Scale invariance, unimodular gravity and dark energy
aa r X i v : . [ h e p - t h ] D ec Scale invariance, unimodular gravity and dark energy
Mikhail Shaposhnikov and Daniel Zenh¨ausern
Institut de Th´eorie des Ph´enom`enes Physiques, ´Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland
Abstract
We demonstrate that the combination of the ideas of unimodular gravity, scale invariance, and the existence of an exactlymassless dilaton leads to the evolution of the universe supported by present observations: inflation in the past, followed by theradiation and matter dominated stages and accelerated expansion at present. All mass scales in this type of theories come fromone and the same source.
Key words: dark energy, non-minimal coupling, unimodular gravity, inflation, Higgs field, Standard Model
PACS:
1. Introduction
The origin of different mass scales in particle physicsis a mystery. The masses of quarks, leptons and inter-mediate vector bosons come from the vacuum expec-tation value (vev) of the Higgs field; the dimensionfulparameters like the QCD scale L QCD or the scales re-lated to the running of all other dimensionless couplingsof the Standard Model (SM) are believed to have noth-ing to do with the Higgs vev. Newton’s gravitationalconstant provides yet another mass scale, very differ-ent from typical particle masses of the SM. The Higgsmass itself – where does it come from?Is it possible that all these mass scales originate fromone and the same source? Indeed, it is not difficultto construct, on the classical level , a theory containinga new singlet field c , which gives masses to all parti-cles and fixes Newton’s constant. Having in mind theSM extended by 3 light right-handed singlet fermions,the n MSM of [1, 2] (this theory – Neutrino Minimal
Email addresses:
[email protected] (Mikhail Shaposhnikov),
[email protected] (Daniel Zenh¨ausern). We will refer to this possibility as “no-scale scenario”.
SM – unlike the SM, can explain neutrino masses andoscillations, dark matter and baryon asymmetry of theuniverse), one can write the Lagrangian realizing thisidea in the following form L n MSM = L SM [ M → ] + L G + ( ¶ m c ) − V ( j , c )+ (cid:0) ¯ N I i g m ¶ m N I − h a I ¯ L a N I ˜ j − f I ¯ N Ic N I c + h . c . (cid:1) , (1)where the first term is the SM Lagrangian without theHiggs potential, N I (I=1,2,3) are the right-handed singletleptons, j and L a ( a = e , m , t ) are the Higgs and leptondoublets respectively, h a I and f I are the matrices ofYukawa coupling constants. The scalar potential is givenby V ( j , c ) = l (cid:16) j † j − a l c (cid:17) + b ( c − c ) , (2)and the gravity part is L G = − (cid:0) x c c + x h j † j (cid:1) R , (3)where R is the scalar curvature. We will only considerpositive values for x c and x h , for which the coefficient This expression is identical to the one in [3], Section 8, but usesdifferent notations.Preprint submitted to Elsevier 14 November 2008 n front of the scalar curvature is positive, whatevervalues the scalar fields take. This is the Lagrangian of“induced gravity” going back to Refs. [4, 5] (see also[6, 7] in the n MSM context).For positive l and b the theory with potential (2)possesses a ground state . It corresponds to the fieldssitting at the minimum of the potential, i.e. c = c , h = h with h = al c and a constant metric de-scribing flat space-time. The field values at the po-tential minimum can be related to the Planck scale as M P = x c c + x h h , M P = . × GeV. Physicsin this theory does not depend on a specific value of c – all dimensionful parameters are proportional to it– and only dimensionless ratios can be measured.Although the aim of having one source for all massscales is achieved by construction of the Lagrangian(1) (we stress that we are still discussing the classicaltheory) the solution is not satisfactory: the absence ofexplicit mass terms for the Higgs field and for singletfermions, and the absence of a gravity scale along withthe introduction of the dimensionful parameter c , re-quired to realize the scenario, are ad hoc and do notfollow from any symmetry principle.The symmetry that forbids (on the classical level)the appearance of any dimensionful parametersis well known – it is the dilatational symmetry.Under dilatations, the scalar and fermionic fieldschange as f ( x ) → s n f ( s x ) ( n = n = / g mn ( x ) → g mn ( s x ) . The action (1) is invariant underthis symmetry, provided c =
0, leading to the absenceof all dimensionful parameters.From now on we will require that a dilatation in-variant theory should possess a ground state. Since thisrequirement is essential for our model, we will furtherdiscuss it in section 4. For the dilatation invariant the-ory to contain massive singlet and doublet fermions,the ground state should be such that c = h = b =
0. Thus,the no-scale scenario can only be realized if b = h − al c =
0, and the theory contains one exactlymassless particle h – a certain mixture of the singlet c and the Higgs field. The requirement b = h appears as a Goldstone boson . By a ground state we mean a constant solution of the equationsof motion including gravity. The existence of such a ground statecould be essential for a consistent quantization of the theory. Spontaneous breaking of the dilatational symmetry also occurs ifthe potential has a flat direction either along c = h = Equivalent arguments were given in [8].The theory (1) with b = : not only is the physics independent of the valueof c = L . So, if we confront it with cosmolog-ical observations, it seems to fail, since the universe isin accelerated expansion, which requires the presenceof dark energy with an equation of state close to that ofthe cosmological constant. This conclusion is certainlycorrect for standard General Relativity (GR), associatedwith the action S E = Z √− gd x L n MSM , (4)where g is the determinant of the metric.The aim of this Letter is to show that the situationis completely different if general relativity in (4) isreplaced by Unimodular Gravity (UG). UG is a verymodest modification of Einstein’s theory: it adds a con-straint g = − g = −
1, adding a constant L to the Einstein-Hilbertaction does not change the equations of motion. Still,the L problem is not solved, since the cosmologicalconstant shows up again, but now as an initial condi-tion for cosmological evolution in UG. We will see thatfor our case of a scale-invariant theory together withUG, the initial conditions lead to a non-trivial run-awayeffective potential for the dilaton rather than to a cos-mological constant, and thus to dynamical dark energy.Moreover, it will turn out that both inflation and accel- a theory with no massive singlet fermions, whereas the second oneis a theory with no electroweak symmetry breaking. We stress that this theory is not invariant under local conformaltransformations. Conformal invariance requires the specific valuesfor x c and x h , x c = x h = − .
2. Scale-invariant unimodular gravity : theclassical theory
In this Letter we want to bring together several apriori separate ideas. One of them is unimodular gravity,which has appeared many times in the literature [9, 10,11, 13, 14, 15, 16]. In unimodular gravity one reducesthe dynamical components of the metric g mn by one,imposing that the metric determinant g ≡ det ( g mn ) takessome fixed constant value. Conventionally one takes | g | =
1, hence the name. Fixing the metric determinantto one is not a strong restriction, in the sense that thefamily of metrics satisfying this requirement can stilldescribe all possible geometries. For pure gravity, thingsare very simple and well known. The analog of theEinstein-Hilbert Lagrangian for unimodular gravity is L EH = − M P ˆ R . (5)Writing quantities with a hat, like ˆ R , we mean thatthey depend on the metric with g = −
1. These quanti-ties transform like tensors under the group of volumepreserving diffeomorphisms, i.e. coordinate transforma-tions x m → x m ( x ) , with the condition ˆ (cid:209) m x m = Do-ing variations of this action that keep the metric deter-minant fixed, since it is not a dynamical variable, yieldsthe equations of motionˆ G mn = − L ˆ g mn , (6) In principle one can fix g = a ( x ) , where a ( x ) is a fixed externalfield, and the results are the same. It is important to distinguish UG from theories constructed on thesimple requirement of invariance under restricted coordinate trans-formations x m → x m ( x ) , with ˆ (cid:209) m x m = where L is an integration constant given by initial con-ditions. Now, these are also the equations for standardEinstein gravity with an added cosmological constant,for a choice of coordinates such that the metric determi-nant is equal to one, which is always possible [9]. There-fore, the two theories are classically equivalent, exceptthat in the standard theory the cosmological constant ap-pears in the action, whereas in unimodular gravity it isan integration constant. It has been shown [9, 13, 14, 15]that if one adds a matter sector that couples minimallyto gravity, and therefore has a covariantly conservedenergy-momentum tensor (cid:209) m T mn =
0, the applicationof UG also results in the appearance of an integrationconstant that plays the role of an additional cosmolog-ical constant. We now want to find a similar statementfor a more general case, in particular the one in whichNewton’s constant is generated dynamically.The action for unimodular gravity and any otherfields, which couple to gravity in an arbitrary way, hasthe following functional dependence: S = Z d x L ( ˆ g mn , ¶ ˆ g mn , F , ¶ F ) , (7)where F stands for all non-gravitational fields. If wewant to derive the equations of motion for this theory,we have to vary the action keeping the constraint on thedeterminant. This is done using the Lagrange multipliermethod. We add an additional variable, whose equationof motion will be the constraint. So, the following La-grangian is equivalent to the former one:˜ L = √− g (cid:16) L ( g mn , ¶ g mn , F , ¶ F ) + L ( x ) (cid:17)| {z } A − L ( x ) | {z } B . (8)Here, apart from the usual symmetry requirement g mn = g nm , g mn is unconstrained (the initial Lagrangianwas multiplied by a factor √− g , which does not changethe theory because of the unimodular constraint).The equations of motion are d A d g mn = , (9) d A d F = , (10) d ( A + B ) d L = = ( √− g − ) . (11)We observe that R d x A ( x ) is invariant under the fullgroup of diffeomorphisms. The infinitesimal transfor-mations are3 mn → g mn + d x g mn , F → F + d x F , L → L + d x L , (12)where d x depends on the nature of the fields, i.e. scalar,vector, etc. If, for instance, we take F to be a scalarfield, the d x ’s are given by d x g mn = (cid:209) m x n + (cid:209) n x m , d x F = ¶ m F x m , d x L = ¶ m L x m . (13)Due to this symmetry, the following relation holds. Z d x (cid:16) d A d g mn d x g mn + d A d F d x F + d A d L d x L (cid:17) = . (14)The coefficients of the first two terms are zero becauseof the equations of motion and the last coefficient yields d A d L = √− g . The equation reduces to Z d x √− g ( ¶ m L ) x m = . (15)Since this holds for all possible functions x m ( x ) , we canconclude that ¶ m L ( x ) = , (16)and hence that L is a constant of motion. Its value canbe determined by the field equations together with theinitial conditions for all fields. Knowing this, let us againlook at the equations (9) d A d g mn = d {√− g (cid:16) L ( g mn , ¶ g mn , F , ¶ F ) + L ( x ) (cid:17) } d g mn = . These equations along with the constraint √− g = L is an integration con-stant. We conclude that the theory given by (7) is clas-sically equivalent to a fully diffeomorphism invarianttheory described by the Lagrangian L diff = √− g (cid:16) L ( g mn , ¶ g mn , F , ¶ F ) + L (cid:17) , (17)apart from the different ways in which the parameter L appears . The quantity L plays the role of a cosmolog-ical constant in the theory with explicit Planck mass.However, as we will see shortly, this is not the case ifNewton’s constant is induced dynamically.We now want to combine the ideas of UG and scaleinvariance. Considering only the gravitational and the In [12] the authors presented a proof of the same statement forTDiff theories a using similar type of arguments. However, to ourunderstanding, this proof is in fact not valid for general TDifftheories. scalar sectors, a general Lagrangian containing scalarfields f i has the form: L = − K i j f i f j ˆ R +
12 ˆ g mn ¶ m f i ¶ n f i − U i jkl f i f j f k f l . (18)The result derived above tells us that the solutions ofUG with this Lagrangian are equivalent to the solutionsof GR with Lagrangian L = √− g (cid:16) − K i j f i f j R + (cid:229) i g mn ¶ m f i ¶ n f i − U i jkl f i f j f k f l − L (cid:17) . (19)Let us finally add to unimodular gravity and scale in-variance the requirement that the scalar potential shouldhave a flat direction. The potential for a theory contain-ing the Higgs field h and an additional scalar field c isthen given by V ( h , c ) = l (cid:16) h − al c (cid:17) . (20)So, our requirements lead us to the scalar and gravita-tional parts of the action (1) with b = L = √− g (cid:16) − ( x c c + x h h ) R + (cid:0) ¶ m c (cid:1) + (cid:0) ¶ m h (cid:1) − V ( h , c ) − L (cid:17) . (21)Now, in order to facilitate the physical interpretationof this Lagrangian, we can do a change of variables(conformal or Weyl transformation) of the followingtype g mn = W ( x ) ˜ g mn . (22)If we choose W such that ( x c c + x h h ) W = M P , theaction (21) in terms of the new metric ˜ g mn reads L E = p − ˜ g (cid:18) − M P ˜ R + K − U E ( h , c ) (cid:19) , (23)and is said to be in the Einstein frame (see e.g. [19]).Here K is a complicated non-linear kinetic term for thescalar fields, given by K = W (cid:18) ( ¶ m c ) + ( ¶ m h ) (cid:19) − M P ( ¶ m W ) . (24)In our case where x c , x h >
0, the kinetic form K is positive-definite, which guarantees the absence ofghosts. The Einstein-frame potential U E ( h , c ) is givenby U E ( h , c ) = M P ( x c c + x h h ) [ V ( h , c ) + L ] , (25)4here the parameter L is related to initial conditions forscalar fields and gravity and does not depend on space-time coordinates. It is not a cosmological constant butrather the strength of a peculiar potential.
3. Dark energy, inflation and cosmological constant
We now want to analyze the cosmological conse-quences of the theory (1), working in the Einstein frame.Although the dynamics of the general system describedby (23) is very complicated due to the non-canonicalform of the kinetic term K , we can gain some insightinto the evolution of the system looking at the potentialpart only.The phenomenologically interesting domain of pa-rameters, explained below, corresponds to x h ≫ x c ≪ l ∼ a ≃ lx c v M P ≪ v ≃
250 GeVis the Higgs vev). For this case the kinetic mixing oftwo fields in K is indeed not essential. The potential U E ( h , c ) for L = , L > L < L = h = ± p a / lc , corresponding to the exact zero mode.As soon as L = L > L < c and h andhas a singularity at the origin.For 0 < L < ∼ M P a typical behavior of the scalar fieldsis as follows. Like in the chaotic inflation scenario [21],it is expected that initially both fields are displaced fromtheir ground state values and are generically larger thanthe Planck scale: the first term in (25) dominates. More-over, by assumption, x h ≫ x c , meaning that for c ∼ h the dynamics is mainly driven by the Higgs field, mov-ing the system towards the valley. This corresponds toinflation due to the Higgs field, suggested in [22]. Whenthe value of the Higgs field becomes of the order ofthe Planck scale, it is trapped by the valley and oscil-lates there, producing particles of the SM (this processis studied in detail in [23]). The correct spectrum of per-turbations is generated if x h ∼ T rh ∼ GeV. This part ofthe evolution is quite similar to the hybrid inflation sce-nario [24]. The later evolution of the universe dependscrucially on the sign of L , which is defined by initial For the system containing just one scalar field the field trans-formation leading to a canonically normalized kinetic term can befound easily (see, e.g. [20]). For multiple fields we did not manageto find the required transformation. conditions in UG. We would expect that with 50% prob-ability the universe was born in the state with L >
0. Inthis case it will evolve along the valley towards a statewith c , h = ¥ with zero cosmological constant. At anyfinite evolution time the universe must contain dark en-ergy.Present cosmological observations allow to pin downthe value of the non-minimal coupling of the field c .For the late time evolution and a ≪ ⇒ h ≪ c ) thedilaton field h with (almost) canonical kinetic term isrelated to c as c = M P exp (cid:18) gh M P (cid:19) , g = q + x c . (26)Its dynamics is practically decoupled from the dynamicsof the Higgs field (the deviation of it from the vev will bedenoted by f in the Einstein frame). The correspondingequations of motion have the form¨ h + H ˙ h + dU h d h = I h , (27)¨ f + H ˙ f + m h f = I f , (28)where m h is the Higgs mass and U h = Lx c exp (cid:18) − gh M P (cid:19) . (29)The source terms I h , f originate from the kinetic mixing K in eq. (24), I h (cid:181) x h (cid:18) ax h lx c (cid:19) (cid:0) ¨ f + H ˙ f (cid:1) , I f (cid:181) (cid:18) ax h lx c (cid:19) (cid:18) ¨ h + H ˙ h − g M P ˙ h (cid:19) , (30)and can be safely neglected. The Hubble constant H isgiven by the standard expression H = M P (cid:18)
12 ˙ h + U h + C g a + C M a (cid:19) , (31)where the two last terms correspond to radiation andmatter contributions to the energy density, and a is thescale factor. It is amazing that here the exponential po-tential, proposed for a quintessence field a long timeago in [25, 26, 27], appears automatically, though with L not being a fundamental parameter but rather a ran-dom initial condition.The dynamics of the universe described by eq. (27)with I h = g > √ iggs DilatonU E Higgs Higgs DilatonU E Higgs Higgs DilatonU E Higgs
Fig. 1. Potential for the Higgs field and dilaton in the Einstein frame. Left: L =
0, middle: L >
0, right L < to scaling solutions. In that case the energy density ofthe scalar field eventually scales like the dominant com-ponent of the universe. Therefore, those models cannotdescribe accelerated expansion. The situation is differ-ent for g < √
3: the scalar component changes slowerthan radiation and matter and eventually starts domi-nating. If g is in this region, the dynamics of h is thatof a ”thawing” quintessence field [29, 30]. In that sce-nario the scalar field at early times is nearly constantand has w ≃ −
1. When the Hubble friction gets weaker,the field starts rolling down the exponential potential.At the same time w grows and moves away, althoughextremely slowly, from w = − x c . The mostconvenient for us is the result of [31], which gives therelationship between the parameter w of the equationof W c , valid for the thawing scenario realized by anexponential potential with g < √
3. It reads [31]:1 + w = g " p W c − (cid:18) W c − (cid:19) log 1 + p W c − p W c . (32)Taking the results of WMAP [32] for the equation ofstate − . < + w < .
12 and identifying W c with thedark energy abundance W DM ≃ .
73, we find that thevalue of x c must be in the interval0 < x c < . . (33)Hence, g . < √ g is indeed in thecorrect parameter region for the thawing scenario.As for the value of L , it cannot be determined un-ambiguously, since the value of the dilaton field is un-known. For typical values appearing in run-away scenar-ios, h ∼ M P log ( tM P ) / p x c [25, 26] ( t is the presentage of the universe), the initial value of L can be aslarge as M P .It has been shown in [26, 30], that in this model (for L >
0) the universe becomes dark energy dominated atlate times, i.e. W c →
1. In this limit the parameter of the equation of state becomes w → g − a ( t ) (cid:181) t / g .An important comment is now in order. The changeof h with time does not lead to any visible time varia-tion of Newton’s constant or of particle masses. In theEinstein frame the dilaton practically decouples fromall the fields of the SM, and the amplitude of the time-dependent corrections to masses, from eq. (28), is ofthe order of f v ∼ I f m h v , (34)which is too small to be tested in any observations.To conclude, the cosmological evolution of a classi-cal scale-invariant theory with unimodular gravity andan exactly massless dilaton typically leads to initial in-flation (large x h ≫ x c ≪
4. Quantum theory
The analysis of the two previous sections was entirelyclassical. Therefore, we will formulate the conditions,which should be satisfied for the results to be valid inthe quantum case as well. Since the theory (21) is notrenormalizable, the discussion in this section will be onthe level of wishful thinking and does not pretend toany rigor (see, however, [33, 34]).As the dilatational symmetry of the theory is easierto see in the Jordan frame, we will use it for the presentdiscussion, which in a number of respects resembles theone in [25, 26, 35]. Clearly, the classical results surviveif the dilatational symmetry remains exact on the quan-tum level and if the dilaton is still massless in full quan-tum field theory. Like in the classical case, the exactdilaton degeneracy of the ground state will guarantee6he absence of the cosmological constant, whereas theunimodular character of gravity would induce, throughinitial conditions, the run-away behavior (for L >
0) ofthe dilaton field at a late time in the expansion of theuniverse. If true, all dimensionful parameters of the SM,including those coming from dimensional transmuta-tion like L QCD will change in the same way during therun-away of the dilaton field. The deviation of dimen-sionless ratios (only they are relevant for physics in ascale-invariant theory) from constants, due to the cos-mological evolution, will be strongly suppressed as in(34) and thus be invisible. The dilaton will only havederivative couplings to the fields of the n MSM, being aGoldstone boson related to the spontaneous breaking ofdilatational invariance and thus evade all the constraints[25, 26, 35] considered for the Brans-Dicke field [36].The required dependence of low-energy parameterssuch as L QCD on the dilaton field would appear if thefollowing strategy is applied to the computation of ra-diative corrections in the constant c , h backgrounds[33, 34] (see also [25, 26]). Use the field-dependent cut-off Q , related to the effective Planck scale in the Jor-dan frame as Q = x c c + x h h , and assume that thevalues of all dimensionless couplings at this scale donot depend on Q . With this prescription the non-trivialdimensionful parameters, appearing as a result of therenormalization procedure, would acquire the necessarydependence on the dilaton field. It is this prescription,which was effectively used in [22] to discuss radiativecorrections to the Higgs-inflaton potential. Note that therenormalization procedure of [37] is completely differ-ent, and there is no surprise that the results and conclu-sions of our present work and of [22] are not the sameas those of [37].The requirement that the dilaton remains exactlymassless on the quantum level, or in other words thatthe quantum effective potential has a flat direction (inclassical theory this corresponds to b =
0) is crucialfor our findings. It is highly non-trivial as it is not aconsequence of scale symmetry. Still, the quantum fieldtheories constructed near the scale-invariant groundstate h h i = h c i = h h i = h c i = (for a review see e.g. [40]). In other words, it does nothave particles at all and thus cannot be accepted as arealistic theory. On the contrary, if the quantum scaleinvariance is spontaneously broken, the theory does If it does, the exact propagators coincide with the free ones andthe theory is likely to be trivial in this case [38, 39]. describe particles and thus may be phenomenologicallyrelevant. These considerations single out the theorieswith degenerate ground state, corresponding to b = .In the cosmological setup the relevance of these ar-guments is not obvious. Indeed, suppose that the flatdirection is lifted, i.e. b > L >
0. A non-zero value of b leads to a positive vacuumenergy in the Einstein frame, E vac ∼ b M P . As in Section3, the scalar field h has a run-away behavior, leading tothe breaking of scale invariance, whereas the universeis expanding exponentially with the Hubble constantdetermined by E vac . Certainly, there is nothing wrongwith the classical solution of this type, except the factthat it is unstable against scalar field fluctuations whichgrow as in the inflationary stage [41]. Whether this pic-ture survives quantum-mechanically, is an open issue.If it does, and if the notion of particles can be defined,we will get a universe with non-zero cosmological con-stant, loosing thus an explanation of its absence. How-ever, the classical picture may happen to be mislead-ing. Indeed, in the full quantum field theory, any stateincluding cosmological solutions, can be considered asa superposition of quantum excitations above the vac-uum state. As no particles can be defined for the scale-invariant theory with scale-invariant ground state , wewould expect that the effective particle-like excitationsabout the classical background can exist only for a fi-nite time, presumably of the order of the inverse Hubbleconstant. If true, the requirement that the scale-invariantquantum theory must be able to describe particles againsingles out the case corresponding to b =
5. Conclusions
In this Letter we showed that the scale-invariant clas-sical SM and the n MSM coupled to unimodular gravityhave all necessary ingredients to describe the evolutionof the universe, including early inflation and late ac-celeration. The requirement of scale invariance and ofthe existence of a massless dilaton leads to a theory inwhich all mass scales, including that of gravity, orig- When quantum effects are included, the flat direction is not neces-sary associated with b =
0. Nevertheless, to simplify the discussion,we will refer to the case with spontaneously broken scale invarianceon the quantum level still quoting the classical value b = The case when particles can be defined corresponds to free fieldtheory, see footnote 10. Then the Lagrangian (1) is simply a verycomplicated way to describe this trivial theory, and the classicalcosmological solution has nothing to do with the exact quantumsolution. b = w for dark energy mustbe different from that of the cosmological constant, butalso that w > −
1, adding an extra cosmological test thatcould rule out the n MSM.
Acknowledgements.
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