Detecting stochastic gravitational waves with binary resonance

LIGO and Virgo have initiated the era of gravitational-wave (GW) astronomy; but in order to fully explore GW frequency spectrum, we must turn our attention to innovative techniques for GW detection. One such approach is to use binary systems as dynamical GW detectors by studying the subtle perturbations to their orbits caused by impinging GWs. We present a powerful new formalism for calculating the orbital evolution of a generic binary coupled to a stochastic background of GWs, deriving from first principles a secularly-averaged Fokker-Planck equation which fully characterizes the statistical evolution of all six of the binary’s orbital elements. We also develop practical tools for numerically integrating this equation, and derive the necessary statistical formalism to search for GWs in observational data from binary pulsars and laser-ranging experiments.

Figure 1

Cartoon illustration of the binary resonance effect. Stochastic fluctuations in the background spacetime geometry due to incoming GWs perturb the trajectories of two orbiting masses, causing cumulative changes to their orbital elements.

Figure 2

Schematic diagram of a Keplerian orbit. The plane of the orbit (shown in blue) is defined relative to the fixed reference frame $(\widehat{\mathit{x}},\widehat{\mathit{y}},\widehat{\mathit{z}})$ (shown in black) by the inclination $I$ and the longitude of ascending node $\mathit{\Omega }$, while the orientation of the orbit within this plane is specified by the argument of pericenter $\omega$. The true anomaly $\psi$ acts as an angular coordinate for the position of the orbit within the orbital plane, measured relative to the pericenter. The polarization tensors describing the effects of an incoming plane GW are described in terms of the basis $(\widehat{\mathit{n}},\widehat{\mathit{u}},\widehat{\mathit{v}})$ (shown in red), which is related to the reference frame by the angles $\vartheta$ and $\varphi$. (Note that the wavelength of the GW is not shown to scale here—our analysis assumes a wavelength much larger than the size of the orbit).

Figure 3

The secular diffusion coefficients $D_{PP}^{(2)}$, $D_{Pe}^{(2)}$, and $D_{ee}^{(2)}$ (first, second, and third rows, respectively) as functions of eccentricity, for various power-law SGWB spectra, $\mathrm{\Omega }(f)\sim f^{\alpha }$. The left column shows positive power-law indices, $\alpha =0,\dots ,2$, while the right column shows negative indices, $\alpha =0,\dots ,-2$.

Figure 4

Stochastic parts of the secular drift coefficients $D_P^{(1)}$ (first row) and $D_e^{(1)}$ (second row) as functions of eccentricity, for various power-law SGWB spectra, $\mathrm{\Omega }(f)\sim f^{\alpha }$. The left column shows positive power-law indices, $\alpha =0,\dots ,2$, while the right column shows negative indices, $\alpha =0,\dots ,-2$.

Figure 5

Contributions to the secular diffusion coefficients $D_{PP}^{(2)}$ (left panel) and $D_{ee}^{(2)}$ (right panel) from different harmonic frequencies, for binaries with various eccentricities $e=0,\dots ,1$. (The subscript “$n$” here indicates that we have extracted the contribution from the $n$th harmonic.) We set $\mathrm{\Omega }(f)=\text{constant}$ here, so that each harmonic receives equal weighting; alternative SGWB spectra will give different weighting to each harmonic. Note that $D_{ee}^{(2)}=0$ when $e\to 1$.

Figure 6

Contributions to the stochastic parts of the secular drift coefficients $D_P^{(1)}$ (left panel) and $D_e^{(1)}$ (right panel) from different harmonic frequencies, for binaries with various eccentricities $e=0,\dots ,1$. (The subscript “$n$” here indicates that we have extracted the contribution from the $n$th harmonic.) We show the absolute values, as the drift coefficients have both positive and negative contributions. Note that $D_e^{(1)}\to +\infty$ as $e\to 0$.

Figure 7

Evolution timescales for the DF of a circular binary with total mass $M=M_{\odot }$ immersed in a scale-invariant SGWB $\mathrm{\Omega }(f)=\text{constant}$. Solid curves show the drift timescale (5.7), which is dominated by GW emission at “short” periods $P\lesssim P_{\mathrm{crit}}$ and by GW absorption from the SGWB at “long” periods $P\gtrsim P_{\mathrm{crit}}$. Dashed curves show the diffusion timescale (5.8). The dotted black curve shows $\tau =P$; the region below this curve corresponds to a timescale faster than the binary period, which breaks our secular averaging assumption and should thus be taken with a grain of salt.

Figure 8

The quasistationary distribution (5.10) for the period $P$ of a binary with mass $M=M_{\odot }$ coupled to a scale-invariant SGWB, $\mathrm{\Omega }(f)=\text{constant}$. The upper and lower cutoffs are due to the age of the Universe and the binary’s ISCO period (5.9), respectively. The dashed vertical lines indicate the value of $P_{\mathrm{crit}}$ for each $\mathrm{\Omega }$, as defined by Eq. (4.9)—this is roughly the peak of the distribution in each case. Binaries with periods $P<P_{\mathrm{crit}}$ decay deterministically through GW emission and are removed from the distribution, whereas binaries with $P>P_{\mathrm{crit}}$ are supported against decay by resonant absorption of the SGWB. As we lower the SGWB intensity $\mathrm{\Omega }$, the resonance becomes weaker, and the binaries must have longer periods to avoid decay.

Figure 9

The covariance matrix $C_{ij}(t)$ of the orbital elements of the binary pulsar $\mathrm{B}1913+16$ (the Hulse-Taylor system) over a 1 kyr interval, assuming a scale-invariant SGWB spectrum $\mathrm{\Omega }(f)={10}^{-5}$, and including the first 400 harmonics. The blue dashed curves show numerical solutions of the evolution equations (6.22), while the pale red curves show the naive solution $t\times {\dot{C}}_{ij}$, which is exact only when ${\dot{C}}_{ij}=\text{constant}$.

Figure 10

Corner plot showing the distribution of the orbital elements of $\mathrm{B}1913+16$ at the end of the 1 kyr numerical integration shown in Fig. 9.