Minimal Conditions for the Creation of a Friedman-Robertson-Walker
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Minimal conditions for the creation of a Friedman–Robertson–Walker universe from a ”bounce” Carmen Molina–Par´ıs+ and Matt Visser++ +Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ++Physics Department, Washington University, Saint Louis, Missouri 63130-4899 (10 September 1998; LATEX-ed September 11, 2018) Abstract: In this Letter we investigate the minimal conditions under which the creation of our universe might arise due to a “bounce” from a previous collapse, rather than an explosion from a big-bang singularity. Such a bounce is sometimes referred to as a Tolman wormhole. We subject the bounce to a general model-independent analysis along the lines of that applied to the Morris–Thorne traversable wormholes, and show that there is always an open temporal region surrounding the bounce over which the strong energy condition (SEC) must be violated. On the other hand, all the other energy conditions can easily be satisfied. In particular, we exhibit an inflation-inspired model in which a big bounce is “natural”. dr2 Oscillating universes [1,2] are alternatives to standard ds2 = −dt2 + a(t)2 + r2(dθ2 + sin2 θdφ2) , big bang cosmology [3–6]. They avoid the big-bang sin- 1 − kr2 gularity and replace it with a cyclical evolution from a (1) previous incarnation of our present universe. Unfortu- nately, many of the older discussions of oscillating uni- with k = +1, 0, or −1 for hyperspherical, flat, or hyper- verses leave the nature of the turnaround quite ambigu- bolic spatial sections, respectively. ous (cusp? angular-momentum barrier?). Interest in os- To have a bounce, there must be some time at which cillating universes largely declined after the development the size of the universe is a minimum. We take this to of the first cosmological singularity theorem [3,4], but be time zero, and use a subscript ∗ to denote quantities we feel that the time is ripe for a reassessment of the evaluated at the bounce t = 0: situation. In this Letter, we model the turnaround by a Friedman–Robertson–Walker (FRW) universe under- a˙ ∗ =0;a ¨∗ ≥ 0. (2) going a “bounce” and ask what the absolute minimum requirements are for such a bounce to occur. Not too This weak inequality ona ¨∗ is not enough for proving our surprisingly, the strong energy condition (SEC) of clas- deeper results. Since we want time zero to be a true sical gravity must be violated [7–9]. (SEC–violation is minimum, we must have a>a∗ for t =6 0, with this now a necessary but not sufficient condition.) More surpris- being a strict inequality. An application of the funda- ingly, for universes with positive spatial curvature, none mental theorem of differential calculus now implies: of the other energy conditions need be violated. We shall present a model-independent analysis of the bounce ∃ t>˜ 0 : ∀t ∈ (−t,˜ 0) ∪ (0, t˜)a> ¨ 0. (3) similar to the model-independent analysis applied to the Morris–Thorne traversable wormholes [10–15], and also This is the analog, for a bouncing baby universe, of show with specific examples how the various cosmolog- the Morris–Thorne “flare-out” condition for traversable arXiv:gr-qc/9810023v1 7 Oct 1998 ical singularity theorems [3,4] and their modern exten- wormholes [10]. (See also [11, equation (11.12), page 104] sions [16–20] can be evaded. Finally we discuss the ex- and [12–15].) Mutatis mutandis, there will be similar tent to which SEC violations are compatible with known open regions surrounding the bounce for which physics, and exhibit an inflation-inspired model for which d2[ln(a)] d2(a3) a big-bounce is “natural”. > 0; and > 0, (4) A bouncing baby universe, or Tolman wormhole, is dt2 dt2 simply a FRW universe that undergoes a collapse, instant with these again being strict inequalities. of maximum compression, and subsequent expansion (as For a FRW universe the Einstein equations reduce to opposed to undergoing a big crunch singularity or ex- hibiting a big-bang singularity). In a model-independent 3 a˙ 2 k analysis, the key idea is to extract as much information ρ = + , (5) 8πG a2 a2 as possible from the energy conditions without making any particular commitment to the equation of state for the matter content of the universe [10–15]. The utility 1 a¨ a˙ 2 k p = − 2 + + . (6) of such an approach has recently been demonstrated in a 8πG a a2 a2 different context: applying the energy conditions to the epoch of galaxy formation [21–23]. Some quantities of interest are, The FRW cosmology is described by the metric [3–6] 1 1 a¨ a˙ 2 k ρ + p = − + + (In particular, any of these three conditions automat- 4πG a a2 a2 ically implies violation of the WEC, SEC, and DEC.) 1 d2 k Thus a bounce in a hyperbolic (k = −1) universe must = − ln a + , (7) violate all the pointwise energy conditions, a bounce in 4πG dt2 a2 a spatially flat (k = 0) universe is on the verge of vio- lating all the energy conditions, and a sufficiently gentle 1 a¨ a˙ 2 k bounce in a hyperspherical (k = +1) universe exhibits ρ − p = +2 +2 4πG a a2 a2 a “window of opportunity” that requires more detailed analysis. 1 1 d2(a3) k = +2 , (8) Secondly, by working in suitable open regions surround- 4πG 3a3 dt2 a2 ing the bounce [and using the strict inequalities (3)–(4) derived above] we obtain the stronger results and 3 a¨ ∃ Bounce + [k =6 +1] ⇒ NEC violated, (20) ρ +3p = − . (9) 4πG a ∃ Bounce ⇒ SEC violated. (21) By the strict inequalities discussed above [equations (3)– (4)], there will be open temporal regions surrounding the Thus the energy condition violations are minimized by bounce for which taking the universe to be hyperspherical (k = +1) and −1 1 k by making the bounce sufficiently gentle:a ¨∗ ≤ a∗ . In ρ + p< . (10) this case it is easy to check that NEC, WEC, and DEC 4πG a2 are satisfied, and only SEC need be violated. Indeed we only need SEC to be violated in some open temporal 1 k region surrounding the bounce, and it is quite possible to ρ − p> . (11) all 4πG a2 satisfy the point-wise energy conditions at sufficiently early and late times: −1 ρ +3p< 0. (12) ∃ Bounce + [k =+1;a ¨∗ ≤ a∗ ] ⇒ NEC, WEC, DEC satisfied; SEC violated. (22) For comparison, the standard point-wise energy con- ditions are the null energy condition (NEC), weak en- This is to be contrasted to the situation for Morris– ergy condition (WEC), strong energy condition (SEC), Thorne traversable wormholes, wherein (for spherically and dominant energy condition (DEC). Their specializa- symmetric wormholes) there is an open spatial region sur- tions to a FRW universe have previously been discussed rounding the throat over which the NEC (and therefore in [21–23]. Basic definitions are given in [3,11]. also the WEC, SEC, and DEC) must be violated [10–13]. Generalization of all these results to spacetimes more NEC ⇐⇒ (ρ + p ≥ 0). (13) general than the FRW universes, along the lines of [14,15] is certainly possible, and we intend to address this issue more fully in a subsequent paper [24]. WEC ⇐⇒ (ρ ≥ 0) and (ρ + p ≥ 0). (14) A simple specific example of a geometry that satisfies NEC, WEC, and DEC but violates SEC is SEC ⇐⇒ (ρ +3p ≥ 0) and (ρ + p ≥ 0). (15) 2 2 2 2 2 ds = −dt + a∗ + β t dr2 DEC ⇐⇒ (ρ ≥ 0) and (ρ ± p ≥ 0). (16) × + r2(dθ2 + sin2 θ dφ2) , (23) 1 − r2 As applied to a bouncing baby universe we have: provided we take β < 1. Note that this is the temporal First, by working at the bounce itself [where we only have analog of the toy model traversable wormhole considered the weak inequality (2)] on page 398 of the Morris–Thorne article [10]. Explicit ∃ Bounce + [k = −1] ⇒ NEC violated, (17) calculation of the stress-energy components yields 2 2 2 2 3 a∗ + β [1 + β ]t ρ = > 0. (24) 2 2 2 2 ∃ Bounce + [k =0;a ¨∗ > 0] ⇒ NEC violated, (18) 8πG (a∗ + β t ) −1 2 2 2 2 2 ∃ Bounce + [k =+1;a ¨∗ >a∗ ] ⇒ NEC violated. −1 a∗[1 + 2β ]+ β [1 + β ]t p = < 0. (25) 2 2 2 2 (19) 8πG (a∗ + β t ) 2 So that rapidly realized that the original black hole singularity 2 2 2 2 2 theorem (which also uses the SEC [3,4]) could be modi- 2 a∗[1 − β ]+ β [1 + β ]t ρ + p = > 0. (26) fied to produce more powerful theorems that used weaker 2 2 2 2 8πG (a∗ + β t ) energy conditions (e.g, the NEC [3,4]). It was commonly believed (in at least some circles) that the cosmological 2 2 2 2 2 singularity theorems could be similarly strengthened, and 2 a∗[2 + β ]+2β [1 + β ]t ρ − p = > 0. (27) there are in fact a number of newer cosmological singular- 2 2 2 2 8πG (a∗ + β t ) ity theorems that use the NEC (but at the cost of adding other rather strong conditions) [16–20]. How does the 2 2 present discussion evade the consequences of these theo- −6 a∗β ρ +3p = < 0. (28) rems? 2 2 2 2 8πG (a∗ + β t ) The theorems of [16,17] are stated using the WEC but really only need the NEC. However the key assumption The condition β < 1 is used to keep ρ+p positive definite made there is that the universe is open in the mathemat- and prevent violations of the NEC. The pressure is not ical sense (which in a FRW universe implies the universe positive in this toy model, (nor does it need to be positive is either hyperbolic, k = −1, or flat, k = 0).