A technical aside about Doppler tracking vs PTAs

I’ve been thinking some about Doppler tracking measurements of GWs (à la Cassini). In reading about this, I found an interesting (at least to me!) issue brought up in Maggiore’s Gravitational Waves textbook (Volume 2). As he explains, usually the equation given for the induced redshift of a photon is:

\[z_\gamma = \frac{\Delta \nu_{\gamma}}{\nu_{\gamma}} = -\frac{1}{2}n^in^j \int_{t_{\rm{em}}}^{t_{\rm{obs}}} dt' \left[ \frac{\partial}{\partial t'} h_{ij}^{\rm{TT}}(t', \mathbf{x}) \right] \; ,\]

where $\mathbf{n}$ is the direction vector to a particular radio emitter that we are observing, $h_{ij}^{\rm{TT}}$ is the GW strain in the TT gauge along the photon trajectory from the emitter.

But this looks like the same equation that pulsar timing arrays (PTAs) actually integrate over to get the pulsar time delays. How can these two things give the same equation? Are they the same thing?

So, as Maggiore explains, they aren’t the same thing and they only have the same equation in a certain limit.

In Doppler tracking, the redshift of a photon is determined by integrating the time derivative of the GW strain $h_{ij}$ along the photon’s path (in the TT gauge). This gives the cumulative change in the photon’s frequency due to the spacetime distortions caused by the GW.

In PTAs, we measure the timing residuals of pulsar signals — essentially, how much earlier or later a pulse arrives compared to when we’d expect it, in the absence of GWs. This is sometimes written in a form like the above, where the timing residual is written as a change in the frequency of the pulses. The frequency change comes from integrating the GW-induced perturbation over the signal’s travel time, which is determined by integrating the time derivative of $h_{ij}$, just like in Doppler tracking. However, this is fundamentally looking at a different observable — the change in frequency of the pulsar pulses rather than the change in frequency of a single photon.

Here’s the key: both methods involve essentially the same integral over the GW strain, but they really are measuring different things. In Doppler tracking, we focus on the redshift of a single photon path. In PTAs, we are comparing the time shift between two different photons paths — each photon path emitted by a different rotation number of the pulsar.

The equations reduce to the same form in the limit where the frequency of the gravitational wave is much smaller than the frequency of the pulsar rotation. This is the regime relevant for PTA observations of nanohertz GWs.

Anyway, long story short: PTAs are measuring slightly different things than Doppler tracking GW measurements, but the difference doesn’t actually matter given the timescales and GW signals we observe.