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Describing the universe

Einstein assumed that the universe was isotropic and homogeneous. Now we know it is.

Since the universe is the same everywhere, the curvature of space can only be either

positive, or
negative, or
the space can be flat.

Respectively, such a universe is called either

closed, or
open, or
flat.

The universe cannot be stationary - it should either expand or contract. A physical quantity that describes this contraction or expansion is called a scale factor. The scale factor describes how a distance between two freely moving points changes relative to the distance at the current moment. The scale factor is usually denoted as a, or more rarely as R (your book uses R).

The scale factor today is of course 1,

$\begin{displaymath}R_{\mbox{NOW}}=1. \end{displaymath}$

It is correct to think about the expanding universe as if the space itself was expanding or contracting.

Inertial observers would participate in this expansion or contraction, but non-inertial do not have to.

$\framebox{\Huge\bf ?}$ We know that the universe actually expands. Is the size of the classroom increasing with time? How about the size of the Milky Way?

$\framebox{\Huge\bf ?}$ If the universe expands, the wavelengths of a bundle of light (or even a single photon) would also increase as the light moves through the universe. Thus, if we observe this light some time after it was emitted, we would see light whose wavelength is larger. How about its frequency?

A: The frequency will decrease
B: The frequency will increase
C: The frequency will stay the same

Cosmological redshift

For a free photon (or a group of photons which is a bundle of light rays), the wavelengths will increase, the frequency will decrease, and we will observe the Doppler redshift, which is called the cosmological redshift. It is normally denoted as z.

$\begin{displaymath}1+z = {R_{\mbox{NOW}}\over R_{\mbox{THEN}}} = {1\over R_{\mbox{THEN}}}. \end{displaymath}$

Thus, z=1 corresponds to the moment when the universe was twice ``smaller'' than the current one.

$\framebox{\Huge\bf ?}$ If the universe is infinite, then at z=1 its size was infinity/2 = infinity. Is there a paradox?

A: Yes, there is a paradox, and the modern cosmology cannot address it.
B: No, there is no paradox, the infinite universe does not expand or contract, since it has nowhere to expand into.
C: No, there is no paradox, the fact that the universe was twice ``smaller' means that all distances between freely moving points were twice smaller, and that's it.