Scale Invariance, Mass and Cosmology 9 E.I. Guendelman 9 9 Physics Department, Ben-Gurion University, Beer-Sheva 84105, Israel 1 n February 7, 2008 a J 4 2 1 v Abstract 7 6 The possibility of mass in the context of scale-invariant, generally 0 1 covariant theories, is discussed. The realizations of scale invariance 0 whichareconsidered,areinthecontextofagravitationaltheorywhere 9 the action, in the first order formalism, is of the form S = L Φd4x 9 1 c/ + L2√−gd4x where Φ is a density built out of degrees ofRfreedom q independent of gravity, which we call the ”measure fields”. For global R - scale invariance, a ”dilaton” φ has to be introduced, with non-trivial r g potentials V(φ) = f eαφ in L and U(φ) = f e2αφ in L . This leads 1 1 2 2 : v to non-trivial mass generation and potential for φ. Mass terms for Xi an arbitrary matter field can appear in a scale invariant form both in L and in L where they are coupled to different exponentials of the r 1 2 a field φ. Implications of these results for cosmology having in mind in particular inflationary scenarios, models of the late universe and modified gravitational theories are discussed. 1 Introduction The concept of scale invariance appears as an attractive possibility for a fundamental symmetry of nature. In its most naive realizations, such a symmetry is not a viable symmetry, however, since nature seems to have chosen some typical scales. 1 Here we will find that scale invariance can nevertheless be incorporated into realistic, generally covariant field theories. However, scale invariance has to be discussed in a more general framework than that of standard generally relativistic theories, where we must allow in the action, in addition to the ordinary measure of integration √ gd4x, another one1, Φd4x, where Φ is a − density built out of degrees of freedom independent of that of g . To achieve µν global scale invariance, also a ”dilaton” φ has to be introduced2. As will be discussed, a potential consistent with scale invariance can ap- pear for the φ field. Such a potential has a shape which makes it suitable for the satisfactory realization of an inflationary scenario3 of the improved type4. Alternatively, it can be of use in a slowly rolling Λ scenario for the − late universe5. Finally, we also discuss how scale invariant mass terms, which lead to phenomenologically acceptable dynamics, can be introduced into the theory. We discuss some properties of such types of mass terms and their implica- tion for the early universe inflationary cosmology, for the cosmology of a late universe filled with matter and for the possibility of obtaining modified gravitational dynamics. 2 The Non Gravitating Vacuum Energy (NGVE) Theory. Strong and Weak Formulations. When formulating generally covariant Lagrangian formulations of gravita- tional theories, we usually consider the form S = L√ gd4x,g = detg (1) 1 µν − Z As it is well known, d4x is not a scalar but the combination √ gd4x is a − scalar. Inserting √ g, which has the transformation properties of a density, − produces a scalar action (1), provided L is a scalar. One could use nevertheless other objects instead of √ g, provided they − have the same transformation properties and achieve in this way a different generally covariant formulation. For example, given 4-scalars ϕ (a = 1,2,3,4), one can construct the den- a sity Φ = εµναβε ∂ ϕ ∂ ϕ ∂ ϕ ∂ ϕ (2) abcd µ a ν b α c β d 2 and consider instead of (1) the action1 S = LΦd4x. (3) 2 Z L is again some scalar, which contains the curvature (i.e. the gravitational contribution) and a matter contribution, as it is standard also in (1). In the action (3) the measure carries degrees of freedom independent of that of the metric and that of the matter fields. The most natural and suc- cessful formulation of the theory is achieved when the connection coefficients arealso treated as anindependent degrees offreedom. This is what isusually referred to as the first order formalism. One can notice that Φ is the total derivative of something, for example, one can write Φ = ∂ (εµναβε ϕ ∂ ϕ ∂ ϕ ∂ ϕ ). (4) µ abcd a ν b α c β d This means that a shift of the form L L+constant (5) → just adds the integral of a total divergence to the action (3) and it does not affect therefore the equations of motion of the theory. The same shift, act- ing on (1) produces an additional term which gives rise to a cosmological constant. Since the constant part of L does not affect the equations of mo- tion resulting from the action (3), this theory is called the Non Gravitating Vacuum Energy (NGVE) Theory1. One can generalize this structure and allow both geometrical objects to enter the theory and consider S = L Φd4x+ L √ gd4x (6) 3 1 2 − Z Z Now instead of (5), the shift symmetry can be applied only on L (L 1 1 → L + constant). Since the structure has been generalized, we call this for- 1 mulation the weak version of the NGVE - theory. Here L and L are ϕ 1 2 a independent. There is a good reason not to consider mixing of Φ and √ g , like for − example using Φ2 (7) √ g − 3 this is because (6) is invariant (up to the integral of a total divergence) under the infinite dimensional symmetry ϕ ϕ +f (L ) (8) a a a 1 → where f (L ) is an arbitrary function of L if L and L are ϕ independent. a 1 1 1 2 a Such symmetry (up to the integral of a total divergence) is absent if mixed terms (like (7)) are present. Therefore (6) is considered for the case when no dependence on the measure fields (MF) appears in L or L . 1 2 In this paper we will see that the existence of two independent measures of integrations asin (6) allows new realizations ofglobalscale invariance with most interesting consequences when the results are viewed from the point of view of cosmology. 3 The Action Principle for a Scalar Field in the Weak NGVE - Theory We will study now the dynamics of a scalar field φ interacting with gravity as given by the following action S = L Φd4x+ L √ gd4x (9) φ 1 2 − Z Z 1 1 L = − R(Γ,g)+ gµν∂ φ∂ φ V(φ) (10) 1 µ ν κ 2 − L = U(φ) (11) 2 R(Γ,g) = gµνR (Γ),R (Γ) = Rλ (12) µν µν µνλ Rλ (Γ) = Γλ Γλ +Γλ Γα Γλ Γα . (13) µνσ µν,σ − µσ,ν ασ µν − αν µσ In the variational principle Γλ ,g , the measure fields scalars ϕ and µν µν a the scalar field φ are all to be treated as independent variables although the variational principle may result in equations that allow us to solve some of these variables in terms of others. 4 4 Global Scale Invariance If we perform the global scale transformation (θ = constant) g eθg (14) µν µν → then (9) is invariant provided V(φ) and U(φ) are of the form V(φ) = f eαφ,U(φ) = f e2αφ (15) 1 2 and ϕ is transformed according to a ϕ λ ϕ (16) a a a → (no sum on a) which means Φ λ Φ λΦ (17) a → ≡ (cid:18) a (cid:19) Y such that λ = eθ (18) and θ φ φ . (19) → − α In this case we call the scalar field φ needed to implement scale invariance ”dilaton”. 5 The Equations of Motion We will now work out the equations of motion for arbitrary choice of V(φ) and U(φ). We study afterwards the choice (15) which allows us to obtain the results for the scale invariant case and also to see what differentiates this from the choice of arbitrary U(φ) and V(φ) in a very special way. Let us begin by considering the equations which are obtained from the variation of the fields that appear in the measure, i.e. the ϕ fields. We a obtain then Aµ∂ L = 0 (20) a µ 1 5 whereAµa = εµναβεabcd∂νϕb∂αϕc∂βϕd. Sinceitiseasytocheck thatAµa∂µϕa′ = δaa′Φ, it follows that det (Aµ) = 4−4Φ3 = 0 if Φ = 0. Therefore if Φ = 0 we 4 a 4! 6 6 6 obtain that ∂ L = 0, or that µ 1 1 1 L = − R(Γ,g)+ gµν∂ φ∂ φ V = M (21) 1 µ ν κ 2 − where M is constant. Let us study now the equations obtained from the variation of the con- nections Γλ . We obtain then µν Φ, Φ, Γλ Γα gβλg +δλΓα +δλgαβΓγ g g ∂ gαλ+δλg ∂ gαβ δλ µ+δλ ν = 0 − µν− βµ αν ν µα µ αβ γν− αν µ µ αν β − ν Φ µ Φ (22) If we define Σλ as Σλ = Γλ λ where λ is the Christoffel symbol, µν µν µν − {µν} {µν} we obtain for Σλ the equation µν σ, g +σ, g g Σα g Σα +g Σα +g g gβγΣα = 0 (23) − λ µν µ νλ − να λµ − µα νλ µν λα νλ αµ βγ where σ = lnχ,χ Φ . ≡ √ g − The general solution of (23) is 1 Σα = δαλ, + (σ, δα σ, g gαβ) (24) µν µ ν 2 µ ν − β µν where λ is an arbitrary function due to the λ - symmetry of the curvature6 Rλ (Γ), µνα Γα Γα = Γα +δαZ, (25) µν → ′µν µν µ ν Z being any scalar (which means λ λ+Z). → If we choose the gauge λ = σ, we obtain 2 1 Σα (σ) = (δασ, +δασ, σ, g gαβ). (26) µν 2 µ ν ν µ− β µν Considering now the variation with respect to gµν, we obtain 1 1 1 Φ(− R (Γ)+ φ, φ, ) √ gU(φ)g = 0 (27) µν µ ν µν κ 2 − 2 − Solving for R = gµνR (Γ) and introducing in (21), we obtain a contraint, µν 2U(ϕ) M +V(φ) = 0 (28) − χ 6 that allows us to solve for χ, 2U(φ) χ = . (29) M +V(φ) To get the physical content of the theory, it is convenient to go to the Einstein conformal frame where g = χg (30) µν µν and χ given by (29). In terms of g the non Riemannian contribution Σα µν µν disappears from the equations, which can be written then in the Einstein form (R (g ) = usual Ricci tensor) µν αβ 1 κ R (g ) g R(g ) = Teff(φ) (31) µν αβ − 2 µν αβ 2 µν where 1 Teff(φ) = φ φ g φ φ gαβ +g V (φ) (32) µν ,µ ,ν − 2 µν ,α ,β µν eff and 1 V (φ) = (V +M)2. (33) eff 4U(φ) In terms of the metric gαβ , the equation of motion of the Scalar field φ takes the standard General - Relativity form 1 ∂ (gµν√ g∂ φ)+V (φ) = 0. (34) √ g µ − ν e′ff − Notice that if V +M = 0,V = 0 and V = 0 also, provided V is finite eff e′ff ′ and U = 0 and regular there. This means the zero cosmological constant 6 state is achieved without any sort of fine tuning. This is the basic feature that characterizes the NGVE - theory and allows it to solve the cosmological constant problem1. It should be noticed that the equations of motion in terms of g are perfectly regular at V +M = 0 although the transformation µν (30)is singular atthis point. Interms ofthe originalmetric g the equations µν do have a singularity at V +M = 0. The existence of the singular behavior in the original frame implies the vanishing of the vacuum energy for the true vacuum state in the bar frame, but without any singularities there. 7 In what follows we will study (33) for the special case of global scale invariance, whichaswewillseedisplaysadditionalveryspecialfeatureswhich makes it attractive in the context of cosmology. Notice that in terms of the variables φ, g , the ”scale” transformation µν becomes only a shift in the scalar field φ, since g is invariant (since χ µν → λ 1χ and g λg ) − µν µν → θ g g ,φ φ . (35) µν → µν → − α 6 Analysis of the Scale - Invariant Dynamics If V(φ) = f eαφ and U(φ) = f e2αφ as required by scale invariance (14), (16), 1 2 (17), (18), (19), we obtain from (33) 1 V = (f +Me αφ)2 (36) eff 1 − 4f 2 Since we can always perform the transformation φ φ we can choose by convention α > O. We then see that as φ ,→V − f12 = const. → ∞ eff → 4f2 providing an infinite flat region. Also a minimum is achieved at zero cosmo- logical constant for the case f1 < O at the point M 1 f 1 φ = − ln . (37) min α | M | Finally, the second derivative of the potential V at the minimum is eff α2 V = f 2> O (38) e′f′f 2f | 1 | 2 if f > O, there are many interesting issues that one can raise here. The first 2 one is of course the fact that a realistic scalar field potential, with massive excitations when considering the true vacuum state, is achieved in a way consistent with the idea (although somewhat generalized) of scale invariance. The second point to be raised is that there is an infinite region of flat potential for φ , which makes this theory an attractive realization of → ∞ the improved inflationary model4. 8 A peculiar feature of the potential (36), is that the integration constant M, provided it has the correct sign, i.e. that f /M < 0, does not affect the 1 physics of the problem. This is because if we perform a shift φ φ+∆ (39) → in the potential (36), this is equivalent to the change in the integration con- stant M M Me α∆. (40) − → We see therefore that if we change M in any way, without changing the sign of M, the only effect this has is to shift the whole potential. The physics of the potential remains unchanged, however. This is reminiscent of the dilatation invariance of the theory, which involves only a shift in φ if g is µν used (see eq. (35) ). This is very different from the situation for two generic functions U(φ) and V(φ) in (33 ). There, M appears in V as a true new parameter that eff generically changes the shape of the potential V , i.e. it is impossible then eff to compensate the effect of M with just a shift. For example M will appear in the value of the second derivative of the potential at the minimum, unlike what we see in eq. (38), where we see that V (min) is M independent. e′f′f In conclusion, the scale invariance of the original theory is responsible for the non appearance (in the physics) of a certain scale, that associated to M. However, masses do appear, since the coupling to two different measures of L and L allow us to introduce two independent couplings f and f , a 1 2 1 2 situation which is unlike the standard formulation of globally scale invariant theories, where usually no stable vacuum state exists. Notice that we have not considered all possible terms consistent with global scale invariance. Additional terms in L of the form eαφR and 2 eαφgµν∂ φ∂ φ are indeed consistent with the global scale invariance (14), µ ν (16), (17), (18), (19) but they give rise to a much more complicated theory, which will be studied in a separate publication. There it will be shown that for slow rolling and for φ the basic features of the theory are the → ∞ same as what has been studied here. Let us finish this section by comparing the appearance of the potential V (φ), which has privileged some point eff depending on M (for example the minimum of the potential will have to be at some specific point), although the theory has the ”translation invariance” (35), to the physics of solitons. 9 In fact, this very much resembles the appearance of solitons in a space- translation invariant theory: The soliton solution has to be centered at some point, which of course is not determined by the theory. The soliton of course breaks thespace translationinvariance spontaneously, just astheexistence of the non trivial potential V (φ) breaks here spontaneously the translations eff in φ space, since V (φ) is not a constant. eff Notice however, that the existence for φ , of a flat region for V (φ) eff → ∞ can be nicely described as a region where the symmetry under translations (35) is restored. 7 Cosmological Applications of the Model SincewehaveaninfiniteregioninwhichV asgivenby(36)isflat(φ ), eff → ∞ we expect a slow rolling (new inflationary) scenario to be viable, provided the universe is started at a sufficiently large value of the scalar field φ. One should point out that the model discussed here gives a potential with two physically relevant parameters f12 , which represents the value of V 4f2 eff as φ , i.e. the strength of the false vacuum at the flat region and α2f12 → ∞ 2f2 , representing the mass of the excitations around the true vacuum with zero cosmological constant (achieved here without fine tuning). When a realistic model of reheating is considered, one has to give the strength of the coupling of the φ field to other fields. It remains to be seen what region of parameter space provides us with a realistic cosmological model. Furthermore, one can consider this model as suitable for the very late universe rather than for the early universe, after we suitably reinterpret the meaning of the scalar field φ. This can provide a long lived almost constant vacuum energy for a long period of time, which can be small if f2/4f is small. Such small energy 1 2 density will eventually disappear when the universe achieves its true vacuum state. For a more detailed scenario which includes the effect of matter other than the dilaton φ see next section. Notice that a small value of f12 can be achieved if we let f >> f . In this f2 2 1 case f12 << f , i.e. a very small scale for the energy density of the universe is f2 1 obtained by the existence of a very high scale (that of f ) the same way as a 2 10