By C. Cattoen
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Additional info for Cosmological Milestones and Gravatars - Topics in General Relativity [thesis]
In this chapter, we use these considerations to derive necessary and sufficient conditions for the existence of cosmological milestones such as bangs, bounces, crunches, rips, sudden singularities, and extremality events. Since the classification is extremely general, the corresponding results are to a high degree model-independent: in particular, we determine, with a minimum of technical assumptions, the necessary and sufficient conditions for the occurrences of all theses cosmological milestones.
If the scale factor 37 possesses a generalized power series in the vicinity of the rip: a(t) = c0 |t − t|η0 + c1 |t − t|η1 + . . 47) with η0 < 0 and c0 > 0. Moreover, the series has been carefully constructed to make a(t ) = ∞. Note the similarity to bangs and crunches, with the only difference being in the sign of the exponent η0 . Sudden singularities “Sudden singularities” are a recent type of cosmological milestones that have appeared in the literature. There could be past or future sudden singularities for which some time derivative of the scale factor diverges at finite time, while the scale factor itself remains finite.
Proof. 4). 3). Lemma 3. 5). Proof. 3). 4). p = − Lemma 4. 4). Proof. 5) to obtain an equation for the pressure p. 3). And as proven for Lemma 2. 4). =⇒ 29 Lemma 5. 4) up to a term proportional to an arbitrary integration constant K: ρ ∼ aK3 . This integration constant has to be eliminated ”by hand”1 . Proof. 23) where K is an integration constant. 2). 4). 1 Physically, this extra term K a3 corresponds to “dust”. 30 Lemma 6. 3) with k arising as an integration constant and being set to k = −1/0/ + 1 by suitably rescaling a(t).
Cosmological Milestones and Gravatars - Topics in General Relativity [thesis] by C. Cattoen