LIGO

LIGO at the Hanford Site

About LIGO

• 2 sites: Livingston, LA; Hanford Site, WA
• Cost: about a billion USD (most expensive/ambitious project ever funded by NSF)
• Each site: Michelson interferometers with 4km arms, $1064 \, \mathrm{nm}$ Nd:YAG laser
• Each arm: Fabry-Pérot cavity (increases path length to equivalent of 280 trips)
• When a GW passes through: arms detune, photons emitted - signal
• Sister project in Europe: VIRGO

LIGO design

First direct detection: GW150914

LIGO noise sources

LISA (and LISA pathfinder)

To look at longer wavelengths, need to go into space!

LISA Pathfinder

Exceeded design expectations by factor of five!

LISA mission profile

• LISA: three arms (six laser links), 2.5 M km separation
• Launch as ESA’s third large-scale mission (L3) in (or before) 2034
• Proposal officially submitted earlier this year 1702.00786

Possible signals

Mission requirements

From the proposal:

Pulsar timing arrays

About pulsar timing arrays

• Millisecond pulsars (MSPs) are very accurate clocks
• PTA projects time tens of MSPs every week or so
• Irregularities in the period of one pulsar might be due to a change in the proper distance to that pulsar because of a passing GW (or a similar effect at Earth).
• Measure correlations between multiple pulsars
  • Disentangle GWs at the pulsar end and at the Earth end
  • Remove effects of 'physics' in the neutron stars
• Study stochastic background at very low frequencies

More about pulsar timing arrays

• They also set the some of strongest constraints on the equation of state of dense nuclear matter.
• Because MSPs are more accurate than atomic clocks, PTAs set the strongest constraints on solar system ephemerides
• Some of the biggest single users of radio telescope time

Electroweak phase transition

• This is the process by which the Higgs became massive
• In the minimal Standard Model it is gentle (crossover)
• It is possible (and theoretically attractive) in extensions that it would experience a first order phase transition

EWPT physics

• Bubbles produced at a thermal phase transition will be the central focus of these lectures
• This would be a significant source of gravitational waves

Where to see it

• The power spectrum is peaked at around the bubble radius, a fraction of the Hubble radius at the time of the transition.
• That corresponds to millihertz today, which means it is ideally placed for space-based detectors like LISA

The end of inflation

• As the inflaton reheated the universe, it created a lot of particles through violent processes.
• The inflation oscillating about the bottom of its potential would excite other particles to oscillate with characteristic frequencies given by their masses.
• These would be an efficient source of gravitational waves, but at high frequencies given by the mass of the field.

Where to see it?

• Some scenarios are potentially observable at earth-based detectors such as advanced LIGO
• Otherwise, must wait for the more ambitious (post-LISA) space-based detectors (DECIGO, BBO, ...)

Cosmic strings

• Extended, long lived, very tense pieces of 'string' formed during symmetry breaking in the early universe
• They produce a scaling network, and lose energy principally by gravitational radiation
• They can also radiate particles, but exactly how much is an open research question!
• Loops of string are long-lived, emitting GWs over many wavelengths as they get smaller with time.

Topological stability

In field theory, cosmic strings are topologically stable:

Can also be modelled by thin Nambu-Goto strings

Cosmic strings movie 1

Where to see them?

• Pulsar timing arrays already place strong constraints on the energy scale on the energy of the strings.
• Improved PTA measurements will constrain $G\mu$ further, meaning the energy scale is lower

Introduction

• The basic tools of general relativity are central ideas from differential geometry:
  • The metric, $g_{\mu\nu}$
  • The connections (also known as Christoffel symbols) $\Gamma^{\mu}_{\nu\rho}$
  • The Riemann tensor (or curvature tensor) $R_{\mu\nu\rho\sigma}$

The Einstein equations

• The Einstein equations are written $$ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 8 \pi G T_{\mu\nu} $$ where
  • The Ricci scalar $R = g^{\mu\nu} R_{\mu\nu}$
  • The Ricci tensor $R_{\mu\nu} = g^{\rho\sigma} R_{\rho\mu\sigma\nu}$
  • The stress-energy tensor is $T_{\mu\nu}$
• 16 equations as written, 6 independent equations
  • symmetry: 6 constraints
  • Bianchi identity: 4 constraints
• A gauge theory where the symmetries are diffeomorphisms (carefully constructed coordinate transformations)

Stress-energy tensor

Note that $T_{\mu\nu}$ looks schematically like $$ T_{\mu\nu} = \left( \begin{array}{c|c} \text{[energy density]} & \text{[energy flux]} \\ \hline & \\ \text{[momentum density]} & \text{[pressure and stress]} \\ & \\ \end{array} \right) $$

Linearised gravity

• As we said, gravity is a gauge theory where the symmetry is diffeomorphism invariance.
• This means it is invariant under coordinate transformations $$ x^{\mu} \to x'^{\mu} (x) $$ which are invertible, and differentiable.
• Under a coordinate transformation the metric transforms as $$ g_{\mu\nu}(x) = g'_{\mu\nu}(x') = \frac{\partial x^\rho}{\partial x'^\mu} \frac{\partial x^\sigma}{\partial x'^\nu} g_{\rho\sigma}.$$

Specialising to linear perturbations

• We now assume that we can write the metric as $$ g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} \qquad \left| h_{\mu\nu} \right| \ll 1 $$ where $\eta_{\mu\nu}$ is the Minkowski metric $$\mathrm{diag}(-1,+1,+1,+1)$$
• Even with this assumption, there is an invariance under transformations of the form $$ x^\mu \to x'^\mu = x^\mu + \xi^\mu(x) $$ provided that $|\partial_\mu \xi_\nu | \leq |h_{\mu\nu}|$

Wave equation

• By substituting $$ g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$$ in the Einstein equations $$ R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = 8\pi G T_{\mu\nu} $$ and retaining only terms at linear order we will obtain a wave-like equation for $h_{\mu\nu}$.

The wave equation, simplified

• We can write it in a fairly clear manner by adopting the trace-reversed metric perturbation $$ \bar{h}^{\mu\nu} = h^{\mu\nu} - \frac{1}{2}\eta^{\mu\nu} h $$
• It then takes the form $$ \begin{multline} \square \bar{h}_{\nu\sigma} + \eta_{\nu\sigma} \partial^\rho \partial^\lambda \bar{h}_{\rho\lambda} - \partial^\rho \partial_\nu \bar{h}_{\rho \sigma} - \partial^\rho \partial_\sigma \bar{h}_{\rho\nu} + \mathcal{O}(h^2) \\ = - \frac{16 \pi G}{c^4} T_{\nu\sigma} \end{multline} $$

Lorenz gauge

• Now we impose Lorenz gauge $$ \partial_\nu \bar{h}^{\mu\nu} = 0 $$
• This is equivalent to Lorenz gauge in electromagnetism: $$ \partial_\mu A^\mu = 0 $$
• The effect is to cancel every term except $$ \square \bar{h}_{\nu\sigma} = - 16 \pi G T_{\nu\sigma} $$ which indeed looks like a wave equation!

Summary (comparison to electromagnetism)

Source: le Tiec and Novak

Transverse traceless gauge

• There are still 6 components in $\bar{h}_{\mu\nu}$, only 2 of which are physical - the two polarisations
• To find these, we consider coordinate transformations that also satisfy Lorenz gauge, meaning $$ x^\mu \to x'^\mu = x^\mu + \xi^\mu(x); \quad |\partial_\mu \xi_\nu | \leq |h_{\mu\nu}| $$ and $\square \xi^\mu = 0 $.
• $\square \xi^\mu = 0$ gives 4 new constraints. First we choose: $$\bar{h} = 0$$ i.e. make $\bar{h}^{\mu\nu}$ traceless [one constraint]
• (Note that with this constraint, the trace of $h^{\mu\nu}$ also disappears, so we no longer need the 'trace reversed' tensor!)

More constraints

• We choose our other 3 constraints to be: $$ h^{i0} = 0 $$ meaning $ \partial_0 h^{00} = 0 $ too, and we can choose $h^{00} = 0$.
• Put another way, our 4 constraints are: $$h^{00} = h^{0i} = 0$$
• The remaining spatial entries of $h_{ij}$ must be transverse $$ \partial_{i} h^{ij} = 0 $$ and traceless $$ h^{ii} = 0. $$

Travelling waves

• A plane wave in the $z$-direction looks like $$ h_{ij}^\text{TT} (t, z) = \left( \begin{array}{ccc} h_+ & h_\times & 0 \\ h_\times & h_+ & 0 \\ 0 & 0 & 0 \\ \end{array} \right)\cos\, \omega \left(t - \frac{z}{c} \right) $$ Note that the polarisation components are perpendicular to the direction of travel.
• For a more general travelling wave with wave vector $\mathbf{k}$, transverse traceless simplifies to $$ \hat{\mathbf{k}}^i h_{ij}^\text{TT} = 0 .$$

Projection

• If we have a general metric perturbation, we can use a tensor version of the usual 'transverse' projector: $$ \Lambda_{ij,lm} \equiv P_{il} P_{jm} - \frac{1}{2} P_{ij} P_{lm} $$
• Here we have $$ P_{ij} = \delta_{ij} - \hat{\mathbf{k}}_i \hat{\mathbf{k}}_j $$
• And this satisfies $$ P_{ij} = P_{ji}, \quad \hat{\mathbf{k}}^i P_{ij} = 0, \quad P_{ij} P^{jk} = P_i^k, \quad P_{ii} = 2 $$
• In other words, it projects out the rotational part of a vector field.

Projection of a general $h_{ij}$

• If we have a general $h_{ij}$ which is in Lorenz gauge but does not otherwise satisfy the transverse traceless (TT) requirements, we can project out the TT parts: $$h_{ij}^\text{TT} = \Lambda_{ij,lm} h^{lm} $$
• Why do we want to do this?
  • We may want to study the polarisation of the gravitational waves (but for stochastic, cosmological sources, these rarely matter)
  • More importantly, as $h_{ij}^\text{TT}$ contains only the propagating degrees of freedom, measures of energy and power must be done with it.

Effective stress-energy tensor

• Step back and consider a more general background: $$ g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}, \quad |h_{\mu\nu}| \ll 1 $$
• What is the background and what is the perturbation?
  • $\bar{g}_{\mu\nu}$ has a scale $L_\mathrm{B}$
  • $h_{\mu\nu}$ have a wavelength $\lambda \ll L_\mathrm{B}$.
• Or, in frequency:
  • $\bar{g}_{\mu\nu}$ only has frequencies up to $f_\mathrm{B}$
  • $h_{\mu\nu}$ has frequencies $f \gg f_\mathrm{B}$.
• The overall effect is that, even if $\bar{g}_{\mu\nu}$ has come curvature, it looks slowly varying to the gravitational waves.

The Isaacson argument

• We can now split the Ricci tensor up as follows: $$ R_{\mu\nu} = \bar{R}_{\mu\nu} + R^{(1)}_{\mu\nu} + R^{(2)}_{\mu\nu} + \ldots $$
  • $\bar{R}_{\mu\nu}$ is the background Ricci tensor
  • $ R^{(1)}_{\mu\nu}$ are high frequency modes at linear order in $h$
  • $ R^{(2)}_{\mu\nu}$ is everything at quadratic order in $h$
• If we average over a volume bigger than $\lambda$ but smaller than $L_\mathrm{B}$ then we can use the Einstein equations to introduce $ t_{\mu\nu} = -\frac{1}{8\pi G} \left< R_{\mu\nu}^{(2)} - \frac{1}{2} \bar{g}_{\mu\nu} R^{(2)} \right> $
• And $ \bar{R}_{\mu\nu} - \frac{1}{2}\bar{g}_{\mu\nu} \bar{R} = 8\pi G \left( \bar{T}_{\mu\nu} + t_{\mu\nu} \right).$

The Isaacson expression

• We end up with (in arbitrary gauge) $$ \begin{multline}t_{\alpha\beta} = \frac{1}{32\pi G} \left< \partial_\alpha \bar{h}_{\mu\nu} \partial_\beta\bar{h}^{\mu\nu} - \frac{1}{2} \partial_\alpha \bar{h} \partial_\beta \bar{h} \right. \\ \left. - \partial_\nu \bar{h}^{\mu\nu} \partial_\beta \bar{h}_{\mu\alpha} - \partial_\nu\bar{h}^{\mu\nu} \partial_\alpha \bar{h}_{\mu\beta} \right> \end{multline}$$ which, for TT, reduces to $$ t_{\alpha\beta} = \frac{1}{32 \pi G} \left< \partial_\alpha h^{\text{TT}}_{\mu\nu} \partial_\beta h_{\text{TT}}^{\mu\nu} \right> $$ which is known as the Isaacson tensor.

Energy density in gravitational waves

• Now that we have the effective stress-energy tensor, the energy density in gravitational waves is $$ \rho_\text{GW} \equiv t_{00} = \frac{1}{32 \pi G} \langle \dot{h}_{ij}^\text{TT} \dot{h}_{ij}^\text{TT} \rangle $$
• By Fourier transforming this expression, define the GW power per logarithmic frequency interval $$ \frac{\mathrm{d} \, \rho_\text{GW}(\mathbf{k})}{\mathrm{d} \, \log k } = \frac{1}{32\pi G} \frac{k^3}{2\pi^2} \left< \dot{h}_{ij}^\text{TT}(\mathbf{k}) \dot{h}_{ij}^\text{TT} (-\mathbf{k}) \right> $$
• These two quantities are what cosmologists most widely quote in papers about gravitational waves, particularly the results of numerical simulations.

Typical assumptions in these lectures

• Minkowski or FRW spacetime: no, or isotropic expansion
  • Physics on timescales much shorter than expansion
  • All gravitational waves sourced by sub-Horizon physics
• Homogeneous, stochastic, isotropic source
  • True for most cosmological sources
  • May not be true for, e.g. cosmic string cusps

Results in context

How does this work in a cosmological simulation?

1. Evolve Lorenz-gauge wave equation in position space $$ \nabla^2 h_{ij} (\mathbf{x},t) - \frac{\partial}{\partial t^2} h_{ij}(\mathbf{x},t) = 8 \pi G T_{ij}^\text{source}(\mathbf{x},t)$$ during simulation, using relevant $T^\text{source}_{ij}$ of 'source system'.
2. Projection to TT-gauge requires expensive Fourier transform, so only project when measurement desired: $$ h^{\text{TT}}_{ij}(\mathbf{k},t_\text{meas}) = \Lambda_{ij,lm}(\hat{\mathbf{k}}) h^{lm}(\mathbf{k},t) $$
3. Measure energy density (or power) in gravitational waves $$ \rho_\text{GW}(t_\text{meas}) = \frac{1}{32 \pi G} \left< \dot{h}_{ij}^\text{TT} \dot{h}_{ij}^\text{TT} \right> $$
4. Redshift to present day.

Other techniques

• Two other approaches are sometimes seen in numerical cosmology:
  • Quadrupole approximation - as we will see, this is a poor approximation for bubble collisions we will be studying, but it still provides insight
  • "Weinberg formula" - this gives a clean time-domain formula where the stress-energy tensor takes simple forms
• We will look at these, and the properties of general sources, next.

Compact, distant sources

• This is also relevant, e.g. for colliding pairs of bubbles.
• Start from the Lorentz-gauge wave equation $$ \square \bar{h}_{\mu\nu} = - 16 \pi G T_{\mu\nu}. $$
• Solve in position space with retarded Green's functions $$ \bar{h}_{\mu\nu}(x) = - 16 \pi G \int \mathrm{d}^4 x' \, G(x-x') T_{\mu\nu}(x'). $$
• If we now specialise to TT gauge and write in terms of the retarded time $t - |\mathbf{x} - \mathbf{x}'|$, $$ \begin{multline} h_{ij}^\text{TT} (t, \mathbf{x}) = \Lambda_{ij,lm} (\hat{\mathbf{n}})\, 4 G \int d^3 x' \frac{1}{|\mathbf{x} - \mathbf{x}'|} \\ \times T_{lm} \left( t- |\mathbf{x} - \mathbf{x}|; \mathbf{x}'\right) \end{multline} $$

Far field approximation

• And if we are also far from the source (where $T_{lm}(t, \mathbf{x}) \neq 0$), $$ |\mathbf{x} - \mathbf{x}'| \approx r - \mathbf{x}' \cdot \hat{\mathbf{n}} $$ and $$ \begin{multline} h_{ij}^\text{TT} (t, \mathbf{x}) \approx \frac{1}{r} \, 4G \, \Lambda_{ij,lm} (\hat{\mathbf{n}}) \int_{\text{source}} d^3 x' \frac{1}{|\mathbf{x} - \mathbf{x}'|} \\ \times T_{lm} \left( t- r + \mathbf{x}'\cdot \hat{\mathbf{n}}; \mathbf{x}'\right) \end{multline}$$
• We will assume(!) that velocities inside the source are non-relativistic
• In other words gravitational waves have lower frequencies $\omega$ than the source diameter: $$ \omega \mathbf{x}' \cdot \hat{\mathbf{n}} \ll 1 $$

Multipole expansion

• We can write the source using a Fourier transform $$ \begin{multline} T_{lm} \left(t - r + \mathbf{x}'\cdot \hat{\mathbf{n}}; \mathbf{x}'\right) \\ = \int \frac{\mathrm{d}^4 k}{(2\pi)^4} T_{lm}(\omega, \mathbf{k}) e^{-i \omega \left( t- r + \mathbf{x}' \cdot \hat{\mathbf{n}} \right) + i \mathbf{k} \cdot \mathbf{x}' } \end{multline} $$ and the leading order term expanding in $\omega \mathbf{x}' \cdot \hat{\mathbf{n}}$ is $$ T_{lm} \left(t - r + \mathbf{x}'\cdot \hat{\mathbf{n}}; \mathbf{x}'\right) \approx \int \frac{\mathrm{d}^4 k}{(2\pi)^4} T_{lm}(\omega,\mathbf{k}) $$ This is the quadrupole term.

Quadrupole source

• To leading order, then, the metric perturbation is $$ h_{ij}^\text{TT}(t,\mathbf{x}) \approx \frac{1}{r} \, 4G \, \Lambda_{ij,lm}(\hat{\mathbf{n}}) \int d^3 x \, T^{lm} \left(t - r, \mathbf{x} \right) + \ldots $$
• This is zero for a spherically symmetric source (or linear superposition of spherically symmetric sources)
  • Vacuum fluctuations at the end of inflation
  • Freshly nucleated bubbles in the early universe
  • Isolated massive objects
• Need some non-spherical dynamics:
  • Particle resonances
  • Bubbles colliding
  • Binary compact massive objects

Why we must go beyond the quadrupole approximation

• In the early universe, velocities within the source are not small, and the gravitational waves are typically the same scale as the bubbles.

Weinberg formula

• So far, we have seen two methods of computing $h_{ij}$
  • Numerically solving the equation of motion $$ \nabla^2 h_{ij} - \frac{\partial}{\partial t^2} h_{ij}= 8\pi G T_{ij} $$ e.g. during a simulation
  • Using the quadrupole approximation if the wavelength of the gravitational waves is long compared to the size of the source(s)
• However, sometimes can simplify the source so that it is simple in Fourier space, and we do not need to do the quadrupole approximation
• Then we can use the Weinberg formula

Using the Weinberg formula

• For radiation in a direction $\hat{\mathbf{k}}$ and frequency $\omega$, the power spectrum per logarithmic frequency interval, per unit solid angle, $$ \frac{\mathrm{d}\rho_\text{GW}}{\mathrm{d} \log \omega \, \mathrm{d} \Omega} = 2 G \omega^3 \Lambda_{ij,lm}(\hat{\mathbf{k}}) T_{ij}^* (\hat{\mathbf{k}},\omega) T_{lm} (\hat{\mathbf{k}},\omega) $$
• One application of this is the 'envelope approximation', which we shall revisit later.

Useful insight 1

Source: Dufaux, Felder, Kofman, Navros

• Begin with the momentum-space Green's function expression (assume source off at $t' < 0$) $$ h_{ij}^\text{TT}(\mathbf{k},t) = 16\pi G \, \Lambda_{ij,lm} \int_0^t \, \mathrm{d} t' \frac{\sin[k(t-t')]}{k} T_{lm}(\mathbf{k},t') $$
• If the source is slowly varying in space at low $\mathbf{k}$: $$T_{lm}(\mathbf{k}) \to \text{const.}; \qquad k \ll k_\text{max} $$ (equivalent to the quadrupole approximation) we get $$ h_{ij}^\text{TT}(\mathbf{k},t) \approx 16\pi G \, \Lambda_{ij,lm} \int_0^t \, \mathrm{d} t' \frac{\sin[k(t-t')]}{k} T_{lm}(0,t') $$

Useful insight 1

• If the source $T_{lm}(0,t)$ varies faster than the $\sin[k(t-t')]$, the equation reduces to $$ h_{ij}^\text{TT}(\mathbf{k},t) \approx 16\pi G \, \Lambda_{ij,lm} \int_0^t \, \mathrm{d} t' T_{lm}(0,t') $$ This gives $$ \frac{\mathrm{d} \rho_\text{GW}(k)}{\mathrm{d} \, \log \, k} \propto k^3 $$
• In other words, at sufficiently long sub-horizon scales, the quadrupole approximation always works and all the matters is how long the source is on for.
• The power law is $k^3$.
• This holds for e.g. first order phase transitions and the end of inflation.

Useful insight 2

• If the source has an intermediate regime where $T_{lm}(0,t)$ varies slower than $\sin[k(t-t')]$, then the source stays in the integral, and there is an additional $1/k$ factor
• Therefore, in some cases we can expect a $k^1$ power law at higher wavenumbers than the $k^3$ is valid
• This is less generally true than the previous $k^3$ regime, so it might not be observed at all.
• In general, though, where a power law is seen in simulation results, it is worth seeing if the underlying physics is amenable to simpliciation!
• We will encounter more power laws later in these lectures...

Useful insight: graphical summary