Make the evolution of non-primary progenitors more self-consistent

[abensonca]

For non-primary progenitors - i.e. halos which will merge into a host and become subhalos - there is inconsistency in how properties are evolved. Consider a non-primary progenitor defined at time $t-\Delta t$, which has its parent at time $t$. The actual merger is assumed to happen at time $t$ (we don't know exactly when in this time interval the merger should happen - making it happen at the end of the interval is just the current convention). The mass of these halos does not evolve up until the time at which they merge. Typically, structural properties, such as the NFW scale radius, also do not evolve - as we have no information on how they should evolve in the merger tree. However, the virial radius does continue to change as the halo evolves toward the time of merging because the virial density evolves.

This leads to some inconsistent behavior - for example, for NFW profiles, the density normalization is computed from the virial mass, virial radius, and scale radius. Since the virial radius evolves during this time period the density normalization will evolve, inconsistent with our choice to keep the scale radius fixed.

The inconsistency will get smaller if trees are built to higher precision (since $\Delta t$ will be reduced in that case).

Some options for better approaches include:

• Evaluate the virial radius at the time at which the halo is last defined (i.e. at $t-\Delta t$), so that it does not evolve between $t-\Delta t$ and $t$;
• Develop some model for the mass growth of the halo between $t-\Delta t$ and $t$;
  • Currently we assume that all unresolved mass is added to the primary halo - we could instead assume that it is added to all progenitors, divided up proportional to their mass at $t-\Delta t$;
  • Then just apply the usual rules for the evolution of the viral radius, scale radii, etc.

Of course, the assumption that merging always happens at $t$ is also not accurate. A better approximation (given that we choose timesteps in ePS trees such that the merger rate is reasonably constant across the step) would be to choose a time uniformly at random between $t+\Delta t$ and $t$ and merge at that time. If mass accretion is being added to the non-primary progenitor we would then have to make this proportional to both the mass of the non-primary and the time prior to it merging.

How much this matters is unclear - probably not very much for most applications?