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40) The Hubble parameter quantifies the “speed” with which the size of the universe is increasing. The value of the Hubble parameter at the present epoch is the Hubble 23 constant,H0 . 42) “Mpc” stands for “megaparsec”, 1 Mpc ∼ = 3 × 1024 cm. The universe is expanding, therefore we know that a˙ > 0. 39) we also know that a ¨ < 0 when assuming that the pressure p and the density ρ are both positive. The universe must have been expanding at a faster and faster rate when going back in time. If we consider that the universe have always been expanding at the present rate, then at the time T = H −1 = a/a˙ ago, the scale factor a would be null, a = 0.

It might be hoped that the perfect symmetry of the FRW universes was responsible for this singularity, but however, the singularity theorems predict (under relatively mild conditions) that any universe with ρ > 0 and p 0 must have begun at a singularity. There are a number of consequences to this view. One consequence is that the universe now is very different than the universe in the past or in the future. The Big Bang theory predicts that at some point, the matter in the universe was hot and dense enough to prevent light from flowing freely in space.

Consider a generalized power series expansion of the scale factor as defined in Definition 1, the deceleration parameter q can be either finite or infinite as the cosmological milestone is approached: • For bangs, crunches, and rips, (η0 = 0) the limit of q is always finite. • For a certain subset of the sudden singularities (η0 = 0) the limit q is infinite. 68) where t⊙ is the the time of the event. Proof. First, we will consider the cosmological acceleration a ¨: a¨(t) = c0 η0 (η0 − 1)|t − t⊙ |η0 −2 + c1 η1 (η1 − 1)|t − t⊙ |η1 −2 + c2 η2 (η2 − 1)|t − t⊙ |η2 −2 + .

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