R ESEARCH A RTICLES 8␲G H2 ϭ A Cyclic Model of the Universe 3 1 1 2 ͩ ␾˙ 2 ϩ V ϩ ␤4 ␳ ϩ ␤4 ␳ ͪ (2) Paul J. Steinhardt * and Neil Turok 2 R M a¨ 8␲G We propose a cosmological model in which the universe undergoes an endless ϭϪ a 3 sequence of cosmic epochs that begin with a “bang” and end in a “crunch.” Temperature and density at the transition remain finite. Instead of having an 1 ͩ␾˙ 2 Ϫ V ϩ ␤4 ␳ ϩ ␤4 ␳ ͪ (3) inflationary epoch, each cycle includes a period of slow accelerated expansion R 2 M (as recently observed) followed by contraction that produces the homogeneity, flatness, and energy needed to begin the next cycle. where a dot denotes a derivative with respect to t and H ϵ /a is the Hubble parameter. The The current standard model of cosmology that the universe is flat, rather than closed, equation of motion for ␾ is (1–4) combines the original big bang mod- based on measurements of the cosmic micro- ˙ 3 ␾¨ ϩ 3H␾ ϭϪV,␾ Ϫ ␤ ␾␤ ␳ el and the inflationary scenario. Inflation, a wave background anisotropy and large-scale , M brief period (10Ϫ30 s) of very rapid cosmic structure. (4) acceleration shortly after the big bang, can In our cyclic model, the universe is infi- and the fluid equation of motion for matter explain the homogeneity and isotropy of nite and flat, rather than finite and closed as (M) or radiation (R) is the universe on large scales (Ͼ100 Mpc), in the oscillatory models. We introduce a ␳ץ ␤ ␳ץ d␳ its spatial flatness, the distribution of gal- negative potential energy rather than spatial i ϭ i ϩ i ϭ ␾ץ ␤ ץ aˆ a axies, and the spatial fluctuations in the curvature to cause the reversal from expan- daˆ a ,␾ i Ϫ ͑␳ ϩ ͒ ϭ cosmic microwave background. However, sion to contraction. Before the reversal, 3 i pi , i M,R (5) the standard model has some cracks. The though, the universe undergoes the usual pe- ϵ ␤ ␾ recent discoveries of cosmic acceleration riod of radiation and matter domination, fol- where aˆ a ( ), pi is the pressure of the ␳ and gravitationally self-repulsive dark en- lowed by a long period of accelerated expan- fluid component with energy density i, and ergy (5–8) were not predicted and have no sion [presumably the acceleration that has “,␾”isd/d␾. The implicit assumption is that 2 particular role in the standard model (1–3). been recently detected (5–8)]. The accelerat- matter and radiation couple to ␤ (␾ )g␮␯ Furthermore, the standard model does not ed expansion, caused by dark energy, is nec- (with scale factor aˆ) rather than the Einstein explain the “beginning of time,” the initial essary to dilute the entropy, black holes, and metric g␮␯ alone (or the scale factor a). Note conditions of the universe, or what will other debris produced in the previous cycle so that the radiation term in Eq. 1 is actually ␾ ␳ ϰ Ϫ4 happen in the long-term future. that the universe is returned to its original independent of (because R aˆ ) so only ␳ ␾ Here, we present a cosmological model pristine vacuum state before it begins to con- M enters the equation of motion. with an endless sequence of cycles of ex- tract, bounce, and begin a cycle anew. We assume the potential V(␾ ) has the pansion and contraction. By definition, following features, as illustrated in Fig. 1: (i) there is neither a beginning nor end of time, Essential Ingredients V(␾ ) must approach zero rapidly as ␾3 Ϫϱ; nor is there a need to define initial condi- As in inflationary cosmology, the cyclic sce- (ii) the potential must be negative for inter- tions. In addition, we explain the role of nario can be described in terms of the evolu- mediate ␾; and (iii) as ␾ increases, the po- dark matter and generate the homogeneity, tion of a scalar field ␾ along a potential V(␾ ) tential must rise to a shallow plateau with a flatness, and density fluctuations without in a four-dimensional (4D) quantum field positive value V0. An example of a potential invoking inflation. theory. The essential differences are the form with these properties is The cyclic aspect of the model is reminis- of the potential and the couplings between the Ϫ ␾ V͑␾͒ ϭ V (1Ϫe c ) F(␾) (6) cent of oscillatory models introduced in the scalar field, matter, and radiation. 0 1930s based on a closed universe that under- The analysis of the cyclic model follows where from this point onward we adopt units goes a sequence of expansions, contractions, from the action S that describes gravity, the in which 8␲G ϭ 1. F(␾ ) is a function we and bounces. The oscillatory models were scalar field ␾, and the matter and radiation introduce to ensure that V(␾ )3 0as␾3 constrained by having to pass through a sin- fluids: Ϫϱ. Without loss of generality, we take F(␾ ) gularity in which the energy and temperature to be nearly unity for ␾ to the right of poten- 1 1 ␾͒2 tial minimum. The detailed manner in whichץ͑ diverge. Furthermore, as pointed out by Tol- S ϭ ͵ d4x ͱ Ϫ g ͩ ᏾ Ϫ man (9, 10), entropy produced during one 16␲G 2 it tends to zero at smaller ␾ is not crucial for cycle would add to the entropy produced in the main predictions of the cyclic model. A the next, causing each cycle to be longer than quantitative analysis of this model potential Ϫ V͑␾͒ ϩ ␤4͑␾͒͑␳ ϩ␳͒ͪ (1) the one before it. Extrapolating backward in M R (11) shows that a realistic cosmology can be Ն time, the universe would have to have origi- obtained by choosing c 10 and V0 equal to nated at some finite time in the past so that where g is the determinant of the metric today’s dark-energy density (ϳ6 ϫ 10Ϫ30 3 the problem of explaining the “beginning of g␮␯, G is Newton’s constant, and ᏾ is the g/cm ) in Eq. 6. time” remains. Furthermore, today we know Ricci scalar. The coupling ␤(␾ ) between ␾ We have already mentioned that the cou- ␳ ␳ ␤ ␾ and the matter ( M) and radiation ( R) den- pling ( ) is chosen so that aˆ and, thus, the sities is crucial because it causes the den- matter and radiation density are finite at a ϭ 1Joseph Henry Laboratories, Princeton University, sities to remain finite at the big crunch/big 0. For example, we consider ␤(␾ ) ϳ eϪ␾/͌6 2 Princeton, NJ 08544, USA. Department of Applied bang transition. as ␾3 Ϫϱ. The presence of ␤(␾) and the Mathematics and Theoretical Physics, Center for ␾ Mathematical Sciences, Wilberforce Road, Cambridge The line element for a flat, homogeneous consequent coupling of to nonrelativistic CB3 0WA, UK. universe is Ϫdt2 ϩ a2dx2, where a is the matter represent a modification of Einstein’s *To whom correspondence should be addressed. E- Robertson-Walker scale factor. The equa- theory of general relativity. Because the sca- mail: [email protected] tions of motion following from Eq. 1 are lar field ␾ evolves by an exponentially small 1436 24 MAY 2002 VOL 296 SCIENCE www.sciencemag.org R ESEARCH A RTICLES amount between nucleosynthesis (t ϳ 1s) sometimes called the visible brane. Particles hypothesis (19) that the branes separate after and today (t ϳ 1017 s), the deviations from on the other brane interact only through grav- collision, so the extra dimension immediately standard general relativity are small enough ity with matter on the visible brane and hence reappears. This process cannot be completely to easily satisfy all current cosmological con- behave like dark matter. smooth, because the disappearance of the straints (11). However, the coupling of matter The scalar field ␾ is naturally identified extra dimension is nonadiabatic and leads to to ␾ produces other potentially measurable with the field that determines the distance particle production; that is, the brane colli- effects, including a fifth force that violates between branes. The potential V(␾ )isthe sion is partially inelastic. Preliminary calcu- ␤ the equivalence principle. Provided (ln ),␾ interbrane potential caused by nonperturba- lations of this effect are encouraging, because ϽϽ 10Ϫ3, for today’s value of ␾, these vio- tive virtual exchange of membranes between they indicate that a finite density of particles lations are too small to be detected (11–13). the boundaries. The interbrane force is what is produced (20). The matching condition, We shall assume this to be the case. Hence, causes the branes to repeatedly collide and Eq. 7, parameterizes this effect. Ultimately, a the deviations from general relativity are neg- bounce. At large separation (corresponding to well-controlled string-theoretic calculation ligible today. large ␾ ), the force between the branes should (11, 18, 20) should determine the value of ␹. The final crucial ingredient in the cyclic become small, consistent with the flat plateau model is a matching rule that determines how shown in Fig. 1. Collision corresponds to ␾3 Dark Energy and the Cyclic Model Ϫϱ ϰ ␥␾ to pass from the big crunch to the big bang. But the string coupling gs e , with The role of dark energy in the cyclic scenario ␾3 Ϫϱ ␥Ͼ The transition occurs as and then 0, so gs approaches zero in this limit is novel. In the standard big bang and infla- rebounds toward positive ␾. Motivated again (18). Nonperturbative effects vanish faster tionary models, the recently discovered dark by string theory (see below), we propose that than any power of gs, for example, as or energy and cosmic acceleration (5–8) are an Ϫ1/g 2 Ϫ1/g some small fraction of the ␾-field kinetic e s,ore s, accounting for the prefactor unexpected surprise with no obvious expla- energy is converted to matter and radiation.