Cosmological Coincidence without Fine Tuning
CCosmological Coincidence without Fine Tuning
Joohan Lee ∗ and James M. Overduin † Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21210, USA
Tae Hoon Lee ‡ Department of Physics, Soongsil University, Seoul 156-743 Korea
Phillial Oh § Department of Physics and Institute of Basic Science,Sungkyunkwan University, Suwon 440-746 Korea (Dated: September 22, 2018)We present a simple cosmological model in which a single, non-minimally coupled scalar field witha quartic potential is responsible for both inflation at early times and acceleration at late times.Little or no fine tuning is needed to explain why the present density of dark energy is comparableto that of pressureless matter. Dark energy is identified with the potential of the scalar field, whichis sourced by the trace of the energy-momentum tensor. This becomes significant when matterhas decoupled from radiation and become fully non-relativistic, so that φ ∝ ρ / m ∝ ρ / m, ( a /a ) ∼ (10 − ) / (10 ) ∼ − and V ∼ φ ∼ − in Planck units, as observed. PACS numbers: 11.30.Pb, 11.30.Qc, 12.60.Jv, 14.80.Hv
I. INTRODUCTION
Both early inflation and the recent accelerated expan-sion of our universe can be explained using scalar fieldswith very flat potentials, raising hopes that the samescalar might be responsible for both. However, the en-ergy densities involved differ vastly, by some 120 orders ofmagnitude. It is a challenge to build a model which caninterpolate smoothly between the two regimes withoutfine-tuning, and without spoiling the success of the stan-dard big bang model. There have been several attemptsto construct a unified theory of inflation and dark energy[1–10], in some cases also including dark energy and/orthe Higgs field in the same framework [11, 12]. A recentreview of what has come to be known as “quintessentialinflation” has been given in Ref. [13].In this paper we present one such model based on gen-eral relativity and ideas from induced-gravity (IG) theory[14, 15], in which gravitation is regarded as arising fromquantum processes within ordinary field theory. One-loop effects in scalar-field theories produce a term corre-sponding to the non-minimal coupling of the scalar fieldto the scalar curvature of the spacetime. In IG theorythis term is taken as the action for gravity. Here, weretain the standard Einstein-Hilbert action and regardthe IG coupling term as an additional part of the ac- ∗ Electronic address: [email protected]; on leave of absencefrom Department of Physics, University of Seoul, Seoul 130-743Korea † Electronic address: [email protected]; also at Department ofPhysics and Astronomy, Johns Hopkins University, Baltimore, MD21218, USA ‡ Electronic address: [email protected] § Electronic address: [email protected] tion. However, we insist that no dimensionful parame-ters other than the Planck mass be introduced. Sec. IIdescribes the model and discusses the general behavior ofthe scalar field in the cosmological context. In the Ein-stein frame the shape of the potential is such that thescalar field initially rolls slowly with energy of order onein Planck units, and later decays rapidly into relativisticparticles, reheating the universe. After reheating is overthe scalar field, having lost all of its energy, is assumedto stay near the ground state.As massive particle species such as baryons or WIMPsdrop out of equilibrium and become non-relativistic, thescalar field begins to grow until balanced by a restoringforce due to the potential gradient. A straightforward nu-merical estimate shows that the potential energy storedin the scalar field at this time is within a few orders ofmagnitude of the present-day density of pressureless mat-ter. The coincidence problem is thus solved if the valueof the scalar field generated at that time remained almostconstant until now. We give an argument showing thatthis is indeed possible. These aspects of the model arediscussed in Sec. III. In Sec. IV we compare our modelto others in the literature, and outline some of the waysin which it can be tested in future work.
II. THE MODEL
Our Lagrangian resembles that of pure IG theory [15]but also retains a role for the Planck mass (as in standardgeneral relativity): L = √− g (cid:20) ( c M P + c M P φ + c φ ) R − K ( φ ) g µν ∂ µ φ∂ ν φ − λ φ (cid:21) . (1) a r X i v : . [ h e p - t h ] N ov We have included K ( φ ) in the kinetic term for generality.A non-canonical form for kinetic energy is characteristicof k-essence cosmology [16, 17], and our choice can beregarded as its low energy limit. In the context of asingle-field model, it can be transformed into the canon-ical form by a redefinition of the scalar field.Note that the c term breaks the symmetry φ → − φ .We rewrite Eq. (1) as L = √− g (cid:26) M P (cid:2) ( φ/M P − α ) + β (cid:3) R − K ( φ ) g µν ∂ µ φ∂ ν φ − λ φ (cid:27) , (2)ensuring that the coefficient in front of the Ricci scalaris always positive. Without loss of generality we alsoassume that α is positive.When the field is large compared to the Planck massone can show that the scale symmetry is restored. Thismeans that the field can be almost stationary at any largevalue. For small values of the field, on the other hand,the scaling symmetry is broken and φ = 0 becomes theunique stable equilibrium configuration. We wish to re-cover standard general relativity in the vicinity of thevacuum, so we require α + β = 1.The dynamics of the the scalar field can be describedmore easily in the Einstein frame. For this purpose wemake a conformal transformation g µν → χ − g µν where χ ( φ ) ≡ ( φ/M P − α ) + β . (3)Then the Lagrangian (2) becomes L = √− g (cid:20) M P R − K E ( φ ) g µν ∂ µ φ∂ ν φ − V E ( φ ) (cid:21) , (4)where K E ( φ ) ≡ K + 6 M P χ (cid:48) χ , (5) V E ( φ ) ≡ λ φ χ − ( φ ) . (6)Apart from a non-canonical kinetic term, this is just theLagrangian for a scalar field with minimal coupling toEinstein gravity and a dilatonic coupling to matter. Theform of the potential is illustrated in Fig. 1. For large neg-ative values of φ the potential becomes flat and asymptot-ically approaches λM P , which is of order one in Planckunits. For smaller negative values the potential dropsrapidly to zero. Near this minimum, the potential is ap-proximately quartic. For positive values of φ the poten-tial has a bump around φ = α − M P whose height is β − times the asymptotic value.Assuming that the scalar field starts with a large nega-tive value, we can divide its evolution into three regimes:(i) φ (cid:28) − M P , (ii) φ ∼ − M P , and (iii) − M P (cid:28) φ < φ ,its potential energy will quickly dominate over kinetic FIG. 1: The effective potential V E ( φ ) in the Einstein framefor several values of α (with φ in Planck units and λ = 1). and other forms of energy and the universe will undergoalmost exponential expansion during regime (i), repre-senting inflation. Similar ideas have been discussed inthe context of “induced-gravity inflation” [18].After sufficient inflation when the scalar field is of theorder of the Planck mass, it will rapidly decay due to thesteep slope of the potential, reheating the universe bytransferring its energy to other forms of matter and ush-ering in the radiation-dominated era. The details of thisand the resulting state of the scalar field are necessarilymodel-dependent [19–21]. In the scheme of Ref. [20], thescalar field decays almost completely, while in other casesit comes into thermal equilibrium with standard-modelfields (as in tracking-type models [22]). In this work wewill simply assume that after reheating the scalar fieldsettles very close to the the minimum of the potential,and concentrate on the behavior of φ in regime (iii) inorder to discuss the cosmological coincidence problem. III. A MECHANISM FOR COSMOLOGICALCOINCIDENCE
The scalar field in this model is sourced by the traceof the energy-momentum tensor, ρ − p . Thus it beginsto grow again as the universe transitions from radiationto matter dominance. This can be seen from the fieldequations, which read as follows (for a flat homogeneousand isotropic metric): M P H = 12 K E ˙ φ + V E + ρ, (7) K E ( ¨ φ + 3 H ˙ φ ) + 12 dK E dφ ˙ φ = − dV E dφ + χ (cid:48) χ ( ρ − p ) , (8)where primes denote derivatives with respect to φ , and ρ and p refer to the energy density and pressure of ad-ditional matter in the Einstein frame, which is not con-served due to conformal coupling of the scalar field. Inprinciple this conformal coupling can lead to severe ex-perimental constraints. Here, however, its effects are un-detectable during the epoch relevant to our main dis-cussion because the value of the scalar field is extremelysmall, as shown below.In what follows, we make a specific ansatz for the ki-netic energy factor in the neighborhood of φ = 0, K ( φ ) ≡ ξ M P φ , (9)where the Planck mass is inserted on dimensionalgrounds. This form of kinetic energy factor is introducedto slow the scalar as it rolls toward zero. In fact, it neveractually reaches zero or becomes positive. This is bestseen using a new field variable σ ≡ − M P √ ξ log( − φ/M P ),in terms of which the model goes over (for small φ ) to onewith a minimal coupling to gravity, conformal couplingto matter, and an exponential potential, V ( σ ) ∼ e − σ .The field value φ = 0 corresponds to σ = ∞ .In the vicinity of the minimum of the potential,Eqs. (7) and (8) can then be approximated as follows,keeping only the leading terms: M P H = ξ M P (cid:32) ˙ φφ (cid:33) + λ φ + ρ, (10) ξ M P φ ddt (cid:32) ˙ φφ (cid:33) = − ξ M P φ H (cid:32) ˙ φφ (cid:33) − ddφ ( λ φ ) − αM P ( ρ − p ) . (11)A few remarks are in order. First, at early times whenall matter was relativistic the last term on the right-handside of Eq. (11) did not directly contribute to the theexpectation value of the scalar field. Second, the frictionterm becomes large as φ approaches zero, so that thescalar field remains near this minimum for a long time.However, as the universe cools and matter decouples fromradiation, the scalar field begins to feel a force due to thegrowing density of non-relativistic matter. It then beginsto be pulled back toward negative values. We assumethat the magnitude of the scalar field reaches a maximumvalue (denoted φ ∗ ) when the potential gradient balancesthe force due to matter. At this point, its velocity iszero, the acceleration term is small, and matter is fullynon-relativistic. (The same results follow as long as theratio of the acceleration term to friction approaches aconstant, as in Ref. [23].)The value of φ ∗ can be estimated using Eq. (11): − λφ ∗ = αM P ρ m, ∗ = αM P ρ m, (cid:18) a a ∗ (cid:19) , (12)where ρ m denotes pressureless matter and the subscripts0 and ∗ denote respectively the present time and thetime when the above balance is reached. The potentialenergy of the scalar field at this time is V ∗ = ( λ/ φ ∗ from Eq. (6), where χ ( φ ) ≈ φ (cid:28) M P .The ratio of this energy relative to the present density ofmatter is then given by V ∗ ρ m, = λ (cid:16) αλ (cid:17) (cid:18) ρ m, M P (cid:19) (cid:18) a a ∗ (cid:19) . (13) To estimate this quantity, we note that ρ m, /M P =24 π (cid:126) G Ω m H /c = 3 × − in physical units (with M P the reduced Planck mass), where we have used WMAPvalues for Ω m and H [24]. If we restrict our attention tomassive particle species originally in thermal equilibriumwith radiation, then a /a ∗ ≈ T ∗ /T ), where the CMBtemperature kT = 0 . T ∗ is determined by theparticle mass m . (As these particles freeze out of equi-librium, they deposit energy into the cosmic plasma sothat product aT does not remain constant [25].) Parti-cles begin to freeze out when kT ∼ mc , but become fullynon-relativistic only when kT ∼ mc /
30 ([26], Fig. 5.1).We thus take a a ∗ ≈ T ∗ T ∼ mc kT = 4 × (14)for baryons in particular. If, however, non-relativisticmatter is dominated by thermal cold dark-matter relics(which also dropped out of equilibrium with radiation),then the scalar field would be sourced primarily by them.There have been recent hints from multiple direct detec-tion experiments that the CDM might consist of super-symmetric WIMPs with m ∼
10 GeV /c [27]; in this caseEq. (14) gives instead a /a ∗ ∼ × .Another strong CDM candidate is the axion, with m ∼ − GeV /c . These particles, however, are non-thermal (they were never in equilibrium with radiation).They are “semi-relativistic” at birth near the QCD phasetransition (i.e., when kT ∼ Λ QCD ≈
150 MeV) but rapidlysettle down to form a zero-momentum condensate as kT (cid:28) Λ QCD ([26], Ch. 10). (There are also thermal axionmodels, but these have been ruled out by astrophysicaland cosmological bounds [28].) Following the same rea-soning as above, we thus take for these particles a a ∗ ∼
110 Λ
QCD kT = 6 × . (15)Eq. (13) with these numerical results implies V ∗ ρ m, ∼ λ (cid:16) αλ (cid:17) ×
200 (axions)4 × (baryons)4 × (WIMPs) . (16)In order to ensure that the potential energy of the scalarfield has remained constant or decayed only slowly since t ∗ consider that during this period the potential gradientis balanced by the Hubble friction so that ξ M P φ H ˙ φ = − ddφ ( λ φ ) = − λφ . (17)From this we can estimate the ratio of kinetic to potentialenergy as(1 / ξM P ( ˙ φ/φ ) ( λ/ φ = λ φ / ξM P H ( λ/ φ = 83 ξ ( λ/ φ H M P (cid:28) . (18)This ratio remains small throughout the subsequent evo-lution of the scalar field, as required.It is thus natural to identify V ∗ with the present-daydark-energy density ρ Λ , . Eq. (16) then shows that lit-tle or no fine tuning of the model parameters α and λ is required to explain the cosmological coincidence that( ρ Λ , /ρ m, ) obsd ≈
3. Taking α ∼ /λ for convenience,we find that α ∼ − , 10 − and 10 − , assuming matterdominated by axions, baryons and WIMPs respectively.(These values could be even closer to unity if the po-tential energy of the scalar field has in fact decreasedslightly since t ∗ .) The implied degree of tuning is negli-gible for axions but possibly significant for WIMPs, sug-gesting that in the context of this model at least, axionsare to be preferred as CDM candidates. IV. SUMMARY AND DISCUSSION
We have constructed a cosmological model withoutsmall parameters in which a single scalar field may beresponsible both for inflation at early times and accel-eration at late ones. We have concentrated here on thepotential of this model to provide a natural explanationfor the present density of the dark energy. Little or nofine tuning appears to be required in either the model pa-rameters or initial conditions, possibly providing a newway to address the coincidence problem.The main argument can be summarized as follows. Asthe universe cools, a growing fraction of its relativisticmatter content gradually becomes non-relativistic. Thispressureless matter exerts a force on the scalar field (lo-cated close to zero at this time), pulling it back towardnegative values until balanced by the potential gradi-ent. By estimating the temperature at which matter(dominated by baryons or cold dark matter in the formof WIMPs or axions) became fully non-relativistic, wehave shown that the potential energy associated withthe scalar field at this time is of order 10 − in Planckunits. This energy changes very little during the sub-sequent evolution of the universe, so it can explain thedensity of dark energy measured today.Our results suggest a new way of looking at dark en-ergy. This picture is different from that in quintessencetheory [29], where a scalar field slowly rolls indefinitelytoward zero, eventually becoming dominant over matter.In these models, some fine tuning in both model param-eters and initial conditions is generally required to ex-plain why the dark-energy density is comparable to thatof matter at the present time. The underlying mecha-nism here is similar in some respects to one in Ref. [30],where non-relativistic matter also “pulls” a scalar fieldout of the ground state at late times. However, our modeldiffers in motivation, in the behavior of the scalar field,and in the degree of tuning required. The scalar fieldin our model may be responsible for inflation as well asacceleration. (It is difficult to satisfy the slow-roll condi- tions for inflation with a canonical kinetic term for thescalar field, as in Ref. [30].) The potential in our modelis everywhere smooth and finite (in the Einstein frame),while that in Ref. [30] becomes singular when the fieldreaches zero. The detailed behavior of the scalar fieldaround the ground state is also different, which affectsthe dynamics both during inflation and at late times.In Ref. [30], the growth of the scalar field at late timesis tied to the epoch at which matter begins to dominateover radiation, rather than the epoch at which it becomesnon-relativistic. Because of the sensitive dependence ofthe value of the scalar potential on this choice of epoch,the two models lead to quite different results.It would be of great interest to test models of this kindvia the power spectrum of large-scale structure [22] orthe magnitude-redshift relation for Type Ia supernovae.The latter requires solutions of the field equations (10)and (11) for the scale factor a over the redshift range0 (cid:54) z (cid:46)
2. We hope to report on constraints of this kindin the near future [31].To test the inflationary aspects of the model, a numer-ical phase-space analysis of the dynamical system wouldbe valuable, varying the scalar field parameters and eval-uating the stability of solutions with respect to the initialconditions. Such a test would however depend stronglyon specific assumptions about the nature of the reheatingprocess, as well as the identity of dark matter.Other tests could come from the spectral index andtensor-to-scalar ratio of CMB fluctuations [32]. We em-phasize that the λφ -term in the potential does not meanthat inflation is of the standard chaotic type, where λ isconstrained by current CMB data to be less than ∼ − [33]. As discussed above, the φ part of the action isrelevant only for small values of the scalar field, whereasinflation takes place when − φ is large and the potential isdominated by the other terms in Eq. (1) and is nearly flat.In this sense our model more closely resembles “new”inflation, and the strong observational constraints thathave been obtained on φ -type chaotic inflation modelsin particular do not apply. Acknowledgments