The Friedmann equation expressed in natural units (##\hbar=c=1##) is given by(adsbygoogle = window.adsbygoogle || []).push({});

$$\left(\frac{\dot a}{a}\right)^2 = \frac{l_P^2}{3}\rho - \frac{k}{R^2}$$

where ##t## is the proper time measured by a comoving observer, ##a(t)## is the dimensionless scale factor, ##l_P=\sqrt{8\pi G\hbar/c^3}## is the reduced proper Planck length, ##\rho(t)## is the proper mass density, the curvature parameter ##k=\{-1,0,1\}## and ##R=R_0a## is the proper spatial radius of curvature.

Now each quantity in this equation has dimensions of powers of ##[\hbox{proper length}]## therefore it seems reasonable to rename it the proper Friedmann equation.

I wish to find the corresponding comoving Friedmann equation that is defined solely in terms of quantities with dimensions of powers of scale-free ##[\hbox{comoving length}]##. I want to then solve this equation to find the comoving mass density ##\rho_0##.

In order to achieve this goal I define conformal or scale-free time ##\eta## using

$$dt=a\ d\eta$$

so that

\begin{eqnarray*}

\frac{da}{dt}&=&\frac{da}{d\eta}\frac{d\eta}{dt},\\

\dot{a} &=& \frac{a'}{a}.

\end{eqnarray*}

The Friedmann equation then becomes

$$\left(\frac{a'}{a}\right)^2 = \frac{l_P^2}{3}\rho a^2 - \frac{k}{R_0^2}.$$

Now the LHS of the equation and the second term on the RHS have dimensions of ##[\hbox{comoving length}]^{-2}## as required. But the remaining term involving ##l_P^2\rho## still has dimensions of ##[\hbox{proper length}]^{-2}##.

In order that it has dimensions of ##[\hbox{comoving length}]^{-2}## we need to express the Planck length squared, ##l_P^2##, in terms of ##[\hbox{comoving length}]^2## and the mass density in terms of ##[\hbox{comoving length}]^{-4}##.

In order to express the proper Planck length in terms of ##[\hbox{comoving length}]## we need to divide by the scale factor so that ##l_P \rightarrow l_P/a##. Finally, in order to describe the mass density in terms of ##[\hbox{comoving length}]^{-4}## we make the substitution ##\rho \rightarrow \rho_0##.

Thus the complete comoving Friedmann equation is given by

$$\left(\frac{a'}{a}\right)^2 = \frac{l_P^2}{3}\rho_0 - \frac{k}{R_0^2}.$$

Does this dimensional reasoning make sense?

I can solve the comoving Friedmann equation for the comoving mass density ##\rho_0## by defining a constant time ##t_0## such that

$$\left(\frac{a'}{a}\right)^2=\frac{1}{t_0^2}$$

which has the solution

$$a(\eta)=e^{\eta/t_0}.$$

By substituting ##a(\eta)## back into the comoving Friedmann equation one finds that the comoving mass density ##\rho_0## is given by

$$\rho_0=\frac{3}{l_P^2t_0^2}\left(1+\frac{k t_0^2}{R_0^2}\right).$$

By substituting the ##a(\eta)## expression into ##dt=a\ d\eta## and integrating I find that the scale factor as a function of proper time ##t## takes the simple linear form

$$a(t)=\frac{t}{t_0}.$$

By substituting ##a(t)## back into the proper Friedmann equation one finds that the proper mass density ##\rho## is given by

$$\rho=\frac{\rho_0}{a^2}.$$

Therefore the comoving Friedmann equation implies a unique functional form for the proper mass density that does not depend on assumptions about the constituents of the Universe.

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# I Comoving mass density leads to linear cosmology?

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