Stochastic gravitational wave backgrounds

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Stochastic gravitational wave backgrounds are formed from the superposition of gravitational-wave signals that are either too weak or too numerous to individually detect. The signals making up a stochastic gravitational-wave background are thus individually unresolvable, unlike the large signal-to-noise ratio binary black hole and binary neutron star merger signals that have been observed by the Advanced LIGO and Virgo detectors. Since the signal corresponding to a stochastic gravitational-wave background is random, it looks much like noise in a single detector. Nonetheless, detecting a stochastic gravitational-wave background is possible if we understand its statistical properties and can separate its effect from instrumental and environmental noise in a single or multiple detectors. The detection of a stochastic gravitational-wave background provides information about the statistical and/or population properties of the sources.


Contents

Recent observations

Pulsar timing array observations: On 30 June 2024, several collaborations around the world announced evidence for a low-frequency "hum" of gravitational waves (GWs) (Agazie et al., 2023, Antoniadis et al., 2023, Xu et al., 2023, Reardon et al., 2023). The frequencies of the GWs were of order \(10^{-9}\) Hz (nanohertz), corresponding to periods of roughly years to decades. The data consisted of the measured arrival times of radio pulses taken over the last 15-20 years from an array of ~100 millisecond radio pulsars in the Milky Way galaxy. The signal was correlated across the pulsars in the array as expected for a stochastic GW background, with the probability that the observed signal was due to noise alone being \(\approx 10^{-4}\). A credible candidate for the source of the GWs is the superposition of signals from pairs of orbiting supermassive black holes (\(\sim\! 10^9\) times more massive than the Sun) in the cores of millions of merging galaxies throughout the visible Universe. Other more exotic sources of GWs are also consistent with the observations.

The two panels in Figure 1 show the measured and expected power spectra (left panel) and measured and expected correlations (right panel) taken from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration 15-year analysis paper (Agazie et al., 2023). Details pertaining to the meaning of the quantities plotted in the figure will be discussed later in this article.

Figure 1: Observations from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) pulsar timing array collaboration. Left panel: Measured power spectra for different signal models for a GW background having correlations following the expected Hellings and Downs pattern (shown in the right panel). The vertical axis shows the contributions of the different signal models to the measured timing residuals in seconds (on a log10 axis). The gray violins are posterior distributions for timing residual perturbations at frequencies i/T, where T is the time span of the total data set and \(i=1,2,\cdots\). The blue curve is the posterior power spectrum (median with 1 and \(2\sigma\) bands) for a power-law signal model with spectral index determined from the data; the dashed black line corresponds to a power spectrum with the median posterior amplitude and an assumed spectral index (-13/3) appropriate for a GW background produced by supermassive black hole binaries. Right panel: Pulsar-pair correlations averaged together in 15 angular separation bins (point estimates \(\pm 1\sigma\) error bars), measured from 2,211 distinct pairings of 67 pulsars assuming maximum-a-posteriori pulsar noise parameters and a GW background having a -13/3 spectral index. The averaging within each bin takes into account correlations between pulsar pairs. The dashed black curve is the expected Hellings and Downs correlation as a function of the angular separation between pairs of pulsars, predicted for a GW background in general relativity.

In order to put the pulsar timing array observations into proper context, we first describe a few more recent observations of GWs and black holes before jumping into specifics about stochastic GW backgrounds.

Binary black hole merger GW150914: On 14 Sep 2015, the two Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) detectors made the first direct detection of GWs. This event, denoted GW150914, consisted of the final inspiral and merger of a pair of black holes, each about 30 times as massive as the Sun. The inspiral and merger took place in a distant galaxy located approximately 1.3 billion light-yrs from Earth. The signal was observed (almost) simultaneously by the two LIGO detectors, arriving at the detector in Hanford, WA approximately 7 msec after it arrived at the detector in Livingston, LA. The signal was observed for 0.15 seconds, sweeping up in amplitude and frequency (from several tens of Hz to approximately 250 Hz) as the two black holes made their final orbits around one another and eventually merged. This characteristic "chirp signal" was the "smoking-gun signature" of a binary black hole merger, which theorists had predicted and data analysts were looking for in the data.

Figure 2: The GW signal associated with GW150914, as observed by the two Advanced LIGO detectors on 14 Sep 2015. Top panel: The GW strain \(h\equiv \Delta L/L\) as a function of time in the two detectors. The top right-hand panel shows the data in both detectors, taking into account the propagation time of the GW from Livingston, LA to Hanford, WA and the relative orientation of the two detectors. Bottom panel: The amplitude and frequency of the measured signal as a function of time in the two detectors.

The difference between GW150914 and that observed by the pulsar timing array collaborations is quite stark: GW150914 was a single event (the final inspiral and merger of two stellar-mass black holes), which was resolvable in the Advanced LIGO detectors. In contrast, the possible GW signal observed by the pulsar timing array collaborations was the superposition of millions of unresolvable (approximately monochromatic) signals associated with the orbital motion of pairs of supermassive black holes in the centers of merging galaxies. So these are two very different types of GW signals.

Binary neutron star merger GW170817: On 17 Aug 2017, the two Advanced LIGO detectors together with their European counterpart Virgo again detected GWs, but this time from the final inspiral and merger of two neutron stars. This event, denoted GW170817, was the first multi-messenger observation of a GW event, as it was also observed in electromagnetic waves. As shown in the left panel of Figure 3, the event was observed almost simultaneously in GWs and via a burst of gamma rays by NASA's Fermi satellite. The difference between the arrival times of the GWs and gamma rays from the merger over the distance (40 Mpc) to the host galaxy constrains the speed of GWs to equal the speed of light to better than one part in \(10^{15}\). The "afterglow" of the inspiral and merger was subsequently observed by several other traditional astronomy telescopes across the electromagnetic spectrum. In the right-hand panel of Figure 3, we show three such observations: (i) a visible light image taken by the Swope telescope, (ii) an X-ray image by NASA's Chandra X-ray satellite, and (iii) an image in radio waves by NRAO's Jansky Very Large Array (VLA). The dark blob is the host galaxy (NGC 4993) of the binary neutron star inspiral and merger, which is located roughly 40 Mpc (or 120 million light-years) from Earth. The small dot in the images is the afterglow of the binary neutron star merger.

Figure 3: Multi-messenger event GW170817 as seen in GWs (left panel, bottom), gamma rays (left panel, top) and via visible, X-rays, and radio waves (right panel, top, middle, bottom). The observations in visible light, X-rays, and radio waves commenced 10.86 hrs, 9 days, and 16.4 days after the initial observations in GWs and gamma rays. This event was also seen by other telescopes and via other forms of light, but images from those observations are not included in this figure. [Figure adapted from Abbott et al., 2017 and https://www.ligo.org/detections/GW170817.php. Credit: NASA GSFC & Caltech/MIT/LIGO Lab]

Unlike black holes which are made of pure spacetime curvature, neutron stars are made of matter; they are basically giant atomic nuclei having roughly the mass of the Sun, packed into a radius the size of a small city. So when the two neutron stars smashed into one another after the final orbits of GW170817, electromagnetic radiation was produced in the process. In fact, some of the constituent neutrons were rapidly converted into heavier elements, most notably gold and platinum. It had been conjectured that binary neutron star mergers were the source of these heavy metals, but it wasn't until this multi-messenger observation that this conjecture could be confirmed.

Event Horizon Telescope images of supermassive black holes M87* and Sgr A*: On 10 Apr 2019 and 12 May 2022, The Event Horizon Telescope (EHT) team released images of the supermassive black holes in the centers of the giant elliptical galaxy Messier 87 and our own Milky Way galaxy ( EHT Collaboration et al., 2019, EHT Collaboration et al., 2022). These images, shown in Figure 4, show the shadow of the event horizon of the black hole (black region) together with the light emitted by material in the vicinity of the black hole (orange region). The images were created using data taken by a network of radio telescopes across the globe, using interferometry to synthesize a radio telescope with a "dish" having a diameter equal to the diameter of the Earth. The mass of the black hole M87* is $6.5\times 10^9 M_\odot$, while that of Sgr A* is \(4.1\times 10^6~M_{\odot}\). The mass of M87* is comparable to the masses of the supermassive black holes that may be contributing to the stochastic GW signal that the pulsar timing array collaborations are seeing.

Figure 4: Event Horizon Telescope images of the supermassive black holes M87* (left panel) and Sgr A* (right panel) taken on 11 Apr 2017 and 7 Apr 2017, respectively. The angular size of the images is roughly \(50 \mu{\rm as}\).


Motivation

These last three observations provide incontrovertible evidence that GWs and black holes exist, and that GWs allow us to observe objects or events in the Universe that are inaccessible to (or complement) traditional electromagnetic telescopes. But, perhaps most importantly, GWs provide a means for astronomers to potentially observe the Universe mere fractions of a second after the Big Bang. This is much earlier than what we can do with light, since photons could only begin to propagate freely ~380,000 years after the Big Bang, when the Universe had cooled enough for neutral hydrogen atoms to form (see Figure 8).

The picture that we have of the Universe at this time of "last scattering" is given by the temperature fluctuations in the cosmic microwave background (CMB), which shows density perturbations in the Universe when it was roughly 1000 times hotter and 1000 times smaller than it is today, see Figure 5. The red regions in the plot, corresponding to over densities in matter at the time of last scattering, may have been the "seeds" around which stars and galaxies would have subsequently formed.

Figure 5: Skymap of temperature fluctuations \(\Delta T/T_0\), with \(T_0=2.73~{\rm K}\), for the cosmic microwave background radiation as measured by ESA's Planck satellite in 2015 (https://www.cosmos.esa.int/documents/387566/425793/2015_SMICA_CMB/). This is the earliest picture we have of the universe as observed with photons, corresponding to roughly 380,000 years after the Big Bang.

The CMB was first detected in 1965 by Arno Penzias and Robert Wilson as blackbody radiation with a temperature of approximately \(T_0=2.73 K\), coming from all directions on the sky. Starting in 1992 with NASA's COBE satellite, deviations from isotropy have been observed with greater and greater angular precision, leading to sky maps like that shown in Figure 5, as measured by ESA's Planck satellite in 2015.

The ultimate goal for stochastic GW background searches is to obtain an analogous sky map for relic GWs produced by the Big Bang. This is arguably the "holy grail" of GW astronomy, as it would allow us to "look" back to a time much less than a second after the Big Bang. Such time scales and energy scales cannot be reached via any other means.

Some background on gravitational waves

We present here a quick review of GWs. If an electric charge is accelerated, electromagnetic radiation is produced. Similarly if a mass is accelerated, gravitational radiation is produced; we will call these GWs. Albert Einstein developed general relativity in 1915, which is the theory that describes gravitation. Quickly Einstein also recognized that there should be GWs, and he published papers in 1916 and 1918 describing them. A GW manifests itself by traveling at the speed of light, and stretching and contracting spacetime perpendicular to its direction of propagation. You can read more about GWs and how they are detected on earth in Gravitational Waves: Ground-Based Interferometric Detectors.

A GW propagating in the \(z\)-direction can be represented as, \( h(z,t) = (h_{0+} + h_{0\times}) e^{i(kz−ωt)} \). Like electromagnetic radiation, GWs have two polarizations. The amplitude of the GW, \( h \), is referred to as the strain. This is because of the stretching and contracting of spacetime that it induces. The two polarizations, \( h_{0+} \) and \( h_{0\times} \), stretch and contract spacetime along axes that differ by \( 45^{o} \) from one another. GWs carry energy, momentum, and angular momentum away from the source, and as explained below, it is the energy density of GWs throughout the universe that has profound importance for cosmology. For the first GW signal detected, GW150914, it was estimated that the two progenitor black holes had masses of \(36 M_{\odot}\) and \(29 M_{\odot}\), the new black hole had a mass of \(62 M_{\odot}\), and that the total energy emitted in GWs was \(3 M_{\odot} c^{2}\)! When the GW signal from GW150914 was detected by LIGO with its 4 km-long interferometer arms, the change in the arm length was about \( 2 \times 10^{-18} \) m.

LIGO and Virgo have announced the observation of more than 90 GW signals from their first three observational runs. LIGO, Virgo, and KAGRA started their fourth period of observation, O4, in May 2023. O4 is scheduled to continue until June 2025. More than 100 signal notices have been sent out in O4. All of these GW signals to date come from the coalescence of compact binary systems; binary black holes, binary neutron stars, and neutron star - back hole binaries. In Gravitational Waves: Science with Compact Binary Coalescences, one can learn more about the detected signals and their scientific importance. LIGO, Virgo, and KAGRA are also looking for other types of GW signals, such as sinusoidal-like continuous wave signals from spinning neutron stars (pulsars), burst signals from supernovae, and stochastic GW signals like those discussed in this article.

Sources

Many different GW sources can give rise to a stochastic GW background. The only condition on the sources is that the GW signals associated with each source be individually unresolvable. In practice, this means that the signals are either too weak or too numerous to be individually detected. As GW detectors improve and become more sensitive to detecting GWs, sources that were previously unresolvable now become resolvable, standing out above the lower levels of instrumental and/or environmental noise.

Potential GW sources are usually classified as either astrophysical or cosmological depending on when during the evolution of the Universe those sources produced GWs. Astrophysical sources typically consist of populations of compact objects like white dwarfs, neutron stars, or black holes. Cosmological sources are associated with events or processes in the early universe (e.g., inflation or phase transitions), which take place well before the formation of stars and galaxies (see Figure 8).

Figure 6 is a schematic representation of possible sources of a stochastic GW background as a function of frequency. The different GW sources are shown in the colored boxes at the top of the figure; relevant detectors are shown in the colored boxes at the bottom of the figure. The horizontal width of the boxes give the range of frequencies relevant for the sources and detectors.

Figure 6: Possible sources of a stochastic GW background across the GW spectrum. The different sources and detectors are shown in the colored boxes. The width of these boxes represents the range of frequency over which these sources and detectors produce and detect signals. Note that the GW signal from cosmological sources such as inflation, cosmic strings, and phase transitions stretch across a broad range of frequencies. The signal from phase transitions peak at basically any frequency depending on the energy scale and time scale over which the phase transition occured, while that from inflationary models and cosmic strings are flat across much of the frequency range shown here.


The strength of a stochastic GW background is typically described in terms of the (dimensionless) energy-density spectrum \[ \tag{1} \Omega_{\rm gw}(f) \equiv \frac{1}{\rho_{\rm crit}}\frac{{\rm d}\rho_{\rm gw}}{{\rm d}(\ln f)} = \frac{f}{\rho_{\rm crit}}\frac{{\rm d}\rho_{\rm gw}}{{\rm d}f} \,, \] where \({\rm d}\rho_{\rm gw}\) is the energy density in GWs contained in the frequency interval \(f\) to \(f+{\rm d}f\), and \[ \tag{2} \rho_{\rm crit}\equiv \frac{3H_0^2 c^2}{8\pi G} \] is the critical energy density needed to close the universe today. Here, \(H_0 \equiv \dot a(t)/a(t)|_{t=t_0}\) is the Hubble expansion rate evaluated today, \(a(t)\) is the scale factor describing the expansion of the universe as a function of time, \(G\) is Newton's gravitational constant, and \(c\) is the speed of light.

There is a relatively simple formula for calculating the energy-density spectrum \(\Omega_{\rm gw}(f)\) produced by a collection of discrete astrophysical GW sources distributed throughout the universe: \[ \tag{3} \Omega_{\rm gw}(f) = \frac{1}{\rho_{\rm crit}}\int_0^\infty {\rm d} z\> n(z) \frac{1}{1+z}\left(f_{\rm s}\frac{{\rm d} E_{\rm gw}}{{\rm d} f_{\rm s}} \right)\bigg|_{f_{\rm s}=f(1+z)}\,, \]

For this expression, one needs only the comoving number density of sources \(n(z)\) as a function of the cosmological redshift \(z\) (defined as \(1+z = a(t_0)/a(t)\)), and the energy spectrum of an individual source \({\rm d} E_{\rm gw}/{\rm d}f_s\) as measured in its rest frame (which is the source frame). The source frame frequency \(f_{\rm s}\) is related to the observed (present-day) frequency \(f\) via \(f_{\rm s}=f(1+z)\). The factor of \(1/(1+z)\) in the integrand is needed to redshift the energy measured in the source frame to that measured today. One can rewrite this equation in terms of the comoving rate density \(R(z)\) by making the substitution \[ \tag{4} n(z) = R(z) \left|\frac{{\rm d}t}{{\rm d}z}\right| = \frac{R(z)}{(1+z)H_0\sqrt{\Omega_{\rm m}(1+z)^3 + \Omega_\Lambda}}\,. \] This relation can be obtained by solving the Friedmann equation (one of the Einstein equations) for the scale factor \(a(t)\), where \(\Omega_{\rm m}\) and \(\Omega_\Lambda\) are the fractional energy densities for matter (ordinary baryonic matter plus dark matter) and dark energy, with numerical values equal to roughly 0.30 and 0.70, respectively. Using (3), one can show that GW sources or processes that are scale-invariant give rise to energy-density spectra that are independent of frequency, \(\Omega_{\rm gw}(f) = {\rm const}\), while a population of inspiraling binaries (e.g., black holes, neutron stars, etc.) gives rise to \(\Omega_{\rm gw}(f)\propto f^{2/3}\). This latter result is a consequence of Kepler's third law \(\omega^2 a^3 = {\rm const}\), \(E_{\rm orb} \propto -1/a\), and \({\rm d}E_{\rm gw}/{\rm d}\omega = -{\rm d}E_{\rm orb}/{\rm d}\omega\), where \(\omega=2\pi f_{\rm orb}=\pi f\) is the orbital angular frequency of the binary, and $a$ is the semi-major axis of the elliptical orbit.

The energy-density spectrum \(\Omega_{\rm gw}(f)\) is related to the power spectrum \(S_h(f)\) of the GW strain \(h(t)\) via \[ \tag{5} S_h(f) = \frac{3H_0^2}{2\pi^2}\frac{\Omega_{\rm gw}(f)}{f^3}\,. \] The strain power spectrum has dimensions of \({\rm strain}^2/{\rm Hz}\). This means that integrating \(S_h(f)\) from frequency $f_1$ to $f_2$ gives the power contained in the signal $h(t)$ in the frequency interval \(\Delta f\equiv f_2-f_1\). Using (5) and the comment made in the last sentence of the previous paragraph, it follows that for a population of inspiraling binaries, \(S_h(f)\propto f^{-7/3}\). Since pulsar timing array observations are typically in terms of timing residuals $\delta t(t)$ (dimensions of time) as opposed to strain $h(t)$ (dimensionless), the associated power spectrum \(S_{\delta t}(f)\) is related to \(S_h(f)\) by a factor of \(f^{-2}\). Thus, for a population of supermassive black hole binaries, which is a credible candidate signal for PTAs, the timing residual power spectrum falls off with frequency as \(S_{\delta t}(f)\propto f^{-13/3}\). This is the reason for the $\gamma=13/3$ spectral index in the left-hand panel of Figure 1.

Astrophysical sources

Stellar-mass binary black hole mergers

As mentioned above, GW150914 was the first direct detection of GWs from the final inspiral and merger of two stellar-mass black holes. Since that first observation on 14 Sep 2015, more than 100 additional binary black hole (BBH) mergers have been detected by the Advanced LIGO and Virgo detectors. All of these mergers have involved pairs of black holes having masses of order 5-100 \(M_\odot\), which is relevant for the frequency range (~10-1000 Hz) of the Advanced LIGO and Virgo detectors.

The detection of these individual binary black hole mergers implies that a population of stellar-mass binary black holes exist, with the more distant and less massive of these sources giving rise to an unresolved stochastic GW background, which the Advanced LIGO and Virgo detectors should eventually be able to detect. But although we are certain that such a stochastic GW background exists, we have not yet detected it with search methods designed to look for the correlations that the combined signal will induce in the outputs of two or more LIGO / Virgo detectors. The estimated energy density from binary black hole mergers at 25 Hz is \( \Omega_{\rm{BBH}}(25~{\rm Hz}) = 4.7^{+1.6}_{-1.4} \times 10^{-10} \). Given current estimates of the stellar-mass binary black hole merger rate and the current sensitivity of the Advanced LIGO and Virgo detectors, it is projected that the stochastic GW background produced by these binary black hole mergers will not be detected until shortly after the these detectors are upgraded to their A+ sensitivity. This is shown in Figure 7 taken from Abbott et al., 2021.

Figure 7: Projected energy density spectra for stellar-mass BBH and binary neutron star (BNS) mergers, based on rate estimates inferred from Advanced LIGO and Virgo's 3rd observation run O3 (Abbott et al., 2021). The solid blue line shows the median estimate of \(\Omega_{\rm gw}(f)\) from BBH and BNS mergers, while the shaded blue band illustrates 90% credible uncertainties. Also shown is a projected upper limit on the total GW background signal (blue dashed line), including an upper limit on the contribution from neutron star-black hole (NSBH) mergers. These projections are to be compared against the \(2\sigma\) detector sensitivity curves (shown in black), for the second observation run O2, the third observation run O3, and as well as projections for the Hanford-Livingston-Virgo network at design sensitivity, and the A+ detectors. If a projected \(\Omega_{\rm gw}(f)\) intersects one of the detector sensitivity curves (as it does for the A+ curve), then the GW signal is detectable with a signal-to-noise ratio of \(\ge 2\).

Part of the reason why it has been hard to observe the stochastic GW background from stellar-mass binary black hole mergers with the Advanced LIGO and Virgo detectors is that this type of signal is "on" only a small fraction of the time. Each binary black hole merger typically lasts less than one second in the frequency band of the LIGO and Virgo detectors, while the average time between successive mergers is of order 5 minutes. The ratio of these two time intervals is called the duty cycle of the signal; for stellar-mass binary black hole mergers it is \(10^{-3}\) - \(10^{-2}\ll 1\). Such a signal is said to be intermittent or popcorn-like. A more sophisticated search method, which takes into account the intermittent nature of this signal, can potentially reduce the time to detection to weeks or months. Such a method is currently under development at the time of writing this article, and has not yet been applied to real data.

Binary neutron star mergers

LIGO and Virgo have observed GWs from the merger of binary neutron stars; these events are labeled as GW170817 and GW190425. Even from only these two events one can make a prediction of the magnitude of the GW background produced by binary neutron star mergers. The estimated energy density from binary neutron star mergers at 25 Hz is \( \Omega_{\rm{BNS}}(25~{\rm Hz}) = 2.0^{+3.2}_{-1.4} \times 10^{-10} \). This binary neutron star background will be difficult to observe with LIGO-Virgo-KAGRA, but will be potentially observable by the future generation of ground based detectors, such as the Einstein Telescope (ET) and Cosmic Explorer (CE).

Super-massive black hole binaries

As mentioned above, a credible candidate for the possible GW signal observed by pulsar timing arrays is the superposition of GW signals from millions of pairs of supermassive black holes orbiting one another in the cores of merging galaxies. How these supermassive black holes formed in the first place is an open question, although a likely scenario is via a sequence of successive mergers of smaller-mass black holes. GW observations of BBH systems across the mass spectrum is a way to potentially answer this question: we can observe stellar-mass black hole binaries (\(\sim 1 \) - \(100 M_\odot\)) with Advanced LIGO and Virgo; intermediate and massive black hole binaries (\(\sim 10^4\) - \(10^6 M_\odot\)) with LISA; and supermassive black hole binaries (\(\sim 10^8\) - \(10^{10} M_\odot\)) with pulsar timing arrays.

We know from electromagnetic observations that most (if not all) galaxies host supermassive black holes at their centers. We also know from electromagnetic observations that galaxies merge over time. Thus, we expect that the supermassive black holes at the centers of these merging galaxies will form binaries, which will inspiral and then eventually merge with one another, generating GWs in the process. But one can show that if GW emission is the sole driver of the evolution of the supermassive black hole binary system, then the two black holes will not get close enough to emit GWs in the nanoHertz frequency band, and they will not merge in less than the age of the universe. Thus, one needs some other mechanism to drive the supermassive black hole binary to merger.

One way of accelerating the merger is via interactions of the supermassive black holes with the environments of their host galaxies. Interactions of the black holes with surrounding stars and gas can bring the black holes closer together, "hardening" the binary and thereby decreasing the time to merger. An accelerated inspiral phase means less power in the lowest frequencies relative to that predicted by the simple \(f^{-13/3}\) power-law dependence for GW-emission as the sole driver of the binary evolution. Indeed, this fall-off at lower frequencies gives a better fit to the actual observed power spectrum, as can be seen in the left panel of Figure 1.

Supermassive BBHs with masses of order \(10^9~M_\odot\) will merge outside the PTA sensitivity band, at frequencies of order \(10^{-6}~{\rm Hz}\). The two supermassive black holes will orbit one another for several million years before they coalesce, once they enter the PTA band at a frequency of \(10^{-9}~{\rm Hz}.\) Thus, the signal produced by such a binary will be approximately monochromatic over the timeframe of an observation, which is of order 10 years to a few decades. The combined signal from a population of millions of supermassive black hole binaries is therefore much different than the intermittent, popcorn-like GW background produced by stellar-mass BBH mergers in the advanced LIGO / Virgo frequency band (~10-1000 Hz).

Galactic white-dwarf binaries

White dwarf binary systems in the Milky Way galaxy that are close enough together to orbit one another with periods of minutes to hours is a guaranteed source of a stochastic GW background for LISA (Laser Interferometer Space Antenna, https://www.lisa.nasa.gov/). LISA is a joint project of ESA (European Space Agency) and NASA (National Aeronautics and Space Administration). LISA is scheduled to be launched in 2035; it will consist of three spacecraft placed at the vertices of an equilateral triangle in orbit around the Sun, trailing the Earth. The three LISA spacecraft will exchange laser light to carefully monitor the distance between the spacecraft, much like what is done with suspended mirrors by the Earth-based laser interferometers such as Advanced LIGO and Virgo. The distance between the LISA satellites is \(2.5\times 10^6~{\rm km}\), which is more than five times the distance between the Earth and the Moon. LISA will be able to detect small perturbations (\(10^{−12}~{\rm m}\)) to the distances between the spacecraft, due to GWs produced by many different types of sources in the Universe.

There are millions of double white-dwarf systems in the Milky Way that will emit GWs in the LISA sensitivity band (\(10^{-4}\) to \(10^{-1}~{\rm Hz}\)). Of those, only ~10,000 such systems will be sufficiently bright to be individually resolvable with LISA. The GW signals from the remaining double white-dwarf systems will interfere with one another, forming an unresolvable "confusion-noise" signal, which is expected to standout above the instrumental noise between \(10^{-4}\) and \(10^{-3}~{\rm Hz}\). This stochastic GW foreground signal will act much like detector noise when trying to detect other GW signals with LISA. Since the LISA constellation orbits the sun once a year, and the orientation of LISA toward the galactic center will consequently change, the foreground GW signal from the galactic binaries will be modulated with a period of one year. This modulation will be used to separate the foreground GW signal from other signals, including the detector noise.

Cosmological sources

Inflation

Probably the most interesting source of GWs, especially for a stochastic GW background, is the "birth" of the Universe, the Big Bang. One can read more about the current interpretation of the early universe in Modern cosmology. As described above, the cosmic microwave background radiation has contributed much to our knowledge of the Universe, but this background was created when the Universe had an age of about 380,000 years, when neutral hydrogen atoms formed and photons could propagate freely (see Figure 8). GWs interact with matter in only a very minor way, hence GWs produced in the first seconds of the universe will escape and not interact with the material in the universe. The GWs will have their wavelength increased due to the expansion of the universe. Cosmologically produced GWs will be our best chance to gain information about the physics of the earliest moments of the Universe.

It is speculated that the Universe initially experience a period of exponential expansion, so-called inflation. Inflation in cosmology is a theory that proposes the Universe underwent a very rapid expansion in its very early stages, a fraction of a second after the Big Bang (see Figure 8). This rapid expansion is thought to have stretched the Universe to be incredibly smooth and flat, laying the groundwork for the large-scale structure we see today. Interestingly, this inflation also leads to the production of GWs. The quantum fluctuations of the spacetime metric amplified during the inflationary period would produce GWs. These GWs would form a stochastic GW background that is essentially flat in energy density across the different frequency bands. Standard inflationary models would produce a background at the level of \( \Omega_{\rm{gw}}(f) \sim 10^{-15}\) to \( 10^{-17} \), which would be difficult to observe. However, modifications to inflationary scenarios might push the energy density to higher values.

First-order phase transitions

A first-order phase transition might be an opportunity to observe cosmologically produced GWs. A first-order phase transition is a dramatic shift in a material's state that occurs with a discontinuity in certain thermodynamic properties. This is in contrast to a second-order phase transition, which is smoother and more gradual. For example, water boiling to produce steam is a first-order phase transition. Electroweak symmetry-breaking in the Standard Model is not thought to be a first-order phase transition, as it is predicted to be a smooth crossover. In some modifications to the Standard Model, however, a first-order phase transition might occur. Hence, the observation of such an effect with a GW background provides information about physics beyond the Standard Model. Such information would then aid in explaining dark matter and baryon asymmetry of the Universe. In a first-order phase transition, the GWs are produced by the collision of bubbles of the (new) stable phase, and then from sound waves and turbulent flows. The expanding bubbles create sound waves, which are the primary source of GWs. Bubble collisions are important with supercooling. An electroweak phase transition producing GWs would be detectable today in the mHz frequency band. Hence LISA offers an important opportunity to make this observation.

Cosmic strings

Cosmic strings, if they exist, would be another source of a stochastic GW background. Cosmic strings are theorized to be one-dimensional topological defects. If they exist, they would have been created in the early Universe. This would happen after a phase transition followed by spontaneously broken symmetries. Cosmic strings would be relics of an earlier more symmetric phase of the universe; they appear in the context of grand unified theories. Cosmic strings decay only via the emission of GWs. They create GWs through oscillations, loop formation, cusps and kinks, and the interactions in a network of strings. The cosmic string tension is usually quantified by the dimensionless tension, \( G \mu/c^2 \). LISA should be able to observe a stochastic GW background from cosmic strings even if the string tension is as low as \( G \mu/c^2 \sim 10^{-16} \) to \( G \mu/c^2 \sim 10^{-15} \).

Figure 8 shows the evolution of the Universe, highlighting several important cosmological events. These include inflation, phase transitions, and the formation of neutral hydrogen allowing for the cosmic microwave background photons to decouple.

Figure 8: Schematic representation of the history of the Universe, showing a few key times and energy scales.

Detection methods

As discussed above, a stochastic background of GWs is described by a random signal, which resembles noise in a single detector. Because of this, standard search techniques like matched filtering (Wainstein & Zubakov, 1970), which correlate the measured data against known, deterministic waveforms (e.g., BBH chirps), will not work when trying to detect a GW background. Instead, we have to consider other possibilities:

(i) One possibility is to know the noise sources in our GW detector well enough (in both amplitude and spectral shape) that we can attribute any unexpected excess "noise" to a GW background. This was basically how the cosmic microwave background was discovered (Penzias & Wilson, 1965). Penzias and Wilson saw an excess noise temperature of \(\sim 3.5~{\rm K}\) in their radio antenna that they could not attribute to any other noise source.

(ii) Another possibility is to use data from multiple detectors. The idea here is to look for evidence of a common disturbance in the multi-detector output that is consistent with the individual detectors' response to GWs.

Currently, (i) is not an option for most GW detectors since, although the individual noise sources may be understood, their amplitude is usually not known precisely enough to attribute any observed excess power to GWs. But (ii) is always an option if more than one detector is available.

Correlation techniques

Cross-correlating data from multiple detectors works for detecting a GW background since, although the signal is random, it is the same signal in the different detectors modulo known differences in the response of the individual detectors due to their intrinsic design, physical location, and orientation with respect to one another. In effect, the random output of one detector can be used as a template for the data in another detector, leading to a signal-to-noise ratio for the cross-correlated signal that grows like the square-root of the observation time. This means that, although the GW signal might be weak relative to the noise, it can still be extracted from a cross-correlation measurement if it is observed for a long enough period of time.

To illustrate the basic idea behind cross-correlation, consider data from two detectors: \[ \tag{6} \begin{align} d_1 &= R_1 h + n_1\,, \\ d_2 &= R_2 h + n_2\,. \end{align} \] Here \(h\) denotes the common GW signal component; \(R_1\) and \(R_2\) denote the responses of the two detectors; and \(n_1\), \(n_2\) denote the corresponding noise contributions. Cross-correlating the data for this case amounts to simply taking the product of the data from the two detectors, \(\hat C_{12}\equiv d_1 d_2\). The expected value of the cross-correlation is then \[ \tag{7} \langle \hat C_{12}\rangle =\langle d_1 d_2\rangle = R_1 R_2\langle h^2\rangle + {\langle h n_2\rangle} + {\langle n_1 h\rangle} + \langle n_1 n_2\rangle = R_1 R_2\langle h^2\rangle + \langle n_1 n_2\rangle\,, \] where \(\langle h n_2\rangle = 0 = \langle n_1 h\rangle\) since the GW signal and instrumental noise are not correlated with one another. If we further assume that the noise in the two detectors is uncorrelated (which is typically a valid assumption if the detectors are widely separated), then \(\langle n_1 n_2\rangle =0\), leaving \[ \tag{8} \langle \hat C_{12}\rangle = R_1 R_2\langle h^2\rangle\equiv \gamma_{12} S_h\,, \] which is proportional to the variance (i.e., power \(S_h\)) in the GW signal. The quantity \(\gamma_{12}\equiv R_1 R_2\) encodes the reduction in sensitivity of the cross-correlation measurement due to the different responses of the two detectors.

Note that global magnetic fields, e.g., Schumann resonances, can produce environmental correlations in widely-separated ground-based detectors like LIGO, Virgo, KAGRA, and thus must be addressed when trying to detect a stochastic GW background signal. Techniques to deal with correlated noise contributions include trying to distinguish the signal from the noise using their expected spectral properties or correlation properties for different pairs of detectors.

Hellings and Downs correlation curve

For pulsar-timing-array searches for GWs, the "smoking gun" signature of a GW background is the Hellings and Downs correlation curve, which is the black dashed curve in the right-hand panel of Figure 1 and the black solid curve in Figure 9.

Figure 9: Hellings and Downs correlation curve (solid black) plus/minus the square-root of the cosmic variance (dotted black). The cosmic variance assumes noise-free measurements and an infinite number of pulsar pairs distributed uniformly on the sky. The “+” symbols are the predicted \(\pm\sigma\) deviations for a binned reconstruction of the Hellings and Downs curve assuming noise-free measurements and using the 15 angular bins and 66 pulsar sky locations from the NANOGrav 15-year analysis (Agazie et al., 2023). For the binned reconstruction, there are roughly 150 pulsar pairs per bin.

The Hellings and Downs curve is the expected correlation in the perturbations to the arrival times of pulses induced by a passing GW, as a function of the separation angle \(\xi_{ab}\) between pairs of pulsars \(a\) and \(b\): \[ \Gamma(\xi_{ab}) = \frac{1}{2} -\frac{1}{4}\left(\frac{1-\cos\xi_{ab}}{2}\right) +\frac{3}{2}\left(\frac{1-\cos\xi_{ab}}{2}\right)\ln\left(\frac{1-\cos\xi_{ab}}{2}\right)\,. \] This expression was originally calculated by Ron Hellings and George Downs in 1983 (Hellings & Downs, 1983), who averaged the product of the GW responses \(R_a\) and \(R_b\) of two fixed pulsars over an isotropic and unpolarized distribution of GW sources. (Isotropic means equal GW power from all directions on the sky; unpolarized means statistically independent but equivalent power in the + and \(\times\) polarization modes.) It was subsequently calculated by Neil Cornish and Alberto Sesana in 2013 (Cornish & Sesana, 2013), where they considered a single fixed GW source, but "pulsar averaged" the product of the GW responses over an infinite number of pulsar pairs having the same angular separation.

This latter procedure of pulsar averaging is the infinite-pulsar limit of the experimental practice of binning a finite number of correlation measurements in several angular separation bins. This is shown, for example, by the blue points and error bars in the right-hand panel of Figure 1. As the number of pulsar pairs per bin increases, the error bars on the binned reconstruction decrease. But they will not decrease to zero, even in the limit of an infinite number of pairs per bin and noise-free measurements, if the GW sources "interfere" with one another, as shown by Bruce Allen (Allen, 2023). This occurs if multiple GW sources radiate in the same frequency bin, and thus cannot be resolved as individual sources. This non-zero cosmic variance implies that the Hellings and Downs curve will never be exactly recovered for interfering GW sources (see Figure 9).

Evidence of the expected Hellings and Downs correlation is what the pulsar timing array collaborations were looking for in the analyses of their most recent data sets (see PTA observations and right-hand panel of Figure 1). It is the stochastic background equivalent of the binary "chirp" waveform that Advanced LIGO was hoping to see--and then saw--on 14 Sep 2015 with GW150914.

B-mode polarization

A much different approach to search for evidence of a stochastic GW background is used by scientists studying the cosmic microwave background (CMB). Primordial GWs produced during inflation may leave an imprint on the CMB radiation via its distinctive B-mode polarization pattern (see Figure 10). B-mode (or curl mode) polarization in the CMB is produced by GWs, which are perturbations in the spacetime metric that "stretch" and "contract" spacetime in the plane perpendicular to their direction of propagation. B-mode polarization is characterized by a divergence-free "swirling" pattern, much like magnetic field lines in electromagnetism. E-mode (or gradient mode) polarization in the CMB is produced by density perturbations, corresponding to regions of over- or under-densities in the primordial plasma, analogous to sound waves in air. Unlike B-mode polarization, E-mode polarization is characterized by a curl-free "radial" or "tangential" pattern, much like electric field lines in electrostatics.

Figure 10: Schematic representation of E-mode and B-mode polarization patterns. Figure taken from (Baumann et al., 2009).

Detecting the B-mode signal and attributing it to primordial GWs is challenging for several reasons: (i) The CMB polarization signal is relatively weak--at the micro-Kelvin level, which is an order of magnitude weaker than CMB temperature fluctuations. (ii) The B-mode signal is at least an order of magnitude weaker than the dominant E-mode signal. (iii) E-modes can be converted to B-modes due to gravitational lensing of CMB photons by matter between us (today) and the surface of last scattering (380,000 years after the Big Bang). (iv) Dust grains in our galaxy can convert E-modes to B-modes. It was this latter effect that led BICEP2 to change an initial claim of detection of inflationary GWs (Ade et al., 2014) to a measurement instead of contamination by interstellar dust (Ade et al., 2015).

Future prospects

Over the next few years, with additional pulsars and longer observation times, the "evidence for" low-frequency GWs announced by several pulsar timing array collaborations in summer 2023 should reach the 5-sigma level. This would then be the first direct detection of a stochastic GW background. During this same time-frame, the Advanced LIGO, Virgo, and KAGRA detectors may detect the GW background from stellar-mass binary black-hole mergers, which we know exists from the 100 or so individually resolvable detections that have already been made. Similarly, at much lower frequencies, CMB polarization experiments may see evidence of stochastic GWs from the very early Universe imprinted in the polarization pattern of the temperature fluctuations in the cosmic microwave background.

In roughly 10 years, during the mid to late 2030's, the space-based interferometer LISA will open up the \(10^{-4}~{\rm Hz}\) - \(10^{-1}~{\rm Hz}\) part of the GW spectrum, most-likely observing new sources of GWs. In addition, proposed 3rd-generation ground-based interferometers, Cosmic Explorer (CE) and Einstein Telescope (ET), will start coming on line. Having longer arms (10's of km) and using advanced technology (higher-power lasers, improved mirror coating, cryogenic cooling, etc.), these detectors will be able to resolve individual GW sources out to very large distances (redshifts). In fact, CE and ET should be able to resolve all the stellar-mass binary black hole mergers out to the edge of the visible Universe, effectively removing the unresolved (stochastic) component of this particular source. This may ultimately help us detect weaker stochastic GW signals from the very early Universe (e.g., from inflation, cosmic strings, and possible first-order phase transitions as mentioned above) since there will not be contamination from astrophysical foreground signals.

But simply detecting these signals is just the start of the story. Further improving the observations to better estimate the parameters of the observed signal is needed to both determine and characterize the source of signal. The detection and characterization of the CMB is a good example to keep in mind. As mentioned previously, the CMB was initially detected in 1965 by Penzias and Wilson as excess noise in their radio antenna, which they could not attribute to other known noise sources. The observed radiation followed a blackbody spectrum with a temperature of approximately 2.75 Kelvin. In the following years, the temperature was determined with better precision and the frequency range over which the measurement was made was increased. In the early 1990's, anisotropies in the temperature fluctuations of the CMB were first measured by NASA's COBE satellite (see COBE), resolving angular scales of order 10 degrees. With follow-up missions by WMAP and Planck, the angular scale of these measurements were reduced to sub-degree precision, leading to exquisite sky maps like that shown in Figure 5. This sky map was produced by ESA's Planck satellite in 2015, roughly 50 years after the initial detection of the CMB.

We imagine a similar timeline from detection to characterization for stochastic GW backgrounds. So there is much work still left to be done!

References

Suggested reading

  • Allen, B. (1997). The stochastic gravity-wave background: sources and detection, in "Relativistic gravitation and gravitational radiation", edited by Marck, J. A. and Lasota, J. P. (Proceedings, Les Houches School of Physics: Astrophysical Sources of Gravitational Radiation), Cambridge Contemporary Astrophysics, pp 373-417. https://doi.org/10.48550/arXiv.gr-qc/9604033
  • Allen, B. and Romano, J. D. (1999). Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities. Physical Review D 59, 102001. https://doi.org/10.1103/PhysRevD.59.102001
  • Christensen, N. (2018). Stochastic Gravitational Wave Backgrounds. Reports on Progress in Physics, 82(1). https://doi.org/10.1088/1361-6633/aae6b5
  • Romano, J. D. and Cornish, N. J. (2017). Detection methods for stochastic gravitational-wave backgrounds: a unified treatment. Living Reviews in Relativity, 20(1):2. https://doi.org/10.1007/s41114-017-0004-1
Sponsored by: Dr. Olivier Minazzoli, Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Artemis, Nice, France AND Bureau des Affaires Spatiales, Monaco
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