THE BIG BANG
In relativistic classical field theories of gravitation, particularly general relativity, an energy
condition is one of various alternative conditions which can be applied to the matter content of the
theory, when it is either not possible or desirable to specify this content explicitly. The hope is then
that any reasonable matter theory will satisfy this condition...In GR, energy conditions are often
used (and required) in proofs of various important theorems about black holes, such as the no hair
theorem or the laws of black hole thermodynamics.
These two lengths become equal when m is the Planck mass.
And when this happens, they both equal the Planck length!
velocity=H0 x distance
Hubble’s Original Diagram
k>0 ΩM>1
k=0 ΩM=1
k=0 (ΩM+ΩΛ)=1
q0= - 0.55!
Q: How does |Ω-1| change with cosmic time?
Assume k very close to 0!
Q: Universe is old, “close-to-flat” in unstable, deviation from flatness
increases with time, so why is the Universe close to flat today?
If the Universe were the accidental byproduct of a Big Bang, it is
difficult to imagine how such a fantastic coincidence could occur.
Standard Big-Bang cosmology cannot explain why the matter density
in the universe isn’t greater, causing it to collapse upon itself (closed
universe), or less, causing the universe to rapidly fly apart (open
universe) ➪ INFLATION
Flatness Problem ➪ Fine Tuning of the IC!
As of March 2013, the expansion rate of the
universe appears to be very finely balanced with
the force of gravity, a condition known as
FLATNESS....
Planck+WP+highL+BAO
Ω0-1=0.0005±0.0065 (95%)
Regardless of whether we understand the
physical origin of Ω0≃1, this is an
observed fact. One useful thing it tells us
is that the Universe at early times is very
close to spatial flatness, which means that
it is a very good approximation to set k =
0 in the Friedmann equation when
describing, e.g.,nucleosynthesis...
When Einstein wrote his 1917 paper on the cosmological consequences of GR, it was not yet
established that there are galaxies outside our own Milky Way, to say nothing of the fact that the
Universe is expanding. Einstein realized that a static Universe created difficulties for GR – the
same difficulties Newton had come across when the tried to make a cosmology within a finite
region of space. Newton resolved this problem by making space infinite so that there was no
preferred place for the matter to collapse to, but GR does not allow this possibility......
➪ p=ρ=0 ➪ static universe is empty!
➪ empty static Universe is flat!
Einstein could have solved this problem by postulating an expanding
Universe (which would in retrospect have ranked as perhaps the
greatest of all scientific predictions). Instead, Einstein decided to
change the long–range properties of gravity, and introduced the
cosmological constant Λ [1/l2], dΛ/dt = 0, a new
energy density term.
In 1917, after de Sitter produced equations that could describe a Universe that was
expanding, Einstein wrote to him that "This circumstance irritates me." In another
letter, Einstein added: “To admit such possibilities seems senseless."
➪ w=-1
➪ a positive Λ acts to cause a large scale repulsion. Vacuum acts as a
reservoir of unlimited energy, which can supply as much as is
required to inflate a given region at constant energy density!
What is negative pressure? A piston chamber
filled with ordinary matter will exert a positive
pressure by pushing out against the piston. If it is
filled with Λ instead, it will exert a negative
pressure by pulling in the piston. This is because,
since ρΛc2 = const, the change in system’s energy
is dE = ρΛc2dV, i.e. the system lowers its energy by
volume-contraction. When dV>0 (hence dE>0) we
have to do work to overcome the pulling by Λ.
Energy conservation is maintained in such a
situation because negative pressure is what is
required by the First Law of Thermodynamics
dE=-pdV.
Work is done by the Vacuum!
dE=-pdV
NEWTONIAN COSMOLOGY WITH LAMBDA
(answer to a question by a student)
E. Static
Universe
is closed!
k
ρ0
ΛE
Back in 1922, the Russian mathematician Alexander Friedmann
had published a set of possible mathematical solutions to
Einstein’s field equations that gave a non-static universe. But
through the 1920s, neither Einstein nor anyone else took any
interest in Friedmann's work, which seemed merely an abstract
theoretical curiosity. Most astronomers continued to take it for
granted that the real universe was static. Friedmann could not
stand up for his ideas, for he died of typhoid fever in 1925, only
37 years old.
➪
➪
i.e. the de Sitter Universe
Note that
satisfies the Perfect Cosmological Principle!
In the de Sitter vacuum Universe,
test particles move away from each
other because of the repulsive
gravitational effect of the positive
cosmological constant. In this
cosmology “a” is perfectly finite back
to t=−∞, so there is no initial
singularity: A UNIVERSE WITH NO
BEGINNING!
ΩM=1, ΩM=ΩΛ=0, ΩΛ=1,
ΩM>1, ΩM+ΩΛ=1
NB: ΩΛ(t)