From eccentric binaries to nonstationary gravitational wave backgrounds

Author(s)

Falxa, Mikel, Leclere, Hippolyte Quelquejay, Sesana, Alberto

Abstract

A large population of binary systems in the Universe emitting gravitational waves (GW) would produce a stochastic noise, known as the gravitational wave background (GWB). The properties of the GWB directly depend on the attributes of its constituents. If the binary systems are in eccentric orbits, it is well established that the GW power they radiate strongly depends on their instantaneous orbital phase. Consequently, their power spectrum varies over time, and the resulting GWB can appear nonstationary. In this work, we estimate the amplitude of time-dependent fluctuations in the GWB power spectrum as a function of the eccentricity of the binaries. Specifically, we focus on the GWB produced by a population of supermassive black hole binaries (SMBHB) that should be observable by pulsar timing arrays (PTA). We show that a large population of homogeneously distributed equal SMBHBs produces nonstationary features that are undetectable by current PTA datasets. However, using more realistic and astrophysically motivated populations of SMBHBs, we show that the nonstationarity might become very large and detectable, especially in the case of more massive and eccentric populations. In particular, when one binary is slightly brighter than the GWB, we demonstrate that time fluctuations can become significant. This is also true for individual binary systems with a low signal-to-noise ratio (SNR) relative to the GWB (SNR $\approx$ 1), which standard data analysis methods would struggle to detect. The detection of nonstationary features in the GWB could indicate the presence of some relatively bright GW sources in eccentric orbits, offering new insights into the origins of the signal.

Figures

Variance of $L_{GW}/P_0$ as a function of the eccentricity of the source $e$.

Variance of $L_{GW}/P_0$ as a function of the eccentricity of the source $e$.


Covariance and variance for the first 60 harmonics of a GW spectrum radiated by a binary system with eccentricity $e=0.8$; (upper panel) covariance matrix $\langle P_n P_m \rangle - \langle P_n \rangle \langle P_m \rangle$, (lower panel) variance of the $P_n$, i.e., diagonal terms of the covariance matrix.

Covariance and variance for the first 60 harmonics of a GW spectrum radiated by a binary system with eccentricity $e=0.8$; (upper panel) covariance matrix $\langle P_n P_m \rangle - \langle P_n \rangle \langle P_m \rangle$, (lower panel) variance of the $P_n$, i.e., diagonal terms of the covariance matrix.


Time-dependent fluctuations $\Delta \Omega^2 (f)$ of the GWB spectrum as a function of frequency $f$ and various initial eccentricities $e_0$ and a population of equal binaries with chirp mass $\mathcal{M}_c$.

Time-dependent fluctuations $\Delta \Omega^2 (f)$ of the GWB spectrum as a function of frequency $f$ and various initial eccentricities $e_0$ and a population of equal binaries with chirp mass $\mathcal{M}_c$.


Cross-correlated fluctuations $\Delta \Omega^2 (f, f'$ of the GWB spectrum for frequencies $f$ and $f'$ and initial eccentricity $e_0=0.8$.

Cross-correlated fluctuations $\Delta \Omega^2 (f, f'$ of the GWB spectrum for frequencies $f$ and $f'$ and initial eccentricity $e_0=0.8$.


The distribution of eccentricity $e$ and rest frame chirp mass $\mathcal{M}$ (given in solar masses $M_\odot$) for the populations "POP A" and "POP B".

The distribution of eccentricity $e$ and rest frame chirp mass $\mathcal{M}$ (given in solar masses $M_\odot$) for the populations "POP A" and "POP B".


The time fluctuations $\Delta \Omega^2 (f)$ as a function of frequency $f$ for "POP A" and "POP B". The solid lines correspond to median fluctuations for 50 different realizations of the population and the colored areas show the 16\% and 84\% percentiles. The dashed lines are the median fluctuations obtained by removing the 100 and 1000 brightest binaries.

The time fluctuations $\Delta \Omega^2 (f)$ as a function of frequency $f$ for "POP A" and "POP B". The solid lines correspond to median fluctuations for 50 different realizations of the population and the colored areas show the 16\% and 84\% percentiles. The dashed lines are the median fluctuations obtained by removing the 100 and 1000 brightest binaries.


Time-dependent fluctuations of the GWB spectrum as a function of frequency $f$ in the presence of a single source with $\rho = 1.4$, $e=0.5$ and $f_p=5$ nHz. The GWB is considered stationary with a power-law spectrum and similar properties as those found in \cite{nanograv_gwb, wm3}. The vertical dashed line shows the orbital frequency $f_p$ of the individual binary, and the dots show the position of the harmonics.

Time-dependent fluctuations of the GWB spectrum as a function of frequency $f$ in the presence of a single source with $\rho = 1.4$, $e=0.5$ and $f_p=5$ nHz. The GWB is considered stationary with a power-law spectrum and similar properties as those found in \cite{nanograv_gwb, wm3}. The vertical dashed line shows the orbital frequency $f_p$ of the individual binary, and the dots show the position of the harmonics.


Total time-dependent fluctuations of the GWB spectrum in the presence of a single source for varying $\rho$ and $e$, with $f_p=5$ nHz. The GWB is considered stationary with a power-law spectrum and similar properties as those found in \cite{nanograv_gwb, wm3}. For $e=0$, the fluctuations go to zero because circular binaries do not generate nonstationarities.

Total time-dependent fluctuations of the GWB spectrum in the presence of a single source for varying $\rho$ and $e$, with $f_p=5$ nHz. The GWB is considered stationary with a power-law spectrum and similar properties as those found in \cite{nanograv_gwb, wm3}. For $e=0$, the fluctuations go to zero because circular binaries do not generate nonstationarities.


Total time-dependent fluctuations of the GWB spectrum in the presence of a single source for varying $\rho$ and $e$, using the approximation in \autoref{eq:total_fluct_approx}.

Total time-dependent fluctuations of the GWB spectrum in the presence of a single source for varying $\rho$ and $e$, using the approximation in \autoref{eq:total_fluct_approx}.


Relative error $\epsilon$ between \autoref{eq:total_variance} and \autoref{eq:cov_timefrequency} as a function of eccentricity $e$. We see that around $e=0.8$, the error goes up. This is because in that example, we used only the first 200 harmonics to estimate the numerical integral in \autoref{eq:cov_timefrequency}. For high $e$, the higher harmonics become significant, so the cutoff at $n=200$ affects our precision.

Relative error $\epsilon$ between \autoref{eq:total_variance} and \autoref{eq:cov_timefrequency} as a function of eccentricity $e$. We see that around $e=0.8$, the error goes up. This is because in that example, we used only the first 200 harmonics to estimate the numerical integral in \autoref{eq:cov_timefrequency}. For high $e$, the higher harmonics become significant, so the cutoff at $n=200$ affects our precision.


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