Measuring a cosmological distance-redshift relationship using only gravitational wave observations of binary neutron star coalescences

Chris Messenger^{1}, Jocelyn Read

Detection of gravitational waves from the inspiral phase of binary neutron star coalescence will allow us to measure the effects of the tidal coupling in such systems. These effects will be measurable using 3rd generation gravitational wave detectors, e.g. the Einstein Telescope, which will be capable of detecting inspiralling binary neutron star systems out to redshift z=4. Tidal effects provide additional contributions to the phase evolution of the gravitational wave signal that break a degeneracy between the system's mass parameters and redshift and thereby allowing for the simultaneous measurement of both the effective distance and the redshift for individual sources. Using the population of O(10^{3}-10^{7}) detectable binary neutron star systems predicted for the Einstein Telescope the luminosity distance--redshift relation can be probed independently of the cosmological distance ladder and independently of electromagnetic observations. We present the results of a Fisher information analysis applied to waveforms assuming a subset of possible neutron equations of state. We conclude that the redshift of such systems can be determined to O(10%) for z>1 and in the most optimistic case accuracies of 2% can be achieved.

It's a very interesting paper as it could provide a way to get round a reliance that has been at the heart of various aspects of cosmology - the cosmic distance ladder (you might also want to read Phil Plait's description of this). If you want to know the distance of an object that is very far away you need to be able to measure some aspect of that object that has a well know relation to how far away it is. For example, if you have a 100 W light bulb that's 10m away you know that the flux (energy per second per unit area, or basically how bright it looks) that reaches you will be 100 W / (4π x (10 m)

^{2}) = 0.08 W/m

^{2}. The same light bulb at twice the distance will be four times dimmer as you know that the flux falls off as the square of the distance - the famous inverse square law. In this case you know the actual power output of the bulb, so by measuring the flux that you receive you can easily work out the distance. Things are a bit more difficult for objects in space as unlike for a light bulb, which tells you its power on the box, they don't come with instructions! So a bit more inference is required to make some so called "standard candles" for astronomical distance measurements.

For the Sun, we know its distance very well and we can also measure its brightness, so if we see other stars (much further away) that look a lot like the Sun (things like their spectral type/colour can tell us this) then by measuring their brightness and assuming they emit the same power as the Sun we can infer their distance. There is another way of telling the distance to nearby stars (within a few hundred light years of us) called parallax, which involves geometry - this uses the fact that as the Earth orbits the Sun the position you observe a star at will shift with respect to distant background objects (like how a close object will appear to shift if you close one eye then the other). The size in the shift and the well know size of the Earth's orbit gives you the distance to the star via simple trigonometry. So for these stars you can measure their distance and brightness (giving their intrinsic luminosity or power ouput) and their spectral type. Again you can use these now well defined stars to calibrate a distance scale for further away stars of the same spectral type that are too far away to get a parallax.

These distance measurements work relatively nearby e.g. in our galaxy, but as you get further away it gets harder to resolve individual stars to get their brightness and spectral type. There is a special type of star called Cepheid variables that allow you to measure distances a bit further. These stars are intrinsically bright (you can see them at further distances) and they have a special property - they pulsate - and the period of the pulsation is related to their overall brightness. So, if you can measure the period of some Cepheids nearby where their distance can be measured by looking at other close stars then you can calibrate the relationship and then stretch the distance ladder out further. A type of event that is even brighter and can be seen out to even greater distances is a Type Ia supernova. These events occur when a white dwarf star (the Earth-sized remnant of a star like the Sun after it has run out of fuel to burn) accretes (gravitationally pulls) material off a companion star. White dwarfs are only stable if their mass is below about 1.4 times that of the Sun and if enough material is accreted to exceed this limit they will explode! As all these stars are the same mass when they explode it is assumed that they will all emit the same amount of power. So again, if you can measure some of these events nearby, where other distance measurements are valid, then they can be calibrated to use as a cosmological distance measurement.

This is called the distance ladder because you can see it relies on several steps. There are other things that need to be taken into account, for example extinction (the dimming of light as it passes through the extremely tenuous, but nevertheless present interstellar/intergalactic medium between us and a source), which makes things appear dimmer than the should be and could lead to their distance being overestimated; or the small possibility that the physics of Type Ia supernovae is be different over the course of the universe's history, so they won't all have the same intrinsic luminosity.

So, why is this important for cosmology? Well cosmology is all about working out the history and geometry of the universe, so how the universe evolves over time is essential to this. Edwin Hubble showed that the universe is expanding and the further away you look (from Cepheid distance measurements) the faster that expansion (as given by a redshift) is happening - know as Hubble's Law. [Redshift is causes by the Doppler effect - light, or any wave, will appear to have a longer wavelength (i.e. be

*redder*) if its source is moving away from the observer than if they were stationary with respect to each other.] In Hubble's Law the relationship between the velocity and distance is constant, but working out the Hubble constant's value (or indeed whether it is constant) requires well known distances and redshifts. [Redshift can be measured by looking for certain features in the spectrum of a star and seeing how much they are shifted by, or by seeing how much the overall spectrum is shifted to the red.] This also relates to the geometry of the universe - is it flat, or curved on the largest scales? One of the major recent(ish - well late 90s) discoveries, using observations of Type Ia supernovae, showed that distant supernova appeared dimmer (e.g. further away) that would be expected from the simple Hubble relation, suggesting that the expansion of the universe has started accelerating in its relatively recent history (the last few billion years!)

The above discussion has talked about using electromagnetic radiation to work out the redshift vs. distance relation, but now we'll switch to the crux of the paper I mentioned which instead uses gravitational waves. The paper is trying to provide a way to get the distance vs. redshift relation without relying on the standard distance ladder of Cepheids and supernova. This could either independently confirm the current cosmological models, or show up systematic errors in the distance scale, both of which would be very important to know. The basis is that certain sources of gravitational waves, namely the inspiral of two neutron stars (the ultra dense remnants of massive stars the cores of which collapse during a supernova [described in a bit more detail here]), are "standard sirens" - like standard candles, but a sound-based analogy is generally thought more apt for these gravitational wave sources as they are in the audio frequency range of around 10-1000 Hz. In these systems the pair of neutron stars will orbit around each other gradually losing energy (and shrinking their orbit) via emission of gravitational waves - this was demonstrated to be true by the Nobel prize winning discovery by Hulse and Taylor of a pair of neutron stars behaving in exactly the way predicted. The amount of energy released via gravitational waves increases as the orbit decays, but still would only be observable with planned ground-based gravitational wave detectors in the very final stages of their evolution - when their orbits have decayed so much that they're about to collide and merge together. This signal is useful, because (as shown by Bernard Schutz) the strength and evolution of the signal gives the source distance directly, independently of the distance ladder - the way the frequency evolves gives the system's redshifted mass, whilst the amplitude contains a combination of the redshifted mass and distance, so the former can be used to get the systems distance from the latter.

Unfortunately the signal doesn't directly enode the source's redshift, it gives a combination the the source mass and redshift that cannot be unentangled, so independent measurements of the redshift are required using a regular telescope. Unlike regular telescopes gravitational wave telescopes are omnidirectional, which means they see the whole sky all the time (although their sensitivity is not equal over the whole sky due to their antenna pattern), but on their own they cannot pinpoint a source's position. You need multiple detectors to get positional information by working out the differences in a signal's arrival time between detectors and triangulating the source's position - and many detectors have been built to give such an array. However, even with several detectors the positional information is not great (even if you can pinpoint a small patch of sky, say the angular size of the moon (or about half a degree), it contains a lot of galaxies that could have been the home of the gravitational wave signal), so finding the home of the signal and measuring its redshift is difficult. In some cases if the gravitational wave signal is coincident with a gamma-ray burst then that helps localising the host with follow-up observations, but this will not be the case for all events.

The holy grail is therefore to find a way to get the distance

*and*redshift information from the gravitational wave signal alone, and that is what the paper provides. How you do this comes about by including some extra physics in the how the orbits of the binary neutron stars decay. For the majority of the binary lifetime you can calculate the gravitational wave signal by assuming that the neutron stars are both point particles i.e. all their mass is in a single point. However, neutron stars are not point particles and there will come a point when they get so close that their very strong gravitational fields will start to distort each other and pull the stars apart. This distortion will be encoded in the gravitational wave signal and give information about the stars' masses. As noted before the combined mass and redshift can already be measured from the signal, so this other mass measurement can unentangle these and give you a redshift. So, voila! Distance and redshift can be measured all in one go.

Unfortunately (again!) there are some caveats. For example: the accuracy that you can measure the redshift depends on redshift - due to how the strength of the signal changes, and how the gravitational wave detectors work, there's a sweet spot at about redshift 2; the accuracy depends on what exactly the neutron star is made of (the star might be stiff or soft and therefore harder or easier to distort), and what a typical neutron star is made of needs to be know beforehand (currently there are many theoretical models, but no conclusive evidence for which is/are correct); and finally these measurements require a gravitational wave detector called the Einstein Telescope, which is currently only a design concept although hopefully will be built some time in the 2020s. It's still very interesting work though and I think the above caveats are by no means insurmountable in the not-to-distant future.

*[Update: another similar interesting paper has also been recently posted.]*

*[Update: I should just note that the method works because the*known

*neutron star radius can be used as a reference length in the waveform, which is kind of analogous to knowing the rest wavelength of a particular spectral line and therefore working out the redshift by measuring the difference from the observed wavelength.]*

^{1}Chris was the instigator and lead guitarist in our famous bands

*Corpse Full of Bees*and

*Look Up for Danger*