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Reconstructing Single Field In?ationary Actions From CMBR Data.?

Christopher S. Gauthier? and Ratindranath Akhoury? Michigan Center for Theoretical Physics Randall Laboratory of Physics University of Michigan Ann Arbor, Michigan 48109-1120, USA

Abstract This paper describes a general program for deriving the action of single ?eld in?ation models with nonstandard kinetic energy terms using CMBR power spectrum data. This method assumes that an action depends on a set of undetermined functions, each of which is a function of either the in?aton wave function or its time derivative. The scalar, tensor and non-gaussianity of the curvature perturbation spectrum are used to derive a set of reconstruction equations whose solution set can specify up to three of the undetermined functions. The method is then used to ?nd the undetermined functions in various types of actions assuming power law type scalar and tensor spectra. In actions that contain only two unknown functions, the third reconstruction equation implies a consistency relation between the non-gaussianty, sound speed and slow roll parameters. In particular we focus on reconstructing a generalized DBI action with an unknown potential and warp factor. We ?nd that for realistic scalar and tensor spectra, the reconstructed warp factor and potential are very similar to the theoretically derived result. Furthermore, physical consistency of the reconstructed warp factor and potential imposes strict constraints on the scalar and tensor spectral indices.

arXiv:0804.0420v3 [astro-ph] 26 Apr 2008

1

Introduction

Since the landmark COBE experiment, the study of the cosmos has entered a new age of precision cosmology. For the ?rst time in the history of modern cosmology, direct quantitative measurements of early cosmological observables were available. The data taken by COBE was critical in establishing in?ation as the central paradigm in our theories of the origin of the universe [1]. Thanks to experiments that measured the spectrum of CMBR ?uctuations, we have now con?rmed that the near-scale invariance of large scale ?uctuations that is a prediction of in?ation are in fact borne out in the data. Although observation supports the general theory of in?ation, as of now the data is unable to conclusively determine the mechanism responsible for in?ation.

?

This work was supported by the US department of energy. Electronic address: csg@umich.edu ? Electronic address: akhoury@umich.edu

?

1

The di?cultly in discriminating between di?erent in?ation models lies in fact that all current models of in?ation predict the same near-scale invariant spectrum of scalar ?uctuations. To further narrow down the number of observationally consistent in?ation models, observables independent of the scalar perturbation need to be measured and compared to model predictions. Two additional in?ationary observables are the spectrum of tensor perturbations Ph [2, 3] and the non-gaussianity fN L [4, 5, 6] of the CMBR temperature perturbation spectrum. In recent years greater progress has been made in measuring these quantities directly. Upper bounds on the amplitude of the tensor perturbation spectrum, which is the spectrum of relic gravitational waves, have been determined directly through analysis of the CMBR polarization [7, 8]. The non-gaussianity, which represents the deviation of the curvature perturbation from gaussian statistics, is also being better understood. Analysis of WMAP3 data [9] has found evidence of non-gaussian statistics in the CMBR temperature spectrum. With a better knowledge of these extra observables it becomes possible to better determine which model of in?ation is most likely to have taken place. For example, a large fN L would tend to rule out a single ?eld in?ation model with minimal kinetic terms, while favoring those models that predict a large non-gaussianity. Ultimately, one would like to use the features of the CMB temperature anisotropy to reconstruct the in?aton action directly. It is customary to write the general scalar ?eld action as [10] S= √ d4 x ?g p(φ, X ), 1 X = g?ν ? ? φ? ν φ. 2 (1)

Note that the Lagrangian (density) p is exactly equal to the pressure, which is the motivation behind denoting the Lagrangian by p. In our analysis we will limit ourselves to actions that contain no third or higher derivatives of the in?aton. Throughout this paper we will assume that the curvature of the 3 non-compact space dimensions will be zero. Following the cosmological principle, our only choice for the metric is the FRW metric: g ?ν = diag(1, ?a2 , ?a2 , ?a2 ), where a is a time dependent scale factor. Using (1), the Friedmann equations for the scale factor are

2 3Mpl H 2 = ρ, ˙ = ρ + p, ?2M 2 H pl

(2) (3)

a is the Hubble parameter and ρ is the energy density, which in terms of the where H = d log dt Lagrangian is

ρ = 2Xp,X ? p. In single ?eld in?ation models with a minimal kinetic term the action is S= √ d4 x ?g [X ? V (φ)] .

(4)

(5)

If we assume (5), the only function that needs to be determined from the data is the potential V (φ). Reconstruction of the in?ationary potential for models of the form (5) has been studied 2

extensively [11, 12, 13, 14, 15, 16, 17]. However, by assuming that the action has a minimal kinetic term we neglect a rich class of models such as DBI in?ation [18, 19], k-in?ation [20] and ghost in?ation [21]. In contrast, only a hand full of articles have been written that deal with the reconstruction of in?ationary action with general kinetic terms [22, 23, 24]. In non-minimal kinetic models the speed at which scalar ?uctuations propagate can be different than the speed of light. This can e?ect the temperature anisotropy in two ways. First, if cs < c = 1, scalar ?uctuations have a sound horizon that is smaller than the cosmological horizon, causing curvature perturbations to freeze in earlier than normal. Depending on how the Hubble parameter and sound speed change during the course of in?ation, the temperature anisotropy can develop noticeable signatures of non-minimal kinetic terms. Second, models with non-minimal kinetic terms will in general produce a non-gaussian spectrum. Traditionally, the non-gaussianity is measured by the non-linearity parameter fN L de?ned by the following ansatz for the curvature perturbation: 3 2 . ζ = ζL ? fN L ζL 5 (6)

Here, ζ is the general curvature perturbation and ζL is a curvature perturbation with gaussian statistics. Within the standard canonical action (5), non-gaussianities can be produced by cubic or higher order terms in the in?aton potential or by secondary interactions with gravity [5]. However, non-gaussianities produced by these mechanisms are on the order of the slow roll parameters, and thus small. In contrast, models with non-minimal kinetic terms can have large non-gaussianities, providing a clear distinction from canonical in?ation. The goal of this paper will be to reconstruct an in?ationary action from observables starting with as few initial assumptions as possible. In this paper we take the experimental inputs to be the scalar curvature perturbation Ps , the tensor curvature perturbation Ph , and the non-gaussianity (non-linearity) parameter fN L . Unfortunately, completely reconstructing the o?-shell action is not possible since the observables only carry on-shell information. To understand why the o?-shell action is inaccessible to us, consider the interpretation of the action p(φ, X ) as a surface in the three dimensional space (φ, X, p) [23]. Because the observables are insensistive to the o?-shell behavior of the action, we can only determine the one-dimensional trajectory p = p(φ, X (φ)) of the action on-shell, embedded in the twodimensional surface de?ned by p = p(φ, X ). A one-dimensional trajectory has an in?nite number of surfaces that contain it, each related to one another by a canonical transformation [23]. Therefore we have to be more speci?c about the form of the action that we are trying to ?nd. In this paper we will reconstruct in?ationary actions that have the form p(φ, X ) = P (g1 (X ), ..., gm (X ), f1 (φ), ..., fn (φ)). (7)

Here it is assumed that P (x1 , ..., xm , y1 , ..., yn ) is a known function of the {xi } and {yα }, and the functions {gi } and {fα } are not all known. Once the on-shell trajectory φ = φ(k ) is determined, the action (7) de?nes a surface in the (φ, X, p)-space up to a ?eld rede?nition. The idea will be to use data on the CMBR perturbation spectrum to ?nd the functions {gi (X )} and {fα (φ)}. In a naive comparison with algebraic linear equations, we expect that if there are n unknown functions, ?nding the action requires n experimental inputs. 3

Since we are assuming that there only three observables: Ps , Ph and fN L , we can derive three reconstruction equations, which can determine an action with three or fewer unknown functions. In the case where the number of unknown functions is less than the number of experimental inputs, the reconstruction equations not used to ?nd the action become constraint equations. Since we are interested in solving for functions {gi } and {fα } and not just numbers, we will need to know at least a portion of Ps , Ph and fN L 1 as functions of the scale k . While the scale dependence of Ps is known to be at least approximately power law dependent on k , the scale dependence of the other two observables Ph and fN L is at this point unclear. Although future experiments will be able to clarify some aspects of the tensor and non-gaussianity signals, their exact functional forms will probably not be available for quite some time if at all. Regardless, the method we develop here does have utility outside of reconstruction. This method is well suited to testing how the form of an action depends on the observables. For instance if the scalar perturbation is of the near scale invariant variety: Ps ∝ k ns ?1 (8)

we can use Ps to help derive an action and study its dependence on the index ns . That way if we wish to connect the action derived from (8) to an action derived from theory, we can see if the theoretical action leads to reasonable results for the observables. Furthermore, as we mentioned earlier when there are only one or two unknown functions in (7), the remaining reconstruction equations determine new consistency relations. In this paper most of the examples we deal with have only two unknown functions, which we solve for using the scalar and tensor spectrum data. The reconstruction equation derived from the non-gaussianity will then be a constraint; relating the non-gaussianity to the sound speed, the Hubble parameter and/or their derivatives. Outside of deriving the action we also ?nd a method for quickly obtaining the sound speed as a function of time from the scalar and tensor perturbation spectra. Finding a sound speed di?erent from the speed of light even over a small range of scales would be a powerful indication of non-canonical in?ation. This paper is organized as follows. In section 2 we present the method for reconstructing the action from the scalar, tensor perturbation and the non-gaussianity parameter. We explain how cosmological data can ?nd the Hubble parameter H , and the sound speed cs , and how these in turn can be used to ?nd three unknown functions of the action (7). Once the method has been explained in section 2.1 we carry out a derivation of the action for di?erent functions P (z1 , z2 , z3 ), assuming that the scalar and tensor power spectra both scale like k to some power. In section 3 we apply our method to ?nd the warp factor and potential in a generalized DBI in?ation model. We ?nd the warp factor and potential as functions of the spectral indices and the initial value of the Hubble parameter. The results for these are compared to the theoretically motivated warp factor and potential used in D3-brane DBI in?ation. Finally, in section 4 we review our main results.

1

In this paper fN L represents the equilateral bispectrum, and is therefore a function of a single scale.

4

2

The Reconstruction Equations

We start our derivation of the reconstruction procedure by explaining how the observables are used to ?nd the Hubble parameter H and sound speed cs 2 . Once we have these, the action can be obtained using a set of reconstruction equations that will be shown later. Let us begin by recalling the de?nition of the slow roll parameter3 in terms of the Hubble parameter. The de?nition of implies that dH = ? H 2. (9) dt Since the perturbation spectra and non-gaussianity are functions of k and not time, we wish ? to rewrite this equation for dH into an equation for H = d dH . However, because we are dt log k assuming a general sound speed, we must be careful to di?erentiate between the horizons of scalar and tensor ?uctuations. If the sound speed di?ers from unity (in particular cs < 1), then the horizon size of scalar ?uctuations: (aH/cs )?1 is smaller than that of the tensor ?uctuations: (aH )?1 . This implies that at any given time, the scales ks and kt at which the scalar and tensor ?uctuations leave their respective horizons, will in general be di?erent. For our purposes, we choose to study the dependence of H on the scalar wave number ks . Therefore, the condition for horizon exit is now kcs = aH instead of the more familiar relation: k = aH . Having made clear our choice of wave number, we now set out to express d log k in terms of familiar quantities: dt d log k κ d log k = H (1 ? + ) dt H dt

cs . Solving for where we have de?ned κ = ? c s

?

(10)

d log k dt

above: (11)

H (1 ? ) d log k = . dt 1?κ With equation (11) in hand, the equation for H can now be found from (9): H= ?

? ?

H (1 ? κ) . 1?

(12)

We will use this equation to ?nd cs once we have found H and in terms of the observables. Since depends on the time derivative of the Hubble parameter, H (k ) and (k ) are independent parameters. Since we have two independent parameters, we will likely need two independent observables. The two observables we will use here are the scalar and tensor perturbation spectra. Recall that to ?rst order in the slow roll parameters the perturbation spectra are given by Ps (ks ) =

2 3

H2 2 8π 2 Mpl cs

,

ks cs =aH

(13)

The method described here was inspired by the technique used in [25] The term “slow roll parameter” is taken from chaotic in?ation where in?ation occurs only when the ˙ is small. However, DBI in?ation can still occur for large φ ˙. in?aton “velocity” φ

5

Ph (kt ) =

2 H2 2 π 2 Mpl

.

kt =aH

(14)

The extra source of information can also be garnered from the non-gaussianity parameter fN L . However, going in this route would result in a more complicated solution. The parameter can be found as a function of wave number using (13). Solving for we obtain |ks cs =aH H2 1 = 2 2 8π Mpl Ps (ks ) cs .

ks cs =aH

(15)

As a matter of convenience de?ne Ps = APs , where A is the value of the scalar perturbation at some ?ducial scale k = k0 . If k0 0.002Mpc?1 then present data suggests that A 10?9 . Here, Ps is the normalized scalar perturbation de?ned such that Ps (k0 ) = 1. Furthermore, 2 A. Substituting (15) in for in equation (12) we have let H = αH where α2 = 8π 2 Mpl H |ks cs =aH

?

H3 =? cs Ps ? H2

cs 1+ cs

?

.

ks cs =aH

(16)

We have eliminated from (12), but two independent variables remain. To get an equation for cs we need to ?nd H in terms of the observables. Since the expression for Ph (14) only depends on H, it can be used to ?nd the Hubble parameter directly. In doing so, one ?nd that H2

kt =aH

=

Ph (kt ) 16

(17)

where Ph = A?1 Ph . This gives the Hubble parameter as a function of the tensor mode wave number kt . In order to ?nd H2 as a function of the scalar mode wave number ks , note that the relation between the wave number of tensor and scalar modes that exit the horizon at the same time is kt = ks cs . Therefore, we can obtain H|ks cs =aH by performing the substitution kt → ks cs : H2

ks cs =aH

=

Ph (ks cs ) . 16

(18)

Plugging this in for H into equation (16) we get dPh (kt ) 2Ph (kt )2 =? . d log kt 16cs (ks )Ps (ks ) ? Ph (kt ) Rearranging (19) we ?nd the equation for cs : cs (ks ) = 1 Ph (ks cs ) 16 Ps (ks ) 1? 2Ph (ks cs ) P h (ks cs )

?

(19)

.

(20)

Here, a solid circle over Ph will denote di?erentiation with respect to log kt , not log ks . Once we specify what Ps (ks ) and Ph (kt ) are, we can use the above to solve for cs . Even if we 6

can only show that cs = 1, this will be a signal for non-minimal kinetic terms. Once we use equations (18) and (20) to get H(k ) and cs (k ) we can use (11) to ?nd the relation between the wave number and time and ultimately ?nd H(t) and cs (t). Having successfully found H and cs , the next step will be to use this information to ?nd the action. The most general single scalar ?eld action is a multivariable function of φ and X . However, since we only have H and cs as functions of a single independent variable (in this case k ) the action can only be determined as a function of k : p(k ). To turn p(k ) into p(φ, X ), we need to ?nd φ and X as functions of k , invert them to get k (φ) and k (X ), and substitute into p(k ). However, there is an ambiguity in how we substitute k for φ and X . Whenever k appears in the expression for p(k ), we do not know whether to substitute it with k (φ), k (X ) or some combination of the two. The ambiguity can be partially resolved if at the onset we specify a separation of the action into functions that depend either entirely on φ or entirely on X . In light of this fact we make an ansatz:

2 2 p(φ, X ) = α2 Mpl (?, x) = α2 Mpl P (g1 (x), ..., gm (x), f1 (?), ..., fn (?))

(21)

?1 where ? = Mpl φ and x = (αMpl )?2 X . We have also de?ned a dimensionless action in order to keep the exposition neat and clear. Here it is assumed that P (y1 , ..., ym , z1 , ..., zn ) is a known function of the {yi } and {zα }, and the functions {gi } and {fα } are not all known. We say that (21) only partially resolves the ambiguity, because it is possible in certain circumstances that a ?eld rede?nition can leave the form of (21) unalterated. For example consider the action

(?, x) = f (?)g (x). Under a general ?eld rede?nition ? = h(? ?) the action (?, x) = f (h(? ?))g ((h (? ?))2 x ?). If the function g is such that g (x · y ) = g (x) · g (y ) then ?(? (?, x) = f (h(? ?))g ((h (? ?))2 )g (? x) = f ?)g (? x).

(22)

(23)

(24)

Thus, not all choices for the function P lead to unique separation between the functions of x and ?. The ?rst equation for the reconstructed action will be obtained from the de?nition of the sound speed, which is given by c2 s = p,X 2Xp,XX + p,X (25)

Assuming that the action has the form (21), equation (25) can now be used to ?nd a differential equation for the gi ’s as a function of time. After some work, this equation is given by g ¨P = gP ˙ x ˙ 2 x 2xx ¨ 1 + 2 ? 1 ? gP ˙ g ˙ 2 x ˙ cs 7 (26)

where a dot denotes di?erentation with respect to the dimensionless time τ = αt. Here, we have used the short hand notation:

m m m

g ¨P =

i=1

g ¨i Pi ,

gP ˙ =

i=1

2

g ˙ i Pi ,

gP ˙ g ˙ =

i,j =1

g ˙ig ˙ j Pij ,

(27)

? P ?P and Pij = ?g . Equation (26), however, is incomplete since x where Pi and Pij are Pi = ?g i i ?gj is not known explicitly. To turn (26) into a more usable form, we need to obtain an equation for x and its derivatives in terms of the known quantities and H. To ?nd such a formula lets write out the expression for the energy density of a general single scalar ?eld action (1):

ρ = 2xp,x ? p. As a consequence of the Friedmann equations, ρ + p is proportional to ? dH ρ+p = . 2 dt 2Mpl

dH : dt

(28)

(29)

Solving for ρ + p in (28), and substituting the result into (29) we ?nd that

2 2 Mpl H2 Mpl dH = , x=? p,x dt p,x

(30)

where in the last step we have used the de?nition of the slow roll parameter. With (21) as our assumed form of the action, this algebraic equation for x becomes a ?rst order di?erential equation: H2 x = . x ˙ gP ˙ (31)

Di?erentiating this equation, one can ?nd an expression for x ¨. After substituting the results of these relations, equation (26) becomes

2 2 c2 sH gP ˙ = (1 + c2 s)

1 η ?H ? 2 H ? gP ˙

n

f˙α gP ˙

α=1

,α

,

(32)

˙ where η ?= H and gP ˙ ,α denotes partial di?erentiation of the quantity gP ˙ with respect to fα . We refer to (32) as the sound speed reconstruction equation. The non-gaussianity parameter fN L can also be used to ?nd an equation for the functions gi and fα . Following from the ansatz of the curvature perturbation (6), fN L is determined by the behavior of the curvature three point function:

ζ (k1 )ζ (k2 )ζ (k3 ) = ?(2π )7 δ (3) (k1 + k2 + k3 )Ps2 (K )

3fN L (K ) 10

3 ki 3 i ki i

(33)

where ki = |ki | and K = k1 + k2 + k3 . As we can see from (33), the fN L will depend on the size and shape of the triangle formed by the three scales of the three point function. In [6] 8

the authors found an expression for fN L for general single ?eld actions (1) when the three scales form an equilateral triangle: fN L = where X 2 p,XX + 2 X 3 p,XXX 3 Λ= . Xp,X + 2X 2 p,XX (35) 35 108 1 5 ?1 ? 2 cs 81 1 ? 1 ? 2Λ , c2 s (34)

To get fN L as a function of K , we have to evaluate (34) at the time when the scale K passes outside of the sound horizon: Kcs = aH . The scalar power spectrum depends on a single scale k , which has a one-to-one mapping with the time through the relation kcs = aH . However, since fN L really depends on three di?erent scales, the mapping between time and scale is not as straight forward. When the delta function in (33) is taken into account, the non-gaussianity still depends on three degrees of freedom: the magnitude of two of the scales and the angle between them [26]. To simplify matters, two of these three degrees of freedom will be ?xed, so as to make fN L a univariate function. Since the equilateral con?guration: k1 = k2 = k3 , has been very widely studied [6, 26], we will take fN L to be the non-gaussianity of the equilateral bispectrum. The equilateral non-gaussianity will be a function of kN L , which is the length of the sides of the equilateral triangle. Since the non-gaussianity freezes in when the scale K leaves the horizon, the scales at which the nongaussianity and the scalar perturbation freeze in are not the same but instead related by 3kN L = ks . After some work, one can use (34) and (31) to show that the g ’s and f ’s satisfy the equation 1 16 H3 c2 s gP ˙ = 2 55 1 ? cs ? 972 c2 f 275 s N L c2 H × κ ?+ s 3 gP ˙

n

f˙α

α=1

gP ˙

g ¨P

,α

+ gP ˙ g ˙

,α

? gP ˙

,α

g ¨P + gP ˙ g ˙

,

(36)

? ) c ˙s where κ ? = H = ? κ(1 . This is the non-gaussianity reconstruction equation. Note that cs 1?κ (36) is only well de?ned if the last line is nonzero. If the last line does vanish and the right 972 2 hand side of (32) is nonzero then 1 ? c2 s ? 275 cs fN L = 0, and the non-gaussianity (34) only depends on the functions gi and fα through the sound speed cs . We will discuss such a case in the next section. Finally, another relation between the f ’s and g ’s can be derived by combining the Friedmann equations (2) and (3):

P (g1 , ..., gm , f1 , ..., fn ) = (2 ? 3)H2 .

(37)

The upshot is that we now have four equations: (31), (32), (36), and (37), which when combined can be used to ?nd x(t) (and by extension ?(t)) and three of the functions fα and gi . With more observational inputs it may be possible to determine even more f and g functions, but for now we will be content with what we have. In what follows, we will consider di?erent, speci?c scenarios for the action and show how the action in each can be determined from the observables. 9

2.1

Examples

In the case where the action (21) has only one function of x the equations (36) and (32) take on a much simpler form: gP ˙ g=

2 2 c2 sH (1 + c2 s)

η ?H ? 2 H ?

1 Pg

n

f˙α Pgα ,

α=1

(38)

16 H3 c2 1 s gP ˙ g= 972 2 2 55 1 ? cs ? 275 cs f N L

c2 H κ ? + s3 Pg

n

f˙α [Pg Pggα ? Pgα Pgg ] .

α=1

(39)

As we mentioned earlier not all forms of the action will yield an equation for the functions gi and fα . In particular if the action is such that Pg Pggα = Pgα Pgg for each α, and if the sound speed is constant, then (39) is not well de?ned. To see why, lets assume that cs is constant. As a result, according to the de?nition of the sound speed: c2 s =

,x

2x

,xx

+

?

,x

(?, x) = f1 (?)x

1+c2 s 2c2 s

+ f2 (?),

(40)

where the f1 and f2 are integration constants and in general will be functions of ? only. We already know that with this form of the action Pg Pggα = Pgα Pgg . Thus the term in (39) inside the large parentheses vanishes, however the right hand side of equation (38) does not 972 2 vanish. Therefore, we expect the denominator 1 ? c2 s ? 275 cs fN L in equation (39) to vanish. Indeed if we use the formulas (34) and (35) for fN L we ?nd that fN L = 275 972 1 ?1 . c2 s (41)

This relation between cs and fN L holds regardless of what the functions f1 and f2 in (40) are. It might be argued that if cs is constant then the action (40) can be assumed and the remaining equations can be used to ?nd f1 and f2 . This however is not that case since we have already used the sound speed equation to ?nd g (x). This can be con?rmed if one assumes the action (40). With (40) as our action, equation (32) is equivalent to the time derivative of equation (31). Thus, there are really only two equations: either (31) or (32), and equation (37). Therefore, only one of the two f1 and f2 can be solved for. There is still yet another potential complication that may arise, speci?cally when the action takes the form (?, x) = f (?)g (x). Using (37) to ?nd f˙ in terms of g ˙ , the equations (38) and (39) become 6η ?H3 c2 s , 3(1 + c2 ) ? 2 s 16 κ ? H 3 c2 1 s gf ˙ = . 2 55 1 ? cs ? 972 c2 f 275 s N L gf ˙ = 10 (42) (43)

As with the previous case, (43) is not de?ned when κ ? = 0. Furthermore, the ?rst equation (42) is also unde?ned when η ? = 0. Since this is equivalent to = constant, lets assume that is constant to see which type of action this corresponds to. From (30) = 3xp,x ρ ? 2? 3 x

,x

=

?

(?, x) = f1 (?)x 2 ?3 .

(44)

Notice, that this action is a special case of the cs = constant action (40) with f2 = 0. Thus, = constant implies that cs = constant. However, the converse of this is not true if f2 = 0. Since (38) is a well de?ned equation even with the action (44), we suspect that the denominator in (42) vanishes. After calculating the sound speed with the action (44) one ?nds that 2 ?3 . (45) c2 s = 3 So indeed, the denominator in (42) does vanish. Assuming that the equations (42) and (43) are well de?ned, consistency requires that the right hand sides of these equations be equal, leading to the relation 3? η 8? κ 1 = . 972 2 2 2 3(1 + cs ) ? 2 55 1 ? cs ? 275 cs f N L (46)

This is a consistency relation between fN L , cs and the slow roll parameters. Although this consistency relation only holds for models with the action = f (?)g (x), analogous consistency relations can be found for any model in question. In what follows, we will carry out the full derivation of the g and f functions for two special cases. 2.1.1 Case 1: (?, x) = g (x) ? V (?)

Suppose the action has the form (?, x) = g (x) ? V (?). (47)

This type of action corresponds to the standard scalar ?eld action when g is the identity map: g (x) = x. We refer to g (x) as the kinetic function. Notice that we have replaced what should be f1 in our previous nomenclature with V (?) in order to draw a clear analogy with the potential in the canonical scalar ?eld action. With this type of action the equations (38) and (39) become:

3 2 c2 η?2 ) s H (? , 1 + c2 s 16 κ ? H 3 c2 1 s g ˙= . 972 2 55 1 ? c2 s ? 275 cs fN L

g ˙=

(48) (49)

Here we have two expressions for the derivative of g (τ ). Consistency demands that the right hand sides of these equations be equal, thus we are lead to an analogue of the relation (46): η ??2 8? κ 1 . = 972 2 1 + c2 55 1 ? c2 s s ? 275 cs fN L 11 (50)

Interestingly enough this is the same consistency relation found in [23] for general single ?eld in?ation models. However, while the relation in [23] was only approximate, in our case it is exact. This shows that the consistency relation of Bean et al. is exact in the case when the action is of the form (47). Continuing with the derivation, the equations for V (τ ) and x(τ ) are given by (37) and (31), respectively. They read V (τ ) = g (τ ) ? (2 ? 3)H2 , g ˙ x ˙ 2c2 η?2 ) s (? = = . 3 Hx H 1 + c2 s (51) (52)

With our equations in hand we are almost ready to solve them and ?nd the action. However, we still lack knowledge about the H and cs . In order to go further we need to look back to section 2 and in particular equations (18) and (20). In order for these equations to be of any use we need two observables as inputs. For these we will assume that the two inputs are the scalar and tensor contributions to the CMBR. Presently, it is believed that these spectra are near scale invariant, and over a limited range of scales possess the forms4 Ps (k ) = e(ns ?1) log k/k0 , Ph (k ) = Be(nT ?1) log kcs (k)/k0 cs0 . (53) (54)

Here cs0 = cs (k0 ) where k0 is the ?ducial scale at which Ps = 1. Note that we have assumed that the spectral indices have no running: i.e. ns , nT = constant. Although running spectral indices is an interesting generalization, in order to better demonstrate the usefulness of this procedure we will stick with simpler case of no running. Equation (18) then tells us that the Hubble parameter is simply proportional to the square root of the tensor perturbation: √ Ph (k ) B nT ?1 log kcs (k)/k0 cs0 . (55) = e 2 H(k ) = 4 4 2 De?ning H0 as H(k0 ) = H0 , the constant B is therefore B = 16H0 . As it stands, (55) is not complete since we still do not have an expression for cs (k ). To ?nd cs we turn to equation (20), the solution of which gives us cs = cs 0 cs0 nT ? 1 2 H0 nT ? 3

1 nT ?2

e

?

nT ?ns nT ?2

log k/k0

.

(56)

Since we de?ned cs0 as cs (k0 ) = cs0 , consistency of our de?nition demands that

2 cs 0 = H 0

nT ? 3 . nT ? 1

(57)

Note that if 1 < nT < 3, cs0 is negative: a nonsense result. Therefore, we must restrict nT to be either nT < 1 or nT > 3. With an expression for cs in hand, H(k ) explicitly in terms of k is H(k ) = H0 e

4

(nT ?1) ns ?2 2 nT ?2

log k/k0

.

(58)

Note that the scale invariant tensor spectrum is most commonly de?ned at nT = 0, whereas we have de?ned it at nT = 1. The reason for using a nonstandard de?nition of nT is to establish a parity between the forms of the scalar and tensor perturbation spectra. The relation between the two di?erent nT ’s is simply nT |standard = nT |ours ? 1.

12

Note that

and κ are constant in this case and are given by = nT ? 1 H2 = , cs Ps nT ? 3

?

(59) (60)

cs nT ? ns κ=? = . cs nT ? 2

Since is a constant then η ? = 0, which will simplify matters later when we try solve the reconstruction equations. Solving for log k in (11), we ?nd that the time dependence of log k is ω (61) log k/k0 = log [1 + H0 (τ ? τ0 )] , κ

2(nT ?ns ) 1? where ω = κ 1 = ? (nT . Therefore, the sound speed and Hubble parameter as ?κ ?1)(ns ?2) functions of time are

cs (τ ) = cs0 [1 + H0 (τ ? τ0 )]?ω ,

H(τ ) =

H0 . 1 + H0 (τ ? τ0 )

(62)

These are the expressions for the sound speed and Hubble parameter that will be used throughout this paper. They are completely independent of the form of the action that we are solving for, and are determined only by the in?ationary observables Ps and Ph . Before we go about solving the reconstruction equations we should point out that not all values of the spectral indices lead to realistic in?ationary scenarios. As has been mention before, the sound horizon of the scalar ?uctuations is not the same as the cosmological horizon. As a consequence it is now possible for the size of the sound horizon to increase as time progresses. Thus, the usual expectation that larger scales freeze in at the beginning of in?ation and smaller scales freeze in at the end, is not always guaranteed to hold. Recall that the time dependence of the scale is given by equation (61). It follows that the sound horizon size depends on time like Sound Horizon Size ∝ (γaH )?1 = (k/k0 )?1 = (1 + H0 (τ ? τ0 ))? κ .

ω

(63)

If ω/κ > 0, the sound horizon decreases with time as is normally expected. However, if ω/κ < 0, the size of the sound horizon increases during in?ation, allowing modes the were previously frozen-in behind the horizon to reenter while in?ation is still going on. This would be a disaster since if the scales were to reenter during in?ation they would continue to ?uctuate, destroying the near-scale invariance of the CMB anisotropy. Therefore, the spectral indices must be ?xed such that ω and κ are either both positive or both negative. Since we are considering only those models that allow for in?ation, we need to be sure that the spectral indices are such that an in?ationary phase is allowed. If we refer to the expression for the equation of state w we see that not all values of nT are allowed if we want to have in?ation: w= p p nT ? 7 = =? . ρ 2Xp,X ? p 3(nT ? 3) 13 (64)

Notice that so long as nT < 3 the equation of state is always w < ? 1 , and so in?ation will 3 ?1 occur. Since cs ∝ , in order for cs to be interpreted as a sound speed, must be positive. If we look back to equation (59) we ?nd that not all values of nT will result in a positive value for . Requiring that > 0, we ?nd that nT must be either nT < 1 or nT > 3. Since we have already found that nT > 3 would not lead to an in?ationary solution, we conclude that nT < 1. Recall that in the previous paragraph we found that the sound horizon could expand during in?ation only if the spectral indices were chosen so that ω/κ > 0. If one refers back to the de?nitions of ω and κ in terms of the spectral indices, we can see that if nT < 1 the scalar spectral index is required to be ns < 2. The sound speed (62) can tell us something about the expected range of validity of the scalar (53) and tensor spectra (54). If ω < 0 then at some time τ > τ0 the sound speed will be greater than one, signaling that ?uctuations propagate at superluminal speeds. Likewise, superluminal speeds also occur at times τ < τ0 when ω > 0. Therefore, (53) and (54) can only be considered approximations; reliable within a certain range of scales. Keeping in mind that the sound horizon needs to shrink during in?ation, the wave number k must respect the following bounds if the sound speed is to be less then the speed of light: k/k0 > (cs0 ) |κ| , k/k0 < (cs0 )

1 ? |κ | 1

for ω > 0, for ω < 0. (65)

,

The only way (53) and (54) could be acceptable at all scales is if ω = 0, in which case cs is a constant. If it turns out that the sound speed is not constant, (53) and (54) are most likely too naive. The most recent data from WMAP [8] suggests that the scalar spectral index may have a small but nonzero running, so we should not be surprised that our simple expressions for the perturbation spectra are not exactly correct. Regardless, scalar and tensor spectra with constant spectral indices are still a good approximation to the CMBR data. Our discussion will still be of relevance, as long as we keep in mind that the reconstructed actions are only approximations, valid over a limited range of scales. We will now simplify our discussion by ?xing the sound speed to a constant, which is achieved by setting ω = 0. Although we will be considering only constant sound speeds, we will keep the value of cs arbitrary. This will allow us to ?nd a more general solution to the reconstruction equations, while allowing us to study the limit cs → 1 and see whether the canonical action is recovered. One might object to this choice of ω on the grounds that if cs is constant the non-gaussianity reconstruction equation (49) will be ill-de?ned for reasons discussed in section 2.1. However, we counter that this is acceptable since we are assuming that only two functions g and V are unknown, thereby making the third reconstruction equation (49) unnecessary. It should be pointed out that while the two unknown functions can still be found when ω = 0, the consistency relation (50) is no longer well de?ned. Once we substitute (62) for cs and H into (48) and solve for g (τ ) the result is g (τ ) =

2 2 2 H0 cs 0 1 + c2 s0

1 ? 1 + g0 . (1 + H0 (τ ? τ0 ))2

(66)

14

Using the Friedmann equation (51) we can now ?nd V (τ ): V (τ ) =

2 H0 1 + c2 s0

3(1 + c2 s0 ) ? 2 ? 2 c2 s0 + g0 . (1 + H0 (τ ? τ0 ))2

(67)

Similarly, the equation for x ˙ is given by H0 x ˙ 4 c2 s0 =? , 2 x 1 + cs0 [1 + H0 (τ ? τ0 )] and the exact solution for x is x(τ ) = x0 [1 + H(τ ? τ0 )] Integrating (69) to ?nd ?(τ )

1?cs0 ? ˙ 0 1 + c2 s0 1+c2 s0 ? 1 (1 + H ( τ ? τ )) + ?0 , ?= 0 0 H0 1 ? c2 s0 2

(68)

?

4c 2 s0 1+c2 s0

.

(69)

(70)

√ where ? ˙ 0 = 2x0 . Now that we have x and ? as functions of time, we can invert these and substitute the results into (66) and (67) to ?nd g (x) and V (?). After carrying this out, we can combine the g (x) and V (?) to arrive at the full action:

1+c2 2 2 s0 cs 0 2 H0 2c2 s0 ( x/x ) 0 1 + c2 s0

(?, x) =

?

2 [3(1 H0

+ c2 s0 ) 1 + c2 s0

?2 ]

1 + H0

1? 1+

c2 s0 2 cs 0

? ? ?0 ? ˙0

?

2(1+c2 s0 ) 1?c2 s0

. (71)

Here is the complete action in the case when the sound speed is constant. Note that the ?nal result does not depend on the integration constant g0 . This is a result of the fact that the right hand side of equation (37) is independent of the initial values of the kinetic and potential functions. The only undetermined constants are the initial values of the scalar ?eld and it’s derivative, and due to the attractor nature of in?ation, their exact values are unimportant. Despite what was said earlier in regards to the inde?niteness of the nongaussianity reconstruction equation (49), this action does have a de?nite non-gaussianity given by the result in equation (41). It is worth noting that in the exceptional case where cs → 1:

2 2 (x/x0 ) ? H0 (3 ? )e (?, x) = H0 ?

2 H0 (???0 ) ? ˙0

(72)

we recover the canonical in?ation action with an exponential potential. If we require that 2 the wave function retain the standard normalization then x0 = H0 and the action becomes

2 (?, x) = x ? H0 (3 ? )e? √ 2 (???0 )

,

(73)

which is the action of power law in?ation [27]. This is a reassuring result; it con?rms that in the appropriate limit, we can recover the standard in?ationary action.

15

2.1.2

Case 2:

(?, x) = f (?)g (x) ? V (?)

Let us now take the complexity of the action one step further and assume that there are now three unknown functions: g , f and V . We de?ne as (?, x) = f (?)g (x) ? V (?). (74)

In equation (38) the only nonzero Pgα is the one corresponding to f . Thus (38) reduces to gf ˙ =

2 2 c2 sH (1 + c2 s)

η ?H ? 2 H ?

f˙ f

(75)

Furthermore, with this action the terms with the f˙α ’s in (39) all vanish. The ?nal result is simply: gf ˙ = 1 16 κ ? H 3 c2 s . 972 2 55 1 ? c2 s ? 275 cs fN L (76)

Combining equations (75) and (76), g decouples and we get equation just for f : 8? κ f˙ 1 + c2 s =η ??2 ? . 972 2 Hf 55 1 ? c2 ? cf s 275 s N L (77)

Once we have solved for f here we can substitute the solution into equation (76) and solve for g . With the solutions for these two, V is found using the Friedmann equation (37). The ?nal step is to ?nd ?(τ ) and x(τ ) by solving (31): x ˙= 16? κHc2 x s . 2 55 1 ? cs ? 972 c2 f 275 s N L (78)

Let’s assume that κ ? = 0, so that the reconstruction equations (76) (77) and (78) are well de?ned. We will again assume that the scalar and tensor perturbation spectra are given by (53) and (54). Therefore, H and cs are the same as those that we found earlier (62). However, now that we are using the non-gaussianity reconstruction equation we need to specify fN L . In this example we will take fN L = 0 to simplify the analysis. With these as our inputs, the reconstruction equations become f˙ 8 ω 1 + c2 s = ?2 + , Hf 55 1 ? c2 s gf ˙ =?

3 16 2 ω c2 sH , 55 1 ? c2 s

x ˙ 16 ω c2 s =? . Hx 55 1 ? c2 s

(79)

Here, we have used the fact that η ? = 0 and κ ? = ? ω . Each of these has an analytic solution. They are f (τ ) = f0 c2 s (τ ) c2 s0

8 1 4 ? 55 ω

1 ? c2 s (τ ) 1 ? c2 s0

4 ? 55

8/55

,

(80) (81) (82)

2 55 4 2H0 (1 ? c? s0 ) g (τ ) = g0 + cs0 f0

c2 s (τ ) c2 s0

2 ?2 F (c? s (τ )) ? F (cs0 ) ,

8 55

x(τ ) = x0

1 ? c2 s0 2 1 ? cs (τ ) 16

.

g

0.0010

x Exact Approximate

0.0005

0.45

0.50

0.55

0.60

x

0.0005

Figure 1: Plot depicting the function g (x). This plot was made with cs0 = 0.5, = 0.1, f0 = 1, g0 = 0 and ? ˙ 0 = 1. The exact behavior of g (x) is contrasted against the approximation (85). The behavior of g (x) is very linear except for small deviations for x < x0 . Note that 8/55 at x0 (1 ? c2 ≈ 0.48 the plot of the exact behavior of g (x) stops abruptly as a result of s0 ) the fact that g becomes non-real in this region.

Here we have de?ned F (x) as F (x) = 2 F1 ( 4 63 59 , , ; x) 55 55 55 (83)

where 2 F1 is a hypergeometric function. Note that we can ?nd a complete expression for g (x). We simply have to solve for cs (τ ) in (82) to get c2 s (x), which is

2 c2 s (x) = 1 ? (1 ? cs0 )

x x0

?55/8

,

(84)

and then substitute this for cs (τ ) in (81) to get g (x). Interestingly enough, g (x) is independent of ω , so taking the ω → 0 limit here is trivial. Since the solution for g (x) is in terms of hypergeometric functions, to get a better idea of what g (x) looks like we expand around cs0 = 1, and thus obtain

2 2 H0 x ? x0 H0 (1 ? cs0 ) x x0 g (x) = g0 + + 2145 ? 2209 + 64 f 0 cs 0 x 0 2585f0 cs0 x0 x

47 8

+ O((1 ? cs0 )2 ). (85)

Let’s take a moment to comment on the analytic behavior of g (x). In ?g. 1 the exact functional behavior of g (x) is shown along with the approximate expression (85). As ?g. 1 and the approximation (85) suggest, the behavior of g is nearly linear with respect to x, when cs0 is close to one. However, for reasons that will be clear shortly, the limit cs0 → 1 17

does not necessarily mean that the action will be linear in x. Another interesting feature 8/55 of g (x) is that it becomes non-real for values of x less than x0 (1 ? c2 . This implies a s0 ) lower bound on the values of x, which is a behavior that is observed in the solution (82). 8/55 the sound speed squared This lower bound is a result of the fact that for x < x0 (1 ? c2 s0 ) would be negative according to (84). As for the functions f (?) and V (?), one cannot ?nd analytic expressions for these like we did for g (x). Once we integrate x(τ ) to ?nd ?(τ ), we can see why: 4 √ ? 21 ? 55 2 4/55 ω 2x0 cs0 (1 ? c? c2 s0 ) s (τ ) 2 ?2 Fω (c? (86) ? = ?0 + s (τ )) ? Fω (cs0 ) , ω 2 3 c (1 + 8 H0 ) s0 55 where we have again shortened things by de?ning Fω as 4 1 4 59 1 + , , + ; x). (87) 55 2ω 55 55 2ω Since ? is such a complicated function there is no way to invert (86) to get time as an analytic function of ?. Therefore, we are forced to either evaluate f (?) and V (?) numerically, or make an approximation for τ (?). Since we will be interested in ?nding a correspondence with the example in the previous section, we will approximate ?(τ ) in the ω 1 limit. The result of this approximation is Fω (x) = 2 F1 (

2 ˙ ˙ 0 τ ? 8φ0 ω cs0 [(1 + H0 τ ) log(1 + H0 τ ) ? H0 τ ] + O(ω 2 ). φ(τ ) = φ0 + φ 55 H0 1 ? c2 s0

(88)

To get τ (?) we will drop all ω dependent terms from (88), so that ?(τ ) is a linear function of τ . This approximation turns out to be remarkably accurate even at late times, since the higher order terms in (88) scale only logarithmically with τ . Now that we have at least an approximate expression for τ (?), f (?) can be found by replacing cs (τ ) with cs (?) = cs0 1 + H0 (? ? ?0 ) ? ˙0

?ω

.

(89)

The exact behavior of f (?) was evaluated numerically and the results are shown in ?g. 2. Since we will be taking the ω → 0 limit later, we make a further approximation of f (?) by taylor expanding around ω = 0:

H0 (? ? ?0 )] log[1 + ? f0 8ωf0 1 + c2 ˙0 s0 f (?) ≈ + . 2 H0 H 0 2 2 55 1 ? c (1 + ? (1 + ( ? ? ? )) ( ? ? ? )) 0 0 s 0 ˙0 ? ˙0

(90)

Notice that the second term diverges at cs0 = 1. Therefore, although the higher order terms in (85) vanish when cs0 = 1, when the limit cs0 → 1 is taken the product f g will retain the nonlinear x terms. This is why the action may not be linear in x even when cs0 = 1. To ?nd the potential we use the Friedmann equation (37). Doing so requires us to ?nd g as a function of ?, which we ?nd by replacing cs (τ ) in (81) with cs (?) (89). The potential, taylor expanded around ω = 0, is

H0 2 2 log[1 + ? (? ? ?0 )] g0 f0 ? (2 ? 3)H0 8ω g0 f0 (1 + c2 ˙0 s0 ) ? 2 H0 cs0 V (?) ≈ + . H0 H0 55 1 ? c2 (1 + ? (? ? ?0 ))2 (1 + ? (? ? ?0 ))2 s0 ˙0 ˙0

(91)

18

f

1.4

1.2

? 0.1

1.0

? 10 ? 15

2

0.8

f

0.6

0.4

0.2

0.0

0

20

40

60

80

100

Figure 2: Plot depicting the function f (?) for di?erent values of ω . This plot was made with cs0 = 0.5, = 0.1, ?0 = 0, ? ˙ 0 = 1.0, f0 = 1.0 and g0 = 0. We have also included a plot of the function f (?) in (95) for comparison.

Note that the individual functions g , f and V depend on the arbitrary integration constants f0 and g0 , even though the action that is composed of them does not. If we are interested in just ?nding the action, ?xing f0 and g0 would be a moot point. However, it does raise the matter of how one separates the action into kinetic and potential terms. For example, suppose we separate the kinetic function into a constant and a “x-dependent” piece: g (x) = c + G(x). (92)

The constant c is arbitrary and can be adjusted to any given value by absorbing the di?erence into G(x). Substituting the right hand side of (92) for g (x), the action (74) becomes (?, x) = f (?)G(x) ? V (?) + cf (?). (93)

With the action written in this way, it would make more sense to de?ne G(x) as the kinetic function and de?ne the potential as v (?) = V (?) ? cf (?). (94)

In the case where f is constant (such as the example in the previous section) then the above rede?nition only amounts to a uniform shift in the potential. However, if f is non-constant then the behavior of the potential can change drastically. Although none of the CMBR data are sensitive to changes in c, it is possible to ?nd a value for c by requiring that in the appropriate limit, the action becomes equivalent to the canonical action. We will de?ne this as the canonical limit of the action. Before we determine c by this method we need to con?rm that the action (74) is canonically equivalent to the canonical action (73) when the sound speed is constant and equal to one. 19

If we turn our attention back to our approximations for f (?) and V (?), we notice that taking cs0 = 1 leads to divergent results. These divergence are understandable since the reconstruction equations (79) are divergent when cs = 1. However, if we set ω = 0 in (90) and (91), it’s possible to take cs0 = 1 and still obtain a well de?ned result. Doing so results in the following for the functions g , f and V : g (x) ≈ g0 +

2 x ? x0 H0 , f0 x0

f (?) ≈

f0 , H0 2 ( ? ? ? )) (1 + ? 0 ˙0

V (?) ≈

2 f0 g0 ? (2 ? 3)H0 , (95) H0 2 ( ? ? ? )) (1 + ? 0 ˙0

2 where now H0 = since cs0 = 1. We refer the reader to ?g. 2 for a comparison of f (?) in (95) and f (?) for general values of ω and cs0 . The action in the cs0 → 1 limit when ω = 0 is

(?, x) =

2 2 H0 x (3 ? )H0 . ? H0 H0 (? ? ?0 ))2 x0 (1 + ? (? ? ?0 ))2 (1 + ? ˙0 ˙0

(96)

This action can be related to the canonical action (73) through the ?eld rede?nition de?ned by √ H0 ??? ?0 ) 1+ (? ? ?0 ) = e 2 (? . (97) ? ˙0 Under this rede?nition, the new action is

2 ? (?, ? x ?) = x ? ? (3 ? )H0 e √ 2 (? ??? ?0 )

.

(98)

This is the same as the canonical action (73) found in the ?rst example. To ensure a smooth transition to the canonical action we must separate the kinetic function as we did in (92) so that G(0) = 0 in the ω → 0 and cs0 → 1 limits. Upon inspection of g (x) in (95) we see that the rede?ned kinetic function G(x) is

2 f0 g0 ? H0 . G(x) = g (x) ? f0

(99)

In doing so the potential is rede?ned according to (94) as

2 g0 f0 ? H0 v (?) = V (?) ? f (?). f0

(100)

There is a subtlety in this analysis that should be addressed. In order to reclaim the canonical action we needed to take the limits ω → 0 and cs0 → 1 simultaneously. In our case we took that limit by setting ω = 0 and then letting cs0 approach one. However, this is by no means the only way to take the limit. For example, we could have approached the limit by setting ω = 1 ? c2 s0 and then take the limit as cs0 goes to one. Had we taken the limit from a di?erent direction it is possible that the action that resulted could have been di?erent from the canonical action (73). After some inspection, it can be shown that under a ?eld rede?nition ? = h(? ?) such that f ?1 (?) = (h (? ?))2 , the potential in the canonical limit is given by

2 V (? ?) = (3 ? )H0 e ?

2 H0 (? ??? ?0 ) ? ˙0

2 + g0 f0 ? H0

f (? ?) . f0

(101)

20

Here f (? ?) = f (h(? ?)) is the function f when the canonical limit is taken. It is simple to show that the canonical limit of f is not unique, which means that the potential is also not unique. However, if we rede?ne our potential according to (100) instead, the new potential v is unique, and the action that results is canonically equivalent to (73). 2.1.3 Case 3: f1 (?) and f2 (?) Unknown

We now bring up a case that will be of particular interest to reconstructions of the DBI action. We start by assuming that the action (?, x) has the form (?, x) = P (f1 (?), f2 (?), x), (102)

where f1 and f2 are unknown functions of ?. Unlike the previous cases, the functional dependence of the action with respect to x is assumed to be known exactly. In this case it is possible to obtain a set of algebraic equations of the two unknowns f1 and f2 . The ?rst of these equations can be most easily obtained by going back to the original de?nition of the sound speed (25): c2 s = p,x 2xp,xx + p,x ? P,xx = 1 2x 1 ? 1 P,x . c2 s (103)

This equation together with (31) and the Friedmann equation (37) are enough to ?nd f1 (?) and f2 (?) in terms of the observables. In the next section we will see explicitly how the equations (103), (31) and (37) come together to reconstruct the DBI action from the power spectrum data.

3

DBI in?ation

In realistic string and M-theories, the number of space-time dimensions is 10 or 11 dimensions. The extra 6 or 7 dimensions are compacti?ed to small sizes, leaving the e?ective theory at low energies a theory of physics in four dimensions. The various moduli that control the shape (complex structure moduli) and size (K¨ ahler moduli) of the internal space, also determine the nature of the four-dimensional low-energy e?ective theory. Therefore, ?xing these moduli is an important step in establishing a connection between string theory and the standard model. In recent years, much attention has been paid to ?ux compacti?cations as a potential means of stabilizing string moduli5 . In a ?ux compacti?cation, various ?uxes wrap around closed cycles in the internal manifold creating a potential for the complex structure moduli. The best known of the these takes place in type IIB string theory. Here the internal space is six-dimensional Calabi-Yau and the 3-form ?uxes F3 and H3 create a superpotential that ?xes the complex structure [29]. These 3-form ?uxes source a warping of the geometry of the internal manifold. In the type IIB ?ux compacti?cation, the ansatz of the line element is taken as

?1/2 ds2 (y )g?ν dx? dxν + h1/2 (y )gmn dy m dy n . 10 = h

5

(104)

For a review of ?ux compacti?cations see [28].

21

Here h is the warp factor which is sourced by the ?uxes and varies only along the dimensions of the internal manifold. In DBI in?ation, which we will be considering in this section, the local geometry of the internal manifold is a Klebanov-Strassler throat geometry [30], and is described by the metric gmn dy m dy n = dr2 + r2 ds2 X5 . (105)

Here ds2 X5 is the line element of a ?ve-dimensional manifold X5 , which forms the base of the KS throat. The coordinate r runs along depth of the throat. For our purposes we will only consider motion along r and integrate over the base manifold X5 6 . The warping of the internal space creates a natural realization of the Randall-Sundrum model [33], and has also provided model builders with a new approach to developing string theory based models of in?ation. The most popular in?ation model that makes use of this warping is DBI in?ation, which is the primary focus of this section. In the simplest DBI in?ation models [18, 34, 35] a D3 brane travels along the r direction, either into or out of the KS throat. The D3 brane extends into the 3 non-compact space dimensions and is point like in the internal manifold. The standard DBI action for the D3 brane is SDBI = ? √ d4 x ?g f ?1 (φ) 1 ? 2f (φ)X ? f ?1 (φ) + V (φ) . (106)

√ Here φ = T3 r (where T3 is the D3 brane tension) is a rescaling of the coordinate r and will play the role of the in?aton. The quantity f ?1 = T3 h?1 is the rescaled warp factor. The metric g that appears in (106) is the metric on the 3 + 1 dimensional non-compact subspace which describes the geometry of our familiar 4 dimensional space-time. We will continue to assume that the geometry of the 3 + 1 dimensional subspace is described by the FRW metric with zero curvature. The energy density and pressure in the non-compact subspace due to the brane are given by ρ = f ?1 (γ ? 1) + V, p = (γf )?1 (γ ? 1) ? V. (107) (108)

Here γ is a new parameter, not found in the standard canonical in?ation. In terms of the quantities in the DBI action, γ is de?ned as γ= 1 1 ? 2f (φ)X . (109)

The γ de?ned here is analogous to the Lorentz factor in special relativity, and will henceforth be referred to as the Lorentz factor. The Lorentz factor places an upper limit on the speed of the brane as it travels through the KS throat. Since the kinetic energy of the brane is limited, this allows one to get a su?cient amount of in?ation even with potentials that would be considered too steep to use in standard canonical in?ation.

Fluctuations of the brane position along the transverse directions of the KS throat have been mentioned as a possible source of entropy perturbations [31, 32]. These could serve as a further constraint on the form of the action.

6

22

In our study of the DBI model we will be assuming that the scalar and tensor spectra are approximately (53) and (54), respectively. With these as our in?ationary observables, we found that was constant (59). The fact that is a constant indicates that in?ation will not end on it’s own, and instead some other mechanism such as D3-D3 annihilation [36] must be used to provide a graceful exit. Since our study is concerned more with the physics during in?ation, this topic will not be addressed further. We will now present a generalized DBI action, and show how it is reconstructed from the in?ationary observables.

3.1

A Generalized DBI Model

Having sketched out the general method for reconstructing di?erent types of in?ationary actions in section 2, it is now time to apply these methods to a DBI-type action given by (?, x) = P (x, F (?), V (?)), where

?1 P (z1 , z2 , z3 ) = ?z2

(110)

√

1 ? 2z2 z1 ? 1 ? z3 .

(111)

2 f (?) is the (dimensionless) warp factor in the throat, and V (?) = Here F (?) = α2 Mpl ?2 (αMpl ) V (?) is the (dimensionless) potential. In the KS throat geometry the warp factor is taken to be F ∝ ??4 . The potential V is assumed by many to be quadratic in ?. For the purposes of this study we will not assume a priori any form for the functions F and V , and instead allow the in?ationary observables to determine them. Now that we have established the general form of the action, we can use the procedure outlined in section 2.1.3 to ?nd F and V . Turning to equations (103) and (31) we ?nd the relations

F 1 P,xx = = P,x 1 ? 2F x 2x Solving for F and x: F (?) =

1 ?1 , c2 s

√ x = H2 1 ? 2F x.

(112)

1 ? c2 s , 2 H 2 cs

x = H 2 cs .

(113)

Comparing the second equation above with (31) and recalling the de?nition of (109), we ?nd that cs = 1 . γ (114)

This result is characteristic of DBI in?ation and holds regardless of the warp factor and potential used. Having found F , equation (37) tells us what V is: V (?) = 3H2 + 1 F 1 ?1 . cs (115)

23

Having already found the expression for F in (113), we can now write down the important reconstruction equations for V and F in terms of H, cs and : V (k ) = H2 3 ? 2 1 + cs 1 ? c2 s F (k ) = . 2 cs H2 , (116) (117)

To turn F and V into functions of ? we need to integrate our solution for x (113) to ?nd ?. Taking equation (31) to ?nd an expression for ? ˙ , we ?nd that in the case of DBI in?ation √ (118) ? ˙ = ± 2 cs H. The sign of the right hand side of the equation is ambiguous, due to the square root taken to get this equation from (31). The sign is left arbitrary for now and will be speci?ed later based on the requirement that ? be positive. Once we solve for ?(τ ) in (118) and invert to get τ (?), we can then ?nd a solution for V (?) and F (?). Now that we have laid the ground work for generating the functions of the generalized DBI action, the next section will show how the perturbation spectra are used to obtain explicit expressions for F (?) and V (?).

3.2

The Warp Factor and Potential in DBI In?ation

In this section we will now use the program that was laid out at the end of the previous section to ?nd an exact solution for the warp factor and potential in the action (110). We will again assume that the scalar and tensor spectra have a power-law dependence with respect to the scale k . Therefore, the sound speed, Hubble parameter and are the same as those found in section 2.1.1. Thus, the potential as a function of time is V=

2 H0 (1 + H0 (τ ? τ0 ))2

3? 1+

2

2 H0

(1 + H0 (τ ? τ0 ))?ω

,

(119)

and the warp factor as a function of time is F= (1 + H0 (τ ? τ0 ))ω+2 4 2H0 1?

4 H0 2

(1 + H0 (τ ? τ0 ))?2ω .

(120)

Since we want in?ation to occur we need to be such that < 1, which according to equation (59) implies that nT ≤ 1. However, nT = 1 can be dismissed as physically unreasonable since that would imply cs0 ∝ 1 → ∞. After substituting our solutions for H(τ ) and cs (τ ) into the equation of motion for ?(τ ) we get √ 2 ω ? ˙ = ± 2H0 (1 + H0 (τ ? τ0 ))? 2 ?1 . (121) Once we integrate this expression we can obtain an answer for ?(τ ). As a matter of convenience we will set the value of the integration constant that results to zero. Later, once we have found F and V , we will see that this choice allows for a correspondence between the 24

reconstructed functions and their theoretically derived counterparts. After integrating (121) we ?nd that √ τ 2 2H0 (1 + H0 (τ ? τ0 ))?ω/2 . (122) ?(τ ) = ?dτ ˙ =? ω Since we are interested in eventually connecting the reconstructed action with the standard DBI model we need to keep the in?aton, which is just a rescaled radial coordinate, positive. The sign that we choose in (122) will therefore depend on the sign of ω . We can write the general solution as ?(τ ) = ?0 (1 + H0 (τ ? τ0 ))?ω/2 where √ H0 ?0 = 2 2 ω (124) (123)

In the case where ω > 0 the ?eld ? decreases monotonically to zero as time passes, which implies that the brane is falling into the throat. This corresponds to the UV DBI scenario. If on the other ω < 0, then ? increases monotonically with time and the brane falls out of the throat, which corresponds to IR DBI in?ation. Solving for time in terms of ? 1 + H0 (τ ? τ0 ) = ? ?0

2 ?ω

.

(125)

Substituting this for 1 + H0 (τ ? τ0 ) in the expressions we found for the potential and warp factor we ?nd that V as a function of ? is ? ? V=

2 H0

? ?0

4 ω

? ?3 ?

2 1+

2 H0

? ?0

2?,

?

(126)

and the warp factor as a function of ? is 1 F= 4 2H0 ? ?0

4 ?2? ω

1?

4 H0 2

? ?0

4

.

(127)

Furthermore, when γ is expressed as a function of ?, it takes on a very simple form: γ = γ0 ?0 ?

2

=

8 1 , ω 2 ?2

(128)

where γ0 = c1 . It is interesting to note that (128) is the same as the approximate results s0 found in the theoretically inspired DBI model [34]. The potential and warp factor derived here are the same as those found in [37]. There the authors reconstructed the potential p and warp factor by assuming that the equation of state w = ρ was a constant and that 25

? ∝ τ ?ω/2 . In contrast, we have reconstructed the potential and warp factor under the assumption that the scalar and tensor perturbations are (53) and (54). The non-gaussianity in this DBI model is the same result that one comes across in the literature [6]: fN L = 35 108 1 ?1 . c2 s (129)

This particularly simple result is a general feature of DBI, and is independent of the warp factor and potential. This result for the non-gaussianity (129) also follows from consistency of the reconstruction equations (32) and (36). Thus, (129) can be viewed as a consistency relation analogous to those found in (46) and (50). An interesting generalization to consider is (?, x) = P (g (x), F (?), V (?)) (130)

where P (z1 , z2 , z3 ) is de?ned by (111). As a consistency check, one can easily show that if the non-gaussianity is equal to (129), and the potential and warp factor are given by (119) and (120), respectively, then g (x) = x is a solution to the reconstruction equations (32) and (36). Having found the potential and warp factor as functions of the in?aton, we can now say that our task is at an end. Amazingly enough, despite the complicated form of the reconstruction equations an exact solution for V and F was available even for semi-realistic scalar and tensor spectra. In the next section we will discuss the properties of the reconstructed action, and its correspondence with the theoretically derived action of DBI in?ation.

3.3

Discussion

In section 2.1.1 we found that not all values of the spectral indices lead to in?ationary and/or physically sensible actions. Speci?cally, we showed that unless nT < 1 the matter described by the action was unable to drive an in?ationary phase. Furthermore, when this constraint on nT was considered in conjunction with the requirement that the sound horizon decrease as in?ation occurs, the scalar spectral index had to be bounded like ns < 2. These results hold for any reconstructed action that was derived assuming the observational inputs (53) and (54). However, even if these constraints are satis?ed, it is not guaranteed that the reconstructed action is physically sensible when interpreted in the context of a given theoretical construction. For example, if we are to interpret the action reconstructed in this section as a DBI action, then γ > 0. Doing so would be contrary to its de?nition (109) within the context of DBI in?ation. In this case, since γ = c1s the fact that we already have enforced the constraint cs > 0 automatically keeps γ positive. We will see later, however, that the constraints found in 2.1.1 are not su?cient for our reconstructed action to be interpreted as a DBI action. First, let’s consider the warp factor (127) and what constraints it places on the observables. Suppose nT < ns . In this case the exponent of ? in front of the square brackets in (127) will always be negative. Therefore, for small values of ? the leading order behavior of F will go 26

nT

1

Κ 0

0.5 -2 -1 1 2

ns

-0.5

-1

Κ 0

-1.5

-2

Figure 3: Plot depicting the regions of the ns -nT parameter space such that F diverges at the origin. The region in the upper left hand corner is the region of IR DBI in?ation while the corner at the bottom right hand corner corresponds to UV DBI. Only those values of ns and nT with ns < 2 and nT < 1 were considered, since any point outside that region would lead to an unphysical and/or non-in?ationary action.

like ??a where a > 0. Thus, the warp factor increases as we fall into the throat, which is what we would expect for a warped compacti?cation in string theory. On the other hand, if nT > ns the leading order behavior of F will be ? to some positive or negative power, depending on the relative di?erence between ns and nT . If the di?erence between nT and ns is too small, then to leading order, F will scale like ? to some positive power. This indicates that the warp factor gets smaller as we reach the bottom of the throat, which is a scenario that is di?cult to embed into a string theory compacti?cation. However, If the di?erence between ns and nT is large enough, then it is possible to get a more sensible solution where F ? ??a . In general, the condition that F increases as we approach the bottom of the throat implies that ?1 ? 2 <0 ω ? ω < ?2 or ω > 0. (131)

The regions in the ns -nT parameter space where the condition (131) is satis?ed are shown in ?g. 3. The region shown in light grey in ?g. 3 is de?ned by ω > 0, or equivalently nT < ns . This region corresponds to the UV phase of DBI in?ation. The region in dark grey is de?ned by ω < ?2, or equivalently nT > ? ns2 , and corresponds to the IR phase. ?3 As ?g. 3 illustrates, if ns is restricted to the presently favored value ns ? 0.96, then the value of nT is tightly constraint in the IR region but relatively unrestricted in the UV phase. Furthermore, since nT < 1, only those models in the UV phase can have blue-tilt. Recent CMBR data favors a blue-tilted spectrum, but only if there is a running spectral index [8]. Although running spectral indices would be an interesting extension of this analysis, we will leave this topic to future studies. An unpleasant feature of the warp factor (127) is that it becomes negative when ? > √ ?0 H0 / . This is particularly distasteful since the metric (104) depends on f 1/2 , which 27

means that at su?ciently large ? the metric is imaginary. The values of τ where the warp factor is positive are given by

1

cω ? 1 τ ? τ 0 > s0 H0

1

for ω > 0 (UV), for ω < 0 (IR). (132)

cω ? 1 τ ? τ0 < s0 H0

One can show using equation (61) that this is equivalent to the bounds in (65). This is no coincidence; it is a result of the fact that F is proportional to 1 ? c2 s (113). Therefore, for the same reasons that were explained in section 2.1.1, the perturbations (53) and (54) can only be used as approximations. The inequalities in (132) tell us that F is a valid warp factor towards the end of in?ation in the case of UV DBI, and at the beginning of in?ation in IR DBI. The time τ0 at which the initial conditions are speci?ed should be at the beginning of in?ation in the UV scenario, and at the end in the case of IR DBI. If we choose τ0 in this manner then the approximations for the perturbation spectra (53) and (54) will lead to realistic sound speeds for the entire duration of the in?ationary episode. It is worth asking if the F that we have derived in (127) can approximate the AdS warp factor derived from theory. It is clear from (127) that this can be achieved if and only if ω = 2. However, the only way we can get ω = 2 is if either i.) nT = 2 ii.) nT → ∞ or iii.) ns = 1. As we have already seen, case i.) is unphysical, and case ii.) is di?cult to imagine taking place. While case iii.) is unlikely to be true exactly, it is nevertheless the more realistic of the three, especially when you consider that observation suggests that ns ≈ 0.96. If we do set ns = 1, the warp factor and potential become F (?) = 1 2 1 ? 2, 4 ?4 2 (133)

V=

3 2 32 2 ? ? ? . 2 1 + 2 ?2

(134)

In the case where the ?eld range is small7 these are approximately

2 2Mpl 1 f (φ) ≈ 2 4 4 , α φ

(135)

2

V (φ) ≈

α2 (3 ? 2 ) 2 φ. 2

(136)

Where we have reverted back to the standard, dimensionful f , V and φ for clarity’s sake. It is a bit of a surprise that it in process of trying to recover the AdS warp factor we have stumbled upon the commonly used potential in UV DBI in?ation. If we take (135) and

In [38] it was shown that DBI in?ation is only consistent when the magnitude of the in?aton ?eld is sub-planckian: ? 1.

7

28

demand that it is consistent with the theoretical result we can arrive at a condition on terms of the D3 charge. Recall that in the KS throat f (φ) is given by f (φ) = where T3 =

1 1 (2π )3 gs (α )2

in

2T3 R4 , φ4

(137)

and R4 = 4πgs N (α )2 π3 . Vol(X5 ) (138)

Consistency with (135) demands that

2 2Mpl πN = 2 4 α Vol(X5 )

?

≈

102 . N 1/4

(139)

In order to get an in?ationary phase N ≈ 1010 , putting us well within the range of validity for the supergravity approximation. While it is interesting that the standard D3 brane DBI model can be recovered from a near scale invariant scalar power spectrum, it has been acknowledged that this in?ation model is problematic. In [38] Baumann and McAllister found that while present bounds on non-gaussianity imply that N 38, primordial perturbations imply that N 108 Vol(X5 ). These two limits are incompatible unless Vol(X5 ) 10?7 . It is not clear that such a space could be naturally embedded into a string theory compacti?cation. More general warp factors and potentials have been considered in [22]. There it was found that models could not simultaneously satisfy bounds on the ?eld range and observational bounds on the non-gaussianity. Therefore, even though our warp factor and potential matches the theoretically based predictions, the problems inherent in the DBI model carry over into its generalizations.

4

Conclusion

In this paper we have presented a method for deriving the actions of single ?eld in?ation models using CMBR data. This method allows one to derive up to three unknown functions of the action using the scalar perturbation Ps , tensor perturbation Pt and the non-gaussianity fN L . After stating the reconstruction equations, we carried out the reconstruction procedure for two simple examples. For the purposes of the reconstruction, we assumed that the scalar and tensor spectra were power-law dependent on the scale k , with the spectral indices kept as free parameters. In the ?rst example we assumed that the action had the form shown in equation (47), and used the reconstruction equations to obtain the action as a function of the spectral indices. In this example there were only two unknown functions, thus the reconstruction equations also lead to a consistency relation (50) between the fN L , cs and the slow roll parameters. However, this consistency relation is only well de?ned when the sound speed is not a constant. In the second example, the action depended on three unknown functions and therefore required all three reconstruction equations. In order to simplify the discussion we took 29

as our input for the non-gaussianity fN L = 0. Although we were unable to express the action in terms of elementary functions we were able to obtain the action numerically and approximately assuming cs0 ≈ 1 and ω ≈ 0. We showed that the action in this example was canonically equivalent to the canonical action derived at end of the previous section. In discussing this example we also pointed out possible ambiguities in the program relating to how one de?nes a separation between the kinetic and potential terms. In section 3, we used the procedure to derive and study the warp factor and potential in a generalized DBI in?ation model. Again, we assumed that both of the perturbation spectra scaled like k to some power. Exact expressions for the warp factor and potential were then derived, each having an explicit dependence on the spectral indices. The demand for a physically sensible DBI in?ation model placed constraints on the spectral indices. In addition we found that the derived action approximates the original UV DBI in?ation model in the case where φ Mpl . Unfortunately, the problems that have plagued UV DBI in?ation are still present in our case. This procedure was shown to be useful in studying how the action of a general in?ation model depends on the observables. For example, we found that if the scalar and tensor perturbation spectra went like k to an arbitrary power, the reconstruction would lead to a realistic in?ationary model only if nT < 1 and ns < 2. Furthermore, to keep the speed of ?uctuations from becoming superluminal, the range of k over which the approximations for the spectra (53) and (54) are taken, had to limited. When we reconstructed a generalized DBI action in section 3, further constraints were needed to keep the action compatible with an interpretation of DBI in?ation. Speci?cally, we found that in order to keep the warp factor positive, the ?eld range had to be limited. Furthermore, in the theoretically motivated DBI model, the warp factor increases as we reach the bottom of the warped throat. In order for this to be true in our reconstruction, the spectral indices needed to satisfy the additional constraints: ω < ?2 or ω > 0. In this paper we have only considered the simplest of the DBI in?ation models, which unfortunately su?ers from several inconsistencies. However, there are many extensions of the D3 brane DBI model that can circumvent some of the problems of the original. Some of these extensions include using wrapped D5 branes [39, 40], multiple D3 branes [41], and multiple throats [19]. Each of these models has its potential advantages and drawbacks. Applying our reconstruction procedure may help to further elucidate their relative strengths and weaknesses. Furthermore, we have limited ourselves to perturbations with simple power law behavior. However, this naive assumption may be incorrect. It is easy to imagine that the spectral indices themselves are also scale dependent. Based on the results of this paper we can predict what kind of e?ect a running spectral index would have on the physics of the underlying models. For instance, in the generalized DBI model it is possible for the spectral indices to change during in?ation in such a way as to pass from the IR to the UV phase8 . Transition between phases would correspond to a completely di?erent physical scenario, one where the brane falls out of one throat and back into another. Therefore, running spectral indices would describe multi-throat DBI in?ation. A model which has so far been shown to

Fig. 3 implies that in?ation can change between UV and IR phases only if it passes through the exactly scale-invariant point: (ns , nT ) = (1, 1).

8

30

be internally consistent [42]. This study has also raised some other questions that may be worth investigation. In particular what is the relation between actions that yield the same observables. It may be possible to de?ne a group of transformations which leave the perturbation spectra and the non-gaussiantiy invariant. Such a set of transformations would allow us to classify actions based on the observables they yield. Another interesting possibility that came out of this study is the idea of using the reconstruction equations as a way to generate consistency relations between fN L , the sound speed cs and the slow roll parameters. These questions will be left for future studies.

5

Acknowledgements

We would like to thank Louis Leblond and Dragan Huterer for helpful discussions. We would also like to give a special thanks to Daniel Chung for his insightful comments on our paper.

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超声波震动膜生物反应创新工艺产品简介一、*CMBR* 创新工艺技术 1.超声波膜生物反应创新工艺的概述 *CMBR* 就是膜分离技术和循环载体生物膜法(生化技术) 、低频超声波...

32 ?32 2.*CMBR*型MBR膜组件规格 表2-1 SJ-*CMBR*浸没式柱状

中国矿业大学,江苏省徐州市,221008)摘要 采用外置式膜生物反应器(*CMBR*)技术处理拜城县众泰煤焦化有限公司焦化 废水,该工艺采用最新的管式膜技术,强化处理能力,能...

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