Pranav Satheesh

Slide 1: This slide contains the Title, author information and the abstract.
The title reads “3PN Frequency Domain Gravitational Waveform for Eccentric Black Hole Binaries including higher modes”. Author: Pranav Satheesh from Indian Institute of Technology Madras. The abstract reads “LIGO’s first-ever detection of gravitational waves was consistent with black holes moving in circular orbits as predicted by General Relativity. The future generation of gravitational waves detectors like LISA will be able to detect signals from binaries that have very small orbital eccentricities when they enter the low-frequency band of such detectors. The gravitational waveform from such systems, modelled using Post-Newtonian methods are used as template banks to match the signals from the detector. The work follows analytical modelling of gravitational waveforms of eccentric binary black holes in the frequency domain. Post-Newtonian waveform analytic models in the frequency domain admit simple structure, allowing computationally efficient data analysis. We use previously computed time-domain waveforms for compact binaries in eccentric orbits to calculate the frequency domain waveform amplitude under small-eccentricity approximation. The gravitational waveform of a merging stellar-mass binary is described at leading order by a quadrupolar mode. However, the complete waveform includes higher-order modes. The following work consists of these higher harmonics in the frequency domain waveform. “

Slide 2: This slide is about the Motivation behind modelling compact binaries.
Gravitational waves carry energy from in spiralling binaries and circularise the orbits. That is, suppose if the orbit has a finite value of eccentricity, once it radiates and shrinks into a circular orbit. Hence usually the inspiralling compact binaries are modelled as quasi-circular orbits. Nevertheless, there is an increased interest in inspiralling binaries with very small eccentricity when they enter the frequency band of Gravitational-wave detectors. If the eccentricity is not
accounted for, it can cause a significant error in the estimated parameters of an inspiralling binary. Figure 2 shows the sensitivity curves for LIGO and LISA detectors. LIGO is a ground-based detector whereas LISA is a future space-borne detector. The sensitivity curve is plotted as the characteristic strain vs the frequency. To describe a binary system, we define some orbital parameters which are shown in figure 1. Figure 1 illustrates an elliptical orbit with “r” and “phi” describing the separation and the angle, respectively. Other important parameters are mean anomaly “u” and true anomaly “v”. A bunch of differential equations determine their dependence which can be solved to get the time evolution of the parameters.
In Post-Newtonian approximation, the equations of general relativity take the form of the familiar Newtonian two-body equations of motion, for velocity v is very less than the speed of light, known as the weak-field limit. Every nth order correction of v over c counts as an n/2 PN correction. At each PN order, we unravel new physics beyond the Newtonian regime. The solution of the two-body problem in general relativity gives us two polarisation state called “h-plus” and “h-cross” (ℎ+,ℎ×). The gravitational waveform has an amplitude and phase part. The total waveform can be decomposed into a multipole expansion where each "modes" are describes by an "l" and "m" value. For black hole binaries characterised by a low mass ratio (q ≤ 4) and a total mass less than 100 solar masses, the “22” modes dominate.


Slide 3: This slide is about Stationary Phase Approximation (SPA) calculations. The time-domain gravitational waveforms are the input, calculated using Post-Newtonian approximation. The goal is to convert this into the Fourier domain to get a nice analytical representation. The expressions involve nonlinear terms in the phase when we include the eccentricity of the orbit. This makes the calculation difficult. We are interested in calculating a Fourier transform integral. This integral is computed using the stationary phase approximation. For large values of f, the integrand oscillates rapidly and causes large cancellations when integrated. So, we take a Taylor series expansion of the phase term and find points where it is stationary (when the derivative of phase is zero). This condition gives us a relation between the instantaneous frequency and orbital frequency. This condition varies for different modes and depends on the eccentricity order in your terms. The figure on this slide shows how a waveform (22 mode) looks like in time-domain. The waveform is also known as the chirp signal: the frequency and the amplitude of the waveform increase.

Slide 4: This slide is about the results. The full analytic form of the waveform expression is described in the frequency domain. Figure 4 depicts the plot of 22 mode frequency domain waveform (amplitude vs frequency) for various values of eccentricity. We see oscillations in the plot that amplifies when eccentricity increases. The waveform is then tested to check up to which order of eccentricity should I expand my terms so that it matches well with the most accurate waveform. The more precise waveform is valid up to sixth order in the expansion of eccentricity. Using this as a target waveform, we compute matches for various eccentricity values. The figure 5 shows the matches plotted against eccentricity values. We find that the waveforms work the best for moderate eccentricity values. My work was done under the guidance of Chandra Kant Mishra, IIT Madras. We are now working on comparing the waveforms with numerical Fourier transform expressions and improve the accuracy.