On large scales ($>$ 100Mpc) the distribution of matter in the universe is isotropic (it looks the same in all directions) and homogeneous (it is isotropic at each point). But only the space is assumed to be isotropic and homogenous. The observed expansion of the universe singles out a special direction in time.⁶²⁶²In other words: The universe is not a maximally symmetric 4-dimensional manifold, but can be depicted as maximally symmetric 3-dimensional spacelike sheets in a 4-dimensional spacetime. The metric on such a manifold is the Robertson-Walker-metric.

The physical distance on large scales⁶³⁶³This is a very important point. If the space would also expand on small scales we couldn’t measure the expansion, because everything including our distance measurement device would expand. But on small scales the universe is not homogenous. On small scales the metric of the universe is not a Robertson-Walker metric, but more like a Schwarzschild metric, which is isotropic, but not homogenous. between two points in such an expanding universe varies with time like

$r_i=a(t)x_i.$

(H.1)

The factor $a$ is a dimensionless scale factor greater than zero, which must be the same for each component of the distance vector because of the assumed isotropy. The scale factor can only depend on the time $t$ and not on the position $x_i$ because of the assumed homogeneity of space.

The change of the distance with time in an expanding universe is then

${\dot{r}}_i=\dot{a}{x}_i+a{\dot{x}}_i.$

(H.2)

The global velocity of a particle $v_i={\dot{r}}_i$ which does not move relative to the expanding space (${\dot{x}}_i=0$) is

${\dot{v}}_i=\dot{a}{x}_i={\displaystyle \frac{\dot{a}}{a}}{r}_i=H(t)r_i$

(H.3)

where $H$ is the so called Hubble parameter. Is a particle moving relative to the expanding space (${\dot{x}}_i\ne 0$) then we measure the additional local (also called proper) velocity $u_i=a{\dot{x}}_i$ of the particle. This local velocity can, according to special relativity, be never greater than the speed of light $c$. Nevertheless, the global velocity (e.g. the measured escape velocities of galaxies at great distances) can be greater than $c$ (Davis and Lineweaver, 2004). Generally the physical velocity of a particle is the sum of global and local velocity

$v_i=\dot{a}{x}_i+u_i(x_i,t).$

(H.4)

From the definition of the distance $r_i$ and the velocity $v_i$ in comoving coordinates we get

	${\displaystyle \frac{\partial }{\partial r_i}}$	$={\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_i}},$		(H.5)
	${\displaystyle \frac{\partial }{\partial r_i}}{v}_i$	$={\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_i}}{u}_i+3{\displaystyle \frac{\dot{a}}{a}},$		(H.6)
	${\displaystyle \frac{\partial }{\partial r_i}}{v}_j$	$={\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_i}}{u}_j+{\displaystyle \frac{\dot{a}}{a}}{\delta }_{ij},$		(H.7)
	${\left({\displaystyle \frac{\partial A}{\partial t}}\right)}_r+v_j{\displaystyle \frac{\partial A}{\partial r_j}}$	$={\left({\displaystyle \frac{\partial A}{\partial t}}\right)}_x+{\displaystyle \frac{1}{a}}{u}_j{\displaystyle \frac{\partial A}{\partial x_j}},$		(H.8)
	$A(r_i,v_i,t)$	$\ne A(x_i,u_i,t).$		(H.9)

With the help of transformation (H.6) and (H.7) we can transform the stress tensor for a newtonian fluid in cartesian coordinates

${\sigma }_{ij}^{\prime }=2\eta \left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial r_j}}{v}_i+{\displaystyle \frac{\partial }{\partial r_i}}{v}_j\right)-{\displaystyle \frac{1}{n}}{\delta }_{ij}{\displaystyle \frac{\partial }{\partial r_k}}{v}_k\right]+\zeta {\delta }_{ij}{\displaystyle \frac{\partial }{\partial r_k}}{v}_k$

(H.10)

into the stress tensor for a newtonian fluid in comoving coordinates

	${\sigma }_{ij}^{\prime }=$	$2\eta \left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}{u}_i+{\displaystyle \frac{\dot{a}}{a}}{\delta }_{ij}+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_i}}{u}_j+{\displaystyle \frac{\dot{a}}{a}}{\delta }_{ji}\right)-{\displaystyle \frac{1}{n}}{\delta }_{ij}\left({\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_k}}{u}_k+n{\displaystyle \frac{\dot{a}}{a}}\right)\right]$		(H.11)
		$+\zeta {\delta }_{ij}\left({\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_k}}{u}_k+n{\displaystyle \frac{\dot{a}}{a}}\right)$		(H.12)
	$=$	$2\eta \left[{\displaystyle \frac{1}{2a}}\left({\displaystyle \frac{\partial }{\partial x_j}}{u}_i+{\displaystyle \frac{\partial }{\partial x_i}}{u}_j\right)+{\displaystyle \frac{\dot{a}}{a}}{\delta }_{ij}-{\displaystyle \frac{n}{n}}{\displaystyle \frac{\dot{a}}{a}}{\delta }_{ij}-{\displaystyle \frac{1}{na}}{\delta }_{ij}{\displaystyle \frac{\partial }{\partial x_k}}{u}_k\right]$		(H.13)
		$+\zeta {\delta }_{ij}\left({\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_k}}{u}_k+n{\displaystyle \frac{\dot{a}}{a}}\right)$		(H.14)
	$=$	${\displaystyle \frac{1}{a}}\left\{2\eta \left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial x_j}}{u}_i+{\displaystyle \frac{\partial }{\partial x_i}}{u}_j\right)-{\displaystyle \frac{1}{n}}{\delta }_{ij}{\displaystyle \frac{\partial }{\partial x_k}}{u}_k\right]+\zeta {\delta }_{ij}\left({\displaystyle \frac{\partial }{\partial x_k}}{u}_k+n\dot{a}\right)\right\}.$		(H.15)

With the help of the relations described in the last section, we can write down the fluid dynamic equations in explicit comoving form

	${\displaystyle \frac{\partial }{\partial t}}\rho +{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}(u_j\rho )=$	$-3{\displaystyle \frac{\dot{a}}{a}}\rho ,$		(H.16)
	${\displaystyle \frac{\partial }{\partial t}}(\rho u_i)+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}(u_j\rho u_i)=$	$-{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_i}}p+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}{\sigma }_{ij}^{\prime }+\rho g_i^{*}-4{\displaystyle \frac{\dot{a}}{a}}\rho u_i,$		(H.17)
			(H.18)

with Newtonian Gravity in comoving coordinates (Poisson Equation)

${\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_i}}{g}_i^{*}=4\pi G,$

(H.19)

where $g_i^{*}=-\frac{1}{a}\frac{\partial \phi }{\partial x_i}$ and the gauge transformed newtonian potential $\phi =\varphi +\frac{1}{2}a\ddot{a}{x}_i^2$.

The energy equation is the sum of the equation for the kinetic energy and the internal energy

	${\displaystyle \frac{\partial }{\partial t}}(\rho e_k)+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}(u_j\rho e_k)$	$=-{\displaystyle \frac{1}{a}}{u}_i{\displaystyle \frac{\partial }{\partial x_i}}p+{\displaystyle \frac{1}{a}}{u}_i{\displaystyle \frac{\partial }{\partial x_j}}{\sigma }_{ij}^{\prime }+{\displaystyle \frac{1}{a}}\rho u_i{g}_i^{*}-5{\displaystyle \frac{\dot{a}}{a}}\rho e_k,$		(H.20)
	${\displaystyle \frac{\partial }{\partial t}}(\rho e_{int})+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}(u_j\rho e_{int})$	$=-{\displaystyle \frac{1}{a}}p{\displaystyle \frac{\partial }{\partial x_j}}{u}_j-{\displaystyle \frac{1}{a}}{\sigma }_{ij}^{\prime }{\displaystyle \frac{\partial }{\partial x_j}}{u}_i-3{\displaystyle \frac{\dot{a}}{a}}(\rho e_{int}+p).$		(H.21)

A even simpler form of the equations of fluid dynamics in comoving coordinates can be found by expressing density and pressure in comoving coordinates. The connection between the density in physical coordinates $r_i=a(t)\cdot x_i$ and the comoving coordinates $x_i$ is given by

$\rho (r_i)={\displaystyle \frac{dM}{dV}}={\displaystyle \frac{dM}{d{r}_{1}d{r}_{2}d{r}_3}}={\displaystyle \frac{1}{a{(t)}^3}}{\displaystyle \frac{dM}{d{x}_{1}d{x}_{2}d{x}_3}}={\displaystyle \frac{1}{a{(t)}^3}}\rho (x_i)={\displaystyle \frac{1}{a{(t)}^3}}\tilde{\rho }$

(H.22)

and in analogy for the pressure

$p(r_i)={\displaystyle \frac{1}{a{(t)}^3}}p(x_i)={\displaystyle \frac{1}{a{(t)}^3}}\tilde{p}.$

(H.23)

Because

${\displaystyle \frac{\partial }{\partial t}}\rho ={\displaystyle \frac{\partial }{\partial t}}{\displaystyle \frac{1}{a{(t)}^3}}\tilde{\rho }={\displaystyle \frac{1}{a{(t)}^3}}{\displaystyle \frac{\partial }{\partial t}}\tilde{\rho }-3{\displaystyle \frac{\dot{a}}{a}}\rho$

(H.24)

the source term on the right hand side of the momentum equation (H.17) and energy conservation equation (H.18) is reduced and even vanishes in the mass conservation equation (H.16), so that we can write the system of equations for fluid dynamic in comoving coordinates like

	${\displaystyle \frac{\partial }{\partial t}}\tilde{\rho }+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}(u_j\tilde{\rho })=$	$0,$		(H.25)
	${\displaystyle \frac{\partial }{\partial t}}(\tilde{\rho }{u}_i)+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}(u_j\tilde{\rho }{u}_i)=$	$-{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_i}}\tilde{p}+{\displaystyle \frac{1}{a}}{\displaystyle \frac{\partial }{\partial x_j}}{\sigma }_{ij}^{\prime }+\tilde{\rho }{g}_i^{*}-{\displaystyle \frac{\dot{a}}{a}}\tilde{\rho }{u}_i,$		(H.26)
			(H.27)