Answer 1:
There is no proof in the sense that we know
this for an absolute fact, but the following is
the best approximation we can give, which is based
on strong evidence but there are a few possible
problems. You might want to brush up on your
knowledge of special relativity,
specifically how the relativistic Doppler
effect works, as it is necessary to understand
how we know how old the universe is.
The entire sky has a glow called the cosmic
microwave background (CMB), which is the
afterglow of the big bang: it is the light from
the primordial plasma that the universe came out
of as it expanded from the big bang. The
temperature of this plasma when we are able to see
it is the minimum temperature by which a plasma
becomes opaque, i.e. light cannot pass through it,
which is about 3000K (if light could pass through
it, we could see all of the way to the big bang
itself). However, because the plasma is moving
away from us so fast, it has been redshifted by
the relativistic Doppler effect to have a
temperature of about 3K, which is why we see it as
a microwave background in the sky instead of a
visible light background. By measuring the
apparent temperature of the CMB, we can calculate
that it has been redshifted by roughly 1000x. You
can then use Einstein's relativistic equations to
calculate how fast it is moving away from us (it's
very close to the speed of light).
The next thing you need to know is how fast
the universe is expanding. We do this through
what are called "standard candles"  events
that have a certain known brightness that we can
use to determine how far away they are based on
how bright they appear to us in the sky. One of
these are typeIa supernovae, which are
what happens when a white dwarf gets overloaded by
mass from a companion star and goes supernova.
This has a finite mass because it's the threshold
at which point white dwarf stars are no longer
possible (it's about 1.44 times the mass of the
sun). Using these, we calculate that the rate
at which the universe is expanding is about 70
kilometers/second per million parsecs. This
number is called the Hubble constant. So,
for example, a galaxy located ten million parsecs
away from us will be moving (on average) about 700
kilometers per second away from us.
By dividing the speed that the CMB is moving
away from us by the Hubble constant, we can
calculate how far away the CMB is. The reason why
the CMB is the distance that it is is because
light has had only a finite time to travel. Thus,
by dividing this distance by the speed of
light, we calculate the age of the universe.
It so happens that the distance to the CMB is
about 4.3 billion parsecs, or 13.7 billion light
years. Thus, the universe is roughly 13.7
billion years old.
All of the above is calculated based on known
physics, but there are a few places that we don't
know the exact details, which is why I objected to
your use of the word "proof" in your question.
These are:
1. We don't know exactly how bright TypeIa
supernovae are. If they are brighter, then they
are further away, so the CMB is further away and
the universe is older. If they are fainter, then
the universe is younger.
2. The rate at which the universe is
expanding is changing with time. We would have to
integrate the rate of change in the expansion rate
of the universe over the universe's age in order
to get a "true" value. We only approximately know
how the rate of expansion is changing.
3. We can really only measure the age of the
primordial plasma itself. Our theories imply that
the universe was very young when the primordial
plasma became traversable to light, but if some
kind of exotic physics was taking place back then
(and it may have been), then it is possible that
our estimate of the age of the universe could be
wrong because of it.
4. Related to #3 above, general relativity
predicts that there was a singularity at the
beginning of the universe: a point of zero volume
and infinite density, in which time literally
began. The Heisenberg uncertainty principle
from quantum mechanics states that this is
impossible. Obviously, one of the two theories
must break down at this point as they cannot both
be right, but we have no experimental verification
of which of these otherwise welltested theories
that might be.
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