# I LIGO and speed of propagation of gravity

1. Feb 12, 2016

### bcrowell

Staff Emeritus
Before the recent LIGO result, there was already not much doubt that gravitational effects propagated at c, but the evidence was indirect. To what extent does the LIGO result test this directly, and how will this be improved in the future?

The H1 and L1 instruments are separated by 3002 km, which corresponds to $T=10$ ms at the speed of light. The time delay for a signal propagating at speed $v$ and arriving from an angle $\theta$ with respect to the H1-L1 line would be $T(c/v)\cos\theta$. The actual time delay in the GW150914 events was 7 ms (caption to fig. 1 in the PRL paper). So I guess this doesn't actually test anything. If the time delay had been *longer* than $T$, then it would have falsified GR's prediction of $v=c$ by proving $v<c$.

I suppose if they see a statistically significant number of events, then comparing the probability distribution of the time delays with a statistical prediction of the curve should give a measurement of $v$.

They localized the source of GW150914 to a certain arc (less than a full circle) in the sky. I haven't seen any details of how this was done, but my guess is that they assume $v=c$ to determine$\theta$, and probably polarization constraints eliminate part of the resulting circle...?? Once the European detectors come online, are there good prospects for doing triangulation, so that we would get an independent measurement of $\theta$?

The LIGO result puts strict limits on dispersion, so in that sense it gives us a direct determination of possible *differences* between values of $v$ for waves of different frequencies.

If future gravitational wave events can be correlated with electromagnetic signals such as gamma-ray bursts, I guess that would also be pretty clear evidence.

2. Feb 12, 2016

### Staff: Mentor

This would be my guess as well, based on other discussions of the subject that I have read (for example, Thorne's description of how LIGO would work in Black Holes and Time Warps).

3. Feb 12, 2016

Staff Emeritus
I posted this on another thread.

The problem is not the local accuracy of clocks - the problem is the exact time stamp of the signal. I don't think you can do a whole lot better than 10ms identification of any structure. Maybe one could get 5ms statistically. You need a 3rd station, and more importantly, that third station has to be far away.

4. Feb 12, 2016

### bcrowell

Staff Emeritus
The caption to Fig. 1 of the PRL paper gives the time delay as $6.9^{+0.5}_{-0.4}$ ms.

5. Feb 12, 2016

### bcrowell

Staff Emeritus
Strictly speaking, it's consistent with a speed of 1 m/s, but if the speed were 1 m/s, the probability of a delay as small as 7 ms would be very small.

6. Feb 12, 2016

Staff Emeritus
Aha! I think I know how they did that. They have about 12 maxima, 12 minima and 24 zeros, so each signal of this size - which was freakishly loud - has ~50 measurements. So they do 7 times better than any single measurement. Probably even better, since these are not independent.

I also think you might want a 4th station to do a precision measurement of the speed of gravity waves. The wave has a speed and two independent direction cosines. Three stations give you two delta t's - so it doesn't fully constrain the system. While you can move the uncertainty into one combination of angles, I think you'd have a much better measurement of c if you didn't have that degree of freedom. A 5th station would get you an uncertainty on the velocity.

7. Feb 12, 2016

### phyzguy

8. Feb 14, 2016

### bcrowell

Staff Emeritus
Another possible method occurs to me by which they could have restricted the direction of the source to an arc rather than a full circle. The two interferometers are not oriented the same way. Therefore the ratio of their gains will depend on the direction from which the signal is coming.

9. Feb 14, 2016

Staff Emeritus
And the initial polarization. How do you get both numbers from a single measurement?

10. Feb 14, 2016

### bcrowell

Staff Emeritus
Clearly the observations are going to depend on both factors, polarization and direction of propagation. Evidently they were able to restrict the solution space enough to rule out part of the circle in the sky that was inferred from propagation delay.

It doesn't seem to me to be impossible in principle to get both polarization and direction from the same signal. For example, suppose that the time delay had been very close to 10 ms, and the ratio of the gains of the two detectors had been such that $I_{H1}\ll I_{L1}$. Then we could conclude that the direction lay along the H1-L1 line, and also that the polarization was very nearly the one that would make H1 have zero gain. But that's a special case.

In general, it seems like you have two numbers you measure: the time delay and $I_{H1}/ I_{L1}$. If the waves are highly polarized, then there is a three-dimensional space for direction and polarization, so we expect these two observables to restrict us to a one-dimensional subset of that three-dimensional space. That may be what happened here, since the arc they draw on the sky is one-dimensional; if so, then it seems based on counting d.f. that they would also know the polarization for this event...?

Last edited: Feb 14, 2016
11. Feb 14, 2016

### phyzguy

On page 4 of this paper, they say, "This reconstructs sky position using a combination of information associated with the triggers: the times, phases and amplitudes of the signals at arrival at each detector." If the signal were a single frequency, the time delay and phase delay would be degenerate, but since the signal chirps in frequency, I think they can separately determine the time delay and phase delay, at least within limits. If you look at Figure 1 from the PRL paper, it is clear that the time delay between the different peaks is not constant between H1 and L1. I think it is these slight differences in time delay as a function of frequency that they are using to tease out both a time delay and a phase difference.

12. Feb 21, 2016

### aabottom

The authors consider the minimal value of the time delay within two-sigma deviation from the mean. They get
cgw <= 1.7 c as an upper bound on the speed of gravitational waves, cgw.

13. Feb 21, 2016

### bcrowell

Staff Emeritus
That's pretty silly.

The upper bound if 1.7c is obvious if you just read the LIGO paper.

They also give a lower bound, but the lower bound is (a) just repeating a result from someone else's previous paper, and (b) model-dependent.

14. Feb 22, 2016

### S.Daedalus

Another paper using the same coincidence between the gamma ray signal and the LIGO event by John Ellis et al. claims to constrain the difference between the speed of light and that of gravity to within a factor of $10^{-17}$ (!).

15. Mar 4, 2016

### Staff: Mentor

The dispersion relation for gravitons has been studied by LIGO, with an upper limit of m<1.2*10-22 eV. Combine it with the known upper limits on a photon mass (10-18 eV quite model-independent, 10-26 with some model-dependence) and you get even better constraints. And those constraints do not rely on Fermi data and the common origin of the events.

16. Mar 4, 2016

### pervect

Staff Emeritus
I would think the best estimate of the delay would be to look at the auto-correlation coefficient for both signals, i.e. if our two signals are f(t) and g(t), we find the value of $\tau$ that maximizes $\int f(u)g(u+\tau)du$

I don't know how the error bars are calculated, exactly, the simplest calculation would assume there is a signal plus white gaussian noise (better cacluations might have a better/different noise model, such as noise with a different frequency spectrum, conceptually on processes white Gaussian noise through some frequency shaping filter), and then find the statistical variation in the autocorrelation function due to the noise.

The applicable theory would be in engineering terms "random processes".