Stability of stealth magnetic field in de Sitter spacetime

A detailed stability analysis is presented for the de Sitter solution with a homogeneous magnetic field that was recently found in the context of a $U(1)$ gauge theory nonminimally coupled to scalar-tensor gravity. The magnetic field is `stealth' in the sense that the corresponding stress-energy tensor is of the form of an effective cosmological constant and thus is isotropic despite the fact that the magnetic field has a preferred spatial direction. We study the stability of the solution against linear perturbations in the subhorizon and superhorizon limits. We then present some explicit examples that satisfy all stability conditions. The stable de Sitter solution with a homogeneous magnetic field opens up a new possibility for inflationary magnetogenesis, in which magnetic fields in the Universe at all scales may originate from a classical, homogeneous magnetic field sustained during inflation.


I. INTRODUCTION
The origin of magnetic fields in the Universe at various scales is one of the mysteries in modern cosmology. There thus have been a large number of attempts to find a theoretical framework in which magnetic fields in the Universe are generated in the course of the cosmic history (see Refs. [1][2][3][4] for reviews). The author recently proposed a new scenario of inflationary magnetogenesis in which magnetic fields in the Universe at all scales may originate from a classical, homogeneous magnetic field sustained during inflation [5]. It was found that a Uð1Þ gauge theory nonminimally coupled to scalar-tensor gravity admits a cosmological attractor solution that represents a de Sitter universe with a homogeneous magnetic field, fully taking into account the backreaction of the magnetic field on the geometry.
In the standard Einstein-Maxwell theory, a homogeneous magnetic field would introduce anisotropies in the geometry through the stress-energy tensor, making it impossible to admit a de Sitter solution. One thus needs to modify the standard Einstein-Maxwell theory one way or another to make a homogeneous magnetic field and an isotropic expansion of the Universe consistent with each other. In the solution found Ref. [5], a homogeneous magnetic field has a preferred spatial direction, but its stress-energy tensor is isotropic and is of the form of an effective cosmological constant. In this sense, the magnetic field in the solution is "stealth." The basic idea behind the model of Ref. [5] is as follows. For simplicity, let us first consider an action for a Uð1Þ gauge field A μ of the form where F μν ¼ ∂ μ A ν − ∂ ν A μ is the field strength and g μν is the spacetime metric. By taking the variation of the action with respect to g μν , one obtains the corresponding stressenergy tensor as If f 0 ðwÞ ¼ 0, then the stress-energy tensor is proportional to g μν and thus admits an exact de Sitter solution. For example, this would be the case if fðwÞ ¼ f 0 þ f 2 ðw − w 0 Þ 2 with f 0 , f 2 , and w 0 constant and if w ¼ w 0 . However, in Friedmann-Lemaître-Robertson-Walker (FLRW) backgrounds (including the de Sitter spacetime), w for a homogeneous magnetic field decays as 1=a 4 , where a is the scale factor, and thus the only constant value of w that is consistent with the expansion of the Universe is zero. For this reason, this simple action does not work. Let us next consider an action of the form where ϕ is a scalar field and it is understood that some kinetic terms for ϕ are also added to the action. In this case, the corresponding stress-energy tensor is again proportional to g μν if f 0 ðWÞ ¼ 0. The main difference from the previous case is that W for a homogeneous magnetic field is proportional to e 2ϕ =a 4 in FLRW backgrounds and thus is constant if e ϕ ∝ a 2 . This is the basic idea behind the model of Ref. [5]. The model action of Ref. [5], however, includes not only a nonlinear function of W but also other terms such as Horndeski's nonminimal vector coupling and the Horndeski scalar terms. This is because, while the simple action (1.3) supplemented by kinetic terms for ϕ in principle admits a de Sitter solution with a homogeneous magnetic field, the stability of such a solution requires inclusion of additional terms in the action.
The purpose of the present paper is to show a detailed stability analysis for the solution representing a de Sitter spacetime with a stealth homogeneous magnetic field. In particular, we study the stability of the solution against linear perturbations in the subhorizon and superhorizon limits. We show that the stability of the solution requires inclusion of Horndeski's nonminimal vector coupling as well as the Horndeski scalar terms. We then present some explicit examples that satisfy all stability conditions. The rest of the present paper is organized as follows. In Sec. II, we briefly review the model and the solutions studied in Ref. [5], focusing on those without the electric field. In Sec. III, we formulate the linear perturbation analysis and study the stability of the de Sitter solution with the stealth magnetic field against subhorizon perturbations. In Sec. IV, we then study the stability of the solution against superhorizon perturbations. In Sec. V, after showing a couple of classes of models that are unstable, we present explicit models that satisfy all stability conditions. Section VI is then devoted to a summary of the paper and some discussions. Appendixes A-D show explicit expressions of some matrices and coefficients.

II. REVIEW OF THE MODEL
In this section, we briefly review the model and the de Sitter solution with a homogeneous magnetic field studied in Ref. [5].

A. Action
The model consists of a metric g μν , a Uð1Þ gauge field A μ and a scalar field ϕ, described by the action where L ¼ LðX; W; Y; ZÞ is an arbitrary function of are shift-symmetric Horndeski scalar terms [6,7]; and is a simple modification (F μν →F μν ) of the Horndeski's nonminimal coupling of the Uð1Þ gauge field to the Riemann tensor R μ νρσ of the metric g μν [8]. Here, the scalar field ϕ is normalized so that its mass dimension is zero, G 3;4;5 ðXÞ are arbitrary functions of X, the subscript X denotes derivative with respect to X, and ξ is an arbitrary constant.
The action is diffeomorphism invariant and enjoys the Uð1Þ gauge symmetry where λ is an arbitrary function. Furthermore, the action respects the global symmetry where ϕ 0 is an arbitrary constant. We assume that the function LðX; W; Y; ZÞ is even with respect to Y, LðX; W; Y; ZÞ ¼ LðX; W; −Y; ZÞ: ð2:8Þ This assumption ensures that the equations of motion admit a solution without the electric field [5].

B. Bianchi I solution with magnetic field
We first consider a Bianchi I spacetime where an overdot represents a derivative with respect to t. We further assume that the scalar field is homogeneous, ϕ ¼ ϕðtÞ, and has a constant "velocity" as and that the Uð1Þ gauge field represents a homogeneous magnetic field as where B is a constant. Assuming that not only H 0 , Σ 0 , B, and _ ϕ=N but also χ ≡ e ϕ =a 2 are constant [see the sentences just after (1.3) for the motivation for this extra assumption] and rescaling the spatial coordinates so that χ ¼ 1, the equations of motion are reduced to where b ≡ B=H 0 and s ≡ Σ 0 =H 0 . Hereafter, subscripts W, Y, and Z represent derivatives with respect to them. Since Y ¼ 0 for the ansatz (2.12) and we have assumed (2.8), the first equation (2.13) is automatically satisfied. Therefore, we have three algebraic equations (2.14) to be solved with respect to the three parameters (H 0 , b, s) of the ansatz. Generically, they admit a solution (or a set of solutions).

C. De Sitter solution with magnetic field
By fine tuning one of parameters in the action, one can then take the limit s → 0 so that the spacetime becomes a de Sitter, g μν dx μ dx ν ¼ −NðtÞ 2 dt 2 þ aðtÞ 2 ðdx 2 þ dy 2 þ dz 2 Þ: ð2:15Þ Then, under the assumption (2.8), the independent equations of motion are These are three algebraic equations. One of the three equations (or one combination of them) simply represents the fine-tuning of one of parameters in the action, and the remaining two equations can generically be solved with respect to the two parameters (H 0 , b) of the ansatz. If we relax the fine-tuning without abandoning the discrete symmetry (2.8), then the solution goes back to the Bianchi I solution with the three parameters (H 0 , b, s) described in the previous subsection.

D. Attractor condition
Reference [5] found the necessary and sufficient condition under which the de Sitter solution with a homogeneous magnetic field introduced in Sec. II C is stable against homogeneous linear perturbations, under the assumption (2.8). The stability condition, or the attractor condition, is A=N > 0, where We shall later see that the absence of ghosts implies that N > 0. Therefore, the attractor condition is ð2:20Þ

III. LINEAR PERTURBATION
In this section, we consider inhomogeneous linear perturbations around the de Sitter attractor solution with a homogeneous magnetic field (H 0 > 0, s ¼ 0, b ≠ 0) and seek the no-ghost conditions and the sound speeds in the subhorizon limit.

A. Decomposition of linear perturbation
Around the general background introduced in Sec. II B, we introduce the components Φ, V i and h ij (i, j ¼ 1, 2, 3) of the metric perturbation δg μν , the components δA a (a ¼ 0; …; 3) of the perturbation of the vector field δA μ , and the perturbation π of the scalar field ϕ as and the Einstein's summation rule is employed. We then decompose each component of the perturbation into spatial Fourier modes. For this purpose, we first introduce scalar harmonics Y n (n ¼ cc, cs, sc, ss) as Y cc ¼ cosðk x xÞ cosðk y y þ k z zÞ; Y cs ¼ cosðk x xÞ sinðk y y þ k z zÞ; Y sc ¼ sinðk x xÞ cosðk y y þ k z zÞ; Y ss ¼ sinðk x xÞ sinðk y y þ k z zÞ; ð3:3Þ and then define odd and even vector harmonics Y odd=even n;p (n ¼ cc, cs, sc, ss; p, q ¼ 2, 3) as , and odd and even tensor harmonics Y odd=even n;pq The components of the perturbations are then decomposed as 3 ðh odd 1;n Y odd n;p þ β n Y even n;p Þ; 3 ðA odd n Y odd n;p þ A even n Y even n;p Þ; 3 ðh odd n Y odd n;pq þ E n Y even n;pq þ τ n Y n δ pq Þ; ð3:6Þ where p, q ¼ 2, 3 and the Einstein's summation rule for n ¼ cc, cs, sc, ss is understood. We fix gauge freedom associated with the spacetime coordinate transformation and the Uð1Þ gauge transformation as

B. General quadratic action
Substituting the decomposition of perturbations to the action, imposing the gauge condition, and expanding the action up to the quadratic order in perturbations, we obtain the quadratic action for the linear perturbations around a general background introduced in Sec. II B as where the superscript ⊺ represents the transpose operation, represents dynamical degrees of freedom, , and the Einstein's summation rule for n ¼ cc, cs, sc, ss is again understood. Important points are that the matrices K, M, V, A, B, and C are common for all n ¼ cc, cs, sc, ss and that there are no couplings between (Y n , Z n ) and (Y n 0 , Z n 0 ) with n ≠ n 0 . In general, however, even and odd sectors within (Y n , Z n ) couple with each other.
Integrating out the nondynamical variables Z n , one obtains the quadratic action for the dynamical variables Y n as It is then straightforward to diagonalize the kinetic matrixK as ð3:15Þ . Correspondingly, the friction matrixM and the mass matrixV transform as The quadratic action is now written as where the new variablesỸ n are related to the original variables Y n via Y n ¼ UỸ n .
C. Quadratic action around de Sitter background with magnetic field In the de Sitter background with a homogeneous magnetic field, the matrices in (3.8) are greatly simplified, and all their components are written in terms of p ⊥ ≡ k ⊥ =a, p x ≡ k x =a, b, g h , g 4 , and ζ α (α ¼ 1; …; 7), where g h and ζ α are defined in (2.18) and Here, l z and l yy are defined as where it is understood that the left-hand sides of (3.20) are evaluated at the de Sitter solution with a homogeneous magnetic field. It is convenient to decompose the matrices into submatrices as where explicit expressions for the submatrices are given in Appendix A.
If b ¼ 0 (we have already set s ¼ 0), then, since L is assumed to be even with respect to Y in (2.8), the theory and the background respect the parity invariance, and as a result, the mixing matrices M mix , V mix , A mix I , A mix II , B mix , and C mix vanish. In general, they do not vanish, and thus the even and odd perturbations do mix.
The quadratic action for the dynamical variables Y n is then given by the formulas (3.11)-(3.13). After diagonalization of the kinetic matrix, one obtains (3.18) with (3.14)-(3.17).

D. Subhorizon limit
For the theoretical consistency, one needs to analyze the stability of the background in the UV, i.e., in the subhorizon limit This ensures the absence of instability of which the timescale is parametrically shorter than the cosmological timescale ∼1=H 0 . The stability of the solution against perturbations with p 2 ≪ H 2 0 shall be studied in the next section.
In the subhorizon limit p 2 ≫ H 2 0 , introducing a small bookkeeping parameter ϵ so that H 2 0 =p 2 ¼ Oðϵ 2 Þ, components of the matrices in the quadratic action (3.18) are simplified as

E. No-ghost conditions
All componentsK 1;2;3;4;5 of the diagonalized kinetic matrix in the subhorizon limit are positive for ∀ p ⊥ and ∀ p x such that p 2 ≠ 0, if and only if N a > 0 (a ¼ 1; …; 7), where and Here, the positivity of the last three N 5;6;7 is necessary and sufficient for c 1 p 4 ⊥ þ 2c 2 p 2 ⊥ p 2 x þ N p 4 x to be positive for ∀ p ⊥ and ∀ p x such that p 2 ≠ 0.
One can actually show that Therefore, the positivity of N 6 and N 7 follows from other conditions, and the no-ghost condition is simply N a > 0; ða ¼ 1; …; 5Þ: ð3:27Þ

F. Sound speeds
The squared sound speeds c 2 s of the five modes are determined by and S 1;…;11 are given in Appendix C.

IV. LONG WAVELENGTH PERTURBATIONS
In the previous section, we studied the stability of the de Sitter solution with a homogeneous magnetic field in the UV limit, i.e., the limit where the wavelengths of perturbations are sufficiently shorter than the size of the de Sitter horizon. In this section, we study the stability of the solution in the opposite limit, namely, the limit where the wavelengths of perturbations are sufficiently longer than the size of the horizon.
As argued in Ref. [9], instabilities for modes with p 2 ¼ OðH 2 0 Þ can be as harmless as the standard Jeans instability. It is nonetheless necessary to impose the stability for those modes with p 2 ≪ H 2 0 if one wants the background under investigation to be realized dynamically as an attractor of the system. In general relativity, the standard Jeans instability prevents the matter-dominated FLRW background from being an attractor of the system, but this is not a problem since it is not the matter-dominated epoch but the inflationary epoch that sets the initial condition of our local patch of the Universe. On the other hand, for the system under investigation in the present paper, we would like the homogeneous magnetic field to be realized dynamically and to be sustained during a de Sitter phase representing the inflationary epoch. For this reason, we require the solution to be an attractor of the system. Namely, we require the solution to be stable against perturbations with p 2 ≪ H 2 0 . For this purpose, we send p ⊥ =H 0 and p x =H 0 to zero after obtaining equations of motion for properly normalized dynamical variables. Since the vector harmonics Y odd=even n;p (n ¼ cc, cs, sc, ss; p, q ¼ 2, 3) vanish in the limit and Y odd=even n;p =k ⊥ remain finite, we introduceh odd The stability of the background solution against anisotropic perturbations with long wavelengths can then be analyzed by studying the equations of motion for the dynamical variables (h odd 1;n , A 1;n , π n , ψ n ,Ã odd n ) in the limit where p ⊥ =H 0 and p x =H 0 are sent to zero, after eliminating nondynamical variables. For simplicity, we set NðtÞ ¼ 1 in the rest of this section.
The equation of motion forh odd 1;n in the long wavelength limit does not involve other variables and ish where an overdot represents a derivative with respect to the time variable t. This gives the solutioñ where C 1 and C 2 are constants. The first solution, which is constant in space and time, is actually a pure gauge. Therefore,h odd 1;n is stable. The equation of motion for A 1;n in the long wavelength limit also does not involve other variables and is which gives the solution where C 3 and C 4 are constants. Therefore, A 1;n is stable. The equations of motion for π n and ψ n in the long wavelength limit are coupled but can be solved easily by setting where π 0 n and ψ 0 n are constants. The general solution is of course a linear combination of solutions of this form for all allowed values of Γ. By substituting (4.5) to the equations of motion for π n and ψ n , one obtains where A and N are the same as those defined in (2.17) and thus in particular N ¼ N 5 [see (3.24)]. For Γ ¼ 0, π 0 n ¼ 0, and ψ n is a pure gauge. Therefore, π n and ψ n are also stable, provided that the attractor condition (2.20) and the no-ghost condition (3.27) are satisfied.
The equation of motion forÃ odd n in the long wavelength limit does not involve other variables and isÃ where A 1;2;3 are given in Appendix D. Therefore,Ã odd n is stable if and only if A a > 0; ða ¼ 1; 2; 3Þ: ð4:8Þ

V. EXAMPLES
In this section, we first show a couple of classes of models that always violate either (3.27) or (3.32). After that, we show some concrete examples of stable models with specific choices of parameters, which satisfy not only (3.27) and (3.32) but also (3.31), (2.20), and (4.8).

A. Unstable classes of models
Let us first consider the case without Horndeski's nonminimal vector coupling (ξ ¼ 0). Setting g h ¼ 0, we obtain S 2 ¼ 0. This implies that α 2 ¼ 0 and thus one of the five modes has vanishing sound speed.
These two unstable examples imply that the stability requires both Horndeski's nonminimal vector coupling and Horndeski scalar terms.

B. Stable models
Our strategy here is to find a set of parameters that satisfies (3.27) and (3.32) and then to check if (3.31), (2.20), and (4.8) are also satisfied.
We have already assumed (2.8). For simplicity, in all the examples considered in this subsection, we further set ð5:1Þ In each of the following examples, we specify (g h , g 4 , g 4x , g 3x , b, l xx , l ww ). We then solve the background equations of motion (2.16) with respect to (l, l x , l w ), where and it is understood that the left-hand sides of (5.2) are evaluated at the de Sitter solution with a homogeneous magnetic field. After that, we confirm that (3.27) and (3.32) as well as (3.31), (2.20), and (4.8) are satisfied. For the set of parameters (g h , g 4 , g 4x , g 3x , b, l xx , l ww ; l, l x , l w ) and the assumptions (2.8) and (5.1), one can easily reconstruct the Lagrangian by assuming a simple ansatz and noting that X ¼ 2H 2 0 and W ¼ −b 2 H 2 0 =2 on the de Sitter solution parametrized by (H 0 , b). For example, if we assume that L is quadratic in X and W and independent of Y and Z, then ðl xx X 2 þ l ww W 2 Þ: ð5:3Þ The simplest choice of G 3;4;5 is The value of ξ is ð5:5Þ

Example 1
Let us consider the following choice of parameters, for which the background equations of motion (2.16) give In this case, we have After substituting (5.9) into (3.29) to obtain explicit expressions of α 1;…;5 , it is straightforward to compute the three expressions on the left-hand sides of the inequalities in (3.31) and to show that they all have the form A=B, where A and B are polynomials of p 2 ⊥ and p 2 x with positive coefficients. Therefore, Eq. (3.31) is also satisfied.

Example 2
Let us consider the following choice of parameters.
for which the background equations of motion (2.16) give the same values of l, l x , and l w as shown in (5.7).
In this case, we have After substituting (5.18) into (3.29) to obtain explicit expressions of α 1;…;5 , it is straightforward to compute the three expressions on the left-hand sides of the inequalities in (3.31) and to show that they all have the form A=B, where A and B are polynomials of p 2 ⊥ and p 2 x with positive coefficients. Therefore, Eq. (3.31) is also satisfied.

Example 3
Let us consider the following choice of parameters, for which the background equations of motion (2.16) give In this case, we have After substituting (5.13) into (3.29) to obtain explicit expressions of α 1;…;5 , it is straightforward to compute the three expressions on the left-hand sides of the inequalities in (3.31) and to show that they all have the form A=B, where A and B are polynomials of p 2 ⊥ and p 2 x with positive coefficients. Therefore, Eq. (3.31) is also satisfied.

Example 4
Let us consider the following choice of parameters, After substituting (5.23) into (3.29) to obtain explicit expressions of α 1;…;5 , it is straightforward to compute the three expressions on the left-hand sides of the inequalities in (3.31) and to show that they all have the form A=B, where A and B are polynomials of p 2 ⊥ and p 2 x with positive coefficients. Therefore, Eq. (3.31) is also satisfied.

VI. SUMMARY AND DISCUSSION
We have presented a detailed stability analysis for the de Sitter solution with a homogeneous magnetic field that was recently found in Ref. [5] in the context of a Uð1Þ gauge theory nonminimally coupled to scalar-tensor gravity. The magnetic field is stealth in the sense that the corresponding stress-energy tensor is of the form of an effective cosmological constant and thus is isotropic despite the fact that the magnetic field has a preferred spatial direction. We have studied the stability of the solution against linear perturbations in the subhorizon and superhorizon limits and have shown some explicit examples that satisfy all stability conditions. Stable models include both Horndeski's nonminimal vector coupling and the Horndeski scalar terms.
The stable de Sitter solution with a homogeneous magnetic field opens up a new possibility for inflationary magnetogenesis, in which magnetic fields in the Universe at all scales may originate from a classical, homogeneous magnetic field sustained during inflation. Toward such a new scenario of inflationary magnetogenesis, an important step forward is to show a graceful exit from the de Sitter solution with constant values of (X, W, Y, Z, χ). In particular, the scalar field ϕ should be stabilized around a local minimum of its potential with a sufficiently large mass at late time after inflation in order to recover the standard Einstein-Maxwell theory. (To be more precise, what is recovered is the Einstein-Maxwell theory with the Horndeski's nonminimal vector coupling.) Also, as already pointed out in Ref. [5], a field other than ϕ needs to be introduced as the main source of curvature perturbation, i.e., either an inflaton or a curvaton, in order to avoid too large statistical anisotropies and non-Gaussianities. One eventually needs to establish the stability of the whole system all the way from the inflationary epoch to the standard expansion history at late time through the graceful exit. While the stability in the superhorizon limit is not necessarily required after inflation (see the second paragraph of Sec. IV), the stability in the subhorizon limit is required not only during but also after inflation. The present paper has established the stability of the inflationary epoch and thus can be considered as the first step toward the new inflationary magnetogenesis scenario based on a classical, homogeneous magnetic field sustained during inflation.
The recent multimessenger detection of binary neutron stars put a strong constraint on the deviation of the propagation speed of gravitational waves from that of light at the present [10,11]. If the Horndeski scalar-tensor theory is used as a model of the late-time acceleration of the Universe then the functions G 4 and G 5 are strongly constrained. On the other hand, the propagation speed of gravitational waves in the early Universe such as the inflationary epoch is not constrained at all by such observations. Therefore, the model considered in Ref. [5] and in the present paper is consistent with the multimessenger detection of binary neutron stars as far as the scalar field is stabilized around a local minimum of its potential with a sufficiently large mass at late time after inflation, as required anyway for the recovery of the Einstein-Maxwell theory (with Horndeski's nonminimal vector coupling).
After inflation and the stabilization of ϕ around a local minimum of its potential, the homogeneous magnetic field is no longer stealth, and thus the background spacetime is then described by a Bianchi I geometry (instead of a FLRW one). It is therefore important to investigate phenomenology of the Bianchi I universe with a homogeneous magnetic field and the standard content of the universe (radiation, matter, and the cosmological constant) in the context of the Einstein-Maxwell theory with Horndeski's nonminimal vector coupling [12].

ACKNOWLEDGMENTS
The author thanks Axel Brandenburg, Ruth Durrer, Yiwen Huang, Tina Kahniashvili, and Sayan Mandal for helpful discussions. A part of the work was completed during the author's visits to Simon Fraser University, University of Alberta, and University of Victoria. The author is therefore grateful to Andrei Frolov, Valeri Frolov, Werner Israel, Maxim Pospelov, and Andrei Zelnikov for their warm hospitality. The work was supported in part by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grants No. 17H02890 and No. 17H06359 and by World Premier International Research Center Initiative (WPI), MEXT, Japan. ðA16Þ where