These are the Einstein equations. They
belong to general relativity our current
theory of the behavior of macroscopic
objects as well as gravity. But they didn't
always look like this. The Einstein
equations were originally going to look
like this. But for a very specific reason
this term was added. Spoiler alert!!
It was the acceleration of the expansion
of the universe.
Let me explain. While Einstein was
developing general relativity he was
asking astronomers at the time many
questions, and something all astronomers
agreed on was that the universe were
static and unchanging.
There was no start to the universe and
there will be no ending this is how the
universe has always been and how it's
always gonna be. There is no global
motion this is where the astronomical
objects have always been. There may be
local motion, the earth is going around
the Sun, but in the grand scheme of
things the universe is static. But this
was bothering Einstein a lot. You see if
there was really no start to the
universe, then his theory predicted that
gravity would eventually pull all matter
into an infinitesimal point. So he
thought: But if every astronomer tells me
the universe static it's more probable
that my theory is wrong instead of
every astronomer in the world to be
wrong. So in order to fix his theory he
had this term to his equations. Lambda
times G mu nu. This lambda is called the
cosmological constant. Its purpose back
then was to counter the gravitational
collapse. That way his theory would be
consistent with the astronomical
observations. But in 1930 thinks changed.
Edwin hubble using the most powerful
telescope at the time, made a
revolutionary discovery. He found that
the universe wasn't static. It was
expanding. The further away he pointed
his telescope the faster the galaxies
were moving away from us. So Einstein
thought lambda wasn't needed anymore and
he removed it from his equations. But now
lambda is back. Why? To explain dark
energy. We now know that the universe is
not only expanding, but it does owe at an
ever-increasing rate. And the reason
behind the acceleration of the expansion
is dark energy. Mathematically dark
energy is expressed using lambda, the
cosmological constant. A positive value
of lambda suggests an exponential
expansion of the universe. The current
data that we have supports the idea of
a constant positive cosmological
constant across the whole universe. Such
a thing would counter all the mass and
pressure from the matter which would
gravitationally slow down the expansion.
According to the second Freedman
equation with just Einstein's
equations solved for the whole universe.
because of the cosmological constant we
have a positive acceleration of the
scale factor, a double dot. Which means
that the universe's expansion rate is
increasing. Without lambda the a double
dot term would be negative, because the
density Rho and the pressure P of the
universe both slow down the expansion.
But overall lambda wins and we have an
accelerated expansion of the universe.
It's easy to underestimate the
significance of the cosmological
constant and hard to overestimate it.
Since lambda is dark energy's number and
dark energy is 68.3% of the universe. The reason why the
universe is how it is is mostly due to
the cosmological constant. Lambda is the
reason the universe is flat. We derive
that from the first Freedman equation.
The first Freedman equation is also a
byproduct of the einstein's equations
solved for the whole universe and if you
don't take into account lambda then the
first Freedman equation predicts that
the universe has a negatively curved
hyperbolic spatial curvature. Because
there is not enough matter, or I should
say density Rho, to stop the current
acceleration of the universe that we
measure. Even without dark energy. If you
include dark energy particles though you
find that there is enough stuff to
flatten the universe. By including lambda
into Einstein's equations and as a
consequence into the first Freedman
equation we get zero in both sides of
the equation. Because cosmological
constant exactly cancels the right side
from being negative. And that is
consistent with what we observe a flat
universe, an Omega, the number which
describes the special shape of the
universe, equal to one or more accurately
something between one point zero 28 and
0.9 72. So dark energy has two roles that
work against each other. First it
introduces an anti-gravity effect which
fights and wins gravity from all the
matter and results in an accelerated
expansion. But second it increases the
density of the universe which helps
matter pull gravitationally the universe
together by adding more stuff to the
universe. The former role is the reason why
the expansion accelerates because
otherwise it will slowly decelerate but
never reach zero. And the lateral role is the
reason the density of the universe is
high enough to reach the critical
density and result in a flat geometry. So
what is the value of the cosmological
constant? haha good question!
Experimentally by measuring the
expansion rate of the universe and by
analyzing the CMB, we have found that in
SI units lambda is 1.5 times 10 to the
minus nine joules of energy per cubic
meter the average of one point 52 times
ten to the minus nine and one point
forty eight times ten to the minus nine.
This is the amount of dark energy a
cubic meter of vacuum has. So we know
what lambda does we know what's the
value of lambda but what is lamda? What is
vacuum energy? Well the leading
interpretation of dark energy suggests
that this energy comes from quantum
fluctuations described by quantum field
theory. But if you apply the quantum
field theory prediction of how much
energy a cubic meter of vaccum contains
just from quantum fluctuations you
conclude that lambda is 10 to the power
of 113 joules per cubic meter.
See the problem? Experimentally we know
that lambda is 1.5 times 10 to the minus
9 but QFT predicts a vacuum energy of 10
to the power of 113. This is one of the
biggest mysteries of physics. It is
called the cosmological constant problem.
When we say that we don't fully
understand dark energy that's what we
mean. And the worst part is we have no
explanation. But it may lead us the way
to another revolution in physics.
