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Constraining exotic compact stars composed of bosonic and fermionic dark matter with gravitational wave events

We investigate neutron star-black hole (NS-BH) merger candidates as a test for compact exotic objects. Using the events GW190814, GW200105 and GW200115 measured by the LIGO-Virgo collabration, which represent a broad profile of the masses in the NS mass spectrum, we demonstrate the constraining power for the parameter spaces of compact stars consisting of dark matter for future measurements. We consider three possible cases of dark matter stars: self-interacting, purely bosonic or fermionic dark matter stars, stars consisting of a mixture of interacting bosonic and fermionic matter, as well as the limiting case of selfbound stars. We find that the scale of those hypothetical objects are dominated by the one of the strong interaction. The presence of fermionic dark matter requires a dark matter particle of the GeV mass scale, while the bosonic dark matter particle mass can be arbitrarily large or small. In the limiting case of a selfbound linear equation of state, we find that the vacuum energy of those configurations has to be similar to the one of QCD.


I. INTRODUCTION
Ever since the first direct observation by the LIGO and Virgo collaborations, gravitational waves (GWs) have opened a new opportunity to further the research of astronomical objects and the universe. While the first observation GW150914 [1] not only provided direct proof of the last remaining prediction of Albert Einsteins theory of general relativity, the event also offered deep insight into compact objects and their merging process. For GW150914, the generally accepted explanation of the nature of the compact objects were black holes, as the calculation of the masses, final orbital separation and velocities confidently ruled out the possibility of a merger of neutron stars [1]. However, more exotic, yet so far undetected and thus unverified compact objects, for example boson stars, could not be excluded by the detection [2], especially since no electromagnetic counterpart was found to coincide with the merger. This was in stark contrast to the event GW170817 [3], which was accompanied by a series of multi-messenger events spanning the electromagnetic spectrum and also non-electromagnetic messengers, such as high-energy cosmic rays and neutrinos [4]. From this it was concluded that the most likely explanation for the merging compact objects was a system of binary neutron stars, which merged into a hypermassive neutron star remnant that collapsed within 1.7 seconds into a black hole. This event has provided deep insight into the inner structure of neutron stars and the equation of state (EOS) describing them. Another event, similarly exciting and groundbreaking as the former observations, was the GW190814 event where the LIGO/VIRGO collaboration [5] reported a compact binary coalescence involving a 22 There has been much debate about the nature of this compact object, and some have argued that the secondary was most likely a black hole [11][12][13]. For example, Fattoyev et al. [14] have shown that a stiff equation of state supporting such a super-massive neutron star is inconsistent with constraints from heavy-ion collisions or from the low deformability of medium-mass stars. Sedrakian et al. [15] examined static and fast rotating configurations of hypernuclear stars, yet found that the resulting maximal masses are well below the lower bound of the mass of the secondary in GW190814, Tews et al. [16] determined that if GW190814 was a BH-NS merger, it would impose strong constraints on the typical radius of neutron stars, while the assumption of a BH-BH merger is consistent with current knowledge of the neutron star EOS. Additionally, Nathanail et al. [17] conducted a statistical analysis of GW events and astronomical observations, showing that all measurements are in agreement if the maximum neutron star mass M T OV = 2.2 +0.12 −0.13 M , while M T OV > 2.4 M leads to an underproduction of ejected mass.
However, Abbott et al. [5] did not rule out the possibility of the secondary partner to be an exotic compact object, for example a gravastar, quark star or boson star. This has lead to several hypothesis being proposed to explain the secondary object. For example, Most et al. have shown [18] that rapidly rotating neutron stars are in fact able to explain the signal, even without collapsing into a black hole shortly before the merger. Similarly, Dexheimer et al. [19] considered the possibility of a massive rapidly-rotating neutron star with exotic degrees of arXiv:2110.12972v3 [astro-ph.HE] 20 Jan 2022 freedom, such as quarks and hyperons, while Bombaci et al. have shown that quark stars can reach masses comparable to those of GW190814 [20]. Additionally, Zhang et al. [21] have concluded that the secondary could have been a super-fast spinning pulsar. Other works have assumed modified gravity, where the modified TOV equations give rise to higher neutron star masses than possible in general relativity. Examples of this are Scalar-Vector-Tensor-Gravity as suggested by Moffat [22] or f (R)-gravity [23]. However, the measurements of the LIGO and Virgo collaborations have already put a strong bound on the parameter spaces of modified gravity and the possible upper mass of the graviton [24,25].
In 2021, the LIGO-Virgo collaboration reported two new NS-BH merger events, GW200105 and GW200115 [26], again without an electromagnetic counterpart, but at 1.9M and 1.5M with vastly different secondary masses than in the case of GW190814. We will investigate if the respective secondary compact objects could have been hypothetical stars comprised of dark matter, and use these three events, which cover nearly the entire mass range for neutron stars, to test our models against. Chang et al. [27] have demonstrated that fermion stars can be formed from primordial density perturbations of dark matter. While they found that not all possible stable solutions to the TOV equation can be formed in realistic scenarios, they nevertheless showed that such exotic objects could very well have formed and survived until today. Two new GW events, GW191219 and GW200210 with respective secondary masses of 1.17 +0.07 −0.06 M and 2.83 +0.47 −0.42 M were reported in the Gravitational-Wave Transient Catalog 3 (GWTC-3), collected during the O3 run of LIGO, Virgo and KAGRA [28]. Those events do lie in a fairly interesting mass range, however, as we will discuss later, the constraining power of those signals is low due to their primary companion stars masses.
We consider several different scenarios: First, we discuss the case of stars made up entirely of either bosonic or fermionic dark matter. In this scenario, we assume that the pressure and energy density are proportional to the coupling strength y and square of the number density n, i.e. ∝ y 2 · n 2 , and analytically determine the limiting cases. In the second scenario, we assume the stars to be a mix of bosonic and fermionic dark matter, coupled by a point-like interaction. Similar to the first scenario, we recover an energy density and pressure proportional to the number density squared. After solving the Tolman-Oppenheimer-Volkoff (TOV) equation numerically, we determine bounds on the masses of the dark matter particles and coupling strength. We assume this star to be uncharged with respect to a dark charge so that the number densities of bosons and fermions are balancing each other. In the third scenario, we consider a selfbound linear equation of state, and constrain the respective parameter spaces for the necessary vacuum energy.

II. SELF-INTERACTING BOSON/FERMION STARS
It is known that an object exceeding the Roche-limit and breaking apart due to tidal disruption leaves a clearly distinct signal in the wave event, where the amplitude is affected by the number of fragments as 1/N [29]. Since such a signal was not observed during GW190814, GW200105 and GW200115, we can conclude that the respective secondaries did not break apart, in the following we use the Roche-radius as an estimate for the radii of the objects. From the masses of the objects, we can estimate the Roche-radius of the secondary partner by where M and m are the respective masses of the larger and smaller partner binary objects. This is a nonrelativistic expression, and it has been numerically shown that the fully relativistic equivalent to the Roche-limit is spin dependent and considers effects from e.g. the innermost stable circular orbit (ISCO), whereas the critical radius could be above as well as below the ISCO (see Pannarale et al. [30] and references therein). As a working hypothesis we consider this non-relativistic approximation to investigate how restrictions of the radius constrict the possible parameter spaces of compact objects consisting of exotic dark matter. Such radius constraints could be measured by next generation gravitational wave detectors.
In the case of purely self-interacting fermionic or bosonic dark matter, we consider an energy denstiy of the form and a pressure where n is the number density and c is a constant with dimensions (1/mass) 2 . This yields a noninteracting Fermi gas in the non-relativistic limit, while in the ultra-relativistic limit it approaches the limiting causal EOS P = ε, which has a speed of sound equal to the speed of light, the maximal causal value. Such an equation of state can also be motivated field-theoretically (see Appendix A).
For this EOS, analytical expressions for the maximum mass and the critical radius can be found for arbitrary values for the interaction strength y = √ c/m f , where m f is the mass of the fermionic dark matter particle. For stars consisting of self-interacting fermionic dark matter  [31], these are as discussed in [32], where m b is the mass of the bosonic dark matter particle and y = √ c/m b . Note that the maximum compactness for those boson stars is then C = 0.215. As the secondaries in the LIGO/Virgo events GW200105 and GW200115 were reported with critical compactness of C = 0.22 and C = 0.26 respectively (see Table I), they cannot have been constituted by purely self-interacting fermionic or bosonic dark matter, while the secondary in event GW190814 can still be explained by this model as the compactness is only C = 0.14. Such bosonic dark matter stars may be formed by gravitational instabilities in the early universe, see e.g. Khlopov et. al [33].
Assuming M max > 2.5M and R crit < 23 Km for GW190814, this constrains the fermionic interaction strength and mass to a narrow region in the parameter space with a minimum at m f = 0.81 GeV and f = m/y = 0.16 GeV (see Figure 1), which corresponds to y = m/f = 3.6. For the purely bosonic case, the interaction strength and mass are constrained to a slightly wider region and can both be arbitrarily small. Therefore, all hypothetical objects which meet the bosonic condition are eligible, while values close to f i ≈ √ 2f π are required for the fermionic case, where f π = 92M eV is the pion decay constant. From this, it is unlikely that the secondary in GW190814 was an exotic boson star purely consisting of Peccei-Quinn-Axions [34] with a mass of ≈ 1 keV, as this boson mass would imply an interaction strength y ≈ 10 −11 or a decay width f a ≈ 10 5 GeV. Similarly, since the crucial scaling relation for QCD-like axions is f i · m i ≈ f π · m π [35,36], stars consisting of these axions would not be compatible within our working hypothesis with the measured constraints for radius and mass. It is also interesting to note that in the case of neutrons comprising fermionic matter, the interaction strength f is required to lie in the range of √ 2f π , 2f π , of the order of interaction strengths in neutron star matter.
For the event GW191219, even though the secondary mass is in a very interesting range, the constraining power is low, since the mass of its primary results in a high estimate for the Roche-radius, thus leaving a large range of possible parameters. For GW200210, both maximum mass and Roche-radius are very similar to GW190814, not offering any more insight into constraining the parameter spaces.

III. COMPACT STARS CONSISTING OF BOSON-FERMION DARK MATTER
We now consider uncharged objects consisting of a mixture of bosonic and fermionic dark matter. Similar configurations are realized in nature in the form of white dwarfs, where the nuclei are comprising the role of the bosonic matter, while electrons are fulfilling the part of fermionic matter, since the contribution of the lattice of nuclei in the interior has a negligible effect on the EOS. Another example of such a configuration are pion stars proposed by [37], where a charged condensate of pions and a gas of charged leptons and neutrinos fulfill the roles of bosonic and fermionic matter, respectively. We will assume those dark matter stars to be globally uncharged stars while the number of fermions and bosons are equal, which requires the existence of a dual electromagnetism mediating between the two particles via dark photons. One such proposed and well motivated mechanism is the one of mirror dark matter [38,39], for example.
Similarly to section II, we assume a dimensionless energy density and pressure proportional to the square of the interaction strength and number density.
where y is the interaction strength and n B is the number density of bosons. Such an equation of state can be motivated field-theoretically by a coupling of fermionic and bosonic complex fields, resulting in an EOS that can be rewritten in terms of the squares of the number densities of fermions and bosons (see Appendix A). The assumption of charge-neutrality requires the number densities of fermionic and bosonic dark matter to be equal, i.e. n F = n B .
Solving the Tolman-Oppenheimer-Volkoff (TOV) equation and assuming M max > 2.6M and R crit < 26 km in the case of GW190814, we find solutions for the coupling strength f and the respective dark matter particle masses (see Figure 2). One can see that the coupling strength y is constrained to a range of [2.5 , 22.5]. Interestingly, a five-fold increase in y can result in a change of two orders of magnitudes in the respective masses. Also, one of the two dark matter particle masses is always of the order of the GeV-scale, regardless of the coupling strength. This is in stark contrast to the findings in section II, where the bosonic masses can be arbitrarily small.
In the case of GW200105 and its high compactness, a similar behaviour can be seen (see Figure 3), however, the allowed parameter space is much more restricted compared to GW190814. For masses of the bosonic particle to be smaller than 10 4 MeV, the interaction strength is constrained to values in the range of y ≈ [10 , 20]. In the case of the even more compact GW200115 secondary, no possible solutions in the parameter space can be found.
where ε nm = 140 MeV fm −3 is the energy density of saturated nuclear matter. Given a maximum mass, these expressions analytically provide an upper bound for ε 0 .
In the case of a minimum radius, the TOV-equations have to be solved numerically to constrain ε 0 from below (the results are shown in Tables II, III and Figure 4.) In the case of GW190814, these required values are of the order of the standard MIT bag value ≈ 57.5 MeV fm −3 . As one can see, the necessary ε 0 is in the range of the QCD scale. This suggests that such and object would be dominated by QCD-like interactions, and that weak interactions do not play a significant role in such self-bound stars, regardless of their actual microscopic composition. This is however in stark contrast to GW200105 and GW200115, which require higher values for ε 0 to fulfill the imposed conditions. Especially in the case of GW200115 for s=1, the required values for ε 0 can be as high as 1 GeV fm −3 . This is still in the typical range of QCD energy densities, as e.g. the one of the QCD crossover transition. However, it is not possible to find values for ε 0 that can fit all signals for a given fixed speed of sound. Hence, selfbound stars could generally be ruled out in our adopted scenario.

V. DISCUSSION
In this work we investigated NS-BH merger candidates to constrain compact exotic object mergers, by assuming as a working hypothesis that these objects have radii no larger than the Roche-radius. This allows us to demonstrate the constraining power of future radius measurements of merger events for exotic compact objects comprised of dark matter. We studied for this purpose generic models of self-interacting fermion and boson stars. We have shown that in all the studied cases, the characteristic scales necessary to explain GW190814 are very close to the QCD-scale, suggesting that weak interactions do not play a significant role in hypothetical dark matter stars with such high masses and large radii. Additionally, while in the case of self-interacting stars the bosonic dark matter particle mass can be arbitrary, the fermionic dark matter particle mass is required to be in the GeV range. Interestingly, in the case of a mixture of bosonic and fermionic dark matter one of both particle masses is also always in the GeV scale. This implies that if the secondary object in GW190814 was comprised of dark matter with a dark photon mechanism, then the possible dark QCD scale must be similar to the QCD scale. In the limiting case of a selfbound linear equation of state, we find again that the vacuum energy needs to be of the scale of QCD. However, in the case of masses and radii which are common for neutron stars, the applied models are either constrained to very small parameter spaces, or allow for no solutions at all, as demonstrated in the case of the very compact object of GW200115. We start with the Lagrangian density for a bosonic field and an interaction term: From this, the conserved Noether-Current is as follows: Applying the Euler-Lagrange-Equations for φ and φ provides us with the following equations of motion: (A6) and conservation of the Noether-Current, i.e.
We can now use the equations of motion for the scalar fields to finally conclude J µ φ J φµ = (φ ∂ µ φ) (φ∂ µ φ ). Since the number density n B is the corresponding zeroth component of the Noether Current, i.e. n B = J 0 φ and we consider the spatial components of the current to be equal to zero, i.e. J i φ = 0, we arrive at a Lagrangian density that can be written proportional to the interaction strength c and number densities squared:

Fermionic field
Similar to the previous section, we begin with the Lagrangian density for a fermionic field: From this, the conserved Noether-Current is as follows: Again, the fermionic number density is the zerothcomponent of the Noether Curent, n F = J 0 Ψ and we consider the spatial component of the current to be equal to zero. Thus, we immediatly find L =Ψ (iγ µ ∂ µ − m Ψ ) Ψ + g 1 Ψ γ µ Ψ Ψ γ µ Ψ = L kin + g 1 J µ Ψ J Ψµ = L kin + g 1 J 0 Ψ 2 = L kin + c · n 2 F (A11)