Gravitational Landau Damping for massive scalar modes

We establish the existence of gravitational Landau damping for massive scalar modes in a non-collisional gas of particles, whose distribution function, assumed homogeneous and isotropic, is governed by a Vlasov equation. Then, by describing the medium with the J\"uttner-Maxwell distribution, we find that damping occurs when the wave phase velocity in the medium is subluminal, which constraints the mass of the scalar mode to be smaller than the proper frequency of the medium. The damping rate, however, calculated in the limit of wave phase velocity much larger than the particle thermal velocity, appears to be negligible on the temporal and spatial scales achievable by present instruments.


INTRODUCTION
One of the most surprising features of a plasma, considered as a dielectric medium, is the attenuation of electromagnetic waves even when collisions can be neglected. This phenomenon, known as the "Landau damping" [1], is essentially due to the presence of a long range interaction, affecting a large number of particles contained in the so called Debye sphere [2]. This situation resembles to some extent the interaction of gravitational waves with a material medium, despite two basic properties are here missing with respect to the electromagnetic counterpart. In particular, we refer to the neutralization of the background, which for a plasma is provided by the ion distribution, and to longitudinal excitations, which appear by virtue of the effective mass acquired by photons when crossing a plasma [3,4]. The first of these discrepancies can be locally overcome, since the role of the neutralizing background can be played by inertial forces. Nearby a spacetime event, indeed, Christoffel symbols associated to the background, which enter the Vlasov equation, can be made almost vanishing by choosing a local inertia frame. In the linear theory considered below, this fundamental point of view is phenomenological summarized by applying the model to a region of space where homogeneity and isotropy of the medium can be assumed, see [5][6][7] for pioneering treatments. By other words, we are considering the characteristic spatial scale of the background much larger than the wavelength of the gravitational perturbation. Concerning the absence of longitudinal modes for gravitational waves, instead, we observe that when gravitational subsystems are properly treated as molecular media [8,9] or following a hydrodynamic [10][11][12][13] and kinetic approach [14], additional polarizations can be typically excited. Although this effect is conceptually very relevant, it is in reality very small and associated to peculiar wavelengths. More intriguing, therefore, is to look at the large number of modified theories of gravity which allow for the emergence of an additional massive scalar modes, as for instance Horndesky gravity [15][16][17][18], hybrid metric-Palatini approaches [19][20][21][22][23][24][25] or massive gravity [26][27][28][29]. In this case, in fact, together with a breathing polarization in the transverse plane, scalar fields are responsible for a longitudinal polarization as well, that we expect could interact with particles of the medium. Now, it is a well established result that ordinary transverse gravitational waves are not absorbed by non-collisional massive media [30][31][32][33] (dissipation of waves in a medium of massless particles is treated in [34][35][36]) and damping is only possible if viscosity is present, as for instance in [10,37,38]. Then, having in mind that restoring longitudinal modes could imply the gravitational analog of Landau damping, we analyze, according a kinetic theory approach, the interaction of scalar waves with a particle distribution, where collisions are absent. Our original conjecture is confirmed by the below analysis and we are able to calculate the gravitational Langmuir dispersion relation, i.e. self-consisting fluctuations in the medium, as well as the amount of the imaginary part of the frequency, determining the damping. We find that such impressive feature of what makes now sense to call a "gravitational plasma" requires that the phase velocity of the scalar mode be subluminal, otherwise the typical poles inducing the Landau damping fall out the allowed domain. In particular, this condition reflects in a specific phenomenological inequality, relating the traversed medium with parameters describing the model taken into account. Moreover, we show that this is just the reason for which standard transverse gravitational wave are not absorbed, violating the condition above. Finally, we remarkably show that the damped scalar mode naturally decouples from the non-damped transverse tensor polarizations, as it emerges from linearized Vlasov system.

EVALUATION OF THE DAMPING RATE
We select a metric perturbation on a Minkowski background, i.e. g µν = η µν + h µν with η µν = diag(−1, 1, 1, 1), where h µν encloses both the tensor and the scalar excitations. In particular, we consider theories where the scalar component propagates at the linear order according the Klein-Gordon equation with both the mass of the scalar mode and the effective gravitational coupling κ ′ = 8πG ′ depending on the specific model taken into account. Here T (1) represents the perturbation of O(h) induced on the trace of the stress energy tensor by the perturbation itself. In these models [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] the transverse and traceless component of the metric perturbation can be always decoupled from the scalar excitation by introducing the generalized trace reverse tensor Here, the parameter α varies according the theory (see [18,39,40]), h = η µν h µν is the trace and the equation of motion forh µν can be rearranged in the well-known form where κ ′′ is the coupling for the tensorial degrees, which can in general differ from (1), and we can restrict the analysis to spatial indices. The medium is described by the distribution function f ( x, p, t), which evolves in time according to the Vlasov equation that we write in terms of the covariant spatial components of the momentum p m , following [33]. Hence, the variation in time of p m is simply given by being p 0 = µ 2 + g ij p i p j the particle energy and µ the particle mass. Then, in the absence of the gravitational wave, we assume the distribution function be some isotropic equilibrium solution f 0 (p) of the unperturbed equation, where where f ′ 0 (p) ≡ df0 dp . At any later time, the gravitational wave induces a dynamical perturbation δf ( x, p, t) = O(h), and we deal with the perturbed distribution function f ( x, p, t) + δf ( x, p, t). Therefore, by ignoring all contributes of O(h 2 ) we obtain the linearized Vlasov equation for the perturbation δf ( x, p, t): The dynamical system is now fully determined once the sources of (1) and (3) are evaluated in terms of f ( x, p, t), namely where d 3 p = dp 1 dp 2 dp 3 . Then, choosing the z axis to be coincident with the direction of propagation of the gravitational wave, we pursue the analysis in the Fourier-Laplace space. Specifically, we perform a Fourier transform on spatial coordinates accompanied with a Laplace transform on time coordinate t. This allows us to solve algebraically the Vlasov equation for the perturbation δf ( x, p, t), i.e.
where integrals have been conveniently restated in cylindrical coordinates by performing the substitution (p 1 , p 2 ) → (ρ, χ), with ρ 2 = p 2 1 + p 2 2 and tan(χ) = p2 p1 . By analogy with the electromagnetic theory we introduced the complex dielectric functions and the settling of damping can be inferred by the behaviour of the real and imaginary part 3 of ǫ (φ,h) (k, ω) ≡ ǫ (φ,h) (k, s = −iω), where we defined the complex frequency ω = ω r +iω i . In particular, when the condition |ω r | ≫ |ω i | holds (i.e. weak damping scenario [1, 41]), the oscillation period is much smaller than the damping time, and we properly deal with a damped wave rather than a transient. In this case, thus, the dispersion relation ω r = ω r (k) can be derived by solving while the damping factor is obtained from We assume that the equilibrium distribution function is the Jüttner-Maxwell distribution where n is the density of particles, Θ is the temperature in units of the Boltzmann constant k B and K 2 (·) is the modified Bessel function of the second kind. Then, in the presence of such a distribution relations, the dielectric functions (15)-(16) assume the form: where v p ≡ ωr k is the phase velocity. Now, by virtue of the condition |ω r | ≫ |ω i | and provided v p < 1, integrals in (20)- (21) are featured by a pair of poles on the real axis, located at the points p 3 = ± v 2 p 1−v 2 p (µ 2 + ρ 2 ). That guarantees the existence of the imaginary part for dielectric functions, stemming from the integration around poles and evaluable with the residue theorem, ultimately responsible for the appearing of the damping factor (18). We note, however, that condition v p < 1 is not a priori ensured and its attainability has to be indeed verified by calculating ω r from (17). As usually done in plasma physics [41], we calculate the Langmuir dispersion relation by expanding the denominator contained in the integrals up to the second order in p 3 , under the assumption µ 2 +ρ 2 ≪ 1. In particular, explicit calculations for tensor modes lead to where we introduced the dimensionless quantity can be seen as the proper phase velocity of the medium for tensor excitations (see [9] for a comparison). Now, the solution of (17) is given by which can be easily demonstrated to be identically greater than unity. It implies that tensor modes cannot be damped when travelling throughout the medium, as we could expect from the very beginning by virtue of their purely transverse nature and in the absence of coupling with the scalar mode in linearized equations (13)- (14). Therefore in the following we will restrict our analysis to scalar perturbations. In this case the dielectric function real part turns out to be with v 2 φ defined by analogy with the tensor case, i.e. v 2 φ ≡ ακ ′ nµ 6k 2 , and v p referring solely to the scalar polarization. Hence, solving (17) for (24), we obtain and in order to have v p < 1 the following constraint must hold This condition, by relating model depending parameters with physical quantities describing the medium, allows to select theories of gravity sensitive to damping from non-interacting ones. In this respect, it can be useful to give some realistic estimates for the cut-off implied by inequality (26): we look at the electronic fraction of the intracluster medium [42], hence we set the value 9.11 · 10 −31 kg for the particle mass, a temperature of 5.00 · 10 8 K and a density n = 1.00 · 10 −2 cm −3 . From this data we calculate for the quantity x the value 11.86, which implies for γ a value 0.89. Therefore, by taking κ ′ ≃ κ and α ≃ 1, inequality (26) gives an upper limit for the mass of the scalar mode m < 2.27 · 10 −33 eV , which is far below the more recent constraints calculated from gravitational waves observational data [43,44]. It has to be noticed that dispersion relations (23)- (25) imply superluminal group velocities in specific regions of k. In many contexts group velocity is associated with energy and information transport, therefore a superluminal behavior would imply causality violation, but this is not in our case: We recall that we have perturbed the medium with a wave which is non-null at the initial time t = 0 but identically vanishes for any negative time. The discontinuity at the initial time causes the emergence of a front wave, namely a high frequency Sommerfeld precursor [45], propagating at the speed of light. It is a well established result [46] that, in such cases, energy and information transport is associated to the front wave propagation. Then, we calculate by means of the residue theorem the imaginary part of the dielectric function, resulting in which inserted in (18) gives us finally ω i , that is We emphasize that by virtue of (25) the condition ω i < 0 identically holds, and the theory is devoid of instabilities attributable to enhancement phenomena. Now, if we expand formula (25) in the limit of short wavelengths, i.e.
and we require that the damping time constant τ = |ω i | −1 be much smaller than the age of the Universe we get that the wavenumber must obey k ≪ 10 −28 m −1 , which implies a wavelength of the signal about 10 orders of magnitude larger than the largest wavelength observable with current instruments, namely pulsar timing arrays [47,48].

CONCLUDING REMARKS
The analysis above establishes a new point of view about the interaction of gravitational waves with physical media: In the presence of a longitudinal scalar mode, even non-collisional ensembles of particles are able to damp the additional polarizations typical of extended models of gravity. In this respect, however, we stress that theories affected by such a Landau damping are indeed characterized by very low massive modes and the damping rate is actually negligible for realistic astrophysical scenarios. That is attributable to the specific regime considered, i.e. phase velocities much larger than thermal ones, where analytic treatment is feasible but damping exponentially depressed. We expect, therefore, numerical investigations around thermal velocity to highlight more robust effects, where quasilinear phenomena should take place and the decay of massive mode be observable. This suggests, moreover, to look at primordial stages of cosmological settings, where the possibility of having simultaneously high temperature and density could lead to a damping rate for sub-horizon modes comparable with the Universe age.