Graviton to photon conversion in curved space-time and external magnetic field

The suppression of relic gravitational waves due to their conversion into electromagnetic radiation in a cosmological magnetic field is studied. The coupled system of equations describing gravitational and electromagnetic wave propagation in an arbitrary curved space-time and in external magnetic field is derived. The subsequent elimination of photons from the beam due to their interaction with the primary plasma is taken into account. The resulting system of equations is solved numerically in Friedman-LeMaitre-Robertson-Walker metric for the upper limit of the intergalactic magnetic field strength of 1 nGs. We conclude that the gravitational wave conversion into photons in the intergalactic magnetic field cannot significantly change the amplitude of the relic gravitational wave and their frequency spectrum.


Introduction
The transformation between gravitons and photons in external magnetic field was considered in multitude of papers starting from 1961 [1] - [8].The problem acquired particular importance in connection with the possible transformation of relic gravitational waves (GW) produced at the inflationary stage into electromagnetic waves (EMW) in primordial magnetic fields.However, in all previous works see e.g.[8,9] the calculations have been always done in Minkowski space-time, though the curvature effects in the very early universe could be quite essential.
In the present work we go beyond the flat space-time restriction and consider graviton and photon propagation in arbitrary curved backgroud.The propagation of gravitational waves in curved space-time was almost always considered in Friedman-LeMaitre-Robertson-Walker metric (FLRW), see e.g.textbooks [10,11], except for some Bianchi types metrics and our recent paper [12], where an arbitrary background metric was allowed.
Here we derive the propagatoin equations for the coupled system of photons and gravitons in an arbitrary background.Next we will turn to the Friedman-LeMaitre-Robertson-Walker (FLRW) space-time, that is a good approximation to the real universe.However, deviations from FLRW could be essential and lead to interesting observable effects.
Already for a century the Friedman equations serve as a basement for the conventional cosmological model.They perfectly well describe the early universe, that is homogeneous and isotropic to a very good approximation.They are operative also in the contemporary universe on very large scales.Friedman cosmology allows for description of cosmological dark matter and what's more surprising dark energy, though the physical nature of the latter is not yet established.
The propagation of gravitational waves in curved space-time was almost always considered in FLRW metric, except for some Bianchi types metrics and our recent paper [12], where an arbitrary background metric was allowed.
Here we derive the coupled equations of motion for metric perturbations and electromagnetic waves over an arbitrary cosmological background in the external cosmic magnetic field.The metric perturbations and EMW are treated in the first order of the perturbation theory.
We introduce the full electromagnetic field Āµ as the sum of an external classical component of the electromagnetic field A µ and a small quantum fluctuation f µ , which is considered as a perturbation, Āµ = A µ + f µ . (1.0.1) Then, the stress tensors of Āµ , A µ and f µ are introduced accordingly: (1.0.3) (1.0.4) The full metric tensor ḡµν is expanded around the metric tensor of the background space-time g µν as ḡµν = g µν + h µν , ḡµν = g µν − h µν (1.0.5) with h µν being a small perturbation of the metric.The properties of the metric tensor g µν are specified by: the orthogonality condition g µν g µλ = δ λ µ , where δ λ µ is the Kronecker delta-symbol; rising and lowering of the indices of the tensors h µν and f µν by the background metric tensor g µν .Note, that the indices of the full and classical stress tensors of the electromagnetic fields are raised and lowered with the full metric tensor ḡµν .
The corrections to the metric determinant ḡ can be found from the first order expansion of an arbitrary non-degenerate matrix M: (1.0.6) So we obtain: It is assumed usually that tensor perturbations are traceless: h ≡ g µν h µν = 0.
(1.0.8)However, we see in what follows that the corrections to the Maxwell energy-momentum tensor are not traceless, see e.g.eqs.(5.4.2, 6.3.1) and a nonzero trace of the gravitational field source leads to the nonzero h, so det[ḡ µν ] = det[g µν ] (1 + h).
The initially derived equations are supposed to be valid in an arbitrary space-time metric, but ultimately we assume that the background metric has the 3D-flat FLRW form: where a(t) is the cosmological scale factor.The Hubble parameter is expressed through it in the usual way as H = ȧ/a.The curved metric reduces to the flat one when a → 1.
The paper is organized as follows.We start in Sec.2 from a brief reminding of the equation for metric perturbations over arbitrary space-time.In Sec.3 we recall the expansion of metric perturbations in terms of helicity eigenstates.After that in Sec. 4 we show that the scalar and tensor modes can mix in general case of inhomogeneous space.Further, in Sec.5 we consider propagation of metric perturbation over FLRW space-time in external magnetic field.In Sec.6 the propagation of electromagnetic waves in magnetic field is considered.In both sections we start from the classical Maxwell and Hilbert-Einstein actions ignoring for a while the Heisenberg-Euler (HE) [13] corrections, the quantum trace anomaly and matter effects.They are taken into account step by step in the subsequent subsections.On the way we discuss the definition of physical magnetic fields through the electromagnetic field tensor F µν in cosmological background (subsection 5.3) and the impact of the HE-corrections to the electromagnetic wave propagation expressed through the physical magnetic field B (subsection 6.2).In Sec. 7 we analyze the full set of differential equations (SDE) for (g − γ)-coupled system, choose a reference frame, and simplify the system for the choice made.Next, in Sec.8, we divide the task into two cases: k||B and k⊥B, and find out that the conversion effect is present only for the second case.In the last Sec.9we divide SDE into two independent subsystems and solve the first of them numerically.In conclusion we summarize the obtained results and formulate the prospects for the future research.

Metric perturbations in general case
In [12] we obtained equation (23) for the propagation of metric perturbations in arbitrary space-time.Let's write it for two lower indices: where G is the gravitational constant, D 2 = D α D α and D α is the covariant derivative, R αµνβ , R µν , R are the Riemann tensor, Ricci tensor and scalar curvature respectively.The equation contains additional terms that disappear in the special cases of Minkowski and FLRW spaces.These extra terms could have significant effects on the GW and EMW propagation over background metric that differs from the FLRW one.
Let us note the agreement between equation (2.0.1) and equation (2.33) from the work [14], published after our work [12].The apparent difference with our result disappears in the Lorentz calibration In this article, the authors obtained the same equation using a double variation of the action, while we obtained it by expanding the Einstein equation to first order in perturbation.

Helicity decomposition and choice of gauge
Now it is worth recalling the formalism of the expansion of the perturbation h µν in terms of helicity states.The generally accepted approach is that (along with the vectors C i , G i and the traceless tensor D ij ) four scalars A, B, E, F are introduced, through which the components of the metric perturbation are expressed: One can impose gauge conditions such that two scalars turn to zero.The so called synchronous gauge corresponds to the choice E = 0 and F = 0.Under this gauge there still remains some more freedom, that may allow to simplify algebra in a specific problem.The second well-known type of gauge is the Newtonian gauge, where B = 0, F = 0, E ≡ 2Φ, and A ≡ −2Ψ.The choice of this gauge better fits our task, so for the scalar sector we will use the Newtonian gauge.
In addition to the gauge in the scalar sector, the Lorentz gauge (2.0.2) is usually imposed on the entire tensor perturbation of metric.This calibration naturally arises in the case when the so-called harmonic Fock coordinates are used.It allows to obtain a simpler expression for the Ricci tensor and, as a consequence, to simplify the equation for the propagation of metric perturbations.Formerly in our paper [12], we have only used the Lorentz gauge.

Mixing of metric perturbation modes
Note that from the expression for the trace of the equation (2.0.1) it turns out that in the general case, for an arbitrary form of the Ricci tensor, there appears a mixing of scalar and tensor modes of metric perturbations.In general case it is impossible to separate the equations for these two sectors.Indeed, taking trace of eq.(2.0.1), we obtain the following expression where ∂ 2 = ∂ µ ∂ µ , and from which it is clearly seen that the second term in the left hand side includes both terms from the scalar and the tensor sectors.
In addition, it is important to pay attention to the trace from the source in the right hand side of the equation.As it will be shown below for the problem of the graviton conversion into photons in an external magnetic field, the trace from the correction to the EMT contains a convolution of the background electromagnetic tensor and the tensor perturbation to the metric, h µν , which also leads to mixing between scalar and tensor modes.This result is evident, because the expansion of metric perturbations in polarizations is valid for a problem with axial symmetry: in this problem there is only one specific direction -the direction of the wave vector k of the metric perturbation.If space is for some reason unisotropic (as, for example, in the case of an external magnetic field or in the presence of an anisotropic stress tensor), this symmetry disappears.

Equation in the FLRW metric
Recall that in the FLRW metric We write the trace of the GW tensor in the following form: where the notation h i i = − (h xx + h yy + h zz ) /a 2 was introduced: Let us write down the system of equations (2.0.1) for the case of the FLRW metric.To do this, we will use the expressions (5.1.1-5.1.4).We get where the Latin indices are the spatial ones (vary from one to three).For a medium where the perturbation propagates, we will consider a model of an ideal fluid.The total energy-momentum tensor in this case is determined through the full metric as follows where ϱ is the energy density, p is the pressure, u µ is the four-speed.Then the right side of equation (5.1.5)can be rewritten as where the first term in the last equality is obtained by expanding equation (5.1.6)to the first order in perturbation at u j = 0 (index j varies from one to three) and the second term is responsible for the perturbation of the EMT due to the presence of an external magnetic field.
The factor before the last term on the left side of the equation (5.1.5)is exactly Thus, equation (5.1.5)can be simplified: Now, for brevity, we omit the expressions for the components of the Riemann tensor and the Christoffel symbols in the covariant derivative and write the final equations for 00, 0j, and ij components separately: where notation (5.1.4)is used, and the last equation is written in terms of mixed components, since then it looks more consistent with the equation for the 00 component.

Corrections to the energy-momentum tensor (EMT)
Corrections to the EMT are due to the presence of an external electromagnetic field.We will find them in accordance with the definition of EMT of matter:

.2.1)
There are two contributions to the EMT perturbation: from the Maxwell action and from the Heisenberg-Euler action.The gravity of the background magnetic field is negligible compared to the background of matter and we ignore its contribution to the EMT corrections.

Corrections to EMT emerging from the Maxwell action
The Maxwell action is written as follows: where F 2 = Fµν F µν = ḡµα ḡνβ Fµν Fαβ .Hence the energy-momentum tensor is or for the mixed components: Clearly this tensor is conserved and its trace is zero: The vanishing of the EMT trace in Maxwell electrodynamics is a consequence of the conformal invariance of the Maxwell action (5.2.2).This is not so for higher order quantum corrections (trace anomaly), see subsecton 6.3.The trace of the zero order term with mixed components: is also zero.Note that moving indices up or down in this equation is done by the background metric, e.g.F µ .α F .α ν = g µσ g αλ F σα F νλ and F 2 = F αβ F σλ g ασ g βλ .The zero order term is presumably small in comparison with the total cosmological energy-momentum tensor and can be neglected in what follows.
For the first order term with mixed components we obtain: where = 0, as is expected.

Heisenberg-Euler (HE) Lagrangian
The second origin of EMT corrections is Heisenberg-Euler effective Lagrangian [13].It describes quartic self-interaction of electromagnetic field and is induced by the loop of virtual electrons with four external electromagnetic legs.In the weak field limit, and low energies, much smaller than the electron mass, m e , the corresponding action has the form: (5.2.8) Here because the tensor quantity is √ −g ϵ αβµν but not just ϵ αβµν , see e.g.chapter 83 from textbook [15].
In what follows we apply this action to photon propagation in external magnetic field B and the weak field limit means B ≪ m 2 e .
We need to generalize the Heisenberg-Euler action (5.2.8) to high energies/temperatures and curved FLRW space-time.To do that let us start from the canonical action of photons and electrons written in terms of the conformal metric: where is the Minkowski metric tensor, and a(τ ) is the scale factor as a function of conformal time τ (t) = dt/a(t).
For FLRW metric it has the form Introducing conformally transformed spinor χ = ψ/a 3/2 , we arrive to the action: This is essentially the same action as it is in flat space-time with rescaled mass and charge: m → am and e → ae, so formally C 0 ∼ e 4 /m 4 does not change, but since we plan to go to very high temperatures, even above the electroweak phase transition, when all bare masses of charged particles vanish, we have to substitute the high temperature value of the mass, to sum over all charged particles, and to take the high temperature value of the electromagnetic coupling α.So where q j is the charge of the contributing to the loop particles in the electron charge units, e.g. for down or up quarks q = −1/3 or 2/3.The integrand in the expression for the action A HE is a scalar with respect to the general coordinate transformation, so we can use for it the same expression as (5.2.8) in arbitrary metric.
In the early universe at high temperatures the Heisenberg-Euler action keeps the same form as (5.2.8) with substitution of C(T ) instead of C 0 : where F 2 = F µν F µν , F F = Fµν F µν , and we have returned bar over F µν and to the metric determinant in accordance with expansion (1.0.5).
The HE action given by eq. ( 5.2.15) leads to the following contribution to the energymomentum tensor: Here the over-bars are eliminated to simplify notation but we keep in mind that this expression will be used with the non-expanded complete quantities, see Eq. (1.0.5).
An explanatory comment may be in order here, namely, the second term containing the dual Maxwell tensor, Fµν , depends upon metric only through the factor ( √ −g) −2 , so with the account of the integration measure the action depends on metric as ( √ −g) −1 instead on ( √ −g).Hence this gives the contribution to T µν from ( F F ) 2 proportional to (+g µν ) instead of the usual one proportional to (−g µν ).
One can see that the trace of tensor (5.2.16) is non-vanishing: It is instructive to check conservation of the energy-momentum tensor (5.2.16), though it surely must be true, since it was obtained by the variation of a scalar function over metric.Still, at least the verification of the conservation would indicate that the calculations are correct.Let us note that the conservation condition should be fulfilled only if C = const.Evidently the energy-momentum tensor (5.2.16) is non-conserved for a non-constant C(T ) because the dependence on temperature appears due to interaction and an exchange of energy with external system.
It would be more convenient to express the square of the dual electromagnetic tensor through F. It enters the action in the form, see eq. (5.2.10): Expressing the product of epsilons through the Kronecker symbols and properly contracting the indices we obtain: where We can verify result (5.2.19) expressing the Maxwell tensor through electromagnetic fields B and E coming to the well known relation The first part of the action (5.2.15), proportional to F 4 , leads to the following contribution to the energy-momentum tensor (5.2.20) The same contribution, up to a numerical factor, comes from the first term in eq. ( 5.2.19),So to find the total EMT we need to find the variation of the second term of eq. ( 5.

2.19).
Eventually the remaining part of the energy-momentum tensor is Bringing together eqs.(5.2.20) and (5.2.21) and raising one index we obtain for the total energy-momentum tensor, originating from the Heisenberg-Euler action, the following expression: Now let us check the conservation of the obtained energy-momentum tensor (5.2.22) in the case of constant C(T ).We consider T (1) µν and T (2) µν separately.
where semicolons mean covariant derivatives in the background metric.The last term in this equation is zero according to the equation of motion corresponding to the Lagrangian L = F 4 .The first two ones can be rewritten using the relation Renaming some dummy indices we come to Here we proved EMT conservation law for those parts of the action which contain F 4 .It must be analogous for the EMT part originated from (5.2.26) where we used the Maxwell equation F µν ;µ = 0. Considering the part inside square brackets and taking into account the equation of motion F µλ (F αβ F λα ) ;µ = 0, that follows from the part of the action: (see eq. ( 5.2.15) and (5.2.19)), we arrive to For the transition to the third term of these equalities we used eq.(5.2.24).The conclusion for this section is that EMT originated from the Heisenberg-Euler action with C = const is conserved It is noteworthy that EMT (5.2.22) is not traceless.Indeed it is equal to (5.2.30)

Corrections to EMT emerging from the HE action
Now making the usual perturbation expansion (1.0.5) we find the following first order correction to the energy-momentum tensor: This expression can be simplified, because in the absence of background electric field F F = 0 and we get: The trace of this expression is nonvanishing: It is usually demanded in FLRW space-time that the source term T (1)µ ν for gravitational wave equation (5.1.10)must be traceless.To this end one may separate the traceless part out of Eq. (5.2.32) subtracting g µν T α HE 1 α /4 out of it.However, this prescription would break the conservation of the source and, as is shown in Sec.6 of paper [12], it would lead to a violation of the transversality conditions D µ ψ µ ν = 0. Indeed, in [12] we used the condition D µ T µ ν = 0 to prove a compatibility of the Einstein equations in the first perturbation order with gauge fixing conditions (6.1.9).
Note that the energy-momentum tensor (5.2.16) is non-conserved for a non-constant C(T ) because the dependence on the temperature appears due to interaction with external system.So anyhow EMT is not formally conserved.

Summary
As conclusion of this subsection, we write the result for the correction to the EMT from the Maxwell action and from the Heisenberg-Euler action, respectively: where

Maxwell tensor and cosmic magnetic and electric fields
Eqs. (5.2.34-5.2.35) look quite complicated.Further we simplify this equations and express them in terms of physical magnetic field.To understand the physical meaning of the different components of F µν , F µν , or F ν µ , let us start from the geodesic equation for a charged particle in external electric and magnetic field (see e.g.book [15].Eq. (90.7)): where u α = dx α /ds is the particle four-velocity.From this equation it is clear that physical electric field is the Maxwell tensor with mixed components, E j = F j 0 , and physical magnetic field is expressed through the Maxwell tensor F i j as: or in compact form B i = ϵ ijl F j .m δ ml .The first pair of Maxwell equation has the same form as in flat space-time: ( If the background electric field is absent, i.e.F tj = 0, then Hence F ij remains constant in the process of cosmological expansion and correspondingly physical magnetic field behaves as: In other words, physical magnetic field drops as 1/a 2 , the well known result.If electric field is absent and only external magnetic field is non-zero, then the dual Maxwell tensor (5.2.10) has only space-time components.The quantity D tj = Ftj / √ −g = (1/2)ϵ tjlm F lm is expressed through magnetic field as (5.3.6) In flat space-time varying magnetic field induces electric field according to In curved space-time the analogue of this equation is Eq.(5.3.3) with λ = t or Eq.(5.3.4),so if originally electric field was absent, it would not be induced by time-varying magnetic field, in the case that the time variation is created by the cosmological expansion (5.3.5).In terms of physical magnetic field B the product F µν F µν with indices lifted by the background metric g µν is (5.3.8)

Scalar and tensor mode mixing in external magnetic field
Using equations (5.1.1-5.1.4),we rewrite equation (4.0.1) for the case of the FLRW metric as: α . (5.4.1) The EMT perturbation originating from the Maxwell action is traceless, while that from the Heisenberg-Euler action has a non-zero trace.Indeed, (5.4.2) Now one could naively divide the source into a traceless part and a non-zero trace part (simply subtract the trace multiplied by the background metric).To this end let's look at Eq.(5.4.1) and explicitly substitute T α EM (1) α into the right-hand side.We get We see that the equation contains both scalar and tensor parts.Thus it is impossible to write a separate equation for each mode.To make this even more obvious, let us fix the coordinates so that the magnetic field is directed along the x axis.Then the following components of the electromagnetic field tensor will be non-zero: (5.4.4) (5.4.5) The trace from the correction to the EMT can then be rewritten taking into account the following relations: (5.4.9) And in equation (5.4.3) we have (5.4.10) The diagonal components can be written as the sum of scalar and tensor quantities of the helicity expansion (5.4.12) And, substituting the helicity expansion into the complete equation, we obtain the final equation, which shows the mixing of scalars Φ, Ψ and tensor D ij : where h = 2Φ + 6Ψ.
In addition, we note that there may also be implicit mixing through the term with f y .z in the above equation, since the equation for the electromagnetic wave contains various convolutions of tensors with tensor h µν ( see below, eq.(7.2.5)).
As was noted in the Sec.4, the result is quite evident, since the external magnetic field gives, in addition to the GW propagation vector, another preferred direction in space.This leads to the loss of axial symmetry and to mixing of the scalar and tensor modes of metric perturbations.

Electromagnetic wave propagation in external magnetic field
In this section, we will derive the equation for the propagation of electromagnetic waves in curved space-time and in the presence of an external magnetic field, thereby completing the derivation of the system of differential equations (SoDE) for the metric-EMF perturbation system.We will briefly call this system g − γ, by g we mean a graviton with any possible polarization -0, 1, 2.

Equation of motion from the Maxwell action
Variation of the Maxwell action from eq.(5.2.2) over δA ν leads to the equation of motion Dµ F µν = Jν , where Dµ is the covariant derivative in the full metric ḡµν .Due to antisymmetry of F µν this equation is reduced to: Below we assume that neither electric charge density nor electric current are present, i.e.Jν = 0. Substituting expansions (1.0.5), (1.0.4), and (1.0.7) into Eq.(6.1.1)we obtain The external electric field is supposed to be zero and only background magnetic field is present, so F tβ = 0. Thus the zero order term, which is the equation of motion for the background magnetic field, has the form; This is the analogue of the equation div B = 0 in flat space-time.
In FLRW metric the metric determinant is expanded as: and so equation (6.1.2) takes the form: where we took into account that g tα = δ tα and F tβ = 0. We also assume that f t = f t = 0 and impose the transversality condition on the propagating photon modes: which for f t = 0 leads to ∂ j f j = 0. Thus the first order expansion of eq. ( 6.1.5)has the form: where we introduced a new quantity Q ν to describe contribution from different additional terms such as Heisenberg-Euler corrections, matter effects, etc to be considered below.
To derive the first order equation for f j we multiply eq. ( 6.1.7)by g νj = −a 2 δ νj (latin indices are always supposed to be the space ones, e.g.j = 1, 2, 3) and recall that F tµ = 0 and f t = 0.So we get finally: where Q j = g νj Q ν = −a 2 δ νj Q ν and ∆ is the flat space Laplacian.
To proceed further we have to fix certain gauge conditions on metric perturbations h µν .We will follow our paper [12], where it is shown that the following conditions can be imposed in arbitrary background metric: Since in FLRW metric the only non-zero components of the Christoffel symbols are: the covariant derivative of h i j is reduced to the ordinary derivative and So using Eq.(6.1.3),(6.1.9),and the absence of external electric field, F tj = 0, we obtain: The terms proportional to ∂ m h cancel out because Here we have introduced the new notation: to be used in what follows.

Equation of motion from the Heisenberg-Euler action
The variation of A HE (5.2.15) over δA ν results in the following contribution to the electromagnetic field equation: where the first term originated from the variation of the Maxwell action (see previous subsection), while the second is the contribution from A HE (5.2.15) and has the following form where F 2 = F µν F µν and (F F ) = Fµν F µν .We have shown in Sec.5.3 that the free external magnetic field is not constant, but rises backward in time with the decreasing scale factor as 1/a 2 .
Let us return to Eq. (6.2.2) and make perturbative expansion according to Eq. (1.0.5).We start from consideration of the first term in square brackets, which with account of the zeroth and first orders terms takes the form where f F = f µν F µν , (F F h) = h α σ F αβ F βσ , and indices are shifted up or down with the background metric.
The zero order term in this equation somewhat changes the equation of motion of the background magnetic field in FLRW metric leading to: which is not of much importance for the evolution of B j .The terms proportional to the time derivatives of √ −g, C, and F 2 do not appear if F tµ = 0 and h t α = 0.The first order part of expression (6.2.3) is equal to: The term proportional to B 2 C in this expression, has the form: where M j [f ] is defined in the equations (6.1.12,6.1.13).The factor in the brackets in the l.h.s. of the above equation coincides with the l.h.s. of Eq. (6.1.12),except for the last term Q j , so it can be absorbed into Eq.(6.1.12)changing the overall coefficient from 1 to (1 − 16CB 2 ).
In addition to the terms proportional to B 2 , the first two terms in Eq. (6.2.5) give the following contribution of the first part of the HE action to the photon propagation equation: So all the terms in Eq (6.2.5), except for those absorbed into Eq.(6.1.12)and containing h µν , turn into: The contribution of the terms containing h µν in Eq. (6.2.5) can be written as Finally, for the total Q HE1 j we obtain where M j [f ] is defined in eq.(6.1.13).
It is convenient to introduce auxiliary vector field through the equation To decipher the last term in Eq. (6.2.10) we use the identity: and Eq.(6.2.11).So F F h and F µ .j∂ µ (f F ) turn into where summation over repeated indices is performed.The variation of the second term in the HE action, Eq. (6.2.2), is equal to: In the case that the background electric field is absent and only magnetic field is nonzero, the r.h.s. of the equation above vanishes in the zeroth perturbation order because F αβ is non-zero only for space-space components, while Fαβ is non-zero for space-time components.Hence Fαβ F αβ = 0. Accordingly, expression (6.2.15) multiplied by g jν can be expanded as where we obtain from Eq.(6.2.16): where the summation over the repeated space indices is made with Kronecker delta and it is taken into account that B j ∼ 1/a 2 .So using Eqs.(6.1.12,6.1.13,6.2.10, 6.2.16) we obtain: and come to the almost final equation for photons We remind that M j [f ] is defined in eq.(6.1.13).

Conformal anomaly effect
Quantum corrections to the energy-momentum tensor of electromagnetic field T (em) µν in curved space-time background lead to the the well known conformal anomaly, for a review see ref. [16], resulting in the nonzero trace of the electromagnetic energy-momentum tensor: where G µν is the gauge field stress tensor, α is the fine structure constant and β is the first coefficient of the beta-function expansion for the gauge group of rank N : with N F being the number of the fermion species.
There are additional contributions into the trace proportional to the products of the Riemann, Ricci tensors, and curvature scalar which are generally nonlocal [17,18].We will not consider them in this work.
The trace anomaly allows for photon production by the conformally flat gravitational field [19,20] in contrast to the Parker theorem [21].
The Fourier transform of the amplitude of the photon propagation in the gravitational field has pole at q 2 = 0, where q is the four-momentum transfer to gravitational field.According to the result of paper [19] the anomalous part of the energy-momentum tensor has the form: It is evidently conserved and has non-zero trace.
As is shown in Ref. [19], conformal anomaly (6.3.1)leads to an additional contribution to eq. ( 6.1.8)or, which is essentially the same, to eq. (6.1.12).
The first term here is the usual charge renormalization and the second one is the anomaly giving rise to photon production in conformally flat metric.This metric allows for the transformation to the conformal time leading to the Minkowski metric proportional to the common scale factor.The canonical Maxwell equation, without the anomalous term, transforms in this metic into the free Maxwell equation in flat space-time, while the additional anomalous term does not allow that.

Plasma interaction effects
Photons propagating in the primeval plasma interact with the plasma particles and as a result they acquire an effective mass, the so called plasma frequency, Ω pl , so the relation between photon frequency, ω, and momentum, k, changes as ω 2 − k 2 = Ω 2 pl .Waves with ω < Ω pl do not propagate in plasma.
In the canonic theory the effective action describing the plasma frequency term is usually written as This term is proportional to the square of the small amplitude f µ of the electromagnetic wave and seemingly should be neglected in our first order approximation.However, this is not so because to obtain the first order equation one has to take the action in the second order in small quantities.The first order terms are absent in the action since f µ satisfies the equation of motion that are realised at the extremal value of action for which δA/δf µ = 0.The corresponding energy-momentum tensor is is quadratic in f and can be disregarded in our approximation.It is similar to a scalar field with small amplitude ϕ that has energy density proportional to m 2 ϕ ϕ 2 , so its energymomentum tensor is quadratically small but non-zero mass, m ϕ is essential for propagation of ϕ waves.
All electrically charged particles contribute to plasma frequency.If the particle mass is larger than the temperature of the relativistic cosmological plasma, m > T , the contribution to plasma frequency from such nonrelativistic charged particles is where n is the number density of particles with charge e, e 2 = 4πα.Note that the number density in this case is exponentially suppressed, n ∼ exp(−m/T ) [22].
On the other hand relativistic particles, with m < T , contribute as: 6.4.4)where the summation is done over all relativistic charged particles with charges e j .The electric charge, e, depends on temperature due radiative corrections [23].Plasma frequency is determined by the photon Green's function in the limit of vanishing photon momentum.More rigorous treatment of the problem of the photon propagation in plasma demands determination of the proper Green's function.Simple derivation of these expression including the Green's function can be found in ref. [24].However, in this paper we will use simplified approximation describing the plasma effects by the plasma frequency only.
We need also take into account the loss of coherence of the photons produced by gravitons.We describe this phenomenon introducing damping term into equation of motion for photons in the form Γ ḟj , where we approximate Γ as Γ = vσn, (6.4.5)where n = 0.1g * T 3 is the density of charged particles in plasma, g * = 10 − 100 is the number of charged particle species, v ∼ 1 is the relative velocity of "our" photon and the scatterer in plasma, and σ = α 2 /T 2 is the scattering cross-section.So the final equation for photons propagating in an arbitrary curved space-time background and external magnetic field in cosmic plasma with an account of the photon colli-sions with plasma particles has the form: It is assumed that f t = 0 and we used eqs.( 6 7 Defining g and γ system of differential equations (SoDE) In total, we have ten equations for the components of tensor h µν and three equations for the components of vector f j .In fact, only six equations for gravitational waves are linearly independent.
In the case considered here we assume that the vector modes do not arise.The first vector G i in eqs.(3.0.1-3.0.3 vanishes due to the gauge condition h 0i = 0.The second vector C i is not zero because the corrections to EoM contain spacial derivative of electromagnetic potential ∂ µ f ν .However the vector modes decay as a −2 and thus they do not play an essential role in cosmology.It worth adding that an account of one more polarisation state would lead to a considerable complication of the system of equations.So in this work we confine ourselves only to scalar and tensor modes.
Finally let us mention that the solution for tensor h i j does not contain pure tensor mode, but a mixture of tensor and scalar modes.Nevertheless the solution represents them qualitatively correctly, including the behaviour of the tensor mode that we are interested in.
Assuming an absence of vector mode we obtain two components less in the EoM.More specifically, we have two scalars, Φ and Ψ (note that in the deal fluid model, i.e. without taking into account dissipation, Φ = Ψ) and two polarisations of the tensor wave, that in total gives four independent equations for metric perturbations.
In the subsequent subsections (7.1, 7.2) the SoDE is simplified for the specific choice of the reference frame, where an external magnetic field is directed along the x axis.Next, one of two independent subsystems is solved numerically in Sec. 8.

Simplification of SoDE for metric perturbations.
To derive the system of equations for the metric perturbations, that is solved below, it remains to simplify the right-hand side of equations (5.1.10).Let us rewrite equation (5.2.34-5.2.35) for an external magnetic field is directed along the x axis.For individual It will now be useful to write down the spatial components separately.After reducing similar terms we get System of equations (5.1.10)is now rewritten as equations(7.1.9,7.1.10,7.1.14-7.1.19).

Simplification of SoDE for electromagnetic waves
Let us simplify the system of equations for an electromagnetic wave for the case when the external magnetic field is directed along the x axis.For spatial components equation (6.4.6) was derived in general form.We assume that C(T ) = const, so the time derivative is Ċ = 0. Also, to begin with, let's omit the last three terms in the left side of the equation, that take into account the interaction of photons with plasma.Now let's write the convolutions with the background tensor of the electromagnetic field in terms of field B: The last equality follows from the fact that the only non-zero component of the dual electromagnetic tensor in the case when the magnetic field is directed along the x is F0x = −aB/2 ̸ = 0. So, let's write the resulting equation (we omit the terms from the interaction with the plasma): By analogy with the equations for gravitational waves, we write for x, y, and z components, respectively: Similar ones can be given, taking into account that B ∼ 1/a 2 (meaning Ḃ = −2HB) and that f t = 0.For x components we get: For y component: For z component: Next we would like to show the validity of the requirement f t = 0.In general, due to the homogeneity of the magnetic field (depending only on time), we arrive at the following equation for the time component: Now we need to select a calibration.If our problem can be called magnetostatic, in such cases the Coulomb gauge ∂ µ f µ = 0 is usually introduced, then we get: From the initial conditions of electrical neutrality we find that this constant is equal to zero.

Two examples of gravitational wave directions
For any initial direction of the gravitational (tensor) wave propagation, we can decompose it into a parallel and perpendicular component relative to the external magnetic field.Note that we consider the case when an initial pure tensor plane wave propagates from vacuum into a region with a magnetic field (and, in the future, with plasma).
It is shown below that, for k||B the scalar mode of metric perturbations is not excited, and the electromagnetic wave is not excited as well.For the perpendicular component k⊥B the situation is different -the scalar mode of metric perturbations and both polarizations of the electromagnetic wave are excited.Until now, we have not taken into account dissipation and loss of coherence for photons due to their interaction with plasma.But even without taking these phenomena into account, it is already possible to detect a change in the amplitude of the initial tensor GW due to the transition to the scalar mode of metric disturbances and to an electromagnetic wave.We will consider both of these cases in more detail in the next two subsections.

k||B
Let us write down the basic relations that allow to simplify the system of differential equations for metric perturbations and for an electromagnetic wave (EMW): Taking into account what was written above, we write the system of equations in terms of h + , h × .From equation (7.1.9,7.1.10,7.1.14-7.1.19) we obtain x = 0, (8.1.9)In equation (8.1.15-8.1.17)there are no terms related to the gravitational wave, therefore, if the electromagnetic wave was not initially present, it does not arise for the case when the wave vector is parallel to the external magnetic field.Hence we obtain that From the remaining non-zero components of EMT eq.(8.1.12-8.1.14)we see that the GW configuration is preserved: it remains tensorial and no scalar modes arise.

k⊥B
Similar to the previous subsection, we write down the main relations that will help to simplify the system of differential equations for perturbations of metric and for electromagnetic waves.Let's direct the wave vector along the z axis (it is always possible to rotate the coordinate system so that k y = 0).k = (0, 0, k z ) , (8.2.1) Taking into account what was written above, we will write the system of equations in terms of h + , h × .From equation (7.1.9,7.1.10,7.1.14-7.1.19)we obtain .9) T x(1) z = 0, (8.2.11) T y(1) z = 0, (8.2.13) ) it follows that an electromagnetic wave with polarization along the x axis is generated by the polarization h × of the GW, and an electromagnetic wave with polarization along the y axis -polarization h + of the gravitational wave.Also from the equations (8.2.7-8.2.14) we clearly see the emergence of a scalar mode from the equations for the 00 and zz components of the EMT (see equation (5.1.10)).
It is important to note that the expressions for EMT in terms of h + , h × are valid only at the moment of time immediately following the initial moment of GW entry into the region with a magnetic field.Further, the wave ceases to be purely tensorial, and it is impossible to assert that h x x = −h y y .To find a solution, it is necessary to express all the quantities precisely in terms of h x x and h y y (not in terms of h + ) or in terms of expansion in helicity, introducing Φ and Ψ.

System solution in the case k⊥B
Let us write out the system of equations completely, taking into account the conclusions of the previous section that f z , h z x , h z y components that are absent at the beginning do not arise during the conversion of tensor GW into photons and scalar perturbations of the metric.
Let us draw the reader's attention to the fact that we write the equations in terms of the electromagnetic potential with the superscript f µ and the gravitational wave potential with mixed indices h ν µ .In this case, we use the following expansion in helicity states for perturbation of the metric We also make the Fourier expansion in terms of momentum, and accept the law of the scale factor variation with time, corresponding to the stage of radiation dominance a(t) ∼ t 1 2 .To further search for a numerical solution, it would be convenient to introduce dimensionless quantities.To do this, let us change the notation Due to the last change, the tensor h µν also becomes dimensionless.
Let us assume that at the present-day Universe a 0 = 1.This is just a choice of reference point and this choice does not influence the solution, because the constant factor in front of the scale factor function has no physical meaning.The condition is convenient in our problem to recalculate magnetic field strength using the present day magnitude.Using the scale factor dependence during matter dominance epoch a = (t/τ tot ) 2/3 we obtain for the coefficient τ 0 : ≈ 35 τ tot , (9.0.9) where τ tot = 13.8 × 10 9 years is the age of the Universe, t eq = 3.3 × 10 5 years is the moment when radiation and dust energy densities were equal.
We accept also that the scale factor a varies in the interval 10 −9 ≦ a ≦ 10 −4 .The selected interval lies inside the radiation dominance epoch (from the hadronic to the recombination).
After the Fourier transform over momentum, it will be clear that the system of equations contains both imaginary and real terms, therefore, to solve the SDE numerically, it will be necessary to decompose each of the required quantities into real and imaginary parts.For example h + = Re(h + ) + iIm(h + ) and so on.For brevity, we write down systems of equations without dividing into real and imaginary parts.To obtain a more universal result, it is convenient to write the system in terms of a(t).The first independent system has the following form: where the prime denotes the derivative with respect to the scale factor, and we introduced the attenuation of the electromagnetic wave due to its interaction with the plasma using the damping factor Γ ∝ α 2 T (a) and the plasma frequency ω 2 pl ∝ αT (a).Let us recall that T (a) ∝ 1/a.
For all solution interval terms with the multiplier αβ ln a can be neglected.Therefore the resulting system is It can be seen that the initial conditions give nontrivial solution.
Let us stress here that the chosen initial conditions just allow us to formulate a simple problem to solve and to obtain the effect in order of magnitude, i.e. to obtain a representative result.That is the first step of the investigation.In future works we are going to take step by step more close to the real physics conditions.
It is important to note that there are poles at a 4 = 16B 2 0 C 0 in the first equation.Let us remind that the effective Heisenberg-Euler action eq.(5.2.8) is correct under the assumption of a weak external electromagnetic field.In our case that means that B 0 /a 2 ≪ m 2 e .This restirction is valid in the selected interval of the variation of the scale factor: a ∈ [10 −9 , 10 and means that the correction to the Maxwell action proportional to α 2 is sufficiently accurate for our consideration.
Another important question to be solved in future work is to what minimum value of the scale factor should the solution be expanded?The solution to this question should be looked for in the theories of cosmological magnetogenesis, that study the epoch when the cosmological magnetic field was generated.We should also keep in mind that as the scale factor approaches the pole, higher order corrections will be excited [13], thereby removing any potential pole.
The second subsystem, which involves the quantities {Φ, Ψ, f y , h + }, is larger, more complex and requires solving many sub-problems.For example, the question of whether scalar metric perturbations propogate is quite nontrivial and requires careful analysis.In order not to make the article too cumbersome, in this work we will concentrate on solving only the subsystem {f x , h × }.
In order to numerically solve the system we need to divide both parts of the equation by a 2 H 2 and to introduce two new functions to lower the order of the equation in order to make it look like: y ′ = f (x, y).
Two new functions and a system 4 × 4 to be solved: 0.17) We use fifth-order implicit Runge-Kutta method which is algebraically stable and allows solving stiff systems of differential equations, see for more [25].

Method of solution validation
Before solving the system for a non-zero magnetic field strength, we must check whether the method for the SoDE solving works correctly for the case when it is absent.Equation of motion for tensor gravitational waves in the approximation kη ∼ 1, where η is a conformal time, can be solved analitically.The solution has the form where h init is an initial magnitude of tensor perturbation, and ϕ 0 is a constant phase.The last two parameters are defined from the matching with the constant mode, obtained from the EoM solution in the approximation kη ≪ 0 (see Sec. 3.2 in Ref. [11]).
In the Fig. 1a the numerical and analytical solutions are presented for the two conformal time values η 1 = 13.2, η 2 = 105.6 and corresponding to them frequencies k 1 = 0.076, k 2 = 0.0095 Hz satisfying the condition kη ∼ 1.Here we see a significant discrepancy, but after the correct phase selections, ϕ 1 0 = −0.22,ϕ 2 0 = −0.00065,we obtain a coincidence with an accuracy of four orders of magnitude (Fig. 1b, 2).In the Fig. 2 an absolute difference between the numerical and the analytical solutions is presented for the two considered cases.
Eventually we can conclude that the method of numerical solution works correctly, and the results obtained for a non-zero value of magnetic field strength are reliable.9.0.12).The code was written in Python using the solve-ivp package.
Of particular interest in the long-wave range are the wavelengths that have left their mark on the CMB.We found a solution for frequencies 10 −18 − 10 −16 Hz.We use an implicit Runge-Kutta method of order five to solve the SoDE for B 0 = 1 nGs and for these frequencies.The scale factor interval is [10 −9 , 10 −4 ], and it lies within the radiation dominance (RD) epoch.For comparison, a solution to the system in the absence of a magnetic field was also found.
The results are the following: by the end of the RD era, amplitude of GW with the selected frequencies is suppressed of about 0.01 percent.Thus, we can conclude that the considered effect of converting GWs into photons in a cosmological magnetic field has an extremely small effect on the amplitude of long-wavelength relict GWs.
It is instructive to say a few words about the physical reason for the suppression of the GW amplitude.In the problem we are considering, where the magnetic field is still not strong enough, the main contribution to the damping comes from the classical Maxwell action.Neglecting the loop correction in the equation of motion for the metric perturbation h y x in eqs.(9.0.11) we obtain the second term in the brackets k 2 a 2 + 16πGB 2 0 a 4 h × , which works similarly to the plasma frequency for photons propagating in the plasma [27].This term suppresses the low frequency end of the GW spectrum.Indeed, for the quantities B 0 = 1 nGs, a 1 = 10 −9 we obtain for the boundary value of the momentum less than the above mentioned analogue of the plasma frequency for GW:

Discussion
In the presented work we have derived a coupled system of equations for gravitational and electromagnetic wave propagation in external magnetic field.After that the simplification of the differential equation system was done for the FLRW background metric and for the case of homogeneous magnetic field directed perpendicularly to the initial gravitational wave vector.Finally we have solved the system numerically for h × -polarization putting B 0 = 1 nGs.The resulting estimate of the effect, without taking into account the inhomogeneity of the magnetic field, is about 0.01% suppression of the amplitude for a relic GW with a frequency of 10 −18 Hz at the recombination.
It is worth to note that the results are obtained under a large list of simplifying assumptions and the research demands deeper investigation (for example the assumption about the magnetic field homogeneity is rather crude).Despite that the results make sence and one can conclude that the considered phenomenon of GW conversion into photons in the intergalactic magnetic field cannot significantly suppress relic gravitational wave amplitude.
Let us emphasize that this result was not obvious at the beginning of the research.The smallness of the second-order corrections to the Maxwell action do not yet mean the smallness of the relic GW suppression effect.It is also necessary to take into account the interaction of the generated by GW photons with the primordial plasma, as well as the fact that the conversion occurs over a long period of time during the evolution of the Universe.A crucial point is also the dependence of the cosmological magnetic field amplitude on the scale factor according to the law B = B 0 /a 2 , which in the early stages of the evolution of the Universe could lead to a rather high magnetic field strength, and therefore to a noticeable conversion effect.
In the future works we plan to solve the second independent part of the SoDE, paying a special attention to the following question: do the emerged scalar perturbations run?After that we want to expand the solution interval up to the end of the matter dominance epoch and to take into account the magnetic field inhomogeneity.
It is worth stressing that the stochastic nature of the relic GW direction and the magnetic field direction should have a large impact on the magnitude of the suppression effect, and a more accurate analysis of this phenomenon is also very important.We plan to perform such analysis in order to present quantitatively the dependence of the full relic GW spectrum suppression on the intergalactic magnetic field strength.
Future research is not only of academic interest, but can also be applied to similar problems of converting gravitational waves into photons near astrophysical sources of strong magnetic fields.Of course, the background metric must be modified to suit the specific task conditions, but the inference structure and some of the qualitative findings discussed .0.6) f y /m pl → f y , (9.0.7) and introduce τ 0 to make the scale factor dimensionless a = t τ 0 .(9.0.8)

Figure 1 :
Figure 1: Verification of the numerical solution (blue line) by the analytical solution (red line) for two frequencies k 1 (left panel) and k 2 (right panel)

Figure 2 :
Figure 2: Absolute difference between the numerical solution and the analytical solution for two frequencies k 1 (left panel) and k 2 (right panel)
In equation(8.2.17)there are no terms associated with a gravitational wave.So, as expected, longitudinal EMW does not arise.From equations(8.2.15,8.2.16