folding branes

PHYSICAL REVIEW D 71, 126004 (2005)

Folding branes

Omid Saremi,1,* Lev Kofman,2,3,† and Amanda W. Peet1,3,‡

1Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A72Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7

3Cosmology and Gravity Program, Canadian Institute for Advanced Research, Canada(Received 17 May 2005; published 15 June 2005)

*omidsar@†kofman@c‡peet@phys1We do not

theory literatu

1550-7998=20

We study classical dynamics of a probe Dp-brane moving in a background sourced by a stack ofDp-branes. In this context the physics is similar to that of the effective action for open-string tachyoncondensation, but with a power-law runaway potential. We show that small inhomogeneous ripples of theprobe brane embedding grow with time, leading to folding of the brane as it moves. We give a fullnonlinear analytical treatment of inhomogeneous brane dynamics, suitable for the Dirac-Born-Infeld+Wess-ZuminoWess-Zumino theory with arbitrary runaway potential, in the case where the sourcebranes are Bogomol’nyi-Prasad-Sommerfield (BPS). In the near-horizon geometry, the inhomogeneousbrane motion has a dual description in terms of free streaming of massive relativistic test particlesoriginating from the initial hypersurface of the probe brane. We discuss limitations of the effective actiondescription around loci of self-crossing of the probe brane (caustics). We also discuss the effect of branefolding in application to the theory of cosmological fluctuations in string theory inflation.

DOI: 10.1103/PhysRevD.71.126004 PACS numbers: 11.25.2w, 98.80.Cq

I. INTRODUCTION

In this paper we investigate generic inhomogeneoussolutions describing a probe Dp-brane or anti-Dp-branefalling in the near-horizon D-brane background geometry.A particularly interesting case occurs for D3-branes, wherethe background involves anti de Sitter space. The back-ground geometry is produced by the stack of N sourceDp-branes. We will show that dynamics of brane motioncan be reduced to that of the Dirac-Born-Infeld (DBI) type,with the action

S � �Z

dp�1xV�T��1� @T@T

q: (1)

Here T�x� is a scalar field (dimensionless: in units of �0)and V�T� is its runaway potential with no minima. Theaction should be understood in the truncated approxima-tion. While first derivatives can be arbitrarily large, theaction is valid only in the regime where second and higherderivatives are small. The action (1) is identical to theeffective field theory action of the open-string theorytachyon describing unstable non-BPS D-branes as foundin e.g.1 [1–3]. The similarity between the DBI effectiveaction for the tachyon and that of a probe D-brane nearNS5-Branes, for homogeneous solutions, was recentlystudied by Kutasov [4]. Some other recent works on similartopics (again for the homogeneous case) include [5–9].

Previous works [10,11] on the generic inhomogeneoussolution in the theory (1) were focused on the open-string

physics.utoronto.caita.utoronto.caics.utoronto.caattempt to be exhaustive referencing the string fieldre here.

05=71(12)=126004(9)$23.00 126004

theory tachyon effective action, with runaway potential ofexponential asymptotic form: V�T� ’ exp��T�. A genericanalytic solution was found [11] which, in particular,shows the growth of T inhomogeneities resulting intachyon energy fragmentation and folding of T. In ourpaper here, we extend the method of [11] to treat genericinhomogeneities for a test D-brane or anti-D-brane freelyfalling in a background D-brane geometry. Folding of thefield T in this context acquires a transparent interpretation,as the folding of the brane embedded in the higher dimen-sional spacetime background. Folding of an inhomogene-ous probe D-brane should also transpire for the backgroundgeometry generated by source NS5-Branes.

The setup of moving a brane in a background spacetimegenerated by many branes is often an ingredient of stringtheory cosmological construction including models of in-flation, as in e.g. [12,13]. Usually generation of inflationaryperturbations in these models are treated in the frameworkof four-dimensional effective theory. The effect of thebrane folding which we find here is based on the fulltreatment of the probe-brane dynamics. Thus it should beincorporated in the complete treatment of fluctuations incosmological applications.

Before proceeding with the story of an inhomogeneousprobe D-brane in the background geometry, let us recall thestory of the inhomogeneous rolling tachyon. Study of therolling homogeneous tachyon in classical string field the-ory has shown that, at late times, the system is described bya pressureless fluid ‘‘tachyon matter,’’ with no perturbativeopen-string modes at the bottom of the potential. Turningon gs would lead to subsequent decay of the unstablebrane, mainly to massive closed string modes as in [14].According to the proposal of [15], this process has a dualdescription purely in terms of the rolling open-stringtachyon.

-1 2005 The American Physical Society

2Clearly, this behavior differs from that of the usual open-string tachyon potential, which is exponentially suppressed forlarge values of tachyon field.

3Apart from its coupling to the 4D gravity, that is.

OMID SAREMI, LEV KOFMAN, AND AMANDA W. PEET PHYSICAL REVIEW D 71, 126004 (2005)

It is difficult to find exact solvable conformal fieldtheories (CFT) describing open-string tachyon dynamicsfor a generic inhomogeneous tachyon profile. Someprogress in this regard was made concerning evolution ofthe string theory tachyon with a plane wave tachyon profile[16]. In this case, the tachyon decays into equidistantplane-parallel singular hypersurfaces of codimension one,which were interpreted as kinks. This inhomogeneousprofile is atypical, though, in the sense that fragmentationbetween kinks does not occur. In the general case, based oncosmological intuition, we expect both types of structures:weblike fragmentation and topological defects.

Quite apart from progress in the CFT approach, theaction (1) has in fact proven to capture many strikingfeatures of tachyon condensation. The relatively simplerformulation of tachyon dynamics, in terms of the effectiveaction (1), has therefore triggered significant interest ininvestigation of the field theory of the tachyon, the possiblerole of tachyons in cosmology, and so forth. Indeed, theend point of string theory brane inflation is annihilation ofD-branes and anti-D-branes, which leads to the formationand subsequent fragmentation of a tachyon condensate[10,11]. So the potential role of the tachyon in cosmologycannot be understood without first understanding itsfragmentation.

In [10], it was found that the evolution of the tachyonfield T�t; ~x� can be viewed as a mapping T�t0; ~x0� !T�t; ~x�, that evolves to become multivalued, with singular-ities at caustics generated. This seemed to be a genericbehavior for runaway potentials. More recently, a study[11] considered the generic inhomogeneous tachyon field,in the region where it rolls down one side of the potential.Formation of sharp features in the tachyon energy density,due to fragmentation, was observed. The tachyon energydensity pattern is reminiscent of the illumination pattern atthe bottom of a swimming pool, or the weblike large scalestructure of the Universe. The similarity is not coinciden-tal: the underlying mathematics has common features in allthree cases. In the free-streaming approximation of [10]this is just focusing of particle trajectories corresponding tohigher density concentrations and, further, to the formationof caustics at the loci where trajectories cross. Thesefeatures, which are related to the convergence of character-istics of the field T, are distinguished from topologicaldefects. The full picture must incorporate both effects:formation of kinks and tachyon fragmentation in the spacebetween them. In the context of brane inflation ending withannihilation of D-branes and anti-D-branes, the tachyon isa complex field and strings will be created. The weblikestructure of fragmentation which will be superposed withthe network of strings.

In this paper, we look at another situation, concerningthe radial motion of a probe brane, in particular, stringtheory backgrounds. Interestingly, moving probe D-branesin the background of Ramond-Ramond (RR)-charged or

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solitonic branes of various dimensions gives rise to effec-tive probe dynamics in which the DBI part, after some fieldredefinition, has the same structure as (1)—with a runawaypotential of power-law type, i.e.

V�T� � cT�� (2)

for some � > 0.2 For example, in the recent work [4],dynamics of a BPS test D-brane in the vicinity of stackof NS5-branes was considered, and for radial motion of theprobe an action of type (1) arises, with a ‘‘tachyon‘‘potential falling off exponentially for large value of the‘‘tachyon‘‘ field. In general, the DBI part of the probeaction for BPS Dp-branes probing a stack of other branesin the near-horizon region of the background has dynamicsof type (1), but with a runaway potential (2) with index� 2�7� p�=�5� p�, which is positive, and hence sen-sible, for 0 p < 5.

Clearly, in these cases, there is a spacetime interpreta-tion of the ‘‘condensation‘‘ of the radial ‘‘tachyon‘‘ mode,in terms of motion of the probe brane in a backgroundgeometry. Analysis of such may open a window to a betterunderstanding of the full open-string tachyon condensationitself [4].

In cosmological applications, the radial distance r to theprobe brane plays the role of the inflaton field [17]. In aparticular cosmological setup, cosmological scenariosbased on the motion of a test anti-D3-brane in the throatregion of a large number of D3-branes have recently beenstudied [12,13]. Dynamics of the radial mode3 is given byan action of the above type (1) with a power-law runawaypotential with � � 4.

Inhomogeneous fluctuations of this field during inflationgenerate primordial cosmological fluctuations. Therefore itis important to know the time evolution for not onlyhomogeneous, plane-parallel brane motion in the anti deSitter or more general background geometry, but genericinhomogeneous branes.

Understanding of the generic solution of the DBI typeeffective action may give insight to the structure of thetheory.

The plan of our paper is the following. In Section II weintroduce the supergravity description of the probe branefreely falling in the background geometry. We will showthat the brane dynamics is reduced to the compact form ofthe DBI plus Wess-Zumino (WZ) effective action for asingle scalar T with the power-law potential. In Section IIIwe make the first approximation towards the inhomoge-neous solution of the nonlinear equation of motion for T,the free-streaming approximation. Yet, the free-streamingapproximation is not enough to calculate the stress-energytensor of the brane. In Section IV we construct the full

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FOLDING BRANES PHYSICAL REVIEW D 71, 126004 (2005)

analytic solution in terms of asymptotic series expansionfor T. Most importantly, the next-to-leading term allows tocalculate the stress-energy of the brane. The answer istransparent and allows a conjecture of a duality betweena probe brane freely falling into the near-horizon back-ground geometry (such as the AdS throat of D3-branes)and a collection of massive relativistic particles, seeSection V. In this final section we also discuss variousaspects of the physics of the folding branes and its cosmo-logical applications.

II. PROBE BRANE IN BACKGROUND BRANEGEOMETRY

The physics of a probe Dp-brane or anti-Dp-branemoving in the background of N source Dp-branes is givenby the usual string-frame DBI plus Wess-Zumino action,

Sprobe � SDBI � SWZ

� ��p

Zdp�1�e��

�� det�P�g��

q� �p

ZP�C�p�1��; (3)

where � denotes � for probe antibranes or � for branes,and P is for pullback.

For a (nonextremal)Dp-brane spacetime in string frame,in a standard coordinate system,

e��H�3�p�=4;

ds2�H�1=2��Kdt2�dx2k��H1=2�dr2=K� r2d�2

8�p�;

C�p�1� � ��1�1�H�1�dt^dx1 � � �^dxp: (4)

The harmonic function is given by

N sourceD−branes

r

Background geometry

FIG. 1. Sketch of probe brane mov

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H � 1� �gsN‘7�p

s

r7�p 1��Rr

�7�p

; (5)

where r2 �P

i�xi?�

2. The event horizon is signalled byK�r� ! 0, with

K�r� � 1�r7�pH

r7�p 1� �2x��H � 1�; (6)

where

x 1

2

�rH

R

�7�p

��1� �2�2�

or � ��1� x2

p� x: (7)

� 2 �0; 1� encodes the degree of nonextremality.It follows easily that

Sprobe � ��Z

dp�1�1

H

�

� �� det�H �gij@�xi

?@�xj? � �g"@�x

k@�x"

k�

q

� �@�xk�@��

; (8)

where for convenience we use an auxiliary metric � �g"�

��K; 1; . . . 1; 1=K; r2 . . . r2�, which collapses to the flatmetric in the BPS limit. In static gauge, where �a � xa

for a � 0; . . . ; p, this expression simplifies in the usualway to give

S�static�probe ��

Zdp�1xk

1

H

� ��1�H �gij@�xi

?@�xj? �g

��q

� �:

(9)

A sketch of the probe brane in the background sourced by astack of N Dp-branes is shown in Fig. 1.

D−branemoving test

x ||S

ing in the background geometry.

-3


Now let us restrict to configurations in which the trans-verse motion is purely radial. In this case, the physicssimplifies to that of just one field describing the motionof the test (probe) brane. This can be seen by performing afield redefinition to give a ‘‘tachyon‘‘

T �R

�p � 7�

ZdH

��H

K�H�

s�H � 1��8�p�=�7�p�: (10)

Physically, this expression (10), in combination with theexpression (5) for H�r�, gives the explicit connection ofthe radial coordinate r and the field T. In other words, Thas clear geometrical interpretation in terms of the radialposition of the brane embedding as r � r�t; ~xk�.Mathematically, in general, the relation (10) gives T interms of hypergeometric functions of H.

The field redefinition yields

S�static;radial�probe ��

Zdp�1xkV�T�

� ��1� �g��@�T@�T

q� �

;

(11)

where the indices run only over �; � � 0; . . . ; p and theonly nontrivial component of �g is the time component. Inthis probe action (11) for radial-only motion, the tachyonpotential is given implicitly by

V�T� 1

H�T�: (12)

To obtain the explicit form of V�T�, it is necessary to invertthe T�H� expression (10) to get H�T�. For Dp-brane ge-ometries where the probe is close to the stack of sourcebranes (e.g. in the decoupling limit in the string theorycontext)

H � 1 � 1; (13)

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the above inversion can be done relatively easily. Weassume this holds in the following.

For the general case for source branes (BPS or not), withthe definition

� 2�7� p��5� p�

(14)

we obtain the exact expression for T�r� in terms of hyper-geometric functions:

T�r� ��5� p�2R

H�r�1=�2 F1

�1

2;1

�; 1�

1

�; 1� K�r�

�:

(15)

Mathematically, this expression is monotonic in r, and isconvergent for K�r� 2 �0; 1� (including at the endpoints)because � > 0.

When the probe is far from the horizon, in the sense thatK�r� � 1�, the series expansion of the hypergeometricfunction can be employed; this is obviously also validwhen there is no horizon (BPS source branes). We then get

Tjbps ��5� p�2R

�Rr

�2=�5�p�

: (16)

Hence, large-T corresponds to small coordinate distancefrom the source branes. Since we are considering thelarge-H limit (13), this is always our regime of interest.The potential for the tachyon

V�T�jbps ��

2R�5� p�

T��

(17)

exhibits power-law runaway behavior.In the non-BPS case, in order to obtain the near-horizon

K�r� � 0� expansion of (15), we use a standard identity forhypergeometric functions to get

2F1

�1

2;1

�; 1�

1

�; 1� K

�� 1� 1=��

��%

p

�1=2� 1=�� 2F1�1=2; 1=�; 1=2;K� � 2 �1� 1=�� 1=��

��K

p2F1�1=2� 1=�; 1; 3=2;K�

(18)

and then, again, employ the series expansion. The upshot isthat, in the near-horizon limit, the entire expression is finiteall the way to the horizon. The second term in (18) issubleading at small-K, while the first term has a coefficientas a ratio of two finite -functions. Amusingly, then, thenon-BPS expressions for T�H�, and hence V�T�, end upbeing identical to the BPS case, up to a finite numericalmultiplicative constant. This system therefore exhibits thesame qualitative physics, as far as the runaway tachyonpotential is concerned. The physically relevant qualitativedifference for non-BPS branes is, of course, that the aux-iliary metric on the brane world volume, � �g�� diag��K�T�; 1; � � � ; 1� features a nontrivial time slowdownnear the horizon. In the following, we will choose to stick

to the BPS case, because it can be solved exactly in ananalytic asymptotic series expansion.

III. FREE-STREAMING APPROXIMATION

The equation of motion for the tachyon field followsfrom the action (11)

T�@@"T

1�@�T@�T@T@"T�

V;T

V

�1� �

��1�@�T@�T

p �� 0:

(19)

This equation is an example of a nonlinear, partial differ-ential equation which, as we will show, admits a relativelysimple, general, inhomogeneous solution.

-4

4One can try asymptotic series with respect to other of S.However the final answer shall be independent of the choice.


The energy density of the tachyon field & � T00 is

& �V�T��

1� @T@Tp _T2 � V�T�

��1� @T@T

q; (20)

where V � V�T� is given above in (17). As has beenobserved in [10] by solving (19) numerically, if we definean operator P�T� � 1� @T@T the field T rapidly ap-proaches to a regime in which P�T� � 0.

Importantly, this means that in the same regime, theRamond-Ramond coupling is essentially irrelevant physi-cally. In particular, neither the � sign (whether the probe isa brane or an antibrane) nor the size of � (the nonextre-mality parameter) matters. This conclusion is somewhatsurprising, naively, because one would think that theRamond-Ramond fields would be very important near thesource branes. The point is that the gauge fields becomeessentially irrelevant for this test brane falling close in tothe source branes, because of a strong kinetic suppression.We will nonetheless keep track of the R-R contribution tothe theory on the probe.

A summary of the free-streaming tool of [10] goes asfollows. In the leading approximation,

T�X� � S�X�; (21)

where S satisfies the equation

_S 2 � � ~rxS�2 � 1; (22)

where we use t � X0, dot is @t and the spatial derivativesare with respect to the p spatial coordinates ~x � X on thebrane. This equation is the Hamilton-Jacobi equation forthe evolution of the wave front function of free-steamingmassive relativistic particles. In this particle description, atsome initial time t0 we can label the position of eachparticle with a vector ~q; equivalently, we can say that ~qparametrizes the different particles. The initial (covariant)velocity of the particle is given by @S0. If we furtherdefine the proper time � along each particle’s trajectory, wecan switch from coordinates (t, ~x) to (��t; ~x�, ~q�t; ~x�) andobtain an exact parametric solution to (22) [10]

~x � ~q � ~r ~qS0�; t ��1� jr ~qS0j2

q�;

S � S0 � �:(23)

Notice that the coordinates (�, ~q) are the proper coordi-nates of the probe brane. The interpretation of the solution(23) is very simple and intuitive. It tells us that the field Spropagates along the trajectories of the massive relativisticparticles, growing linearly with proper time. The slope ofeach characteristic depends only on the initial gradients ofS0 on that characteristic.

In the free-streaming approximation, the denominator ofthe first term in the tachyon energy density expression (20)is zero, because the leading term in _T2 is unity. Given thisfact, it is clear that in order to compute the energy density

126004

near the asymptotic runaway regime, it is necessary to gobeyond the free-streaming approximation (22). In the nextsection, we compute the energy density of the tachyon byfinding corrections beyond the free-streaming approxima-tion by using a working ansatz in an asymptotic seriesexpansion.

IV. THE FULL SOLUTION

We are interested in the energy density: something thefree-streaming approximation cannot give us. So let usbegin here by describing the energy density qualitatively.Looking at Eq. (20) we see that the potential pieces aregrowing small, as an inverse power law, as are the argu-ments of the square roots. The second term in (20) will thusrapidly become irrelevant. What we need to consider isonly the competition in the first term in (20) between thesmall potential in the numerator and the kinetic square-rootterm in the denominator, all evaluated near the runawayasymptotic region of the potential. In a previous work [11],the asymptotic series solution for the exponentiallydecaying potentials was proposed to be T�x� �

S �P

1n�0 fne��n�1�S. Here we generalize this result by

conjecturing the following asymptotic expansion for thefield T4

T�x� � S �X1n�0

fnS��n�1�,: (24)

In the next to the leading approximation, we have

T�X� � S � f1G1; (25)

where G1 is of the following form:

G1 � S�,; (26)

where , > 0 and

� �, � 1

2: (27)

Plugging the expansion (25) into Eq. (19) and keeping onlyterms linear in f1, gives the following equation for thefunction f1

� S@S@"S�@@"f1� � �2S�S � 2, � 2��@S�@f1�

� ��2�, � ,�, � 1��S�1 � 2,�S�f1 � 0; (28)

where �S � @"@"S � � #S �r2~xS. This equation can be

dramatically simplified by changing from (t, ~x) coordinatesto (�, ~q) coordinates, as defined by the characteristics of Sin (23). In these coordinates, f1 has no spatial derivativesand Eq. (28) reduces to

Sf1;�� 2S�S � , � 1�f1;� � 2,�Sf1 � 0; (29)

-5


where using (27) we have got rid of the term proportionalto S�1. Note that �S can be calculated either with respectto (t, ~x) or (�, ~q) coordinates. We can further simplify thisequation by introducing a new variable y

y � ,f1 � Sf1;�: (30)

The second-order differential Eq. (29) can be rewritten interms of y satisfying a first-order differential equation

y;� � 2�Sy � 0; (31)

which can be immediately integrated to give y��; ~q� �y0� ~q� exp��2

R� d�0�S�. From this and (30)

f1��; ~q� � f1i� ~q�S,Z �

d�0S��0; ~q��,�1e�2R

�0 d�00�S:

(32)

(The integration constant that one gains after integrating(30) is irrelevant as it can be absorbed into S.)

To proceed further we need to estimate (32). It can beshown that exp��2

R� d�0�S� is a polynomial of order 2p

with respect to � such that f1G1 is small in the asymptoticregion. We start by referring back to the free-streamingparametric solution (23). Using the fact that T ’ S � f1G1,this shows the validity of the asymptotic series expansion(25) for runaway power-law potentials. We discuss second-order corrections in the Appendix A; these are also shownto preserve the validity of the asymptotic series expansion.Comments on arbitrary runaway potentials can be found inthe Appendix B.

To compute the energy density, we need more informa-tion. Also, proper coordinates (�, ~q) are not so easilyinterpreted in the spacetime picture, where the brane posi-tion in static gauge is given by X�. Therefore, let uscompute the Jacobian that transforms ��; ~q� ! �t; ~x�. Tosee its role, let us use the following trick. Plugging the

expansion (25) into the energy density (20) we find & �

1=��2�,f1 � S@S@f1�

q. The denominator here is none

other than��2y

p. Therefore,

& �1��2y

p � &0� ~q� exp�Z �

d�0�S�: (33)

This is precisely the expression for the energy density offree-streaming relativistic particles which obey the relativ-istic continuity equation @S�&@S� � 0. In fact, (33) isthe solution of this continuity equation in the coordinates(�, ~q). However, there is another form of the energydensity, which is equivalent to (33)

& � &0� ~q�= @~x

@ ~q

; (34)

where the denominator is the Jacobian of the ~x ! ~q trans-formation. Note that we can compare expressions (33) and(34) for & to find a precise expression for �S and then, via(32), for f1. Indeed, from conservation of the energy

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density in a differential volume we have dp ~q&0� ~q� �dp ~x&, which leads to the formula (34). It is now straight-forward to calculate the Jacobian from the formulas (23).In p dimensions it is a polynomial in � of order p. This canbe seen by using ~x � ~q � �rqS0. We can define the sym-metric tensor Da

b rqarqbS0; then the Jacobian can be

written in terms of the eigenvalues of D, f1i; i � 1 . . .pg,as the product �1� 11�� . . . �1� 1p�. Using (34) for &, ittherefore becomes clear that there is a critical proper timewhere the energy density blows up. In detail, at leadingorder in the asymptotic series the energy density is givenby

& � &0� ~q�Ypi�1

1

�1� 1i��: (35)

Corrections to this expression are suppressed by powersof S.

A similar behavior in the context of exponential poten-tials was observed in [11]. We now discuss this phenome-non in our new context of power-law runaway potentials,relevant in string theory for D-brane motion, in the nextsection.

V. INTERPRETATION AND DISCUSSION

Consider the probe brane or antibrane in the backgroundgeometry. Suppose the generic embedding is not exactlylocated in the hyperplane ~xk, but rather given by thedistance function r � r�xk�. Then the following pictureemerges of how the brane is moving in the backgroundgeometry. First, very quickly the brane hypersurface be-comes moving in the regime of free streaming, when eachpoint of the brane is ballistically propagating. By veryquickly we mean that the test brane is in the vicinity ofthe near-horizon throat. Simple formulas (23), resemblingthe Huygens principle of geometric optics, are describingthe evolution of the brane hypersurface T; in other words,its radial position r. Geometrical rules and pictorial illus-trations of the free-streaming approximation are given indetail in the previous papers [10,11].

The most important feature of brane evolution we seehere is unbounded bending of its hypersurface, in otherwords the growth of its inhomogeneities. A simple rulestands, that the brane regions around local maxima of T aregetting flatter while regions around minima are sharpening.This illustrated by the Fig. 2 which shows edge-on profileof the folding brane segment around the minima of T withthe most dramatic reconstruction. At some moment thebrane profile acquires a discontinuity where its secondderivative blows up. This corresponds to caustic formation.Formally, if we allow the free-streaming approximation towork further, the brane hypersurface is self-crossing, and Tgoes multivalued as shown on the figure.

One may think that the multivaluedness of the field T isrelated to the failure of the parametrization of the world

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23

1

FIG. 2. Folding of the brane segment from initial profile 1 through the instance 2 of the first caustic formation to typical multiplevalued configuration 3.


volume of the probe Dp-brane in static gauge. We arguethat brane self-crossing is real, and accompanied by localblowup of the stress-energy tensor. Indeed, we went be-yond the free-streaming approximation to describe evolu-tion of T. While next-to-leading order terms in the seriesT ’ �S �O�1=Sn�� have essentially negligible correctionto the free-streaming propagation of T, they are fullyresponsible for the stress-energy tensor. However, thevery simple and transparent expression for stress-energytensor (33) and (34), shows that at a certain critical timeand location the energy density becomes singular. Theformula (33) and (34), is given in terms of proper timeand proper brane coordinates, and does not relate to thestatic gauge. We conclude that formally the effectiveDBI�WZ theory for generic inhomogeneous solutionsfor radial probe-brane motion leads to multivalued solu-tions, with caustics where the stress-energy tensor turnssingular.

Certainly, the effective field theory description of thebrane dynamics is limited by finite field gradients andstress-energy components. Therefore, we have to discountthe formal results beyond the points where the field gra-dients are large (say, larger than unity in string units, whichwe used here). What happens with the probe brane arounda region about to develop a caustic? Here, we only partlyaddress the issue and suggest some directions for futurethoughts. One possibility is related to the classical physicsof extended objects with sharp kinklike features. Sharprestructuring of the brane hypersurface may lead to sig-nificant radiation similar to gravitational wave flashes fromcosmic string kinks. This would result in local dissipationof the stress-energy into classical radiation. Another pos-sibility is better couched in terms of string theoretic phys-ics. Suppose the multivalued configuration as shown in

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Fig. 2 begins to form. The horizontal segment of brane inconfiguration 3 has orientation opposite to that of the restof the brane. That is to say, this segment now effectivelydescribes a piece of antibrane. Brane-antibrane segmentsin close proximity are unstable due to the open-stringtachyon—corresponding to fundamental string stretchedbetween the segments—which results in their annihilation.

Next, we discuss implication of our effect for cosmo-logical applications.

The interbrane moduli field r is often exploited as aninflaton in the string theory constructions of the braneinflation. We mention one of the realizations of this ideawhich is close to the setup of paper, namely, the infla-tionary model of [12]. The high dimensional constructionincludes a D3 brane freely falling in the background offive-dimensional anti de Sitter geometry generated by alarge stack of source D3-branes. The radius r plays the roleof the moduli field in the effective four-dimensionaldescription. Under certain conditions, again, in the 4deffective description of gravity plus that moduli field infla-tionary regime can be realized.

One of the most important predictions of inflation isgeneration of scalar metric perturbations, usually associ-ated with quantum fluctuations of the inflaton. Thereforewe are interested in spatial inhomogeneities of the modulifield. In the context of our setup, these correspond to thespatial inhomogeneities of the brane embedding.

As we learned, even at the classical level brane inhomo-geneities are unstable and grow. It is therefore interestingto consider combining probe-brane dynamics with infla-tion in 3� 1 dimensions. At the level when 4-dimensionalgravity is introduced phenomenologically (correspondingto adding an Einstein-Hilbert term in the action by hand),in the DBI effective brane action we just have to introduce

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four-dimensional (quasi-) de Sitter geometry. In the equa-tion of motion for T, this will just lead to the friction term3H _T and redshifting of all spatial gradients @ ! 1

a @. Wecan conjecture that apparently folding of the brane will besuppressed at scales of the de Sitter radius 1=H. However,it can persist at larger scales. A more quantitative descrip-tion of this is left for the future.

ACKNOWLEDGMENTS

The authors wish to thank Gary Felder and EvaSilverstein for useful discussions. L. K. and A. W. P. alsowish to acknowledge useful conversations with variousparticipants in the BIRS ‘‘New Horizons in StringCosmology’’ meeting in June 2004. Financial supportfrom an OGS of Ontario (OS), CIAR (LK, AWP), andthe Alfred P. Sloan Foundation (AWP) is acknowledged.

APPENDIX A

We can use a trick to derive the second-order terms in theperturbation about free-streaming. Namely, we write

f1G1 ! G1�f1 � f2G2�; (A1)

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where G2 is a power-law function G2 / S�1, whose index1 to be determined.

The above trick contains nontrivial information. Indeed,for any nth order perturbation fn we will need to use theequation of motion for the lower-order perturbation(s),which in general yield an equation for fn which is notidentical to the equation for f1. Let us see this explicitly.The equations give

�@@"f2��@S��@"S�S � �@f2��@S��2� � 2, � 21�

� 2S�S�f2

�2�, � 1�S �

1�1 � ,�

S2

� 0: (A2)

Changing coordinates to (�, ~q) gives

Sf2;�� 2S�S � �1� , � 21��f2;�

�

�2�, � 1�S �

1�1 � ,�

S2

f2 � 0: (A3)

Defining

y �Sf2;�G2 � �, � 1�f2G2 (A4)

we find that again the equation becomes integrable:

f2 � f2i� ~q� exp�Z �

d�0�, � 1�S��0�

��Z �d�00 exp

��Z �00

d�0 �1 � ,�S��0�

� 2Z �00

d�0�S�

1

S��00��1�1

�: (A5)

At long-time, S � �, and this equation can be integratedeasily. It is then clear that f2G2 is of the same order as f1.In other words, the extra piece just renormalizes f1. Thisfact holds true for any choice of 1.

We can obtain higher-order equations in perturbationtheory by repeating our trick iteratively: substitutingfn�1 ! Gn�1�fn�1 � fnGn� and using lower-orderequations of motion. We conjecture that at general orderadditional perturbations either correspond to a reparamet-rization of S, or a renormalization of lower-order terms, orpieces which are down by additional powers of S in anasymptotic expansion.

APPENDIX B

It is interesting to ask if the analysis at first order beyondthe free-streaming approximation can be done for arbitraryrunaway tachyon potential. Indeed, we show how to do thishere.

We start by writing

T�x� � S � 2��; q�: (B1)

Next, we plug this expression into the general nonlinearequation of motion (19) for the tachyon.

In an asymptotic series expansion, 2 should be muchsmaller than S, this will prove to be a self-consistentapproximation. Keeping only lowest-order terms in 2,

and using the equation of motion for S, we find a surprise:the equation for 2 involves only derivative terms.

The next simplification occurs upon changing to propercoordinates (�, ~q). The resulting equation of motion for 2becomes

�2;�� 2;�

��2�S � 2

�V;T

V

� T�S

� 0 (B2)

which can be easily integrated (twice) to give

2��; ~q� � 2i� ~q�Z �

d�00 exp�2Z �00

d�0 V;S

V

�

� exp��2

Z �00

d�0�S�: (B3)

In the long-time limit, or when initial inhomogeneitiesare small (which is valid here) we can do the integralinvolving the potential to give

2��; ~q� � 2i� ~q�Z �

d�00�V�S��00��2 exp��2

Z �00

d�0�S�:

(B4)

As is clear from the above equation, in the long-time limit,the suppression of the next-to-leading term is controlled bythe square of the tachyon potential, as was suggested insection IV. For runaway potentials, this is always small inthe regime of validity of the asymptotic series expansion.

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In particular, note that our results here agree with the caseof the power-law worked out in the body of the paper.

The energy-momentum tensor can then be written as

&��; ~q� �V�S��%

p ; (B5)

where5

5In proper coordinates, of cours �

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��%

p�

��2@�S@�2

q� �higher order terms�: (B6)

We have @�S@�2 � V2 exp��2R

� d�0�S�. Substitutingthis into Eq. (B5) we obtain Eqs. (33) and (34). Thus wefind out that Eqs. (33) and (34) for the energy density arevalid even for arbitrary runaway potentials. This quantity��%

pcontrols the degree of kinetic suppression, of e.g. the

Ramond-Ramond terms in the D-brane probe action.

e, @ S@�2 � �22;�.

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folding branes

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