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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!).

Photons, neutrinos, and gravitational-wave astronomy

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 17^th and 19^th century astronomy
was unified with physics (universal law of gravitation and
interpretation of spectra). In the 20^th century, the wavelength
range accessible to telescopes greatly increased (from high-energy
g-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
give some elements necessary to understand the astrophysical problem
of the formation of (stellar-mass) GW sources for ground and
space-based detectors.

GR and GW basics

General relativity is a geometric theory of gravity that treats
space-time as dynamical entity described by a tensor, the metric \(g_
{\mu\nu}\). 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:

\begin{equation}\label{eq:EFE} G_{\mu\nu} + \Lambda g_{\mu\nu} = T_{\
mu\nu} \ \ , \end{equation}

where \(g_{\mu\nu}\) is the a priori unknown metric that acts as the
functional variable of this equation, \(T_{\mu\nu}\) is the
stress-energy tensor that describes the distribution of matter/energy
that shapes space-time (i.e., the source term which determines \(g_{\
mu\nu}\)), and \(G_{\mu\nu}=R_{\mu\nu} - 0.5Rg_{\mu\nu}\) is
Einstein's tensor (\(R=R^{\mu}_{\mu}\) is the trace of the Ricci
tensor \(R_{\mu\nu}\) which describes how different from flat is the
space-time described by \(g_{\mu\nu}\) and it is a function of \(g_{\
mu\nu}\) itself). The indexes \(\mu\) and \(\nu=0,1,2,3\) span the
four space-time dimensions (but gravity can "mingle" these). Eq. \ref
{eq:EFE} is the one that describes simultaneously how matter
(represented by the stress-energy tensor \(T_{\mu\nu}\)) "bends"
space-time (represented by the metric tensor \(g_{\mu\nu}\)) and
viceversa how the structure of space-time shapes the motion of matter
along the geodesics defined by the curvature of space-time (see any
textbook on general relativity for more information).

The fact that there is a finite speed \(c\) for the propagation of
interaction fields (including space-time itself) allows for the
existence of gravitational fields unbound from matter. An oscillating
unbound gravitational field is by definition a gravitational wave.

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 \(g_{\
mu\nu}\) describing space-time far away from any mass, and the
perturbation is often indicated with \(h_{\mu\nu}\). Say that we know
a solution \(g_{\mu\nu}\) (e.g., \(g_{\mu\nu} = \eta_{\mu\nu} = \
mathrm{diag}(1, -1, -1, -1)\) Minkowski's metric describing a flat
space-time), we can apply a small perturbation to it substituting \
(g_{\mu\nu}\rightarrow g_{\mu\nu} + h_{\mu\nu}\). Keeping only terms
linear in \(h_{\mu\nu}\) (which is our new functional variable) we
can rewrite Eq. \ref{eq:EFE} as:

\begin{equation}\label{eq:GW_wave} \left( \nabla^{2} - \frac{\partial
^{2}}{\partial t^{2}}\right) h_{\mu\nu} \propto T_{\mu\nu} \ \ . \end
{equation}

which is a wave equation in three dimensions for \(h_{\mu\nu}\) with
speed of propagation \(c\), source term proportional to \(T_{\mu\nu}
\). In vacuum (\(T_{\mu\nu}=0\)), and consequently \(h_{\mu\nu}\)
admits oscillating solutions \(h_{\mu\nu} = A_{\mu\nu}\exp(i k_{\
alpha}x^{\alpha})\)! These are the "ripple in space-time" (i.e., in
the metric tensor describing the properties of space-time in general
relativity) as GWs are often described.

Sources of GWs

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" \(M/R\) with \(M\) mass of a source and \(R\) its
linear dimension. Note that in natural units (\(G=c=1\) typically
used to simplify the formalism in GR), this is a dimensionless
number.

For low values of \(M/R\), General Relativity reduces to Newtonian
gravity - as expected - and in Newtonian gravity the gravitational
field is fixed and any change propagates instantly (at infinite
velocity): there are no gravitational waves.

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 be important, \(M/R\) needs to be
"large": either extremely large masses regardless of the scale (see
the very first ideas of what today we call a black hole from Michell
1784 and Laplace 1799), or for very dense matter limited to a very
small linear scale \(R\). In the "stellar regime", we expect the
densest stars, also known collectively as "compact objects" to be
involved, namely white dwarfs (WD), neutron stars (NS), and black
holes (BHs).

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 \(T_{\mu\nu}\)).

  * Q: what is the lowest order source term for electromagnetic
    radiation?

By analogy with electromagnetism (EM), let's consider the spatial
momenta of \(T_{\mu\nu}\) assuming the mass distribution of the
source to be finite in extent, that is multiply by (possibly more
than one factor) \(x^{\alpha}\) 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: the monopole
term of the matter/energy distribution does not generate 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 (for
intuition, think \(\sim m \times r\)) has for time-derivative the
total momentum of the source (\(\sim m \times v\)). That is also
conserved: there is also 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:

\begin{equation}\label{eq:quad} h_{\mu\nu}(r) = \frac{2}{c^{4}}\frac
{G}{r}\frac{d^{2} Q_{\mu\nu}}{d t^{2}} \ \ , \end{equation}

where \(r\) is the luminosity distance and

\begin{equation} \label{eq:quadrupole_def} Q_{\mu\nu} = \int d^{3} x
T_{00}x_{\mu}x_{\nu} \ \ , \end{equation}

is the quadrupole of the mass distribution, and \(T_{00}\equiv\rho\)
is the mass density with an appropriate choice of reference frame.

From Eq. \ref{eq:quad} we can see several important facts:

 1. the amplitude of GWs scales with \(1/r\), as opposed to \(1/r^{2}
    \) 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.
 2. 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).

Indirect detection of GWs

Pulsars are astrophysical radio sources repeating with very high
precision interpreted physically as a neutron star rotating fast
(down to milliseconds!).

Hulse & Taylor 1975 discovered the first pulsar 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 must be another
(unseen) compact object, of mass compatible with another neutron
star.

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!

GW-decay.jpg Figure 1: Dots are the measured cumulative time shift in
periastron passage w.r.t. a Keplerian orbit with constant period for
PSR B1913+16. The solid line is the prediction assuming the period is
changing due to GW emission as predicted by general relativity. Note
that this is not a fit! From Weisberg & Huang 2016

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).

N.B.: This is possible because the timing of arrival of a pulse is
extremely stable and thus predictable. A fast spinning neutron star
in a binary system is used as a clock to demonstrate that the binary
orbit is decaying in time because of the emission of GWs!

Minimum orbital separation for significant GW emission

Besides its historical importance, the "Hulse-Taylor pulsar" allows
for the introduction of an other important thing which requires GR to
demonstrate properly: what should be the orbital separation of a
binary for it to emit detectable GWs?

N.B.: The orbital separation \(a\) is a linear scale. From the fact
that GR matters for large \(M/R \sim M/a\) we can also expect that
the mass of a binary \(M\) will play a role in this answer (cf. also
Eq. \ref{eq:quad} and Eq. \ref{eq:quadrupole_def}).

For the Hulse-Taylor pulsar, we have:

  * \(M_{1} = 1.441M_{}\) for the mass of the detectable
    radio-pulsar
  * \(M_{2} = 1.387M_{}\) 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 \(a\simeq2.8R_{}\), which
corresponds to a periastron distance \(a(1-e)\simeq 1.09R_{}\) and
apastron distance \(a(1+e)\simeq4.5R_{}\).

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 \(\le\mathrm{few} \times 10R_{}\) to generate
significant amounts of GWs.

The amount of energy that goes in emitted GWs is a strong function of
the orbital separation \(a\), the orbital eccentricity \(e\), and the
masses of the systems in a binary: one can also ask what should the
separation be such that the timescale to shrink the orbital
separation to zero by GW emission (that is: how long it takes to
obtain a GW-driven inspiral and merger) is shorter than the age of
the Universe. Using the Peters 1964 formulae (which assume the
compact objects to be point-masses), one can again estimate that the
separation at the formation of the second compact object in a binary
needs to be below a few tens of \(R_{}\) to obtain GW-driven mergers
within the age of the Universe.

  * 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?

Direct detection

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 GW emission.

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 21^st century, the first direct
detection came from ground-based interferometric observations
performed by the Laser Interferometry gravitational observatory
(LIGO) laboratory - after \(\sim50\) years of continued effort.

On September 14^th 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 (first independently as a short g ray
burst GRB170817A 1.7s later than the GW signal, and even later as a
lower energy multi-wavelength transient AT 2017gfo). Unfortunately no
neutrinos have been detected from this event.

N.B.: Because EM interactions are stronger than gravitational ones,
light has to filter through the ejecta produced by the merger, which
delays it (speed \(c\rightarrow c/\tau\)), while GWs are unimpeded,
and thus arrive on Earth faster.

GW150914.png Figure 2: Detection of the first GW signal from the
inspiral and merger of two BHs from LVK collaboration 2016. Each
column corresponds to a separate and independent detector (one in
Washington and one in Louisiana): two are needed to make sure the
signal is not a fluke, but appears in both at the same time modulo
the light-travel time from one detector to the other. The top panels
show the strain \(h=\Delta L/L\), that is the relative change in size
of the detector caused by the passage of the GW. Note the scale! For
LIGO \(L\simeq4\) km, corresponding to \(\Delta L\sim 10^{-16}\) cm,
smaller than a nucleus! The second row show the prediction from
numerical relativity calculations (i.e., the solutions of Eq. \ref
{eq:EFE} obtained on a computer), the third row shows the residuals
between the observation and the models in the second row. The third
panel shows how the frequency of the signal changes in time, showing
the characteristic "chirping" behavior (as time passes, the signal
increases in frequency and becomes louder). Note also that we observe
the final half-second of the life ot the system (how long the signal
is within the detector band depends also on the masses involved, it
is up to ~10 seconds for a BNS)

The quest for a direct detection of GWs was such a long process
because it required pushing the limits of technological capabilities
(on multiple fronts). Without entering in the details of the
detection strategy, a successful detection requires to measure a
change of \(\Delta L\le10^{-16}\) cm in the travel path of laser
beams bouncing between mirrors \(L\sim4\) km apart. \(\Delta L\)
produced by the passage of the GW is is \(\le1/1000\) of the
characteristic size of a nucleus!

Today, while observations continue, we know of \(\sim100\) BBH
mergers, a couple of BNS merger, and we have a few BH-NS mergers (but
unfortunately all BNS except GW170817 and all BH-NS mergers have been
too far for the detection of EM counterparts): we already know more
stellar-mass BH from GWs than any other EM signature!

GWTC-3_stellar-graveyard.jpg Figure 3: "Stellar graveyard" as of the
publication of the third Gravitational wave catalog (GWTC3), see LVK
collaboration 2023 and LVK Collaboration 2023b. The spread in the
horizontal direction is just for clarity but contains no information,
while the vertical position indicates the mass in \(M_{}\) units.
Red points are known BHs in X-ray binaries, yellow points are (an
incomplete) census of NS known as pulsars or accretors in X-ray
binaries, orange points are NS detected in GW-driven inspiral and
mergers, and blue dots are BBH mergers (two dots for the pre-merger
BHs in the binary and one dot for the resulting BH).

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 \(\gg 10M_{}\) 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.

The problem: how do compact objects get so close to each other?

  * 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)

Isolated evolution channels

Tauris17.png Figure 4: Cartoon of the various steps in the evolution
of a massive binary system evolving to be a GW-driven BNS merger,
from Tauris et al. 2017. Many qualitative variations (e.g., NS - BH,
SN explosion - failed explosion, RLOF - Common envelope) have been
proposed to explain BNS and BBH with various properties, see also
review by Marchant & Bodensteiner 2024.

Dynamical channels

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 ($s [?] a^2
≫ R^2) 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.

Multiple dense stellar environment have been proposed:

  * stellar clusters: if sufficiently dense they can produce large
    rates of BBH (while they don't work well for systems involving NS
    that get a kick at birth and whose progenitors are less massive
    and live in the cluster outskirts preventing them from entering
    the most dense part of the cluster where most interactions occur)
  * nuclear star cluster: these are clusters around the supermassive
    BH at the center of a galaxy. This makes the escape velocity from
    these higher, and increases the chances of retaining the merger
    products and get multiple "generations" of mergers
  * Active Galactic Nucleus disk: in the accretion disk of the
    supermassive BH in the center of a galaxy there can be stars and
    compact objects. They can interact dynamically (possibly with the
    gas of the disk playing an important role), and this has been
    proposed as a possible site for GW mergers.
  * (Newtonian) Gravitational dynamics in triples and quadruple
    systems are in a sense a "transition" class between dynamical and
    isolated evolution channels.

N.B.: This brief overview is far from complete: this is a relatively
new and extremely active field, and by the time I finish writing any
comprehensive summary there would be much more to summarize already!

The future of GW astronomy

GWdetectors.png Figure 5: Sensitivity curves of present and planned
GW detectors: Ground based detectors are on the right (highest
frequencies), space-based in the middle, and galaxy-size
pulsar-timing based on the left. Resonances in the detectors can make
them "blind" to specific narrow frequencies: this can cause narrow
"spikes" in the black curves at specific frequencies, which are
removed here for clarity. From Moore et al. 2015

Ground-based detectors

GW_timeline.png Figure 6: LIGO, VIRGO, and KAGRA are three
ground-based GW observatories. This figure shows the predicted uptime
for each detector with the horizon out to which they are predicted to
be able to detect a BNS merger. From https://observing.docs.ligo.org/
plan/.

The current rate of GW detection within the frequency range of LIGO,
VIRGO, KAGRA is very high, and by the end-of-life of these
most-precise machines ever built, we expect to have a sizeable
population of GW-driven mergers of compact objects.

Future plans for so-called 3^rd generation GW detectors (the 2^nd
generation are the space-based detectors discussed below) are being
discussed. These will likely be interferometers with much longer arms
(\(L\simeq 4\) km - \(L\simeq 40\) km) and buried underground to
limit the high-frequency noise from micro-earthquakes. Multiple
competing plans exist at this point (e.g., Cosmic Explorer in the US
and Einstein Telescope in Europe), and are expected to be able to
detect BBH merger up to redshift ~20, much before the formation of
the first stars!

Space-based detectors

  * Mention DWD here

Galaxy-wide detectors: Pulsar Timing Arrays

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Last updated on 6 April 2025
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