[sjastro]

It’s been awhile since I have posted here.
While we don’t have a sample size of fifty binary neutron stars to determine the Hubble constant, the current sample size of one allows us to gather information on the redshift.
How does the redshift of the gravitational wave source GW170817 formed by the neutron star merger compare to the cosmological redshift z calculated by “conventional” means?

By conventional means using the formula:
z = (λ – λ0)/ λ0

λ is the wavelength say of hydrogen Balmer lines in the spectrum of a distant galaxy.
λ0 is the wavelength of the Balmer lines in a rest frame such as a laboratory spectrum.
GW170817 fortunately originated in the galaxy NGC4993 the redshift of which is calculated by conventional means.

Determining the cosmological redshift of a gravitational wave is anything but straightforward.
Firstly we don’t have the benefit of a catalogue of gravitational wave laboratory spectra that serves as our rest frame.
Secondly the gravitational waves measured by LIGO and Virgo don’t explicitly measure a redshift in frequency as there is degeneracy issue.
This issue will be made clear a little later on.

We need to define the effective mass of a binary system to understand how the redshift is calculated.
Intuitively one might think this is simply the sum of the masses of the neutron stars or black holes making up the binary system but it is somewhat more complicated.
We are interested in the mass-energy of the binary system which also includes the energy contribution from the gravitational waves.

The effective mass or “chirp mass” CM for a binary system is defined as.

CM= (m1.m2)^0.6/(m1+m2)^0.2

m1 and m2 are the masses of the neutron stars.


Since the chirp mass includes energy which is a function of frequency that can be redshifted, the chirp mass itself can be redshifted according to the formula.

CM(redshifted) = (1+z)CM(rest frame)

The gravitational frequency data from LIGO and VIRGO allows us to calculate the redshifted chirp mass (CM(redshift)).
Here lies the problem, we don’t know the contributions of z and CM(rest frame) to the redshifted chirp mass.
This problem is known as the M, Z degeneracy.
If the CM(rest frame) can be determined then the redshift z can be calculated as CM(redshifted) is already known.

This is where LIGO stepped into to the picture.

sjastro said:
Congratulations on this significant discovery.
Has this discovery broken the mass-redshift degeneracy allowing the calculation of the redshift of GW170817?
If so how does it compare to the redshift of NGC4993?

Christopher Berry LIGO said:
Not directly, but you could if you are willing to assume a neutron star mass.

Since (i) we get masses consistent with known neutron stars, and (ii) we overlap with the distance estimates for NGC 4993, you’d end up with a redshift consistent with that of the galaxy. For precise numbers you’d need to factor in the peculiar velocity of the binary.

Using this piece of advice and data from the PRL paper on GW170817 the redshifted chirp mass in the detector frame (CM(redshift)) was 1.1977 solar masses, while the chirp mass in the rest or source frame (CM(rest)) for typical neutron star binaries averages 1.188 solar masses.

Using the formula CM(redshift)=(1+z)CM(rest) and substituting in these mean values, the redshift for GW170817 is found to be z=0.008.
From the SIMBAD astronomical database NGC 4993 has a redshift z=0.009787.

This doesn’t factor in the peculiar velocity of GW170817 which is the velocity of the binary in space rather than due to the expansion of the Universe.
There is some promise even with this rough back of the envelope calculation…………….