Gravitational wave echoes from interacting quark stars

We show that interacting quark stars (IQSs) composed of interacting quark matter (IQM), including the strong interaction effects such as perturbative QCD corrections and color superconductivity, can be compact enough to feature a photon sphere that is essential to the signature of gravitational wave echoes. We utilize an IQM equation of state unifying all interacting phases by a simple reparametrization and rescaling, through which we manage to maximally reduce the number of degrees of freedom into one dimensionless parameter $\bar{\lambda}$ that characterizes the relative size of strong interaction effects. It turns out that gravitational wave echoes are possible for IQSs with $\bar{\lambda}\gtrsim10$ at large center pressure. Rescaling the dimension back, we illustrate its implication on the dimensional parameter space of effective bag constant $B_{\rm eff}$ and the superconducting gap $\Delta$ with variations of the perturbative QCD parameter $a_4$ and the strange quark mass $m_s$. We calculate the rescaled GW echo frequencies $\bar{f}_\text{echo}$ associated with IQSs, from which we obtain a simple scaling relation for the minimal echo frequency $f_\text{echo}^{\rm min}\approx 5.76 {\sqrt{B_{\rm eff}/\text{(100 MeV)}^4}} \,\,\, \rm kHz$ at the large $\bar{\lambda}$ limit.


INTRODUCTION
The recent observations of gravitational wave (GW) signals from compact binary mergers by the LIGO and Virgo collaborations [1][2][3][4][5][6][7] have greatly moved our understanding of black holes and compact stars forward. The detected binary black hole merger events inspired many studies on black hole mimickers termed exotic compact objects (ECOs), whose defining feature is their large compactness: their radius is very close to that of a black hole with the same mass while lacking an event horizon. While some non-GW probes of ECOs have been studied [8], most studies are on the distinctive signatures from gravitational wave echoes in the postmerger signals [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], in which a wave that falls inside the gravitational potential barrier travels to a reflecting boundary before returning to the barrier at the photon sphere after some time delay.
Considering the detected binary neutron star merger events, we want to explore the possibility of GW echoes also being signature of realistic compact stars. Generating GW echoes requires the star object to feature a photon sphere at R P = 3M , where M is the object's mass. For compact stars, the minimum radius should be above the Buchdahl's limit R B = 9/4M [29]. Therefore, GW echo signals are possible if R B < R < R P .
This compactness criterion excludes the realistic neutron stars [30,31]. This motivates the exploration of other more compact star objects such as quark stars composed of quark matter.
It was proposed by Bodmer [32], Witten [33] and Terazawa [34] that quark matter with comparable numbers of u, d, s quarks, termed strange quark matter (SQM), might be the ground state of baryonic matter at zero pressure and temperature. However, it was demonstrated in a recent study [35] that u, d quark matter (udQM) can be more stable than SQM and the ordinary nuclear matter at a sufficiently large baryon number beyond the periodic table. The SQM hypothesis and udQM hypothesis, as mentioned above, allow the possibility of bare quark stars, such as strange quark stars (SQSs) [36,37] that consist of SQM or up-down quark stars (udQSs) [38,39] that consist of udQM. In the context of recent LIGO-Virgo events, there are a lot of studies on the related astrophysical implications of SQSs [40][41][42][43][44][45] and udQSs [38,[46][47][48][49], many of which involve interacting quark matter (IQM) that includes the interquark effects induced by strong interaction, such as the perturbative QCD (pQCD) corrections [50][51][52] and the color superconductivity [53][54][55] In order to achieve a large compactness for stars to arXiv:2107.09654v2 [hep-ph] 5 Nov 2021 generate GW echoes, people commonly assumed ad-hoc exotic equations of state (EOS) [31,59,60] or special semiclassical treatment of gravity [61]. Here we demonstrate that the physically motivated interacting quark stars (IQSs) composed of IQM can have GW echo signatures within the classical Einstein gravity framework.
Referring to [48,57], we first rewrite the free energy Ω of the superconducting quark matter [56] in a general form with the pQCD correction included: where µ and µ e are the respective average quark and electron chemical potentials. The first term represents the unpaired free quark gas contribution. The second term with (1 − a 4 ) represents the pQCD contribution from one-gluon exchange for gluon interaction to O(α 2 s ) order. To phenomenologically account for higher-order contributions, we can vary a 4 from a 4 = 1, corresponding to a vanishing pQCD correction, to very small values where these corrections become large [51,57,58]. The term with m s accounts for the correction from the finite strange quark mass if applicable, while the term with the gap parameter ∆ represents the contribution from color superconductivity: The corresponding equation of state was derived in Ref. [48]: where Note that sgn(λ) represents the sign of λ. One can easily see that a larger λ leads to a stiffer EOS which results in a more compact stellar structure that is more likely to have GW echoes. Thus, for this study, we need to explore only positive λ space.
As shown in Ref. [48], one can further remove the B eff parameter by doing the following dimensionless rescaling: so that the EOS (2) reduces to the dimensionless form Asλ → 0, Eq. (6) or, equivalently, p = ρ − 2B eff , using Eq. (4). We see that strong interaction effects can reduce the surface mass density of a quark star from ρ 0 = 4B eff down to ρ 0 = 2B eff and increase the quark matter sound speed c 2 s = ∂p/∂ρ from 1/3 up to 1 (the light speed) maximally.

GW ECHOES FROM IQS
To study the stellar structure of IQSs, we first rescale the mass and radius into dimensionless form in geometric   From Eq. (5), we see that the echo criterionλ 10 maps to the constraint on dimensional parameters The characteristic echo time is the light time from the star center to the photon sphere [10][11][12], where dΦ dr = − 1 ρ + p dp dr .
We can also do the dimensionless rescalinḡ such that Eq. (11) can also be calculated in a dimensionless approach. After obtaining the echo time, we directly get the GW echo frequency from the relation [10][11][12] and similarly, we can rescale it into the dimensionless formf echo via the relation In Fig. 3 where the coefficients c 1 ≈ 13.0361, c 2 ≈ 12.0661, c 3 ≈ 12.6916, and c 4 ≈ 10.3398 are the best-fit values, with an error only at the 0.1% level. The first termf min echo ≈ 2.3046 is the smallest echo frequency value achieved at theλ → ∞ limit, which maps to the largest compactness.
After rescaling back with Eq. (15), we obtain a relation between the minimal echo frequency for a given λ and the effective bag constant (17) Among all f low echo for differentλ, the minimal value is achieved at theλ → ∞ limit as f min echo = f low echo |λ →∞ ≈ 5.03 B eff 10 MeV/fm 3 kHz, (18) where B eff is in units of MeV/fm 3 , or, equivalently