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Physics & Astronomy International Journal

Research Article Volume 4 Issue 1

Dark matter structure formation in the presence of gravitational fluctuations

Catarina Bastos, Hugo Tercas

1GoLP/Instituto de Plasmas e Fusao Nuclear, Portugal
2Instituto de Plasmas e Fusao Nuclear, Portugal
3Instituto de Telecomunicacoes, Portugal

Correspondence: Catarina Bastos, GoLP/Instituto de Plasmas e Fusao Nuclear, Lisbon, Portugal

Received: December 20, 2019 | Published: January 7, 2020

Citation: Bastos C, Tercas H. Dark matter structure formation in the presence of gravitational fluctuations. Phys Astron Int J. 2020;4(1):5‒9. DOI: 10.15406/paij.2020.04.00196

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Abstract

We present the coupling between a gravitational wave in a Minkowski spacetime with dark matter modelled by a self-interacting complex scalar field. In flat spacetime, quantum fluctuations in dark matter, as described as a Bose-Einstein condensate (BEC), are stable and display a relativistic Bogoliubov dispersion relation. In the weak gravitational field limit, both relativistic and nonrelativistic models self-gravitating dark matter suggest the formation of structures as the result of a dynamical (Jeans) instability. In this work we show that in the presence of spontaneous symmetry breaking of the dark matter field, the gravitational wave is damped for wave-lengths larger than the Jeans length. Such energy is converted to the Bogoliubov modes of the BEC that in their turn become unstable and grow, leading to the formation of structures even in the absence of expansion. Remarkably, this compensated attenuation/amplification mechanism is the signature of a discrete PT symmetry-breaking of the system.

PACS numbers: 70.-a, 42.50.Nn, 42.50.Wk, 71.36.+c

Keywords: Dark matter, Bose-Einstein condensate, spontaneous symmetry breaking, gravitational wave, Jeans length

Abbreviations

KG, klein-gordon; KGE, klein-gordon-einstein; DM, dark matter; GPP, gross-pitaevskiipoisson; SP, schrodinger-poisson; BECs, bose-einstein condensates; CDM, cold dark matter

Introduction

Scalar field theories in curved spacetimes are on the basis of modern advances in cosmology and astrophysics,1–3 as they constitute important candidates to explain the behavior of dark-matter.4–7 Their relativistic dynamics is governed by the Klein-Gordon (KG) equation,8 historically emerging as a first attempt to unify quantum mechanics with the special relativity theory, to obtain a unified theory to explain our universe. The Klein-Gordon-Einstein (KGE) equations, that involve a coupling between the KG equations and gravity, were first used to study boson stars.9 Although, the first models of these stars do not assumed that bosons could have a self-interaction potential, it was shown then that self-interaction can significantly change the physical dimensions of the boson stars and make them clearly more interesting as an astrophysical object.10 Furthermore, models of dark matter (DM) halos were proposed based on scalar fields that are described by KGE equations.6 These DM halos can be explained through Schrodinger-Poisson (SP) equation or Gross-PitaevskiiPoisson (GPP) equations, since the Newtonian limit is valid at the galactic scale. In this sense, we can think about DM halos as gigantic quantum objects made of Bose-Einstein condensates (BECs).

Furthermore, these models are a tentative of solving the problem of Cold Dark Matter (CDM), as the wave properties of bosons can stabilize the system against gravitational collapse. At the cosmological level it is quite important to study the implications of these scalar field models. It was shown by Matos et al.11 that when a spatially homogenous interacting real scalar field competes with baryonic matter, radiation and dark energy in terms of cosmological evolution, these real scalar fields can reproduce quite well the cosmological predictions of the Λ-CDM model. A perturbative analysis then showed the formation of structures corresponding to DM halos. Finally, Chavanis4 has considered the case of a complex self-interacting scalar field in the context of Newtonian cosmology and based on the GPP equations. The formation of structures has been recently studied through the Jeans instability of an homogeneous self-gravitating BEC in a static background.13 Basically, the so-called BECDM have shown that perturbations grow faster than in a Λ-CDM model. Some relativistic models have been then analyzed.4,14

In the last year, gravitational waves (i.e. fluctuations in the metric) generated by accelerated mass distributions, like massive black holes, were finally detected by the LIGO collaboration.15 The theory of general relativity predicts that the amplitude of these gravitational waves is extremely small, which harnessed their detection for a long time. Although gravitational waves produced by black hole collisions could be detected, finding experimental evidence of primordial gravitational waves remains elusive. Nevertheless, the advent of table-top, high-sensitivity devices based on quantum technologies revived their interest. Also, ESA is now developing the eLiSA, a space-based interferometer that will be used to detect gravitational waves in other range of frequencies, the low-frequency band.16 It is expected that it could detect waves coming from other sources rather than merging black holes. On the other hand, a lot of attention has been drawn to the study of space-time effects in quantum systems as, for instance, in phononic fields.17,18 It was shown in that gravitational waves can create phonons in a BEC,18 a features that is motivating a new generation of gravitational-wave detectors using matter waves, which may become a reality in a medium-term timescale.

In this work, we investigate the self-consistent dynamics of an interacting complex scalar field (BEC) - described by a nonlinear Klein-Gordon equation - evolving in a fluctuating space-time. In particular, we show that gravitational waves (obtained from Einstein’s equation in the weak field limit) can couple to scalar field fluctuations, leading the dynamics of the latter unstable. Remarkably, the present instability mechanism appears to be associated with the violation of the discrete parity-time (PT) symmetry. The latter, initially proposed as a concept in quantum mechanics,19 is now being extensively studied in optics20–22 and, more recently, in acoustics.23 In such systems, any spatial region with a loss is mirrored by a region of gain. Therefore, the processes of light (or sound) absorption and amplification can be compensated, and the frequencies of the eigen optical (acoustic) modes can be real.

When PT-symmetry is broken, the eigenmodes appear in complex conjugate pairs. In our case, complex eigenmodes appear for hybrid modes made of the mixture between gravity and Bogoliubov (sound) BEC modes. This suggests that the formation of structures, as described by the Jeans self-gravitating instability, due to primordial gravitational waves is a consequence of the breaking of the U(1)×PT symmetry. Our findings show that the gravitational wave is damped for wave lengths larger than the Jeans length and the energy is converted to the Bogoliubov modes of the BEC, which grow in time. This will turn the system unstable, leading to the formation of primordial cosmological structures even in the absence of an expanding universe. Moreover, we argue that this particular form of the space-time−field interaction may be an important mechanism preventing the detection of primordial gravitational waves, as their energy is transferred to the matter field originating structures in the universe. Minimally coupled theory. The dynamics of a complex scalar field (SF) ϕ(xµ) in a curved spacetime of curvature R is governed by the following minimal-coupling action

S= c 4 16πG d 4 x g R+ d 4 x g Lφ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaeyypa0tcfa4aaSaaaOqaaKqzGeGaam4yaKqbaoaaCaaabeqaaKqz adGaaGinaaaaaOqaaKqzGeGaaGymaiaaiAdacqaHapaCcaWGhbaaaK qbaoaapeaakeaajugibiaadsgajuaGdaahaaqabeaajugWaiaaisda aaqcLbsacaWG4bqcfa4aaOaaaOqaaKqzGeGaeyOeI0Iaam4zaaWcbe aajugibiaadkfacqGHRaWkjuaGdaWdbaGcbaqcLbsacaWGKbqcfa4a aWbaaeqabaqcLbmacaaI0aaaaKqzGeGaamiEaKqbaoaakaaakeaaju gibiabgkHiTiaadEgaaSqabaqcLbsacaWGmbGaeqOXdOMaaiilaaWc beqabKqzGeGaey4kIipaaSqabeqajugibiabgUIiYdaaaa@5F70@   (1)

where x μ =( ct,x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG4b qcfa4aaSbaaeaajugWaiabeY7aTbqcfayabaqcLbsacqGH9aqpjuaG daqadaqaaKqzGeGaeyOeI0Iaam4yaiaadshacaGGSaGaamiEaaqcfa OaayjkaiaawMcaaaaa@44E7@ is the four vector, g μυ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGNb qcfa4aaSbaaeaajugWaiabeY7aTjabew8a1bqcfayabaaaaa@3D59@ is the metric tensor, g= g μ μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGNb Gaeyypa0Jaam4zaKqbaoaaDaaabaqcLbmacqaH8oqBaKqbagaajugW aiabeY7aTbaaaaa@4069@  denotes its trace, and Lϕ is the SF Lagrangian

L φ = 1 2 g μν μ φ * ν φV( | φ | 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadYeajuaGdaWgaaqaaKqzadGaeqOXdOgajuaGbeaacqGH 9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadEgadaahaaqabeaaju gWaiabeY7aTjabe27aUbaajuaGcqGHciITdaWgaaqaaKqzadGaeqiV d0gajuaGbeaacqaHgpGAdaahaaqabeaajugWaiaacQcaaaqcfaOaey OaIy7aaSbaaeaajugWaiabe27aUbqcfayabaGaeqOXdOMaeyOeI0Ia amOvamaabmaabaWaaqWaaeaacqaHgpGAaiaawEa7caGLiWoadaahaa qabeaajugWaiaaikdaaaaajuaGcaGLOaGaayzkaaGaaiOlaaaa@5FD9@   (2)

Here, V( | φ | 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb qcfa4aaeWaaOqaaKqbaoaaemaakeaajugibiabeA8aQbGccaGLhWUa ayjcSdqcfa4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcacaGLPa aaaaa@4240@ contains the KG rest mass term and the self-interaction potential,

V( | φ | 2 )= m 2 c 2 2 2 | φ | 2 + 1 4 λ | φ | 4 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb qcfa4aaeWaaOqaaKqbaoaaemaakeaajugibiabeA8aQbGccaGLhWUa ayjcSdqcfa4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcacaGLPa aajugibiabg2da9Kqbaoaalaaakeaajugibiaad2gajuaGdaahaaWc beqaaKqzadGaaGOmaaaajugibiaadogajuaGdaahaaWcbeqaaKqzad GaaGOmaaaaaOqaaKqzGeGaaGOmaiabl+qiOLqbaoaaCaaaleqabaqc LbmacaaIYaaaaaaajuaGdaabdaGcbaqcLbsacqaHgpGAaOGaay5bSl aawIa7aKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey4kaSsc fa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaI0aaaaiabeU7aSL qbaoaaemaakeaajugibiabeA8aQbGccaGLhWUaayjcSdqcfa4aaWba aSqabeaajugWaiaaisdaaaqcLbsacaGGSaaaaa@6AFB@   (3)

where λ=8π a s m/ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH7o aBcqGH9aqpcaaI4aGaeqiWdaNaaiyyaKqbaoaaBaaabaqcLbmacaGG ZbaajuaGbeaacaWGTbGaai4laiabl+qiOnaaCaaabeqaaKqzadGaaG Omaaaaaaa@44DF@ is the coupling constant, as is the scattering length and m is the field mass. The minimization of Eq. (1) with respect to φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAaaa@3842@ provides the EulerLagrange equation μ [ L φ ( μ φ ) * L φ φ * ]=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0juaGdaWgaaqaaKqzadGaeqiVd0gajuaGbeaadaWadaqaamaalaaa baGaeyOaIyRaamitamaaBaaabaqcLbmacqaHgpGAaKqbagqaaaqaai abgkGi2oaabmaabaGaeyOaIy7aaSbaaeaajugWaiabeY7aTbqcfaya baGaeqOXdOgacaGLOaGaayzkaaWaaWbaaeqabaqcLbmacaGGQaaaaa aajuaGcqGHsisldaWcaaqaaiabgkGi2kaadYeadaWgaaqaaKqzadGa eqOXdOgajuaGbeaaaeaacqGHciITcqaHgpGAdaahaaqabeaajugWai aacQcaaaaaaaqcfaOaay5waiaaw2faaiabg2da9iaaicdacaGGSaaa aa@5F1D@ which in turn yields the following generalized KG equation

gμ+V'( | φ | 2 ), φ * =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqWIHw YvcaGGNbGaeqiVd0Maey4kaSIaaiOvaiaacEcajuaGdaqadaGcbaqc fa4aaqWaaOqaaKqzGeGaeqOXdOgakiaawEa7caGLiWoajuaGdaahaa WcbeqaaKqzadGaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGaaiilaiab eA8aQLqbaoaaCaaabeqaaKqzadGaaiOkaaaajugibiabg2da9iaaic daaaa@4FF2@   (4)

where g g μν μ ν g μν Γ μν α α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqWIHw YvjuaGdaWgaaqaaKqzadGaam4zaaqcfayabaGaeyyyIOBcLbsacaWG Nbqcfa4aaWbaaeqabaqcLbmacqaH8oqBcqaH9oGBaaqcLbsacqGHci ITjuaGdaWgaaqaaKqzadGaeqiVd0gajuaGbeaajugibiabgkGi2Mqb aoaaBaaabaqcLbmacqaH9oGBaKqbagqaaKqzGeGaeyOeI0Iaam4zaK qbaoaaCaaabeqaaKqzadGaeqiVd0MaeqyVd4gaaKqzGeGaeu4KdCuc fa4aa0baaeaajugWaiabeY7aTjabe27aUbqcfayaaKqzadGaeqySde gaaKqzGeGaeyOaIyBcfa4aaSbaaeaajugWaiabeg7aHbqcfayabaaa aa@6899@ is the generalized d’Alembert operator and Γ μν α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHto WrjuaGdaqhaaqaaKqzadGaeqiVd0MaeqyVd4gajuaGbaqcLbmacqaH Xoqyaaaaaa@4094@ denotes the Christoffel symbol.24 In Eq. (4), we made use of the parallel transport of the metric μ g μν =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaey4bIe 9aaSbaaeaajugWaiabeY7aTbqcfayabaGaam4zamaaCaaabeqaaKqz adGaeqiVd0MaeqyVd4gaaKqbakabg2da9iaaicdacaGGUaaaaa@4447@  A similar minimization procedure with respect to the metric g μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaacEgajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaG beaaaaa@3D69@ leads to the Einstein field equations

R μν 1 2 g μν R=k T μν , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb qcfa4aaSbaaeaajugWaiabeY7aTjabe27aUbqcfayabaqcLbsacqGH sisljuaGdaWcaaqaaKqzGeGaaGymaaqcfayaaKqzGeGaaGOmaaaaca WGNbqcfa4aaSbaaeaajugWaiabeY7aTjabe27aUbqcfayabaqcLbsa caWGsbGaeyypa0Jaam4AaiaadsfajuaGdaWgaaqaaKqzadGaeqiVd0 MaeqyVd4gajuaGbeaajugibiaacYcaaaa@5484@   (5)

with k=8πG/ c 4  and  T μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb Gaeyypa0JaaGioaiabec8aWjaadEeacaGGVaGaam4yaKqbaoaaCaaa beqaaKqzadGaaGinaaaajugibabaaaaaaaaapeGaaiiOaiaadggaca WGUbGaamizaiaacckacaWGubqcfa4aaSbaaeaajugWaiabeY7aTjab e27aUbqcfayabaaaaa@4C68@ is the energy-momentum tensor

T μν = 1 2 ( μ φ * ν φ+ ν φ * μ φ ) g μν [ 1 2 g αβ α φ * β φV( | φ | 2 ) ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaG beaajugibiabg2da9KqbaoaalaaabaqcLbsacaaIXaaajuaGbaqcLb sacaaIYaaaaKqbaoaabmaabaqcLbsacqGHciITjuaGdaWgaaqaaKqz adGaeqiVd0gajuaGbeaajugibiabeA8aQLqbaoaaCaaabeqaaKqzad GaaiOkaaaajugibiabgkGi2MqbaoaaBaaabaqcLbmacqaH9oGBaKqb agqaaKqzGeGaeqOXdOMaey4kaSIaeyOaIyBcfa4aaSbaaeaajugWai abe27aUbqcfayabaqcLbsacqaHgpGAjuaGdaahaaqabeaajugWaiaa cQcaaaqcLbsacqGHciITjuaGdaWgaaqaaKqzadGaeqiVd0gajuaGbe aajugibiabeA8aQbqcfaOaayjkaiaawMcaaKqzGeGaeyOeI0Iaam4z aKqbaoaaBaaabaqcLbmacqaH8oqBcqaH9oGBaKqbagqaamaadmaaba WaaSaaaeaajugibiaaigdaaKqbagaajugibiaaikdaaaGaam4zaKqb aoaaCaaabeqaaKqzadGaeqySdeMaeqOSdigaaKqzGeGaeyOaIyBcfa 4aaSbaaeaajugWaiabeg7aHbqcfayabaqcLbsacqaHgpGAjuaGdaah aaqabeaajugWaiaacQcaaaqcLbsacqGHciITjuaGdaWgaaqaaKqzad GaeqOSdigajuaGbeaajugibiabeA8aQjabgkHiTiaadAfajuaGdaqa daqaamaaemaabaqcLbsacqaHgpGAaKqbakaawEa7caGLiWoadaahaa qabeaajugWaiaaikdaaaaajuaGcaGLOaGaayzkaaaacaGLBbGaayzx aaqcLbsacaGGUaaaaa@A093@   (6)

Perturbative analysis

We now assume a perturbation around the Minkowski space time of the form g μν = η μν + h μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaacEgajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaG beaajugibiabg2da9iabeE7aOLqbaoaaBaaabaqcLbmacqaH8oqBcq aH9oGBaKqbagqaaKqzGeGaey4kaSIaamiAaKqbaoaaBaaabaqcLbma cqaH8oqBcqaH9oGBaKqbagqaaaaa@4EBA@  where h μν η μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaG beaacqWIQjspjugibiabeE7aOLqbaoaaBaaabaqcLbmacqaH8oqBcq aH9oGBaKqbagqaaaaa@46D9@ is the spacetime ripple and η μν =diag( ,+,+,+, ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeE7aOLqbaoaaBaaabaqcLbmacqaH8oqBcqaH9oGBaKqb agqaaiabg2da9iaadsgacaWGPbGaamyyaiaadEgadaqadaqaaiabgk HiTiaacYcacqGHRaWkcaGGSaGaey4kaSIaaiilaiabgUcaRiaacYca aiaawIcacaGLPaaacaGGUaaaaa@4B67@  To first order in h μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaG beaaaaa@3D6B@ Eq. (4) reads

φ +V' ( | φ | 2 ) ,φ* + h μν μ ν φ η μν γ μν α α φ=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiablgAjxLqbaoaaBaaabaqcLbmacqaHgpGAaKqbagqaaKqz GeGaey4kaSIaaiOvaiaacEcajuaGpaWaaeWaaOqaaKqbaoaaemaake aajugibiabeA8aQbGccaGLhWUaayjcSdqcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaajuaGdaWgaaqaaKqzadGaaiilai abeA8aQjaacQcaaKqbagqaaKqzGeGaey4kaSIaamiAaKqbaoaaCaaa beqaaKqzadGaeqiVd0MaeqyVd4gaaKqzGeGaeyOaIyBcfa4aaSbaae aajugWaiabeY7aTbqcfayabaqcLbsacqGHciITjuaGdaWgaaqaaKqz adGaeqyVd4gajuaGbeaajugibiabeA8aQjabgkHiTiabeE7aOLqbao aaCaaabeqaaKqzadGaeqiVd0MaeqyVd4gaaKqzGeGaeq4SdCwcfa4a a0baaeaajugWaiabeY7aTjabe27aUbqcfayaaKqzadGaeqySdegaaK qzGeGaeyOaIyBcfa4aaSbaaeaajugWaiabeg7aHbqcfayabaqcLbsa cqaHgpGAcqGH9aqpcaaIWaGaaiilaaaa@8387@   (7)

where γ μν α = 1 2 ( α h μν + ν h ν α ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzjuaGdaqhaaqaaKqzadGaeqiVd0MaeqyVd4gajuaGbaqcLbmacqaH XoqyaaqcLbsacqGH9aqpjuaGdaWcaaqaaKqzGeGaaGymaaqcfayaaK qzGeGaaGOmaaaajuaGdaqadaqaaKqzGeGaeyOaIyBcfa4aaWbaaeqa baqcLbmacqaHXoqyaaqcLbsacaWGObqcfa4aaSbaaeaajugWaiabeY 7aTjabe27aUbqcfayabaqcLbsacqGHRaWkcqGHciITjuaGdaWgaaqa aKqzadGaeqyVd4gajuaGbeaajugibiaadIgajuaGdaqhaaqaaKqzad GaeqyVd4gajuaGbaqcLbmacqaHXoqyaaaajuaGcaGLOaGaayzkaaqc LbsacaGGSaaaaa@663B@ with h μ ν = η αν h μα . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aa0baaeaajugWaiabeY7aTbqcfayaaKqzadGaeqyVd4gaaKqz GeGaeyypa0Jaeq4TdGwcfa4aaWbaaeqabaqcLbmacqaHXoqycqaH9o GBaaqcfaOaamiAamaaBaaabaqcLbmacqaH8oqBcqaHXoqyaKqbagqa aiaac6caaaa@4E4F@  Making use of the trasnverse-traceless (TT) gauge, μ h ν μ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHci ITjuaGdaWgaaqaaKqzadGaeqiVd0gajuaGbeaajugibiaadIgajuaG daqhaaqaaKqzadGaeqyVd4gajuaGbaqcLbmacqaH8oqBaaqcLbsacq GH9aqpcaaIWaaaaa@46DF@ and h h μ μ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb GaeyyyIORaamiAaKqbaoaaDaaabaqcLbmacqaH8oqBaKqbagaajugW aiabeY7aTbaajugibiabg2da9iaaicdaaaa@437D@ the last term in Eq. (7) vanishes and the KG equation explicitly reads

φ + m 2 c 2 2 φ+λ | φ | 2 φ+ h μν μ ν φ=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiablgAjxLqbaoaaBaaabaqcLbmacqaHgpGAaKqbagqaaKqz GeGaey4kaSscfa4aaSaaaeaajugibiaad2gajuaGdaahaaqabeaaju gWaiaaikdaaaqcLbsacaWGJbqcfa4aaWbaaeqabaqcLbmacaaIYaaa aaqcfayaaKqzGeGaeS4dHGwcfa4aaWbaaeqabaqcLbmacaaIYaaaaa aajugibiabeA8aQjabgUcaRiabeU7aSLqbaoaaemaabaqcLbsacqaH gpGAaKqbakaawEa7caGLiWoadaahaaqabeaajugWaiaaikdaaaqcLb sacqaHgpGAcqGHRaWkcaWGObqcfa4aaWbaaeqabaqcLbmacqaH8oqB cqaH9oGBaaqcLbsacqGHciITjuaGdaWgaaqaaKqzadGaeqiVd0gaju aGbeaajugibiabgkGi2MqbaoaaBaaabaqcLbmacqaH9oGBaKqbagqa aKqzGeGaeqOXdOMaeyypa0JaaGimaiaacYcaaaa@735F@   (8)

where we made use of the property h μν = h μν . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGdaahaaqabeaajugWaiabeY7aTjabe27aUbaa juaGcqGH9aqpjugibiaadIgakmaaBaaaleaadaahaaadbeqaaKqzad GaeqiVd0MaeqyVd4gaaaWcbeaakiaac6caaaa@45B5@ Similarly, the weak-field limit of Eq. (5) describes spacetime radiation (gravitational waves) in the presence of matter

h μν =2k T μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIHwYvjugibiaadIgakmaaBaaaleaadaahaaadbeqaaiab eY7aTjabe27aUbaaaSqabaqcLbsacqGH9aqpcqGHsislcaaIYaGaam 4AaiaadsfajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaGbeaa aaa@488B@   (9)

with =( 1/ c 2 ) t 2 + 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqWIHwYvjugibiabg2da9iabgkHiTKqbaoaabmaabaGaaGym aiaac+cacaWGJbWaaWbaaeqabaqcLbmacaaIYaaaaaqcfaOaayjkai aawMcaaiabgkGi2oaaDaaabaqcLbmacaWG0baajuaGbaqcLbmacaaI YaaaaKqbakabgUcaRiabgEGirpaaCaaabeqaaKqzadGaaGOmaaaaju aGcaGGUaaaaa@4E44@ In what follows, we introduce quantum fluctuations around the homogeneous scalar field (i.e. the vacuum expectation value φ = n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaaadaqaaiabeA8aQbGaayzkJiaawQYiaiabg2da9maakaaa baGaamOBamaaBaaabaqcLbmacaaIWaaajuaGbeaaaeqaaaaa@3ED1@ spontaneously breaking the continuous U(1) symmetry) in the form

φ( x μ )= n 0 e iμt/ [ 1+ k ( uk e i k μ x μ + υ k * e i k μ x μ ) ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeA8aQLqbaoaabmaabaqcLbsacaWG4bqcfa4aaSbaaeaa jugWaiabeY7aTbqcfayabaaacaGLOaGaayzkaaqcLbsacqGH9aqpju aGdaGcaaqaaKqzGeGaamOBaKqbaoaaBaaabaqcLbmacaaIWaaajuaG beaaaeqaaKqzGeGaamyzaKqbaoaaCaaabeqaaKqzadGaeyOeI0Iaam yAaiabeY7aTjaadshacaGGVaGaeS4dHGgaaKqbaoaadmaabaqcLbsa caaIXaGaey4kaSscfa4aaabuaeaadaqadaqaaKqzGeGaamyDaiaadU gacaWGLbqcfa4aaWbaaKqbafqabaqcLboacaWGPbGaam4AaKqbaoaa BaaajuaqbaqcLbiacqaH8oqBaKqbafqaaKqzGdGaamiEaKqbaoaaCa aajuaqbeqaaKqzacGaeqiVd0gaaaaajugibiabgUcaRiabew8a1Lqb aoaaDaaabaqcLbmacaWGRbaajuaGbaqcLbmacaGGQaaaaKqzGeGaam yzaKqbaoaaCaaabeqcfawaaKqzaeGaeyOeI0IaamyAaiaadUgajuaG daWgaaqcfawaaKqzGcGaeqiVd0gajuaybeaajugabiaadIhajuaGda ahaaqcfawabeaajugOaiabeY7aTbaaaaaajuaGcaGLOaGaayzkaaaa baqcLbmacaWGRbaajuaGbeqcLbsacqGHris5aaqcfaOaay5waiaaw2 faaKqzGeGaaiilaaaa@8868@   (10)

where μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBaaa@383B@  is the chemical potential of the condensate. By dividing the metric fluctuation into its time and space components, h μν = h 00 + h ij 2ϕ/ c 2 + h ij , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaG beaacqGH9aqpjugibiaadIgajuaGdaWgaaqaaKqzadGaaGimaiaaic daaKqbagqaaiabgUcaRKqzGeGaamiAaKqbaoaaBaaabaqcLbmacaWG PbGaamOAaaqcfayabaGaeyyyIORaaGOmaiabew9aMjaac+cacaGGJb WaaWbaaeqabaqcLbmacaaIYaaaaKqbakabgUcaRKqzGeGaamiAaKqb aoaaBaaabaqcLbmacaWGPbGaamOAaaqcfayabaGaaiilaaaa@5A49@ we can obtain purely transverse solutions satisfying the condition k i h ij =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbWaaWbaaeqabaqcLbmacaWGPbaaaKqbakaadIgadaWg aaqaaKqzadGaamyAaiaadQgaaKqbagqaaiabg2da9iaaicdaaaa@40C7@ as

( m eff 2 c 2 2 ) h ij =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaqaaKqzGeGaeSyOLCLaeyOeI0scfa4aaSaaaeaajugi biaad2gajuaGdaqhaaqaaKqzadGaamyzaiaabAgacaqGMbaajuaGba qcLbmacaaIYaaaaKqzGeGaam4yaKqbaoaaCaaabeqaaKqzadGaaGOm aaaaaKqbagaajugibiabl+qiOLqbaoaaCaaabeqaaKqzadGaaGOmaa aaaaaajuaGcaGLOaGaayzkaaqcLbsacaWGObqcfa4aaSbaaeaajugW aiaadMgacaWGQbaajuaGbeaajugibiabg2da9iaaicdacaGGSaaaaa@56CA@   (11)

where m eff 2 =2 2 kV( n 0 ) c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaad2gajuaGdaqhaaqaaKqzadGaamyzaiaabAgacaqGMbaa juaGbaqcLbmacaaIYaaaaKqzGeGaeyypa0JaaGOmaiabl+qiOLqbao aaCaaabeqaaKqzadGaaGOmaaaajuaGcaWGRbGaamOvamaabmaabaGa amOBamaaBaaabaqcLbmacaaIWaaajuaGbeaaaiaawIcacaGLPaaaju gibiaadogajuaGdaahaaqabeaajugWaiaaikdaaaaaaa@503A@ is the square of the effective graviton mass. Assuming plane-wave solutions of the form h ij = k χij e i k μ x μ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGdaWgaaqaaKqzadGaamyAaiaadQgaaKqbagqa aKqzGeGaeyypa0tcfa4aaabuaeaacqaHhpWycaWGPbGaamOAaiaadw gadaahaaqabKqbGfaacaWGPbGaam4AaKqbaoaaBaaajuaybaqcLbka cqaH8oqBaKqbGfqaaiaadIhajuaGdaahaaqcfawabeaajugOaiabeY 7aTbaaaaaajuaGbaGaam4AaaqabiabggHiLdGaaiilaaaa@5250@ we obtain the dispersion relation

ω 2 = ω p 2 + c 2 k 2 ,   ω p 2 = m eff 2 = c 4 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeM8a3LqbaoaaCaaabeqaaKqzadGaaGOmaaaajugibiab g2da9iabeM8a3LqbaoaaDaaabaqcLbmacaWGWbaajuaGbaqcLbmaca aIYaaaaKqzGeGaey4kaSIaam4yaKqbaoaaCaaabeqaaKqzadGaaGOm aaaajugibiaadUgajuaGdaahaaqabeaajugWaiaaikdaaaqcLbsaca GGSaGaaiiOaiaacckacqaHjpWDjuaGdaqhaaqaaKqzadGaamiCaaqc fayaaKqzadGaaGOmaaaajugibiabg2da9KqbaoaalaaabaqcLbsaca WGTbqcfa4aa0baaeaajugWaiaadwgacaqGMbGaaeOzaaqcfayaaKqz adGaaGOmaaaajugibiabg2da9iaadogajuaGdaahaaqabeaajugWai aaisdaaaaajuaGbaqcLbsacqWIpecAjuaGdaahaaqabeaajugWaiaa ikdaaaaaaKqbakaac6caaaa@6D63@   (12)

The latter is obtained by making use of the equation of state obtained at the zeroth order, which fixes the chemical potential of the BEC as

μ= m 2 c 4 + 2 λ n 0 c 2 =m c 2 1+ c s 2 c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeY7aTjabg2da9KqbaoaakaaabaqcLbsacaWGTbqcfa4a aWbaaeqabaqcLbmacaaIYaaaaKqzGeGaam4yaKqbaoaaCaaabeqaaK qzadGaaGinaaaajugibiabgUcaRiabl+qiOLqbaoaaCaaabeqaaKqz adGaaGOmaaaajugibiabeU7aSjaad6gajuaGdaWgaaqaaKqzadGaaG imaaqcfayabaqcLbsacaWGJbqcfa4aaWbaaeqabaqcLbmacaaIYaaa aaqcfayabaqcLbsacqGH9aqpcaWGTbGaam4yaKqbaoaaCaaabeqaaK qzadGaaGOmaaaajuaGdaGcaaqaaKqzGeGaaGymaiabgUcaRKqbaoaa laaabaqcLbsacaWGJbqcfa4aa0baaeaajugWaiaadohaaKqbagaaju gWaiaaikdaaaaajuaGbaqcLbsacaWGJbqcfa4aaWbaaeqabaqcLbma caaIYaaaaaaaaKqbagqaaaaa@67BE@   (13)

where c s =/( mλ n 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbWaaSbaaeaajugWaiaadohaaKqbagqaaiabg2da9iab l+qiOjaac+cadaqadaqaaiaad2gacqaH7oaBcaWGUbWaaSbaaeaaju gWaiaaicdaaKqbagqaaaGaayjkaiaawMcaaaaa@44FC@ is the BEC sound speed. The dispersion relation in Eq. (12) is analogous of that of an electromagnetic wave propagating in a charged medium characterized by a plasma frequency ω p , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHjpWDdaWgaaqaaKqzadGaamiCaaqcfayabaGaaiilaaaa @3BF3@ where the photon also acquires an effective mass. Consequently, the KG equation decouples from Einstein’s equations and the Bogoliubov modes for the scalar field can be obtained from Eqs. (8) and (10). Their dispersion relation can be found from plugging Eq. (10) in Eq. (8) and separating it into its particle (anti-particle) coefficients e i k μ x μ ( e i k μ x μ ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHDi sTcaGGLbGcdaahaaWcbeqaaiaadMgacaWGRbWaaSbaaWqaamaaCaaa beqaaiabeY7aTbaaaeqaaSGaamiEamaaCaaameqabaGaeqiVd0gaaa aajuaGdaqadaqaaKqzGeGaeyyhIuRaamyzaKqbaoaaCaaabeqcfawa aKqzaeGaeyOeI0IaamyAaiaadUgajuaGdaWgaaqcfawaaKqzGcGaeq iVd0gajuaybeaajugabiaadIhajuaGdaahaaqcfawabeaajugOaiab eY7aTbaaaaaajuaGcaGLOaGaayzkaaGaaiOlaaaa@5439@  The resulting secular equation contains two real solutions (and the corresponding “anti-mode” solutions ω ± * =ω± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeM8a3LqbaoaaDaaabaqcLbmacqGHXcqSaKqbagaajugW aiaacQcaaaqcfaOaeyypa0JaeyOeI0IaeqyYdCNaeyySaelaaa@44E4@

ω ± 2 =2 ω 0 2 + c 2 k 2 ±2 ω 0 4 + c 2 k 2 ω 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeM8a3LqbaoaaDaaabaqcLbmacqGHXcqSaKqbagaajugW aiaaikdaaaqcLbsacqGH9aqpcaaIYaGaeqyYdCxcfa4aa0baaeaaju gWaiaaicdaaKqbagaajugWaiaaikdaaaqcLbsacqGHRaWkcaWGJbqc fa4aaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaam4AaKqbaoaaCaaabe qaaKqzadGaaGOmaaaajugibiabgglaXkaaikdajuaGdaGcaaqaaKqz GeGaeqyYdCxcfa4aa0baaeaajugWaiaaicdaaKqbagaajugWaiaais daaaqcLbsacqGHRaWkcaWGJbqcfa4aaWbaaeqabaqcLbmacaaIYaaa aKqzGeGaam4AaKqbaoaaCaaabeqaaKqzadGaaGOmaaaajugibiabeM 8a3LqbaoaaDaaabaqcLbmacaaIWaaajuaGbaqcLbmacaaIYaaaaaqc fayabaaaaa@6D3B@   (14)

where ω 0 =m c 2 ( 1+ β 2 )/, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeM8a3LqbaoaaBaaabaqcLbmacaaIWaaajuaGbeaacqGH 9aqpcaWGTbGaam4yamaaCaaabeqaaKqzadGaaGOmaaaajuaGdaqada qaaiaaigdacqGHRaWkcqaHYoGydaahaaqabeaajugWaiaaikdaaaaa juaGcaGLOaGaayzkaaGaai4laiabl+qiOjaacYcaaaa@4AFE@ with β 2 =3 c s 2 /( 2 c 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGydaahaaqabeaajugWaiaaikdaaaqcfaOaeyypa0Ja aG4maiaacogadaqhaaqaaKqzadGaam4CaaqcfayaaKqzadGaaGOmaa aajuaGcaGGVaWaaeWaaeaacaaIYaGaam4yamaaCaaabeqaaKqzadGa aGOmaaaaaKqbakaawIcacaGLPaaacaGGSaaaaa@4A01@ is the cut-off frequency. In the long wavelength limit kξ( 1+ β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGRbGaeSOAI0JaeqOVdG3aaeWaaeaacaaIXaGaey4kaSIa eqOSdi2aaWbaaeqabaqcLbmacaaIYaaaaaqcfaOaayjkaiaawMcaaa aa@4212@ - with ξ=/m c s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH+oaEcqGH9aqpcqWIpecAcaGGVaGaamyBaiaadogadaWg aaqaaKqzadGaam4Caaqcfayabaaaaa@3FF8@ denoting the healing length - the lower (Goldstone) mode is gapeless, ω c s ( 1+ β 2 )k+ξ c s k 2 /2 ( 1+ β 2 ) 3 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHjpWDcqGHsislcqWIdjYocaWGJbWaaSbaaeaajugWaiaa dohaaKqbagqaamaabmaabaGaaGymaiabgUcaRiabek7aInaaCaaabe qaaKqzadGaaGOmaaaaaKqbakaawIcacaGLPaaacaWGRbGaey4kaSIa eqOVdGNaam4yamaaBaaabaqcLbmacaWGZbaajuaGbeaacaWGRbWaaW baaeqabaqcLbmacaaIYaaaaKqbakaac+cacaaIYaWaaeWaaeaacaaI XaGaey4kaSIaeqOSdi2aaWbaaeqabaqcLbmacaaIYaaaaaqcfaOaay jkaiaawMcaamaaCaaabeqaaKqzadGaaG4maaaajuaGcaGGSaaaaa@5CA4@ reducing to the usual Bogoliubov dispersion in the non-relativistic limit β0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHYoGycqGHsgIRcaaIWaGaaiOlaaaa@3B9E@ 25,26 The gapped mode, corresponding to the massive Higgs mode of mass M= ω 0 / c 2 =m( 1+ β 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbGaeyypa0JaeS4dHGMaeqyYdC3aaSbaaeaajugWaiaa icdaaKqbagqaaiaac+cacaWGJbWaaWbaaeqabaqcLbmacaaIYaaaaK qbakabg2da9iaad2gadaqadaqaaiaaigdacqGHRaWkcqaHYoGydaah aaqabeaajugWaiaaikdaaaaajuaGcaGLOaGaayzkaaGaaiilaaaa@4C47@ reads ω + M 2 c s 4 / 2 + c s 2 k 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeM8a3LqbaoaaBaaabaqcLbmacqGHRaWkaKqbagqaaKqz GeGaeS4qIStcfa4aaOaaaeaajugibiaad2eajuaGdaahaaqabeaaju gWaiaaikdaaaqcLbsacaWGJbqcfa4aa0baaeaajugWaiaadohaaKqb agaajugWaiaaisdaaaqcLbsacaGGVaGaeS4dHGwcfa4aaWbaaeqaba qcLbmacaaIYaaaaKqzGeGaey4kaSIaam4yaKqbaoaaDaaabaqcLbma caWGZbaajuaGbaqcLbmacaaIYaaaaKqzGeGaam4AaKqbaoaaCaaabe qaaKqzadGaaGOmaaaaaKqbagqaaKqzGeGaaiOlaaaa@5B73@  Although with a different notation, the dispersion modes of Eq. (14) have first been discussed in Ref.27

The situation changes if we consider perturbations in the time-time components of Einstein’s equations (5), i.e., for gravitational waves of the form h μν h 00 =2ϕ/ c 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadIgajuaGdaWgaaqaaKqzadGaeqiVd0MaeqyVd4gajuaG beaajugibiabloKi7iaadIgajuaGdaWgaaqaaKqzadGaaGimaiaaic daaKqbagqaaKqzGeGaeyypa0JaaGOmaiabew9aMjaac+cacaWGJbqc fa4aaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaaiOlaaaa@4D86@  This amounts to generalize the usual self-gravitating problem, as described by the Klein-GordonPoisson system,13 to the study of propagation of gravitational radiation in a symmetry-broken quantum vacuum. As we are about to see, the formation of structures emerges in this case as a consequence of the violation of the discrete Pτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadcfajugGbiabes8a0jabgkHiTaaa@3B1B@ symmetry. Putting Eqs. (9) and (8) together, and keeping terms to the first order in the Fourier components of the vector Vk=( uk,υk,ϕk ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAfacaWGRbGaeyypa0tcfa4aaeWaaeaajugibiaadwha caWGRbGaaiilaiabew8a1jaadUgacaGGSaGaeqy1dyMaam4Aaaqcfa OaayjkaiaawMcaaKqzGeGaaiilaaaa@46A2@ we obtain the eigenvalue problem L k , V k =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadYeajuaGdaWgaaqaaKqzadGaam4AaaqcfayabaqcLbsa caGGSaGaamOvaKqbaoaaBaaabaqcLbmacaWGRbaajuaGbeaajugibi abg2da9iaaicdacaGGSaaaaa@4345@ where

L k =[         ε 1 2           m 2 c 2 c s 2        2 μ 2        m 2 c 2 c s 2            ε 2 2                 0 = k ˜ ( μω ) 2   k ˜ ( μ+ω ) 2     ε 0 2 ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadYeajuaGdaWgaaqaaKqzGeGaam4AaaqcfayabaqcLbsa cqGH9aqpjuaGdaWadaqcLbsaeaqabKqbagaajugibiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacqaH1oqzjuaGdaqhaaqc faAaaKqzadGaaGymaaqcfayaaKqzadGaaGOmaaaajugibiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiab gkHiTiaad2gajuaGdaahaaqabeaajugWaiaaikdaaaqcLbsacaWGJb qcfa4aaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaam4yaKqbaoaaDaaa baqcLbmacaWGZbaajuaGbaqcLbmacaaIYaaaaKqzGeGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiabgkHiTiaaikdacqaH 8oqBjuaGdaahaaqabeaajugWaiaaikdaaaqcLbsacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckaaKqbagaajugibiabgkHiTiaad2ga juaGdaahaaqabeaajugWaiaaikdaaaqcLbsacaWGJbqcfa4aaWbaae qabaqcLbmacaaIYaaaaKqzGeGaam4yaKqbaoaaDaaabaqcLbmacaWG ZbaajuaGbaqcLbmacaaIYaaaaKqzGeGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiabew7aLLqb aoaaDaaabaqcLbmacaaIYaaajuaGbaqcLbmacaaIYaaaaKqzGeGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaic daaKqbagaajugibiabg2da9iqacUgagaacaKqbaoaabmaabaqcLbsa cqaH8oqBcqGHsislcqWIpecAcqaHjpWDaKqbakaawIcacaGLPaaada ahaaqabeaajugWaiaaikdaaaqcLbsacaGGGcGaeyOeI0Iabi4Aayaa iaqcfa4aaeWaaeaajugibiabeY7aTjabgUcaRiabl+qiOjabeM8a3b qcfaOaayjkaiaawMcaamaaCaaabeqaaKqzadGaaGOmaaaajugibiaa cckacaGGGcGaaiiOaiabew7aLLqbaoaaDaaabaqcLbmacaaIWaaaju aGbaqcLbmacaaIYaaaaaaajuaGcaGLBbGaayzxaaGaaiilaaaa@DE65@   (15)

with k ˜ =8πG n 0 / c 4 , ε 1,2 2 = ( μ±ω ) 2 2 c 2 k 2 m 2 c 4 2 m 2 c s 4 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiqadUgagaacaiabg2da9iaaiIdacqaHapaCcaWGhbGaamOB aKqbaoaaBaaabaqcLbmacaaIWaaajuaGbeaajugibiaac+cacaGGJb qcfa4aaWbaaeqabaqcLbmacaaI0aaaaKqzGeGaaiilaiabew7aLLqb aoaaDaaabaqcLbmacaaIXaGaaiilaiaaikdaaKqbagaajugWaiaaik daaaqcLbsacqGH9aqpjuaGdaqadaqaaKqzGeGaeqiVd0MaeyySaeRa eS4dHGMaeqyYdChajuaGcaGLOaGaayzkaaWaaWbaaeqabaqcLbmaca aIYaaaaKqzGeGaeyOeI0IaeS4dHGwcfa4aaWbaaeqabaqcLbmacaaI YaaaaKqzGeGaam4yaKqbaoaaCaaabeqaaKqzadGaaGOmaaaajugibi aadUgajuaGdaahaaqabeaajugWaiaaikdaaaqcLbsacqGHsislcaWG Tbqcfa4aaWbaaeqabaqcLbmacaaIYaaaaKqzGeGaam4yaKqbaoaaCa aabeqaaKqzadGaaGinaaaajugibiabgkHiTiaaikdacaWGTbqcfa4a aWbaaeqabaqcLbmacaaIYaaaaKqzGeGaam4yaKqbaoaaDaaabaqcLb macaWGZbaajuaGbaqcLbmacaaI0aaaaKqzGeGaaiilaaaa@7FF2@ and ε 0 2 = 2 ( ω 2 c 2 k 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabew7aLLqbaoaaDaaabaqcLbmacaaIWaaajuaGbaqcLbma caaIYaaaaKqzGeGaeyypa0JaeS4dHGwcfa4aaWbaaeqabaqcLbmaca aIYaaaaKqbaoaabmaabaqcLbsacqaHjpWDjuaGdaahaaqabeaajugW aiaaikdaaaqcLbsacqGHsislcaWGJbqcfa4aaWbaaeqabaqcLbmaca aIYaaaaKqzGeGaam4AaKqbaoaaCaaabeqaaKqzadGaaGOmaaaaaKqb akaawIcacaGLPaaajugibiaac6caaaa@54A7@ Nontrivial solutions are obtained by solving the secular equation det  in respect to ω, for which we obtain six solutions (three for positive-energy and other three for negative-energy excitations). For zero gravity-matter coupling (G = 0), the positive-energy modes are the BEC Goldstone Bogoliubov and the Higgs, as described in Eq. (14), and the gravity mode ω=ck. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeM8a3jabg2da9iaadogacaWGRbGaaiOlaaaa@3C02@ In this situation, all modes are real and therefore dynamically stable (Figure 1A). In the presence of gravity, however, the Bogoliubov and the gravity modes hybridize and collapse, exhibiting an imaginary part for k−modes below the Jeans wave vector kJ that satisfy the condition ( ω )=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabggribNqbaoaabmaabaGaeqyYdChacaGLOaGaayzkaaGa eyypa0JaaGimaiaacYcaaaa@3E78@ for which we obtain the equation.

4 k J 4 +2 2 k J 2 m 2 c s 2 k ˜ m 4 ( c 2 + c s 2 )2=0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabl+qiOLqbaoaaCaaabeqaaKqzadGaaGinaaaajugibiaa dUgajuaGdaqhaaqaaKqzadGaamOsaaqcfayaaKqzadGaaGinaaaaju gibiabgUcaRiaaikdacqWIpecAjuaGdaahaaqabeaajugWaiaaikda aaqcLbsacaWGRbqcfa4aa0baaeaajugWaiaadQeaaKqbagaajugWai aaikdaaaqcLbsacaWGTbqcfa4aaWbaaeqabaqcLbmacaaIYaaaaKqz GeGaam4yaKqbaoaaDaaabaqcLbmacaWGZbaajuaGbaqcLbmacaaIYa aaaKqzGeGaeyOeI0Iabm4AayaaiaGaamyBaKqbaoaaCaaabeqaaKqz adGaaGinaaaajuaGdaqadaqaaKqzGeGaam4yaKqbaoaaCaaabeqaaK qzadGaaGOmaaaajugibiabgUcaRiaadogajuaGdaqhaaqaaKqzadGa am4CaaqcfayaaKqzadGaaGOmaaaaaKqbakaawIcacaGLPaaajugibi aaikdacqGH9aqpcaaIWaGaaiOlaaaa@70A7@   (16)

Remarkably, the imaginary part of the Bogoliubov and the gravity modes have opposite signs, suggesting that the formation of dark-matter cosmological structures (triggered by the long-wavelength dynamical instability) is accompanied by the damping of space-time perturbations. In other words, the gravitational waves transfer their energy to the BEC modes so the latter can grow. Because Re( ω )>0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadkfacaWGLbqcfa4aaeWaaeaacqaHjpWDaiaawIcacaGL PaaacqGH+aGpcaaIWaGaaiilaaaa@3EBC@  a I 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGjbWaaSbaaeaajugWaiaaicdaaKqbagqaaaaa@3A09@ -type of instability28 is responsible for the formation of large structures in flat spacetimes. Also, we observe that the positive and negative Bogoliubov modes are not symmetric, i.e. ω ω * , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHjpWDcqGHsislcqGHGjsUcqGHsislcqaHjpWDdaqhaaqa aiabgkHiTaqaaKqzadGaaiOkaaaajuaGcaGGSaaaaa@4208@ indicating violation of the PT-symmetry. These features are depicted in Figure 1B).

Figure 1 Dispersion relation of the various modes present in the dynamics. Top panel: mode dispersion in the absence of coupling ( G=0 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaqaaKqzGeGaam4raiabg2da9iaaicdaaKqbakaawIca caGLPaaajugibiaac6caaaa@3D17@ We observe that the Bogoliubov-Goldstone modes are PT -symmetric. Bottom panel: when the gravity is switched on, the Goldstone modes lose their symmetry. The gravity mode is damped to favour unstable (growing) modes in the BEC. The imaginary part of the frequencies goes to zero at the Jeans mode kJ, exhibiting the usual signature of PT− symmetry breaking. For illustration purposes, we use c s =c/3. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaacogajuaGdaWgaaqaaKqzadGaai4CaaqcfayabaqcLbsa cqGH9aqpcaGGJbGaai4laiaaiodacaGGUaaaaa@3F8C@ .

In order to illustrate how the -symmetry breaking affects the formation of structures, we perform onedimensional simulations of Eqs. (8) and (9) for the early stages of the Jeans instability. As depicted in Figure 2, an initial linear superposition of plane gravitational and Bogoliubov waves (Figure 2A)) lead to the formation of 1d structures in the BEC sector. Short after the onset the instability, the long wavelength structures of the BEC start to grow, leading to the formation of structures of typical size λ J =2π/kJ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeU7aSLqbaoaaBaaabaqcLbmacaWGkbaajuaGbeaacqGH 9aqpcaaIYaGaeqiWdaNaai4laiaadUgacaWGkbGaaiOlaaaa@4236@  Simultaneously, the gravitational modes in the same wavelength modes attenuate, eventually vanishing out for longer times. We notice that our calculations are valid near the onset of the instability only, i.e for t/m c 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeSOAI0JaeS4dHGMaai4laiaad2gacaWGJbWaaWba aeqabaqcLbmacaaIYaaaaKqbakaacYcaaaa@3FF7@ for which a quasi-linear approximation of Eqs. (8) and (9) is valid. A more accurate, quantitative discussion of our results would involve taking into account saturation effects.

Figure 2 One-dimensional illustration of the structure formation dynamics at early stages of the instability onset. Panel a) shows the initial ( t=0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaqaaKqzGeGaamiDaiabg2da9iaaicdaaKqbakaawIca caGLPaaaaaa@3C03@ plane-wave superposition solution for the gravitational wave ϕ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabew9aMLqbaoaabmaabaGaamiEaaGaayjkaiaawMcaaaaa @3B81@ (black line) and the BEC φ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeA8aQLqbaoaabmaabaGaamiEaaGaayjkaiaawMcaaaaa @3B76@ (lighter line). Panels b), c) and d) depict their evolution at t=1.5/m c 2 ,t=4.5/m c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyypa0JaaGymaiaac6cacaaI1aGaeS4dHGMaai4l aiaad2gacaWGJbWaaWbaaeqabaqcLbmacaaIYaaaaKqbakaacYcaca WG0bGaeyypa0JaaGinaiaac6cacaaI1aGaeS4dHGMaai4laiaad2ga caWGJbWaaWbaaeqabaqcLbmacaaIYaaaaaaa@4BBF@ and t=5.5/m c 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG0bGaeyypa0JaaGynaiaac6cacaaI1aGaeS4dHGMaai4l aiaad2gacaWGJbWaaWbaaeqabaqcLbmacaaIYaaaaKqbakaacYcaaa a@41D3@ respectively. The shadowed region represent the Jeans length λ J =2π/kJ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH7oaBdaWgaaqaaKqzadGaamOsaaqcfayabaGaeyypa0Ja aGOmaiabec8aWjaac+cacaWGRbGaamOsaiaac6caaaa@41A7@ We use c s =c/3. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGJbWaaSbaaeaajugWaiaadohaaKqbagqaaiabg2da9iaa dogacaGGVaGaaG4maiaac6caaaa@3E71@

Conclusion

In this work, we have studied the coupling between a gravitational wave in a Minkowski spacetime with dark matter modelled by a self-interacting complex scalar field (Bose-Einstein condensate). Considering perturbations in the spatial components of the metric only, the gravitational wave dispersion relation is analogous to that of an electromagnetic wave propagating in a charged medium characterized by a plasma frequency ωp, where the photon also acquires an effective mass. In this case, the two modes (the gravity mode and the Bogoliubov mode) are decoupled. However, when we consider perturbations in the temporal component of the metric, the gravity and the Bogoliubov modes hybridize and become dynamically unstable. Because of the local breaking of the Pτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadcfajugGbiabes8a0baa@3A2E@ - symmetry, the modes form conjugate pairs, in such a way that there is a transfer of energy from the gravitational wave (damping) to the BEC field (growth). In short, this means that the instability mechanism triggering the formation of large dark-matter structures is accompanied by the breaking of the U( 1 )×PT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadwfajuaGpaWaaeWaaOqaaKqzGeWdbiaaigdaaOWdaiaa wIcacaGLPaaajugib8qacqGHxdaTcaWGqbGaamivaaaa@3F86@ symmetry. Remarkably, our findings may also constitute an alternative explanation why primordial gravitational waves are quite hard to detect: they just vanish and give away their energy to the formation of large-scale structures. In a near future, our work could strongly benefit from numerical GR tools, both in weak and strong gravity scenarios, which could correctly describe the saturation at later stages due to the nonlinearity in the KleinGordon equation and, eventually, the effects of curvature due to the presence of massive objects.

Acknowledgments

One of the authors (H. T.) acknowledges the Security of Quantum Information Group for the hospitality and for providing the working conditions, and financial support from Fundac¸˜ao para a Ciˆencia e a Tecnologia (Portugal) through grant number SFRH/BPD/110059/2015. The work of C.B. is supported by the European Research Council (ERC-2010AdG Grant 267841).

Conflicts of interest

The author declares there is no conflict of interest.

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