Cosmology in Poincaré gauge gravity with a pseudoscalar torsion

A cosmology of Poincaré gauge theory is developed, where several properties of universe corresponding to the cosmological equations with the pseudoscalar torsion function are investigated. The cosmological constant is found to be the intrinsic torsion and curvature of the vacuum universe and is derived from the theory naturally rather than added artificially, i.e. the dark energy originates from geometry and includes the cosmological constant but differs from it. The cosmological constant puzzle, the coincidence and fine tuning problem are relieved naturally at the same time. By solving the cosmological equations, the analytic cosmological solution is obtained and can be compared with the ΛCDM model. In addition, the expressions of density parameters of the matter and the geometric dark energy are derived, and it is shown that the evolution of state equations for the geometric dark energy agrees with the current observational data. At last, the full equations of linear cosmological perturbations and the solutions are obtained.


Introduction
The discovery of the accelerated expansion of the universe motivates a large variety of theoretical works to explain it. In order to account for the acceleration the Einstein equation has to be modified and then two approaches are developed. One is to introduce "dark energy" in the right-hand side in the framework of general relativity (see [1][2][3][4][5][6][7][8][9][10] for recent reviews). The another is to modify the left-hand side of the equation, called modified gravitational theories, e.g., f (R) gravity (see [11][12][13][14][15][16][17][18][19][20] for recent reviews). A large amount of literature in every approach has been accumulated in the past years. However, none of them offers a convincing explanation of the observed results, and most of them were introduced to explain the acceleration phenomenologically, rather than emerging naturally out of fundamental physics principles. The most popular model in the fist approach is the Λ cold dark matter (ΛCDM) model which is plagued by the cosmological constant problem and the coincidence and fine tuning problem. Meanwhile, there is not the enough evidence on the validity of this model. It is shown that the thermal and mechanical stability requirement provides an evidence against the dark energy hypothesis [21]. Adding dark energy to the content of the Universe may not be the answer to the cosmic acceleration problem. In the second approach the Einstein-Hilbert Lagrangian is usually generalized to a function f of the Ricci scalar R. However, at present there are no fully realized and empirically viable f (R) theories that explain the observed level of cosmic acceleration. Furthermore, the f (R) theories suffer from a long-standing controversy about which frame (Einstein or Jordan) is the physical one [22][23][24][25][26]. It should be noted that although we have strong observational evidence for accelerated cosmic expansion but no compelling evidence that the cause of this JHEP05(2016)024 acceleration is really a new energy component. At the same time we do not have enough independent data yet to clarify the nature of dark energy. This provides further motivation for a deeper investigation of the nature of dark energy or the origin of the accelerated cosmic expansion. In the framework of f (R) gravity, the field equation can be written as the Einstein equation with an effective energy-momentum tensor that contains all the modifications and the energy-momentum tensor of matter fields. The contributions of the modifications of gravity can be identified with some kind of geometric dark energy. This is specially advantageous since one can define an equation of state associated with such dark energy and compare it with the ΛCDM model [27][28][29]. However, the function f (R) is not known a priori, none introduces a new fundamental principle that can be used as a guiding line, it is usually constructed by trial and error. In fact, as a geometric theory a modified gravity should be formulated in a gauge theoretical framework. A famous example is the Poincaré gauge theory of Gravity [30][31][32][33][34][35][36][37]. Some works have been done to develop a model of geometric dark energy in the Poincaré gauge theory framework [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56]. In [38][39][40][41][42][43] the effect of torsion is to introduce an extra-term into matter density and pressure which gives rise to an accelerated behavior of the universe. However, the torsion contributes only a constant density, it is not possible to solve the coincidence and fine tuning problem. The torsion model in [44][45][46][47][48] contributes an oscillating aspect to the expansion rate of the universe and displays features similar to those of only the presently observed accelerating universe. In [49][50][51][52][53][54][55][56] the Lagrangian involves too many terms and indefinite parameters, which make the field equations complicated and difficult to solve and the role of each term obscure. In order to simplify the field equations, some restrictions on indefinite parameters have to be imposed. Under these restrictions, especially, all the higher derivatives of the scale factor are excluded from the cosmological equations.
In fact, starting from a well behaved Lagrangian 1 2 R + αR 2 + βR µν R µν in quadratic gravity [57][58][59][60][61][62][63] and string theory [64] and adding a quadratic term of torsion γT µ νρ T µ νρ a good toy model can be obtained [65], where we derive to give the gravitational field equation and the cosmological evolutional equations. When the macroscopic spacetime average of the spin vanishes, the cosmological equations are found to split into two families. Each of them is related with only one torsion function, the scalar or the pseudoscalar torsion function. It has been argued [44][45][46][47][48] that only these two scalar torsion modes are physically acceptable and no-ghosts. This model has a free-ghost dynamics. It has a well posed initial value problem without any ghost or tachyonic propagation. Also, the field equations are allowed to contain higher derivatives in [65], which is different from [49][50][51][52][53][54][55][56], where some restrictions on indefinite parameters are imposed in order to exclude higher order derivatives. In addition, for the first family we solve the cosmological equations corresponding to the scalar torsion function by using the dynamical system approach in ref. [65]. In this paper, we study the second-family cosmological equations corresponding to the pseudoscalar torsion function in detail, with using a totally different way. Some meaningful consequences can be inferred, such as the geometrical interpretation of cosmological constant is investigated, the cosmological constant problem and the coincidence and fine tuning problem are solved naturally, the state equation of the geometrical dark energy is derived and its evolution is consistent with the current observations, the analytic solution a = a(t) in cosmology is obtained and the perturbation analysis is given, etc.

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In section 2, starting from the Poincaré gauge principle and the simple Lagrangian 1 2 R+αR 2 +βR µν R µν +γT µ νρ T µ νρ , which is the sum of the Starobinsky Lagrangian [66] and Yang-Mills type terms of the local rotation and translation field strength, we introduce the main equations in this Poincaré gauge cosmology, including the gravitational field equations and the cosmological equations, etc. In order to evade any unnecessary discussion regarding frames (i.e. Einstein vs. Jordan) the theory is treated using the original variables instead of transforming to a scalar-tensor theory in contrast to f (R) theories. Furthermore, a set of cosmological equations corresponding to the pseudoscalar torsion function are discussed, where it is found that although we do not introduce a cosmology constant in the action it automatically emerges in the derivation of the cosmological equations and then is endowed with intrinsic character. The dark energy is identified with the geometry of the spacetime and is a function of the density and the pressure of the matter. It includes the cosmological constant but can not be identified with it. It is nothing but the intrinsic torsion or curvature of the vacuum universe. In section 3, the analytic expressions of the state equation and the density parameters of the matter and the geometric dark energy are derived and used to determine the values of α, β and γ. Then a theoretical value of the cosmological constant is computed and compared with the observed datum. The cosmological constant problem and the coincidence and fine tuning problem are solved naturally. In section 4 an analytic integral of the cosmological equation is obtained and used to evaluate the age of the universe which can be compared with observed data. In section 5 the full equations of linear cosmological perturbations and the solutions are obtained. In addition, the behavior of perturbations for the sub-horizon modes relevant to large-scale structures is discussed. It is shown that our model can be distinguished from others by considering the evolution of matter perturbations and gravitational potentials. Section 6 is devoted to conclusions.

Cosmological equations
The discussion in this paper are entirely classical. We consider a Poincaré gauge theory of gravity , in which there are two sets of local gauge potentials, the orthonormal frame field (tetrad) e I µ and the metric-compatible connection Γ IJ µ associated with the translation and the Lorentz subgroups of the Poincaré gauge group, respectively. We use the Greek alphabet (µ, ν, ρ, . . . = 0, 1, 2, 3) to denote (holonomic) indices related to spacetime, and the Latin alphabet (I, J, K, . . . = 0, 1, 2, 3) to denote algebraic (anholonomic) indices, which are raised and lowered with the Minkowski metric η IJ = diag (−1, +1, +1, +1). The field strengths associated with the tetrad and connection are the torsion T λ µν and the curvature R µν λτ . We use the geometrized system of units in which 8πG = 1, c = 1, and start from the action where L m denotes the Lagrangian of the source matter including baryonic matter, cold dark matter and radiation, α and β are two parameters with the dimension of [L] 2 , γ is a dimensionless parameter. The vales of α, β and γ can be determined by experiment and JHEP05(2016)024 observational data. The terms 1 2 R and γT µ νρ T µ νρ represent weak gravity, while αR 2 and βR µν R µν represent strong gravity with the dimensionless strong gravity constant α and β according to Hehl et al. [30][31][32][33][34][35][36][37].
The variational principle yields the field equations for the tetrad e I = e I µ ∂ µ and the connection Γ IJ = Γ IJ µ dx µ [30][31][32][33][34][35][36][37]: with the covariant derivatives (D) of the translation excitation H I and the Lorentz excitation H IJ , the gauge currents of energy-momentum T (g)I and spin s (g)IJ and the canonical matter currents of energy-momentum T I and spin (angular momentum) s IJ . The reduced explicit form of these field equations are [65]: tensor and spin tensor of the source matter, respectively, while and are the energy-momentum and the spin of this kind of "geometric dark energy" corresponding to the terms αR 2 + βR µν R µν + γT µ νρ T µ νρ in (2.1). Note that the energy-momentum tensor T νµ of type (0, 2) should not be confused with the torsion tensor T λ µν of type (1,2). If α = β = γ = 0, these equations become the field equations of Einstein-Cartan-Sciama-Kibble theory. Furthermore, for T λ µν = 0 we come back to General Relativity. The eqs. (2.2)-(2.5) can be rewritten in terms of the covariant derivative. Here given that they are convenient in the concrete calculation for deriving the following cosmological equations, we exhibit these expressions (2.2)-(2.5) directly.
For the spatially flat Friedmann-Robertson-Walker (FRW) metric and are the density and the pressure of the geometric dark energy. Eqs. (2.12) and (2.13) indicate that the geometric dark energy is just the gravitational field itself described by h, H and f . The source matter is a fluid characterized by the energy density ρ = T 00 , the pressure p = T ij (i = j) and the spin s IJ µ . Eqs. (2.8) and (2.9) lead to ·· a a = − 1 6 (ρ + ρ g + 3p + 3p g ) . (2.14) It is easy to see that when α = β = γ = 0 and h = f = 0, (2.8), (2.9) and (2.14) reduce to the Friedmann cosmology. Eqs. the Raychaudhuri equation respectively, while (2.14) is the acceleration equation, which represent the Einstein frame of the theory.
Since the spin orientation of particles for ordinary matter is random, the macroscopic spacetime average of the spin vanishes, we suppose s IJ λ = 0, henceforth. Then, the equation (2.11) has the solutions f = 0, (2.15) and .
The solution (2.15) has been investigated in [65]. We concentrate on the equation (2.16) now. Differentiating (2.16) gives Substituting it and (2.16) which further give

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is the geometric cosmological constant coming from the terms βR µν R µν + γT µ νρ T µ νρ in (2.1). It is easy to see that the higher order derivative ·· H and · f in (2.10) disappear. We also note that although we do not introduce a cosmological constant Λ in the action (2.1), it automatically emerges in these equations. In (2.18) the pseudoscalar torsion f is a function of ρ and p rather than a constant, in contrast to [27][28][29]. It should be noted that although (2.8)  h , and f as indicated by (2.12) an (2.13). In other words, this is a different model from the ΛCDM model essentially.
Eqs. (2.12) and (2.13) become now which mean that the geometrical dark energy includes the cosmological constant Λ but can not be identified with it. The cosmological constant Λ is really a constant determined by β and γ as indicated by (2.20) while the geometrical dark energy ρ g is a function of the density ρ and the pressure p of the matter. The cosmological constant problem and the coincidence and fine tuning problem are relieved naturally, as shown in the next section.
The source matter includes ordinary baryon matter, dark matter and radiation: The equation (2.8) can be written as where are the dimensionless density parameters of the matter, the radiation and the geometrical dark energy, respectively. Suppose Eqs. (2.22), (2.23) and (2.24) become

5)
and ·· a a give ) Since  Figure 1 plots the evolution history of w g (a) given by (3.18).
In observation and experiments it is conventional to phrase constraints or projected constraints on w(z) in terms of a linear evolutional model [70]: where w 0 is the value of w at z = 0 (a = 1), and w p is the value of w at a "pivot" redshift z p . For typical data combinations, z p ≈ 0.5. To this end we give the linear approximation of (3.18). When a = 1, w g0 = −1,

An exact analytic solution of cosmological equation
The equation (2.19) can be solved in two cases as follows. In the radiation dominated era ρ r = ρ r a 4 , ρ r = ρ r,a=1 = const, p r = 1 3 ρ r ,

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and can be rewritten Its integration gives In the matter dominated era and then Its integration gives ln . where the time t is in cm. This is a new exact analytic cosmological solution which resembles but differs from the ΛCDM solution [72]. Now we can evaluate the age of the universe using (3.13), (4.3) and (4.7). In the radiation dominating era z 3000 [72], we have the equation Choosing z = 3000, then a = 1/3001, this equation gives t = 3. 298 × 10 23 cm = 3. 4895 × 10 5 Y ears. In the matter dominating era z 3000, we have the equation Choosing z 1 = 3000, a 1 = 1 3001 ,

Perturbation theory
In order to discriminate lots of dark energy models, it is interested to seek the additional information other than the background expansion history of the universe [73][74][75][76]. Now we discuss the dynamics of linear perturbations and the structure growth of universe.

Cosmological perturbations of gravitational potentials and the torsion
The perturbed equations can be derived by straightforward and tedious calculations, following the approach of [77][78][79][80][81][82]. The computer software Maple has been applied to work out the lengthy calculations. We focus on the scalar perturbations, since they are sufficient to reveal the basic features of the theory, allowing for a discussion of the growth of matter overdensities. The perturbed vierbein reads in which we have introduced the scalar modes φ and ψ as functions of t. This induces a metric perturbation of the known form, namely in the longitudinal gauge and the conformal time η.
In order to preserve the global homogeneity and isotropy of the spacetime the perturbations are assumed to be small. It has been argued [44][45][46][47][48] that only two scalar torsion modes h and f are physically acceptable and no-ghosts. On the basis of the above theoretical tests (e.g., "no-ghosts" or "no-tachyons"), we use (2.7) to give the linear perturbation of the nonvanishing torsion components δT ij0 = δ ij a 2 δh, δT ijk = 2ǫ ijk a 3 δf, i, j, k, . . . = 1, 2, 3.
In the case (2.16), h = 0, we have δT ij0 = 0, δT ijk = 2ǫ ijk a 3 ξ, i, j, k, . . . = 1, 2, 3. where ξ = δf . The unperturbed field equation (2.2) can be written as where G µ ν is the Einstein tensor, T µ ν is the energy-momentum of the ordinary matter and the radiation, T (g) µ ν is the energy-momentum of the "geometric dark energy" given by (2.4). The equations of motion for small perturbations linearized on the background metric are For scalar type metric perturbations with a line element given in (5.2) (in conformal time), the perturbed field equations can be obtained following the approach of [77][78][79][80][81][82].

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The cosmic fluid includes radiation, baryonic matter and dark matter, we suppose Then we have The equation takes the form where the growth of the baryonic matter density perturbation δ := δρ b /ρ b , H := a ′ /a = aH, prime denotes derivative with respect to the conformal time η. The equation where i = 1, 2, 3. The equation 14) JHEP05(2016)024 In commoving orthogonal coordinates, the three-velocity of baryonic matter vanishes, V i b = 0 [83]. For the potential V of the three-velocity field of the dark matter, the perturbed conservation law δ (∇ µ T µ ν + ∇ µ T g µ ν ) = 0, (5.16) leads to the equation (5.18) In the Fourier space k, from (5.11) and (5.13) we obtain the equations of φ and ψ, agreeing with GR but in contrast to f (R) theory [84]. Then we have the equations of ψ: One of the methods to measure the cosmic growth rate is redshift-space distortion that appears in clustering pattern of galaxies in galaxy redshift surveys. In order to confront the models with galaxy clustering surveys, we are interested in the modes deep inside the Hubble radius. In this case we can employ the quasistatic approximation on sub-horizon scales, under which, ∂/∂η ∼ H ≪ k. Then the perturbation equations (5.22), (5.23) give The equation (5.24) gives the expression of gravitational potential ψ. In the case it reduces to the Poisson equation where is the effective gravitational coupling constant. In the framework of GR, G eff is equivalent to the gravitational constant G = 1.

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Up to now, the complete set of equations that describes the general linear perturbations has been obtained. It provides enough information about the behaviors of the perturbation and can be compared with the results of the ΛCDM model.

JHEP05(2016)024 6 Conclusions
A cosmology of Poincaré gauge theory has been developed. We focus on the case including a pseudoscalar scalar torsion function f as suggested by Baekler, Hehl and Nester [98]. The gravitational field equation and the two-family cosmological equations has been obtained in ref. [65]. In this paper, we focus on studying the second family cosmological equations corresponding to the pseudoscalar torsion function. It is found that although we do not introduce a cosmological constant in the action it automatically emerges in the derivation of the cosmological equations and then is endowed with intrinsic character. It is nothing but the intrinsic torsion and curvature of the vacuum universe. The dark energy is identified with the geometry of the spacetime. Now we are returning to the original idea of Einstein and Wheeler: gravity is a geometry [99,100]. The cosmological constant puzzle and the coincidence and fine tuning problem are solved naturally. The point is that the dark energy is the functions of the density and pressure of the cosmic fluid and includes the cosmological constant but can not be identified with it. The analytic expressions of the state equation and the density parameters of the matter and the geometric dark energy are derived and used to determine the values of α , β and γ. Then a theoretical value of the cosmological constant is computed and compared with the observed datum. An analytic integral of the cosmological equation is obtained and used to evaluate the age of the universe which can be compared with observed data. The full equations of linear cosmological perturbations and the solutions are obtained. In addition, the behavior of perturbations for the sub-horizon modes relevant to large-scale structures is discussed. This model can be distinguished from others by considering the evolution of matter perturbations and gravitational potentials.

Acknowledgments
We thank the anonymous referee for his/her very instructive comments, which improve our paper greatly. The research work is supported by the National Natural Science Foundation of China (11205078).

A The growth of structures in linear perturbation theory
In the following, we give the derivation of the equation (5.38): Letting