Propagation Constraints and Classical Solutions in K-essence Like Theories
aa r X i v : . [ h e p - t h ] J un Propagation Constraints and Classical Solutions inK-essence Like Theories
Ratindranath Akhoury and Christopher S. Gauthier
Michigan Center for Theoretical PhysicsRandall Laboratory of PhysicsUniversity of MichiganAnn Arbor, Michigan 48109-1120, USA
Abstract
We consider two examples of solutions of the equations of motion ofscalar field theories with higher derivatives. These are the cosmology of therolling tachyon and static spherically symmetric solutions of the scalar fieldin flat space. By requiring that the field equations always be hyperbolic andthat the speed of propagation of the small fluctuations are not superluminal,we find constraints on the form of the allowed interactions in the first caseand on the choice of boundary conditions in the latter. For the rollingtachyon we find a general class of models which have the property that atlarge times the tachyon matter behaves essentially like a non-relativistic gasof dust. For the spherically symmetric solutions we show how causalityinfluences the choice of boundary conditions and those which are finite atthe origin are shown to have negative energy density there. Introduction
Scalar theories with higher derivatives play an essential role in the effective field theoryapproach (for reviews see [1]). An example is provided by the chiral lagrangian whichprovides a good description of the strong dynamics at low energies. Applications of higherderivative theories to cosmology have also become popular in the last few years: exampleshere are effective field theories of the rolling tachyon [2], DBI inflation [3], and k-essence [4]which attempts to provide a dynamical explanation of the so called coincidence problem andthe accelerated expansion of the universe. Recently [5], such higher derivative actions havebeen shown to enhance the non-gaussian fluctuations in the cosmic microwave background.Theories with higher derivatives have the possibility of modifying the dispersion relationsand hence may potentially lead to superluminal propagation. This aspect has been studied indetail in [6] where it was shown that causality and the absence of superluminal propagationrequire certain coefficients of the effective lagrangian to be positive definite which in turn hasconsequences for phenomenology [6], [7]. Thus the constraints of causality and hyperbolicityof the equations of motion play a particularly important role in such theories. Another recentstriking example is the no-go theorem proved in [8]. Here it was shown that in the context ofthe original k-essence theories [4], it is impossible to simultaneously resolve the coincidenceproblem and the accelerated expansion of the universe without violating causality.In this paper we apply the constraints [9, 10, 11, 12] that the equation of motion of the scalarfield has a well defined initial value problem and there is no superluminal propagation ofthe small fluctuations around classical solutions in higher derivative theories. In particularwe discuss in sections 3 and 4 respectively the case of the rolling tachyon and the staticsolutions to the equations of motion of a general scalar theory with higher derivatives. Forthe case of the tachyon we consider a general lagrangian of the form L = V ( φ ) K ( X ), with V the potential for the tachyon and X = g µν ∂ µ φ∂ ν φ and find the constraints on K ( X ) andthe potential such that the the energy density is finite but the equation of state parametergoes to zero at large times up to small corrections. We find that in order achieve this it isnot nescessary for K ( X ) to vanish as ˙ φ → K are obtained which allows for a more general framework for the rolling tachyon thanwas previously considered. The only constraint on the potential is that a V → a is the scale factor. The physical motivation is that the tachyon could thenbe considered as a possible candidate for dark matter [13]. In section 4 we discuss the staticspherically symmetric solutions to the equations of motion for the most general scalar fieldlagrangian with higher derivatives in flat space which are consistent with hyperbolicity andcausality. We find the interesting result that for scalar field solutions which are finite atthe origin, causality requires its first derivative to vanish there, and even though the totalenergy is positive, the energy density for such solutions is negative at the origin. A physicalmotivation for this study arises from the possibilty that such scalar fields could describe thedark matter halos around galaxies [12]. In section 2 we set up the problem and review someresults concerning the criteria for superluminal propagation and hyperbolicity of the scalarfield equations. In the concluding section we discuss the results.2 Preliminaries
In this paper we will be interested in scalar field theories with a lagrangian of the generalform L = F ( X, φ ). Here, X = ∂ µ φ∂ µ φ We will first discuss the case of flat space time andat the end comment on the general case in the presence of gravity.The equations of motion for the scalar field are given by: G µν ∂ µ ∂ ν φ = 12 { L φ − XL Xφ } (1) G µν = L X η µν + 2 L XX ∂ µ φ∂ ν φ. (2)Throughout this paper we will be using the notation, L X = ∂L∂X ; L φ = ∂L∂φ (3)and so on. In (2), G µν plays the role of an effective metric in which the scalar field propagates.For an equation of this type to have a well defined initial value problem and to satisfy globalhyperbolicity, the following conditions must hold [9, 10, 11] L X > , L X + 2 XL XX > u = 0 is the characteristic surface and n µ = ∂ µ u then the speed of propagation of thesmall disturbances is given by solving L X n + 2 L XX ( n µ ∂ µ φ ) = 0 . (5)From this, one deduces the maximum speed to be n | ~n | = W (cid:16) ~n. ~W | ~n | (cid:17) + r W − (cid:16) ~n. ~W | ~n | (cid:17) W (6)where, W µ = q L XX L X ∂ µ φ . The two cases discussed in this paper are the time-like spatiallyhomogenous and static spherically symmetric ones. The expressions for the propagationspeeds in the two cases are respectively, n | ~n | = { L X L X + 2 XL XX } , X = ˙ φ (7) n | ~n | = { L X + 2 XL XX L X } , X = − φ ′ . (8)From these it is easy to see that there is superluminal propagation when L XX <
0. Insummary, the conditions of hyperbolicity and no superluminal propagation may be statedas: L X > , L X + 2 XL XX > , L XX ≥ . (9)3or future reference we note that in the static spherically symmetric case when there is nosuperluminal propagation, L X L X + 2 XL XX ≥ . (10)When gravity is included, the effective scalar metric now becomes: G µν = L X g µν + 2 L XX ∂ µ φ∂ ν φ (11)We require this metric to be Lorentzian. In particular in order to have a consistent definitionof temporal and spatial directions the largest eigenvalue of (11) must be positive whilethe other three must be negative. This can be shown to be true [10, 11] only if the firsttwo conditions in (9) are satisfied while the last one once again avoids [9, 6] superluminalpropagation. Sen [2] has discussed the qualitative dynamics of a tachyon field in the background of anunstable D-brane system and conjectured that the simplest description within an effectivefield theory framework can be provided by the following lagrangian, L = V ( φ ) K ( X ) with, φ the scalar field (dot denotes derivative with respect to t ) and, K ( x ) = − q − ˙ φ V ( φ ) = V exp( − φ ) . (12)The cosmology of this model in the FRW background has been studied in [2], [13], anda particularly surprising result is the existence of solutions with exponentially vanishingpressure at large times but a non-zero energy density. Since there is no compelling reasonfor the precise forms Eq.(12), in this section we keep K and V arbitrary (apart from themild assumptions below) and determine from the constraints of causality the conditionsunder which the equation of state for tachyonic matter becomes ω = 0 at large times up tosmall corrections. The tachyon could then be considered a dark matter candidate in a widerclass of models than originally envisioned.Consistent with the fact that we are dealing with the case of a rolling tachyon, we willmake the following assumptions about the potential V and the kinetic term K : (1) K ≤ φ is bounded. We will take the upper limit of ˙ φ to be 1 in appropriateunits. (3) K is bounded as ˙ φ → V ( φ ) is positive, has a maximum at φ = 0 and monotonically decreases to zero at φ = ∞ at large times where it is a minimum.The equations of motion for the scalar field and the scale factor a ( t ) are in units πG = 1:¨ φ = − H L X L X + 2 XL XX ˙ φ − ∂ǫ t ∂φ L X + 2 XL XX H = ρ = ǫ t + ǫ m . (13)4or the homogenous FRW background X = ˙ φ > ǫ t = 2 XL X − L is the tachyon energydensity and ǫ m is that of the rest of matter and ρ is the total energy density. H = ˙ aa is theHubble factor. Note that from the inequality Eq.(9 ) and L <
0, it is easy to see that ǫ t > L X >
0. The equation of state parameter for the tachyon is, ω t = L XL X − L = K XK X − K (14)Using the φ field equation of motion it is straightforward to show that˙ ǫ t = ddt (2 XL X − L ) = − HXL X = − H (1 + ω t ) ǫ t (15)The inequality in Eq. (9) then implies that the tachyon energy density is a monotonicallydecreasing function of time and ω t > −
1. The Hubble factor itself is monotonically decreasingin time as can be seen from ˙ H = −
32 ((1 + ω m ) ǫ m + (1 + ω t ) ǫ t ) , (16)Defining y = √ X and using the factorized form for the tachyon lagrangian, the tachyonequation of motion may be written as, dydt = − yK y − KK yy H { K y yK y − K } + ∂V∂φ V ! . (17)The constraints given in Eq. (9) for the initial value problem to be well defined and theabsence of superluminal propagation are expressed in terms of the new variable as: K y > , K yy > , K yy > K y y . (18)Note that whenever V → K y → ∞ such that L X >
0. As we will see below, it is thissimple fact that guarantees that the tachyon energy density is nonzero and positive in thelimit t → ∞ , while ω t vanishes. Let us consider Eq. (17) at large times. We first discuss theconditions on the potential under which the second term in the brackets is dominant. Letus define the term inside the curly brackets in this equation as g , then dgdy = − KK yy ( yK y − K ) . (19)Since K < dgdy >
0. The maximum value of g is thus at y = 1which is g max ≤
1. Moreover, H is monotonically decreasing. Let us now write for largetimes φ = t + θ ( t ) with θ ( t ) ≪ t . Then it is easy to check from the above results that thesecond term inside the brackets in Eq.(17) dominates over the first for large times as longas V → a as t → ∞ . This condition on the potential will reappear below.Since the overall factor outside the brackets in (17) is negative, and since ∂V∂φ < y is monotonically increasing as it goes to 1 atlarge times. In addition, since y is bounded at y = 1, ˙ y = 0 at y = 1. Therefore we concludethat as y → K y K yy = 0; KK yy = 0 (20)As mentioned earlier, K y → ∞ for large times while K is bounded. Thus the second of theabove conditions is not a new requirement since the first implies that as y → K yy > K y . Itshould be noted that the condition for the absence of superluminal propagation only impliesthat K y K yy < y, (21)Thus the condition (20) is much stronger.Let us now expand the equation (17) about the point y = 1 by writing, φ = t + θ ( t ) with˙ θ < θ ≪ t . Using (20), it is straightforward to get,¨ θ = − ˙ θλ { H + ∂V∂φ V } (22) λ = (cid:18) − K yyy K y K yy (cid:19) ( y = 1) . (23)Integrating this we get (taking ˙ φ ≈ θ = − αa − λ V − λ . (24)where α is a constant and consistency with the boundary conditions require λ <
0. Since θ ≪ t , we see that with a negative λ , ˙ θ vanishes like a V → t → ∞ , which is exactlythe condition derived earlier for the term involving the potential V to dominate over the firstone in Eq. (17). λ < y = 1, K yyy > K yy is then a new constraint on theallowed forms of K .We are now in a position to prove that the equation of state parameter vanishes at y = 1up to small corrections. From Eq.(14) we can obtain, dω t dy = yK y − KK y − yKK yy ( yK y − K ) . (25)Since K ≤
0, and K y and K yy are both positive, we see that ω t is a monotonically increasingfunction of y . Its maximum is therefore at y = 1. Near y = 1 we can write ω t ≈ K (1) + ˙ θK y (1) K y (1) . (26)However we have argued above that K y (1) is infinite, thus ω t = 0 apart from correctionswhich vanish like a V at large times. 6 Background Static Solutions Consistent With Causal-ity
We next consider the static spherically symmetric solutions to the equations of motion ofthe scalar field in flat spacetime (prime denotes the derivative with respect to r ). φ ′′ + 2 r { L X L X + 2 XL XX } φ ′ + 12 { L φ − XL Xφ L X + 2 XL XX } = 0 . (27)In the above, X = − φ ′ and from section 1 the combined constraints of hyperbolicity andabsence of superluminal propagation now give the following bound for the coefficient of the φ ′ term for all r (see Eq. (10)): δ = L X L X + 2 XL XX ≥ . (28)Here we consider only solutions to (27) which are finite at the origin. We first want todetermine the appropriate boundary condition for φ ′ at r = 0. We will use a series expansionmethod for φ near r = 0 to guide us to the correct choice. Even though the coefficient δ isnot a constant but dependent on φ , we know that independent of r , δ ≥
1, so to find theindical equation, which is all we are interested in to determine the boundary condition for φ ′ , we may treat it as such. The same applies for the last term in (27) as long as we restrictourselves to solutions which are finite at the origin. These two complications do not affectthe indical equation. With this in mind, let us look for a series solution of the form: φ = r s ( c + c r + c r + c r + .... ) (29)From this we get the indical equation s ( s − δ ) = 0. Since causality requires δ ≥ r we see that for φ to be finite at the origin only the solution with s = 0 is allowed.Substituting this expansion into (27) we see by matching equal powers of r that c = 0.Thus the boundary condition for this problem which is consistent with causality and thefiniteness of φ at the origin is φ ′ = 0 , r = 0. We now consider the analog of Eq. (15). Let usdefine γ = − XL X + L . Then using the equation of motion we obtain dγdr = − r φ ′ L X . (30)Since L X >
0, we see that γ is a monotonically decreasing function of r . The minimum of γ is therefore at infinity. As r → ∞ , the solutions to the equations of motion must be suchthat γ → r in order to keep the total energy content finite. This implies thatat r → γ >
0. From the boundary condition on φ ′ at r = 0 we see that here, γ = L > H = − L . Thus we concludethat at r = 0, the energy density H is negative. It is easy to see that the total energy in thestatic limit is, however, always positive: E = − π Z r drL = − π Z ( γ − φ ′ L X ) r dr. (31)7onsider the integral over γ . Integrating by parts and using the fact that γ vanishes fasterthan r as r approaches infinity, we get Z γr dr = − Z dγdr r dr = 43 Z φ ′ L X r dr, (32)where the last equality follows from (30). Combining everything we see that E = 8 π Z φ ′ L X r dr, (33)which is manifestly positive definite.When such a theory is coupled to the Schwarzschild metric, we can look for solutionsto the combined equations for both gravity and scalar matter. Such a situation could berelevant for understanding the formation of dark matter halos around galaxies [12]. Thoughthe above analysis has been performed in flat space-time, our considerations indicate thatat least solutions of the scalar field equations which are finite at the origin should not berelevant to such a scenario. The detailed question of the solutions of the scalar field in thepresence of gravity needs further investigation. Nevertheless it is interesting that the modelwe have considered in this section has solutions which violate the weak energy condition atthe origin. Using the requirement that the field equations are always hyperbolic (and hence the cauchyproblem is well defined) we have obtained a set of consequences for two different problemsof physical interest.For the case of the rolling tachyon in a homogenous FRW background, we have obtainedconstraints on the forms of the potential and the kinetic terms such that the equation ofstate of the tachyon vanishes at large times up to small corrections. The tachyon could thenbe considered a dark matter candidate. The key observation here was that what is requiredfor this to happen is that K remains bounded but not necessarily zero at large times, but K y goes to infinity. The latter in fact guarantees that the energy is non vanishing in thislimit. Other requirements are given by Eqs.(20), the potential V is such that a V → λ defined in (23) be negative. It is easy to check that the choice (12)does in fact satisfy all the requirements, but is not unique. The class of models is thus largerthan the original.We have also looked quite generally at the problem of the static spherically symmetricsolutions to the equations of motion of the scalar field described by the lagrangian of section1 and found that if we require the finiteness of the scalar field at the origin then the solutionsconsistent with causality have the property that the energy density becomes negative at theorigin. This example brings out very clearly the role that causality plays in the choice ofboundary conditions. 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