- #1
jcap
- 166
- 12
The Friedmann equation expressed in natural units (##\hbar=c=1##) is given by
$$\left(\frac{\dot a}{a}\right)^2 = \frac{l_P^2}{3}\rho - \frac{k}{R^2}$$
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.
$$\left(\frac{\dot a}{a}\right)^2 = \frac{l_P^2}{3}\rho - \frac{k}{R^2}$$
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.
Last edited: