The Photon Spectrum of Asymmetric Dark Stars

CERN-TH-2019-070 CP3-Origins-2019-20 DNRF90 The Photon Spectrum of Asymmetric Dark Stars

Andrea Maselli,1, ∗ Chris Kouvaris,2, 3, † and Kostas D. Kokkotas4, ‡ 1Dipartimento di Fisica, Sapienza Universitá di Roma & Sezione INFN Roma1, P.A. Moro 5, 00185, Roma, Italy 2CP3-Origins, Centre for Cosmology and Particle Physics Phenomenology University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark 3Theoretical Physics Department, CERN, 1211 Geneva, Switzerland 4Theoretical Astrophysics, IAAT, University of Tüebingen, Tüebingen 72076, Germany (Dated: December 8, 2020) Asymmetric Dark Stars, i.e., compact objects formed from the collapse of asymmetric dark matter could potentially produce a detectable photon flux if dark matter particles self-interact via dark photons that kinetically mix with ordinary photons. The morphology of the emitted spectrum is significantly different and therefore distinguishable from a typical black-body one. Given the above and the fact that asymmetric dark stars can have masses outside the range of neutron stars, the detection of such a spectrum can be considered as a smoking gun signature for the existence of these exotic stars.

I. INTRODUCTION Moreover, DM self-interactions could resolve another potential problem related to supermassive black holes at Today there is strong evidence about the existence of high redshifts. Present models of stellar evolution do not dark matter (DM) from a variety of different sources and seem able to explain how these objects can grow to their scales, such as rotational curves of individual galaxies, current mass within the age of the Universe, if they start clusters of galaxies such as the bullet cluster [1], and the as typical stellar remnants. However, collapsing DM can Cosmic Microwave Background [2]. The so-called Col- provide the seeds for such supermassive black holes [10]. lisionless Cold Dark Matter (CCDM) paradigm is well Given the above, if DM possesses sufficiently strong consistent with the observed large scale structure of the self-interactions and is of asymmetric nature, it could Universe. However, this picture changes at small scales. potentially form compact objects. Asymmetric DM is CCDM suffers from several issues that have currently a viable alternative to the thermally produced DM sce- not been resolved within the CCDM scenario. One of nario. An asymmetry in the population between DM them is the core-cusp problem of dwarf galaxies, which particles and antiparticles in the early Universe can lead is related to the fact that dwarf galaxies are observed to to the depletion of the antiparticles via annihilations, re- have flat density profiles in their central regions [3,4], sulting to the survival of the component in excess. Once while cuspy profiles for collisionless DM are predicted by annihilations deplete the antiparticle population, no fur- N-body simulations [5]. The latter also predict dwarf ther annihilations can take place simply because DM con- galaxies with masses that are too large to have not pro- sists only of particles and not antiparticles. Therefore duced stars inside. However such dwarf galaxies have not in such a case, collapsing DM with the appropriate self- been observed yet [6], leading to the so called too-big- interactions needed to evacuate efficiently the energy, will to-fail problem. Furthermore a diversity problem does result in the formation of neutron star-like dark matter exist: galaxies with the same maximum velocity disper- objects. The possibility of asymmetric DM forming “dark sion can differ significantly in the inner core [7,8], in stars" was first explored in the context of fermionic [11] contrast with the results obtained from N-body CCDM and bosonic DM [12] where star profiles and basic prop- simulations. The resolution of such problems can be at- erties were determined. In addition, the possibility of tributed to different factors: e.g. the inclusion of the forming admixed DM-bayonic stars [13, 14] or asym- baryonic feedback could potentially resolve the core-cusp metric DM cores inside neutron stars (NS) [15, 16] also arXiv:1905.05769v3 [astro-ph.CO] 7 Dec 2020 problem, while statistical deviations from mean values exists. In standard cosmological scenarios, the forma- could explain the too-big-to-fail problem. However, an tion of asymmetric dark stars requires an efficient en- attractive alternative solution to the CCDM inconsisten- ergy evacuation mechanism. Such a mechanism was de- cies is the existence of DM self-interactions (see [9] and scribed and studied in [17], in which asymmetric DM is reference therein for a review). Self-interactions of the or- assumed to feature self-interactions mediated by a dark der of 1cm2/g are sufficient, for example, to disperse DM photon.Collapsing DM can evacuate energy and shrink particles and flatten the density profile in dwarf galaxies. further via dark Bremsstrahlung. In this paper was also shown that the collapsing DM cloud can fragment and eventually form dark stars of different masses (depend- ∗ [email protected] ing on the DM and mediator masses and the dark photon † [email protected] coupling). ‡ [email protected] Asymmetric dark stars could in principle be detected 2 via gravitational waves produced from mergers of such almost anywhere in the bulk of a dark star and, as long as objects. The signal can be distinguished from similar their mean free path is larger than the stellar radius, they mass black hole binaries [18–23] or NS ones [24], exploit- can escape. On the contrary, photons produced by NS are ing the effect of tidal interactions during the orbital evo- exclusively emitted from their surface, as the mean free lution. path is very small compared to the radius. This volume Although it seems that it will be difficult to detect vs surface effect has also another dramatic impact on the dark stars without gravitational wave observations, in observed flux. Photons produced at different radii inside this paper here we investigate another possibility given a dark star are redshifted, as they escape, by a different by the emission of a photon flux. Our model is simply amount due to the gravitational potential. As we show described by the following Lagrangian: in the next sections, these features lead to a very peculiar photon spectrum, that is completely different than black- ¯ µ ¯ 1 0 0µν L = Xγ DµX − mXXX − F F body radiation, and thus represent a unique smoking gun 4 µν signal for the discovery of asymmetric dark stars. 1 2 0 0µ 1 2 µ κ 0 µν + m 0 A A + m A A + F F + ... , Throughout the paper we use natural units, in which 2 A µ 2 D µ 2 µν (1) c = ~ = kB = 1. where X is the asymmetric DM particle, which couples 0 to a dark photon Aµ of mass mA0 , and possesses a ki- II. DARK STARS STRUCTURE netic mixing with the ordinary photon κ. We assume that the dark photon acquires its mass via an unspec- In this work we consider fermionic asymmetric dark ified Higgs mechanism. The details of this mechanism stars. We summarise here the basic features of the DM are not important for the purposes of this work. Dµ is 0 equation of state (EoS), referring the reader to [24] (and the usual covariant derivative Dµ = ∂µ − igAµ (g being reference therein) for more details on the stellar models the coupling to the dark photon). We have omitted the and their macroscopic properties. Standard Model part of the Lagrangian. Dark/hidden The fermionic DM particles of mass mX feel the re- photons mixing with ordinary photons have been studied pulsive dark photon interaction which due to the dark thoroughly in literature especially in the context of their photon mass mA0 behaves as a Yukawa potential of the effect on stars [25–30]. Eq. (1) contains also a mass term form for the ordinary photon. Although this is unacceptable αX in the vacuum, we will discuss later on how such a mass V = e−mA0 r, (3) r term can arise in the medium. In the absence of the afore- mentioned mass term in eq. (1), by shifting appropriately with αX being the dark fine structure constant. The the photon field Aµ, one can eliminate the kinetic mixing stellar pressure originates from Fermi exclusion principle term from the Lagrangian, inducing a suppressed cou- and from the short-range interaction (3). Pressure and pling between dark photon and charged Standard Model energy density are computed in the mean field approxi- (SM) particles, while leaving the dark sector intact, i.e., mation [11] as a function of the DM Fermi momentum no coupling between X and Aµ. However we will argue x = pF/mX: later that in the context of a dark star, there are ways ξ(x) 2 m2 to generate such a photon mass term. Shifting the pho- 4 X 6 ρ = mX 2 + 3 αX 2 x , (4a) 2 0 µ 8π 9π mA0 ton field will induce a mixed mass term κmDAµA which 2 upon diagonalization of the mass matrix gives 4 χ(x) 2 mX 6 p = mX 2 + 3 αX 2 x . (4b) 8π 9π m 0 ¯ µ ¯ 1 µν 1 µν A L = Xγ DµX − mXXX − FΦµν FΦ − FΦ0µν FΦ0 , 4 4 The two functions ξ and χ encapsulate the effect of the (2) Fermi-repulsion [31] and read: 0 where Φ and Φ are the new diagonal fields which are p p linear combinations of the old A and A0. The covariant ξ(x) = x 1 + x2(2x2 + 1) − ln x + 1 + x2 , 0 2 2 derivative now reads Dµ = ∂µ − igΦ − igκm /(m 0 − µ D A p 2 p 2 χ(x) = x 1 + x2(2/3x − 1) + ln x + 1 + x2 . mD)Φµ. To leading order Aµ ' Φµ and therefore SM particles couple mainly to Φ. Therefore there is an At low (high) densities eqns. (4a)-(4b) reduce to a poly- effective coupling of the DM X to ordinary photons γ 2 2 2 tropic EoS P = Kρ with index γ ' 5/3 (γ ' 1). ∼ gκmD/(mA0 − mD). Under these assumptions, asymmetric dark stars can produce a spectrum of detectable To build the star’s structure, we solve Einstein’s equa- photons via dark Bremsstrahlung. tions for a spherically symmetric, stationary and static Although the overall luminosity of such objects is sup- spacetime, supplied by the fermionic EoS. We assume the pressed by a power of κ, its magnitude could still be following ansatz for the non-rotating star: significant, and not dramatically smaller than the lumi- 2 µ ν ds = gµν dx dx nosity of a standard NS. This somewhat unexpected re- 2ν(r) 2 2λ(r) 2 2 2 sult can be easily understood. Photons can be produced = −e dt + e dr + r dΩ , (5) 3 in the Schwarzschild coordinates xµ = (t, r, θ, φ), where where e−2λ(r) = 1 − 2Gm(r)/r. The latter leads to the rela- 4 3 3 tivistic stellar equations: Y d pi d k 1 dΠ = , dΠγ = , fi = , (2π)32E (2π)32ω Ei−µ i=1 i e T + 1 dp (ρ + p) m(r) + 4πr3p = −G , (6a) (8) dr r[r − 2Gm(r)] with fi being the Fermi function, and µ and T the DM dm chemical potential and the star’s temperature, respec- = 4πr2ρ , (6b) dr tively. For sake of simplicity, we introduce the variable Ei−µ dν 1 dp Xi = , and recast the phase space volume element = − , (6c) T dr p + ρ dr as: 3 2 which we solve with appropriate boundary conditions at d pi = pi dpidΩi ' pFpidpidΩi, (9) the center of the star, up to the surface, where r = R, where dΩ is the solid angle specified by ~p , and p is m(r) = M and p(R) → 0. The mass-radius profiles of i i F the dark particle Fermi momentum, such that EF = all the configurations considered in this paper, are shown 2 pF/(2mX). in Fig. (1), for different values of the DM mass, m 0 = A The latter can be computed from the star’s energy den- (10MeV, 1MeV) and α = (10−1, 10−2). X sity profile ρ, obtained by solving the relativistic stellar structure equations, namely

Z ∞ 3 Z pF III. BREMSSTRAHLUNG EMISSION g d p 1 2 ρ = 3 E E−µ ' 2 Ep dp , (10) (2π) 0 e T + 1 π 0 We consider the emission of Bremsstrahlung photons µ p 2 2 with momentum k = (ω,~k), produced in a process of q where E = p + mX, and g = 2 reflects the two spin two DM fermions labeled as 1 and 2 scattering to states states of the X. We assume a system of fermions at low 3 and 4 (see Fig.2 for the Feynmann diagram of this temperaturesuch that the system is degenerate i.e., the µ chemical potential is much larger than the temperature, process). Each state is defined by the 4-momentum pi = (Ei, ~pi), where i runs from 1 to 4. The density number µ T . Equation (10) allows to compute the Fermi mo- of emittedphotons per unit time is given by: mentum as a function of the radius inside the star, i.e. pF(r) = pF[ρ(r)]. As an example, in Fig.3 we show the d2N Z values of pF versus the central energy density correspond- = dΠγ dΠ|M|2f f (1 − f )(1 − f )(2π)4 dV dt 1 2 3 4 ing to all the stellar configurations displayed in Fig.1. (4) Replacing the expressions (8)-(9) into the master equa- × δ (p1 + p2 − p3 − p4 − k) , (7) tion (7) then yields:

4 d2N dΠγ Y Z = p4 T 4 dΩ |M|2δ(3)(~p + ~p − ~p − ~p − ~k)× dV dt 16(2π)8 F i 1 2 3 4 i=1 1 Z 1 1 1 1 × dXi δ(X1 + X2 − X3 − X4 − ω/T ) , (11) T eX1 + 1 eX2 + 1 e−X3 + 1 e−X4 + 1

where we have made the assumption that particles are elastic scattering amplitude [32] as: nonrelativistic i.e., E ' m 1. Momentum conservation i X ! requires that ~q = ~p − ~p , and therefore q2 = 2p2 (1 − 1 (~q · ~k)2 1 3 F |M|2 = c|M |2 ~q 2 − = cos θ ) (θ being the angle between ~p and ~p ). The el 2 2 2 13 13 1 3 mXω ω scattering amplitude can be factorized in terms of the 2 2 2pF 2 = c|Mel| 2 2 (1 − cos θ13)(1 − cos θqk) , (12) mXω where

2 2 c = 2(g) = 8παX , (13)

1 ~ The assumption is a posteriori veriﬁed. As it can be inferred θqk the angle between ~q and k, and g the coupling be- from Fig.3, the Fermi energy is always much smaller than m . 2 X tween DM and dark photons (αX = g /(4π)). Here 4

○ ○ ○ ▲ ▲ ○ ○ ▲ ▲ ▲ 100 ● ▲ □ □●□ ● ○ 10 3 = 0.2 □ ● ● ■ ■ □ ○ ■△ △ ■ ▲ △ △ ● △■ △ △ △ ■ ■ ■ △ -2 -1 ● mX 100MeV _α 10 ● ● ▲ ● mX 500MeV _α 10 ■ ■ □ ● ○ X ● x

] △ ] ● ● △ -2 ■ m 250MeV 10 -1 ⊙ m 250MeV α 10 ⊙ □ α ▲ ■ ■ X _ X □ □ ■ X _ x ▲ ▲ □ ● ○ □ ● △

M 1 M 10 ▲ -2 □ - mX 500MeV _α 10 ▲ mX 100MeV _αx 10 ▲ X [ [ ▲ ■△ ● ■ ○ ○ ○ ● ○ -1 2 □ △ -2 □ ○ mX 100MeV _αX 10 10 ○ ○ mX 500MeV _αx 10

M ○ M = 0.1 ▲ △■ ● m 250MeV α 10 -1 □ m 250MeV α 10 -2 □ X _ X ○ □ X _ x ▲ ■ □ = 0.1 -1 -2 △ mX 500MeV _α 10 mX 100MeV _α 10 △ X ○ □ △ x 1 ▲ ■ = 0.01 ■ ○ □ ▲ 10 10 2 10 3 10 3 10 4 R [km] R [km]

FIG. 1. Mass-radius profiles for the fermionic EoS. The configurations are identified by the dark photon mass mA0 = 10MeV −1 −2 (left panel) and mA0 = 1MeV (right panel), by the DM mass mX = (100, 250, 500)MeV, and by the constant αX = (10 , 10 ). Dashed straight lines identify stellar configurations with constant compactness C = M/R.

-1 -2 αX=10 αX=10

△ 200 ▲ ○ mX =100MeV □ ● mX =100MeV 75 m =250MeV □ X ■ mX =250MeV

△ mX =500MeV ▲ mX =500MeV 150 △ ▲ □ ] ] 50 ■ MeV ○ △ MeV ▲ [ 100 □ [ F F ■ p △ p ● ○ □ ▲ FIG. 2. Feynmann diagram of the Bremsstrahlung process we 25 ■ ● ○ △ ▲ consider in this paper. Note that the photon can be attached □ 50 ■ 0 △ ● ▲ ○ □ ■ to any DM leg. In the diagonal basis the DM exchange a Φ ▲ ○ □ ●■ ○ ● ■ and emit a Φ. ● ■

10 9 10 10 10 11 10 12 10 9 10 10 10 11 10 12 10 13 ρ[ g /cm 3] ρ[ g /cm 3] specifies the effective mixing between dark the ordinary photons. As mentioned earlier, the kinetic mixing term FIG. 3. The DM Fermi momentum as a function of the central of eq. (1) can be formally removed by shifting the photon energy density for the three different stellar configurations shown in Fig.1, i.e., m = (100, 250, 500)MeV, and m 0 = field and therefore in such a case no coupling between DM X A 10MeV. The right end of the lines corresponds to the central and photons exists. However, in the case where there is a energy density of the stellar configuration with the maximum nonzero photon mass inside the dark star, the aforemen- mass for fixed DM and dark photon masses as well as αX . tioned shift will remove the kinetic mixing term, while 2 0µ introducing a term κmDAµA , where mD is the photon mass inside the dark star. Upon diagonalization of ical potential of electrons 2. Despite the fact that the the mass matrix, there is an induced DM-photon cou- amount of electrons inside the star might be negligible 2 2 2 pling with strength −gκmD/(mD − mA0 ). Clearly if the to DM, this contribution to the Debye mass comes from photon remains massless inside the dark star, no cou- the direct coupling of photons to electrons (or protons) pling between DM and photons exists. There are (at and is not suppressed by κ. iii) Photons might acquire a least) three potential scenarios and reasons for a photon medium induced mass via a Higgs mechanism. One can to have a nonzero mass inside the dark star: i) DM is envision a Higgs-like scalar field φ coupled to photons and in a degenerate state and the vacuum polarization dia- DM via an interaction e.g. φXX¯ . The nonzero DM den- gram of the photon gets a correction via a loop of DM. sity which translates to a nonzero hXX¯ i sources in turn This is the usual Debye mass contribution which in the a nonzero expectation value for φ, effectively providing case of mD > mA0 (and within the hard-dense-loop ap- a mass for the photon in the DM infested medium (via proximation) takes the simple form mD = gκµ/π, where µ is the chemical potential of DM. In this case = κ. ii) the existence of even small quantities of protons and electrons inside the dark star will also induce a photon 2 Protons will also be present to maintain overall electric neutral- ity. Debye mass of the order of eµe/π where µe is the chem- 5 a Higgs-like mechanism), while photons remain massless perature of the dark star, the kinetic mixing will always outside of the star. We leave for future work the pre- induce a Bremsstrahlung process to real photons. cise study of all three scenarios. In this paper we are going to assume that the photon acquire a mass via any The factor of 2 in eq. (13)comes from the fact that the of the above mechanisms and we focus on the estimate photon can be emitted by any of the two DM particles of the star’s luminosity. We should mention at this point that scatter. The amplitude of the elastic process is given that as long as the photon mass is smaller than the tem- by

2 2 2 2 2 2 4 s + t s + t 2s |Mel| = 2g 2 2 + 2 2 + 2 2 , (14) (u − mA0 ) (t − mA0 ) (t − mA0 )(u − mA0 )

2 2 2 where s = (p1 +p2) , t = (p1 −p3) and u = (p1 −p4) are integration using the energy delta function. The integral the usual Mandelstam variables. To proceed and simplify I1 can be computed analytically. Since all ~pi are on the things we will consider two distinct and orthogonal cases. Fermi surface, the only non-trivial phase space comes In the ﬁrst case we assume that mA0 >> pF , while in the from back-to-back scattering, i.e. ~p1 ' −~p2 and ~p3 ' −~p4. second case we will consider the opposite limit mA0 << In this case pF . In the ﬁrst case the amplitude becomes simple based 2 2 1 Z 32π on the fact that m 0 will always dominate over t and u 2 A I1 = dΩ1dΩ3(1 − cos θ13)(1 − cos θqk) = , 2 p3 3p3 in the denominators of eq. (14), and since s ' 4mX , F F s >> t and s >> u, the elastic scattering matrix now (18) reads reads: and therefore

4 4 2 3 3 64g m d N cT 2 pF 3 |M | ' X , (15) = |Mel| I2 d ω , (19) el 4 dV dt (2π)9 6m2 ω3 mA0 X where the numerical prefactor comes from considering where I2 = I2(ω). all possible diagrams [33]. A symmetry factor of 1/2 Combining eqns. (13), (15), and (19), we have has been taken into account based on the fact that the 2 3 2 3 scattering particles are identical. The previous equation d N 8 T mX pF 3 2 3 = 6 4 3 αX I2 d ω . (20) is strictly valid when mA0 >> pF. To obtain this result dV dt 3 π mA0 ω we have also assumed that the DM particles are non- Given that the photon emission is isotropic and d3ω = relativistic within the star i.e., mX >> pF . The rate of 2 photon production is now given by 4πω dω, we can rewrite eq. (20) as:

2 3 6 d2N 32 m2 p3 d N cT 2 pF 3 = T 3 X F α3 2I dω . (21) = |Mel| I1 × I2 d ω , (16) 5 4 X 2 11 2 3 dV dt 3π m 0 ω dV dt 16(2π) mXω A where we have split the two integrals: In the detector reference frame, time interval and energy of emitted photons are redshifted by the source’s √ √ 4 Z gravitational ﬁeld, i.e. ω∞ = g00ω and dt∞ = dt/ g00, Y (3) I1 = dΩiδ (~p1 + ~p2 − ~p3 − ~p4) where the subscript ∞ identify redshifted quantities, and i=1 g00 is the time-time component of the metric tensor in- 2 × (1 − cos θ13)(1 − cos θqk) , side the star. Note that the latter depends on the radial (17a) coordinate r, and it matches the analytic Schwarzschild expression at the stellar surface, where g = 1 − 2M/R. Z dX dX dX 1 00 1 2 3 Taking into account the redshift and using eq. (21), we I2 = X X −X −X −X +X + ω . e 1 + 1 e 2 + 1 e 3 + 1 e 1 2 3 T + 1 can obtain the ﬂux of photons arriving on an Earth based (17b) detector:

3 √ 3 2 3 The integration of Xi is performed within the interval d N 8 g00 T m p µ µ X F 3 2 = αX I2dω∞ , (22) − T , ∞ ; however, since µ T , we can assume T → 6 2 4 dV dSdt∞ 3 π d mA0 ω∞ ∞, so the integrals can be taken from [−∞, ∞]. In I1 we omit ~k from the delta function since the magnitude where dS is the detector differential area. To get the flux of the latter should be of the order of the temperature we have divided by a factor of 4πd2 (d being the distance which is much smaller than ~pi which are practically equal between Earth and dark star). The reader should keep in magnitude to pF . In I2 we have performed the X4 in mind that g00 is a function of r the distance from the 6 center of the star where the photon was emitted. In gen- for this term in the opposite limit i.e., mA0 >> pF . Here eral, some of the produced photons inside the dark star we are interested in light dark photons i.e., γ >> 1. As might not make it out, due to the possibility of convert- mentioned earlier, the amplitude becomes dependent on ing back to dark photons via dark Compton scattering. the angle θ13 and is not constant anymore. This means This happens when the mean free path of the photon be- that the photon production rate will still be given by 2 comes smaller than the distance it must cross within the eq. (16), with |Mel| being the angle independent piece star in order to get out. We take this absorption effect of the amplitude i.e., the bracket term of eq. (26) and the 0 into account by suppressing the photon flux by an expo- angle integral I1 substituted by an I1 which includes the nential factor of the optical depth of the stellar medium, angle dependent part of eq. (26) and reads ∼ e−τ(r), with Z 2 0 1 (1 − cos θ13)(1 − cos θqk) R Z I1 = 3 dΩ1dΩ3 2 . (27) √ p (1 + γ + γ cos θ13) τ(r) = grrnX(y)σcdy , (23) F r 0 At the limit where γ << 1, one recovers I1 ' I1 (see where nX is the number density of DM particles inside eq. (18)). The angle θqk does not depend on the θ13 the star, σc is the photon-DM scattering cross section and we can simplify things by taking into account that 2 which is given by the dark Compton one multiplied by cos θqk = 1/3. Using that, rotating the coordinate sys- 2 4 2 2 a factor of , i.e., σc = g /(6πmX), and grr is the rr tem so particle 1 is along the z−axis and performing first component of the metric tensor. By including this factor, the trivial integrations over Ω3 and φ1 and then the in- we can now integrate over the volume of the whole dark tegration over θ1, gives to leading order in 1/γ star 2 2 √ 16π m 0 d3N 32 T 3 m2 α3 2 Z R g g I0 = A . (28) = X X 00 rr p3 I r2dr. 1 3p5 5 2 4 τ(r) F 2 F dSdt∞dω∞ 3π d mA0 ω∞ 0 e (24) As mentioned above, the new version of eq. (16), is with Finally, the total energy flux arriving at the detector |M |2 being the angle independent the bracket term of 2 el F = d E is: eq. (26) and the angle integral I substituted by I0 . The dSdt∞ 1 1 photon production rate now reads 4 2 Z ∞ Z R 3 2 32 T mX 3 2 √ pFr F = α I2dz g00grr dr, 2 3 2 5 2 4 X τ d N T 3 2 mX pF 3 3π d mA0 0 0 e = 6 αX 2 3 I2d ω, (29) (25) dV dt 3π mA0 ω where we have introduced the dimensionless variable3 where we have used eq. (13). eq. (29) is the new version z = ω /T . Recall that p is a function of r. of eq. (20) at the limit mA0 << pF . Performing the ∞ F 3 2 Now we would like to consider the opposite limit trivial angular integral of d ω and dividing by 4πd as in eq. (22) we get mA0 << pF . This is slightly more complicated because the denominator of each of the three terms in eq. (14) is 3 3 2 4 d N T √ 3 2 mX pF not simply ∼ m 0 and Mel becomes angle depended. In A = 6 2 g00αX 2 I2dω∞. (30) this case the factorization of the angle integrals via the I1 dV dSdt∞ 3π d mA0 ω∞ and I2 of Eqs. (17) is not valid anymore. Recall that at To complete the set of equations, the mA0 << pF limit the previously examined limit m 0 >> p , M was in- A F el of Eqs. (24) and (25) are given by dependent of angles. At the limit where mA0 << pF one can easily show that the second and third term of eq. (14) √ d3N 4 T 3 m2 α3 2 Z R g g give contributions of the order of ∼ g4m4 /p4 , while the = X X 00 rr p I r2dr X F 5 2 2 τ(r) F 2 4 2 2 dSdt∞dω∞ 3π d m 0 ω∞ e first one of ∼ m /(p m 0 ) which dominates since in this A 0 X F A (31) limit mA0 << pF . Therefore we will keep only the contribution from this term in the following derivation. Within and this approximation the amplitude becomes 4 2 Z ∞ Z R 2 4 T mX 3 2 √ pFr F = 5 2 2 αX I2dz g00grr τ dr. 4 3π d m 0 e 2 4 mX 1 A 0 0 |Mel| = 16g 4 2 , (26) (32) mA0 (1 + γ + γ cos θ13)

2 2 where γ = 2pF /mA0 . In the above we have taken into account the symmetry factor of 1/2 due to identical scat- IV. NUMERICAL RESULTS tering particles. The term within the bracket is the result Based on the discussion above we are now in position to estimate the photon flux and luminosity that can be produced by asymmetric dark stars. At first sight, what con- 3 √ Note that I2 = I2 z/ g00 . trols the photon production is the coefficient . Clearly 7 too small values of will lead to undetectable photon for heavy dark photons within the allowed range of κ signal. As we mentioned earlier an induced photon mass and despite the fact that the star is not opaque to the 2 2 2 will lead to an = κmD/(mD − mA0 ). The kinetic mix- produced photons i.e., the mean free path of the photons ing κ is constrained experimentally [34, 35]. For heavy is much larger than the size of the star and therefore all dark photon masses e.g. of the order of mA0 = 1GeV, photons make it out from the star, the produced photon κ < 10−5. Despite the fact that smaller dark photon spectrum is too small to be detected within current ob- masses are more constrained, they can provide a much servational capabilities. As shown in Fig.4 the situation larger luminosity in dark stars. The reason is twofold. changes once we consider light dark photons, because as Firstly the mass of the dark photon appears in the de- we mentioned not only they lead to larger mixing with nominator of the flux equation (see e.g. eq. (32)) and the ordinary photons, but they also give an overall higher therefore lighter dark photons produce larger luminosi- amplitude. ties. The second reason is also that light dark photons (for example lighter than the medium acquired photon 4 2M m 10MeV _m 250MeV mass m ) saturate to κ providing significant amount ⊙_ A X D 10M _mA10MeV _mX 250MeV 4 ⊙ 20M m 10MeV _m 250MeV of mixing . The value of or κ can also affect the photon 0 ⊙_ A X

200M _mA1MeV _mX 250MeV production in two different ways: generally large values ] ⊙ 500M m 1MeV m 250MeV - 1 ⊙_ A _ X of correspond to larger photon production since the lat- s 500M m 1MeV m 100MeV ⊙_ A _ X 2 - 2 ter scales as ∼ . However in principle, too strong can -4 allow re-conversion of produced photons at the center of cm the star back to dark photons due to large cross section - 1 -8 keV of a dark Compton process. If photons produced at the [ center are reabsorbed by the star, then the emitted pho- N tons will come from layers close to the surface leading to -12 an overall reduction. This represents a major difference with respect to the emission from standard NS where the -3 2 -1 2 bulk of the star does not contribute due to the very short 10 10 - 10 1 10 10 photon mean free path inside the core. As a result, the ω[ keV ] spectrum of photons produced via Bremsstrahlung inside dark stars is qualitatively different from that of NS. In FIG. 4. Flux of received photons as a function of the energy for different dark stars at a distance d = 1kpc, with tem- the latter case, one should expect the usual black-body 8 spectrum (sometimes deformed by the star’s atmosphere) perature of T = 5 × 10 K. Coloured curves refer to different values of the dark particle, dark photon and stellar masses. peaking roughly at a frequency close to the surface tem- For m0 = 10MeV (m0 = 1MeV) we fix κ = 10−4 (κ = 10−8). perature of the star. On the contrary, the potential spec- A A trum from dark stars is not solely produced from photons Figures5 show the photon energy flux (25) produced produced on the surface of the star, but it rather comes by a large variety of dark stars with various masses and from the whole bulk redshifted appropriately depending EoS. We consider sources at a prototype distance of on the depth produced. Therefore although the photon d = 1kpc. However, since F is proportional to 1/d2, spectrum in dark stars is 2 suppressed, this is to some these results can be immediately rescaled to any loca- extent counterbalanced by the fact that the whole vol- tion. In Fig.5 in particular we draw F as a function of ume of the star participates in the photon production. 7 the normalised temperature T7 = T/10 K for dark stars In addition, the fact that the produced photons are not 0 0 with mA = 10MeV and mA = 1MeV. We use two pre- in thermal equilibrium with the medium leads to a dif- −4 −8 ferent shape in the spectrum compared to photon pro- scriptions for κ, choosing κ = 10 and κ = 10 for duction from the surface of a NS where the photons are the heavy and the light dark photon, respectively. The latter plays a key role in determining the amplitude of always in thermal equilibrium with the nuclear matter 0 the flux, which is overall favoured by small values of mA. and therefore the spectrum is that of black-body. The 4 photon spectrum from dark stars in shown Fig.4, where Note that F in eq. (32) is proportional to T , so these results can be extrapolated to any temperature by the we plot the flux of received photons per energy, namely 8 d3N/(dSdt dω ) in eq. (24). For energies smaller than appropriate rescaling. For T . 5 × 10 K all the stel- ∞ ∞ lar models considered lead to values of F smaller than ∼ T , the spectrum is determined by the characteristic −2 −1 −1 1eV cm s . The Bremsstrahlung fluxes grow with the ω∞ of the Bremsstrahlung rate, while the rate drops 2 −2 −1 faster above ∼ T due to a thermal exponential suppres- temperature, and can be as high as F ∼ 10 eV cm s sion. Overall, we have tried different parameter values for massive stellar configurations with M & 50M and T = 109 K. Although the flux in eq. (32) depends di- 2 rectly on the mass of the DM particles as F ∼ mX, it also depends indirectly on mX because the latter affects 4 In fine tuned situations where mD ' mA0 can be become in the stellar density profiles, namely the stiffness of the principle much larger than κ. We will not examine such fine fermion star. Indeed for fixed particle parameters (i.e., tuned cases here. DM and dark photon mass, κ and coupling αX ), more 8 massive stars have higher luminosities. black holes which seem in need of seeds other than typical At this point we can make a comparison between the stellar remnants. luminosity of an asymmetric dark star with that of a NS. Up to now, it was considered that asymmetric dark Assuming a thermally cooling weakly magnetised NS, stars could be detected only via gravitational wave sig- the flux produced at the surface is proportional to the nals. In fact in previous work, we had investigated and 4 2 2 stellar surface temperature, FNS = σBTsurR /d , where computed the tidal deformabilities of such stars and the σB is the Stefan-Boltzmann constant and Tsur the sur- prospects of their discovery via gravitiational waves [24]. face temperature of the NS. A typical NS of 1.5M and In this paper we demonstrate that in some cases dark R ∼ 11km at d = 1kpc, with a temperature within the stars might also be detected by direct observation of a 7 9 5 13 range [10 , 10 ]K, provides a flux of FNS ∈ [10 , 10 ]eV photon spectrum produced inside such a star via a dark cm−2s−1 on Earth. As we already mentioned the lu- Bremsstrahlung provided that dark photons mix kineti- minosity of a dark star scales also as T 4 (see eq. (32)). cally with ordinary photons. We calculated the explicit Therefore for given DM parameters and mass of the dark form of the photon flux in terms of the stellar and EoS star, its photon energy flux will always be a specific frac- parameters. We numerically computed the emitted flux tion of that of a typical NS with the same temperature for a variety of model parameters. Obviously the over- located at the same distance from the Earth. On the all process is strongly affected by the distance between other hand, larger values of κ, or smaller distances may the Earth and the dark star and by the kinetic mixing. change this picture. Let’s take for example the two pul- We found that the numbers of emitted photons is en- sars J0437-4715 and J0108-1431, which are about 140pc hanced for heavier/larger dark stars. Depending on the and 130pc away from the Earth, and have surfaces tem- stellar temperature, the Bremsstrahlung flux can be as peratures of ∼ 105K[36, 37]. All the dark stars shown high as 100 eV cm2s−1 for galactic sources. Although in the left panel of Fig.5 with M & 50M at the same this process is in general smaller than the energy emitted distance, would produce a flux higher than J0437-4715 by standard NS due to black-body radiation, the dark and J0108-1431, provided that their temperature is larger photon spectrum features a spectrum morphology, which than T ∼ 107K. is completely different from a thermal black-body com- A couple of comments are in order here. NS evac- ponent, thus providing a distinct discovery signature for uate energy from the bulk via neutrino emission. In dark stars. Finally it will be interesting to determine the fact the modified Urca process is the main mechanism of rate of cooling for such asymmetric dark stars which will NS cooling that dominates over surface photon emission be dominated by emission (via dark Bremsstrahlung) of in temperatures above ∼ 108K. A process that would dark photons. In case where DM is admixed with bary- be somewhat analogous to our dark Bremsstrahlung is onic matter and electrons, it will also be interesting to the neutrino Bremsstrahlung emission inside a NS stud- study how these particles affect the overall cooling and ied in [38]. However this process scales parametrically luminosity of the dark star. In addition, compact asym- differently with temperature from our case, since they metric dark stars can be composed not only of fermionic are two thermal particles produced (a pair of neutrino- DM as in the case studied here but also of bosonic DM antineutrino) instead of one (the photon) in our case. A (e.g. studied in [12]). In this case the Bremsstrahlung second point is that the dark star luminosity (as it can mechanism can work in a similar way as in our current be seen in Fig.5) can be significantly higher if the value study. However, the emissivity will be different due to the of αX is larger. For a given DM and dark photon mass, fact that in the bosonic case there is no Fermi surface. the value of αX has an upper bound due to constraints In our present study only DM particles sitting within on DM self-interactions from the bullet cluster. However, ∼ T from the Fermi surface can interact and produce this constraint does not apply if X does not account for photons. This Pauli blocking is not present in the case the whole DM abundance but only a fraction of it. of bosonic stars and therefore the amount of produced photons could potentially be larger than the one studied here. Therefore the calculation is completely different. V. CONCLUSION We will address all the above in future work. Acknowledgements.— We would like to thank M. Tyt- Asymmetric DM is an attractive alternative to the gat for useful discussions. A.M. acknowledge sup- thermally produced WIMP paradigm. Due to an inher- port from the Amaldi Research Center funded by the ited asymmetry between particles and antiparticles, DM MIUR program “Dipartimento di Eccellenza” (CUP: annihilations are absent once the population of antiparti- B81I18001170001). CK is partially funded by the Danish cles is depleted. Therefore in case such a DM candidate National Research Foundation, grant number DNRF90, possesses an effective mechanism of evacuating energy, and by the Danish Council for Independent Research, it has been recently demonstrated [17] that such a DM grant number DFF 4181-00055. A.M. and K.K would component could collapse and form compact objects. In like to acknowledge networking support by the COST fact this is also desired in the view of the supermassive Action CA16214. 9

2 -2

] -4 ] - 1 - 1 s

s -2 - 2 - 2 -6 -4 Vcm eV Vcm eV 2M m 10MeV_m 250MeV -8 -6 ⊙_ A X [ ℱ

2M⊙_mA10MeV_mX 500MeV [ ℱ

10 10M⊙_mA10MeV_mX 250MeV 10M m 10MeV_m 500MeV 10 -8 ⊙_ A X 200M _mA1MeV_m 250MeV Log -10 X 50M⊙_mA10MeV_mX 100MeV ⊙ Log 200M m 1MeV_m 500MeV 100M⊙_mA10MeV_mX 100MeV ⊙_ A X 20M m 10MeV_m 250MeV 500M _m 1MeV_m 100MeV -10 ⊙_ A X ⊙ A X 50M m 10MeV_m 250MeV 500M m 1MeV_m 250MeV ⊙_ A X -12 ⊙_ A X

1 25 50 75 100 1 25 50 75 100 T/10 7 K T/10 7 K

FIG. 5. Bremsstrahlung photon energy flux produced by dark stars located at a distance d = 1kpc away from the detector with mX = (100, 250, 500)MeV and mA0 = 10MeV (left panel), mA0 = 1MeV (right panel), as a function of the star’s temperature. −4 −8 −1 We consider different values of the stellar mass, fixing κ = 10 for mA0 = 10MeV, κ = 10 mA0 = 1MeV, and αX = 10 . The dashed horizontal line corresponds to F = 10eV cm−2s−1.

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