Eccentricity evolution of spinning binaries

We study the evolution of the eccentricity of an eccentric orbit with spinning components. We develop a prescription to express the evolving eccentricity in terms of initial eccentricity and frequency. For that purpose we considered the spins to be perpendicular to the orbital plane. Using this we found an analytical result for the contribution of spin in eccentricity evolution. As a result, we expressed orbital eccentricity in a series of initial eccentricity and gravitational wave frequency. The prescription developed here can easily be used to find arbitrarily higher-order contributions of initial eccentricity. With the eccentricity evolution at hand, we computed the evolving energy and angular momentum fluxes for eccentric orbit with spinning components. This result can be used to construct the waveforms of spinning compact objects in an eccentric orbit.Comment: arXiv admin note: text overlap with arXiv:2305.0377

Black holes immersed in dark matter: energy condition and sound speed

In this work, we study the impact of the environment around a black hole in detail. We introduce non-vanishing radial pressure in a manner analogous to compact stars. We examine both isotropic and anisotropic fluid configurations with and without radial pressure respectively. Our focus extends beyond just dark matter density to the vital role of the energy condition and sound speed in the spacetime of a black hole immersed in matter. In cases of anisotropic pressure with vanishing radial pressure, all profiles violate the dominant energy condition near the BH, and the tangential sound speed exceeds light speed for all dark matter profiles. In our second approach, without assuming vanishing radial pressure, we observe similar violations and superluminal sound speeds. To rectify this, we introduce a hard cutoff for the sound speed, ensuring it remains subluminal. As a consequence, the energy condition is also satisfied. However, this results in increased density and pressure near the BH. This raises questions about the sound speed and its impact on the density structure, as well as questions about the validity of the model itself. With the matter distribution, we also compute the metric for different configurations. It reveals sensitivity to the profile structure. The metric components point towards the horizon structure

Horizon fluxes of binary black holes in eccentric orbits

I compute the rate of change of mass and angular momentum of a black hole, namely tidal heating, in an eccentric orbit. The change is caused due to the tidal field of the orbiting companion. I compute the result for both the spinning and non-spinning black holes in the leading order of the mean motion, namely $\xi$. I demonstrate that the rates get enhanced significantly for nonzero eccentricity. Since eccentricity in a binary evolves with time I also express the results in terms of an initial eccentricity and azimuthal frequency $\xi_{\phi}$. In the process, I developed a prescription that can be used to compute all physical quantities in a series expansion of initial eccentricity, $e_0$. This result was only known in the leading order while ignoring the contribution of the spin on the eccentricity evolution. Although the eccentricity evolution result still ignores the spin effect in the current work, the prescription can be used to compute higher-order corrections of initial eccentricity post-leading order. Using this result I computed the rate of change of mass and angular momentum of a black hole in terms of initial eccentricity and azimuthal frequency up to $\mathcal{O}(e_0^2)$.Comment: arXiv admin note: text overlap with arXiv:1605.00304 by other author

On the Consistency of Maximum Likelihood Estimation of Probabilistic Principal Component Analysis

Probabilistic principal component analysis (PPCA) is currently one of the most used statistical tools to reduce the ambient dimension of the data. From multidimensional scaling to the imputation of missing data, PPCA has a broad spectrum of applications ranging from science and engineering to quantitative finance. Despite this wide applicability in various fields, hardly any theoretical guarantees exist to justify the soundness of the maximal likelihood (ML) solution for this model. In fact, it is well known that the maximum likelihood estimation (MLE) can only recover the true model parameters up to a rotation. The main obstruction is posed by the inherent identifiability nature of the PPCA model resulting from the rotational symmetry of the parameterization. To resolve this ambiguity, we propose a novel approach using quotient topological spaces and in particular, we show that the maximum likelihood solution is consistent in an appropriate quotient Euclidean space. Furthermore, our consistency results encompass a more general class of estimators beyond the MLE. Strong consistency of the ML estimate and consequently strong covariance estimation of the PPCA model have also been established under a compactness assumption.Comment: 15 pages, 1 figure, to appear in NeurIPS 2023. Update: included minor typographical correction

Relativistic tidal properties of superfluid neutron stars

We investigate the tidal deformability of a superfluid neutron star. We calculate the equilibrium structure in the general relativistic two-fluid formalism with entrainment effect where we take neutron superfluid as one fluid and the other fluid is comprised of protons and electrons, making it a charge neutral fluid. We use a relativistic mean field model for the equation of state of matter where the interaction between baryons is mediated by the exchange $\sigma$, $\omega$ and $\rho$ mesons. Then, we study the linear, static $l=2$ perturbation on the star to compute the electric-type Love number following Hinderer's prescription.Comment: Accepted for publication in Physical Review