Scale-invariant scalar field dark matter through the Higgs portal

Journal of High Energy Physics, May 2018

Abstract We discuss the dynamics and phenomenology of an oscillating scalar field coupled to the Higgs boson that accounts for the dark matter in the Universe. The model assumes an underlying scale invariance such that the scalar field only acquires mass after the electroweak phase transition, behaving as dark radiation before the latter takes place. While for a positive coupling to the Higgs field the dark scalar is stable, for a negative coupling it acquires a vacuum expectation value after the electroweak phase transition and may decay into photon pairs, albeit with a mean lifetime much larger than the age of the Universe. We explore possible astrophysical and laboratory signatures of such a dark matter candidate in both cases, including annihilation and decay into photons, Higgs decay, photon-dark scalar oscillations and induced oscillations of fundamental constants. We find that dark matter within this scenario will be generically difficult to detect in the near future, except for the promising case of a 7 keV dark scalar decaying into photons, which naturally explains the observed galactic and extra-galactic 3.5 keV X-ray line.

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Scale-invariant scalar field dark matter through the Higgs portal

Received: March Scale-invariant scalar eld dark matter through the Catarina Cosme 0 1 2 Jo~ao G. Rosa 0 1 2 O. Bertolami 0 1 2 0 of the Universe 1 Campus de Santiago , 3810-183 Aveiro , Portugal 2 Rua do Campo Alegre 687 , 4169-007, Porto , Portugal We discuss the dynamics and phenomenology of an oscillating scalar eld coupled to the Higgs boson that accounts for the dark matter in the Universe. The model assumes an underlying scale invariance such that the scalar eld only acquires mass after the electroweak phase transition, behaving as dark radiation before the latter takes place. While for a positive coupling to the Higgs eld the dark scalar is stable, for a negative coupling it acquires a vacuum expectation value after the electroweak phase transition and may decay into photon pairs, albeit with a mean lifetime much larger than the age We explore possible astrophysical and laboratory signatures of such a dark matter candidate in both cases, including annihilation and decay into photons, Higgs decay, photon-dark scalar oscillations and induced oscillations of fundamental constants. We nd that dark matter within this scenario will be generically di cult to detect in the near future, except for the promising case of a 7 keV dark scalar decaying into photons, which naturally explains the observed galactic and extra-galactic 3.5 keV X-ray line. Beyond Standard Model; Cosmology of Theories beyond the SM 1 Introduction 2 Dynamics before electroweak symmetry breaking 2.1 In ation 2.2 2.3 Radiation era Condensate evaporation eld Dynamics after the electroweak symmetry breaking 3.1 3.2 Negative Higgs-portal coupling Positive Higgs-portal coupling Phenomenology 4.1 Astrophysical signatures 4.2 Laboratory signatures 4.1.1 4.1.2 4.2.1 4.2.2 4.2.3 Dark matter annihilation Dark matter decay Invisible Higgs decays into dark scalars Light shining through walls Oscillation of fundamental constants Cosmological implications of the spontaneous symmetry breaking 2.3.1 2.3.2 Higgs annihilation into higher-momentum particles Perturbative production of particles by the oscillating background HJEP05(218)9 Dark matter is one of the most important open puzzles of modern cosmology and funda Universe and later decoupled to yield a frozen-out abundance. The so-called \WIMP miracle", where the relic WIMP abundance matches the present dark matter abundance for weak-scale cross sections, makes such scenarios quite appealing, with a plethora of candidates within extensions of the Standard Model at the TeV scale. However, the lack of experimental evidence for such WIMPs and, in particular, the absence of novel particles at the LHC, strongly motivates looking for alternative scenarios. An interesting candidate for dark matter is a dynamical homogeneous scalar eld that is oscillating about the minimum of its (quadratic) potential, and which can be seen as a condensate of low-momentum particles acting coherently as non-relativistic matter. Scalar is the Higgs portal, where dark matter only interacts directly with the Higgs eld, H. A coupling of the form g2j j2jHj2 should generically appear for any complex or real scalar eld, since it is not forbidden by any symmetries, except for the QCD axion and analogous pseudo-scalars where such an interaction is forbidden by a shift symmetry. The Higgs portal for dark matter has been thoroughly explored in the context of scalar WIMP-like candidates [1{14], but only a few proposals in the literature discuss the case of an oscillating scalar condensate [15{19]. In this work, we aim to ll in this gap and consider a generic model for scalar eld dark matter where, like all other known particles, the dark scalar acquires mass exclusively through the Higgs mechanism, i.e. no bare scalar mass term in the Lagrangian is introduced for dark matter. While the Standard Model gauge symmetries forbid bare masses for chiral fermions and gauge bosons, this is not so for scalars, since j j2 is always a gauge-invariant operator. Scalar mass terms are, however, forbidden if the theory is scale-invariant (or exhibits a conformal invariance). This has arisen some interest in the recent literature, with the possibility of dynamically generating both the Planck scale and the electroweak scale through a spontaneous breaking of scaleinvariance. In fact, with the inclusion of non-minimal couplings to gravity allowed by scale-invariance, one can generate large hierarchies between mass scales from hierarchies between dimensionless couplings and naturally obtain an in ationary period in the early Universe, as shown in refs. [20{23].For other scenarios with scale-invariance and viable dark matter candidates, see also refs. [24{29]. In this work we will pursue this possibility, considering a model of scalar- eld dark matter with scale-invariant Higgs-portal interactions. We assume that scale invariance is spontaneously broken by some unspeci ed mechanism that generates both the Planck scale, MP , and a negative squared mass for the Higgs eld at the electroweak scale, thus working within an e ective eld theory where these mass scales are non-dynamical. The important assumption is that scale-invariance is preserved in the dark matter sector, such that the dark scalar only acquires mass after the electroweak phase transition (EWPT). In addition, our scenario has also a U( 1 ) symmetry (or Z2 symmetry for a real scalar, as we { 2 { discuss later on) that, if unbroken, ensures the stability of the dark scalar. Its dynamics are thus fully determined by its interaction with the Higgs boson and gravity, as well as its self-interactions, all parametrized by dimensionless couplings. The relevant interaction Lagrangian density is thus given by: where the Higgs potential, V (H), has the usual \mexican hat" shape, g is the coupling between the Higgs and the dark scalar and is the dark scalar's self-coupling. The last term in eq. (1.1) corresponds to a non-minimal coupling of the dark matter eld to density is an extension of the model that we considered in ref. [18] where self-interactions played no role in the dynamics, giving origin to a very light dark scalar, m and therefore a very small coupling to the Higgs boson, g 10 16. Such feeble interactions make such a dark matter candidate nearly impossible to detect in the near future, and in this work we will show that the inclusion of self-interactions allows for heavier and hence O 10 5 eV , more easily detectable dark scalars. On the one hand, if the Higgs-dark scalar interaction has a positive sign, the U( 1 ) symmetry remains unbroken in the vacuum and the dark scalar is stable. On the other hand, if the interaction has a negative sign, the U( 1 ) symmetry may be spontaneously broken, which can lead to interesting astrophysical signatures, as we rst observed in ref. [19]. In this work we wish to provide a thourough discussion of the dynamics and phenomenology in both cases, highlighting their di erences and similarities and exploring the potential to probe such a dark matter both in the laboratory and through astrophysical observations. This work is organized as follows. In the next section we explore the scalar eld dynamics from the in ationary period to the EWPT, where the Higgs-portal coupling plays a negligible role. In section 3 we discuss the evolution of the scalar eld after the EWPT, analyzing separately the cases where the Higgs-portal coupling is positive or negative, and computing the present dark matter abundance in both scenarios. We explore the phenomenology of these scenarios in section 4, discussing possible astrophysical signatures and experiments that could test these scenarios in the laboratory. We summarize our conclusions and discuss possible avenues for future work in this subject in section 6. 2 Dynamics before electroweak symmetry breaking Before the EWPT, the dynamics of the eld does not depend on the sign of the coupling to the Higgs, since it plays a sub-leading role. This allows us to describe the behavior of the eld without making any distinction between the two cases. We will rst explore the dynamics during the early period of in ation, where the non-minimal coupling to gravity plays the dominant role, and then the subsequent evolution in the radiation era, where the eld dynamics is mainly driven by its quartic self-coupling. { 3 { We consider a dark scalar eld non-minimally coupled to gravity, with the full action being S = Z p 1 2 g d4x MP2 l f ( ) R 1 2 (r ) 2 V ( ) ; (2.1) p where = = 2 and f ( ) = 1 eld component in a at FRW Universe is thus: 2=MP2 l.1 The equation of motion for the homogeneous  + 3H _ + V 0 ( ) + R , such that the dynamics during in ation is mainly driven by the non-minimal coupling to gravity. Since during in ation R ' 12 Hi2nf , where the nearly constant Hubble parameter can be written in terms of the tensor-to-scalar ratio r: the e ective eld mass is: with m > Hinf for > 1=12. Hinf (r) ' 2:5 1013 r 0:01 1=2 GeV ; m ' p12 Hinf ; The mass of the dark scalar has to exceed the Hubble parameter during in ation, since otherwise it develops signi cant uctuations on super-horizon scales that may give rise to observable cold dark matter isocurvature modes in the CMB anisotropy spectrum, which are severely constrained by data [30]. This implies that the classical eld is driven towards the origin during in ation, but its average value can never vanish due to its de Sitter quantum uctuations on super-horizon scales. Any massive scalar eld exhibits quantum uctuations that get stretched and ampli ed by the expansion of the Universe. For m =Hinf > 3=2 ( > 3=16), the amplitude of each super-horizon momentum mode is suppressed by the mass of the dark scalar eld, yielding a spectrum [31]: 2 j kj ' Hinf 2 2 Hinf m Hinf 2 2 1 p12 ; where a(t) is the scale factor. Integrating over the super-horizon comoving momentum 0 < k < aHinf , at the end of in ation, the homogeneous eld variance reads: 1Since since MPl and . 1, f ( ) ' 1 and there are no signi cant di erences between the Jordan and Einstein frames. In addition, is not the in aton and, therefore, the gravitational dynamics during in ation is not altered by , regardless its non-minimal coupling to curvature. { 4 { (2.3) (2.4) (2.5) (2.6) setting the average amplitude of the eld at the onset of the post-in ationary era, inf : inf = p h 2i ' < 3=16, m =Hinf < 3=2, the spectrum is given by [31]: (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) + g 1 2 4 r V ( ) R 2 + 2 2R ; (2.15) leading to a cold dark matter isocurvature power spectrum of the form: with the corresponding dimensionless power spectrum: PI (k) ' k3 the end of in ation, we obtain for the eld variance: . As above, integrating over all super-horizon modes at HJEP05(218)9 Isocurvature perturbations are bounded by the Planck collaboration in terms of the ratio iso (k) = R 2 (k) + I2 (k) ; where 2R ' 2:2 lower bound: 109 is the amplitude of the adiabatic curvature perturbation spectrum generated by the in aton eld. Since the uctuations in and in the in aton eld are uncorrelated, we may use iso (kmid) < 0:037, for kmid = 0:050 Mpc 1, corresponding to the upper bound on uncorrelated isocurvature perturbations imposed by Planck [30]. For 55 e-folds of in ation, this yields . 1:3, implying that m & 0:75 Hinf and setting a which thus gives for the average homogeneous eld amplitude at the end of in ation: inf = p h 2i ' Hinf is the Ricci tensor. The energy density and pressure of the eld are thus, The eld is in an overdamped regime until the e ective eld mass, m the Hubble parameter in this era. At this point, the eld begins oscillating about the = p3 origin with an amplitude that decays as a 1 / T and radiation. a 4, thus behaving like dark The temperature at the onset of the post-in ationary eld oscillations can be found by equating the e ective eld mass with the Hubble parameter: Trad = 1=4 p inf MPl 270 2 g 1=4 ; below the reheating temperature, Trh its amplitude decreases with T : where g is the number of relativistic degrees of freedom. The temperature Trad is thus pMPl Hinf . Since the eld behaves like radiation, Using _ m for an underdamped eld, we can easily see that so that the dark scalar will not a ect the in ationary dynamics even for large values of its non-minimal coupling to gravity. Note that this assumes also that the Higgs eld does not acquire a large expectation value during in ation, which is natural since de Sitter uctuations also generate an average Higgs value . Hinf [9]. After in ation, the eld will oscillate about the minimum of its potential and its e ective mass m H, and we show in appendix A that, for . 1, all modi cations to the dark eld's energy density and pressure due to its non-minimal coupling to gravity become sub-dominant or average out to zero, thus recovering the conventional form for = 0. In addition, since R = 0 in a radiation-dominated era and R non-minimal coupling's contribution to the eld's mass also becomes negligible. Thus, we may safely neglect the e ects of the non-minimal coupling to gravity in its post-in ationary evolution, hence its only role is to make the eld su ciently heavy during in ation so to prevent the generation of signi cant isocurvature modes in the CMB anisotropy spectrum. O(H2) in subsequent eras, the 2.2 Radiation era After in ation and the reheating period, which we assume for simplicity to be \instantaneous", i.e. su ciently fast, the Universe undergoes a radiation-dominated era for which R = 0. Above the electroweak scale, the potential is then dominated by the quartic term, V ( ) ' 4 4 : rad (T ) = Trad inf T = 2 g 270 1=4 1=2 inf MPl T 1=4 : { 6 { a a . Hi4nf for (2.16) in the vicinity of the origin, with average thermal uctuations h 2 T 2, as shown e.g. in ref. [32]. We can use this to show that the Higgs-dark scalar eld interactions play a subdominant role before the EWPT. In particular, in the parametric regime that we are interested in (see eqs. (3.13) and (3.22)), g 1; since h 2 MTP53l according to ref. [32]. the Higgs The dark scalar continues to behave like radiation until the EWPT, at which point eld acquires its vacuum expectation value, H = h=p2 = v=p2, generating a mass for the dark scalar. The EWPT transition will be completed once the leading thermal contributions to the Higgs potential become Boltzmann-suppressed, which occurs approximately at TEW mW , where mW is the W boson mass. Comparing the quadratic and quartic terms in the dark scalar potential at TEW, we nd: where we used that g S ' 86:25 for the number of relativistic degrees of freedom contributing to entropy at TEW, and de ned EW rad(TEW). Since r . 0:1 [30], in the parametric regime g & 10 4 1=4, . 1 , we conclude that the quadratic term is dominant at TEW, implying that the eld starts behaving as non-relativistic matter already at the EWPT. We will verify explicitly below that this is, in fact, the parametric regime of interest for the eld to account for all the present dark matter abundance. For T < TEW the dynamics of the eld is di erent depending on whether the Higgsportal coupling is positive or negative, so that we will study these cases separately. We need, however, to ensure that the coherent behaviour of the homogeneous scalar eld is preserved until the EWPT, as we explore in detail below. 2.3 Condensate evaporation If its interactions are su ciently suppressed, the dark scalar eld behaves as a long-lived oscillating condensate, i.e, a set of particles with zero (or actually sub-Hubble) momentum that exhibit a collective behavior, and is never in thermal equilibrium with the SM particles. Thereby, we must analyze the constraints on g and to prevent its thermalization and evaporation into a WIMP-like candidate, the phenomenology of which was studied in ref. [9]. There are two main processes that may lead to condensate evaporation, as we describe in detail below - Higgs annihilation into higher-momentum particles and the perturbative production of particles by the oscillating background eld. 2.3.1 Higgs annihilation into higher-momentum particles Higgs bosons in the cosmic plasma may annihilate into pairs, with a rate given by, for T & TEW: hh! = nh h vi ; { 7 { (2.22) (2.20) (2.21) For T < TEW, the Higgs bosons decay into Standard Model degrees of freedom and the production of stops, so that we must require hh! of the dark scalar condensate for T > TEW. Since hh! . H to prevent the thermalization / T and H / T 2 , the strongest constraint is at TEW, which leads to an upper bound on g: g . 8 corresponding to an upper bound on the dark scalar's mass m Another possibility for the condensate's evaporation is the production of particles from the coherent oscillations of the background condensate. Particle production from an oscillating background eld in a quartic potential has been studied in detail in refs. [17, 33, 34]. For T > TEW, particles are massless and interact with the background eld. The coupling between the background eld and particle uctuations, , is: Lint = 3 2 2 For a quartic potential, the oscillating condensate evolves as [34]: (t) = p 5 4 3 4 rad 1 X n=1 ei(2n 1)!t + e i(2n 1)!t e 2 (2n 1) 1 + e (2n 1) ; with ! = 12 p 6 ( ) ( 3 ) ticle production rate is then given by [34]: and m = p3 rad before the EWPT. The corresponding parwhere v c 1 and nh is the number density of Higgs bosons in the termal bath, Before the EWPT, the momentum of the Higgs particles is of the order of the thermal bath temperature, jpj T , and the cross-section for the process is nh = ( 23 ) T 3 : g 4 64 1 1 + m2 h m2 : The energy density, , can be averaged over eld oscillations: ! = ' 8:86 { 8 { We have computed this numerically, using eq. (2.27), and obtained: implying a particle production rate: 1 most stringent constraint is at TEW, where rad = EW, which places an upper bound on the dark scalar's self-coupling: If the constraints (2.25) and (2.32) are satis ed, the dark scalar is never in thermal equilibrium with the cosmic plasma, behaving like an oscillating condensate of zero-momentum particles throughout its cosmic history. We will see below that eq. (2.32) yields the most stringent constraint. 3 Dynamics after the electroweak symmetry breaking At the EWPT, the eld starts behaving di erently depending on the sign of the Higgsportal coupling. If this coupling is negative, the eld may acquire a vacuum expectation value and thus become unstable, although su ciently long-lived to account for dark matter. On the other hand, if the coupling is positive, the U( 1 ) symmetry remains unbroken after the EWPT and the eld never decays. In this section, we explore the eld dynamics in each case. 3.1 Negative Higgs-portal coupling This case has been explored in ref. [19], and here we intend to give a more detailed description of the eld dynamics. The relevant interaction potential is: V ( ; h) = 2 h2 + 4 + h 4 h 2 v~ As soon as the temperature drops below the electroweak scale, both the Higgs eld and the dark scalar acquire non-vanishing vacuum expectation values for g4 < 4 h, which we take to be the relevant parametric regime: h0 = 1 g 2 4 g 4 4 h 0 4 v~ 1=2 g 2 p g3 2 { 9 { where v = 246 GeV. This induces a mass-mixing between the \ avour" basis and h elds, described by the squared mass matrix v; 0 = g v p2 ; p 2 2 h For small mixing, the mass eigenvalues are approximately given by: The mass eigenstates can then be written in terms of the avour-basis elds as: ~ = h ; h~ = h + This mixing is extremely relevant for the direct and indirect detection of the dark scalar as we discuss in section 4.2. In this scenario, di erentiating the Lagrangian twice with respect to , yields the mass of the eld: m = g v : As we have seen in section 2.2, the eld starts behaving as cold dark matter at TEW mW . The amplitude of the oscillations at this stage is given by eq. (2.19), which can be rewritten as: EW ' From eq. (3.9), in the parametric regime g & 10 4 1=4, EW . 0, for & 1. This implies that the potential minimum will move smoothly from the origin to 0, where the eld starts to oscillate with an amplitude DM = xDM 0, with xDM . 1. In fact, a rigorous study of the dynamics of the eld for TEW < T < TCO, where TCO corresponds to the electroweak crossover temperature, would necessarily involve numerical simulations that are beyond the scope of this paper. We may nevertheless estimate the uncertainty associated to the eld's amplitude. Since TEW . TCO by an O ( 1 ) factor, and given that T while behaving as radiation and T 3=2 while behaving as non-relativistic matter, the eld's amplitude might decrease by at most an O ( 1 ) factor as well. Therefore, we expect the eld's amplitude to be smaller than 0 at TEW by xDM . 1. Notice that xDM is thus not an additional parameter of the model, but only a theoretical uncertainty in our analysis that does not a ect the order of magnitude of the dark matter abundance and lifetime, as we shall see below. (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) As soon as the eld reaches 0, it stops behaving like dark radiation and starts to oscillate as cold dark matter. The eld's amplitude of oscillations about 0 evolves according to At this point, the number of particles in a comoving volume becomes constant: number density and g S ' 86:25 at TEW. Using this, we can compute the present dark matter abundance, ;0 ' 0:26, obtaining the following relation between the latter and m : m = (6 ;0)1=2 g S where g S0, T0 and H0 are the present values of the number of relativistic degrees of freedom, CMB temperature and Hubble parameter, respectively. We thus obtain the following relation between the couplings g and : This expression satis es the constraint g4 < 4 h and EW . 0 . Given this relation between couplings, eq. (2.32) yields, as anticipated, the strongest constraint, limiting the viable dark matter mass to be . 1 MeV. It should be emphasized that the properties of our model depend strongly whether the U( 1 ) symmetry is global or local. In particular, if the symmetry is global, it leads to tight constraints on the dark scalar mass. We study the cosmological implications of the spontaneous symmetry breaking (both local and global) in section 5. 3.2 Positive Higgs-portal coupling As we have seen before, for T > TEW, the eld is oscillating as radiation. If the coupling between the Higgs and the SDFM has a positive sign, as soon as the EWPT occurs, the only eld that undergoes spontaneous symmetry breaking is the Higgs eld. The corresponding interaction Lagrangian is: Lint = + g 2 4 2 v2 + 4 4 + h 4 h 2 v The interactions with the Higgs are responsible for providing the dark scalar mass, which is given in this case by m = p g v ; 1 2 (3.10) (3.11) (3.12) (3.13) (3.14) (3.15) HJEP05(218)9 + = g so that the number of particles per comoving volume reads: n s = 45 1 4 2 g S m 2 inf 1 Tr2ad TEW : Analogously to the negative coupling case, we may compute the eld's mass, taking into account the present dark matter abundance, m = p6 ;0 H0 MPl inf g ' 9 di ering from the mass of the negative coupling case, eq. (3.8), by a factor of p2 . Although there is no spontaneous U( 1 ) symmetry breaking, there is in practice an e ective massmixing between the Higgs and dark scalar elds, given that the latter's value is oscillating about the origin but does not vanish exactly, yielding an e ective mixing parameter: where denotes the time-dependent eld amplitude. Since the dark matter energy density depends on the amplitude of the eld, HJEP05(218)9 Eq. (3.16) can be rewritten as At TEW, the eld starts to behave as cold dark matter, and its amplitude varies as: leading to the following relation between g and : Once again, eq. (2.32) yields the strongest constraint and, as in the case of the negative coupling, the dark scalar's mass must be smaller than 1 MeV to account for the present dark matter abundance. It must be highlighted that, in this scenario, once the dark matter abundance is xed, the value of g depends on all other parameters of the model, , r and , contrary to the negative coupling case, where the Higgs-portal coupling is only related to as given in eq. (3.13). In other words, in the positive coupling case the eld's mass depends on the initial conditions set by in ationary dynamics, whilst in the case with spontaneous symmetry breaking the nal dark matter abundance is e ectively independent of the initial conditions, and all dynamics depends essentially on the vacuum expectation value, 0 which sets the amplitude of eld oscillations. Having discussed the dynamics of the dark scalar throughout the cosmic history and determined the parametric ranges for which it may account for the present dark matter abundance, in this section we discuss di erent physical phenomena that may be used to probe the model. This includes searching for direct signatures in the laboratory, as well as for indirect signals in astrophysical observations. Given the smallness of the Higgs portal coupling required for preventing the scalar condensate's evaporation before the EWPT, it is likely that any signatures of such a dark matter candidate require large uxes of particles or the particles it interacts with, which may be di cult to produce in the laboratory. Astrophysical signatures may, however, be enhanced by the large dark matter density within our galaxy and other astrophysical systems, and thus we shall explore in this section the possibility of indirectly detecting the dark scalar through its annihilation or decay. Dark matter annihilation galactic center. tion is given by: Over the past decades, the emission of a 511 keV -ray line has been observed, by several experiments, in a region around the galactic center (see e.g. ref. [37] for a review). The 511 keV line is characteristic of electron-positron annihilation, as has been shown by the INTEGRAL/SPI observations [38{40] and has a ux of GC = 9:9 This photon excess in the galactic bulge is intriguing because, although it is common to nd positron sources in the galaxy (namely, from core collapse of supernovae and low-mass X-ray binaries), most of these astrophysical objects are localized in the galactic disk rather than in the galactic center [37]. An alternative possibility to explain this line is considering dark matter annihilation into an e e+ pair. Although light WIMPs are practically excluded as a possible explanation [42], other plausible dark matter candidates could predict this photon excess. In this context, we study the annihilation of particles into e e+ and into photons, presenting the predictions for the photon ux associated with our dark matter candidate in the In general, the ux of photons from an angular region from dark matter annihilaann = 2m2 N h vi Z 1 4 l:o:s d ; Z where is the dark matter density for a given pro le, r is the radial distance from the galactic center, is the angle between the direction of observation in the sky and the galactic center, the integral is evaluated along the line-of-sight (l.o.s) and N is the number of photons produced by the annihilation [43]. We can split eq. (4.1) into two parts: one that only depends on the particle physics, 2m2 N h4 vi 1 ; (4.1) (4.2) and another one depending only on the astrophysical properties: Z d ; Z and analyze each one separately. jpf j = m q 1 Let us rst focus on the particle physics component. Consider the case in which the particles annihilate via virtual Higgs exchange, H , and the latter decays into Standard Model particles. Since m . 1 MeV (by the constraint imposed on , eq. (2.32)), the possible nal decay products are only electron-positron pairs and photons for non-relativistic dark scalars. Following eq. (4.2), it is necessary to compute the cross section of the process. In the center-of-mass frame, using E ' m , the 4-momentum of H is ph = (2 m ; 0), implying that the invariant mass of the virtual Higgs is mH = 2 m . By energy conservation, if the nal state is a fermion-antifermion pair, the momentum of each particle reads mf2 =m2 , where mf is the fermion's mass. Using this, the cross section for dark matter annihilation into a virtual Higgs and its subsequent decay into fermions reads: h vrelifermions ' 8 Nc D v2 m4 h 4m2 h h where Nc is the number of colors (Nc = 1 for leptons and Nc = 3 for quarks), h = 4:07 10 3 GeV [35] is the total Higgs decay rate, and D = 1 for the case where the U( 1 ) symmetry is spontaneously broken and D = 4 otherwise. In the case where we have a pair of photons in the nal state, the momentum of each photon is jp j = m . The cross-section for this process is: (4.3) (4.4) (4.5) (4.6) (4.7) HJEP05(218)9 where GF = 1:17 constant and F = X f Nc Qf2 A1H=2 ( f ) + A1H ( w) ' 3 11 is a factor that takes into account the loop contributions of all charged fermions and the W boson to H ! , with i = 4mi2=m2 1 for all particle species involved, for m MeV [44]. We point out that for the decay of a virtual Higgs all charged fermions give essentially the same loop contribution, whilst for an on-shell Higgs boson only the top quark contributes signi cantly. Concerning the astrophysical part, one needs to consider the pro le that best describes the dark matter distribution in a particular region of the Universe to compute the so-called J -factor in eq. (4.3). In the case of the galactic center, the most appropriate models are the cuspy pro le ones. Following ref. [35], using a generalized Navarro-Frenk-White pro le (NFW) to describe the dark matter distribution in the galactic center, h vreli ' 32p2 3 v2 D m4 GF QED m2 2 4m2 + m2 2 h h F 2 ; 10 5 GeV 2 is Fermi's constant, QED ' 1=137 is the ne structure (r) = s r rs 1 + r rs 3+ ; where r is the spherical distance from the galactic center, rs = 19:5 kpc; hence, the corresponding J -factor is: = 1:2, s = 0:74 GeV=cm3 and Using the last expressions, the predicted ux of photons originated by the annihilation of particles into a virtual Higgs and the decay of the latter into an electron-positron pair, for the galactic center, is given by: h 4m2 h h m v 2 me 2 v 1 me2 !3=2 m2 : & 0:511 MeV so that it can annihilate into an electron-positron pair, we can simplify this to give: !e e+ ' 1 which is clearly insu cient to explain the galactic center excess but shows how the electronpositron ux varies with the mass of the dark scalar and hence the Higgs-portal coupling. The ux coming from the annihilation of into a virtual Higgs and its decay into photons reads: ! ' 10 31 D m MeV 4 cm 2 s 1 : The uxes we have found for the annihilation of particles into a virtual Higgs and its decay into an electron-positron pair or photons are thus extremely suppressed, which makes its indirect detection through annihilation a very di cult challenge. 4.1.2 Dark matter decay Our dark matter candidate exhibits a small mixing with the Higgs boson, as we have seen in section 3 and may, therefore, decay into the same channels as the Higgs boson, provided that they are kinematically accessible, and the decay width is suppressed by a factor 2 relative to the corresponding Higgs partial width, !pp = 2 H !XX , where X corresponds to a generic particle. The partial decay width of a virtual Higgs boson into photons is [44]: where F is de ned in eq. (4.6). This yields for the dark scalar's lifetime: H ! = GF QED m3 2 in the case with spontaneous symmetry breaking and (+) 7 keV m 5 sec ; (4.8) (4.9) for the positive coupling. As expected, the case with no spontaneous symmetry breaking yields an e ectively stable dark matter candidate, with the probability of decay into photons being tremendously suppressed. In the negative coupling case, the dark scalar's lifetime is larger than the age of the Universe, but there is a realistic possibility of producing an observable monochromatic line in the galactic spectrum. In fact, the XMM-Newton X-ray observatory has recently detected a line at 3:5 keV in the galactic center, Andromeda and Perseus cluster [45{48]. The origin of the line is not well-established, although dark matter decay and/or annihilation are valid possibilities [48{53]. There are other astrophysical processes that might explain the 3:5 keV line [54]. However, some independent studies suggest that those processes cannot provide a satisfactory explanation [55{57]. In addition, some groups state that such a photon excess is not present in dwarf galaxies, such as Draco [58], while others assert that the line is only too faint to be detected with the technology that we have at our current disposal, not ruling out a possible dark matter decay interpretation [59]. According to refs. [47, 59], the line present in the galactic center, Andromeda and Perseus can be explained by the decay of a dark matter particle with a mass of 7 keV with a lifetime within the interval (6 9) 1027 sec, which also explains the absence of such a line in the blank-sky data set. Concerning the positive coupling scenario, the dark scalar's lifetime (eq. (4.14)) is not compatible with the required range. However, the scenario where undergoes spontaneous symmetry breaking yields a lifetime (eq. (4.13)) that matches the above-mentioned range, up to an uncertainty on the value of the eld amplitude after the EWPT parametrized by xDM . 1, for m = 7 keV. Furthermore, for this mass, we have g ' 3 10 8 and, from eq. (3.13), constraints in eqs. (2.25) and (2.32), as we had already presented in ref. [19]. We should emphasize the uniqueness of the result for the spontaneous symmetry breaking scenario. In this case, our model predicts that the decay of into photons produces a 3.5 keV line compatible with the observational data, with e ectively only a single free parameter, either g or . Recall that the initial conditions of the eld depend on the parameters r and , the rst one determining the initial oscillation amplitude in the radiation era, whilst the latter is responsible for suppressing cold dark matter isocurvature perturbations. However, after the EWPT, the eld oscillates around 0, which only depends on g and . Its amplitude is of the order of 0, meaning that r and do not play a signi cant role below TEW. Therefore, we have three observables that rely on just two parameters (g and ) - the present dark matter abundance, the dark scalar's mass and its lifetime. However, assuming that the dark scalar eld accounts for all the dark matter in the Universe imposes a relation between g and (eq. (3.13)), implying that m and depend exclusively on the Higgs-portal coupling. Hence, the prediction for the magnitude of the 3.5 keV line in di erent astrophysical objects is quite remarkable and, as far as we are aware, it has not been achieved by other scenarios, where the dark matter's mass and lifetime can be tuned by di erent free parameters. 4.2 . MeV, the coupling between the dark scalar and the Higgs boson, g, is extremely small, which hampers the detection of dark matter candidates of this kind. However, the small h mass mixing may nevertheless lead to interesting experimental signals that one could hope to probe in the laboratory with improvements in current technology. In this section, we explore examples of such signatures, namely invisible Higgs decays, photon-dark scalar oscillations in an external electromagnetic eld and dark matter-induced oscillations of fundamental constants, such as the electron mass and Invisible Higgs decays into dark scalars The simplest and cleanest way to test the Higgs-portal scalar eld dark matter scenario in collider experiments is to look for invisible Higgs decays into dark scalar pairs, with a HJEP05(218)9 decay width: where mh is the Higgs mass. The current experimental limit on the branching ratio for Higgs decay into invisible particles is [36]: h! = 1 g4v2 s 8 4 mh 1 4m2 m2 ; h Br ( h!inv) = h!inv h + h!inv Assuming that h!inv = h! following bound on the Higgs-portal coupling g: and using h = 4:07 which is much less restrictive than the cosmological bound in eq. (2.25). Conversely, for m < MeV to ensure the survival of the dark scalar condensate up to the present day, we obtain the following upper bound on the branching ratio: Br ( h!inv) < 10 19 : This is, unfortunately, too small to be measured with current technology, but may serve as motivation for extremely precise measurements of the Higgs boson's width in future collider experiments, given any other experimental or observational hints for light Higgsportal scalar eld dark matter, such as, for instance, the 3.5 keV line discussed above. 4.2.2 Light shining through walls The dark scalar exhibits a small coupling to photons through its mass mixing with the Higgs boson, which couples indirectly to the electromagnetic eld via W -boson and charged fermion loops [44]. This will allow us to explore experiments that make use of the coupling to the electromagnetic eld to probe dark matter candidates, as the case of \light shining through a wall" experiments (LSTW). These experiments are primarily designed to detect WISPs - Weakly Interacting Sub-eV Particles, such as axions and axion-like particles (4.15) (4.16) (4.17) (4.18) (ALPs), taking advantage of their small coupling to photons. Therefore, we intend to apply the same detection principles to our dark scalar eld, given its similarities with ALPs, in terms of small masses and couplings. In LSTW experiments, photons are shone into an opaque absorber, and some of them may be converted into WISPs, which traverse the absorber wall. Behind the wall, some WISPs reconvert back into photons that may be detected. This can be achieved using an external magnetic (or electric) eld, which works as a mixing agent, making the photonWISP oscillations possible and allowing for the conversion of photons into WISPs and vice-versa. Using the paradigmatic case of a pseudo-scalar axion/ALP, the interaction term between the latter and photons is of the form: its dual and a the axion/ALP eld. For where F is the electromagnetic eld tensor, F~ the QCD axion, the coupling ga reads with Fa ' 6 and K O (1 1011 ma with the electromagnetic eld is given by [63, 65]: La = ga 4 a F ~ F = ga a E:B ; ga = QED K 2 Fa 10 15K ma GeV 1 ; In the speci c case of the QCD axion, for L mately given by: 4!=m2a, the LSTW probability is approxiP !a! = 6 and, using the parameters of the LSTW experiment ALPS (\Any Light Particle Search experiment) at DESY [63] (! = 2:33 eV, Bext = 5 T and L = 4:21 m), the probability of La a F F = ga a B 2 E In both cases, the presence of an external magnetic eld, Bext, gives rise to a mass mixing term between a and photons, thus inducing photon-ALPs oscillations. In the relativistic limit, the WISP wavenumber is ka t ! Using this, considering the conversion of a photon into a WISP in a constant magnetic eld of length L, in a symmetric LSTW setup, the conversion probability of a photon into a WISP, P !a, is the same as for the inverse process, Pa! , being given by [63, 64]: 2! m2a , where ! is the photon angular frequency [63]. Therefore, the total conversion probability along the path is simply the square of g 4 Be4xt !4 m8a sin4 m2a L 4 ! : : 1 = ga 4 GeV denoting the axion decay constant, ma the axion mass 10) is a model-dependent factor [60{62]. For scalar WISPs the interaction (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) with ALPS 2010 results [64, 65]. the Lagrangian [66]: where the coupling gh has the form shining a photon through a wall via an intermediate axion with ma P !a! ' 10 53 and the coupling to photons is ga 10 5 eV is about 10 15 GeV 1 . The upgrade of LSTW experiments intends to achieve a sensitivity in the photon-WISP coupling of ga < 10 11 GeV 1, corresponding to an improvement of 3 orders of magnitude comparing In our case, the Higgs coupling to two photons is expressed by the following term in with F given by eq. (4.6). Using the mixing h in eq. (3.7) we can write and we may de ne the e ective coupling of the dark scalar eld to photons, g , as: Lh = gh 4 h F F ; gh = QED F v ; L gh 4 ~ F F ; = gh ; = g (4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.31) where = former, we nd for the negative coupling between and h, and = + otherwise. For the g ( ) 10 26 m xDM 0:5 1 GeV 1 ; while for the case without spontaneous symmetry breaking we obtain: g(+) ' 8 m where we have used the cosmological value of the dark matter energy density. Hence, even for the most promising case with spontaneous symmetry breaking the e ective coupling to photons of our dark scalar is 11 orders of magnitude below the corresponding axionphoton coupling for the same mass, with the probability for LSTW thus being suppressed by 10 44 with respect to the axion. This makes our dark scalar much harder to detect in LSTW experiments, and hence requiring a very substantial improvement in technology. 4.2.3 Oscillation of fundamental constants A light dark scalar, 10 22 eV < m < 0:1 eV, behaves as a coherently oscillating eld on galactic scales, which may cause the variation of fundamental constants, namely the electron's mass, me, and the ne structure constant, The Standard Model Yukawa interactions can be generically written as: Lhff = pgf f f h ; 2 where gf is the Yukawa coupling. Using the mass after the EWPT is thus given by: h mass mixing in eq. (3.7), the electron's me ' p gf v 2 1 p dme CDM ; where CDM corresponds to the present amplitude of the scalar eld and dme is a dimensionless quantity that works as an e ective \dilatonic" coupling and is normalized by the Planck mass, such that [68, 69]: In the case where undergoes spontaneous symmetry breaking, the dilatonic coupling d(me) is d(me) ' 6 v : = m 10 15 eV d(m+e) ' 2 m m m xDM 0:5 1 ; CDM xDM 0:5 ; 1 ; : : (4.33) (4.32) (4.34) (4.35) (4.36) (4.37) (4.38) Similarly, we may compute the variation of the ne structure constant due to the oscillating dark scalar eld. We have seen that the interaction with the Higgs boson introduces a small coupling to the electromagnetic eld (eq. (4.27)), which will induce a variation on the ne structure constant of the form [74]: where, after the EWPT, and The dilatonic couplings dme and d have thus comparable values, both in the case with and without spontaneous symmetry breaking, which is a distinctive prediction of our model. Despite the smallness of these dilatonic couplings, there are a few ongoing experiments designed to detect oscillations of fundamental couplings, using, for instance, atomic clock spectroscopy measurements and resonant-mass detectors [69{77]. For instance, atomic spectroscopy using dysprosium can probe the dme coupling in the mass range 10 24 eV . m . 10 15 eV, for 10 7 . dme . 10 1. In turn, the current AURIGA experiment may reach 10 5 . dme ; d . 10 3 in the narrow interval 10 12 eV . m whereas the planned DUAL detector intends to achieve, for 10 12 eV . m . 10 11 eV, . 10 11 eV, sensitivity to detect 10 6 . dme ; d . 10 2. From eqs. (4.34){(4.38), we conclude that current technology is not enough to probe our model, which may nevertheless serve as a motivation to signi cantly improve the sensitivity of such experiments. We should also point out that ultra-light scalars with mass below 10 10 eV can lead to superradiant instabilities in astrophysical black holes that may lead to distinctive observational signatures, such as gaps in the mass-spin Regge plot as rst noted in [78]. However, these instabilities should only be able to distinguish our Higgs-portal dark matter candidate from other ultra-light scalars, such as axions and axion-like particles, if non-gravitational interactions also play a signi cant dynamical role (see e.g. ref. [62]). 5 Cosmological implications of the spontaneous symmetry breaking Throughout our analysis, we have assumed that the model exhibits a U( 1 ) symmetry which, in the case of a negative sign for the coupling between the Higgs and the dark scalar, is spontaneously broken at the EWPT. The implications of such symmetry breaking depend on whether it is a global or a local symmetry. The Lagrangian density is of the form: The third term on the right-hand-side of eq. (5.3) allows for the decay of into two massless particles, which could imply the complete evaporation of the dark scalar condensate. The corresponding decay width is then: where Lint is given by eq. (1.1). First, let us suppose that the Lagrangian density is invariant under a global U( 1 ) symmetry, i.e. that the complex dark scalar is invariant under the following transformation: where is a constant parameter. Expanding the kinetic term for = introducing the rescaled eld = 0 , we nd the following kinetic and interaction terms between the dark scalar, , and the associated Goldstone boson, : p ei = 2 and such that, using the relation between g and in eq. (3.13), corresponding to = 1 64 m3 2 ; 0 ! 0:5 xDM 1=2 0:5 1=2 5=4 sec 1 ; 5=4 sec : Lint ; ! e i ; 2 2 0 ! 1021 (5.1) (5.2) (5.4) (5.5) (5.6) (5.7) 1 2 y = (5.3) 0 The eld is su ciently long-lived to account for dark matter if placing an upper bound on : ! > tuniv 4 1017 sec, ; and thus restricting the viable range for the dark matter mass to: Therefore, if the U( 1 ) symmetry were global, there would be a stricter constraint on the value of , and our dark matter model could not, in particular, explain the 3.5 keV line. However, we may consider a local U( 1 ) gauge symmetry, where the Lagrangian is invariant under which can be achieved by introducing a gauge eld and covariant derivative such that: This remains nite even in the limit e0 ! 0 and, in fact, it tends to the value in eq. (5.4), since in this limit the symmetry is global and the decay into massless Goldstone bosons is allowed. On the other hand, for m 0 > m =2, the dark scalar's decay into dark photon pairs becomes kinematically forbidden. This requires: e0 > p2 ; where e0 denotes the associated gauge coupling. In this case, expanding the scalar kinetic term in the unitary gauge, where the Goldstone boson is manifestly absorbed into the longitudinal component of the \dark photon" gauge eld, we get: D D y = 1 2 2 1 e02 A A 2 + 2 where the second and the third terms correspond to dark scalar-dark photon interactions and the last term gives the mass of the dark photon, 0: For m 0 < m =2, the dark scalar may decay into two dark photons with a decay width m 0 = e0 0 : ! 0 0 ' 16 1 v u ut1 4 m20 m2 m40 m 2 0 3 + 1 m4 ! A + i e0 A which does not pose a signi cant constraint, given the magnitude of the scalar self-couplings considered in our analysis. Note that even if this condition is satis ed the scalar selfcoupling is stable against gauge radiative corrections, since = self-coupling is typically very small in the scenarios under consideration. e04= & and the It is important to mention that, similarly to the case studied in section 2.3.2, before the EWPT, the dark scalar's oscillations may also lead to the production of dark photons, through the second term in eq. (5.12). This leads to an upper bound on the squared gauge (5.8) (5.9) (5.10) (5.11) (5.12) (5.13) (5.14) (5.15) coupling comparable to that obtained for the scalar self-interactions in eq. (2.32), and thus nevertheless compatible with the stability condition eq. (5.15). Notice that the spontaneous breaking of a U( 1 ) symmetry can lead to the generation of cosmic strings at the EWPT. The ratio between the energy density of such cosmic strings, s, and the background density, c, reads [79] s c G where G is Newton's gravitational constant and is the string's energy per unit length. Nevertheless, as we reported in ref. [19], 0 instance, 0 is always smaller than 1016 GeV for 1016 GeV even for very suppressed (for > 10 66), implying that the ratio eq. (5.16) is negligible and that, therefore, this does not pose additional constraints on the model. Finally, we note that all the dynamics and predictions of our model could be achieved, however, by considering only a real scalar eld with a Z2 symmetry. Spontaneous breaking of such a symmetry then leads to the generation of domain walls at the EWPT, which could have disastrous consequences for the cosmological dynamics. However, it has been argued that a statistical bias in the initial con guration of the scalar eld could e ectively yield a preferred minimum and thus make the domain wall network decay [80]. In particular, according to ref. [80], in ation itself may produce such a bias through the quantum uctuations of the scalar eld that become frozen on super-horizon scales. Since our dark scalar never thermalizes with the ambient cosmic plasma, such a bias could survive until the EWPT and therefore lead to the destruction of any domain wall network generated during the phase transition. A detailed study of the evolution of domain wall networks in the context of the proposed scenario is beyond the scope of this work, but this nevertheless suggests that a real scalar eld, with no additional undesired degrees of freedom, may yield a consistent cosmological scenario for dark matter with spontaneous symmetry breaking. In summary, we conclude that the cosmological consistency of the scenarios with spontaneous symmetry breaking requires either a complex scalar eld transforming under a gauged U( 1 ) symmetry with su ciently large gauge coupling or possibly a real scalar eld transforming under a Z2 symmetry. 6 Conclusions In this work we have analyzed the dynamics and the phenomenology of a non-thermal dark matter candidate corresponding to an oscillating scalar eld that, as all other known elementary particles, acquires mass solely through the Higgs mechanism. The model assumes an underlying scale invariance of the interactions that is broken by an unspeci ed mechanism to yield the electroweak and Planck scales. The dynamics of the scalar eld may be summarized in the following way. During in ation, the eld acquires a Hubble-scale mass through a non-minimal coupling to gravity that drives the classical eld towards the origin, while de Sitter quantum uctuations generate a sub-Hubble eld value on average. The Hubble scale mass also suppresses potentially signi cant cold dark matter isocurvature modes in the CMB anisotropies spectrum. After in ation, this non-minimal coupling plays no signi cant role in the dynamics, which is driven essentially by the scalar potential. After in ation, the eld oscillates in an quartic potential, behaving as dark radiation, until the electroweak phase transition. At this point the eld acquires mass through the Higgs mechanism, and starts behaving as non-relativistic matter. If the Higgs portal coupling is positive, the dark scalar oscillates about the origin in a quadratic potential until the present day, while for a negative coupling it undergoes spontaneous symmetry breaking and oscillates about a non-vanishing vacuum expectation value. Whereas in the former case the present dark matter abundance depends on all model parameters, including the non-minimal coupling to gravity and the scale of in ation, in the latter scenario it is determined uniquely the the Higgs-portal coupling and the dark scalar's self-interactions. The suppression of particle production processes that could lead to the oscillating condensate's evaporation places strong constraints on the value of these couplings, and we generically nd that the dark scalar's mass must lie below the MeV scale. It should be pointed out that the aim of this paper is not to explain the smallness of the dark matter couplings. In fact, in the parametric regime that we are interested in (g= 1=4 10 3 10 2), both g and are technically natural, since their relation is not signi cantly a ected by radiative corrections, which makes them as natural as the electron Yukawa coupling. Nevertheless, small couplings can be naturally achieved in theories with extra dimensions, as we have shown in ref. [18]. While there are several phenomenological consequences of the Higgs-portal interactions of the dark scalar that could allow for its detection, the required suppression of the latter makes this rather challenging in practice. Possible laboratory signatures include invisible Higgs decays, dark scalar-photon oscillations and induced oscillations of the ne structure constant and the electron mass. Indirect astrophysical signatures are also possible, namely dark matter annihilation or decay into photons. All these processes could lead to a robust identi cation of our proposed dark matter candidate, but unfortunately they lie generically below the reach of current technology, even for the case with spontaneous breaking that is generically easier to detect experimentally. An interesting exception that we have already reported in ref. [19] is the decay of a 7 keV dark scalar in the case with spontaneous symmetry breaking, whose decay into photons may naturally explain 3.5 keV emission line observed in the galactic centre, the Andromeda galaxy and the Perseus cluster. In summary, the proposed oscillating scalar eld is a viable dark matter candidate with distinctive observational and experimental signatures, constituting a promising alternative to WIMPs, which have so far evaded detection. While it is certainly amongst the \darkest" dark matter candidates available in the literature, there may already be astrophysical hints for its existence, and we hope that this work may motivate future technological developments that may allow for testing its implications in the laboratory. Acknowledgments C.C. is supported by the Fundac~ao para a Ci^encia e Tecnologia (FCT) grant PD/BD/114453/2016. J.G.R. is supported by the FCT Investigator Grant No. IF/01597/ 2015 and partially by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904 and by the CIDMA Project No. UID/MAT/04106/2013. We would like to thank Igor Ivanov and his colleagues from the CFTP, IST-Lisbon for interesting discussions. A E ects of the non-minimal coupling to gravity on an oscillating scalar eld Consider a homogeneous scalar eld that is oscillating about the minimum of a potential V ( ) much faster than Hubble expansion, i.e. in the regime where the e ective eld mass m H. In this regime we then have that: in a stationary con guration for quantities averaged over the oscillating period, where we have used the equation of motion  = expansion. We thus obtain the virial theorem: V 0( ) discarding the sub-leading e ects of (A.1) (A.2) m2 (A.3) (A.4) (A.5) (A.6) h _2i = h V 0 ( )i = nhV ( )i; where the second line is valid for a potential of the form V ( ) n. Since R H2 and _ m , we may discard the terms that include R or H explicitly in the expressions for the eld's energy density and pressure given in eq. (2.16), assuming . 1. We may then write these quantities approximately as: It is easy check, using the virial relation (A.2), that the term proportional to vanishes on average, thus yielding the usual expressions for the energy density and pressure of a homogeneous scalar eld in minimally-coupled general relativity. 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Catarina Cosme, João G. Rosa, O. Bertolami. Scale-invariant scalar field dark matter through the Higgs portal, Journal of High Energy Physics, 2018, 129, DOI: 10.1007/JHEP05(2018)129