Questions for the Day 3 lectures (May 17, 2023) can be posted here

How can we generate these different movies ?

To my knowledge, these are generated from the data of Numerical Relativity simulations. You can explore more of such movies here: Movies - SXS - Simulating eXtreme Spacetimes

In the Metropolis-Hastings algorithm which distribution does it converges too? Is it the prior distribution of the parameter or a conditioned probability given the data?

The Metropolis-Hastings converges to the posterior, the point is that usually the posterior should involve the calculation of too expensive (computationally speaking) integrals, so MH is used for faster convergence to the posterior

Good question! So we aim to sample from the posterior distribution. Prior distribution is our prior belief which we are trying to update using the given data and likelihood estimate.

Do we also sample for the “Luminosity distance” just like other parameters like masses ? (asking this in context of Hubble tension). i.e Can we use this distance reliably for the H_0 parameter estimation?

Are parameter estimation and noise reduction related in any way?

The answer is yes. Gravitational Wave sources are the only type of source for which you can directly measure the luminosity distance. So yes, for every event you have the luminosity distance…but you don’t have the redshift! The challenge to study cosmology is to get the redshift of these sources.

Can you explain bayesian inference in layman’s terms?

Yes, they are related through the likelihood, i.e. the probability of observing a data set x given a signal h. This probability includes the power spectral density, that quantifies your noise fluctuation.

Can you explain again why we create the random points?

Yes we do. The posterior should not depend upon our prior belief if their is sufficient information present in the data. For example, your errors in the posterior distribution are typically inversely proportional to the SNR, so your posterior distributions should be reliable for a high enough SNR event.

yes, an electromagnetic counterpart is a big problem (hopefully not so much in future). I just wanted to know If we have any secondary way to double check on ‘distance’ . I work in the Hubble tension regime and hence I am interested to test cosmology with whatever few redshift available data we have.

Thank you for your answer.

You can not directly sample from your posterior distributions because that is not even a PDF unless you have a way of estimating evidence before hand, or your PDF can also be very complex to sample directly from. So to sample, you resort to an easier way, i.e. randomness. If you random naively and independently (for example using rejection sampling), that is also inefficient as the number of rejections would increase exponentially with dimensionality. But, if you can make your random samples communicate, like in MCMC sampling where next proposal is dependent on the current one, or in nested sampling where all the nlive points communicate to exclude the lowest likelihood point, your sampling becomes much more efficient which you can actually use for parameter estimation in higher dimensional space.

Unfortunately, we only have one GW+EM observation, which is GW170817. To infer H0, currently LVK use two methods for redshift estimation of dark sirens(no EM info): galaxy catalog and redshifted mass distribution. You can check this paper: https://arxiv.org/abs/2111.03604.

However, for BNS or NS-BH detection with no EM counterpart observation, one can estimate redshift from NS equation of state (in particular mass-tidal deformability posterior). The basic idea is illustrated here: [1107.5725] Measuring a cosmological distance-redshift relationship using only gravitational wave observations of binary neutron star coalescences

For the case of GW170817, was the redshift computed from the EM counterpart information contrasted with any of the other methods of finding the distance to the event?

I have been studying the first two methods you mentioned but not the last one yet.

However, I wonder If the ‘black box’ which is estimating the distances also includes the cosmology (Om_m,Om_\lambda etc) . ? I mean the space between the source and the observer must contain some cosmology which has to be included in the ‘black box’?

Sorry , that I’m currently completely unaware of the GW parameter estimation black box

Thank you.

Yes, for GW170817, EM counterpart allows us to identify the host galaxy, NGC 4993. You can check more details in this paper: https://arxiv.org/abs/1710.05835

GW observation provides detector-frame parameters, for example redshifted mass-pairs and luminosity distance. In parameter estimation, we directly sample in luminosity distance.