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Phantom Energy and Cosmic Doomsday

Robert R. Caldwell,1 Marc Kamionkowski,2 and Nevin N. Weinberg2

1Department of Physics & Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 037552Mail Code 130-33, California Institute of Technology, Pasadena, CA 91125

Cosmologists have long wondered whether the Universe will eventually re-collapse and end witha Big Crunch, or expand forever, becoming increasingly cold and empty. Recent evidence for aflat Universe, possibly with a cosmological constant or some other sort of negative-pressure darkenergy, has suggested that our fate is the latter. However, the data may actually be pointing towardan astonishingly different cosmic end game. Here, we explore the consequences that follow if thedark energy is phantom energy, in which the sum of the pressure and energy density is negative.The positive phantom-energy density becomes infinite in finite time, overcoming all other forms ofmatter, such that the gravitational repulsion rapidly brings our brief epoch of cosmic structure toa close. The phantom energy rips apart the Milky Way, solar system, Earth, and ultimately themolecules, atoms, nuclei, and nucleons of which we are composed, before the death of the Universein a Big Rip.

PACS numbers: 98.80.-k

Hubbles discovery of the cosmological expansion,crossed with the mathematical predictions of Friedmann

and others within Einsteins general theory of relativ-ity, has long sparked speculation on the ultimate fateof the Universe. In particular, it has been shown thatif the matter that fills the Universe can be treated asa pressureless fluid, which would be the case for galax-ies, then the Universe expands forever (if it has a Eu-clidean or hyperbolic spatial geometry) or eventually re-collapses (if its spatial geometry is that of a 3-sphere).Evidence from supernova searches [1, 2] and the stun-ning cosmic microwave background (CMB) results fromballoon and ground experiments [5, 6, 7, 8, 9, 10] andnow from WMAP [3, 4] that indicate an accelerating cos-mological expansion show that this simple picture is notenough; the Universe additionally consists of some sortof negative-pressure dark energy.

The dark energy is usually described by an equation-of-state parameter w p/, the ratio of the spatially-homogeneous dark-energy pressure p to its energy density. A value w < 1/3 is required for cosmic acceleration.The simplest explanation for dark energy is a cosmolog-ical constant, for which w = 1. However, this cosmo-logical constant is 120 orders of magnitude smaller thanexpected from quantum gravity. Thus, although we canadd this term to Einsteins equation, it is really only aplaceholder until a better understanding of this negativepressure arises. Another widely explored possibility isquintessence [11, 12, 13, 14, 15, 16], a cosmic scalar field

that is displaced from, but slowly rolling to, the minimumof its potential. In such models, the equation-of-state pa-rameter is 1 < w < 1/3, and the dark-energy densitydecreases with scale factor a(t) as Q a

3(1+w).

Fig. 1 shows constraints to the w-m parameter space(where m is the pressureless-matter density in units ofthe critical density) from the cluster abundance, super-novae, quasar-lensing statistics (see Refs. [17, 18] and ref-erences therein), and the first acoustic peak in the CMBpower spectrum (values taken from Ref. [4]). As the

FIG. 1: Current constraints to the w-m parameter space.The red solid curves show the age (in Gyr) of the Universe to-day (assuming a Hubble parameter H0 =70 km sec

1 Mpc1).The light shaded regions are those allowed (at 2) by theobserved cluster abundance and by current supernova mea-surements of the expansion history. The dark orange shadedregion shows the intersection of the cluster-abundance and su-pernova curves, additionally restricted (at 2) by the locationof the first acoustic peak in the cosmic-microwave-backgroundpower spectrum and quasar-lensing statistics.

Figure shows, w seems to be converging to w = 1.

But what about w < 1? Might the convergence tow = 1 actually be indicating that w < 1? Whyrestrict our attention exclusively to w 1? Matterwith w < 1, dubbed phantom energy [19], has re-ceived increased attention among theorists recently. Itcertainly has some strange properties. For example, theenergy density of phantom energy increases with time.It also violates the dominant-energy condition [20, 21],a cherished notion that helps prohibit time machines

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FIG. 2: Same as in Fig. 1, except extended to w < 1. Here,the blue dot-dash curves show for phantom-energy (w < 1)models the time (in Gyr) remaining in the Universe (assuminga Hubble parameter H0 = 70 km sec

1 Mpc1).

and wormholes. However, it is hard to see how timemachines and wormholes would arise with phantom en-ergy. Although sound waves in quintessence travel atthe speed of light, it does not automatically follow thatdisturbances in phantom energy must propagate fasterthan the speed of light; in fact, there are already sev-eral scalar-field models for phantom energy in which thesound speed is subluminal [19, 22, 23, 24, 25, 26]. It istrue that these models feature unusual kinetic terms intheir Lagrangians, but such terms may arise in supergrav-ity [27] or higher-derivative-gravity theories [28]. Theo-

rists have also discussed stringy phantom energy [29] andbrane-world phantom energy [30]. Connections with thedS/CFT correspondence have also been made [31]. Tobe sure, phantom energy is not something that any the-orist would have expected; on the other hand, not toomany more theorists anticipated a cosmological constant!Given the limitations of our theoretical understanding, itis certainly reasonable to ask what empirical results haveto say.

In Fig. 2 we generalize the analysis of cosmologicalconstraints to a parameter space that extends to w < 1.As indicated here, there is much acceptable parameterspace in regions with w < 1; see also Refs. [32, 33].With certain prior assumptions, the best fit is actuallyat w < 1.

As we now show, if w < 1 persists, then the fateof the Universe is quite fantastic and completely differ-ent than the possibilities previously discussed. To begin,let us review these other fates. In a flat or open Uni-verse without dark energy, the expansion continues for-ever, and the horizon grows more rapidly than the scalefactor; the Universe becomes colder and darker, but withtime the co-moving volume of the observable Universeevolves so that the number of visible galaxies grows. If

the expansion is accelerating, as a consequence of darkenergy with 1 w < 1/3, then the expansion againcontinues forever. However, in this case, the scale factorgrows more rapidly than the horizon. As time progresses,galaxies disappear beyond the horizon, and the Universebecomes increasingly dark. Still, structures that are cur-rently gravitationally bound, such as the Milky Way andperhaps the Local Group, remain unaffected. Thus, al-

though extragalactic astronomy becomes less interesting,Galactic astronomy can continue to thrive.[37]

With phantom energy, the Friedmann equation govern-ing the time t evolution of the scale factor a(t) becomesH2 (a/a)2 = H20 [m/a

3 + (1 m)a3(1+w)], where

H0 is the Hubble parameter, and the dot denotes a timederivative. If m 0.3, then the Universe is alreadydark-energydominated, and for w < 1 it will becomeincreasingly dark-energydominated in the future. Wethus approximate the subsequent evolution of the scalefactor by neglecting the first term on the right-hand side.Doing so, we find that the scale factor blows up in a timetrip t0 (2/3)|1 + w|1H

10 (1 m)

1/2 from the cur-

rent time t0. For example, for w = 3/2 and H0 = 70km sec1 Mpc1, the time remaining before the Universeends in this Big Rip [31] is 22 Gyr.

As in a cosmological-constant Universe, the scale fac-tor grows more rapidly than the Hubble distance H1

and galaxies will begin to disappear beyond the horizon.With phantom energy, the expansion rate H grows withtime, the Hubble distance decreases, and so the disap-pearance of galaxies is accelerated as the horizon closesin on us. More intriguing is that the increase in the dark-energy density will ultimately begin to strip apart gravi-tationally bound objects. According to general relativity,the source for the gravitational potential is the volumeintegral of + 3p. So, for example, a planet in an orbit ofradius R around a star of mass M will become unboundroughly when (4/3)( + 3p)R3 M. With w 1,(+3p) is decreasing with time so if (4/3)(+3p)R3

is smaller than M today, then it will remain so ever after.Thus, any system that is currently gravitationally bound(e.g., the solar system, the Milky Way, the Local Group,galaxy clusters) will herafter remain so.

With phantom energy, ( + 3p) increases, and soat some point in time every gravitationally bound sys-tem will be dissociated. With the time evolution of thescale factor and the scaling of the phantom-energy den-sity with time, we find that a gravitationally-bound sys-tem of mass M and radius R will be stripped at a time

t P

2|1 + 3w|/[6|1 + w|], where P is the period ofa circular orbit around the system at radius R, beforethe Big Rip (see Table I). Interestingly, this time isindependent of H0 and m.

Thus, for example, for w = 3/2, the interval ist 0.3 P before the end of time. In this case, clusterswill be stripped roughly a billion years before the end oftime. In principle, ifw were sufficiently negative, the An-dromeda galaxy would be torn from the Local Group be-fore it could fall into the Milky Way; however, given cur-

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TABLE I: The history and future of the Universe with w =3/2 phantom energy.

Time Event

1043 s Planck era 1036 s InflationFirst Three Minutes Light Elements Formed 105 yr Atoms Formed 1 Gyr First Galaxies Formed 15 Gyr Todaytrip 1 Gyr Erase Galaxy Clusterstrip 60 Myr Destroy Milky Waytrip 3 months Unbind Solar Systemtrip 30 minutes Earth Explodestrip 10

19 s Dissociate Atomstrip = 35 Gyrs Big Rip

rent upper limits to w, this is unlikely. For w = 3/2,the Milky Way will get stripped roughly 60 million yearsbefore the Big Rip. Curiously, when this occurs the hori-zon will still be 70 Mpc, so there may still be other

observable galaxies that we will also see stripped apart(although given the time delay from distant objects, wewill see the Milky Way destroyed first). A few monthsbefore the end of time, the Earth will be ripped from theSun, and 30 minutes before the end the Earth will fallapart. Similar arguments also apply to ob jects bound byelectromagnetic or strong forces. Thus, molecules andthen atoms will be torn apart roughly 1019 secondsbefore the end, and then nuclei and nucleons will getdissociated in the remaining interval. In all likelihood,some new physics (e.g., spontaneous particle productionor extra-dimensional, string, and/or quantum-gravity ef-fects) may kick in before the ultimate singularity, but

probably after the sequence of events outlined above.The end of structure, from cosmic, macroscopic scales

down to the microscopic, leads us to remark that ourpresent epoch is unique from the viewpoint that at noother time are non-linear structures possible. When thephantom energy becomes strong enough, gravitationalinstability no longer works and the Universe becomeshomogeneous. Eventually, individual particles becomeisolated: points separated by a distance greater than3t(1 + w)/( 1 + 3w) at a time trip t cannot communi-

cate before the Big Rip. Therefore, the dominance of thephantom energy signals the end of our brief era of cosmicstructure which began when the non-relativistic matteremerged from the radiation. In such a Universe, certaincosmic questions have new significance. It is natural tofind ourselves or more generally, non-linear structure living close to the onset of acceleration if the struc-ture is soon destroyed and the Universe does not survivemuch longer afterwards [31]. A Big Rip renders the whynow?, or question of cosmic coincidence, irrelevant.

The current data indicate that our Universe is poisedsomewhere near the razor-thin separation between phan-tom energy, cosmological constant, and quintessence. Fu-

ture work, and the longer observations by WMAP, willhelp to determine the nature of the dark energy. In themeantime we are intrigued to learn of this possible newcosmic fate that differs so remarkably from the re-collapseor endless cooling considered before. It will be necessaryto modify the adopted slogan among cosmic futurologists Some say the world will end in fire, Some say in ice[36] for a new fate may await our world.

Acknowledgments RRC thanks the UCSB KITP forhospitality. This work was supported at Caltech byNASA NAG5-9821 and DoE DE-FG03-92-ER40701, atthe KITP by NSF PHY99-07949, and at Dartmouth by

NSF grant PHY-0099543. NNW was supported by a NSFgraduate fellowship.

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