . PD PM + 3 4W λ i=1 i i 2W 3 3 2 1  2 2 a Pa → Pτ P P + D M 4W 3 λ i=1 i i 2W 3 Appendix A: The variation of the gravitational action in the f (R, T ) gravity theory δS = (179)   δT 1 f R (R, T )δ R + f T (R, T ) μν δg μν 16π δg  √ 1 1 δ( −gL m ) √ −gd4 x. − gμν f (R, T )δg μν + 16π √ 2 −g δg μν (A1) For the variation of the Ricci scalar we obtain  λ λ δ R = δ(g μν Rμν ) = Rμν δg μν + g μν (∇λ δŴμν − ∇ν δŴμλ ). (A2) (180) Since λ δŴμν = Finally, the transformation − Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.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 SCOAP3 . By varying the gravitational action (1) of the f (R, T ) theory with respect to the metric tensor we obtain first (178) ∂ N ai2 Pa2i = a˙i Pai = −iai , i = 1, 2, 3, 3  ∂a 2W λ i Acknowledgments We would like to thank to the anonymous referee for comments and suggestions that helped us to significantly improve the manuscript. T. H. would like to thank the Yat Sen School of the Sun Yat Sen University in Guangzhou, P. R. China, for the kind hospitality offered during the preparation of this work. S.-D. L. thanks the Natural Science Foundation of the Guangdong Province for support (No. 2016A030313313). (177) then we have − will allow us the introduce the time dependence of the Wheeler–de Witt equation for anisotropic Bianchi type I geometries in the f (R, T ) gravity theory. Note that here we have assumed that a1 (t), a2 (t), a3 (t), W (t) are independent variables. Hence we can safely conjecture that in the anisotropic case we can still introduce a cosmological quantum time in the Wheeler- de Witt equation of f (R, T ). On a qualitative level the overall results of the anisotropic case will be very similar to the ones obtained for isotropic and homogeneous geometries. In the present paper we have introduced some basic theoretical tools that could be used for the investigation of the quantum properties of the gravitational interaction, and of the evolution and origin of the very early Universe, in which the complex interaction of geometry and matter give birth to time, entropy, and irreversibility. 1 λσ g (∇μ δgνα + ∇ν δgαμ − ∇α δgμν ), 2 (A3) we finally obtain (181) δ R = Rμν δg μν + gμν δg μν − ∇μ ∇ν δg μν . (A4) 123 449 Page 18 of 19 Eur. Phys. J. C (2016) 76:449 −2Tμν + pgμν . Hence we find   ∇ μ Tμν = ∇ μ ǫ + ∇ μ p Uμ Uν + (ǫ + p)∇ μ Uμ Uν Therefore we obtain the variation of the action as δS = 1 16π   f R (R, T )Rμν δg μν + f R (R, T )gμν δg μν + (ǫ + p)Uμ ∇ μ Uν + ∇ μ pgμν . δ(g αβ Tαβ ) μν δg − f R (R, T )∇μ ∇ν δg μν + f T (R, T ) δg μν √  Since 1 1 δ( −gL m ) √ − gμν f (R, T )δg μν + 16π √ −gd4 x. 2 −g δg μν (A5) We define the variation of T with respect to the metric tensor as δ(g αβ Tαβ ) δg μν = Tμν + μν , (A6) δTαβ . δg μν (A7) where μν ≡ g αβ After partially integrating the second and the third term in the variation of the action, we obtain the field equations for our model as 1 f R (R, T )Rμν − f (R, T )gμν + (gμν  − ∇μ ∇ν ) f R R, T 2 = −8π Tμν − f T (R, T )Tμν − f T (R, T )μν . (A8) Appendix B: The energy balance equations in f (R, T ) gravity With the use of the mathematical identity   1 μ f R Rμν − f gμν + (gμν  − ∇μ ∇ν ) f R ∇ 2 1 ≡ − f T ∇ μ T gμν , (B1) 2 by taking the covariant divergence of the field equations (4) we obtain first 1 − f T ∇ μ T gμν = 8π ∇ μ Tμν + ∇ μ f T Tμν + f T ∇ μ Tμν 2 − ∇ μ f T pgμν − f T ∇ μ pgμν . (B2) Therefore we have μ ∇ Tμν  fT ( pgμν − Tμν )∇ μ I n f T = 8π + f T  1 + ∇ μ p − ∇ μ T gμν . 2 (B4) In the comoving reference frame Uμ = (N (t), 0, 0, 0), U μ = (−1/N (t), 0, 0, 0), L m = p. and therefore μν = 123 fT 8π + f T  − (ǫ + p)Uμ Uν ∇ μ I n f T  1 μ + ∇ (ǫ − p)gμν , 2 (B6) we obtain  μ  ∇ ǫ + ∇ μ p Uμ Uν + (ǫ + p)∇ μ Uμ Uν fT + (ǫ + p)Uμ ∇ μ Uν + ∇ μ pgμν = 8π + f T   1 × − (ǫ + p)Uμ Uν ∇ μ ln f T + ∇ μ (ǫ − p)gμν . 2 (B7) Multiplying U ν , and by taking into account the geodesic equation U nu ∇ μ Uν = 0, we have Uμ ∇ μ ǫ + (ǫ + p)∇ μ Uμ   fT 1 (ǫ + p)Uμ ∇ μ ln f T + Uμ ∇ μ (ǫ − p) . =− 8π + f T 2 (B8) Appendix C: The components of the Ricci tensor and the Ricci scalar in the FRW geometry By simple calculations we obtain the components of the Ricci tensor for the FRW metric (9) as Ṅ ȧ ä +3 , a Na  ȧ gii ä Rii = 2 +2 N a a R00 = −3 (C1) 2 +  2k N 2 Ṅ ȧ , i = 1, 2, 3. − a2 Na (C2) Therefore for the Ricci scalar we obtain  6 ä ȧ 2 k N 2 Ṅ ȧ + R= 2 + 2 − . N a a a Na (C3) (B3) In the following we adopt for the energy-momentum tensor the perfect fluid form, Tμν = (ǫ + p)Uμ Uν + pgμν . ∇ μ Tμν = (B5) Appendix D: The canonical momenta of the cosmological action in f (R, T ) gravity The canonical momenta associated to the cosmological action Eq. (11) of the f (R, T ) gravity are given by Pa = −2 6 6 ˙ λ, a ȧ λ − a 2 N N PA = − B 6 2 , a ȧ 1 − N A (D1) Page 19 of 19 449 Eur. Phys. J. C (2016) 76:449 PB = − 3BZ Z 6 2 a ȧ − + N A A2 , PC = − 6 2 a ȧ N   BZ R B(T − 4 p) 6 , − PD = − a 2 ȧ − N A A2   6 6 BZ(T − 4 p) PE = − a 2 ȧ , PF = − a 2 ȧ N N A2 PG = − 6 2 3BZ a ȧ − 2 N A , PR = − BZ E BD 6 + , PT = − a 2 ȧ − N A A2 4D B 6 4BZ E Pp = − a 2 ȧ . − N A A2 BR A , (D2) (D3) 3B A , (D4)  6 2 BC BZ D , a ȧ − N A A2  (D5) (D6) (D7) References 1. R.A. Hulse, J.H. Taylor, Astrophys. J. Lett. 195, L51 (1975) 2. B.P. Abbott, R. Abbott, T.D. Abbott, M.R. Abernathy et al., Phys. Rev. Lett. 116, 061102 (2016) 3. A.G. Riess et al., Astron. 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