Inferring the DM-z relation statistically by associating poorly-localized FRBs with galaxy catalogs
Some relevant papers/links:
- McQuinn 2013
- Macquart et al. 2020
- Xu et al. 2021
- CHIME catalog
- Shin et al. 2022
- James et al. 2022a
- James et al. 2022b
- James et al. 2022c
The relevant FRB observables are the DM and the sky location
So the single-event likelihood for each FRB
Looking at the CHIME catalog, it seems "poorly localized" is a (1-sigma) error of 0.2-0.3 degrees in RA and Dec each. The fractional uncertainty on DM is always less than 1/1000, it's essentially perfectly measured.
Following Macquart et al. (2020),
$$\mathrm{DM}(z_i) = \mathrm{DM}\mathrm{MW} + \mathrm{DM}\mathrm{cosmo}(z_i | \lambda) + \mathrm{DM}_\mathrm{host}(z_i | \theta_i)/(1 + z_i)$$
where the MW contribution is a sum of the ISM and halo contributions,
For some reasonable values for the ionized gas, the average cosmological contribution is approximately (Eq. 4 of Xu et al. 2021): $$\mathrm{DM}\mathrm{cosmo}(z_i) \approx 807,\mathrm{pc},\mathrm{cm}^{-3} \int_0^{z_i} \frac{(1+z)dz}{(\Omega_m(1+z)^3 + \Omega\Lambda)^{1/2}}$$
However the electron column density has some scatter due to large-scale structure. According to cosmological simulations, this scatter in terms of fractional standard deviation of
Meanwhile, without specific information about the host galaxy, we can assume the host galaxy contribution is drawn from a log-normal distribution, giving us two additional population parameters: the mean and standard deviation of the underlying normal distribution.
The MW terms can be assumed to be fixed.
The galaxy catalog gives us a prior
Assuming galaxy redshifts and sky positions are perfectly measured, this prior is:
The term
$$p({d_i }_N, { \mathrm{DM}_i }_N, {\Omega_i }_N, {z_i }_N, {\theta_i }N | \lambda) = \prod{i=1}^N \frac{p(d_i | \mathrm{DM}_i, \Omega_i) p(\mathrm{DM}i | z_i, \lambda, \theta_i)p(\Omega_i, z_i, \theta_i)}{P\mathrm{det}(\lambda)}$$
Each term in the numerator is $$p(d_i | \mathrm{DM}i, \Omega_i) p(\mathrm{DM}i | z_i, \lambda, \theta_i) [\sum\mathrm{gal} w\mathrm{gal} \delta(z_i - z_\mathrm{gal})\delta(\Omega_i-\Omega_\mathrm{gal})p_\mathrm{gal}(\theta_i) ]$$
Marginalizing over the single-event parameters, we get that each term in the numerator is:
$$\sum_\mathrm{gal} w_\mathrm{gal} \int p(d_i | \mathrm{DM}i, \Omega\mathrm{gal}) p(\mathrm{DM}i | z\mathrm{gal}, \lambda, \theta_i) p_\mathrm{gal}(\theta_i) d\mathrm{DM}_i d\theta_i $$
If the DM is perfectly measured for each FRB, this becomes:
$$\sum_\mathrm{gal} w_\mathrm{gal} \int p(d_i | \Omega_\mathrm{gal}) p(\mathrm{DM}i | z\mathrm{gal}, \lambda, \theta_i) p_\mathrm{gal}(\theta_i) d\theta_i $$