I cant understand here why is receding acceleration is taken equal to gravitational acceleration? Here is the full document link
https://www.lapasserelle.com/cosmology/
Click on lecture 1 notes for the document


Now my question is why the acceleration with galaxies is receding is taken equal to gravitational acceleration? Is it just die to the fact that because stay fixed in fictious moving grid?
 A: In eqn. $16$ only magnitude of gravitational acceleration is equated to receding acceleration. And it is not Newton’s gravitational acceleration but  (cosmological) time  dependent Susskind’s gravitational acceleration. Second derivative of $D$ in the eqn. $16$  is equivalent to second derivative of $r$ in Newton’s inverse square law.
Equation $15$ is more meaningful when object of mass $m$ is assumed at origin. But eqn. $15$ is not Newton’s inverse square law of gravitation which is (cosmological) time   independent. Eqn. $15$ is Susskind’s inverse square law of gravitation which is time dependent. Following examples can explain this.
Assume two points $a$  and $b$ in space at astronomical distance $R$ which is Euclidean distance. Place two massive objects at these points. If you calculate gravitational force between these two objects in year $2020$ and compare it with the gravitational force between same objects at same points at the time of Newton’s birth using Newton’s inverse square law of gravitation, you will get almost the same result. But if you want to make similar comparison between year $2020$ and $1$ billion year ago, then Newton’s inverse square law will not work. Then you need time dependent Susskind’s inverse square law of gravitation given by eqn. $15$.
