Rafia Sarwar

Numerous galaxies host a supermassive black hole (SMBH) in their galactic center, encoding rich informative of their evolutionary epoch. The strain sensitivity of the Laser Interferometer Space Antenna (LISA) detector appears to be centralized at 3 mHz, the most anticipated sources for such frequency domain are extreme mass ratio inspirals (EMRIs) formed when a compact object (CO i.e. black hole, neutron star, and white dwarf) comes under the strong gravitational influence of the SMBH. CO sets onto the highly eccentric, relativistic, bound orbits dwindling to the last stable orbit, depends on the parameters at the instant of capture with the shrinking orbits closer to the SMBH. Inspiralling dissipates orbital energy and angular momentum, emitting continuous gravitational radiations of the order of millihertz, sensitive to LISA.
We have computed event rates for detectable EMRIs using the mission lifetime of LISA 2 years, we will require the number density of comoving SMBHs and intrinsic rate of probable EMRIs per SMBH. SMBH’s spin remains highly uncertain, hence, the integrating probability of spin distribution is normalized to 1 with a uniform range of spins ranging from 0 to 1, considering the prograde spin orbits.
We reduced the problem for the Milky Way galaxy using the Dirac delta function.
17 physical parameter fully describes the CO-SMBH system that is further reduced to 14 parameters by ignoring the spin of the compact object. We have used the Analytical Kludge model based on post-Newtonian formulae. The parameter space reduces to 10 parameters by constraining the mass of the SMBH, distance to the source and source location, and orientation angles for well-calibrated source Sgr A*.
Statistical density distributions scaling and two-dimensional posteriors for the set of parameters are shown in figure 2. With the individual scatter plots initial orbital frequencies relate to the well-behaved negatively skewed converging distribution centered at some higher level. Also, cosine angles of orbital inclination i and azimuth angle of spin φk take the bimodal distribution to be the evidence of sinusoidal behavior of angles. The true value is symmetrically confined to the first and fourth quadrant adjacent to the angles 0 and 2π. Other parametric distributions are non-gaussian and non-consequential. The figure shows that parameters have no multicollinear relation with a substantial degree of accuracy.
We developed the equation of back-of-envelope estimates for EMRIs in the Galactic Center (GC). This power law was fitted making use of multiple linear regression methods with (forward and backward)-step-wise approach compared to the null model selected with the least variance inflation factor (VIF) of each parameter indicating the absence of multicollinearity. The best fit line was drawn using a generalized linear model based on gaussian error distribution. R-squared and adjusted R-squared indexes of the regression model with the p-value approaching zero. Individual predictor parameters semi-major axis a, the mass of CO μ, initial orbital eccentricity e, (cosine–) of orbital inclination cos ι and spin of SMBH ã contributes to our best fit model illustrated in Fig. 3. EMRI signal sensitively varies with each parameter of the model.
We estimated EMRI event rates N = 1.43 per year for a 2-years LISA mission. The decimal outcome for the occurrences of EMRIs remains unrealistic in practice, event rate of 1.43 per year is averaged for ten thousand mission realizations. The distribution is in agreement with the Poisson distribution in figure 4. Event rates remain low for the Milky Way galaxy over the mission duration. Detection of even a single inspiraling source from GC will be robust to the understanding of the demographic mapping of the GC and testing General Relativity.