Effects of dark matter on the in-spiral properties of the binary neutron stars

In the extended RMF (E-RMF) formalism, the $\sigma-$meson is mainly responsible for the strong attraction at the intermediate range of the nuclear force, but the self-interaction of this scalar meson produces a weak repulsion at long-range (Biswal et al., 2015). Because of the self-interaction of the $\sigma-$meson, the incompressibility of nuclear matter at nuclear saturation $K_{\infty}$ comes down to the empirical range of $K_{\infty}=210\pm{30}$ MeV (Blaizot, 1980). The $\omega-$meson responsible for the strong hard-core repulsion of the nuclear potential and its self-interaction generates attraction at very short range and makes the nuclear EoS softer (Bodmer, 1991; Gmuca, 1992; Sugahara & Toki, 1994; Bunta & Gmuca, 2003). We take one RMF set NL3 (Lalazissis et al., 1997) and two E-RMF sets such as G3 (Kumar et al., 2017) and IOPB-I (Kumar et al., 2018) to calculate the NS properties in the present work.

Inside the NS, the neutron decays to proton, electron and anti-neutrino. For the stability of the system, the inverse $\beta$-decay takes place to maintain both $\beta$ equilibrium and charge neutrality. Hence, the EoS of NS is calculated from the contributions of both nucleons and leptons. The energy density and pressure of a NS is written as (Das et al., 2020, 2021):

Where ${\cal{E}}_{NM}$ and $P_{NM}$ are the energy density and pressure of the nucleon; ${\cal{E}}_{l}$ and $P_{l}$ are for leptons, respectively.

After adding the DM to the NS, the total Lagrangian density is written as (Das et al., 2019, 2020):

where ${\cal{L}}_{NS}$ is the NS Lagrangian. The $\psi$ is the nucleon field and $\chi$ is the DM wave function. The parameters $y$, $f$ and $v$ are the DM-Higgs coupling, nucleon-Higgs form factor and the vacuum expectation value of Higgs field, respectively. The values of $y$, $f$ and $v$ are 0.07, 0.35 and 246 GeV, respectively (Das et al., 2020, 2021).

The two parameters $y$ and $f$ play significant role on the effect of DM in NS. Therefore, we constraint these two parameters using available experimental data. The expression of scattering cross-section for the DM is written as (Bhat & Paul, 2020):

where $\mu_{r}$ is the reduced mass $\frac{M_{nucl.}\times M_{\chi}}{M_{nucl.}+M_{\chi}}$ and $M_{\chi}$ is the mass of the DM particle. The DM detection experiments, like XENON (Aprile et al., 2016), PANDA-II (Tan et al., 2016) and LUX (Akerib et al., 2017) have put up their upper limit for the spin-independent scattering cross-section $\sigma_{SI}$ as $1.6\times 10^{-47}$ cm^-2 ($M_{\chi}=50$ GeV), $1.6\times 10^{-47}$ cm^-2 ($M_{\chi}=40$ GeV) and $1.6\times 10^{-47}$ cm^-2 ($M_{\chi}=50$ GeV), respectively. The PANDA-II experiment has also put a limit on the mass of the WIMP as $M_{\chi}=5-1000$ GeV. The LHC has put a limit on the WIMP-nucleon scattering for Higgs portal DM in the range $10^{-40}-10^{-50}$ cm^-2 (Tan et al., 2016; Aad et al., 2015). We calculate the $\sigma_{SI}$ for three different masses of DM $M_{\chi}=50$, 100, and 200 GeV and their corresponding cross-sections are found to be 9.43, 9.60 and 9.70 in the order of ($10^{-46}$ cm²) respectively. These data confirmed that the predicted values are consistent with the data within 90% confidence level. This model also satisfies the LHC limit. The value of $f$ is reported in Ref. (Djouadi et al., 2012) using both lattice QCD (Czarnecki et al., 2010) and MILC results (Toussaint & Freeman, 2009) whose value is $0.33_{-0.07}^{+0.30}$ (Aad et al., 2015) and our used $f$ (= 0.35) is well within the region.

The ${\cal{E}}$ and $P$ for NS with DM can be calculated by solving the Eq. (2) (Das et al., 2020, 2021)

and

where $k_{f}^{DM}$ and $M_{\chi}^{*}$ are the Fermi momentum and the effective mass of the DM respectively. $M_{h}$ is the mass of the Higgs equal to 125 GeV, and $h_{0}$ is the Higgs field calculated by applying the mean-field approximation (Das et al., 2019). In the present calculations, the mass of the DM is taken as 200 GeV.

A large number of forces are available in the literature (Dutra et al., 2012; Kumar et al., 2017, 2018). Among them, NL3, G3 and IOPB-I are very much used both for finite nuclei as well as for infinite nuclear matter (Kumar et al., 2017, 2018). The NL3 is one of the stiffest parameter set, which predicts a NS of mass 2.775 $M_{\odot}$ and the G3 is the softest one with a predictive mass of 1.979 $M_{\odot}$. The values of the models are available in Ref. (Kumar et al., 2018). Once the EoS has solved, the M-R profile of a single NS is calculated by solving the Tolmann-Oppenheimer-Volkoff (TOV) equation. When we consider a BNS system, the individual NS gets distorted due to the external field created by its companion. The tidal deformability parameter ($\lambda=\frac{2}{3}k_{2}R^{5}$) created by the deformation depends on the structural/macroscopic properties. The dimensionless Love number $k_{2}$ can be calculated by solving two coupled differential equations as it is done in Refs. (Hinderer, 2008; Hinderer et al., 2010; Kumar et al., 2018). The values of $R$ and $k_{2}$ are fixed for a given mass by the EoS for the NS. The dimensionless tidal deformability $\Lambda$ is defined as ($\Lambda=\lambda/M^{5}$).

As it is stated earlier, the three RMF EoS, namely NL3 (stiff) (Lalazissis et al., 1997), IOPB (less stiff) (Kumar et al., 2018) and the G3 (soft) (Kumar et al., 2017) for the core part are calculated. The well known BCPM EoS (Sharma et al., 2015) is used uniformly for all these sets to include the effect of the lower dense regions, i.e. the crust part and the NS properties are calculated with these unified EOS. We add the DM inside the NS, which interacts with nucleons via SM Higgs (see Sub-Sec. 2.1). As a representative case, the EoS of IOPB-I parameter set is chosen with DM momentum, $k_{f}^{DM}$= 0, 0.03 and 0.05 GeV, which in short, written as IOPB-I+DM0¹¹1same as IOPB-I because DM0 means no DM inside the NS, IOPB-I+DM3, and IOPB-I+DM5 respectively. The NS properties such $R_{1.35}$, $\lambda_{1.35}$, $\Lambda_{1.35}$, $k_{2,1.35}$ are given in Table 1. The predicted mass-radius for DM admixed NS with different DM momenta are also given for comparison. The maximum mass decreases with the increase of $k_{f}^{DM}$. The total EoS of the NS is depicted in Fig. 1. From the right panel of Fig. 1, it is clear that with the increase of DM momentum, the EoS becomes softer, which is consistent with our previous results (Das et al., 2020).

The $M$-$R$ relation of a star is calculated by solving the TOV equations with EoS as the main input with the boundary conditions $P=P_{c}$ at $r=0$ and $P=0$ at $r=R$. The masses and radii with IOPB-I parameter sets are calculated for DM admixed NS, which are shown in Fig. 2. Recently measured massive pulsar PSR J0740+6620 (Cromartie et al., 2019), PSR J0348+0432 (Antoniadis et al., 2013) and NICER (Miller et al., 2019; Riley et al., 2019) data are also depicted. With the addition of DM, the maximum mass $M_{max}$ and radius $R_{max}$ decrease. For $k_{f}^{DM}$=0.03 GeV, the $M_{max}$ satisfy the constraint given by Cromartie et al. (Cromartie et al., 2019) as well as Antoniadis et al. (Antoniadis et al., 2013) and the radius constraint given by NICER (Miller et al., 2019; Riley et al., 2019). However, with $k_{f}^{DM}$= 0.05 GeV, the $M_{max}$ and $R_{max}$ become very less, which doesn’t satisfy both massive pulsar and NICER data. Therefore, one can constraints the DM percentage inside the NS with the help of the observational data.

The tidal deformability of the NS decreases with DM’s addition, as shown in Fig. 3. The same trend also obtained for $\Lambda$. This is because, with the addition of DM, the EoS becomes softer and the mass and radius change accordingly, as shown in Fig. 2. Therefore, the softer EoS gives lower $\lambda$ as compare to the stiffer one. The vertical double-headed line represents the constraints from GW170817 with $\Lambda_{1.4}=190_{-120}^{+390}$ (Abbott et al., 2018). The values of $\Lambda_{1.4}$ are 473.22, 169.644 for DM momentum 0.03 and 0.05 GeV, respectively. This implies that the value of $\Lambda_{1.4}$ with these DM momenta satisfies the constraints given by GW170817.

In this sub-sec., we describe the BNS’s in-spiral properties using the PN formalism applied to quasi-circular orbit. For simplicity, we restrict our calculations only to EMBNS. Again, we take IOPB-I as a representative EoS for the DM admixed NS since it predicts the mass of the NS 2.149 $M_{\odot}$ which is consistent with PSR J0740+6620 (Cromartie et al., 2019). Here, we calculate the polarization waveforms and the GW properties such as amplitude, frequency, orbital phase for the EMBNS. The observables are calculated at retarded time defined by (Hotokezaka et al., 2013):

where $r_{*}$ is the tortoise coordinate defined as

where $r_{A}=\sqrt{A/4\pi}$ and A is the proper surface area of the sphere. In a simplify manner, $r_{A}$ is written as (Hotokezaka et al., 2013)

where $r\approx 200M_{t}$ is the finite spherical-coordinate radius.

Fig. 6 shows the waveforms, frequencies, phases, and $x$ for NL3, G3 and IOPB-I sets. The stiff EoS, like NL3, predicts higher mass and tidal deformability as compared to the softest EoS like G3. Therefore, the massive NS easily deformed by the presence of tidal fields created by its companion star. Hence, a less massive BNS system sustains a longer time than a more massive BNS system in the in-spiral phase. This is because the tidal interactions accelerate the orbital evolution in the late in-spiral phases due to increased interactions between two NSs. Some significant effect has seen in Fig. 4 with and without tidal interactions.

To have a quantitative and comparative estimation for the in-spiral properties of the BNS merger, we enumerate the numerical value of $t_{ret}$, $\Phi$, $f$ and $x$ in Table 2 using NL3, IOPB-I and G3. The last orbital time period is 0.0583 sec. for NL3, which is less than from other two sets and confirms the model-dependent prediction. There is significant influence seen for $t_{ret}$, $\Phi$ and $f$, except the value of $x$. The frequency and phase are also increased for the BNS of soft EoS. The pictorial representations of $f$, $\Phi$ and $x$ are compared in Fig. 6 with and without DM contains.

With the addition of DM to the NS EoS, the calculated macroscopic properties ($M$, $R$ and $\lambda$) decreases. As a result, the tidal interaction between the two NSs becomes weaker as compared to the normal BNS combination. For example, $M=2.149M_{\odot}$ without DM and it is 2.051 $M_{\odot}$ with $k_{f}^{DM}=0.03$ GeV with IOPB-I set. Similarly, the radius of the star is $R_{1.35}=13.168$ km without DM, and it is 12.047 km when DM is included (see Table 1). The calculated values for $f$, $\Phi$ and $x$ are tabulated in Table 2 for IOPB-I parameter set as a representative case with DM momenta $k_{f}^{DM}=0.03$ and $0.05$ GeV. The time period $t_{ret}$ is found to be 0.0603 sec. with IOPB-I+DM5, which is larger than the other two values obtained with IOPB-I and IOPB-I+DM3. These results infer the more sustainable time in the in-spiral orbits of the BNS system in the presence of DM particles (see Fig. 7). The other quantities such as $f$ and $\Phi$ also affected by DM, as shown in Fig. 7.

In general, with the addition of DM, the EOS becomes softer, which affects the macroscopic properties of the BNS, i.e. the mass of the NS decreases with DM. However, a lot of ambiguity prevails in the post-merger remnants of BNS. This has been settled partially after the analysis of GW170817 and GW190425 events. Some of the possible pathways can be found in the Ref. (Sarin & Lasky, 2021). The likely theoretical probabilities for a post-merger NS remnant basically depend on (i) the remnant mass ($M_{R}$), and (ii) the EOS of the NS which dictates TOV mass ($M_{TOV}$);

1. 1.

   if $M_{R}$ $\gtrsim 1.5\ M_{TOV}$, the system promptly collapses to a black hole.

2. 2.

   for $1.2\ M_{TOV}\lesssim M\lesssim 1.5\ M_{TOV}$, a hypermassive neutron star survives the collision, but collapses to form a black hole on dynamical timescales.

3. 3.

   in the case, $M_{TOV}<M_{R}\lesssim 1.2\ M_{TOV}$, a supramassive NS is survived the collision and will collapses to form a black hole on secular timescales.

4. 4.

   if $M_{R}\leq M_{TOV}$, a stable NS is formed after the merger.

We refer the readers to see Fig. 1 of this Ref. (Sarin & Lasky, 2021) for a clear picture.

The post-merger remnant of DM admixed BNS merger depends on the percentage of DM inside the NS. In Ref. (Ellis et al., 2018), it is reported that for a contribution of DM $\sim$ 5-10% of the NS mass, an extra peak in the power spectral density of the GW signal is found. Also, for a self-interacting DM of about $\lesssim 10\%$, leads to the early formation of a black hole (Pollack et al., 2015), which can be subsequent seed for supermassive black hole. Therefore, to understand the nature of post-merger remnant of DM admixed BNS, one has to know (i) the exact nature of DM particles and (ii) the percentage of DM present inside the NS.