Modeling the universe

Einstein was not pleased with his cosmological model, but he could not invent anything better.

In 1924 Russian physicist Alexander Friedmann found all possible theoretical models for the isotropic and homogeneous universe in GR without a cosmological constant. None of them was stationary. Einstein first claimed that Friedmann made a mistake in his calculations, but later admitted that Friedmann was right.

If the universe expands with time, it is larger now than it was some time ago, i.e. we can use the cosmological redshift as a measure of cosmic time: z=1 does not mean that the universe is twice younger, but there is one-to-one correspondence between the age of the universe and the redshift.

Friedmann found out that

• The universe either expands or contracts.
• All three types of the universe (open, flat, and closed) begin at the state with infinite density and zero size (i.e. a singularity) and expand from there. This process of beginning from the infinite density is now called a Big Bang (the term itself appeared in 50s).
• The flat and open universes expand forever.
• The closed universe initially expands, reaches the maximum size, and then contracts to the singularity again (Big Crunch). Thus, the closed universe is also ``closed in time'' (it exists only a finite time).
• It takes the closed universe precisely the same time to reach the singularity from a point where it stops expanding as to reach this point from the beginning of the Big Bang.

The curvature of the universe (positive, negative, or zero) is usually labeled by a quantity called k:

closed universe

has k=+1 (positive curvature).

flat universe

has k=0 (zero curvature, flat).

open universe

has k=-1 (negative curvature).

The fate of the universe depends on the amount of matter in the universe:

• A lot of matter: closed universe.
• Just right amount of matter: flat universe.
• Too little matter: open universe.

The amount of matter is quantified by a cosmological parameter called ``Omega'' ($\Omega$). In general, $\Omega$ changes with time, so we will use a symbol $\Omega_0$ to denote it at the current epoch.

• $\Omega_0>1$: closed universe.
• $\Omega_0=1$: flat universe.
• $\Omega_0<1$: open universe.

If $\Omega$ changes with time, how can we measure $\Omega_0$? If, say, today we measure that

$$\Omega_0 = \Omega(\mbox{today}) = 0.32,$$

then tomorrow (literally) shouldn't we get something like that:

$$\Omega_0 = \Omega(\mbox{tomorrow}) = 0.31\mbox{\,?}$$

How about this:

$$\Omega_0 = 0.316782120856286435\mbox{\,?}$$

Important conclusion

Cosmological parameters like H₀ and $\Omega_0$ change with time. They only have definite values up to a given precision. For example, the 18th decimal place in both those numbers changes every second.

But since both H₀ and $\Omega_0$ change significantly only on a cosmological time-scales, over the periods of billions of years, for any practical purposes these two parameters have definite values.

The density parameter always stays on the same side of unity:

• if $\Omega_0>1$, then $\Omega>1$ always.
• if $\Omega_0=1$, then $\Omega=1$ always.
• if $\Omega_0<1$, then $\Omega<1$ always.

In other word, an open universe is always open, a flat universe is always flat, a closed universe is always closed. This is because the current moment, when we are measuring $\Omega$, is not a special moment in history of the universe.

Historically, scientists also use a quantity called the deceleration parameter q (q today is again q₀). It is related to the density parameter:

$$q_0 = {\Omega_0\over 2}.$$