Cosmic strings in $f\left(R,L_m\right)$ gravity

We consider Kasner-type static, cylindrically symmetric interior string solutions in the $f\left(R,L_m\right)$ theory of modified gravity. The physical properties of the string are described by an anisotropic energy-momentum tensor satisfying the condition $T_t^t=T_z^z$; that is, the energy density of the string along the $z$-axis is equal to minus the string tension. As a first step in our study we obtain the gravitational field equations in the $f\left(R,L_m\right)$ theory for a general static, cylindrically symmetric metric, and then for a Kasner-type metric, in which the metric tensor components have a power law dependence on the radial coordinate $r$. String solutions in two particular modified gravity models are investigated in detail. The first is the so-called"exponential"modified gravity, in which the gravitational action is proportional to the exponential of the sum of the Ricci scalar and matter Lagrangian, and the second is the"self-consistent model", obtained by explicitly determining the gravitational action from the field equations under the assumption of a power law dependent matter Lagrangian. In each case, the thermodynamic parameters of the string, as well as the precise form of the matter Lagrangian, are explicitly obtained.


I. INTRODUCTION
The recently released Planck satellite data of the 2.7 degree Cosmic Microwave Background (CMB) full sky survey [1] have generally confirmed the standard Λ Cold Dark Matter (ΛCDM) cosmological model. Observations of high redshift type Ia supernovae have also confirmed that our universe is presently undergoing a period of accelerated expansion, for which the most natural explanation would be the presence of some form of dark energy, with an equation of state parameter w = −1.018 ± 0.057 for a flat universe [2]. From a theoretical point of view, the necessity of explaining dark energy, as well as the second dominant component of the universe, dark matter [3], raises the fundamental question of whether the standard Einstein-Hilbert action (S = (R/2 + L m ) √ −gd 4 x, where R is the scalar curvature, and L m is the matter Lagrangian density), in which matter is minimally coupled to geometry, can give an appropriate quantitative description of the universe on all scales, from the boundary of the solar system to the edge of the horizon.
A gravitational theory with an explicit coupling between an arbitrary function of the scalar curvature and the Lagrangian density of matter was proposed in [4]. The gravitational action of this theory is of the form S = {f 1 (R) + [1 + λf 2 (R)] L m } √ −gd 4 x, where λ is a constant, and f 1 (R) and f 2 (R) are arbitrary functions of the Ricci scalar. In this model an extra force acting on massive test particles arises, and the motion is no longer geodesic. Moreover, in this framework, one can also explain dark matter [4]. The initial "linear" geometry-matter coupling introduced in [4] was extended in [5], and a maximal extension of the Einstein-Hilbert action with geometrymatter coupling, of the form S = d 4 x √ −gf (R, L m ), was considered in [6]. Geometry-matter couplings in the presence of scalar fields were discussed in [7] and the cosmological and astrophysical implications of the f (R, L m ) type gravity theories were investigated in [8]. For a recent review of the f (R, L m ) model see [9]. In the current work, we specifically investigate the role of cosmic strings embedded in a spacetime governed by an f (R, L m ) modified gravity theory. Topological defects, including strings, monopoles and domains walls, are expected to have formed during in phase transitions in the early Universe [10][11][12][13]. In particular, magnetic monopoles are generic features of Grand Unified Theories (GUTs) [14,15], with expected mass-scales in the region of m ∼ m GUT /α GUT ∼ 10 17 GeV, where m GUT ∼ 10 15 GeV is the expected grand unification scale and α GUT is the coupling constant.
Though not completely ruled out by existing data, the mass and number densities of monopoles and domain walls, direction, r (such as the Abelian-Higgs string [52]), the equivalent condition is T t t = T z z , or ρ(r) = −p z (r), where ρ(r) and p z (r) denote the mass density and thermodynamic pressure per unit volume within the string core. Since, for a cylindrically-symmetric distribution of matter, these quantities depend only on the radial coordinate, integration over r yields the effective mass per unit length,m = T tt √ −gdr, and tension T = T zz √ −gdr, of the string-type field configuration, which thus also satisfy T = −m [53]. For the sake of simplicity, we restrict our attention to vacuum strings throughout the following analysis and impose the condition T t t = T z z on the matter sector. The structure of this paper is then as follows. In Section II, we give a brief outline of the general f (R, L m ) gravity theory, and derive the general field equations and conservation equations from the action. The form of these equations is determined for a general static, cylindrically-symmetric source in Section III A. Specific Kasner type solutions [47,48] of the field equations are presented in Section III B. In Section IV, a specific choice is made for the function f , and the "exponential" modified gravity theory, f (R, L m ) = Λ exp [(1/2Λ)R + (1/Λ)L m ], which reduces naturally to Einstein gravity in the low energy limit, is chosen as a toy model. Solutions with self-consistent matter Lagrangians are derived in Section V. That is, the self-consistent form of f (R, L m ) is determined by imposing the condition T t t = T z z on the matter sector, which may be seen as a physical requirement corresponding to the existence of vacuum/Nambu-Goto strings, and by invoking the correspondence principle. This states that the standard general relativistic limit of the modified gravity action (i.e. the Einstein-Hilbert action), must exist for appropriate choices of the arbitrary parameters and functions of the model. Specific, Kasner-type solutions with self-consistent matter Lagrangians are investigated in Section VI, for a simple choice of matter Lagrangian dependence in f , f (R, L m ) ∝ L q m , q = constant = 1, and two distinct string-type solutions are found to exist. These correspond to two self-consistent sets of constraints, which may be imposed on the free parameters of the Kasner metric. Section VII contains a brief summary of the conclusions and a discussion of the main results of this analysis, together with suggestions for future work.
In this Section, we first briefly summarize the f (R, L m ) theory of gravity and introduce the gravitational action, the field equations and the conservation equations, respectively. The f (R, L m ) theory is based on the assumption that the gravitational Lagrangian is given by an arbitrary function f of the Ricci scalar R and of the matter Lagrangian L m [6]. This theory represents a maximal extension of the Hilbert-Einstein action, and the action takes the form [6] where √ −g is the square root of the determinant of the metric tensor. We require that the function f be analytic in both R and L m ; that is, f must possess a Taylor series expansion about any point (R, L m ). The energy-momentum tensor of the matter is defined by [54] Thus, by assuming that the Lagrangian density of the matter, L m , depends only on the metric tensor g µν , and not on it's derivatives, we obtain T µν = g µν L m − 2 ∂L m /∂g µν . By varying the action with respect to the metric we obtain the following field equations for the f (R, L m ) gravity theory [6], where we have denoted f R (R, L m ) = ∂f (R, L m ) /∂R and f Lm (R, L m ) = ∂f (R, L m ) /∂L m , respectively. Alternatively, Eq. (3) can be written as where we have denotedP and The differential operators ∇ µ ∇ ν and are defined according to and where F x λ is an arbitrary function of the coordinates x λ , and the Γ α µν , defined according to the convention are the the Christoffel symbols associated to the metric.
With the choice f (R, L m ) = R/2 + L m (the Hilbert-Einstein Lagrangian), we recover the standard Einstein field equations of general relativity, where f 1 and f 2 are arbitrary functions of the Ricci scalar, and G is a function of the matter Lagrangian density, we may also re-obtain the field equations of the (non-minimally coupled) modified gravity theory introduced in [4], for an appropriate choice of G.
The contraction of Eq. (3) gives the following relation between the Ricci scalar R, the matter Lagrangian density L m , and the trace of the energy-momentum tensor, T = T µ µ ; R By eliminating the f R (R, L m ) term using Eqs. (3) and (11), we obtain the gravitational field equations in an alternative form as Taking the covariant divergence of Eq. (3), with the use of the mathematical identity [55], then gives the following equation for the divergence of the energy-momentum tensor T µν , The explicit derivation of Eq. (14) is also presented in Appendix A. The requirement that the energy-momentum tensor for matter be conserved, ∇ µ T µν = 0, then yields an effective functional relation between the matter Lagrangian density and the total Lagrangian f (R, L m ), Thus, once the matter Lagrangian density is known, by an appropriate choice of f (R, L m ), one can construct, at least in principle, conservative gravity models with arbitrary curvature-matter coupling.
In the present Section we consider the gravitational field equations describing explicit string solutions in the f (R, L m ) gravity theory. We assume a cylindrically-symmetric geometry in which the source term is represented in the form of an anisotropic fluid with three distinct pressures components along the r-, φ-and z-axes (in cylindrical polar coordinates, (t, r, φ, z)). The simplest model for a cosmic string is the so called "gauge string" [56], which is an idealized cylindrical mass distribution with a finite radial extension having T t t = T z z as the only non-vanishing components of the energy-momentum tensor. In the following we restrict our analysis to a similar configuration, but with a more general pressure distribution, for which T r r = 0 and T φ φ = 0, respectively.
A. String geometry and field equations in f (R, Lm) gravity We assume a cylindrical-symmetry metric, giving a line element of the form [57] where N (r), L(r) and K(r) are arbitrary functions of the radial coordinate r. The nonzero Christoffel symbols associated to the metric (16) are given by where a prime denotes the derivative with respect to r. The operator is then given specifically via while for the covariant derivatives of an arbitrary function f (r) we obtain With the line element (16), the components of the Ricci tensor are [57] The source term in the field equations is given by the energy-momentum tensor with the following components [57] Therefore the field equations describing cylindrically-symmetric string type solutions in f (R, L m ) gravity can be written as Generally, the regularity of the geometry on the symmetry axis is imposed via the initial conditions, We define the string radius R s as the value of r at which the radial pressure p r (r) vanishes, p r (R s ) = 0. Additionally, we may also wish to impose the conditions p φ (R s ) = 0, together with Rs r=0 p r (r)dr = 0, and Rs r=0 p φ (r)dr = 0, respectively. The physical reasoning behind these conditions is that, if they do not hold, the string core must either expand or contract. However, as we shall see in Sect. VI E, for some string models these conditions may not hold, thus allowing the possibility of a nonzero surface energy density and tangential pressure.
Next we consider the relations that follow from the "conservation" of the energy-momentum tensor, given by Eq. (14). Since all the components of the energy-momentum tensor and of the function f (R, L m ) are independent of the coordinates (t, φ, z), we have Furthermore, the divergence of the T µ ν can be obtained in a general form as [54] It follows that Eq. (27) is identically satisfied, with the only potentially nonzero component of the divergence being given by With the use of Eq. (29), the "conservation equation" for the Abelian Higgs string in the f (R, L m ) gravity theory takes the form and, since p z = −ρ, this is equivalent to To obtain a description of the physical properties of the string we introduce the Tolman mass per unit length M of this system, defined as [58] along with the parameter which can be related to the angular deficit of the space-time via the "angular" Einstein equation [58]. With the use of the field equations Eqs. (22)- (25) we obtain and respectively.
B. The string gravitational field equations for the Kasner metric We now investigate the possibility of the existence of Kasner type solutions for static, cylindrically-symmetric strings in f (R, L m ) gravity. In standard general relativity, by neglecting the effect of the cosmological constant, the unique, static, cylindrically-symmetric vacuum solution of the Einstein field equations is given by the Kasner metric [47,48], where k sets the length-scale and β is a constant, related to the deficit angle of the conical space-time. The Kasner metric is characterized by two free parameters which, for the unique vacuum solution, satisfy the Kasner conditions [47,48], For the reader's convenience, the derivation of the Kasner metric (36), in standard general relativity, is reviewed in Appendix B. The physical interpretation of the free parameters a, b, c is of fundamental importance in the context of the description of cosmic strings, as is the relation between these parameters and the internal properties of the matter distribution. The simplest model for a cosmic string, the "gauge string" [56], belongs to a very simple class of Kasner type solutions, with a = c = 0, b = 1, which is locally flat. However, as one can see from Eq. (36), in the gauge string model, there is an important global non-trivial effect namely, that the string geometry is that of a cone. From a geometrical point of view the parameter β in the Kasner solution is directly related to the conic angular deficit [59,60], which, in turn, is also related to the mass distribution of the source [61][62][63][64][65].
In the case of a gauge string (in general relativity), there is an explicit relation between the angular deficit ∆φ = 2π(1 − β) and the "inertial mass" (per unit length)m, given by ∆φ = 8πGm. This relation was first obtained by using a linear approximation for the case of an infinitesimally thin source [66]. The same relation was obtained by solving the full nonlinear Einstein equations around a uniform source (T t t = constant) with a finite radius [67,68], and in the case of a non-uniform source [69]. Since the spacetime around a gauge string is locally flat, the angular deficit is the only geometrical evidence of its existence, but allows for the possibility of observationally detecting the presence of a cosmic string via lensing effects [70]. Historically, the next step in the study of gravitating cosmic strings required an analysis of the more realistic, Abelian-Higgs type models. In this context, the analysis of the full coupled field equations for the gravitational and matter fields was performed in [71].
Hence, for a Kasner type metric, the gravitational field equations for the cylindrically-symmetric static string in f (R, L m ) gravity, Eqs (22)-(25), take the form The conservation equation Eq. (31) becomes and the Ricci scalar is given by The Tolman mass per unit length M and the W parameter, which controls the angular deficit of the string geometry, are obtained as and respectively.
As a first example of string type cylindrically-symmetric solutions in f (R, L m ) gravity we consider a Lagrangian density of the form [6] f where Λ > 0 is an arbitrary constant.
That is, we recover the full Hilbert-Einstein gravitational Lagrangian with a cosmological constant. With this choice of Lagrangian density the gravitational field equations take the form In the case of weak gravitational fields and of small particle velocities, the exponential curvature-matter coupling induces an extra-acceleration of massive test particles, which is proportional to the gradients of both the Ricci scalar and the matter Lagrangian, a = − (1/Λ) [∇ (R/2) + ∇L m ]. In the case of pressureless dust, the extra-force is proportional to the gradient of the matter density ρ only.
As one can see from Eqs. (48), the gravitational field equations of the exponential gravity model contain a new background term, proportional to the metric tensor g µν , which depends on both the constant Λ, and on the physical parameters of the matter. Thus, through the geometry-matter coupling, the exponential model introduces an effective, time dependent "cosmological constant".
From Eqs. (6) and (7), we have Then, for a Kasner type metric, the field equations (38)-(41) describing the static, cylindrically-symmetric string in exponential gravity are where we have also used the explicit form of the Ricci scalar for the Kasner metric, given by Eq. (43). Eqs. (50) and (53) immediately give the following differential equation, which is satisfied by the matter Lagrangian; By introducing a new dependent variable l m (r), defined via L ′ m (r) = Λl ′ m (r)/l m (r), L m (r) = Λ ln l m (r), Eq. (54) takes the form of a linear differential equation for l m (r), given by (55) By introducing a new independent variable ξ = 1/r, we have l ′ m = (dl m /dξ) (dξ/dr) = −ξ 2 dl m /dξ, and l ′′ m = 2ξ 3 (dl m /dξ) + ξ 4 d 2 l m /dξ 2 , respectively. Therefore Eq. (55) takes the form where we have denoted β = 4R 0 /Λ, s = 3a + 2b + c, µ = β(s − 6)/8, and m = (a + b + c − 1)(c − a)/2. Eq. (56) has the general solution where U (a, b, z) is the confluent hypergeometric function, defined as U (a, b, z) = [1/Γ(a)] ∞ 0 e −zt t a−1 (1 + t) b−a−1 dt, L a n (x) is the generalized Laguerre polynomial, satisfying the differential equation xy ′′ + (a + 1 − x)y ′ + ny = 0, and where we have denoted and The constants c 1 and c 2 are arbitrary constants of integration. Hence, the thermodynamical parameters of the Kasner string in the exponential type f (R, L m ) gravity can be expressed in terms of the function l m (r) as In the limit of small ξ, corresponding to large r, Eq. (57) can be approximated as l m (ξ) ∝ ξ m2 , or, equivalently, l m (r) ∝ r −m2 . Thus it follows that, at large radial distances, L m (r) ≈ −m 2 Λ ln r and l ′ m (r)/l m (r) ≈ −m 2 /r. By using these approximations for L m (r) and l m (r) in Eqs. (63)-(65), we can easily determine the behavior of the energy density and thermodynamic pressures of the Kasner string in the exponential f (R, L m ) gravity theory.
In the same order of approximation we obtain the expressions for the gravitational Lagrangian density and its derivative with respect to R, respectively. By using the results above we can now estimate the mass per unit length and angular deficit parameter of the string in the exponential model. We assume that the string core extends between r = 0 and r = R s , and thus we avoid the singularity at the upper integration limits in the integrals. The string radius can be estimated from the equation p r (R s ) = 0, which gives By neglecting the term Λ [1 + m 2 ln R s ] in Eq. (67), we obtain the follow second order algebraic equation for the string radius; Thus, if the Kasner conditions hold, at least approximately, so that a + b + c ≈ 1 and a 2 + b 2 + c 2 ≈ 1, the string radius can be approximated as R s ≈ −R 0 /m 2 Λ.
Hence, for the mass and the angular deficit parameter, we obtain and where χ(a, b) , respectively. In order to avoid any divergence at the origin, the parameters a and b must satisfy the condition 2a + b > 6.
By assuming that the parameter Λ in the exponential model is the cosmological constant, we may assume for it a numerical value of the order of Λ ≈ 10 −56 cm −2 . Then we can estimate the string radius as R s ≈ (−R 0 /m 2 ) 1/2 × 10 28 cm. This relation imposes the constraint R 0 /m 2 < 0. However, this value of the radius is obtained in the quasi-Kasner approximation, which implies that R 0 may have very small values, R 0 = ǫ << 1, while m 2 still can have values of the order of unity. Therefore, the radius of the string can be written as R s ≈ √ ǫ × 10 28 cm. To obtain physically reasonable values of the radius, very small deviations from the exact Kasner regime are required. For example, with ǫ = 10 −40 , we can obtain more realistic values of order R s ≈ 10 8 cm. By taking into account the approximate expression of R s in Eq. (69), we can estimate the mass per unit length of the string as M ≈ 2β c 2 /G (kR s ) 2a+b−1 ≈ 2.7 × 10 28 × β × (kR s ) 2a+b−1 g/cm. Besides the parameters a and b, the mass of the string depends on the other two parameters β and k of the Kasner metric. For k ≈ 1/R s , and β = 10 −4 , we obtain for the mass per unit length of the string M ≈ 10 24 g/cm. Therefore, ultra-long cosmic strings, with lengths of the order of 1 kpc = 3 × 10 21 cm, could reach masses of the order of M ≈ 1.5 × 10 12 M ⊙ in the exponential f (R, L m ) gravity theory. Of course these estimates are strongly dependent on the values of beta, k and the parameters of the Kasner metric, whose exact values can be obtained only by fitting with the above models astrophysical data. Though highly worthwhile, for the sake of brevity, such an analysis lies beyond the scope of the present work, and is therefore deferred to a future publication.

V. STRING SOLUTIONS IN f (R, Lm) GRAVITY WITH SELF-CONSISTENT MATTER LAGRANGIAN
In the present Section we consider some explicit solutions of the field equations (22)-(25), without imposing a priori the functional form of f (R, L m ).
From Eqs. (38) and (41) it immediately follows that the function f R satisfies the second order linear differential equation which has the general solution where C ′′ 1 and C ′′ 2 are arbitrary constants of integration and α 1 and α 2 are solutions of the algebraic equation given by Using R = R 0 /r 2 , where R 0 is given by Eq. (43), we can rewrite Eq. (72) as a function of R as where where n 1 = 1 − α 1 /2, n 2 = 1 − α 2 /2, C 1 = C ′ 1 / (1 − α 1 /2), C 2 = C ′ 2 / (1 − α 2 /2), and g (L m ) is an arbitrary integration function of the matter Lagrangian L m . This encompasses the solution obtained in [32] (which corresponds to the condition f (R = 0) = 0, giving f (R) as a superposition of powers R k , k ≥ 1)), when both sets of Kasner conditions,(37), are satisfied and g(L m ) = 0. However, it is clearly more general and holds for R = 0, g(L m ) = 0.
The components of the string energy-momentum tensor can then be obtained as By requiring that the generalized gravitational Lagrangian density, given by Eq. (76), must have the standard general relativistic limit, that is, by requiring it to reduce to the standard Hilbert-Einstein Lagrangian for an appropriate choice of the free parameters of the model, we can impose the conditions which, according to Eq. (76), require α 1 = 0. This criterion is satisfied if the coefficients a, b, c satisfy the condition Then, for α 2 we obtain α 2 = 1 − 3a − 2b − c, giving the expression for n 2 = 1 − α 2 /2, where, for simplicity, we relabel n 2 as n from now on. Therefore, the self-consistent gravitational Lagrangian for a string in f (R, L m ) gravity takes the form which reduces to the Hilbert-Einstein Lagrangian when C 2 = 0 and g (L m ) → L m .

A. The energy conservation equation
In order to completely solve the problem of the determination of the generalized gravitational action for a string-like configuration, one must determine the dependence of the function f (R, L m ) on the matter Lagrangian L m . This can be achieved by substituting the expressions for the energy momentum tensor, given by Eqs. (77)-(80) into the energy conservation equation, Eq. (42). The resulting equation completely determines the functional form of f , as well as the dependence of the matter Lagrangian on the coordinate r. Taking into account the conditions given by Eq. (82) we have to consider two cases independently.
After substituting the expressions for the string energy density and pressures into the conservation equation Eq. (42), and by taking into account the conditions a + b + c = 1, c = a, we obtain the following differential equation, giving the dependence of the function g (L m (r)) on the matter Lagrangian and on the radial coordinate r, The second case in which the condition given by Eq. (82) is satisfied is when the coefficients a, b, c satisfy the conditions a + b + c = 1 and c = a. For these values of the Kasner coefficients the energy conservation equation Eq. (42) takes the form In the present Section we present some explicit, exact, string type solutions of the gravitational field equations in f (R, L m ) gravity, for which the source term satisfies the condition T t t = T z z and the metric is assumed to be of the Kasner form. The solutions are obtained by imposing specific conditions on the coefficients a, b, c, or by assuming a specific functional form for the integration function g (L m (r)).

A. Solutions satisfying both Kasner conditions
We first consider string type solutions that satisfy both the Kasner conditions given in Eqs. (37). In this case R t t = R r r = R φ φ = R z z = 0, and R = 0, respectively. The conservation equation Eq. (85) reduces to the condition g ′′ (L m ) = 0, giving g (L m ) = constant + L m . Then, by taking into account the fact that f R (R, L m ) = 0, from the gravitational field equations (3), we obtain T µν = 0, implying that L m = 0. We therefore re-obtain, in the framework of f (R, L m ) gravity theory, the Kasner vacuum solution of standard general relativity.
B. Kasner type string solutions with g (Lm) = g0L q m (r) As a second example of a string solution in f (R, L m ) gravity we consider the simple case in which g (L m ) has a power law dependence on L m , so that g (L m ) = g 0 L q m (r), where g 0 and q are constants. With this choice of the matter term the conservation equation Eq. (85) and (86) gives the r-dependence of the matter Lagrangian, allowing us to obtain, explicitly, the full solution of the gravitational field equations for the Kasner string. In the following we consider separately the cases a + b + c = 1, c = a and a + b + c = 1, c = a, respectively.
If the values of the constants a, b, c satisfy the constraints a + b + c = 1, c = a, then from Eq. (85), for g (L m ) = g 0 L q m (r), we first obtain the differential equation satisfied by the matter Lagrangian as with the general solution given by where c 1 is an arbitrary constant of integration, and we have denoted Hence we obtain the Lagrangian gravitational density explicitly as Imposing 2a + b = 2 or, equivalently, c = a − 1, and setting C 2 = 1/Λ, we obtain the model f (R, L m ) = 1 2 R + R 2 /Λ 2 + g 0 L q m , whose f (R) version with matter minimally-coupled to geometry is commonly studied [32]. Hence, static, string-type solutions do exist in the f (R, L m ) cosmology, but their self-consistent description requires R ∝ r −2 = constant within the string core.
If the parameters of the Kasner metric satisfy the conditions a + b + c = 1, c = a, then for g (L m ) = g 0 L q m , from the conservation Eq. (86), it follows that the matter Lagrangian satisfies the differential equation with the general solution given by where c 2 is an arbitrary integration constant, and we have denoted The gravitational Lagrangian of this model takes the form Imposing a = c = 1 4 (3 − 2b), C 2 = 1/Λ, we again obtain the model f (R, L m ) = 1 2 R + R 2 /Λ 2 + g 0 L q m , whose f (R) limit corresponding to q = 1 and g 0 = 1 was studied in [32]. Therefore, in general, it is clear that in f (R, L m ) gravity theories two different families of self-consistent Kasner-type string solutions exist.

D. Quasi-Kasner solutions of the field equations
Finally, we consider the case in which both Kasner conditions, given by Eqs. (37) hold approximately, so that a + b + c ≈ 1, and a 2 + b 2 + c 2 ≈ 1, but R 0 = a 2 + b 2 + c 2 + (a + b + c)(a + b + c − 2) = 0, implying that the Ricci scalar R is again nonzero inside the string. For the function g (L m ) we adopt again a power law form, with g (L m ) = g 0 L q m (r). Under these assumptions the energy-conservation equation Eq. (42) becomes with the general solution for L m given by where c 3 is an arbitrary constant of integration.
The thermodynamic parameters of the quasi-Kasner string are obtained as With the use of Eqs. (6) and (7) we obtain for G ef f and Λ ef f the general expressions and respectively. Again, we may recover the common model with R and R 2 terms for an appropriate choice of the Kasner parameters, but the important point to note is that, for all cases in which self-consistent Kasner-type string solutions exist, other more general models are also possible. In particular, there exist two distinct families of non-vacuum Kasner type solutions for which the Kasner conditions are not satisfied inside the string core, and specific relations between the parameters a, b and c must be imposed in order to satisfy the correspondence principle, embodied in Eq. (82).
However, an additional family of string-type solutions also exists in which the Kasner conditions are satisfied approximately, but no further conditions relating a, b and c need to be imposed to ensure the existence of the general relativistic limit. Setting C −1 2 = R 0 = 2Λ and n = 2 in Eq. (84) is one way to obtain an effective cosmological constant (to leading order for large r) at the level of the field equations, which is naively consistent with the results of the previous work in which Linet-Tian type solutions (corresponding to the exterior of a cosmic string embedded in a space-time with Λ = 0 [49]) were found to exist in f (R) gravity [32,33].
Thus, it may be hoped that future studies, in which the local EOM for field-theoretic variables in the matter Lagrangian are explicitly solved in conjunction with the gravitational field equations, recover the solutions obtained in [32,33] as T t t (r) = T z z (r) → 0 for r → 0; that is, as the string energy density tends asymptotically to zero. For small r, one would also hope to be able to relate the parameters in the Kasner metric to the fundamental field theory parameters (e.g. the symmetry-breaking energy scale, η, scalar and vector coupling constants, λ and e, and topological winding number, |n| in the Abelina-Higgs model), directly, via expressions such as Eq. (84).

E. The mass and angular deficit of the string
Again, for the sake of brevity, though at the expense of a comprehensive treatment, we limit the following analysis of the mass density and angular deficit of the string to the quasi-Kasner case discussed in the previous section. That is, we assume that the Kasner parameters obey the conditions a + b + c ≈ 1, a 2 + b 2 + c 2 ≈ 1, but that R 0 = 0. The radius R s of the string can be obtained from the condition p r (R s ) = 0, and is given by the solution of the equation In the string Lagrangian, given by Eq. (96), we now have a sum of three terms, the first varying as a function of r as 1/r 2+2a+b , the second as 1/r 2 , and a third constant term. Therefore in the large distance limit the dominant term is 1/r 2 , and we can adopt the approximate expression Then, for the radius of the string, we obtain For this value of the radius both the energy density and the tangential pressure p φ have non-zero values at the string surface.
The Tolman mass per unit length of the string is then where, in order to avoid the singularity in the origin, we have to impose the conditions b + 2(c + 1/q) < 3, and b + 2(c + n + 1/q) < 5, respectively. In the same approximation for the W parameter, which determines the angular deficit, we obtain where, again, in order to avoid the singularity at r = 0, we must impose the conditions given above on the values of b, c, n and q.
By assuming that C 2 = C 0 /Λ, where Λ is the cosmological, assumed to be of order of Λ = 10 −56 cm −2 , the radius of the quasi-Kasner string in self-consistent f (R, L m ) gravity (with power law matter coupling), can be estimated as R s ≈ 2 [4n(n − 1) − R 0 ] /nR 2 0 (1 + 2g 0 /n) 1/2(n−1) × 10 28/(n−1) cm. Thus, the string radius decreases rapidly with increasing n, which, in turn, determines the R dependence of the gravitational action. The numerical value of R s also depends on the coefficients n and n 0 . In particular, very small values of 4n(n − 1) − R 0 can further reduce the string radius and, since n = 1 + a + b/2, very small values of a and b can significantly reduce the radius of the string. In the leading order of approximation, the mass-radius relation of the quasi-Kasner string is given as M (R s ) ∝ (kR s ) 5−b−2c−2/q−2n . Thus, the mass of the string also depends on q, which determines the dependence of the gravitational action on the matter Lagrangian. On the other hand, the mass of the string depends on the coefficient R 0 , which, due to the impositions of the approximate Kasner conditions, may have a very small value. Hence, depending on the adopted values of the model parameters, a very wide range of radii and masses can be generated for cosmic strings in f (R, L m ) gravity.

VII. DISCUSSIONS AND FINAL REMARKS
In the present paper we have considered static, cylindrically-symmetric, solutions of the gravitational field equations for a string-like distribution of matter in the f (R, L m ) modified gravity theory, in which the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar and matter Lagrangian. Having determined the gravitational field equations for a general cylindrically-symmetric metric, with arbitrary dependence of the metric tensor components on the radial distance, r, we have restricted our subsequent analysis to a specific case by adopting the Kasner metric for the string interior, in which the components of the metric tensor are proportional to powers of r.
Two distinct models have been investigated. In the first, the gravitational Lagrangian was fixed in the form of an exponential function, f (R, L m ) = (1/Λ) exp (R/2Λ + L m /Λ), which obviously reduces to the Einstein-Hilbert Lagrangian in the limit R/2 + L m ≪ Λ. In the second, the function f (R, L m ) was self-consistently determined from the field equations using the string condition, T t t = T z z , or equivalently, ρ = −p z , where ρ is the mass per unit volume of the string and p z is the thermodynamic pressure, together with the correspondence principle, which requires the Einstein-Hilbert limit to exist for appropriate choices of the free model parameters.
In the first case, from the field equations one can obtain a second order linear differential equation for the matter Lagrangian, whose solution can be expressed in terms of the hypergeometric and Laguerre functions. The knowledge of the matter Lagrangian fully determines the solution of the field equations, allowing the determination of the energy density, and anisotropic pressures inside the string. Crucially, in this case, the Kasner parameters are required to satisfy the relation 2a + b > 6, in order to avoid divergences in the Tolman mass density (i.e. the relativistic mass per unit length of the string), M , and "W " parameter, which determines the angular deficit [58], as r → 0.
In the second case, by imposing the string conditions for the field equations in the Kasner metric we obtain a second order differential equation for f (R, L m ), which allows the determination of the gravitational Lagrangian density. In this case f (R, L m ) has an additive structure, being the sum of a Ricci scalar dependent function, and of a matter Lagrangian dependent function g (L m (r)), which is an arbitrary function of integration. The dependence on the Ricci scalar can be simplified with the use of the correspondence principle, by requiring the standard general relativistic limit of the modified gravity action (i.e. the Einstein-Hilbert action) to exist for appropriate choices of the arbitrary parameters and functions of the model. In order to determine the matter Lagrangian dependence one must use the conservation equation of the theory, which gives a strongly nonlinear differential equation for g (L m (r)). In order to obtain some explicit solutions of the field equations we have assumed the simple case in which g (L m (r)) = g 0 L q m (r), q = 1. This choice allows the immediate determination of the matter Lagrangian from the conservation equation, and of the general solution of the field equations, for a gravitational model described by a Lagrangian density of the form f (R, L m ) = R/2 + (C 2 /n) R n + g 0 L q m , q = 1, where n = n(a, b, c) depends on the parameters a, b, c of the Kasner metric. In this case, the function n(a, b, c) may take two different forms, corresponding to two distinct sets of conditions imposed on the Kasner parameters, a, b, c, in order to satisfy the correspondence principle. Thus, depending on the choice of conditions for a, b and c, two distinct string-type models can be obtained. An additional family of solutions, which we refer to as "quasi-Kasner" solutions also exists, in which both sets of Kasner conditions (i.e. those defining both the the Kasner sphere and Kasner plane [47]) are approximately satisfied, but for which the Ricci scalar is nonzero inside the string core.
Importantly, for all sets of self-consistent Kasner-type string solutions obtained in our study, the Ricci scalar is of the form R = R 0 (a, b, c)/r 2 , where R 0 (a, b, c) = 0 when both sets of Kasner conditions are satisfied. Thus we recover, in the context of the f (R, L m ) theory of gravity, the unique R = 0 vacuum solution obtained in general relativity [47,48] and, previously, in f (R) gravity [32,33]. However, the existence of vacuum solutions in the f (R) theory for which R = const. = 0, also found in [32,33], and which correspond to Linet-Tian (LT) type solutions [49] under the identification of R = const. with the cosmological constant term in the gravitational field equations, imply that the Kasner metric fails to "capture" LT type solutions in both f (R) and, more generally, f (R, L m ) gravity.
The physical interpretation of this is that, although cylindrically-symmetric "Kasner type" solutions, for which R = 4Λ = constant may be identified with the cosmological constant, exist in general relativity [50], no such solutions exist for the Kasner metric in f (R) and f (R, L m ) gravity. Hence, since the (non-Kasner) solutions exhibiting cylindrical symmetry so far obtained in f (R) gravity correspond to vacuum solutions for which R = constant = 0 and, since the unique R = 0 vacuum solution obtained in [32,33] is equally well expressed in terms of the Kasner metric, as shown in the present study, we may conclude that non-Kasner metrics are required to capture all additional static, cylindrically-symmetric, vacuum solutions (for which R = 0) in both the f (R) and f (R, L m ) modified gravity theories.
In standard general relativity the vacuum gravitational field equations satisfy the condition R ν µ = 0 and R = 0, respectively. For a general static cylindrically-symmetric metric of the form (16), the components of the Ricci tensor are given by Eqs. (20). The equations R t t = R φ φ = R z z = 0 can be immediately integrated to give where C 1 , C 2 , C 3 are arbitrary constants of integration. From the above equations we obtain where α 1 = C 1 /C 3 , β 1 = C 1 /C 3 , and γ 1 = C 2 /C 3 . The condition R r r = 0 then gives the following equation for L(r), (α 1 γ 1 + γ 1 + 1) L(r)L ′′ (r) − (α 1 γ 1 + 1) L ′2 = 0, which has the general solution where c 2 is an arbitrary integration constant and where we have taken, without any loss of generality, an arbitrary integration constant to be zero, and have denoted b = 1 + α 1 + 1/γ 1 . In the last equality in Eq. (B4) we have rescaled the arbitrary integration constants. By denoting a = α 1 b and c = b/γ 1 , we obtain N (r) = N 0 (γ 1 r) a , K(r) = K 0 (γ 1 r) c , where N 0 and K 0 are also arbitrary integration constants. Then, by using the above expressions for the metric tensor components, Eqs. (B1) impose the restriction a + b + c = 1 on the constants a, b, c. Finally, the condition R = R 0 /r 2 = 0 requires R 0 = 0 for the vacuum solution, and with the use of the explicit form of R 0 , given by Eq. (43), it follows that in order for the Kasner metric to represent a static general relativistic vacuum solution of the gravitational field equations, the coefficients a, b, c must also satisfy the second Kasner condition a 2 + b 2 + c 2 = 1.