Dynamic wormhole geometries in hybrid metric-Palatini gravity

In this work, we analyse the evolution of time-dependent traversable wormhole geometries in a Friedmann-Lema\^{i}tre-Robertson-Walker background in the context of the scalar-tensor representation of hybrid metric-Palatini gravity. We deduce the energy-momentum profile of the matter threading the wormhole spacetime in terms of the background quantities, the scalar field, the scale factor and the shape function, and find specific wormhole solutions by considering a barotropic equation of state for the background matter. We find that particular cases satisfy the null and weak energy conditions for all times. In addition to the barotropic equation of state, we also explore a specific evolving wormhole spacetime, by imposing a traceless energy-momentum tensor for the matter threading the wormhole and find that this geometry also satisfies the null and weak energy conditions at all times.

In this work, we analyse the evolution of time-dependent traversable wormhole geometries in a Friedmann-Lemaître-Robertson-Walker background in the context of the scalar-tensor representation of hybrid metric-Palatini gravity. We deduce the energy-momentum profile of the matter threading the wormhole spacetime in terms of the background quantities, the scalar field, the scale factor and the shape function, and find specific wormhole solutions by considering a barotropic equation of state for the background matter. We find that particular cases satisfy the null and weak energy conditions for all times. In addition to the barotropic equation of state, we also explore a specific evolving wormhole spacetime, by imposing a traceless energy-momentum tensor for the matter threading the wormhole and find that this geometry also satisfies the null and weak energy conditions at all times.

I. INTRODUCTION
The explanation of the accelerated expansion of the Universe is one of the most challenging problems in modern cosmology [1,2]. From the mathematical point of view, the simplest way to treat this problem is to consider the cosmological constant term [3]. Nonetheless, this model faces some difficulties such as the coincidence problem and the cosmological constant problem. The latter dictates a huge discrepancy between the observed values of the vacuum energy density and the theoretical large value of the zero-point energy suggested by quantum field theory [3]. There are some alternative models proposed to overcome these problems such as modified gravity [4][5][6][7][8], mysterious energy-momentum sources [9][10][11], such as quintessence [12][13][14][15] and k-essence [16][17][18] fields, and complex equations of state storing the missing energy of the dark side of the Universe [19,20]. In general, models with varying dark energy candidates may be capable of overcoming all the mathematical and theoretical difficulties, however, the main underlying question is the origin of these terms. One proposal is to relate this behavior to the energy of quantum fields in vacuum through the holographic principle which allows us to reconcile infrared (IR) and ultraviolet (UV) cutoffs [21] (see Ref. [22] for a review on various attempts to model the dark side of the Universe).
In this paper, we study the evolution of traversable wormholes in a FLRW universe background in the scalartensor representation of hybrid metric-Palatini gravity. Furthermore, we explore the energy conditions for matter which threads these wormhole geometries. The paper is organized in the following manner: In Sec. II, we briefly present the action and field equations of hybrid metric-Palatini gravity, we consider the spacetime metric and explore a barotropic equation of state for the background fluid. In Sec. III, we analyse evolving traversable wormhole geometries for specific values of the barotropic equation of state parameter, as well as evolving wormholes with a traceless energy-momentum tensor (EMT), and study the energy conditions for the solutions obtained. Finally, in Sec. IV, we summary our results and conclude.

A. Action and field equations
Here, we briefly present the hybrid metric-Palatini gravitational theory. The action is given by where κ 2 ≡ 8πG, R is the metric Ricci scalar, and the Palatini curvature is R ≡ g µν R µν , with the Palatini Ricci tensor, R µν , defined in terms of an independent connection,Γ α µν , given by and S m is the matter action. However, the scalar-tensor representation of hybrid metric-Palatini gravity provides a theoretical framework which is easier to handle from a computational point of view [24], where the equivalent action to (1) is provided by (3) Note a similarity with the the action of the w = −3/2 Brans-Dicke theory version of the Palatini approach to f (R) gravity. However, the hybrid theory exhibits an important and subtle difference appearing in the scalar field-curvature coupling, which in the w = −3/2 Brans-Dicke theory is of the form φR [24].
By varying the action (3) with respect to the metric provides the following gravitational field equation where T µν is the standard matter energy-momentum tensor, and is the energy-momentum tensor of the scalar field of the theory.
Varying the action with the scalar field yields the following second-order differential equation, The above equation shows that in hybrid metric-Palatini gravity the scalar field is dynamical. This represents an important and interesting difference with respect to the standard Palatini case [23].

B. Spacetime metric
The metric of a time-dependent wormhole geometry is given by where Φ (r) and b (r), respectively, are the redshift and shape functions, which are r-dependent, and a (t) is the time-dependent scale factor, and dΩ 2 = dθ 2 + sin 2 θdϕ 2 is the linear element of the unit sphere.
To correspond to a wormhole solution, the shape function has the following restrictions [49]: (i) b(r 0 ) = r 0 , where r 0 is the wormhole throat, corresponding to a minimum radial coordinate, (ii) b(r) ≤ r, and (iii) the flaringout condition given in the form rb ′ (r)−b (r) < 0. The latter condition is the fundamental ingredient in wormhole physics, as taking into account the Einstein field equation, one verifies that this flaring-out condition imposes the violation of the NEC. In fact, it violates all of the pointwise energy conditions [49,50,53,54,107]. In order to avoid the presence of event horizons, so that the wormhole is traversable, one also imposes that the redshift function, Φ(r), be finite everywhere. However, throughout this work, we consider a zero redshift function, Φ = 0, which simplifies the calculations significantly.
We also consider an anisotropic matter energymomentum tensor given by T µ ν = diag(−ρ, −τ, p, p), where ρ, τ and p are the energy density, the radial tension (which is equivalent to a negative radial pressure) and the tangential pressure, respectively. Using the metric (7), the gravitational field equations (4) provide the following energy-momentum profile: respectively, where H =ȧ/a and φ = φ (t), the overdot and prime denote derivatives with respect to t and r, respectively, and for notational simplicity, we have considered κ = 1. One recovers the standard field equations of the Morris-Thorne wormhole [49], by fixing the scale factor a to unity and excluding the background timedependent evolution and the scalar field φ contribution. Furthermore, we will also analyse the null and weak energy conditions for the solutions obtained below. The weak energy condition (WEC) is defined as T µν u µ u ν ≥ 0, where u µ is a timelike vector, and is expressed in terms of the energy density ρ, radial tension τ and tangential pressure p as ρ ≥ 0, ρ − τ ≥ 0 and ρ + p ≥ 0, respectively. The last two inequalities, i.e., ρ − τ ≥ 0 and ρ + p ≥ 0 correspond to the NEC, which is defined as T µν k µ k ν ≥ 0, where k µ is any null vector.
The scalar field equation of motion (6) yields where the trace of the energy-momentum tensor T = T (t, r) = T µ µ = −ρ − τ + 2p is given by In order to solve the differential equation (11) for φ (t), the trace T (t, r), given by Eq. (12), should be independent of r. This condition leads to b ′ /r 2 = CH 2 0 where H 0 is the present value of the Hubble parameter and C is an arbitrary dimensionless constant. Thus, the shape function is given by b(r) = r 0 + CH 2 0 (r 3 − r 3 0 )/3, which satisfies b(r 0 ) = r 0 , and the flaring-out condition at the throat imposes CH 2 0 r 2 0 < 1. To solve Eq. (11) numerically for φ, it is useful to rewrite it in terms of dimensionless functions of the scale factor a. Thus, we consider the definitions: φ =äφ ′ (a) +ȧ 2 φ ′′ (a),φ =ȧφ ′ (a), a = Hȧ +Ḣa,Ḣ =ȧH ′ (a),ȧ = Ha .
Note that in the above definitions, the prime denotes a derivative with respect to the scale factor. In addition to this, consider U = V /3H 2 0 and E = H/H 0 , so that Eq. (11) finally takes the following form

C. Barotropic equation of state
To solve the differential equation (13) for φ (a), we need to deduce E. To this effect, it is also useful to define the following background quantities: and consider a background barotropic equation of state given by τ b = −ω b ρ b . From this condition, we find E = a −3(ω b +1)/2 . Furthermore, we consider a power law potential for scalar field U (φ) = φ α and for simplicity set C = 0, so that b(r) = r 0 . Thus, taking into these conditions and using the dimensionless definitions given above, Eqs. (8)-(10) lead to the following relations respectively. It is interesting to note that ρ is independent of the radial coordinate r. To keep the terms dimensionless, we will consider the wormhole throat as In what follows, we set A to unity.
In the following, we shall consider specific solutions for the parameters ω b = (−1, 1/3, 0), respectively. These parameters range tentatively correspond to the inflationary, radiation and matter epochs.

III. SPECIFIC EVOLVING WORMHOLE SOLUTIONS AND THE ENERGY CONDITIONS
In this section, we analyse evolving traversable wormhole geometries for the specific parameters ω b = (−1, 1/3, 0) considered above, as well as evolving wormholes with a traceless EMT, and study the NEC and WEC for the solutions obtained.   Here, we consider the specific case Figure 1(a) depicts the behaviors of φ and φ ′ with respect to the redshift z The behaviors of φ, φ ′ (a) and E vs z for the traceless EMT case with U (φ) = φ 2 . Note that both horizontal and vertical axes are logarithmic, except for the vertical axis of Fig. 5(a). See the text for more details. (= 1/a − 1). In Fig. 2, the behaviors of ρ, ρ − τ and ρ + p versus z for different values of r are shown. As one can see, ρ− τ is negative throughout the entire evolution. We emphasize that we have imposed different possible initial values and obtained the same result. Note that one may justify this outcome in the following manner. For this case with ω b = −1, Eq. (16) reduces to Since a ≤ 1, the term with a coefficient 1/a 2 is dominant. This term is therefore negative, rendering ρ − τ negative. Thus, for this specific case, both the NEC and the WEC are violated.
B. Specific case: ω b = 1/3 Figure 1(b) illustrates the behavior of φ and φ ′ versus z for the specific case ω b = 1/3. For this case, E = a −2 , a(t) ∝ t 1/2 and we consider U (φ) = φ 2 . The behaviors of ρ, ρ−τ and ρ+p with respect to the redshift z for different values of r are displayed in Fig. 3, which shows explicitly that these quantities decrease as time evolves. However, they remain positive at all times and consequently the NEC and WEC are always satisfied. This occurs for the wormhole throat as well as for other wormhole radii. It is interesting to note that at a specified time/redshift, the quantity ρ − τ increases for increasing values of the radius, and the minimum value corresponds to the throat, as depicted by Fig. 3(b). On the other hand, ρ + p decreases for increasing values of the radius, and has a maximum at the throat (Fig. 3(c)).
C. Specific case: ω b = 0 Here we intend to study the energy conditions for the case ω b = 0, for which E = a −3/2 and a(t) ∝ t 2/3 . One can verify the behaviors of the scalar field φ and its derivative with respect to a as functions of the redshift z in Fig. 1(c), where we consider the power law potential corresponding to scalar field as U (φ) = φ 2 . The energy conditions are depicted in Fig. 4, where one verifies that ρ, ρ − τ and ρ + p are positive at all times and for different values of spatial coordinate r. These quantities are also decreasing (increasing) functions with respect to time (redshift). Thus, the NEC and WEC are always satisfied for all wormhole radii including the wormhole throat. The behaviors of ρ − τ and ρ + p at a specified time/redshift are analogous to the previous case with ω b = 1/3, namely, the quantity ρ − τ (ρ + p) increases (decreases) for increasing values of the radius (see Figs. 4(b) and 4(c)).

D. Wormholes with traceless EMT
In this subsection, we consider the traceless energymomentum tensor, with T = −ρ − τ + 2p = 0, which taking into account the dimensionless quantities defined above, provides the following differential equation Here, we solve the system of coupled differential equations (13) and (19), for φ and E, numerically. The behaviors of φ, φ ′ and E versus z, taking into account the choice U (φ) = φ 2 , are shown in Fig. 5. Note that E tends to unity at the present time (z → 0), as expected ( Fig. 5(c)). In Fig. 6, the behaviors of ρ, ρ − τ and ρ + p, with respect to z for different values of r are displayed. Note that ρ, ρ − τ and ρ + p are decreasing (increasing) functions of time (redshift) and are positive for all radii, as time evolves. Thus, the NEC and WEC are satisfied at all times. Moreover, as shown in Figs. 6(b) and 6(c), the qualitative behaviors of ρ − τ and ρ + p with respect to r is analogous to the previous two cases, namely, ω b = 1/3 and ω b = 0, where the quantity ρ − τ (ρ + p) has a minimum (maximum) value at throat.

IV. DISCUSSION AND CONCLUSION
In the present paper, we have studied the evolution of dynamic traversable wormhole geometries in a FLRW background in the context of hybrid metric-Palatini gravity. This theory, which recently attracted much attention, consists of a hybrid combination of metric and Palatini terms and is capable of avoiding several of the problematic issues associated to each of the metric or Palatini formalisms (for more details, we refer the reader to Refs. [25][26][27]). For the evolving wormholes, we presented the components of the energy-momentum tensor that supports these geometries in terms of the model's functions, namely, the scalar field, the scale factor and the shape function (we considered a zero redshift function, for simplicity). Furthermore, we found specific wormhole solutions by considering a barotropic equation of state for the background matter, i.e., τ b = −ω b ρ b , and considered particular equation of state parameters.
More specifically, we showed that for the specific cases of ω b = 1/3 and ω b = 0, the entire wormhole matter satisfies the NEC and WEC for all times. The latter cases are similar to the wormhole geometries analysed in the presence of pole dark energy [70], however, there the WEC is violated at late times, i.e., the energy density becomes negative. Thus, the present results outlined in this work strengthen the varying dark energy models and may suggest that hybrid metric-Palatini gravity is a rather more promising model to explore. In addition to the barotropic equation of state, we also studied evolving wormhole geometries supported by the matter with a traceless EMT. For this specific geometry, we discovered that both the NEC and the WEC are satisfied at all times as well. These results are extremely promising as they build on previous work that consider that the energy conditions may be satisfied in specific flashes of time [51,52].