Mass assembly history of dark matter halos in the light of $H_0$ tension
  The Hubble tension may introduce a new course of action to revise the
standard $\Lambda$CDM model to unravel dark energy and dark matter physics. The
Hubble parameter can be reconstructed by late-time observations of the
background evolution model independently. We relate the reconstructed Hubble
parameter to the structure formation and large scale structure observables in
this work. We use the excursion set theory to calculate the number density of
dark matter halos and the mass function of progenitors. We obtain the results
for both the Markov and non-Markov extension of the excursion set theory in the
context of spherical and ellipsoidal collapse. We show that the number density
of dark matter halos in the reconstructed model has approximately $\sim2\sigma$
difference in comparison to the Planck-2018 $\Lambda$CDM in the mass range of
$M\gtrsim10^{12}M_{\odot}$. We also compare the dark matter halo progenitor
mass function with the pair-galaxy statistics and their mass assembly history
from observational data of the HST, CANDEL survey. Due to complications to
distinguish the ratio of accretion and merger in mass assembly, our result on
pair fraction is for illustration only. However, a $\sim5$ times more accurate
observations will be promising to distinguish the reconstructed model and the
Planck-2018 $\Lambda$CDM.
