$f(R)$ gravity solutions for evolving wormholes

Scalar-tensor $f(R)$ theory of gravity is considered in the framework of a simple inhomogeneous space-time model. In this we use the reconstruction technique to look for possible evolving wormhole solutions within viable $f(R)$ gravity formalism. These $f(R)$ models are then constrained so that they are consistent with existing experimental data. Energy conditions related to the matter threading the wormhole are analysed graphically and are in general found to obey the null energy conditions (NEC) in regions around the throat, while in the limit $f(R)=R,$ NEC can be violated at large in regions around the throat.

Wormhole is a hypothetical object which acts like a bridge or tunnel to connect asymptotic regions of a single universe or two distinct universes. This imaginary, intuitive concept is one of the most popular and intensively studied area of research in general relativity. The nature of wormholes studied so far can be classified as static wormholes and dynamic wormholes. Although there are wormhole solutions [1][2][3][4][5] in the literature since 1973, but pioneering work related to static wormholes is due to Morris and Thorne [6]. Such a wormhole is characterized by the metric: with c = G = 1. Here φ(r) is called the red-shift function. This red shift function help us to identify the event horizon, since g 00 = −e 2φ → 0 on the event horizon. The function b(r) is called the shape function and indicative of the spatial shape of the wormhole. Normally an exotic fluid (which violates Null Energy Conditions (NEC)) is required for the formation of static wormholes [7][8][9][10][11][12], while for asymptotically flat space time the violation of NEC is due to topological censorship [13,14]. These wormholes are considered traversable as they can facilitate two way passage through them. For a wormhole to be traversable, e 2φ(r) should be finite everywhere (no horizon condition) and at the throat r = r 0 the condition b(r 0 ) = r 0 holds. Since superluminal travel is possible through such a wormhole, they can be conceived of as hypothetical time machines [7,[15][16][17][18][19].
Generalizing the original Morris-Thorne metric (1) by inducting the scale factor a(t) gives us the evolving relativistic wormholes. Evolving relativistic wormholes or rather dynamical wormholes are not very popular in the literature, and hence there is only limited understanding on the topic. Various different work on dynamical wormholes speculated their existence with matter satisfying WEC and DEC [20][21][22][23]. While, there exists other works [24,25] speculating that matter threading the wormhole would violate NEC. Recently dynamical wormholes have been treated with two fluid system in [26,27], important from the perspective of accelerated expansion of the universe [28][29][30][31] and also in [32] with dissipation due to particle creation mechanism in non-equilibrium thermodynamics.
In this work we extend the analysis of dynamic wormholes in f (R) modified theories of gravity. Traversable wormholes have already been treated in modified gravity (for example in Einstein-Gauss-Bonnet gravity, in higher dimensional Lovelock theories [33][34][35][36] and in f (R) gravity [37,38]) without any exotic matter. However such work is limited for their dynamic counterparts. f (R) modified theory of gravity can be considered to be an extension of Einstein's General Relativity. This theory was developed by adopting a modification in the Einstein-Hilbert gravitational Lagrangean density involving an arbitrary function of the Ricci scalar R [39] and essentially, it is an higher order (4 th ) gravity theory. In this context it can be stated that, since Einstein's formulation of General Relativity, several extensions of it have been introduced but with quite disparate reasons. Weil [40] initiated such attempt with the motivation of unifying electromagnetic and gravitational interaction geometrically, also there were modifications, namely to the gravitational action in analogy to quantum field theory in curved space-time, to have fundamental unification schemes and to obtain complete geometric explanation of the dark sector phenomenology [41]. f (R) gravity theory was probably one of the simplest such extension, and can act as natural model for inflation [42]. Also recently it has been shown to act as a geometrical dark energy model [43]. Hence f (R) gravity is an alternative framework based on general relativity that does not need any new scalar field to explain the accelerated expansion of the universe. The action in f (R) theory is given by: where κ = 8πG, R is the usual Ricci Scalar, f (R) is an arbitrary function of R and L m is the matter Lagrangian density with the corresponding integral denoting matter action with metric g µν coupled with collective matter field ψ. Now variations of the above action with respect to the metric coefficient gives the fourth order field equations where R ν µ is the Ricci tensor, T ν µ is the stress energy tensor of matter part and F (R) = ∂f ∂R . Here we are interested in constructing dynamical wormholes having viable f (R) gravity solutions within the framework of standard cosmological assumptions. Normally, it is seen that wormhole geometries are not dictated by Einstein field equations, rather they are constructed by supposing the red shift function, shape function or the scale factor (i.e. space-time metric) and then using the Einstein field equations the corresponding matter part is evaluated. However the matter part so obtained satisfies the local conservation equation due to Bianchi identities and violates the NEC. In this work we take a somewhat similar approach to obtain the f (R) gravity solutions with prescribed matter content. We consider the spherically symmetric inhomogeneous space-time metric given by It is evident from the metric that, we have considered the red shift function φ(r) = constant, i.e. we are considering zero-tidal force wormhole solutions, for it is well known that zero tidal force wormholes must be filled by an anisotropic type fluid [44], which is true for an inhomogeneous space-time model. Also zero tidal force class of wormholes are interesting and most often studied because of their tractability, which allows one to study important properties of wormhole geometry. Now taking trace of the field equations (3), (after simple algebraic manipulations) we get with and Here the matter threading the wormhole is anisotropic, T = −ρ + p r + 2p t and it satisfies the conservation equations (T ν µ;ν = 0) given by For the metric(4) we get the explicit form of the field equations (5) as: where an over dot indicates differentiation w.r.t. t and prime indicates differentiation w.r.t. r, and ρ = ρ(r, t), p r = p r (r, t), p t = p t (r, t) are respectively the energy density and thermodynamic radial and transverse pressure of a perfect fluid that should satisfy the conservation equations (8) and (9). Further ρ g , p rg and p tg the energy density, radial and transverse pressure of a hypothetical curvature fluid satisfies the following: Note that the field equations (10) can be interpreted as the Einstein field equations with non-interacting two-fluids. Now the scalar curvature for the space-time metric (4) has the expression i. e. R = R(r, t). In order to find out expressions for f (R) we shall make some specific choices of the shape function b(r) as follows: For this choice of b(r) and a(t) we get such that F (R) is now a function of the t co-ordinate alone, i. e. F (R) = ψ(t) and this gives F = F = 0. So from (11) we have Now from the field equations (10) we can get so that from the conservation equation (8) we get the following expression Using (16) together with the field equations (10) we get the following : Here we note that the conservation equation (9) is identically satisfied with the above values of p r , p t . Also with these values of ρ, p r and p t we get The above solutions of energy density for the choice of b(r) represents a dynamical wormhole solution with throat radius d 0 . Comparing the equation (14) with the above equations (18) we get the following differential equation: This has a solution of the form: I α being the modified Bessel's function of the first kind.
For two specific values of n given by n = 1/2 and n = 2/3 we get the following solutions for F (R) respectively and (It can be noted that the above two solutions of F (R) corresponds to the cosmological periods of radiation and matter dominated epoch of the universe). Here R 0 = 6b0 For n = 1 we have the simple power law form of solution given by with m 1,2 = ± 3 2 1 + 4R0

27R1
In the above solutions we note that viable f (R) solutions involving exponential of curvature R have been explored in [45,46] and have been successfully used to explain cosmological dynamics. Also solutions involving the negative powers of curvature may be obtained from some time-dependent compactification of string/M-theory [47], further quantum fluctuations in nearly flat space-time may also include such terms [48]. Also the negative power terms can serve as gravitational alternative to Dark Energy and produce cosmic acceleration [50,51]. Moreover such modified gravity theories do not suffer from instabilities and may pass the solar system tests for scalar-tensor gravity [49]. (17) and (18) and using (14) we get the following equation: which has the solution: where R 2 = 6n(2n − 1). As is evident in this case for n = 1 2 we get f (R) = R and then the system reduces to a case of Einstein's GR corresponding to the radiation epoch.
For n = − 1 2 we get the solution Thus equations (25) and (26) represents the possible F (R) gravity solutions corresponding to the evolving wormhole metric having constant throat radius d 0 .
We shall consider another special case with b(r) = constant for which we can get f (R) gravity solution for some special values of ρ, p r and p t . From the field equations we get: which further gives the solution and T = T µ µ = 0. We note that conservation equations (8) and (9) are identically satisfied. Now using (28) in (14) we get the following equation: Hence for any n we have the solution , n > 1 2 (30) with n 1,2 = ∓ √ n 2 + 10n + 1 and R 2 defined as before. Therefore we get F (R) gravity solution for the above evolving wormhole with constant throat radius b 0 , although matter here my not be considered real.
Considering the viability of the f (R) solutions we note that solution for f (R) contains both positive and negative powers of the scalar curvature. It can describe cosmic evolution from (a curvature driven) inflation through a radiation dominated era where modified Friedmann equations can be effectively identical to the standard one, followed by a matter dominated era and then a late-time era, in which the universe undergoes accelerated expansion [50,51].

Admissibility of the obtained F (R) solutions:
We shall now examine whether the present f (R) models are viable or not i.e. whether the above f (R) models are consistent with existing experimental and observational data.
It is commonly known that in metric formulation f (R) gravity theory is equivalent to scalar-tensor gravity [52][53][54][55] with coupling parameter ω = 0 [54] (as in Brans-Dicke Theory). As solar system tests rule out small values of ω, so f (R) theory is not a realistic one. However, it is known [56,57] that in strongly massive region, the non-minimally coupled scalar degree of freedom can be largely suppressed due to acquisition of excess mass and hence one cannot make such simple equivalence between the two gravity theories. Thus one should impose a set of constraints on f (R) model to pass all of the tests. The following are the conditions that any viable f (R) models must satisfy [57][58][59][60][61]. a) To prevent the sign change of the effective Newton's constant: G ef f = G F (R) , F (R) should be positive for all finite R. Also in microscopic level, it forbids the graviton to turn to ghost models [61].
b) The existence of a stable high-curvature regime i.e. matter dominated universe implies dF dR > 0 for higher values of R. Quantum mechanically it ensures that scalaron models are not Tachyonic. c) F (R) < 1 due to tight constraints from big bang nucleosynthesis and cosmic microwave background. d) Recent galaxy formation surveys have constrained |F (R) − 1| to be smaller than 10 −6 . As galaxy formation has not yet being studied in F (R) model using N −body simulation so the above statement is yet to be confirmed.
Thus for our solutions in equations (21), (23), (25) and (30) the above constraints restricts the arbitrary parameters as: The above constraints imposes the restriction that should be negative. . Then from definition of m 1 and m 2 we get α 2 < 0, α 1 > 0. Thus we rewrite the equation as: Then the restriction is:Ĉ 1,2 > 0 and 0 The restriction is: |C1| Let β 1,2 = (n+1)∓n1, 2 4 , then β 1 > 0 and β 2 < 0. Therefore the restriction can be stated as: C 1,2 > 0 and Energy Conditions: In wormhole physics, one of the basic ingredient is the violation of null energy condition. However in modified gravity theory the situation changes due to extra terms (curvature related in f (R) gravity) in the field equations. In Einstein gravity, validity of null energy condition can be interpreted geometrically through Raychaudhuri equations as follows: a congruence of null geodesics focus within a finite value of the parameter labelling points on the geodesics ensures through Raychaudhuri equation that R µν κ µ κ ν ≥ 0, where κ µ is a null vector tangent to the null geodesics. Thus in turn from Einstein field equations we have the null energy condition. However, in modified gravity theories if the field equations are written as G µν = T (ef f ) µν then focusing of congruence of null geodesics will imply T (ef f ) µν κ µ κ ν ≥ 0. Hence for wormhole configuration we must have de-focusing of null geodesics and consequently in modified gravity theory this implies T (ef f ) µν κ µ κ ν < 0. In principle one might impose the condition that T (m) µν κ µ κ ν ≥ 0 for normal matter threading the wormhole. In f (R) modified gravity theories we therefore infer that it is the higher order curvature term that is interpreted as the gravitational fluid sustaining the non-standard wormhole geometries. For the present wormhole geometry we have considered the energy conditions graphically as follows: The matter threading the wormhole has components given by (15) and (17) for case 1 and by (27) for case 2. We check for possible WEC violations related to both pressures. Here in the following examples the scale factor is assumed proportional to radiation and matter dominated epoch in a FLRW metric. F (R) solutions as obtained in equations (21), (22) and (25) are used to evaluate ρ − p r , ρ − p t , ρ + p r , ρ + p t . In the plots below for the radiation era (n = 1/2) t t0 is assumed to range from 10 −3 upto 10 −1 and for matter dominated epoch (n = 2/3) t t0 is assumed to range from 10 −1 upto 10 2 . For all these plots d 0 (the throat radius) is taken such that 0 < d 0 < r and other arbitrary constants are assumed maintaining the validity of F (R) solutions. Using these we obtain the following figures: Here we note that for the choice b = b 0 r r0 3 + d 0 the matter component always satisfies the energy conditions for both n = 1/2 and for n = 2/3 as in Figures 1 and 2. However for the choice b(r) = d 0 matter component will not always satisfy the energy conditions (figures 3 and 4). The energy conditions are satisfied in some restricted region, conforming that in those restricted regions matter component will always behave as a normal matter (also we note that for n = 1/2 our choice reduces to Einstein's GR in the radiation epoch where NEC is violated in regions around the throat as is expected). Thus we affirm that wormhole solutions inf (R) gravity are possible with normal matter as claimed in the literature.
Note that we have not considered the energy conditions corresponding to our Case 2 because matter considered there is exotic and hence the case is treated only from a theoretical aspect. Assuming a inhomogeneous space time metric in a f (R) modified gravity formalism, we have found evolving dynamical wormholes using known forms of the shape function corresponding to specific cosmological epochs (i.e. radiation and matter dominated eras of the universe in Einstein gravity). The wormhole solutions are also constrained in conformation to existing experimental tests and conditions of viability obtained. Graphical analysis of normal matter (in conformity to viable F (R) solutions) confirms that it satisfies the WEC in some specific regions around the throat. In fact for case 1 in both, radiation and matter dominated epoch WEC is found to be satisfied always, however for the special case of Case 1 WEC would be satisfied only in some specific regions. We have seen that that for the case f (R) = R as expected the NEC (Fig: 3) would be violated at large in regions around the throat.
Therefore we assert that dynamical wormhole solution is possible in f (R) gravity theory without violation of null energy condition.

ACKNOWLEDGMENTS
SB is thankful to UGC for financial support under their Faculty Recharge Program. SC thanks IUCAA for their research facility.