Estimating Stochastic Gravitational Wave Backgrounds with Sagnac Calibration

TL;DR

The paper presents a noise-calibration strategy for space-based detectors like LISA using the symmetrized Sagnac observable $\zeta$ to distinguish instrumental noise from a broadband stochastic gravitational-wave background. By formulating estimators such as $E_k = \eta_k^2 - D(f)|\zeta_k|^2$ (and high-/low-frequency variants like $E_k = \eta_k^2 - [S_{ave}/S_\zeta]_{est}|\zeta_k|^2$) and exploiting bandwidth averaging with $B \approx f/2$, the approach can substantially enhance sensitivity to isotropic GW backgrounds beyond traditional single-frequency estimates. The method has direct implications for astrophysical backgrounds from galactic/extragalactic binaries and primordial backgrounds, predicting improvements in detectable $h_{rms}$ levels by up to an order of magnitude in pivotal bands and offering insights into star-formation history, binary evolution, and early-Universe physics. The paper also discusses a high-frequency follow-on mission with shorter arms and superior noise performance, which could extend Sagnac-calibrated measurements to higher frequencies (up to ~0.1 Hz) and tighten constraints on primordial backgrounds, potentially rivaling cross-correlation methods with a single antenna.

Abstract

Armstrong et al. have recently presented new ways of combining signals to precisely cancel laser frequency noise in spaceborne interferometric gravitational wave detectors such as LISA. One of these combinations, the symmetrized Sagnac observable, is much less sensitive to external signals at low frequencies than other combinations, and thus can be used to determine the instrumental noise level. We note here that this calibration of the instrumental noise permits smoothed versions of the power spectral density of stochastic gravitational wave backgrounds to be determined with considerably higher accuracy than earlier estimates, at frequencies where one type of noise strongly dominates and is not substantially correlated between the six main signals generated by the antenna. We illustrate this technique by analyzing simple estimators of gravitational wave background power, and show that the instrumental sensitivity to broad-band backgrounds at some frequencies can be improved by more than an order of magnitude over the standard method, comparable to that which would be achieved by cross-correlating two separate antennas. The applications of this approach to studies of astrophysical gravitational wave backgrounds generated after recombination and to searches for a possible primordial background are discussed.

Figures (4)

• Figure 1: Illustration of the signal combinations discussed in the text. The numbers labelling each pair of arrows correspond to the subscript labels of signals in the notation of Armstrong et al.: "12,3" for example refers to $y_{12,3}$, the signal traveling on the side opposite spacecraft 1, received by spacecraft 2 (from spacecraft 3), with a time delay corresponding to the light travel time along the side opposite 3. The $\beta$ and $\gamma$ observables correspond to cyclic permutations of the indices for $\alpha$. The symmetrized Sagnac observable $\zeta$ is very similar to the round-trip-difference observables $\alpha,\beta,\gamma,$ except that for $\zeta$ all the signals are compared with almost the same time delays, leading to a minimal sensitivity to low-frequency gravitational waves. The $X$ observable is based on a Michelson interferometer using only two sides, but is the difference in signals at two times separated by approximately the the round trip travel time on one arm. The $Y$ and $Z$ observables are equivalent to $X$ but based on the other spacecraft pairings.
• Figure 2: Instrument sensitivity in terms of rms strain per $\sqrt{\rm Hz}$, to broad band backgrounds, assuming a one year integration. The "standard" S/N=1 levels in one frequency resolution element, for LISA and for the shorter-baseline follow-on mission described in the text, are shown as lighter lines. The sensitivity is shown for both the (standard) Michelson observable $X$ and the symmetrized Sagnac observable $\zeta$. The levels theoretically attainable with Sagnac calibration and averaging over bandwidth $f/2$ are shown in bold lines. The Sagnac estimator loses its advantage at high frequencies where $\zeta$ is no longer insensitive to gravitational waves; the analytic form for the estimator discussed here is also inefficient at frequencies where the proof mass noise and optical path noise are comparable. At low frequencies where proof mass noise dominates, another analytic form yields a significant improvement in sensitivity, which allows the confusion background to be measured to lower frequencies. Estimated astrophysical backgrounds are shown for Galactic binaries, extragalactic white dwarf binaries, and extragalactic neutron star or black hole binaries.
• Figure 3: Noise levels are shown in terms of the equivalent energy density of an isotropic stochastic background. Units are the energy density per factor $e$ of frequency, in units of the critical density, normalized for Hubble constant $h_0=1$. Where applicable, a one-year integration is assumed. The sum UB+WUMa+GCWDB represents the estimated confused background from the sum of unevolved Galactic binaries, W Ursa Majoris binaries and white dwarf binaries. These estimates are uncertain by about a factor of 10 in $\Omega$. The confusion noise level drops abuptly above the frequency where almost all Galactic binaries can be fitted out. Extragalactic white-dwarf binaries "XGCWDB" create a stochastic confusion noise which cannot be eliminated. At still higher frequencies above about 0.1 Hz, the white dwarfs coalesce, leaving only the confusion background from extragalactic neutron star binaries and stellar-mass black hole binaries (XGNSB). The LISA instrument noise limit (S/N=1) after one year is shown, both the traditional narrow-band sensitivity and the broad-band sensitivity allowed by Sagnac calibration and discussed here. The shaded regions show the main areas for improvement possible from using Sagnac calibration. The Sagnac technique allows a significantly improved measurement of a low resolution spectrum of the confusion background with LISA both at low frequencies $\approx 0.1$mHz and at higher frequencies to $\ge$ 20 mHz, including an accurate measurement with LISA of the extragalactic white dwarf binary confusion background. The Sagnac sensitivity limit for the smaller-baseline follow-on mission is shown for the parameters discussed in the text; in this case the Sagnac technique offers a more substantial overall improvement in sensitivity.
• Figure 4: Regions of new parameter space for primordial backgrounds opened up by proposed experimental setups and data analysis strategies. Scale on the top axis shows the cosmic temperature for which classical waves generated at the Hubble frequency and redshifted to the present yield the observed frequency on the bottom axis. Several characteristic energy densities are shown: Classical primordial gravitational wave background limit (PGWB) shows the sum of energies of photons and massless neutrinos, the maximal level expected for primordial backgrounds; "SBBN" denotes the maximum level consistent with Standard Big Bang Nucleosynthesis (both of these for a background with $\Delta f= f$); and "inflation" denotes a typical, untilted, scale-free inflation-generated spectrum, at the maximum level consistent with the background radiation anisotropy. Shaded regions lie above both instrument noise and binary confusion backgrounds, where primordial backgrounds can be detected. The darker-shaded regions show the extra benefit (for primordial background measurements) of Sagnac calibration with both missions.