A Study of Structure Formation and Reheating in the D3/D7 Brane Inflation Model

A Study of Structure Formation and Reheating in the D3/D7 Brane Inflation Model

Robert H. Brandenberger rhb@hep.physics.mcgill.ca    Keshav Dasgupta keshav@hep.physics.mcgill.ca    Anne-Christine Davis A.C.Davis@damtp.cam.ac.uk 1) Department of Physics, McGill University, Montréal, QC, H3A 2T8, Canada 2) DAMTP, Centre for Math. Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, U.K.
July 19, 2019
Abstract

We study the spectrum of cosmological fluctuations in the D3/D7 brane inflationary universe with particular attention to the parametric excitation of entropy modes during the reheating stage. The same tachyonic instability which renders reheating in this model very rapid leads to an exponential growth of entropy fluctuations during the preheating stage which in turn may induce a large contribution to the large-scale curvature fluctuations. We take into account the effects of long wavelength quantum fluctuations in the matter fields. As part of this work, we perform an analytical analysis of the reheating process. We find that the initial stage of preheating proceeds by the tachyonic instability channel. An upper bound on the time it takes for the energy initially stored in the inflaton field to convert into fluctuations is obtained by neglecting the local fluctuations produced during the period of tachyonic decay and analyzing the decay of the residual homogeneous field oscillations, which proceeds by parametric resonance. We show that in spite of the fact that the resonance is of narrow-band type, it is sufficiently efficient to rapidly convert most of the energy of the background fields into matter fluctuations.

pacs:
98.80.Cq
preprint:

I Introduction

In recent years a lot of interest has focused on the interface area between superstring theory and cosmology. The reasons are twofold. Firstly, cosmology can provide a possible arena to test string theory observationally. Secondly, superstring theory may be able to resolve the conceptual problems from which the current realizations of scalar field-driven inflation suffer (see e.g. RHBrev () for a discussion). In particular, a lot of attention has recently been devoted to attempts to obtain periods of inflationary expansion of space in compactifications of superstring theories to four space-time dimensions. Since such compactifications contain a large number of scalar fields and since some of those remain massless before supersymmetry breaking, string theory gives rise to promising candidates for the inflaton, the scalar field driving the inflationary expansion of space (see e.g. Linderev (); Cliffrev (); JCrev (); EvaLiam () for recent reviews of attempts to obtain inflation from string theory compactifications).

D-branes have played a particularly important role in the recent approaches to obtaining inflation from string theory Dvali (); Stephon (); Shafi (); Cliff (); JuanGB (); KKLMMT (). A particularly promising model is the D3/D7 brane inflation model, in which the inflaton is the separation between the two branes in the directions transverse to both branes DHHK (). This model features a shift symmetry shift () for the inflaton field which ensures that at the classical level, the potential is flat along the inflaton direction (this shift symmetry is broken under quantum corrections kors ()). Inflation is induced by supersymmetry breaking terms. Various aspects of this model have been studied previously herd (); D37strings (); previous ().

The large number of light moduli fields in string inflation models 111These moduli in general could be the complex or Kähler structure moduli of the internal space. They could also be the moduli of the branes in our theory. Once the Kähler and complex structure moduli are fixed by fluxes GVW (); DRS (); GKP (), they could be the remaining brane moduli. In this paper, since we are not fixing any of the moduli, the light fields could come from all the above moduli., however, leads to a new danger: light fields which are not the inflaton are potential entropy modes and can give rise to entropy fluctuations. The “curvaton” Mollerach (); SY (); LM (); Moroi (); LW (); Sloth () and “modulated reheating” DGZ (); Lev (); MR (); Uzan (); Vernizzi () scenarios are examples of this effect. Non-parametric generation of fluctuations from a global symmetry breaking (also involving an key isocurvature field) has been investigated in KRV (); Matsuda ().

As pointed out in BaVi (); FB2 (), these entropy modes may undergo parametric instability during the initial stages of reheating. In particular, this instability occurs on super-Hubble (but sub-horizon) scales. This is possible FB1 ()since the background fields carry the causal information to scales larger than the Hubble radius. The entropy fluctuations, in turn, will seed a curvature mode (which we will call “secondary” mode). There are two field scalar field toy models in which this secondary curvature mode dominates over the primary mode, the pure adiabatic linear perturbation theory mode. In fact Zibin () the secondary mode can grow to be larger than unity before back-reaction shuts off the parametric instability which drives the growth of the entropy fluctuation. In this case, the model is ruled out by the observed (small) amplitude of curvature fluctuations on large cosmological scales.

The D3/D7 brane inflation model is a prototypical example in which there is an entropy mode which can undergo resonance during the initial stages of reheating, the preheating JB1 (); KLS1 (); JB2 (); KLS2 () phase. In this paper, we study the growth of this entropy mode and calculate the magnitude of the induced secondary curvature fluctuations. It turns out that for reasonable particle physics parameters, the secondary fluctuations remain in the linear regime at the end of the reheating phase. However, their amplitude can be (in the absence of back-reaction effects) larger than the amplitude of the primary mode. This would necessitate changing the parameters of the model in order to reproduce the observed magnitude of the large-scale cosmological perturbations.

Note that a similar conclusion was recently found Larissa () in another model of brane inflation, the KKLMMT model KKLMMT (). In that model, it had already been observed that due to their large phase space enhancement, second order fluctuations may give a dominant contribution BC2 (). We should emphasize that the effect we are calculating in this paper is an effect which arises in linear cosmological perturbation theory. Both the linear effects described here and the quadratic processes discussed in BC2 () and elsewhere (e.g. Losic (); Patrick ()) can be operational at the same time. In fact, the enhanced growth of the linear perturbations which is the focus of this work will increase the strength of the quadratic processes. We expect that effects similar to those discussed in BC2 () will also arise in the D3/D7 system.

The structure of this paper is as follows: we first revisit the calculation of the amplitude of the spectrum of cosmological perturbations in the D3/D7 inflationary model based on the magnitude of the primary adiabatic mode alone. Next, we give an approximate but analytical study of reheating in the D3/D7 brane inflation scenario. The initial stages of reheating are governed by a tachyonic decay channel. The same tachyonic resonance channel leads to exponential growth of super-Hubble entropy modes, which then in turn induce a secondary curvature mode. The last major section of this paper focuses on the determination of the amplitude of this mode.

Our analysis of the reheating phase provides results which are interesting in their own right. A new aspects of our analysis is that we allow for an initial offset in the value of the “waterfall” field. Such an offset could arise as a consequence of the back-reaction of long wavelength 222Long means longer than the Hubble radius. fluctuations of the matter fields. Such an offset will suppress the formation of domains on small scales. If the offset were sufficiently large, it could prevent the onset of the tachyonic instability. However, if we take the offset to be generated by the abovementioned fluctuations, we find that for most of the realistic space of parameters of the D3/D7 inflationary model, the resulting values of the field lie in the configuration space region for which the tachyonic resonance channel tachyonic () is effective. Thus, the initial stages of the transfer of the energy in the inflaton field to matter fluctuations will proceed via the tachyonic instability in a time interval which is less than the characteristic time it takes for the matter fields to oscillate about their minima. A substantial part of the initial inflaton energy is converted to localized nonlinear fluctuations during the period of tachyonic resonance. Neglecting the back-reaction of these fluctuations on the homogeneous background, we see that a fraction of order unity of the initial inflaton energy density still remains in the background when the tachyonic instability shuts off. We are interested in demonstrating analytically that the transfer of energy from the inflaton to fluctuations is effective in rapidly draining almost all of the energy from the homogeneous fields. The dominant energy conversion processes are those related to the interactions of the nonlinear fluctuations. These have been studied numerically in great detail in tachyonic (). Since our aim is to demonstrate analytically that within less than a Hubble time the energy can be effectively drained from the homogeneous background, we will neglect the nonlinear interactions and focus on the evolution of the homogeneous component of the fields once the tachyonic resonance shuts off (due to our approximation scheme, we will thus only obtain an upper bound for the decay time). We show that the decay of the residual homogeneous component of the inflaton field occurs via a parametric resonance instability JB1 (); KLS1 (); JB2 (); KLS2 (). In spite of the fact that the resonance turns out to be of narrow-band type, it leads to an efficient conversion of most of the energy from the background inflaton field to matter fluctuations.

One of the advantages of the D3/D7 brane inflation model is that the matter fields are directly associated with the branes whose separation provides the inflaton field333Although not directly related to the main theme of the paper, we would like to point out that some recent papers beasly () have shown how to get standard model with chiral fermions in a D3/D7 set-up from F-theory. Of course one need to change our background manifold to a certain Del-Pezzo surface to accomodate the standard model. It would be interesting to realise D3/D7 inflationary dynamics with this manifold.. As a consequence, there is a direct coupling between the inflaton field and the matter fields, rendering it easy, as we will see, to obtain efficient reheating. This is a feature the D3/D7 brane inflation model shares with the D4/D6 model in which reheating was analyzed in Easson () and with the brane-antibrane inflation models in which reheating was studied in CFM (); BC () (see also TM ()), but contrasts with the situation in inflationary models in which the inflaton dynamics and the standard model fields live in different throats, in which case efficient reheating is more difficult to obtain (see e.g. BBC (); KY (); DSU (); FMM (); CT (); PL ()).

The outline of this paper is as follows. In the following section, we review the D3/D7 brane inflationary model, with particular emphasis on the form of the potential for the inflaton field. In Section 3 we discuss the vacuum manifold of the model, and in Section 4 we revisit the calculation of the spectrum of cosmological perturbations. Section 5 focuses on the dynamics of the inflaton and of the other relevant scalar fields of the model before the onset of reheating, introduces the conditions for effectiveness of tachyonic resonance and demonstrates that for most of the relevant parameter space of the theory, the tachyonic resonance condition is satisfied. However, the tachyonic instability turns off before most of the energy of the inflaton field is released as matter fluctuations. To obtain an upper bound on the time it takes for the inflaton energy stored in the residual oscillations to further dissipate, we study the later dynamics of the homogeneous field components which proceeds via the less efficient parametric resonance channel (Section 6). Section 7 is devoted to the computation of the secondary curvature fluctuation mode, the mode induced by the entropy fluctuations.

Ii The D3/D7 Brane Inflation Model

The simplest way DHHK () to construct a D3/D7 inflationary model in type IIB theory is to place a D3 brane (which we take to be in the directions) parallel to a D7 brane (which is in the directions) in flat space-time and break the resulting supersymmetry completely by switching on non-primitive gauge fluxes on the D7 brane. The BPS condition is broken, and the D3 brane starts to move towards the D7 brane. The distance between the D3 brane and the D7 brane in the plane appears on the world-volumes of the branes as a complex scalar , whose value changes as the distance between the branes changes. This scalar field is identified with the inflaton, and the motion of the D3 brane towards the D7 brane is the slow-roll of hybrid inflation.

Once the D3 brane comes close to the D7 brane, the open string between the branes become tachyonic because of the non-primitive fluxes on the seven brane. This is the top of the hybrid inflationary scenario where one would expect a tachyonic phase transition. The D3 brane then falls onto the D7 brane and dissolves as a non-commutative instanton DHHK (). The non-commutative instanton is not point-like and therefore the final configuration of a dissolved instanton is non-singular instanton (). One can also argue that in the end supersymmetry can be restored.

This simple picture has many inherent problems. Firstly, being non-compact, the gravitational solution is ten dimensional and a direct dimensional reduction will give us vanishing Newton’s constant in four dimensions. One way out is to compactify the other six dimensions. Secondly, once we compactify, new problems have to be faced. An immediate issue that on a compact space, charge neutrality must be enforced, which requires introducing additional elements to the construction. Next, the problem of moduli stabilization arises, because without stabilizing the moduli, the system will decompactify back to the ten dimensional solution via the so-called Dine-Seiberg runaway DiS (). Thirdly, one needs to avoid an over-production of cosmic strings when the D3 dissolves in the D7 brane.

Ways to solve these problems were addressed in the papers DHHK (); previous () where the proposal was to consider a warped solution of the form

(1)

where are the warp factors and are the coordinates of the internal space. In DHHK () the metric was chosen to be the metric of with being a combination of an orbifold and an orientifold operation (see DHHK () for details). Both the D3 and the D7 are points on the , and the motion of the D3 is trigerred by three form fluxes and 444Observe that once is dualised to form gauge field on the world volume of D7 brane, this would trigger motion of the D3 brane towards D7 brane.. The D3 brane is oriented along and therefore is also a point on the K3 manifold. The D7 brane wraps the K3 and is oriented parallel to the D3 as we discussed before. Supersymmetry is broken either by making

(2)

where , or by switching on ISD (1,2) or (0,3) forms (recall that ISD (1,2) fluxes also break susy. For more detailed analysis of this, see the recent paper DFKS ()). The Hodge star is on the internal space and (2) gives rise to non zero supersymmetry breaking potential because of its IASD property.

Once supersymmetry is broken, there would be non-trivial potential between the branes. The complete picture can be worked out easily from the low energy type IIB supergravity lagrangian in the presence of D-brane sources in the following way (see also D37strings ()):

(3)

where the dotted terms involve axio-dilaton plus other higher form interactions, is a constant and with . where is the gauge field on the D3 brane, is the gauge field on the D7 brane, and () are the corresponding scalars on these branes. For more details see D37strings ().

The first term in eqn. (II) when reduced to four dimensions give rise to the kinetic terms for the complex and Kähler structure moduli. The second term is the potential term for the moduli:

(4)

and therefore fixes (at least) all the complex structure moduli and the axio-dilaton (the term goes away for the no-scale models).

The second line in eqn. (II) is related to the kinetic terms for the gauge fields on the D-branes and the inflaton (if we ignore the scalar ). The other scalar doublet denote the hypermultiplet scalars with the corresponding D-term and F-term potentials being given in the third and the fourth lines of eqn. (II). The potential , whose value will be specified later, denotes the additional attractive potential between the D3 and the D7 brane. Finally, the non-perturbative potentials stabilize the remaining Kähler structure moduli.

The above picture is almost complete except for an important subtlety. The potential that fixes the complex structure moduli in eqn. (4) also gives a large mass to the inflaton field because the Kähler potential in eqn. (4) picks up a dependence on the inflaton witten85 (); KKLMMT (). The potential for the inflaton then becomes very steep and consequently do not allow slow roll to happen in generic models. In the D3/D7 model this could be avoided (at least at the tree level) by allowing a shift symmetry, but this symmetry is broken by the term in (II). Thus a generic model of inflation from string theory is very complicated (see BCDF (); mcall (); krause () for some recent developments) and since our aim here is to make the first step in understanding structure formation and reheating after inflation, we will ignore the subtleties associated with moduli stabilization.

Therefore, from the analysis presented above in eqn. (II) we will only concentrate on the F-term and D-term potentials for the hypermultiplet scalars (which we will henceforth label as and instead of the complex doublet ) and the inflaton , and assume that the contributions from the background fluxes given in eqn. (4) and the non-perturbative contributions fix most of the moduli. Note however that, as we mentioned earlier, there are other contributions to the inflaton potential coming from the Kähler potential in (4). For the simple analysis that we present here, we will ignore this contribution. More details on this will appear elsewhere.

Looking from the D3 point of view, the complex scalar that parametrises the , and a set of complex scalars and arising from the hypermultiplet-string between the branes give rise to the following potential:

(5)

plus the additional term that we will add later. Here we have taken a more generic case by switching on and as the FI terms to break susy.

There is also another subtlety related to the issue of moduli stabilization that we should mention now. Due to background NS and RR fields, one might expect that all the complex structure moduli should be naturally fixed via a GVW potential GVW () (see also DRS (); GKP ()). But our background is time-dependent, so the issue of fixing moduli should be addressed at any given time. Similarly, one might also expect that the supergravity no-scale structure will be broken by allowing a gaugino condensate along the lines of KKLT (); DDFGK (). These problems are not yet solved in a time dependent background, and an attempt to address these issues along the lines of previous () is under way (see also Brax () for another recent study of moduli corrections to D-term inflation).

Finally, the issue of cosmic strings can be addressed by going to the full non-perturbative F-theory picture of the D3/D7 system (see D37strings (); lumps () and the second paper of previous ()). From F-theory it is easy to generate global symmetries, and an global symmetry can convert the cosmic strings to semi-local strings of Achucarro-Vachaspati AV (); UAD () type (see also Burrage ()). In terms of the D3/D7 system, this is simply a doubling of the number of D7 branes. These semi-local strings reduce the severity of the problems generated by cosmic strings UAD () (they do not completely eliminate the problems, as a recent study Jon () has shown).

Iii The Vacuum Manifold of the Model

From the previous section, we see that the potential which describes the dynamics of the complex inflaton field (the separation of the two branes in the 4-5 plane) and of the two complex scalar fields and associated with the lowest energy modes of the strings stretching between the branes is given by (II).

The vacuum manifold is the set of field configurations which minimize the potential. The potential has been normalized such that the minimal value of the potential is . To be at the minimum of the potential, the inflaton field must have relaxed to . A special case which we will focus most of our attention on is the case with . Introducing radial and angular coordinates with each field:

(6)
(7)

we see that the conditions to have become

(8)

The vacuum manifold is given by

(9)

It is the manifold discussed in previous (). There are no non-compact flat directions.

Another interesting special case occurs if and . In this case, setting the potential to zero yields the conditions

(10)

In this case, the vacuum manifold is the circle given by

(11)

There are no non-compact flat directions.

In the general case there are no flat directions, either, and the vacuum manifold is given by

(12)

In the following, we will focus mostly on the special case .

Iv Spectrum of Cosmological Fluctuations

In this section we will review the calculation of the amplitude of the primary adiabatic perturbations. This is the mode which is usually considered in the literature and which is used to fix the model parameters. In Section 7 we will turn to a discussion of the entropy mode.

In the D3/D7 brane inflation scenario, inflation takes place while the branes are widely separated, i.e. while is large. From (II) it follows immediately that for large values of the inflaton, the scalar fields and will vanish at the classical level (their quantum fluctuations will play an important role in the following section).

Before turning on any supersymmetry breaking, and even after introducing the supersymmetry-breaking parameters and at the tree level, the configuration of branes is BPS, there is no force between the branes, and will remain constant. One loop supersymmetry breaking terms will result in an additional one-loop contribution to the potential

(13)

In the language of supersymmetric effective field theory, this term corresponds to D-term supersymmetry breaking Dterm (). We are omitting the quartic contribution to the symmetry breaking potential which corresponds to F-term supersymmetry breaking. In the language of D3/D7 system, the above potential appears naturally from the supergravity solution of a D3-brane separated from a D7-brane by a radial distance as DHHK ():

(14)

where is related to the D7 brane gauge fluxes. Therefore the identification of with some components of the gauge fluxes on the D7 brane is consistent with expectation. We also see that in (13) is related to the string scale in (14).

For small values of the string coupling constant , this extra term in the potential will lead to a slow rolling of the inflaton field towards . Initially, the values of and will vanish. Note that the presence of (13) lifts the flat direction, preferring the value . For a sufficiently small value of , a tachyonic instability of the field will set in. The critical value below which this instability appears is given by

(15)

the condition coming from setting the second derivative of the potential (II) to zero at . From direct open string calculation this is given by

(16)

Again we see that the identification of with some components of the gauge fluxes on the D7 brane is consistent with expectation.

Since there is no instability in the direction, will remain zero, again modulo quantum uncertainties which may become important during the later stages of preheating. We will therefore, for the moment, set and consider the dynamics of the reduced configuration space consisting of the inflaton and .

The inflationary scenario realized here is a specific case of hybrid inflation hybrid (), and more precisely of supersymmetric hybrid inflation Dterm () (see also Pterm ()). Whereas it is the field which is slowly rolling and producing the density fluctuations, it is the potential energy of the field which generates inflation. Fluctuations in general hybrid inflation models have been studied before (see, in particular, Renata (); Pterm ()).

Using the standard theory of cosmological perturbations in inflationary cosmology (see e.g. MFB (); LR () for comprehensive review articles and RHBrev2 () for a pedagogical introduction) we obtain as amplitude of the spectrum of scalar metric fluctuations

(17)

where is the reduced Planck mass. In terms of the D3/D7 model, this could be derived from the fluctuation of the background metric when the D3 brane moves towards the D7 brane. Now, making use of (II) and (13) we obtain (see also D37strings ())

(18)

This is to be evaluated at the value of the inflaton (we will from now on omit the absolute value sign for this field) when modes which are being observed today on large scales exit the Hubble radius during the period of inflation. For high scale inflation this happens Hubble expansion times before the tachyonic instability develops, at a value . Assuming that the acceleration of during inflation is negligible, then is given by

(19)

where is evaluated at (we solve the scalar field equation of motion working backwards in time assuming constant to obtain (19)).

Provided that

(20)

the second term on the right hand side of (19) is smaller than the first. In this case, (18) yields

(21)

On the other hand, if the inequality in (20) is reversed, then the result becomes

(22)

V Scalar Field Dynamics and Conditions for Tachyonic Resonance

The cosmological evolution starts with a separation between the branes which is large on string scale. At large values of , the coupling between and the string fields and constrains the latter to be zero, modulo classical and quantum fluctuations.

Neglecting the classical and quantum fluctuations of and , the cosmological evolution is as follows: the inflaton field rolls down the “valley” of its potential until a tachyonic instability for sets in. This occurs at the value given by (15).

At that point, the evolution will take place in the complex two-dimensional field space of and . There is no instability in direction, and hence will remain zero. At the background level, the term proportional to induces a linear confining potential for .

Let us now focus on the potential in direction for a fixed value of . This potential has a negative effective mass (i.e. a tachyonic instability) for small values of only. For larger values, the effective square mass is positive, leading to oscillations about the minimum of the potential.

The phases of and do not play an important role in most of our considerations, and hence we will set them to zero in the following unless indicated otherwise. This will simplify the notation. We will briefly return to the issue of these phases in the concluding section.

A new aspect of our work is the inclusion of the possibility that the field has an offset from the symmetric point at the end of the period of inflation 555Note that such an offset has been discussed in a different context in Rachel ().. Such an offset in any given Hubble patch will in general be produced by the fluctuations of the field on super-Hubble scales Starob (); BR (). The offset will suppress the ability of small-scale quantum vacuum fluctuations of to produce topological or non-topological defects on small length scales at the beginning of the resonance process 666A quantitative study of the issue is currently in progress.. Such defect formation and the subsequent non-linear interaction of the defects was seen in previous work tachyonic () to be the most important aspect of reheating in hybrid-type inflation models like the one we are considering. In our analytical study of reheating, we will assume that the effect of the small-scale vacuum fluctuations is sub-dominant compared to the effect of the offset. Even if the small-scale fluctuations dominate the reheating dynamics, they will unlikely influence the evolution of the large-scale metric fluctuations which is the focus of most of our interest.

The first question to ask concerning the effects of the offset is whether it can prevent the onset of the “tachyonic” (or “negative coupling”) GB2 (); tachyonic2 () resonance. For the sake of concreteness, we will assume that the offset is generated by the long wavelength fluctuations of which were generated during the period of slow-roll inflation and check to see if they are sufficiently small such that, when reaches the onset of the instability, will in fact be in the region of tachyonic evolution. If this is the case then, as discussed in tachyonic (), the energy transfer from the coherently evolving fields to matter fluctuations (the “preheating” phase of the reheating process) will be very rapid and complete before oscillations of and can set in. If, on the other hand, the fluctuations of are sufficiently large, then there will be no tachyonic preheating phase, and the energy transfer from the inflaton to the matter fields will proceed by the parametric instability channel first discussed in JB1 () (see also DK ()) and discussed comprehensively in KLS2 ().

Before estimating the magnitude of the fluctuations of , we note that the slow rolling of the inflaton field will prevent the onset of the instability of until

(23)

Before this condition is satisfied, the evolution of the dynamical system in the plane corresponds to a horizontal straight line (the solid curve in Figure 1).

Figure 1: Sketch of the evolution of the fields and in configuration space. The solid line represents the evolution of the background fields. The cosmological evolution begins in the region of slow-roll (to the right and below the dashed curve). The slow-rolling ends when the dashed line is reached, after which a period of quasi-periodic oscillations about the ground state and sets in.

We next find the region of phase space where the dynamics is dominated by the instability in the direction. From the potentials (II) and (13) (with ) it follows that the phase space curve where the two derivatives in (23) are equal is given by the following relation:

(24)

Above this curve, the dynamics is dominated by the instability. In the limit where is small, we can simplify the potential such that the equation relating and has no quadratic or cubic powers of . If we also ignore the phase of , and work in the limit (this approximation is good except at the beginning of the resonance) we get the following approximate relation:

(25)

The minimal value of is taken on when , which yields a value of about . The curve where the condition (25) is satisfied is represented by the dashed curve in Figure 1.

The condition for efficiency of tachyonic reheating is

(26)

evaluated at the phase space point where the phase space trajectory of the background dynamics of our system crosses the dashed curve given by (25). This would mean

(27)

From here it is clear that the conditions (26) and (27) correspond to

(28)

which along the trajectory of Fig. 1 is automatically satisfied for small values of if .

We must now check whether the condition (26) is satisfied including the effects of fluctuations of . Thus, we must calculate the magnitude of the root mean square fluctuations of . Since at the beginning of the cosmological evolution, for large values of , the valley of the potential is very steep, and since during inflation all pre-existing classical fluctuations are red-shifted, we expect the classical fluctuations to be negligible. Hence, we focus on the quantum fluctuations. The quantum vacuum fluctuations averaged over a Hubble volume are given by integrating the Fourier space quantum vacuum fluctuations for all comoving modes which are super-Hubble Starob (); BR ():

(29)

where the subscript refers to the fact that we are computing a quantum vacuum expectation value, and the is the comoving Hubble constant. While the field has a value much larger than , the effective mass for the waterfall field will be large, and the vacuum fluctuations small. However, if the effective mass remains smaller than the Hubble rate for several Hubble expansion times before the onset of the tachyonic instability, then the field fluctuations will approach the values obtained for a massless field. It can easily be verified that the above condition for “effective masslessness” will be satisfied if (20) holds. In this case, one obtains

(30)

where the subscript now indicates that we are taking the quantum fluctuations for a massless scalar field. Since in an expanding universe it is the field which has canonical normalization, evaluating the right-hand side of (29) gives

(31)

The above estimate is based on the same ideas which lead to stochastic inflation Starob ().

The Hubble expansion rate at the end of the slow-roll period is

(32)

Plugging the value with the above value of into (28), yields the following condition on the parameters of our model for which the reheating dynamics will go through a phase of tachyonic instability:

(33)

We will initially choose the parameters of the D3/D7 brane inflation model such that the direct adiabatic fluctuations produced during inflation, whose magnitude was discussed in the previous section, provide the primordial fluctuations observed today. It is now straightforward to check, making use of (33) and (21) or (22), whether tachyonic resonance preheating will occur.

Let us consider some characteristic parameter values. Scales which are being observed today leave the Hubble radius about 50 e-foldings before the end of inflation. Hence, we take . In the range of values of for which the inequality (20) is violated, it follows from (22) that for a COBE-normalized COBE () value of , we require

(34)

and that hence the condition for tachyonic reheating is trivially satisfied. For a value of such as , the inequality (20) is violated. However, for sufficiently small values of , there is a transition to the region in which (21) rather than (22) applies. The conclusions concerning the validity of (33), however, remain the same.

To conclude this section, we have shown that the initial stages of reheating will proceed, for reasonable parameters of the D3/D7 brane system, via the tachyonic instability channel. However, once has increased beyond the value (28), the tachyonic decay will terminate. At this point, although most of the energy is contained in the nonlinear local fluctuations, a substantial fraction of the energy may remain in the background fields. To obtain an upper bound it takes on the time for most of the energy in the homogeneous fields to dissipate, we neglect the effects of the nonlinear fluctuations and analyze the subsequent dynamics which then proceeds by the parametric resonance instability during the period when the background fields oscillate about their minima. This is the topic of the following section.

At this point, we must add some comments about the phase of . Neglecting the phase of means that the two branes (point particles in the plane) will be approaching each other in a straight line. This requires fine-tuned initial conditions. Also, if these initial conditions were realized, then the D3 brane would most likely immediately be absorbed by the D7 brane when for the first time. Thus, no oscillations of could take place. In the following, we will be assuming that the impact parameter is sufficiently small such that the background field dynamics for can be approximated by an oscillation of a real-valued field . Oscillations of about are interpreted as the D3 brane performing approximately oscillatory motion about the D7 brane without zero impact parameter being reached.

Vi Reheating in the D3/D7 Brane Inflation Model

As discussed in the previous section, even though the initial stages of reheating in the D3/D7 brane inflation model take place by tachyonic resonance, this process shuts off early, leaving some of the initial energy in the form of oscillations of the background fields. To obtain an upper bound on the time it takes to drain this residual energy from the background fields, we proceed to study the phase when the background fields and are oscillating about their respective minima. Under these conditions, preheating via a parametric resonance instability is expected to be the most efficient mechanism for energy transfer from the inflaton to matter field fluctuations.

We recall that for , the potential of our model reduces to that of supersymmetric hybrid inflation. Reheating in hybrid inflation was first studied in GB2 (), and then in greater depth in tachyonic2 (); tachyonic () where, in particular, the tachyonic instability channel was investigated 777See also Natalia () for an application of tachyonic decay to the cosmological moduli problem.. Some aspects of reheating in supersymmetric hybrid inflation have been investigated in BKS (). In the following, we take up this problem again and study it from a more analytical point of view.

The strength of the preheating instability depends significantly on whether the resonance is “broad” or “narrow”, in the notation of KLS2 (). Broad-band resonance is significantly more efficient. However, even narrow-band resonance can be much more efficient than the lowest order perturbative reheating channels first discussed in DL (); AFW (). Below, we will find that, except possibly for a short period at the beginning of the phase of oscillations, we are not in the broad resonance region. Nevertheless, we will verify that the resonance is sufficiently efficient such that a substantial amount of the inflaton energy is transferred to matter fluctuations in one Hubble expansion time. Following the philosophy first applied in JB1 (), we will neglect the expansion of space and use the efficiency condition just mentioned to argue that the approximation of neglecting the spatial expansion is self-consistent. An improved analysis could be done by going to conformal time and rescaling the fields such that the Hubble damping terms in the equations of motion for the rescaled fields disappear. These equations have the form of Floquet-type equations and show exponential instabilities. In the absence of expansion of space, we obtain a particularly simple Floquet-type equation, namely the Mathieu equation.

The analyses of preheating are based on studying the equations of motion for the field fluctuations semi-classically for a classical background dynamics. Before turning to the discussion of the fluctuation equations, we must specify the background evolution. As pointed out in BKS (), the background trajectories in the plane are typically not oscillatory along a line in the two dimensional configuration space. This introduces complications compared to the usual studies of preheating. However, it was also shown in BKS () that the solutions are typically not too far from a preferred solution to the background equations where and oscillate in phase (i.e. along a line in configuration space). Let us consider a particular Hubble volume in which the fluctuation of has positive amplitude. In this case, we introduce a rescaled field via

(35)

Since the initial value of at the end of the period of slow rolling is approximately whereas the amplitude of is almost vanishing, the approximate linear background trajectory during the resonance is characterized by

(36)

We observe that both fields oscillate with the same frequency which is given by

(37)

The potential of our problem is given by the sum of (II) and (13). For small values of the string coupling constant , the supersymmetry breaking term of (13) is negligible for small values of the inflaton compared to the first term (II). Hence, for the relevant part of the potential becomes

(38)

where we recall that we are neglecting the phase of and thus treating as a real variable.

Now we are ready to derive the equations of motion for the field fluctuations and (to simplify the notation we are dropping the subscript on ). We take the background field

(39)

where is the amplitude of the background field oscillations, and expand the potential to quadratic order in the field fluctuations defined by

(40)

Since we are expanding about a solution of the equations of motion, the terms linear in the fluctuations in the action cancel by the background equations of motion. The quadratic terms in the potential are

Since the resulting equations of motion are linear, it is convenient to work in momentum space. The equations of motion for the fluctuation modes and with comoving wavenumber hence become

(42)
(43)

The amplitude of the background field oscillations starts out of the order but decreases as a consequence of the back-reaction of the fluctuations. As the fluctuations drain energy from the background, decreases. Since we are interested in whether preheating is efficient at draining most of the energy from the background, we will in the following work in the approximation that and neglect terms quadratic in . Inserting the background field expressions (39), the equation for the inflaton fluctuation takes the form

(44)

Making use of the same approximations, the equation of motion for the matter field fluctuations becomes

(45)

Fluctuations along the trajectory given by (36) obey

(46)

For these trajectories, both equations (44) and (45) take the form

(47)

and can be put into the standard Mathieu equation form

(48)

by introducing a rescaled time variable via

(49)

and denoting the derivative with respect to by a prime.

By comparing (44) and (48) we can read off the values of and for the equation of motion for :

(50)

The condition for broad parametric resonance is

(51)

Hence, it follows from (50) that, except right at the beginning of the reheating period when , the broad resonance condition is not satisfied 888Note that the authors of BKS () did not take into account the gravitational back-reaction which will rapidly reduce the value of ..

We thus conclude that the resonance is of narrow-band type. This means LL (); Arnold (); KLS2 () that it will take place for values of in resonance bands centered at half integer multiples of . The lowest instability band is the widest. Its range is

(52)

We will now show that, in spite of the fact that the resonance is narrow rather than broad, it is strong enough to drain a substantial fraction of the energy of the inflaton within one Hubble expansion time, hence justifying neglecting the expansion of space in our analysis of the reheating equations.

The condition for efficiency of the parametric resonance is KLS2 ()

(53)

where is the Hubble expansion rate, evaluated at the beginning of the period of reheating. This condition comes from demanding that the exponential growth of the fluctuations induced by the parametric resonance instability is rapid compared to the Hubble expansion rate. The fluctuations grow exponentially, with the exponent being , where is the so-called Floquet exponent given by

(54)

The condition (53) comes from demanding that for the time interval that a mode initially at the center of the resonance band will remain in the resonance band (recall that due to the Hubble expansion, modes are red-shifting with respect to the center of the resonance bands).

Inserting the value of from (50) and the value of from (32), we find that the condition (53) for effectiveness of the parametric resonance instability becomes

(55)

which is easily satisfied for interesting values of .

We thus conclude that, in spite of the fact that preheating occurs in the narrow resonance region, it is sufficiently effective to drain a substantial fraction of the energy density of the background fields within a Hubble expansion time.

The production of matter fluctuations by the preheating instability will drain energy from the background field oscillations and thus lead to a decrease in the amplitude of these oscillations. The resonance will continue to be efficient until the condition (53) ceases to be satisfied. This will be the case once decreases to the value

(56)

which corresponds to a final amplitude of

(57)

We immediately see that, provided , a substantial fraction of the background energy density will transfer to fluctuations during the time period when the resonance is efficient. This fraction is given by

(58)

We will complete this section with the discussion of two side issues. The first is the role of the mixing terms in the set of fluctuation equations (42). In principle, since the mass matrix for the fluctuations is symmetric, it could be diagonalized. Since the mixing terms are of the order of , the only effect they have would be to change the coefficients of the two values slightly. Our conclusions about narrow versus broad resonance and about the efficiency of preheating are unchanged.

The second issue concerns possible fluctuations in the field produced during the preheating period. In the vacuum, (see Section 3), and this is the value we will expand about. Note also that is real and positive semi-definite. One can give a plausible explanation using duality arguments. Under a U-duality the string connecting the D3 and the D7 branes map to a wrapped D3 brane on a three cycle of a deformed conifold BSV (). This is a massive black-hole in four dimensions and gives rise to a unique charged 4D hypermultiplet Sb (). The expectation value of the scalars in this hypermultiplet gives the 4D black-hole configuration, and therefore it makes sense to take both to be positive definite. This observation removes the apparent non-analyticity in the potential coming from the term proportional to . As a consequence, every fluctuation about must be positive semi-definite in space and hence must have a non-vanishing spatial average. This average is subject to the linear confining potential mentioned above, and hence there is no possibility of a growth in fluctuations of .

Vii Entropy and Induced Secondary Curvature Fluctuations

In this section we turn to the study of entropy fluctuations generated during the initial stages of reheating, and to the calculation of the induced curvature fluctuations. The fact that entropy fluctuations induce a growing curvature mode has been known from the early days of cosmological perturbation theory. An early application in the context of inflationary cosmology is to the computation of curvature fluctuations induced by axion perturbations in the early universe (see e.g. Minos ()). We will use the more recent formalism of Gordon et al. Gordon () for our analysis.

We will focus on the effects of a single entropy mode and thus consider a two field system, the initial inflaton field (which dominates the energy-momentum tensor) and the initial entropy field . The equation which describes the growth of the induced curvature perturbation on super-Hubble scales is Gordon ()

(59)

where is the curvature fluctuation in the comoving coordinate system, is the effective adiabatic field, is the effective entropy field, and is the derivative of the potential with respect to the field . The effective fields and are combinations of the inflaton field and tachyonic field which depend on the background motion:

(60)

and

(61)

where the angle is given by

(62)

During the early stages of the tachyonic instability, the inflaton velocity is larger than the velocity of . Hence, the angle is approximately zero.

The potential of our system is given by (38). Hence, the Hubble constant at the onset of reheating is approximately given by (32).

During the tachyonic resonance period, the derivative entering into our basic equation (59) is approximately given by 999Note that the inflation field has non-vanishing once it hits the instability point . It thus rolls on, and terms in proportional to are negligible.

(63)

Since the Hubble friction is negligible, the equation of motion for the entropy field during this phase is

(64)

which has exponentially growing solutions

(65)

with Floquet exponent given by

(66)

In the above, we set the time at the beginning of the instability to be .

Neglecting the Hubble damping, the equation of motion for the entropy fluctuation on super-Hubble scales becomes

(67)

where the subscripts on the potential indicate which fields the derivative is taken with respect to. While the tachyonic instability condition (28) is satisfied, this equation can be approximated by

(68)

This shows that the entropy fluctuation increases with the same Floquet exponent as the background entropy field:

(69)

The initial values of and are both given by the same large scale quantum fluctuations and will be taken to be of the same order of magnitude.

Having determined the growth of the entropy mode, we can now integrate (59) from the time when the instability starts to the time when the tachyonic instability condition breaks down. We denote the corresponding time by . In the absence of back-reaction, the tachyonic resonance stops when (28) is satisfied. We denote the result of the integration by . An approximate evaluation of the integral (taking to be constant) gives

(70)

In the absence of back-reaction, then from (28)

(71)

and analogously for , we can solve for the duration of the tachyonic instability. We see that the dependence of our result for on the initial value of the entropy mode drops out 101010A similar cancellation was seen in the work Larissa (). and we obtain the result

(72)

Let us now turn to a brief discussion of back-reaction effects. The ”waterfall” field has a dispersion on microscopic scales which are due to its quantum vacuum fluctuations. By integrating up these quantum fluctuations of to its mass scale , we obtain the following result for the initial dispersion (that is the root mean square value of the field) at the time the tachyonic instability sets in

(73)

This dispersion then grows exponentially with the exponent set by the Floquet exponent , and after a time

(74)

the dispersion will have grown to be comparable to the value of at the minimum of the potential. At this time (called the spinodal decomposition time), the field on small scales fragments into domains of typical size . While in itself the formation of nonlinearities on small distance scales does not interfere with the linear growth of fluctuations on cosmological scales, the nonlinearities can induce a positive contribution to the effective square mass of the tachyon field which shuts off the resonance.

Whether back-reaction shuts off the resonance before the entropy fluctuation has had time to fully develop depends a lot on the values of the dispersion and of the quasi-homogeneous mode on the scale of the fluctuation. If the latter is calculated based on energetics, i.e. by setting the energy density in this mode to be comparable to the quantum vacuum energy density

(75)

then the time scales and are comparable and our estimates hold. However, if is estimated by integrating quantum vacuum fluctuations on length scales larger than , and taking into account that the waterfall field is massive during most of the inflationary period and the amplitude of the fluctuations is hence redshifted, then is much smaller than , and hence the growth of the entropy fluctuations is cut off before they can become significant.

Let us for a moment assume that and are comparable. In this case, our result (72) for the amplitude of the secondary curvature fluctuations induced by the entropy mode must be compared to the amplitude of the primary adiabatic mode. In the parameter regime where (20) is satisfied, i.e.

(76)

the primary adiabatic fluctuations are given by (see (21))

(77)

Hence, in the parameter region given by

(78)

the secondary fluctuations dominate over the primary ones.

When the condition (20) is not satisfied, the amplitude of the primary adiabatic modes is given by (see (22))

(79)

and hence the secondary curvature fluctuations dominate over the primary ones unless

(80)

From the above results we can draw two main conclusions. Firstly, we see that the induced curvature fluctuation produced by the entropy mode remains in the linear regime. This is good news for the model. The second conclusion is that it is possible the secondary fluctuations dominate over the primary ones, thus necessitating a change in the model parameters in order to achieve agreement with the observed amplitude of fluctuations on large scales.

Viii Discussion and Conclusions

We have studied reheating and structure formation in the D3/D7 brane inflation model of DHHK (), with particular emphasis on the tachyonic instability of super-Hubble scale entropy fluctuations. These entropy perturbations induce a curvature fluctuations which we call “secondary”. We have found that under certain conditions these secondary fluctuations are larger than the primordial adiabatic ones which have been considered in the past. They do, however, remain in the regime of applicability of linear perturbation theory. This would imply that the parameters of the model have to be changed compared to what is usually assumed in order to obtain agreement with the observed amplitude of the large-scale curvature fluctuations.

Along the way, we have given an extensive analytical analysis of the reheating mechanism in the D3/D7 brane inflation model111111Our analytical analysis is complementary to the numerical work of tachyonic (). We need to make certain approximations and neglect some effects, as discussed in the text. On the other hand, numerical work can only handle a limited range of scales. We are interested in large cosmological scales, whereas the basic physical scale of the system is microphysical. It is not clear that numerical work which is sensitive to the microphysical scale can make reliable predictions for questions involving cosmological scales.. In the low energy field theory limit, the dynamics of this system is a special case of supersymmetric hybrid inflation. We have taken into account the quantum fluctuations in the “waterfall” field . These play an important role for both the reheating dynamics and for the generation of entropy fluctuations. For COBE-normalized values of the string theory parameters and for reasonable values of the string coupling constant we have found that, including the effects of the above mentioned quantum fluctuations, the initial stages of reheating occur via a tachyonic instability. This instability, however, shuts off quite early, leaving some of the initial inflaton energy in the background fields. To obtain an upper bound on the time it takes to drain most of the residual energy from the background homogeneous fields, we have neglected the interactions of the nonlinear fluctuations produced during the initial phase of tachyonic decay, and focused on the residual homogeneous field dynamics. This later dynamics proceeds by the parametric resonance instability of JB1 (); KLS1 (); JB2 (); KLS2 (). In fact, except for at the onset of the reheating process, the system is in the narrow-band region of parameters. Nevertheless, the reheating is sufficiently efficient to convert a substantial fraction of the inflaton energy into matter fluctuations.

We note that efficient reheating in the D3/D7 brane inflation model is easier to achieve than in multi-throat inflation models since the matter fields are directly coupled to the inflaton in the form of strings stretching between the two branes whose separation constitutes the inflaton.

In our work, we have neglected the issue of moduli stabilization. This is a very important caveat to our analysis. Moduli stabilization in our model has been considered in the first reference of previous () and more recently in BBDD (). We plan to study the implications of the corrections to the potential induced by moduli stabilization on the dynamics of reheating in a followup paper.

We have also simplified the dynamics of the background fields. Namely, we have considered in-phase oscillations of the two background fields and , and we have neglected their phases. These phases provide extra low mass entropy modes. It would be interesting to consider their excitation during reheating.

Acknowledgements.
The work of R.B. and K.D. is supported by funds from McGill University, by NSERC Discovery Grants and by the Canada Research Chairs program. The work of ACD is supported in part by PPARC. She wishes to thank the Physics Department, McGill University for hospitality whilst this work was in progress. We are grateful to Neil Barnaby, Jim Cline and Andrew Frey for discussions, and to Renata Kallosh, Lev Kofman and Andrei Linde for comments on an earlier draft. We also thank the referee for his/her comments that helped us to improve the paper.

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