Hubble constant from GW observations

Remya Nair Dept. of Physics Kyoto University

with Sukanta Bose (IUCAA, Pune) &T. D. Saini (IISc, Bangalore)

COSMO21

STANDARD SIRENS [SCHUTZ, NATURE 1986]

Most Promising Source: Compact binary coalescence Neutron star -Neutron star /Neutron star-Black hole/

Black hole-Black hole

Phase measurement redshifted chirp mass

M(z) = (1 + z)(m1m2)3/5

(m1 +m2)1/5

Amplitude measurement ratio of redshifted chirp mass and luminosity

distance

A / M(z)5/3

DL

Constrain the distance redshift relationship

Problem: Distance but no redshift

Easiest fix— Electro-magnetic counterpart measurement

ELECTROMAGNETIC COUNTERPART

Two fold advantage

redshift information to constrain DL-z relation

precise location of the GW source

Candidates - Short Gamma ray bursts

GW170817

ESTIMATING THE HUBBLE CONSTANT

The joint posterior PDF in H0 for a sample of 15 isotropically-oriented NS-NS binaries observed using a three detector network (LIGO Livingston, LIGO Hanford, Virgo).

Nissanke et al (2013)

but….

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Using knowledge of neutron star mass distribution

+ mobs=(1+z) mphys z estimate

Assumption: NS mass distribution is known

NS mass distribution

But, distribution could be bimodal, or observations could be biased towards systems with EM signatures

Markovic (1993)

Taylor, Gair & Mandel (2012)

Using tidal deformation of neutron stars

tidal terms mass dependent, but not paired with z

z estimate

Assumption: detection of BNS or BH-NS events would constrain NS equation of state

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Tidal deformation

The fractional uncertainties in the redshift as a function of redshift, for three representative EOSs

Messenger & Read (2012)

Bayesian inference to include all available information

p(~⌦|E ,H, I) = p(~⌦|H, I)p(E|~⌦,H, I)

p(E|H, I)

E ⌘ (✏1, ✏1....✏n) catalog of gravitational wave events

cosmological parameters

H Cosmological model/ hypothesis

I all other relevant information

p(~⌦|E ,H, I) = p(~⌦|H, I)nY

i=1

p(✏i|~⌦,H, I)

p(✏i|H, I)

~⌦ ⌘ (H0,⌦m,⌦k..)

Walter Del Pozzo (2012)

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Galaxy catalogs

p(✏i|~⌦,H, I) =

Zd~✓ p(~✓|~⌦,H, I) p(✏i|~⌦, ~✓,H, I)

~✓ ⌘ (m1,m2,�c, tc,↵, �, z, ...) intrinsic parameters

p(~✓|~⌦,H, I) = p(m1,m2|I) p(�c|I) p(z,↵, �|I)...

Including other available information:

coincident observation of a GRB and GW

p(z,↵, �|I) = �(z � zGRB) �(↵� ↵GRB) �(� � �GRB)

p(m1,m2|I) ⇠ particular choice of NS mass function

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Galaxy catalogs

quasi-likelihood

Walter Del Pozzo (2012)ALTERNATIVES TO EM - FOLLOW UP

Galaxy catalogs

Observation: direction and distance ~✓ & DL

+

DL(z)

3D error box in ~✓ � z space

galaxy-z survey

statistical estimate of host redshift Macleod and Hogan (2008)

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Galaxy catalogs

GW event likelihood distribution lnL(~✓, DL)

For each eventlnLj = N

�1j ⌃ lnLj(Dj = czi/H0)

lnL(H0) = ⌃⌃ N�1j lnLj(Dj = czi/H0)

For whole sample

Nj : Number of galaxies in each box

Macleod and Hogan (2008)

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Galaxy catalogs

Use the clustering of galaxies in the distance space as the background distribution for the GW sources.

Our proposal

Constrain the distance-redshift relation

Nair, Bose & Saini (2018),

BBH mergers as standard sirens

likelihood function for the source location and distance

PDF for the source (redshift, location) and cosmological (Hubble constant, matter density..) parameter

number density function in distance space

Fiducial cosmology - Flat LCDM

Obtain the probability distribution over distance from GW measurements.

For each GW observation, obtain the galaxy distribution from a sky-patch taken from the SDSS catalog that has support in the BBH distribution

Assume a prior for the cosmological parameters.

Use these three distributions to obtain the posterior over the cosmological parameters.

Combine the posterior distribution obtained from all the GW measurements to obtain the final posterior.

BBH mergers as standard sirens

Mock GW catalog

P (H0 |x) /Z

dD

exp

� (D �D0)

2

2�2D

�⇥ ns(D)Pc(H0)

!

ns(D) /X

⌦i2�⌦

Wi exp

"� (D �DL(zi, H0))

2

2�2Di

#

Cosmological parameters from GW observations

Posterior in H0

Data D obtained from GW Prior on H0

Obtained from galaxy catalog

Updating the posterior with new GW observation

P (H0 |xnew) /Z

dD

exp

� (D �D0)

2

2�2D

�⇥ ns(D)P (H0 |xold)

!

BBH mergers as standard sirens

adLIGO+VIRGO : design sensitivity

SUMMARY

GW measurements provide a new window to see the universe

These measurements will provide information like masses, spins etc. for the binary components, and in addition they will also yield the distance to the sources.

These distances will be independent of (and complimentary to) the distances obtained from the distance ladder measurements (SNeIa).

We proposed a way of using GW measurements to determine cosmological parameters.

Thank you for your attention!