Inflation in DBI models with constant γ

Transcription

1 Preprint typeset in JHEP style - HYPER VERSION arxiv: v1 [astro-ph] 27 Nov 2007 Inflation in DBI models with constant Micha l Spaliński So ltan Institute for Nuclear Studies ul. Hoża 69, Warszawa, Polska. Abstract: Dirac-Born-Infeld scalar field theories which appear in the context of inflation in string theory in general have a field dependent speed of sound. It is however possible to write down DBI models which posess exact solutions characterized by a constant speed of sound different from unity. For this to be possible the potential and the throat function appearing in a DBI action have to be related in a specific way. This paper describes such models in general and presents some examples with a constant speed of sound c s < 1 for which the spectrum of scalar perturbations can be found analytically without resorting to the slow roll approximation. Keywords:. mspal@fuw.edu.pl

2 1 Contents 1. Introduction 1 2. DBI scalar field theories 3 3. Models with constant 4 4. Perturbation spectra 5 5. The case of a constant potential 7 6. Power law inflation 8 7. A DBI analog of Easther s model Conclusions Introduction Nonlinear scalar field theories of the Dirac-Born-Infeld type [1]-[16] have attracted much attention in recent years due to their role in models of inflation in string theory [17, 18]. These scenarios identify the inflaton with the position of a mobile D-brane moving on a compact 6-dimensional submanifold of spacetime (for reviews and references see [19, 20, 21]). This means that the inflaton is interpreted as an open string mode. In consequence, once all other degrees of freedom are eliminated the effective action in four dimensions is of the DBI type. These actions involve a characteristic square root reminiscent of the relativistic Lorentz factor, and depend

3 2 on a function, usually denoted by f, which characterizes the local geometry of the compact manifold on which the D-brane is moving. The Lorentz factor locally imposes a maximum speed on the itinerant D-brane [1]. When this factor is close to unity the action reduces to that of a canonical scalar. The interpretation of the inflaton as an open string mode is interesting in that the effective field theory is rather distinct and is very well motivated by string computations [22]. From a hydrodynamical point of view the presence of the field-dependent Lorentz factor corresponds to a nontrivial, time dependent speed of sound one finds [1] that c s = 1/. Models of inflation with c s 1 were considered previously in [23, 24]. A non-constant speed of sound complicates the field equations considerably. The object of this note is to point out that there is a special class of DBI models which is in many ways as simple as models with a canonical kinetic energy term. These are models where the potential and the function f appearing in the Lorentz factor are related in such a way as to admit solutions with constant (in general different from unity). These models have the constant appearing as a parameter, whose deviation from unity measures the deformation of the kinetic energy from the canonical form. For such -deformed models the equations have the same form as in the canonical case, apart from numerical factors involving. In particular, for potentials of interest for which a solution is available when the kinetic terms take canonical form one can find the corresponding family of solutions in terms of the constant. For cosmology one is interested not only in the homogeneous limit, but also in the perturbed system. The mode equation which determines the spectrum of scalar perturbations is known also when the kinetic energy is non-canonical [24] and is of course more complicated than in the canonical case. However when the speed of sound is constant it simplifies considerably, so the study of perturbations in the class of models with constant is possible at the same level of approximation as required in the case of = 1. In particular, one can write down -deformed models corresponding to known cases where the spectrum of scalar perturbations is known analytically. These examples are the constant potential (i.e. de Sitter space), the exponential potential 1 (leading to power law inflation) and the model introduced 1 This solution was recently discussed by Chimento and Lazkoz [16]

4 3 by Easther [26]. This note is organized as follows: Section 2 sets up some notation and reviews the Hamilton-Jacobi formulation of DBI scalar field theories. Section 3 presents the Hamilton-Jacobi equations for DBI models with constant and the following section discusses the computations of the perturbation spectra. Sections 5, 6, 7 discuss the three examples where the perturbation spectrum for the -deformed models is obtainable analytically. 2. DBI scalar field theories DBI scalar field theories are an example of a wide class of scalar field theories with non-canonical kinetic terms, whose significance in the cosmological context was discussed in [23]. In D-brane inflation models the effective action for the inflaton is the Dirac-Born-Infeld action, which for spatially homogeneous inflaton configurations takes the form [1] S = d 4 x a(t) (f(φ) 3 1 ( 1 f(φ) φ ) 2 1) + V (φ). (2.1) The function f appearing here can be expressed in terms of the warp factor in the metric and the D3-brane tension. Many aspects of DBI scalar field theories have recently been discussed in a number of papers [1]-[16]. Einstein equations for homogeneous fields reduce to ρ = 3H(p + ρ) (2.2) 3M 2 P H2 = ρ, (2.3) where M P is the reduced Planck mass (M 2 P = 1/8πG), the dot indicates a time derivative and H ȧ/a. For the action (2.1) (2.2), (2.3) the pressure and energy density are given by p = 1 f ρ = 1 f V (φ) (2.4) + V (φ), (2.5)

5 4 where = 1 1 f(φ) φ 2. (2.6) It is easily established that is in fact the inverse speed of sound in models of this type: c 2 s = p / ρ φ φ = 1. (2.7) 2 The Hamilton-Jacobi form of the field equations which will be employed in the following is obtained by eliminating explicit time dependence. Using (2.2) (2.3) one finds φ = 2M2 P H (φ). (2.8) This equation can easily be solved for φ which allows to be expressed as a function of φ: Using this in (2.3) gives (φ) = 1 + 4M 4 P f(φ)h 2 (φ). (2.9) 3M 2 P H2 (φ) V (φ) = (φ) 1 f(φ). (2.10) This is the Hamilton-Jacobi equation for DBI inflation [1]. For comparison, the corresponding equation for a canonical inflaton [27] [30] is 3MPH 2 2 (φ) V (φ) = 2M 4 P H 2 (φ). (2.11) This of course follows from (2.10) when tends to unity. 3. Models with constant As discussed in the Introduction, the field dependence of in (2.9) complicates subsequent analysis of DBI models significantly. It is natural to look for models where the speed of sound is constant 2. This is somewhat analogous to considering field theory models for which the ratio w = p/ρ is constant (which leads to models of power law inflation). This is not to suggest that such models are particularly likely 2 A model with a constant speed of sound different from unity is discussed in [23].

6 5 to emerge from string theory or that they are a priori especially suited to describe the real world. They are however interesting as tractable models which facilitate the exploration of cosmology with a low speed of sound. It turns out to be very easy to formulate DBI models with constant. The requirement that be constant imposes a relation between the function f and the potential V. One way to proceed is to set (2.9) to a constant (denoted below by ) to eliminate f from (2.11). This leads to 3M 2 P H2 (φ) V (φ) = 4M4 P + 1 H (φ) 2. (3.1) This has the form of the Hamilton-Jacobi equation for a canonical inflaton (2.11) apart from the constant factor 1/( + 1). This factor cannot be eliminated, so this equation is not actually identical the one valid in the case of a canonical scalar, but the form is the same. Thus, to write down a DBI model with a constant speed of sound for a given potential requires solving (3.1) and putting f(φ) = 2 1 4M 4 P 1 H (φ). (3.2) 2 Alternatively, if one wanted to write down a model with a specific choice of f, then (3.2) could be solved for H(φ), which would then determine the required potential through (3.1). To illustrate these observations it is interesting to consider some explicit examples, specifically solvable models known from studies of inflation in the context of canonical scalar field theories. As discussed in the following section, when the speed of sound is constant the computation of the scalar perturbations is technically the same as for the case c s = 1. Thus it is interesting to consider examples where also the scalar perturbation spectrum can be computed analytically this will be the subject of subsequent sections. 4. Perturbation spectra The scalar perturbation spectrum for a general form of kinetic energy was described by Garriga and Mukhanov [24]. These results were used by a number of authors to

7 6 work out the spectrum for the special case of DBI scalar field theories [2, 3]. The perturbed system is parametrized in the longitudinal gauge as follows: φ = φ(t) + δφ(x, t) (4.1) and ds 2 = (1 + 2Ψ)dt 2 + a(t) 2 (1 2Ψ)dx 2. (4.2) The perturbations can be described in terms of a single scalar degree of freedom ζ = Ψ + Ḣ δφ. (4.3) φ Introducing the scalar density perturbation u = ζz and conformal time τ, dτ = dt/a, the Fourier mode u k as a function of the wave number k satisfies the equation ( d 2 u k k 2 dτ ) d 2 z u 2 z dτ 2 k = 0, (4.4) where z = a c s 2ǫ/3. For DBI models it follows that [2] and then one can show that [3] z = a H 3/2 φ (4.5) z,ττ z = a 2 H ((1 2 + ǫ η σ/2)(2 η σ/2) + ǫ(2ǫ 2η + σ) + (4.6) η(ǫ + σ) + ξ 1 2 σ(2σ + ǫ η) + 1 ) 2 ǫω, where the Hubble slow roll parameters are ǫ = 2M2 P (H H )2 (4.7) η = 2M2 P ξ = 4M4 p 2 σ = 2M2 P ω = 2M2 p H H ( ) H (φ)h (φ) H H 2 (φ) (4.8) (4.9) (4.10) H ( ). (4.11)

8 7 The above equations are valid in any DBI scalar field theory. In models with constant they simplify considerably, since in that case σ = ω = 0. The dependence on enters only via the k 2 term in the mode equation (4.4) and implicitly via the slow roll parameters in (4.6), which reduces to the same form as in models with a canonical inflaton: z,ττ z = a 2 H 2 ( (1 + ǫ η)(2 η) + 2ǫ 2 3ǫη + ξ ). (4.12) In general solving the mode equation still requires the slow roll approximation, but the deviation of the speed of sound from unity does not introduce any new complications. 5. The case of a constant potential For a constant potential V (φ) = V 0 the Hamilton-Jacobi equation (3.1) reads 3M 2 PH 2 (φ) V 0 = 4M4 P + 1 H (φ) 2. (5.1) There are two solutions for H. The first is just H = V 0 /3MP 2, which describes exponential inflation and the inflaton does not change with time. This solution is valid for any function f; it follows from (2.9) that in this case c s = = 1. There is also another solution, which for the case of a canonical scalar was discussed previously by Kinney [31] (see also [32]). This solution is given by V 0 3( + 1) φ H(φ) = cosh( ). (5.2) 2 M P 3M 2 P It describes a scalar which evolves with time according to ( + 1)V0 3( + 1) φ φ = sinh( ). (5.3) 2 M P When the scalar field reaches φ = 0 it stops and so the late-time limit is de Sitter space with H = V 0 /3M 2 P (independently of the value of ). The Hubble slow roll parameters are ǫ = ( + 1) tanh 2 φ ( ) 2 2 M P (5.4) η = (5.5)

9 8 In the canonical case (i.e. = 1) one has η = 3; this model was discussed in [31] as model of non-slow roll inflation with an exactly scale invariant spectrum of scalar perturbations. The DBI generalization discussed here is a family of models parametrized by, which remain non-slow roll even for large : the slow roll parameters (5.4) decrease with increasing, but the most one gets is a factor of 1/2 relative to the canonical result. 6. Power law inflation The second example is the model of power law inflation [25], characterized by an exponential potential: V (φ) = V 0 exp(2b φ M P ), (6.1) where b is a constant parameter. In the case of a canonical scalar it leads to power law inflation when b < 1/ 2. To find a constant DBI model with this potential one has to solve (3.1). The obvious solution is with H(φ) = H 0 exp(b φ M P ), (6.2) H 0 = V 0 3MP 2(1 4b2 ). (6.3) 3(+1) The form of the function f required for constant follows from (2.9): f(φ) = 1 ( ) exp( 2b φ ). (6.4) V 0 4 b 2 M P This solution still describes power law inflation, as was recently discussed by Chimento and Lazkoz [16]. In general, power law inflation in DBI models requires [7] This follows from (2.4) and (2.5) which imply 4M 2 P H 2 = 3(w + 1)H 2 (φ). (6.5) p + ρ = 2 (φ) 1 f(φ). (6.6)

10 9 Equating this to (w + 1)ρ and using (2.9) and (2.10) gives (6.5). For the model discussed in this section (6.5) implies that which is indeed constant. w = 4b2 3 1, (6.7) For the present model the Hubble slow roll parameters are ǫ = 2b2 (6.8) and η = ǫ. Thus inflation occurs when b < 2. (6.9) Thus even a steep exponential potential can be suitable for inflation if the constant is taken large enough. The spectrum of scalar perturbations in this model is computable exactly in the same way as in the canonical case. Since the mode equation can be solved analytically in the case of power law inflation, the present model provides a new solvable example with c s 1. To see this, note that in the present case equation (4.12) reduces to z,ττ z = 2a 2 H 2 (1 1 ǫ). (6.10) 2 This can be expressed in terms of conformal time as τ = 1/aH(1 ǫ), so that (4.4) becomes where ( ) d 2 u k k 2 dτ + 2 ν2 1/4 u 2 τ 2 k = 0, (6.11) ν 2 = 1 (3 ǫ) 2 4 (1 ǫ). (6.12) 2 This is as in the usual calculation for power law inflation [33], except that now ǫ depends on as seen from (6.8), not just on the parameter b of the potential. The mode equation with the Bunch-Davis initial condition is solved in terms of the first Hankel function which leads to the final formula for the spectral density: k P 1/2 R (k) = 3 u k = 2 ν 3/2Γ(ν) 2π 2 z Γ( 3)(1 H 2 ǫ)ν 1/2 2 2π φ k=ah. (6.13)

11 10 The scalar spectral index n s implied by this is given by n s 1 d ln P R d ln k 2ǫ(1 + ǫ). (6.14) The tensor mode spectral density is given by P h = 2H2 M 2 p π2 k=ah, (6.15) so the ratio of power in tensor modes versus scalar modes is given exactly by r = 16ǫ. (6.16) This example is interesting and perhaps important not only because it is a solvable deformation of power law inflation, but it may have some phenomenological appeal, as it shows that taking > 1 can make inflation possible even when the potential is too steep. 7. A DBI analog of Easther s model Easther has set up a model which for which the Mukhanov equation is exactly solvable [26]. It is easy to write down a soluble DBI model of this type with constant. Easther s observation is that the Mukhanov equation reduces to the harmonic oscillator equation if z is independent of the scalar field. This gives a differential equation for H(φ) which turns out to be analytically solvable and the corresponding potential can be identified using the Hamilton-Jacobi equation. This scheme can be carried out in the present context. A simple calculation shows that dz dφ = 2 3 a1/2 ( 2M 2 P + H H (H H )2 ). (7.1) Thus one obtains the differential equation for H(φ) which is of the same form as Easther s equation, apart from the factor of in the first term: 2M 2 P + H H (H H )2 = 0. (7.2)

12 11 The solution is given by H(φ) = H 0 exp( 1 4 ( φ M P ) 2 + aφ), (7.3) where a, H 0 are integration constants. Since a can be eliminated by shifting φ it can be set to zero without loss of generality. The potential and the function f can then be determined from (2.9) and (3.1): V (φ) = 3MP 2 H2 0 ( φ 2 3M 2 P ) exp( 1 2 ( φ M P ) 2 ) (7.4) f(φ) = H 2 0φ 2 exp( 1 2 ( φ M P ) 2 ). (7.5) The slow roll parameter ǫ is given by ǫ = 3 2 φ2. (7.6) Note that in this example increasing acts to shrink the range of φ where inflation takes place. The power spectrum of scalar perturbations comes out as P = 9 16π 2H2 H. (7.7) evaluated at k = ah. A simple calculation then shows that the scalar index in this model is independent of, and given by Easther s result n s = 3. Thus this model remains outside the limits set by observation even when deformed as discussed above. 8. Conclusions DBI scalar field theories follow from the interpretation of the inflaton as an open string mode. Despite difficulties in embedding them in a fully satisfactory way in string theory[4] it is very important to understand their dynamics. This note described a special class of DBI models with an arbitrary constant speed of sound c s < 1. The three examples of this kind which were discussed in the text provide new examples of inflationary scalar field theories for which the spectrum of primordial scalar perturbations can be computed analytically (at the level of linear perturbation theory, but without appeal to any slow-roll approximation).

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