# Gravitational waves and gamma ray burst: how were the error bars determined for this speed of gravity calculation? Was $H_0$ used?

This newly updated answer to How precise are the observational measurements for the speed of gravity? and this answer to How is the most accurate value of 𝐺 measured? cites the November 2017 arXiv preprint Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A which says that these measurements:

constrain(s) the difference between the speed of gravity and the speed of light to be between −3×10−15 and +7×10−16 times the speed of light.

What are the major assumptions and other measurements that went into these error bars? Did they use a range of values for the Hubble constant? Was the dispersion of the interstellar medium at optical frequencies included? Were known limits to the variability of fundamental constants also applied or were those assumed to be constant? Anything else?

We don't often see uncertainties in the 10-15 range in Astronomy! :-)

Potentially related:

• @CarstenS Thank you for point that out! I remember struggling this morning trying to get the title to be under 150 characters as required by SE, and it looks like after several edits it's become unparsable. If I had 240 characters the title would be more like For the recent determination of the speed of gravity based on the timing between the detection of gravitational waves and nearly-coincident gamma ray bursts, how were the error bars determined? Was the Hubble constant used as part of this? If you can think of a better title please feel free to edit. Otherwise I'll have a look tomorrow.
– uhoh
Aug 7 '20 at 15:53
• I need help rewording this unwieldily title
– uhoh
Aug 8 '20 at 3:27
• We don't often see uncertainties in the 10ˉ¹⁵ range in astronomy. Quite. Aug 8 '20 at 14:12

What are the major assumptions and other measurements that went into these error bars?

The error bars in the paper are based on the shortest reasonable distance (to the authors) between the source and the Earth and a zero to ten second lag between gravity wave emission and gamma ray emission.

One key assumption is how long it took for the two signals, gravitational wave and gamma rays, to travel from the source to the receiver (the Earth). The 1.74 second difference in timing of the arrival of those signals would mean the speed of gravity and speed of light would be nearly identical if the signals traveled a long time (long distance), but maybe not so little if the signals traveled a shorter amount of time. The authors of the paper intentionally picked what they deemed to be the shortest reasonable light travel time (shortest reasonable distance) so as to magnify the uncertainty bars.

Another key assumption is that the two signals were emitted very closely in time, with the photon burst following the gravity wave by no more than ten seconds. The authors alluded to fringe theories that had the photon burst preceding the gravity wave emission by a non-trivial amount of time, and other fringe theories that had the photon burst lagging behind the gravity wave emission by much more than ten seconds. The paper mentions these only in passing.

Did they use a range of values for the Hubble constant?

The Hubble constant does not come into play here. The authors used what they deemed to be the shortest reasonable distance (the product of travel time and the speed of light) between the source and the Earth, based on luminosity.

Note that at 26 Mpc, the Hubble constant is not particularly relevant.

Was the dispersion of the interstellar medium at optical frequencies included?

Apparently not; this is a simple calculation. Moreover, dispersion at optical frequencies is rather irrelevant as the observations were of gravity waves and gamma rays.

The source is at a distance $$x \pm \Delta x$$ (assuming the sources are coincident). The lag between the gravitational wave signal being detected and the gamma ray signal being detected was $$t \pm \Delta t$$.

The difference in signal speeds is $$\Delta v = c -\frac{x}{t_0 + t},$$ where $$t_0 = x/c$$. Dividing through by $$c$$ $$r = 1 -\frac{\Delta v}{c} = \frac{x}{x +ct},$$ and the assumption has been made that the two signals were emitted at the same time.

Then, if I have my error propagation formulae correct, the error in this ratio (call it $$\Delta r$$) is $$\Delta r = \frac{c \sqrt{x^2(\Delta t)^2 + t^2 (\Delta x)^2}}{(x+ ct)^2}.$$ If we assume $$ct \ll x$$, then this simplifies to $$\Delta r = \frac{\sqrt{(\Delta t)^2 + t^2(\Delta x/x)^2}}{t_0},$$

where $$t_0$$ is of course very much larger than either $$\Delta t$$ or $$t \Delta x/x$$. This is why $$\Delta r$$ is of order $$10^{-15}$$, though I'm sure the authors did a more complex calculation of the uncertainty.

If the distance to the source were known via the Hubble redshift-distance relation (which is not the case here), then the uncertainty in the Hubble parameter would enter as a contributor to $$\Delta x$$, where $$\Delta x \simeq x\Delta H_0/H_0$$.

I think if you are comparing the speed of light to the speed of gravity waves, you are assuming that space has the same "refractive index" for both. Strictly speaking you are measuring the ratio of the speeds, not the ratio of the speed of GWs to the speed of light. i.e. Where I have used $$c$$ in the formulae above, you can use $$v_{{\rm EM}}$$ where that is the speed that the electromagnetic waves propagate. If the "refractive indices" differ, then of course $$r \neq 1$$.

Now looking at the paper itself (section 4.1), we see that the authors approximate $$\Delta r \simeq v_{\rm EM} \Delta t/x$$ (in my notation). This would appear to neglect any uncertainty in $$x$$, but if one reads on, we see that what they have done is use a minimum distance to define the maximum possible $$|\Delta r|$$ and assuming that the signals were emitted at the same time, so that the observed 1.74s observed delay is because the gamma rays travel slower. This minimum distance is the distance derived from the GW signal itself of $$40^{+8}_{-14}$$ Mpc, which is independent of cosmological parameters (see https://physics.stackexchange.com/questions/235579/how-were-the-solar-masses-and-distance-of-the-gw150914-merger-event-calculated-f).

The lower limit is obtained by assuming that there was some lag between when the EM signal was emitted and when the GWs were generated. This was assumed to be 10 s for the purpose of the lower limit quoted in the paper. The reason for this value is discussed at some length in the paper. Since 10 s is much larger than the observed lag of $$1.74 \pm 0.05$$ s, then the uncertainty in the distance is less importantant (i.e. it becomes an error in the error). They seem to have used the 26 Mpc minimum distance again, in order to arrive at the lower limit of $$-3 \times 10^{-15}$$ for a total lag of 8.26 s.

• The lower and upper range values in the paper are consistent with an observed delay of 1.74 seconds, a distance of 26 Mpc, a generation lag of 0 to 10 seconds. If there was no lag, the speed of gravity is greater than the speed of light by a factor $6.5\times10^{-16}$ (the authors apparently rounded the 6.5 up to 7). If the generation lag is ten seconds, the observed lag of 1.74 seconds means the speed of gravity is smaller than the speed of light by a factor $3.09\times10^{-15}$ (the authors apparently rounded to 3.09 down to 3). Aug 7 '20 at 10:11