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Old December 15th 18, 07:59 PM posted to sci.astro.research
Jonathan Thornburg [remove -animal to reply][_3_]
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Default Cosmological Problems

A few comments about just how one goes about measuring (estimating)
the Hubble constant with gravitational-wave "standard sirens"...

The key physics underlying these measurements is that gravitational-wave
(GW) observations of the final stages of the orbital decay and coalescence
of a compact-object binary system allow it to be treated as a GW "standard
siren".
[In this context "compact" means compact enough so
that we can ignore tidal effects, in practice this
means the individual objects are black holes (BHs) or
neutron stars (NSs).]

[By analogy to the classic phrase "standard candle",
such GW sources are called "standard sirens" (the
pre-coalescence GW signal is roughly a sine wave which
sweeps up in frequency an amplitude as the system gets
closer to coalescence).]

That is, assuming that general relativity can accurately model the system,
from the GW observations alone we can calculate the distance -- more
precisely, the luminosity distance -- to the source in meters.

Unfortunately, the current GW observations don't give a very accurate
sky position -- the error ellipsoids have areas of tens of square degrees.
This should decrease to a few square degrees when additional detectors
come online 5-10 years from now. But that's still a substantial sky
area, containing many many galaxies.

In a classic 1986 paper (Nature 323, 310), Bernard Schutz worked out
that there are two subcases for estimating the Hubble constant:


The first subcase occurs if the coalescence produces a strong
electromagnetic (EM) signal, which we use to accurately localize its
sky position. In practice this (strong EM signal) is true for binary
NS coalescences (which seem to produce a short gamma ray burst, with
strong flux everywhere in the EM spectrum from radio to gamma rays).

Given an accurate sky localization, then we can then use standard
optical/infrared astronomy techniques to measure the redshift of the
galaxy at that position. (We assume, as seems very likely, that the
compact binary is in or nearby a galaxy.)

Combining the measured redshift with the GW luminosity distance then
gives an estimate of the Hubble constant.

I don't recall the exact numbers, but I think a few dozen binary-NS
coalescences observed with the current generation of GW detectors should
yield an estimate of the Hubble constant good to a few km/sec/Mpc (i.e.,
good enough to be useful in discriminating between the supernovae and
CMB estimates).


The second subcase is if the coalescence does NOT produce a detectable
EM signal. In practice this second subcase occurs for binary-BH
coalescences. (We don't yet know whether BH/NS coalescences will
produce a detectable EM signal -- theoretical models suggest that it
will, but there are a lot of uncertainties.)

Schutz described a statistical procedure for estimating the Hubble
constant from this type of observations (GW "standard siren" but no EM
signal & no accurate sky localization), but this would require a larger
number of observations, say on the order of 100 to a few hundered.


The actual event rates for BH/BH, NS/NS, and BH/NS coalescences are
rather uncertain. To date 7, 1, and 0 of these (respectively) have been
observed (https://arxiv.org/abs/1811.12907), and with continued GW-detector
tweaks and improvements) it seems likely that the detection rates will
increase by a factor of 3-10 over the next decade.

--
-- "Jonathan Thornburg [remove -animal to reply]"
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
"There was of course no way of knowing whether you were being watched
at any given moment. How often, or on what system, the Thought Police
plugged in on any individual wire was guesswork. It was even conceivable
that they watched everybody all the time." -- George Orwell, "1984"