7.5: The Friedmann Equation
( \newcommand{\kernel}{\mathrm{null}\,}\)
Up until now, we haven't talked about how to determine the scale factor function a(t). In 1922, Alexander Friedmann combined the FLRW metric with the Einstein Field equations and discovered a way to determine a(t). We now call it the Friedmann Equation
H2(t)=8πρtot(t)3−Ka2(t)
where the Hubble Parameter H(t) is defined as
H(t)≡˙a(t)a(t)
and where ρtot is the mass-energy density of the universe. The constant K determines whether the geometry of the universe is spherical (K>0), flat (K=0), or saddle-like (K<0).
The "dot" over the a(t) in Equation ??? is a shorthand notation for a derivative with respect to time.
By setting K=0, we can determine an expression for the critical density ρcrit(t). A density greater than the critical density will produce a universe with a spherical geometry, and anything less will produce a universe with a saddle-like geometry.
The critical density is the density of the stuff in the universe that would produce a universe with flat geometry.
Setting K=0 in Equation ??? in solving for ρ(t) yields
ρcrit(t)=3H2(t)8π.
As you can see, the critical density changes with time. The critical density today is written as
ρcrit,0=3H208π.
Use the conversion 1 pc = 3.1×1016 m and 1 kg = 7.42×10−28 m to determine the value of the critical density today in kgm3. How many hydrogen atoms per cubic meter is that equivalent to?
- Answer
-
Let's start by converting the Hubble constant.
H0=70km/sMpc×1000 m1 km×1 Mpc106 pc×1 pc3.1×1016 m=2.26×10−181s
Then we square it.
H20=5.1×10−361s2
Now we need to figure out how to get this in units of kgm3. We can start by using the speed of light to convert seconds to meters.
H20=5.1×10−361s2×(1 s3×108 m)2=5.67×10−531m2
Now we can write 1m2 as mm3 and convert the top to kilograms.
H20=5.67×10−53mm3×1 kg7.42×10−28 m=7.6×10−26kgm3
Therefore the critical density is
ρcrit,0=38π(7.6×10−26kgm3)=9.1×10−27kgm3.
The mass of a hydrogen atom is about 1.67×10−27 kg, so the critical density is equivalent to a little over five hydrogen atoms per cubic meter of space. That is a very tiny number, especially considering the fact that the density of everything around us is much bigger than that. On the other hand, space is really big. As we will find out later, it appears as the universe as a whole has a density extremely close to the critical density. That goes to show you just how much empty space there is in the universe.