Materials: Chapter 15 in Tauris & Van den Heuvel 2023 book, Schutz 1984, Landau & Lifschitz vol. 2, LVK Collaboration 2017, Chen et al. 2017, Astrobites post on GW (and references therein!).
Astronomy since its inception lost in history is based on observations of (visible) light from sources in the Sky. As mentioned in at the beginning, between the 17th and 19th century astronomy was unified with physics (universal law of gravitation and interpretation of spectra). In the 20th century, the wavelength range accessible to telescopes greatly increased (from high-energy γ-rays to long radio wavelengths), making astronomy a multi-wavelength science.
With the detection of neutrinos (first solar, then from SN1987A, and most recently very high-energy neutrinos), astronomy became “multi-messenger” (photons+neutrinos), a buzz word that is highly used presently.
The addition of gravitational waves (GW) adds a completely new way to study astrophysical sources, probing optically thick and thus inaccessible regions, and regimes where gravity is strong (i.e., a full general relativistic treatment of the interaction of matter with space-time becomes necessary).
Because the wavelength range for GWs (which are not EM waves nor sound waves!) accessible to presently available ground-base detectors such as LIGO/Virgo/Kagra is in the 10-1000Hz range – corresponding roughly to the auditory range for typical human hears, it is often metaphorically said that GWs allow us to “hear” the Universe. However, this metafore is bound to become obsolete as multi-wavelength GW observations become available (e.g., from “pulsar timing arrays” and space-based GW observatories).
Disclaimer: to give a complete overview of GW physics we would need to have a full course of general relativity (GR) first just as a start. I will not attempt to be complete or exhaustive here, but just to give some elements necessary to understand the astrophysical problem of the formation of GW sources for ground and space-based detectors.
General relativity is a geometric theory of gravity that treats space-time as dynamical entity described by a tensor, the metric $gμ\nu$. The dynamics of this entity is dictated by the distribution of mass (or better energy density) within it, which determines the curvature of space time. In turn, that curvature determines the geodesics that the energy density will follow in absence of other (non-gravitational) forces.
This is formally described by Einstein’s field equation:
where $gμ\nu$ is the a priori unknown metric that acts as the
functional variable of this equation, $Tμ\nu$ is the stress-energy
tensor that describes the distribution of matter/energy that shapes
space-time (i.e., the source term which determines $gμ\nu$), and
$Gμ\nu=Rμ\nu - 0.5Rgμ\nu$ is Einstein’s tensor ($R=Rμμ$ is
the trace of the Ricci tensor $Rμ\nu$ which describes how different
from flat is the space-time described by $gμ\nu$ and it is a
function of $gμ\nu$ itself). The indexes
The fact that there is a finite speed
GWs, as any wave, are the solution of a linear perturbation on a state. In the GW case, the state is the flat space-time metric $gμ\nu$ describing space-time far away from any mass, and the perturbation is often indicated with $hμ\nu$. Say that we know a solution $gμ\nu$ (e.g., $gμ\nu = ημ\nu = \mathrm{diag}(1, -1, -1, -1)$ Minkowski’s metric describing a flat space-time), we can apply a small perturbation to it substituting $gμ\nu→ gμ\nu + hμ\nu$. Keeping only terms linear in $hμ\nu$ (which is our new functional variable) we can rewrite Eq. \ref{eq:EFE} as:
which is a wave equation in three dimensions for $hμ\nu$ with speed
of propagation
Before narrowing down what could be (mathematically and
astrophysically) the sources of GWs, let’s consider when are GR
effects most relevant? A typical quantity to look at the so-called
“compactness”
For low values of
N.B.: Just introducing the postulate that “gravity” has a finite speed of propagation in Newtonian physics, one can build a lot of intuition and quantitative results correct to order of magnitude for GW physics, see Schutz 1984 and LVK Collaboration 2017.
For general relativity effects to matter,
In general, the source term of GWs is going to be related to the term describing the distribution in space-time of matter (the stress energy tensor $Tμ\nu$).
- Q: what is the lowest order source term for electromagnetic radiation?
By analogy with electromagnetism (EM), let’s consider the spatial momenta of $Tμ\nu$ assuming the mass distribution of the source to be finite in extent, that is multiply by (possibly more than one factor) $xα$ and integrate over the spatial volume. Like in EM, the zeroth order momentum of a charge distribution is just the total charge that is conserved, and that does not lead to EM radiation, the same goes for GWs. In EM, the next order give the charge dipole, which if it has a time-dependence creates EM radiation (e.g., Thompson scattering). For gravity, the first order momentum of a mass distribution, assuming the mass to be constant, has for time-derivative the total momentum of the source. That is also conserved: there is no dipole radiation of GWs. The next order is then the quadrupole of the mass distribution: gravitational waves are generated by the time-dependence of the quadrupole distribution of mass at leading order.
One can obtain, at leading order, the so called quadrupole formula:
where
is the quadrupole of the mass distribution, with $T00≡\rho$ the mass density.
From Eq. \ref{eq:quad} we can see several important facts:
- the amplitude of GWs scales with
$1/r$ , as opposed to $1/r2$ for EM waves outside the near-field zone. This means that we can have detectable GWs from regions of the Universe that are too dim and far for EM observations. - the source need to have a non-zero second-time derivative of the quadrupole term of the mass distribution (at least): spherical objects, or objects moving in a straight line don’t produce GWs.
In astrophysical context, what could be the sources? The most common ones considered and searched for are non-spherical rotating compact objects (for example a spinning neutron star with a mountain not aligned to the rotation axis would produce a GW with constant frequency equal to the rotation frequency of the source), binary systems made of compact objects (which would lose energy to GWs and progressively shrink the orbit until a final merger of the two compact objects) and echoes of the Big Bang in GWs (this is a target for pulsar timing arrays and beyond the scope of this course).
Hulse & Taylor 1975 discovered the first pulsar (radio source repeating with very high precision interpreted physically as a neutron star rotating fast) in a binary system, PSR B1913+16 (a.k.a. “Hulse-Taylor pulsar”). They showed a radial velocity curve (recall the lecture on binary orbital motion) which demonstrated the orbit is eccentric and the companion is another compact object.
Monitoring this system, and measuring the delay between periastron passage observed and the periastron passage predicted with a Keplerian orbit, one can see that the period is progressively speeding up, or, in other words, the orbit is shrinking in time: the next periastron passage is earlier than predicted by a Keplerian orbit!
The measured agreement between the period decay of the Hulse-Taylor pulsar and general relativity prediction of the energy loss due to GW emission is considered the first indirect evidence for GW (and was awarded the Nobel prize in physics in 1993).
Besides its historical importance, the “Hulse-Taylor pulsar” allows to introduce an other important thing which requires GR to demonstrate properly: what should the orbital separation of a binary be for it to emit detectable GWs?
From Eq. \ref{eq:quad} and Eq. \ref{eq:quadrupole_def} one can see that the answer should be also dependent on the masses of the binary components.
For the Hulse-Taylor pulsar, we have:
- $M1 = 1.441Mo$ for the mass of the detectable radio-pulsar
- $M2 = 1.387Mo$ for the mass of the unseen object
-
$P=0.323$ days for the orbit -
$e = 0.61$ for the orbit (likely a product of the natal kick of the second-born NS)
Approximating the orbit as Keplerian (which we know is a mistake, but the energy lost to GW in one orbit is fairly small and we are only after one order of magnitude), we obtain $a≅2.8Ro$, which corresponds to a periastron distance $a(1-e)≅ 1.09Ro$ and apastron distance $a(1+e)≅4.5Ro$.
For BHs, which are more massive than NS, we can afford larger orbital separations, while less massive WDs require shorter separations/faster orbital periods.
The take home point is that the compact objects (WD, NS or BH) have to have separations $≤\mathrm{few} × 10Ro$ to generate significant amounts of GWs.
The amount of energy that goes in emitted GWs is a strong function of
the orbital separation
- Q: How big do the stellar progenitors of these compact objects get during their evolution? How does that compare to the loose requirement we have derived above?
Although impressive, the observations of the Hulse-Taylor pulsar (and other systems since then, see for example Table 3 in Weisberg & Huang 2016) only prove that the orbit of this binary NS loses energy at a rate that matches impressively well predictions based on assuming that the energy is lost to the emission of GWs.
From before the discovery of this system and for decades after, the
quest for a direct detection continued – with controversial claims
and rebuttals (see for example Chen et al. 2017 for an historical
overview). Skipping ahead to the 21st century, the first direct
detection came from ground-based interferometric observations
performed by the Laser Interferometry gravitational observatory (LIGO)
laboratory – after
On September 14th 2015, the first direct detection of a binary BH (BBH) merger, GW150914 occurred. And just two years later the first binary NS (BNS) merger was detected in GW first (GW170817), and through followup observations informed on the sky location by the GWs, also in EM observations.
This was such a long process because this detection really pushed the
limits of technological capabilities. Without entering in the details
of the detection strategy, a successful detection requires to measure
a change in position of mirrors bouncing laser light of $≤10-16$ cm
(over
Today, while observations continue, we know of
Thanks to the direct detections of GWs we now know several astrophysical facts that had been hypothesized before, but were lacking empirical confirmation:
- BBH exist!
- stellar mass BHs with masses $\gg 10Mo$ exist!
- BH-NS binary exist!
- we have some constrain on the rate at which these form with a “final” (from the stellar evolution point of view)/”initial” (from the GW-driven inspiral point of view) separation sufficient to merge within the age of the Universe
N.B.: GWs also offer cosmological facts, e.g., the non-detection (as of yet) of a stochastic background, unique constraints on GR in strong gravity (e.g., from the “ring-down” phase just after the merger, when the new formed BH “shakes away its hairs”), and nuclear physics (GW170817 confirmed that BNS mergers are one site for r-process nucleosynthesis and formation of element heavier than Iron). Ultimately, GW astronomy is a completely new way to explore the Universe. The discussion here is far from complete and focused on the aspects related to stellar physics only.
- two classes: isolated systems (binary/triples/quadruples) and dynamical systems (globular clusters, nuclear star clusters) and exotic channels (AGN gas-assisted inspirals, multiple compact objects from a single star)
Alternatively, another way to solve the issue of two compact objects needing to be closer than their parent stars are large to get GW-driven mergers within the age of the Universe is to leverage dynamical N-body interactions.
The core of the idea is that stars could evolve in isolation (or in binaries that might interact, but not necessarily in the way leading to a GW progenitor), and be put together by their (Newtonian) gravitational interaction in a dense stellar system.
N.B.: Binaries are still important! Since the cross section for N body interactions scales with some power of the stellar radius for single stars, and with the orbital separation for a binary ($σ \propto a2 \gg R2) it is much more likely to have a significant gravitational interaction between a binary and a star (or between two binaries) than between two single stars.
The video (from Prof. Carl Rodriguez) below shows a “zoom in” on one of the many N-body interactions that can happen. There is one incoming binary (in orange) that interacts (chaotically) through purely Newtonian gravity with a single star (in cyan). At the end of the interaction one of the initial binary members (statistically the least massive) finds itself alone and shot out at a high velocity, and the new binary has a shorter separation (the kinetic energy of the ejected star comes from the orbital energy of the original binary). This example thus shows that the simplest 3 body system results in the ejection of a “runaway star” and a tighter binary. Iterating this multiple times in a dense stellar system can lead to stellar or compact object binaries tight enough to emit significant amount of GWs and merge within the age of the Universe.