[YETI] Gravitational waves

The following are the (incomplete, possibly erroneous) notes I took during Daniel Figueroa’s talk at the YETI 2017 conference in Durham.

Introduction to Gravitational Waves

First approach to GWs

We can write a metric tensor for our universe in terms of the Minkowski metric $\eta_{\mu\nu}$ and a small perturbation $h_{\mu\nu}$ as :

We have now fixed a frame and thus lost diffeomorphism invariance, one of the tenets of general relativity. However, we retain a residual symmetry :

A quantity of interest is the trace-reverse of $h$ : ; in terms of which we can rewrite the Einstein equation :

Fixing the Lorentz gauge $\partial^\nu\bar{h}_{\mu\nu}=0$, it can be reduced to :

Furthermore, if $T_{\mu\nu}=0$ (outside source) then , which tells us that gravitational waves propagate at the speed of light (for an observer situated far away from the source). Considering the residual symmetry described above and fixing , we get , and ; and we define the transverse-traceless gauge : . We are therefore left with a wave equation with only two degrees of freedom.

Can we ‘gauge away’ GWs?

Having moved from ten degrees of freedom to only two, the question at this point is, ‘can we further gauge away these remaining DOFs?’. Are gravitational waves just a mathematical trick, without physical significance? The answer is obviously ‘no’, since GWs have been detected in 2015-6. Those two DOFs are in fact two polarisations, and we can write $h$ as a superposition of plane waves :

where, in the transverse-traceless gauge

GW modes

It’s important to note that an observer in the transverse-traceless frame wouldn’t see anything! A change of coordinates is needed to compute the physical effects (change in $\mathrm{ds}^2$) of a passing GW in the lab / on Earth.

Second approach to GWs

This time, instead of expanding around the Minkowski metric, we expand over a dynamical background $\bar{g}$ (one might object that the separation is not well defined) :

To solve this separation of metrics issue, we assume that has a characteristic length scale (low frequency), and a lower amplitude / higher frequency one. We write :

Here $\bar{R}$ is the background term (low frequency, large scale), whereas $R^{(1)}\sim\mathcal{O}(h)$ and $R^{(2)}\sim\mathcal{O}(h^2)$ are higher frequency / shorter scale perturbations. Consider now

We take the spacetime average of these quantities, so as to integrate out small scale signatures (and thus make the separation of terms discussed above) :

where the first term is simply the background contribution, and we recognise the second term as an energy-momentum tensor for GWs. Writing

one might think it looks like $t_{\mu\nu}$ has six DOFs, but it can be shown (by a lengthy calculation) that in fact, only the two transverse-traceless DOFs contribute. The GW energy-momentum tensor and power radiated by area are then :

At this point in our discussion, we have shown the effect of a GW on the background; we now turn our attention to the properties of the GW itself :

The first term is effectively negligible, and we can rewrite the LHS as :

Fixing the Lorentz gauge $D_\mu\bar{h}_{\mu\nu}=0$, we get the wave equations for (respectively) the propagation and creation of GWs in curved spacetime :

GWs from inflation

GWs from the early universe decouple from the moment of their production, retaining their spectral form, which makes them particularly advantageous to study when looking for specific high energy processes; on the other hand, detection is difficult because of the weakness of gravity. We recall that inflation corresponds to a period of accelerated expansion of the universe, with $a\sim e^{H_\star t}\gtrsim e^{60}$ ($1$ cm $\to$ $10^{28}$ cm).

Expand the Einstein-Hilbert action to second order and consider only the GW terms :

We analyse these tensor perturbations roughly in the same way as scalar perturbations : quantization, Bunch-Davies equation, power spectrum.

Quantum fluctuations grow large with inflation, become red-tilted and approach a semi-classical behaviour :

Temperature anisotropies, due to $\rho_\gamma+\delta_\gamma$, go as $T_\gamma(\hat{n})=\bar{T}+\delta T(\hat{n})$. Meanwhile, the perturbations $\delta\rho_\gamma$ and $\delta\rho_e$ lead to Thompson scattering and linear polarisation, modeled by the parameters $C_l^E$ and $C_l^B$ in the angular polarisation spectrum. Back in 2014, BICEP2 claimed to have observed such B and E modes. The B modes are particularly interesting, as they are only made of tensor perturbations and so carry the possible imprint of GWs.

More recent results place a bound $r<0.07\, (2\sigma)$ on inflationary GWs. In the future, probing around $r\sim 10^{-2}-10^{-3}$ could probe fundamental physics, as it corresponds to an energy of $\sim 10^{15}$ GeV.

Particle production during inflation

There are however more ways to engance GW production, and we now explore one of them. Consider an axion-inflation model with terms $V(\phi)+\frac{\phi}{\Lambda}F_{\mu\nu}\tilde{F}^{\mu\nu}$ in the action. The rolling inflaton excites the gauge fields and we have :

Thus, $A_+$ is amplified by inflation and gauge field excitations create chiral GWs! Additionally, this model predicts

GWs from (p)reheating

Reheating is defined as the period of perturbative decay of the energy density of the inflaton to SM particles at the end of inflation, whereas preheating is understood as its non-perturbative analoguel; we focus on the latter.

Consider an inflaton with potential $V(\phi)\propto\phi^n$. The scalar field will condensate after inflation, leading to coherent oscillations $\phi(t)\approx\Phi(t)f(t)$, and couple to SM fermions via $y\phi\bar{\psi}\psi$. These oscillations are accompanied by the creation of $\psi$-particles, with mass $m_\psi^2=h^2\phi^2$. For bosons, with coupling , we observe parametric resonance (see e.g. here), which leads to GWs from the decay products but at very high frequencies (way out of reach for current experiments).

GWs from cosmic defects

Cosmic defects are the aftermaths of phase transitions in the early universe, and come in many forms : monopoles, strings, domain walls… Consider a simple $U(1)$ symmetry breaking : it happens differently in causally disconnected regions of space, yet, the field having to be continuous, somewhere there must be a defect where the symmetry is artificially restored. Where the “Higgs” field is zero, the gauge fields are maximum - this gives rise to cosmic string -like defects where the energy density is concentrated.

We can get GWs from the evolution of such defect networks (parameterised below by $F_u$) and they are scale-invariant :

In the case of matter, the scale dependence goes like $(k_{\text{eq}}/k)^2$, and so :


Further reading

Chapters 2 and 3 of Michele Maggiore’s book on GWs.

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