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In $N$-body simulations, there is a Hubble constant $H$ in equation of motion. when we integrate the equation of motion, does this constant H also change with the simulation time? or it is just a constant of present Hubble constant in simulation. Most simulations seems use $H = 100$ h KM/s/Mpc, where normalized h is usually $0.5$; does this mean that the H in the equation (1) is a constant in simulation?

but we know $H$ is dependent on the physical time. A little bit confused here if someone can help! any explanation and references are appreciated.

thanks a lot!

Equations of motion

Cosmological N-body simulations aim to represent the dynamics of galaxies in a large volume (from ~$10$ to $10^3$ Mpc across, where $1\text{ Mpc}=3.26\times10^6$ light years), whose overall motions are dominated by the mean cosmological expansion, described by a scale factor $a(t)$. In the absence of perturbations, two particles separated initially by $\vec r_1$ at time $t_1$, will, at time $t_2$, be separated by $\vec r_2=\vec r_1a(t_2)/a(t_1)$. The relative velocity between any pair of particles then obeys Hubble's law $\vec v = H \vec r$, where $H(t)=\dot a/a$ is the Hubble parameter. It is convenient to remove this zeroth-order motrion by using a uniformly expanding system of "comoing" coordinates $\vec x = \vec r/a(t)$. Written in comoving coordinates, Newton's second law for a particle with comoving trajectory $\vec x(t)$ is: $$\frac{d^2\vec x}{dt^2}+2H\frac{d\vec x}{dt}=-\vec\nabla\phi,\tag{1}$$ where $\phi$ is the perturbed gravitational potential. This Newtonian equation is correct even in general relativity as long as the gravitational fields are weka and the motion in the omoving sytstem are nonrelativistic. The gravitational potential $\phi$ follows from the usual Poisson equation, but with only the spatiall y varying part of the mass density field, $\delta \rho/\rho\equiv[\rho(\vec x, t)-\bar\rho(t)]/\bar\rho(t),$ providing the source:...

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  • $\begingroup$ Simulations of what? The answer is dependent on what is being simulated. Growth of structure in the universe, then yes of course. $\endgroup$
    – ProfRob
    Mar 28 '20 at 8:25
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The Hubble parameter evaluated at this current time (redshift $z = 0$) is the Hubble constant ($ H_0 $), but the Hubble parameter is a function of time (as you have it in the scanned text).

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