The End of Expansion

1 The End of ExpansionCosmin Andreia, Anna Ijjasb, Paul J. Steinhardta, aDepartment of physics , Princeton University, Princeton, NJ 08544, USAbCenter for Cosmology and Particle physics , Department of physics , New York University,New York, NY 10003 USAA bstractIf dark energy is a form of quintessence driven by a scalar field evolving downa monotonically decreasing potentialV( ) that passes sufficiently below zero,the universe is destined to undergo a series of smooth transitions: the currentlyobserved accelerated Expansion will cease; soon thereafter, Expansion will cometo end altogether; and the universe will pass into a phase of slow this paper, we consider how short the remaining period of Expansion can begiven current observational constraints on dark energy. We also discuss howthis scenario fits naturally with cyclic cosmologies and recent conjectures aboutquantum the CDM model, dark energy is assumed to be due to apositive cosmological constant, in which case the current period of acceleratedexpansion will endure indefinitely into the future [1].

2 An alternative is that thecurrent vacuum is metastable and has positive energy density. If it is separatedby an energy barrier from a true vacuum phase with zero or negative vacuumdensity, then accelerated Expansion will be ended by the nucleation of a bubbleof true vacuum that grows to encompass us. Until that moment, cosmologicalobservations will be indistinguishable from the CDM picture. Without ex-treme fine tuning, the time scale before a bubble will nucleate [2] and pass ourlocation can be exponentially many Hubble times in the future (see, for example, Corresponding authorEmail J. Steinhardt)1 [ ] 19 Jan 2022 Refs. [3, 4]). (Here and throughout, Hubble time refers toH 10 14 Gy whereH0is the current Hubble Expansion rate.) Also, the ultra-relativistic bubblewall will likely destroy all observers in its path, so there will be no survivingwitnesses to the end of accelerated Expansion [2].A third possibility, to be considered here, is that the dark energy is a formof quintessence due to a scalar field evolving down a monotonically decreasingpotentialV( ) [5].

3 In this case, since the current value ofV( 0) is extraor-dinarily small today as measured in Planck mass units, there is a wide rangeof forms forV( ) that pass through zero and continue to large negative val-ues whereV( ) V( 0). In this case, the equations of motion of Einstein sgeneral theory of relativity dictate that the universe is destined to undergo aremarkable series of smooth transitions [6, 7].First, as the positive potential energy density decreases and the kinetic en-ergy density comes to exceed it, the current phase of accelerated expansionends and smoothly transitions to a period of decelerated Expansion . Next, asthe scalar field continues to evolve down the potential, the potential energy den-sity becomes sufficiently negative that the total energy density ( H2(t)) and,consequently, the Hubble parameterH(t), reaches zero. Expansion (H >0)stops altogether and smoothly changes to contraction (H <0).

4 More precisely,the transition is to a phase ofslow contraction[6, 7] in which the Friedmann-Robertson-Walker (FRW) scale factora(t) |H| where <1 this paper, we consider how soon these transitions could begin. Thatis, what is the minimal time, beginning from the present (t=t0), before ex-pansion ends and contraction begins given current observational constraints ondark energy and without introducing extreme fine-tuning? One might imag-ine the answer is several Hubble times given how well CDM is claimed to fitcosmological Q-SC-CDM refers to cold dark matter (CDM) modelswith a phase of quintessence-driven (Q) accelerated Expansion transitioning inthe future to decelerated Expansion and subsequently to slow contraction (SC),2where all phases are dominated by a scalar field (x,t) evolving down a potentialV( ).The series of continuous transitions can be understood by tracking the thetotal cosmic equation-of-state TOT(t), including both matter and dark energydensities: TOT 32(1 +pTOT TOT) 3(12 2+12 0ma312 2+V+ 0ma3),(1)wherepTOTand TOTare the total pressure and energy density, respectively;pQ 12 2 Vis the scalar field pressure; Q 12 2+Vis the scalar field energydensity; and 0mis the current (pressureless) matter density.

5 (Dot represents thederivative with respect to FRW time.)AsV( ) approaches zero from above, TOTgrows to be greater than one,which marks the end of accelerated Expansion ( a >0) and the beginning ofdecelerated Expansion ( a <0) according the Friedmann equation: aa=4 G3(1 TOT) TOT,(2)whereGis Newton s constant. The value of TOTcontinues to rise asV( )passes below zero untilV( ) becomes sufficiently negative that TOTreacheszero. According to the Friedmann constraint,H2=( aa)2=8 G3 TOT,(3)the Expansion rateH(t) also reaches zero at that point. (Spatial curvature isnegligible today and throughout these stages.) The Friedmann equations abovecombined with the equation-of-motion for the scalar field + 3H = V, (4)dictate that the field continues to evolve down its potential and thatHcontinuesto decrease. This means thatHpasses through zero; ,the universe neces-sarily begins to contract. (Note that Eq. (3) ensures that TOTcannot becomenegative; rather TOTincreases from zero once contraction begins.)

6 For /mPlV( ) b dec conFigure 1: The Q-SC-CDM scalar field potential in Eq. (5),V( ) (in units ofH20m2Pl) vs. /mPlwithM= As described in the text, the field is fixed byHubble friction near buntil around redshiftz= 3 (t=tb= 1); it then evolvesto = 0 today (t=t0); continues to evolve to dec>0, at which time (tdec) acceleratedexpansion turns to decelerated Expansion ; and then evolves further untilV( ) becomessufficiently negative (att=tcon) that the Hubble parameterHpasses through zero, theexpansion phase ends and slow contraction potentials of interest in this paper that minimize the time until expansionends, TOTbecomes 3, corresponding toa(t) |H| where 1/ Q 3,the condition for slow , the entire sequence of transitions from accelerated Expansion toslow contraction would be sufficiently smooth and slow that observers couldsafely survive and bearing witness to each determine the minimal time before these transitions could occur, weconsider Q-SC-CDM potentials of the formV( ) =V0e /M V1e /m.

7 (5)The initial conditions and parametersV0,1>0 are chosen such that evolvesfrom 0 in the past (t t0) to >0 in the future, as illustrated inFig. 1. The first (positive) potential term dominates during the quintessence-4driven accelerated Expansion phase (which includes the past and part of thefuture; and the second (negative term) dominates beginning at some time inthe initial value of the scalar field at the beginning of the dark energydominated phase, = b, is uniquely determined once the parameters are chosensuch that 0mand 0DE, the ratios of the present dark energy and matter densitiesto the critical density, are in accord with current observational limits. Moreprecisely, extrapolating the Friedmann equations and the equation of motion for back in time beginning from 0= 0, one finds that the scalar field becomesfrozen by Hubble friction (the 3H term in Eq. (4)) as matter dominates overdark energy, which is what sets the value of form forV( ) is well-suited to determining the shortest time beforethe end of Expansion .)

8 The shortest time , the fastest evolution of corresponds to the steepest potential the largest allowed value of|V, /V|orsmallest values ofMandmin Eq. (5) both consistent with observationsand without encountering extreme fine-tuning. As shown in Ref. [8], a positiveexponential potential withM , is the steepest potential compatible(to within 2 ) with current observations, wheremPl= 1/ 8 Gis the reducedPlanck mass. The negative potential term is negligible in the past, somisnot constrained by observations. However,m/Mcan be viewed as the figure-of-merit for judging fine-tuning; we therefore confine our study to values of10 2< m/M < worked 1 illustrates the case wherem= The potentialparametersV0= chosen such that 0m= and 0DE= , within current observational limits [9, 10].For the negative exponential potential term in Eq. (5) that dominates by thetimeHreaches zero and contraction begins, there is an attractor solution witha(t) |H| where = 2(m/mPl)2; for the worked example withm= , = 1/3, the signature of slow 2a compares the evolution of of the Hubble parameterH(t) as a TOT/ H0(a)(b) CDMQ-SC-CDMtconH0(t - t0) 2: (a) The Hubble parameterH(t) for the best-fit CDM model (dotted) and Q-SC-CDM (solid) models.

9 (b) A plot of 1/ the Q-SC-CDM model depictedin Figs. 1 and Fig. 2a. Unshaded regions are periods of decelerated Expansion . The lightgrey shaded region is the phase of accelerated Expansion (H >0 and TOT<1 beginningabout redshiftz= ). The dark shaded region is the phase slow contraction (H <0 and TOT>3) that begins att= of time (in units ofH 10) for the best-fit CDM model [9, 10] and thebest-fit Q-SC-CDM model withm= In the past (negative values oft), the two curves are nearly parallel; the first term in Eq. 5 dominates; andthe Expansion rate is accelerating. The two curves diverge going forward intime. Accelerated Expansion occurs forever in the CDM model but ends andeventually transitions to contraction (att=tconwhereHpasses through zero)in the Q-SC-CDM evolution of the total cosmic equation-of-state TOT(t) is shown inFig. 2b. According to the Friedmann equation (2), a (1 TOT), so, when6 Distance 3: The predicted luminosity-redshift relations (distance modulus ) for the Q-SC-CDM model shown in preceding figures compared to supernovae observations [11], a fit towithin 2.

10 H >0, acceleration corresponds to TOT<1 (light shade region in the middle)and deceleration corresponds to TOT>1 (unshaded regions). From the figure,one can observe when quintessence first dominates sufficiently for acceleratedexpansion to begin and when the accelerated Expansion ends in the future. Dur-ing contraction (H <0), TOT 3, corresponding to slow contraction (darkshaded region). The value of TOTrapidly asymptotes to12(m2Pl/m2) = 50 3after contraction 3 shows the predicted luminosity-redshift relation for the Q-SC-CDMmodel compared to current supernovae observations [11], demonstrating thatthe goodness of fit is 2 . The distance modulus is the conventional way ofparameterizing the apparent luminosity of an object at redshiftz; for standardcandles, the modulus is equal to 5 log10[dp(z)(1 +z)/(10 pc)] wheredp(z) is theluminosity distance. (Because the evolution ofH(t) is so similar in the past tothe CDM model, the Q-SC-CDM model fits no better or worse; so it does notalleviate or exacerbate the current H0problem [12].)