statistical-frb

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

FRB observables

The relevant FRB observables are the DM and the sky location $\Omega$.

So the single-event likelihood for each FRB $i$ is $$p(d_i^\mathrm{FRB} | \mathrm{DM}_i, \Omega_i)$$

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.

DM-z relation

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, $\lambda$ are parameters that are common to every FRB in the Universe (like the background cosmology, including $\Omega_b$, fraction of cosmic baryons in diffuse ionized gas, mass fraction of Helium, scatter in electron column density along different lines of sight, etc.) and $\theta_i$ are parameters that are unique to the host galaxy. The host galaxy contribution is the dispersion in the rest frame of the host galaxy, hence the extra (1 + z) term. Technically some of the parameters in $\lambda$ also depend on the line of sight, if we have more information about the line of sight we can split the cosmo (equivalently, IGM) contribution into background cosmology + line of sight IGM.

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 $\mathrm{DM}_\mathrm{cosmo}$ is $Fz^{-1/2}$ for $z < 1$ (Macquart et al. 2020). $F$ measures the (inverse) strength of baryon feedback.

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.

Galaxy catalog prior

The galaxy catalog gives us a prior $p(z, \Omega, \theta)$ over the redshifts, sky positions, and (possibly) host galaxy properties $\theta$ that affect the host's DM contribution.

Assuming galaxy redshifts and sky positions are perfectly measured, this prior is: $$p(z, \Omega, \theta) = \sum_\mathrm{gal} w_\mathrm{gal} \delta(z - z_\mathrm{gal})\delta(\Omega-\Omega_\mathrm{gal})p_\mathrm{gal}(\theta)$$ where $w$ are optional weights proportional to the probability that galaxy "gal" hosts an FRB.

The term $p_\mathrm{gal}(\theta)$ should be thought of as a posterior on each galaxy's properties $\theta$ -- in other words, it includes a prior term (e.g. a lognormal distribution over DMs) so in the absence of specific information about a candidate host galaxy, it reduces to this prior, which should be the population distribution

Likelihood given $N$ FRB events

$$p({d_i }_N, { \mathrm{DM}_i }_N, {\Omega_i }_N, {z_i }_N, {\theta_i }N | \lambda) = \prod{i=1}^N \frac{p(d_i | \mathrm{DM}_i, \Omega_i) p(\mathrm{DM}i | z_i, \lambda, \theta_i)p(\Omega_i, z_i, \theta_i)}{P\mathrm{det}(\lambda)}$$

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:

$$\sum_\mathrm{gal} w_\mathrm{gal} \int p(d_i | \Omega_\mathrm{gal}) p(\mathrm{DM}i | z\mathrm{gal}, \lambda, \theta_i) p_\mathrm{gal}(\theta_i) d\theta_i $$