Wormholes in viable $f(R)$ modified theories of gravity and Weak Energy Condition

In this work wormholes in viable $f(R)$ gravity models are analysed. We are interested in exact solutions for stress-energy tensor components depending on different shape and redshift functions. Several solutions of gravitational equations for different $f(R)$ models are examined. Found solutions imply no need for exotic material, while this need is implied in the standard general theory of relativity. Simple expression for WEC violation near the throat is derived and analysed. High curvature regime is also discussed, as well as the question of the highest possible values of Ricci scalar for which WEC is not violated near the throat, and corresponding functions are calculated for the several models. The approach here differs from the one that has been common since no additional assumptions to simplify the equations are made, and functions in $f(R)$ models are not taken to be arbitrary functions, but rather a feature of the theory that has to be evaluated on the basis of consistency with observations for the Solar System and cosmological evolution.


I. INTRODUCTION
Modified f (R) gravity represents a possible alternative to Einstein's theory of general relativity which has received increased attention in the last decade. It is based on a generalization of the Einstein field equations that comes as a result of replacing the Ricci scalar curvature, R, with an arbitrary function of the scalar curvature, f (R), in the gravitational Lagrangian density. One of the main reasons for increased interest in modified gravity theories comes from the possibility of explaining accelerating expansion of the universe, that has basically been confirmed by observations from type Ia supernovae [1,2,3], but also from other cosmological observations such as those from large scale structure [4], and cosmic microwave background radiation [5]. An important feature of f (R) gravity is that in its framework, unlike in Λ CDM cosmology based on the standard general relativity, there is no need for postulating dark energy or introducing any kind of new scalar or spinor field to explain the accelerated expansion [6,7]. The action for f (R) theories is given by where k = 8πG, g is a determinant of the metric, and S M AT is the matter action. Depending on the assumptions taken in the variational procedure starting from (1) we can make a distinction between metric, Palatini and metric-affine formalism [8,9]. In metric formalism we proceed from the assumption that the connection is dependent on the metric, namely that it is given by the Christoffel connection. In Palatini formalism the connection is treated independent of the metric and it is also assumed that the matter part of the action is not dependent on the connection. Finally, in metric-affine formalism the matter part of the action now depends on the connection which is metric independent. In this work we will use the metric approach which is the simplest of the above mentioned and also usually used in where f R = df (R)/dR, and we will use an analogous notation for higher derivatives of f (R). Adopting the standard definition: G µν = R µν − 1/2Rg µν , and after some mathematical manipulations, we can obtain a following equation for the Einstein tensor from (2) where T is the trace of the stress-energy tensor. In this work we have analysed wormhole solutions in the framework of viable metric f (R) gravity models which do not violate standard energy conditions. Wormholes are hypothetical tunnels with a throat that connects two asymptotically flat regions of spacetime. In Einstein's general relativity, a construction of a wormhole is possible only by the use of exotic matter i.e. matter that violates usual energy conditions [10,11,12]. The matter threading the wormhole is usually described by the perfect anisotropic fluid T µν = diag(ρ, p r , p t , p t ).
It can be shown that existence of a wormhole in General relativity implies the condition ρ + p r < 0 and according to [10] we shall call the material with this property exotic. This violates the Weak Energy Condition (WEC), which is given by T µν k µ k ν ≥ 0 for any timelike vector k µ [13]. The Weak Energy Condition expresses constraints on a possible matter behaviour in order to guarantee some usual properties, such as possitive energy density. The general question of the WEC violation in modefied gravity still remains open [14].
In modified f (R) theories of gravity, wormholes can be supported by ordinary matter [15,16,17]. Therefore, while interested in WEC non-violation we are exploring solutions that satisfy ρ ≥ 0 and ρ+p r ≥ 0 [15]. Our aim is to analyse, without any additional assumptions, possible wormhole solutions in different viable recently proposed f (R) models that do not imply the existence of exotic material. In Section II. we present wormhole geometry, effective field equations and derive suitable expression for the WEC non violation near the throat.
In Section III. we present and analyse some specific solutions in different models. High curvature regime is considered in Section IV. and we make conclusions in Section V.

II. WORMHOLES IN f (R) GRAVITY
The geometry of a static, spherical symmetric wormhole is given by where ϕ(r) is the redshift function and b(r) is a shape function [10]. Functions ϕ(r) and b(r) are arbitrary functions of the radial coordinate r, which nonmonotonically decreases from infinity to a minimal value r 0 in the throat and increases to infinity. For the throat position r = r 0 ⇒ b(r 0 ) = r 0 the metric tensor component is singular. Nevertheless, the proper distance must be well behaved, from which the following integral must be real and regular outside the throat [10]: from which follows the condition: So, far from the throat in both radial directions space must be asymptotically flat which implies the condition b(r)/r → 0 as l → ±∞ [10].
One of the fundamental wormhole properties is that by definition b(r) must fulfil the flaring-out condition: , where b (r) = db/dr, (in further text prime denotes a derivative with respect to the argument of a function). The second condition which we impose is practical: we demand that a wormhole must be traversable which means the absence of horizons. So ϕ(r) must be finite everywhere.
Using standard mathematical procedure from (5) we obtain the Ricci curvature scalar: While studying wormholes in f (R) modified theories of gravity, in order to simplify equations, it is common to place an additional condition on red-shift function ϕ(r) to be constant [15,18,19]. This condition on ϕ(r), which is assumed without any physical reason, is not justified because the fundamental parameters of a wormhole should not be restricted in such an artificial way. Moreover, wormhole solutions of modified Einstein's equations and the WEC violation will in some cases critically depend on ϕ(r).
Matter is described by the stress-energy tensor of the anisotropic perfect fluid: where U is a four-velocity, ρ the energy density, p t and p r are transversal and radial pressure respectively, and χ µ = 1 − b(r)/rδ µ r . In (1) we select k = 1 for simplicity, and from (4) we obtain modified Einstein's equations for the wormhole geometry Note that field equations (10 -12) are forth order nonlinear differential equations in ϕ(r) and b(r). However, equations (10 -12) at the same time represent the system of algebraic equations for the stress-energy tensor components that, despite it's complexity, have analytic solutions. In our work in specific models of modified f (R) gravity we consider solutions for the components of the stress-energy tensor by exploring different redshift, ϕ(r), and shape functions, b(r). From field equations (10 -12) we can derive a specific form of the WEC for wormhole solutions in f (R) gravity We require that the matter threading the wormhole satisfies WEC, so we demand that inequalities (13) and (14) are fulfilled. Since in Einstein's general relativity, which corresponds to f (R) = R, this is not possible, higher curvature terms in the action support wormhole geometries. We can see that explicit analysis of the equations (13) and (14) is extremely difficult, and that for a specific wormhole geometry WEC violation can critically depend on the redshift functions ϕ(r) and its derivatives. As an important and interesting case we can consider the equation (14) near the throat. This approach simplifies the problem considerably. Near the throat b(r 0 ) r 0 and the equation (14) becomes From the flaring-out condition we must have b (r)r − b(r) < 0, so we finally obtain The last condition is, due to its simplicity, particularly suitable for analysing the influence of modifying the theory of gravity on the question of the WEC violation.
It is obvious that this condition cannot be fulfilled for every choice of f (R). For instance, if we take f (R) = R this condition is not satisfied and this corresponds to the need for exotic matter in Einstein's relativity. In the next section our approach will be to find specific solutions of equations (10 -12) for a given wormhole geometry and then check whether the WEC is satisfied or not, rather than analyse equations (13) and (14).

III. SPECIFIC MODELS AND SOLUTIONS
In some works which analyse wormholes in the context of f (R) gravity [15,18] f (R) is usually treated as an unknown function, which can be derived from modified field equations, or it is considered to have some simple convenient shape. We prefer the approach in which f (R) functions are taken as predetermined characteristic of the theory. In fact, due to the highly hypothetical nature of a wormhole, which is at the moment far away from any empirical observation, we cannot impose conditions on f (R) in a manner stated above. Therefore, the form of f (R) should be consistent with observations for the Solar System and cosmological evolution, so we analyse wormhole solutions in several viable models of f (R) gravity [20,21,22,23,24]: • MJWQ model [25] f where β and R * are free positive parameters of the model.
with three free positive parameters λ, R * and q.
• Exponential gravity model [32,33] f whereR = R/R * , with λ and R * as free positive parameters of the model.
• Tsujikawa model [28,34] f where µ and R * are free positive parameters of the model.
In all models R * = σH 2 0 , where σ is some dimensionless parameter and H 0 is the current value of the Hubble parameter, which is taken to be H 0 = 1.

A. MJWQ model
In MJWQ model (17) we solve field equations (10 -12) to obtain components of the stress-energy tensor and check if the conditions ρ ≥ 0 and ρ + p r ≥ 0 are satisfied. We consider specific red-shift and shape functions given by ϕ(r) = ln(r 0 /r +1) and b(r) = r 0 r 0 /r. Parameters of the model, β and R * , are taken to be close to the values proposed in [20]. Above mentioned solutions are depicted in Fig. 1 and Fig. 2

B. Starobinsky model
Let us consider specific functions b(r) = r 0 ln r/r 0 + r 0 , ϕ(r) = r 0 /r. As in [20] we choose R * = 4.17 with λ and q close to the values λ = 1, q = 2. The solutions are depicted in Fig. 3 and Fig. 4. We see that every combination of parameters implies the need for exotic matter. Moreover, for every considered combination of simple shape and red-shift functions we did not find non-exotic matter solutions in the Starobinsky model.

C. Exponential gravity model
We take the shape and the redshift functions previously considered in the MJWQ model with λ and R * close to the values in [20]. Solutions are presented in   Figure 6: The non-exotic material condition, ρ + p r , in the Exponential gravity model for the specific choice of b(r) = r 0 r 0 /r, ϕ(r) = ln(r 0 /r + 1), where x = r/r 0 .   [35]. In accordance with this demand all considered models share the same mathematical property that in the high curvature limit they have the following form

D. Tsujikawa model
so the models lead to the effective cosmological constant λR * /2, as can be seen by applying (21) to (2) and using the standard definition of the cosmological constant in Einstein's field equations. Therefore the high curvature regime is interesting because we have an interplay of cosmological features of the f (R) models and wormhole solutions. We expect to have the high curvature limit in the vicinity of the throat for a suitable choice of b(r) and ϕ(r) which will lead to R(r) R * .
By rewriting Einstein's field equations (10)(11)(12), with (8) and (21), we can easily obtain the solutions for the stress-energy tensor components in the high curvature It can be seen that these equations are equal to the ones presented in [10] with addition of an effective cosmological constant, as should be expected. By inspecting (16) it is apparent that the WEC is violated near the throat in the high curvature regime. Since all viable f (R) models share the same asymptotic behaviour described by (21) one can question the critical R(r) value for every point near the throat, in a specific f (R) model. By critical value we mean the highest possible R(r) value for which WEC is satisfied at a specific point in space. Let us consider solutions, R critical (r), of the following equation that can be obtained from (16) f RR R critical (r)r + 2f R = 0 near the throat, (25) as well as solutions,R, of the WEC violation inequality f RRR (r)r + 2f R > 0. (26) From the theory of differential inequalities follows in the interval near the throat r 0 < r < r 1 , wherē R(r 0 ) = R critical (r 0 ). Therefore, values of R critical correspond to the critical values of the Ricci scalar in the above mentioned sense. We can solve the equation (25) and obtain R critical in different models of f (R) gravity.
For instance in the MJWQ model we get where R * = 1, β = 2, R(r 0 ) = R 0 . In Exponential gravity for fixed parameters q = 2, λ = 2 we obtain with c = 2r 2 0 e R 0 /2 − r 2 0 . In this way it is possible to considerably simplify the analysis of the wormhole WEC violation in f (R) theories of gravity. For a given R(r) one can compare its values near the throat with values of R critical (r) in a concrete f (R) model and using (27) and (26)  we can see that for the above cases the character of the solutions in the MJWQ, Exponential gravity and Tsujikawa model depends more strongly on the choice of the free parameters than on the choice of a specific model. It is also possible to satisfy non-exotic matter conditions with other simple choices of ϕ(r) and b(r) for each model. For instance b(r) = r 0 e (1−r/r0) , b(r) = r 2 0 /r, ϕ(r) = 1/r, etc. Simple inequality for the WEC violation near the throat is derived, which we demonstrate to be particularly suitable for analysing the influence of modifying theory of gravity on the question of the WEC violation. We have shown that all viable f (R) models must share the same mathematical form in the high curvature regime and that in this limit the WEC is necessar-