These two lengths become equal when m is the Planck mass. And when this happens, they both equal the Planck length!

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2 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.

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4 These two lengths become equal when m is the Planck mass. And when this happens, they both equal the Planck length!

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7 Hubble s Original Diagram velocity=h0 x distance

8 k>0 Ω M >1 k=0 Ω M =1 k=0 (Ω M +Ω Λ )=1 q 0 = !

9 Q: How does Ω-1 change with cosmic time? Assume k very close to 0!

10 Q: Universe is old, close-to-flat in unstable, deviation from flatness increases with time, so why is the Universe close to flat today?

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12 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

13 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± (95%) Regardless of whether we understand the physical origin of Ω 0 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...

14 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...

15 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/l 2 ], 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."

16 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!

17 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 ρ Λ c 2 = const, the change in system s energy is de = ρ Λ c 2 dv, 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

18 NEWTONIAN COSMOLOGY WITH LAMBDA (answer to a question by a student)

19 E. Static Universe is closed! k ρ0 ΛE

20 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.

21 Note that i.e. the de Sitter Universe 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=Ω Λ =0, Ω Λ =1, ΩM>1, ΩM+Ω Λ =1

22 NB: Ω Λ (t)