Modern cosmology

ORSAY-LPT-02-62

PACS: 98.80Cq, 04.25Nx

■he"i)-tli/p206147;

Perturbations on a moving D3-brane and mirage cosmology

Timon Boehm"*! and D.A.Steer^.
(N ■ L

a) Departement de Physique Theorique, Universite de Geneve,
^SJ | 24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland.

b) Laboratoire de Physique Theorique, Bat 210, Universite Paris XI,

Orsay Cedex, France and

Federation de Recherche APC, Universite Paris VII, France.

:<n:

i(N :

>■
:»:

Abstract

i— I .

\Q . We study the evolution of perturbations on a moving probe D3-brane coupled

to a 4-form field in an AdSs-Schwarzschild bulk. The unperturbed dynamics are
parameterized by a conserved energy E and lead to Friedmann-Robertson- Walker
'mirage' cosmology on the brane with scale factor a{r). The fluctuations about
the unperturbed worldsheet are then described by a scalar field <p(r,x). We derive
an equation of motion for cf>, and find that in certain regimes of a the effective
mass squared is negative. On an expanding BPS brane with E = superhorizon
modes grow as a 4 whilst subhorizon modes are stable. When the brane contracts,
all modes grow. We also briefly discuss the case when E > 0, BPS anti-branes

^ ■ as well as non-BPS branes. Finally, the perturbed brane embedding gives rise to

scalar perturbations in the FRW universe. We show that (f> is proportional to the
gauge invariant Bardeen potentials on the brane.

1 Introduction

The idea that our universe may be a 3-brane embedded in a higher dimensional space-
time is strongly motivated by string- and M-theory, and it has recently received a great
deal of attention. Much work has focused on the case in which the universe 3-brane is of
co-dimension 1 [ill, % !3] and the resulting cosmology (see e.g. [4, !5|, B]) and cosmological
perturbation theory (e.g. [7\ K, U, 'lG\ 4l., 42, .13]) have been studied in depth. When there
is more than one extra dimension the Israel junction conditions, which are central to the
5D studies, do not apply and other approaches must be used [14, 45. Hi]. In the 'mirage'
cosmology approach [15, 47.] the bulk is taken to be a given supergravity solution, and our
universe is a test D3-brane which moves in this background spacetime so that its back-
reactions onto the bulk is neglected. If the bulk metric has certain symmetry properties,

*Timon.Boehm@physics .unige . ch
t steerQth . u-psud . f r

J2

the unperturbed brane motion leads to FRW cosmology with scale factor a(r) on the
brane ]15], |18| . Our aim in this paper is to study the evolution of perturbations on such a
moving brane. Given the probe nature of the brane, this question has many similarities
with the study of the dynamics and perturbations of cosmic topological defects such as
cosmic strings |9], |(| 0, ||] .

Though we derive the perturbation equations in a more general case, we consider in
the end a bulk with AdSs-SchwarzschildxSs geometry which is the near horizon limit
of the 10- dimensional black D3-brane solution. In this limit (using the AdS-CFT corre-
spondence) black-hole thermodynamics can be studied via the probe D3-brane dynamics
|23| , |24J1 - As discussed in section |2j], we make the assumption that the D3-brane has
no dynamics around the S 5 so that the bulk geometry is effectively AdS 5 -Schwarzschild.
Due to the generalized Birkhoff theorem |^5| , this 5D geometry plays an important role
in work on co-dimension 1 brane cosmology. Hence links can be made between the unper-
turbed probe brane FRW cosmology discussed here and exact brane cosmology based on
the junction conditions |TB[. Similarly the perturbation theory we study here is just one
limit of the full, self- inter acting and non- ^-symmetric brane perturbation theory which
has been studied elsewhere flO| . Comments will be made in the conclusions regarding
generalizations of this work to the full 10D case.

Regarding the universe brane, the zeroth order (or background) solution is taken to be
an infinitely straight brane whose motion is now constrained to be along the single extra
dimension labelled by coordinate r. The brane motion is parameterized by a conserved
positive energy E |TjJ. In AdS 5 -Schwarzschild and to an observer on the brane, the motion
appears to be FRW expansion/contaction with a scale factor given by o oc r. Both the
perturbed and unperturbed brane dynamics will be obtained from the Dirac-Born-Infeld
action for type IIB superstring theory (see e.g. p6|),

3 / d 4 ayj -det(<y ab + 2ira'F ah + B ab ) - p 3 d A aC 4 . (1.1)

Here a a (a = 0, 1, 2, 3) are coordinates on the brane worldsheet, T 3 is the brane tension,
and in the second Wess-Zumino term p 3 is the brane charge under a RR 4-form field
living in the bulk. We will write

Ps = qT 3 (1.2)

so that q = (— )1 for BPS (anti-)branes. In ( |1 . 1|) 7 a 6 is the induced metric and F^ the
field strength tensor of the gauge fields on the brane. The quantities B ab and C± are is
the pull-backs of the Neveu-Schwarz 2-form, and the Ramond-Ramond 4-form field in
the bulk. In the background we consider, the dilaton is a constant and we set it to zero.
In general the brane will not move slowly, and hence the square root in the DBI part of
( |l~Tl) may not be expanded: we will consider the full non-linear action. Finally, notice
that since the 4D Riemann scalar does not appear in ( |I~1"| ) (and it is not inherited from
the background in this probe brane approach) there is no brane self-gravity Hence the
'mirage' cosmology we discuss here is solely sourced by the brane motion, and it leads to
effects which are not present in 4-dimensional Einstein gravity. The lack of brane self-
gravity is a serious limitation. However, in certain cases it may be included, for instance
by compactifying the background space-time as discussed in []27| (see also @). Generally
this leads to bi-metric theories. Even in that case, the mirage cosmology scale factor a(r)

1

JD3 ~

which we discuss below plays an important role and hence we believe it is of interest to
study perturbations in this 'probe brane' approach.

Deviations from the infinitely straight moving brane give rise to perturbations around
the FRW solution. Are these 'wiggles' stretched away by the expansion, or on the contrary
do they grow leading to instabilities? To answer this question, we exploit the similarity
with uncharged cosmic topological defects and make use of the work developed in that
context by Garriga and Vilenkin |]20| , Guven |^T| and Battye and Carter p2| . The
perturbation dynamics are studied through a scalar field (ft(<r) whose equation of motion
is derived from action (|1 . 1|) . We find that for an observer comoving with the brane, (ft
has a tachyonic mass in certain ranges of r which depend on the conserved energy E
characterizing the unperturbed brane dynamics. We discuss the evolution of the modes
4>k for different E and show that in many cases the brane is unstable. In particular, both
sub- and super-horizon modes grow for a brane falling into the black hole. It remains
an open question to see if brane self-gravity, neglected in this approach, can stabilize the
system.

Finally, we also relate (ft to the standard 4D gauge invariant scalar Bardeen potentials
$ and *_ on the brane. We find that $ oc _$ oc (no derivatives of (ft enter into the
Bardeen potentials).

The work presented here has some overlap with that of Carter et al ]28] who also
studied perturbations on moving charged branes in the limit of negligible self-gravity
Their emphasis was on trying to mimic gravity on the brane, and in addition they included
matter on the brane. Here we consider the simplest case in which there is no matter on the
brane: namely F ab = in ( |Tl|) . Our focus is on studying the evolution of perturbations
solely due to motion of the brane: we expect the contribution of these perturbations
to be important also when matter is included. Moreover, we hope that this study may
more generally be of interest for the dynamics and perturbations of moving D-branes in
non-BPS backgrounds.

The outline of the paper is as follows. In section ^ we link our 5-dimensional metric
to the 10-dimensional black D3-brane solution and specify the unperturbed embedding of
the probe brane. To determine its dynamics from the action ( |1 . 1| ) the bulk 4-form RR field
must be specified. We discuss the normalization of this field. At the end of the section we
summarize the motion of the probe brane by means of an effective potential. Comments
are made regarding the Friedmann equation for an observer on the brane. In section [3|
we consider small deviations from the background brane trajectory and investigate their
evolution. The equation of motion for (ft is derived, and we solve it in various regimes
commenting on the resulting instabilities. In section § we link (ft to the scalar Bardeen
potentials on the brane. Finally, in section || we summarize our results.

2 Unperturbed dynamics of the D3-brane

In this section we discuss the background metric, briefly review the unperturbed D3-
brane dynamics, and comment on the cosmology as seen by an observer on the brane.
The reader is referred to Jl5|, ^j| for a more detailed analysis on which part of this section
is based.

2.1 Background metric and brane scale- factor

For the reasons mentioned in the introduction, we focus mainly on a AdS 5 -SxS 5 bulk
spacetime. This is closely linked to the 10D black 3-brane supergravity solution |30j, [31]. |32|
which describes iV coincident D3-branes carrying RR charge Q = NT3 and which is given
by

ds 2 10 = H~ 1/2 (-Fdt 2 + dx ■ dx) + Hi 12 f^- + r 2 dftl\ (2.1)

where the coordinates (£, x) are parallel to the N D3-branes, dflf is the line element on
a 5-sphere and

Hs(r) = l + ^, F = l-±. (2.2)

The quantity I is the AdSs curvature radius and the horizon r# vanishes when the ADM
mass equals Q. The link between the metric parameters £, r H and the string parameters
N, T 3 is given e.g. in f32f| . The corresponding bulk RR field may also be found in J32|.

The near horizon limit of metric ( |2.1|) is AdS 5 -SxS 5 space time f3l]]. Our universe is
taken to be a D3-brane moving in this background. We make the following two assump-
tions. First, the universe brane is a probe so that its backreaction on the bulk geometry
is neglected. This may be justified if iV » 1. Secondly, the probe is assumed to have no
dynamics around S 5 so that it is constrained to move only along the radial direction r.
This is a consistent solution of the unperturbed dynamics since the brane has a conserved
angular momentum about the S 5 , and this may be set to zero ]15], [18|. In section [3] we
assume that is also true for the perturbed dynamics. Thus in the remainder of this paper
we consider an AdSs-S bulk spacetime with metric

dsl

= -f(r)dt 2 + g{r)dx ■ dx + h{r)dr 2

(2.3)

= g iiv dx il dx v

(2.4)

i-£). *(') = £ Kr) = -±r y

(2.5)

f(r)

(In the limit r H — > this becomes pure AdS 5 .)

More generally, by symmetry, a stack of non-rotating D3-branes generates a metric of
the form ds1 Q = dsl+k(r)dfll, where ds^ is given in (|2.3|) |53|. In this case, since the metric
coefficients are independent of the angular coordinates (9 1 , ■ ■ ■ , # 5 ), the unperturbed brane
dynamics are always characterized by a conserved angular momentum around the S 5
p~5[ ] . As a result of the second assumption above, we are thus effectively led to consider
metrics of the form ( ]2.3| ): hence for the derivation of both the unperturbed and perturbed
equations of motion we keep f,g,h arbitrary and consider the specific form (|2.5|) only at
the end.

The embedding of the probe D3-brane is given by x M = X^{x a ). (We have used
reparametrization invariance to choose the intrinsic worldsheet coordinates a a = x a .)
For the unperturbed trajectory we consider an infinitely straight brane parallel to the x a
hyperplane but free to move along the r-direction:

X a = x a , A 4 = R(t). (2.6)

4

Later, in section [3|, we will consider a perturbed brane for which X 4 = R(t) + 8R(t, x).
The induced metric on the brane is given by

dX» dX v

(where the hat denotes a pullback), so that the line element on the unperturbed brane
worldsheet is

dsl = j ab dx a dx b = -(f(R) - h(R)R 2 )dt 2 + g(R)dx ■ dx = -dr 2 + a 2 {r)dx ■ dx. (2.8)

An observer on the brane therefore sees a homogeneous and isotropic universe in which
the time r and the scale factor a(r) are given by

- = J y/(f - hfc)dt, a{r) = ^gjkj

t)). (2.9)

The properties of the resulting Friedmann equation depend on f(R),g(R), h(R) (i.e. the
bulk geometry) as well as R (the brane dynamics) as discussed in |T5|, [L8| and summarized
briefly below.

2.2 Brane action and bulk 4-form field

In AdS 5 -S, B^ u vanishes, and we do not consider the gauge field F ab on the brane. (For
a detailed discussion of the unperturbed brane dynamics with and without F ab , which
essentially corresponds to radiation on the brane, see [|15[ [H|. Non-zero B^ has been
discussed in |34|.) Thus the brane action ( |1 . 1|) reduces to

3 f d A x^i -p 3 f d A xC 4 (2.10)

where

* , ,- s a ^ dX" dX v dX° dx? , n „.

7 = det( 7afe ), ^ = 0^—-^ ——. (2.11)

and Cpvap are components of the bulk RR 4-form field. The first term in ( |2.10|) is just
the Nambu-Goto action.

In the gauge ( |2.6| ), 7 and C4 depend on t only through R. Thus rather than vary-
ing ( |2.10D with respect to X^ and then integrating the equations of motion, it is more
straightforward to obtain the equations of motion from the Lagrangian

£ = -V^f~C = -y/ fg* - g*hR* - C (2.12)

where C — C(R) — jrC^ = qC^. Since C does not explicitly depend on time, the brane
dynamics are parameterized by a (positive) conserved energy E = |^i? — C from which

Transforming to brane time r denned in equation (|2T9| ) yields

where the subscript denotes a derivative with respect to r.

In order to analyze the brane dynamics in AdSs-S where /, g and h are given in fl2.5| ),
one must finally specify C(i?) or equivalently the 4-form potential C^ U(Tp . To that endQ
recall that the 5D bulk action is

l = ^Jd 5 xV=g(R-2A)-^JF 5 A*F 5

(2.15)

where A is the bulk cosmological constant and F 5 = o?C 4 is the 5-form field strength
associated with the 4-form C 4 . The resulting equations of motion are

R,u = lAg,u + ^^^F/^-^F a ^F a ^g, u y (2.16)

''~ r ' ' ^ f - + 3- + PjF 012U -2F^ 2U )dr = (2.17)

where the prime denotes a derivative with respect to r. In AdSs-S, R^ v = —ikg^u an d
Eq. (^TTD gives

£ 3 /3 \ r 3

-3 ( --P01234 - -P01234 J = =^ -P01234 = c-rg (2.18)

where c is a dimensionless constant (see for example |35[). (Note that this solution
satisfies dF 5 = since the only non-zero derivative is ^4^01234 which vanishes on anti-
symmetrizing.) Integration gives

C0123 = v T - + w (2.19)

where v = c/4 and w are again dimensionless constants. Hence the function C(r) ap-
pearing in Eq. fl2.12|) is

C{r) = qCous = qu T - + qw. (2.20)

In 10 dimensions the constant c (and hence v) is fixed by J *F = Q , and w may be
determined by imposing (before taking the near horizon limit - hence with metric Q2.1|) )
that the 4-form potential should die off at infinity |3^]. This second argument is not
applicable here. Instead, we fix v and w in the following way: consider the motion of the
unperturbed brane seen by a bulk observer with time coordinate t. One can define an
effective potential Vj; ff through

l -R 2 + V* s = E (2.21)

1 For the 10D AdSs-SxSs geometry the solution for the 4-form field is given, for example, in Q. For
completeness, we re-derive the result starting directly from the 5D metric (|2.5|) .

so that on using equation (|2.13|) ,

(see Fig.§) where

and C = C(R) is given in ( |2.20| ). We now use the fact that there is no net force between
static BPS objects of like charge, and hence in this case the effective potential should
be identically zero. Here, such a configuration is characterized by r H = 0, q = 1, E = 0:
imposing that V e ' ff = for all R, forces v = ±1 and, in this limit, w = 0. Second we
normalize the potential such that V* S (E, g = l,i?— >oo) = 0for arbitrary values of the
energy E and r H . This leads to

« = -!> « = +^|- (2-23)

In particular for E 1 = 0, then the brane has zero kinetic energy at infinity. Even in this
case the potential is not flat, unless r H = 0, as can be see in Fig.[|. According to this
normalization

c ^=~4 +q ^ (2 - 24)

as in the 10D case []32]]. Notice that since the combination appearing in the equation of
motion for R is E — C, the constant w only acts to shift the energy. For later purpose
we define the shifted energy E by

E = E-qw = E-q^L. (2.25)

Finally we comment that substitution of ( |2. 18j ) into the equation ( |2.16| ) determines
the bulk cosmologicalQ constant to be given by £ 2 A = —6 — c 2 /4 = —10.

2.3 Brane dynamics and Friedmann equation

We now make some comments regarding the unperturbed motion of the 3-brane through
the bulk, .R(t), as seen for an observer on the brane. This will be useful in section § when
discussing perturbations. Recall that since a(r) = R(r)/£ (see Eq. (|2~^)), an 'outgoing'
brane leads to cosmological expansion. Contraction occurs when the brane moves inwards.
For the observer on the brane, one may define an effective potential by

l -R T 2 + V; e = E (2.26)

2 l£(|iiivalcntry wc could have started from the 10D SUGRA action, used the 10D solution for F
(which is identical to ( |2.24[ ) ) and then integrated out over the 5-sphere. After definition of the 5D
Newton constant in terms of the 10D one, the above cosmological constant term is indeed obtained,
coming from the 5-sphere Ricci scalar.

Figure 1: V* S (E, q, R) for E = 0, q = 1, £ = 4 and different values of r#. For R — > oo
the potential goes to zero according to our normalization. When r# = 0, the potential is
exactly flat.

whence, from Eq. ( (2.14| ),

V; H {E,q,R) =E +

2 \RJ

-{E-Cf

(2.27)

Consider a BPS brane q = +1 (see Fig.|]). As noted above, for r# = E = one has
Kff = so that the potential is flat. For r H ^ 0, V^ s contains a term oc — i?~ 6 , and the
probe brane accelerates towards the horizon which is reached in finite (r-)time. On the
other hand, for a bulk observer with time t, it takes infinite time to reach the horizon
where V* s — E, (see Fig. g).

From equations ( |2.14|) and (|2.20|) it is straightforward to derive a Friedmann-like
equation for the brane scale factor a(r) [|15|, ^]:

[2qE^

t

+ (q 2 ~ 1)

(2.28)

The term in l/o 8 (a 'dark fluid' with equation of state p = 5/3p) dominates at early
times. The second term, in a -4 , is a 'dark radiation' term. As discussed in [|18j, the part
proportional to r H corresponds to the familiar dark radiation term in conventional Z 2 -
symmetric (junction condition) brane cosmology, where it is associated with the projected
bulk Weyl tensor. When E is non-zero, Z 2 -symmetry is brokenP] |18| and this leads to
a further dark radiation term |35|, [3IJ. The last term in Q2.28J ) defines an effective 4-
dimensional cosmological constant A 4 = -^(q 2 — 1) which vanishes if the (anti) brane is

3 When making the link between mirage cosmology and the junction condition approach, E oc M_ —
M + where M± are the black- hole masses on each side of the brane Q .

BPS (i.e. q = ±1). All these terms have previously been found both in 'mirage' cosmology
and conventional brane cosmology [0, [35| .

Notice that the dark radiation term above has a coefficient

^2qE + r f = 2qE- r f( q 2 -l) (2.29)

which is positive for q = +1 (since E > 0). However, for BPS anti-branes q = — 1, the
coefficient (|2.29| ) is negative unless E = 0. Thus when E ^ and q = — 1 there is a
regime of i? for which H 2 is negative. In Fig.|3] this is represented by the forbidden region
where the potential exceeds the total energy E. At VJ S = E the Hubble parameter is
zero and an initially expanding brane starts contracting. On the contrary, we do not
obtain bouncing solutions in our setup, regardless of the values of q and E. Bouncing
and oscillatory universes are discussed in e.g. |37|, [38], |3~9"| .

The Friedmann equation fl2.28Q can be solved exactly. In the BPS case, A 4 = 0, the
solution is

«(r) 4 = of + f{r - nf ± \{r - r % ){& + pffl* (2.30)

where a { is the value of the scale factor at the initial time r,, and the ± determines
whether the brane is moving radially inwards or outwards. In the next section when we
solve the perturbation equations, it will be sufficient to consider regimes in which only
one of the terms in equation ( |2.28| ) dominates. These will be given in section [|

One might wonder whether it is possible to obtain a term oc a -3 (dust) in the Fried-
mann equation, and also one corresponding to physical radiation on the brane (rather
than dark radiation). Physical radiation comes from taking F a b ^ in ( |1 . 1| ) [ |15|| , and a
'dark' dust term has been obtained in the non-BPS background studied in |27j]. Finally,
a curvature term a~ 2 has been obtained in |j40|| .

3 Perturbed equations of motion

In this section we consider perturbations of the brane position about the zeroth order
solution R(t) given in (|2.13| ). Once again we work with the metric (|2li| ), specializing to
AdS 5 -S only at the end. The perturbed brane embedding A 4 = R(t) + 8R(t,x) leads to
perturbations, 8^ ab , of induced metric on the brane and these are discussed in section [|.
Note that these perturbations about the flat homogenous and isotropic solution are not
sourced by matter on the brane, and their evolution will depend on the unperturbed
brane dynamics and hence on E. We now derive an equation for the evolution of the
perturbed brane and try to see if there are instabilities in the system.

3.1 The second order action

Since we consider a codimension one brane, the fluctuations about the unperturbed mov-
ing brane can be described by a single scalar field <fi(x a ) living on the unperturbed brane
world shoot [21]. To describe the dynamics of 4>{x a ) (which is defined below), we use the
covariant formalism developed by |2l| to study perturbed Nambu-Goto walls. (For other

Effective potential in bulk time

Figure 2: V^{E, q, R) for E = 2, r H = 1, £ = 4. For a BPS-brane (g = 1), V^ ^ as
i? — > oo according to our normalization. This should be contrasted with a non-BPS brane
e.g. with q — 1.2. Note that V* S (E, q,R = rn) = E. Any inwardly moving (contracting)
brane takes an infinite amount of t-time to reach the horizon.

Effective potential in brane time

Figure 3: V^(E,q,R) for the same parameters as in Fig.§. A BPS brane has zero
kinetic energy at infinity corresponding to a vanishing cosmological constant on the brane.
Otherwise, the cosmological constant is oc q 2 — 1. A BPS anti-brane is allowed to move
only in a restricted range of R: after having reached a maximal scale factor, the universe
starts contracting. Any inwardly moving brane falls into the black hole in a finite r.

10

applications, see also |20j, ^TJ.) The perturbed brane embedding is given by

X"{t, x) = X"{t) + </>(t, x)n"{t) (3.1)

where X^(t) is the unperturbed embedding, and physical perturbations are only those
transverse to the brane (see also section g). The unit spacelike normal to the unperturbed
brane, n M (t) = n M (X M (t)), is defined through

dX v

gy.v'ri 1 —— = 0, g^n^n" = 1 (3.2)

ox a

n»=\Rj ^^,0,0,0, J f—^ I . (3.3)

Thus for a 5D observer comoving with the brane, (ft (which has dimensions of length) is
the measured deviation from the background solution of the previous section [^(J. For
an observer living on the brane, the perturbations in the FRW metric generated by (ft are
discussed in section [| in terms of the gauge invariant scalar Bardeen potentials.

An equation of motion for (ft can be obtained by substituting ( |3.1|) into the action
( p.lQ| ) and expanding to second order in (ft. The terms linear in (ft give the background
(unperturbed) equations of motion studied in the previous section — now we are inter-
ested in the terms quadratic in (ft which give the linearized equations of motion. A similar
analysis was carried out by Garriga and Vilenkin |2(| for Nambu-Goto cosmic domain
walls in Minkowski space and was generalized by Guven |2l| for arbitrary backgrounds.
For the action ( |2.10[ ), the quadratic term is [^TJ

Sp = -- J d'x^i [(V a 0)(V» - [K a b k b a + R^n^j (ft 2 \ . (3.4)

Here V is the covariant derivative with respect to the induced metric 7„&, and the extrinsic
curvature tensor K ab is given by

*— <^>fPlF (3 - 5)

where V is the covariant derivative with respect to the 5D metric g^ u . Finally, R^ v is
the Ricci tensor of the metric g^. Apart from (ft, all the terms in ( |3~^ ) are unperturbed
quantities. Note that there is no contribution to S*^ from the Wess-Zumino term of
action ( |2.10| ): all terms quadratic in (ft cancel since C0123 is the only non-zero component
of the 4-form field. However C does enter into the term linear in (ft and hence into the
background equations of motion, as analyzed in the previous sections.

Variation of the action (|3.4|) with respect to (ft leads to the equation of motion

V a V a (ft + \Klk h a + R^n v \(ft = (3.6)

or equivalently

V a V a (ft - m 2 (ft = (3.7)

11

where

m 2 = -\ktk h a + R^rf]. (3.8)

To determine the extrinsic curvature contribution to (|3.8| ), it is simpler to calculate
first the five dimensional extrinsic tensor defined by

Kj; = t^Va^ (3.9)

where 7 Am = g x ^ — r^n^ and then use

k%k b a = K^K^.

On defining T by

the non-zero components of K% are

K = ^ 3/2 " 1/2 («-7 A2 + ^ A2 + ^). < 3 - 10 >

A'° = -fht (3.11)

*>■ - Mff /2l 4 =Kl=Kl (3 - i2)

A' 4 4 = -^A'» (3.13)

so that

Arm-^.g)^^^,.

The Ricci term is

JW*" = ~ * f 4-( f 4) +4 9 --^t) (3-15)

4A V / \fJ f 9 f h

31/ /7v s" AA*^'

+ 4T^ 1/ <? "g ' \gj ' gh

Collecting these results gives

3(£-c) 2 f/y o / , K /yy , g'h'_ + ^ a 4/ c

4 /^/j ^ /5 ^ ^y gh 9 E-C 3\E-CJ

In the remainder of this section we try to obtain approximate solutions for 4> from equation
Q3.7|) . Some aspects of this calculation are clearer in brane time r, and others in conformal
time 7] (where 77 = J dr /a(r)). Of course the results are independent of coordinate system.
For these reasons we have decided to present both approaches beginning with brane time.

12

3.2 Evolution of perturbations in brane time r

On using the definition of brane time r in equation (|2j|), the kinetic term in (J3l|) is
given by

V a V a = -<j) TT -3H(j) T + \[<l> x l x l + (f) x 2 x 2 + (j) x 3 x3 ].

(In conformal time the factor of a~ 2 multiplying the spacial derivatives disappears — see
below.) We now change variables to ip = a 3 ^ 2 (f) so that (|3?7|) becomes

Vtt - — [^w + (Pa** + yw] + m 2 {t)p = o

(3.16)

M 2 {t) = m 2 -

K-~( 9 -) Rl + 9 -Rr

4 \gj g

3 (g - cf ( _ i /y ,£_B(2?

4 fg 3 h { 2f g g 4 \g j

±( 2 r_(fX + 3 fl_LVg_^_3(gJ\

4h\ 2 f [fj +3 fg fh +3 g A\ g )

3gf_h!_
" 2g h

(3.17)

ig'h' cf c 4 / a \

~2~g~~h~ ~g E-C~3\E-C)

(3.18)

This expression is valid for any /, g and h. We now specialize to AdSs-S in which case
I I 33 E 2 3 1 / ~ r%\ 25,

M 2 (t)

£ 2

33 E 2 3 1/ -

4 « s

W

33

ff 2

£V 4

g 2 -i

(g 2 - 1)

.— l 2q E +f ) + 2 f

(3.19)

Notice that there are regimes of a in which M 2 < — such as, for instance, for small
a where the a -8 term dominates — and furthermore that the location of these regimes
depends on the energy E of the brane. We also see that since M 2 ~ H 2 , instabilities will
occur for modes with a wavelength greater than H~ l . Figure |4] shows the typical shape
of M 2 as a function of a for fixed energy and different q. In the following, we only discuss
cases with q 2 > 1 as the 4D cosmological constant is positive.
Analysis of equation (|3.16| ) is simpler in Fourier space where

Vk(r]

■) = /*

'x(p(r, x)e

(3.20)

and k is a comoving wave number related to the physical wave number k p by k = ak p .
Thus (|3.16|) becomes

Vk,TT + ^ {k 2 - k 2 c {r)) ip k = (3.21)

13

Effective mass squared

Figure 4: The dimensionless quantity M 2 l 2 as a function of a for E = 1,£ = l,rn = 1.
Here, the effective mass squared is positive in a certain range only for the BPS-brane.
Note that the negative M 2 £ 2 region is not hidden behind the horizon.

vhere the time dependent critical wave number, kl(r), is given by
k 2 c ( T ) = -M 2 (r)a 2 .

(3.22)

One might suppose that for M 2 > all modes are stable. However, due to the
r-dependence of k c this is not necessarily true (as we shall see in equation ( |3.36| )).

Our aim now is to determine the a-dependence of <fk- We proceed in the following
way: notice first that the Friedmann equation ( |2.28| ) and the expression for M 2 (t) in
( p.!9| ) both contain terms in a -8 , a -4 and a . We will focus on a regime in which one of
these terms dominates. Then the Friedmann equation can be solved for a(r) which, on
substitution into (|3.19|) , gives M 2 (t). A final substitution of M 2 (t) into the perturbation
equation fl3.21| ) for ip k enables this equation to be solved in each regime. We consider the
following cases: i) q = +1, ii) q = — 1 and in) q 2 > 1.

3.2.1 BPS brane: q = +1

For a BPS brane, the Friedmann equation (|2.28| ) and effective mass M 2 (t) are given by

M\t)

1

r-

E 2 2E\
a 8 a 4

i

33 E 2 3E

~T^ + 2^

(3.23)
(3.24)

The -E 1 - dependence of these equations slightly complicates the analysis of these equations,
and hence we begin with the simplest case in which E = 0.

Case 1: E =

When E = — the static limit in which the probe has zero kinetic energy at infinity
(see Fig.[i]) — only the term proportional to a -8 survives in ( |3.23| ) and ( |3.24| ). Recall that
when r H vanishes the potential VJ S is flat. Furthermore, since E oc r A H = 0, it follows
from ( |3.24| ) that M 2 (t) = in this limit: as expected, a BPS probe brane with zero
energy in AdS 5 has no dynamics and is completely stable.

When r H ^ 0, M 2 {t) < Vr, and the solution of ( ^23|) is

9a 4
a(r) i = at±-^(T-T i ). (3.25)

Here a t > a H = r H /£ is the initial position of the brane at r = T i7 and the choice of sign
determines whether the brane is moving radially inwards (— ) or outwards (+): this is a
question of initial conditions. Let Rh = l/\Ha\ denote the (comoving) Hubble radius.
Then it follows from ( |3.24|) and the definition of k 2 in ( |3.22|) that

±~\kc(T)\~\Ha\ = ±- (3.26)

where we neglect numerical factors of order 1. Thus the critical wave length is A c ~ Rh-
(Notice that Rh is minimal at an and increases with a.)

For superhorizon modes A 3> Rh or \k\ <C \k c \, and in this limit the perturbation
equation (|3.21| ) becomes

<Pk,TT ~ ^fk = 0. (3.27)

a 1

On inserting solution ( p. 25 ) into k 1 one obtains

<!»< = ^h = A * ai + B * a ~ 3 ( 3 - 28 )

(where the constants A k and B k are determined by the initial conditions). Hence if the
brane moves radially outwards the superhorizon modes grow as o 4 oc r. If the brane
is contracting they grow a -3 . In the near extremal limit, (r H <^C I or) a# <*C 1, the
amplitude of these superhorizon modes can become very large suggesting that they are
unstable. Of course our linear analysis will break down when 4> becomes too large.

Consider now subhorizon modes A <C Rh or |fc| ^> \k c \. Then ( |3.21|) is just (pk, TT +
(k 2 /a 2 ) ipk = 0. However, in this case it is much easier to solve the equation in conformal
time r] where the factor of a~ 2 is no longer present. We anticipate the result from section
O: it is

<f) k = A k + B k . (3.29)

a a

For an outgoing brane a increases and subhorizon modes are stable. For an ingoing

brane a decreases, and the amplitude of the perturbation becomes very large in the

near extremal limit. (Note that as the brane expands, superhorizon modes eventually

become subhorizon, and similarly, on a contracting brane, subhorizon modes become

superhorizon.)

15

To conclude, when r# ^ 0, E = and the brane expands, superhorizon modes are
unstable whilst subhorizon modes are stable. For a contracting brane, and in the near
extremal limit, both super- and sub- horizon modes are unstable.

Case 2: E ^

When the energy of the brane is non-zero the situation is more complicated. Notice
first from Q3.24T ) that M 2 (t) has one zero at a = a c given by

HE 2
2E '

(3.30)

Hence M (r) is negative when a < a c and positive for a > a c (see Fig|5|). However, since
a c is ^-dependent, there may be ranges of E for which the negative mass region is hidden
within the black hole horizon. Indeed we find

a c < an

E-< E <E +

% (13 ±W3).

(3.31)
(3.32)

The situation is shown schematically in Fig.

M 2 >0

M 2 <0

3 H

E

Y E +

1.5
E

Figure 5: The curve represents a c , the zero of M 2 (t), as a function of the energy E
as given in Eq. |3.30| . Below the curve the effective mass squared is negative, above it
is positive. For E < E_ and E > E + the M 2 (t) becomes negative already outside the
horizon, whereas for energies within the interval E_,E + the M 2 (t) < region is hidden
within the horizon. The parameters chosen are q = l,r H = 1 and £ = 1.

Now consider H 2 given in Eq. fl3.23|) . The two terms are of equal magnitude when
a = a c = (E 2 /2E) l / A ~ a c . Thus when a <ti a c (and hence in the regions in which
M 2 < in Fig.||), the dominant term in H 2 is the one proportional to a -8 . The system

16

is therefore analogous to the one considered above when E = 0, and for superhorizon
modes the solution is given in ( |3.28| ): for an outgoing brane cj) k ~ a 4 . When E>E + or
E<E_, these regimes extend down to the blackhole horizon: thus in the near extremal
limit the contracting brane will again be unstable since 4> k ~ a -3 .

When o 3> a c (and hence in the regimes in which M 2 > in Fig|5|), the dominant
term in H 2 is oc a -4 so that

a{r) 2 = a 2 l ±2^^{T-r l ) (3.33)

and

k c(r) = -M 2 (r)a 2 = - 3 -^- 2 (3.34)

On superhorizon scales the mode equation is

V*,«- + ~ Vfc = 0. (3.35)

At first sight one might expect the solution to this equation to be stable since M 2 > 0.
However, surprisingly, it is not. (Indeed, below we will see that in conformal time the
effective mass is actually negative in this regime.) A change of variables to u = a 2 shows
that the solution of fl3.35p is

<p k = A k a z ' 2 + B k a 1 ' 2 (3.36)

which grows as t z / 4 : t 1 / a respectively. Finally

<f> k = A k + B k a-\ (3.37)

For E within the band E_<E<E + , the solution ( |3.37| ) for the modes is valid for all a so
that superhorizon modes grow as a -1 as the brane approaches the black hole horizon.

When E>E + or E<E_ these solutions are valid for a 3> a c . Thus for an expanding
brane 4> k tends to a constant value. For a contracting brane, the term oc a -1 could
become important, though for small enough a the relevant regime is that considered
above in which case the solution is given by ( |3.28| ) and the superhorizon modes grow as
a~ 3 .

For subhorizon modes, the solution is still as given in ( |3.29| ).

3.2.2 BPS anti-branes: q = -1

Now the Friedmann equation ( |2.28| ) and effective mass M 2 (t) become

1 \E 2 2E ,

1 ' (3.38)

„, 2/ . 1 \33E 2 3E\ , n .

M(T) - -¥[t^ + 2^\ (3 - 39)

so that M 2 is always negative, independently of E. Note that H 2 > for a < a c where
a c = (E 2 /2E) 1 / 4: . However, since E = E + o^ H j1 for anti-branes, it follows that a c > an

for all E (i.e. there are no energy bands to consider in the case of anti-branes) . When
a <C a c , H 2 oc M 2 oc a -8 and once again this is analogous to the case studied above for
E = 0: superhorizon modes grow as a 4 , and in the near extremal limit the subhorizon
modes on an ingoing brane are unstable.

3.2.3 Non-BPS branes: q ^ ±1

Here we shall only briefly discuss the case q 2 > 1 for large a. Now, independently of E,
there is a cosmo logical constant dominated regime (see Eq. fl2.28|) ). There the solution
for the scale factor is

a( T ) = a(r i )e ± ^ ;(r - r ' ) where A 4 = — ^. (3.40)

OK

M 2 (t) = — -A 4 (3.41)

In this regime, however, M 2 is negative with

~ 4

and R h = j^ = A 4 " 1/2 a- 1 .

For subhorizon modes (A <C Rh) the solution for ip k is again given by (|3.29|) . For
superhorizonm.od.es, and considering an outgoing brane, there is an exponentially growing

unstable mode

<t> k = A k e^ T ~^ = A k a. (3.42)

Hence, this non-BPS brane is unstable for large a. It is not clear to us why the acceleration
due to the positive cosmological constant does not rather stretch the perturbations away.

3.3 Comments on an analysis in conformal time rj

It is instructive to carry out a similar analysis in conformal time rather than brane time,
and we comment briefly on it here. In conformal time and transformed to Fourier space,
Eq. (|377|) becomes

<j) km + 2H(j) k , v + (k 2 + a 2 m 2 )<\> k = (3.43)

where 7i = aH. The friction term can be eliminated by a change of variables to xf) = acf),
and the above equation becomes

^ v + {k 2 -k 2 ( V ))^k = (3.44)

where

k 2 c ( V ) = -M 2 ( V ).

and

M 2 {rj) = a 2 m 2 - a m /a (3.45)

1 I" g " tf + l ( 9 '\ 2 & 9 ' R

{E-cf (if'9'__f + 3 (g'Y , ig'ti , 5 s' c l f c

/^/j ^2/s 5 VW 2p/» 2gE-C \E - C J

9 (f" _l(f\ 2 Zf'9' _lfh'g" _lg'h>\

+ 2^/ 2\f) + 2f g 2fh + g 2gh)- {6Ab)

Specializing to AdSs-S yields

M 2 ( V ) = -±\^ + 6(q 2 -l)a 2 \ CUT)

Notice that in conformal time and for \q\ > 1, A / I 2 (?7) is always negative indepen-
dently of E. From this, one can immediately see the instability for small k in Eq. fl3.36| ),
even though M 2 (t) can be positive in that case. It is clear that the results on brane
(in) stability must be independent of whether or not the analysis is carried out r] or r
time. We will see that this is indeed the case: the reason is that not only the sign of
the effective mass squared, but also its functional dependence on time determines the
stability properties. We now summarize briefly some of the aspects which differ between
the r and r\ analysis.

Consider the simplest case: q = +1 and E = 0. The solution of the (conformal time)
Friedmann equation is a 3 = a 3 ± Sa 2 H (r] — r]i)/2£, and k c {rj) ~ \H\ = 1/Rh- For superhori-
zon modes, \k\ <^C \k c \, Eq. (|3.44|) reduces to 4>k, vv — k 2 (r])ip k = 0. Given a{rj) and hence
k c (a(r])) it is straightforward to find the solution which is, as expected, exactly that given
in ( |3.28| ). For subhorizon modes, \k\ » |fc c |, the solution was given in ( |3.29| ).

Consider now q = +1, E > 0. Recall that in the r-time analysis both M 2 (r) and H 2
contained terms in a -4 and a~ 8 and, in particular, there was a regime in which M 2 {t)
was positive and proportional to a -4 oc H 2 . In 77-time, however, 7i oc a -6 + a~ 2 with
Ai 2 is always being negative, oc —a -6 . Thus whilst the a <^. a c regime reduces to that
discussed above for E = 0, the a 3> a c regime is a little less clear. There H 2 ~ a(rf)~ 2 ,
but M 2 - -a(r])- 6 . Thus

a( v ) = ai ±^( V - m ) (3.48)

and

k c (v) 2 = -M( V ) 2 = 1 -^. (3.49)

Now \k c (r])\ ~ \H\ 3 £ 2 = ( 2 /Rl, and so one can no longer identify the critical wave-
length with the Hubble radius. For \k\ <C \k c \ Eq. ( |3.44| ) reduces to d 2 ip k /da 2 -
(5E 2 /E)(xlj k /a 6 ) = 0. The solution is expressed in terms of Bessel functions which,
however, show exactly the same behavior as ( |3.37[ ): namely 4> k = ifj k /a tends to a con-
stant as a — > 00. The other limit a — > is not relevant as the above equation is only
valid for a 3> a c .

We do not discuss further the case of q = —1 and g^ 1 since the results obtained in
this approach are exactly as discussed in sections [3.2.2| and |3.2.3| .

1')

X°(t,x)

= t + (°(t,x),

X%x)

= x { + C%x),

X 4 (£,£)

= R(t) + e(t,x)

4 Bardeen potentials

So far we have discussed the evolution of 0, the magnitude of the brane perturbation as
seen by a 5D observer comoving with the brane. For an observer living on the brane, the
perturbed brane embedding gives rise to perturbations about the FRW geometry. Recall
(see Eq. (p.8[)) that for the unperturbed brane

ds 2 = ^ ab dx a dx h

= -{f(R)-h{R)R 2 )dt 2 + g{R)dx-dx

= -n 2 (t)dt 2 + a 2 (t)dx-dx (4.1)

where the bar on 7 denotes that it is an unperturbed quantity. Note that the scale
factors n 2 (t) and a 2 (t) pick up their time-dependence through R(t) — for instance a 2 (t) =
g(R(t)). In this section we calculate S^fab resulting from the perturbed embedding ( |3.1|)
and relate it to the Bardeen potentials.

Initially, rather than using the covariant form ( |3.1|) , let us write more generally

(4.2)
(4.3)
(4.4)

Below we will see that the perturbations C do not enter into the two scalar Bardeen
potentials which correspond to the two degrees of freedom £° and e. This is expected
since perturbations parallel to the brane are not physical and can be removed by a
coordinate transformation [12], Then only right at the end will we set (°/n° = e/n 5 = (p.
We will find that the two Bardeen potentials are proportional to each other and to <fi.
By definition, the perturbed brane embedding is given by

lab = lab + Slab

= g ^(X + 5X)^-(X^ + 5X^^-(X v + 5X v ). (4.5)

ox a ox b

Evaluating 8^ a b to first order for the perturbed embedding ( |4.2|) -( [4~4"1) and the general
bulk metric (|2.3| ), one obtains

£700 = e(-f + tiR 2 ) + 2(-(°f + ehR) J (4.6)

Sloi = -(dtOf + Cg + id^hR, (4.7)

6% = eg'Sij + idiQ + d&g. (4.8)

Note the terms proportional to e come from the Taylor expansion of g^ u (X + 5X) in ( fOD
to first order.

In the usual way, the perturbed line element on the brane is written as

ds\ = -n 2 (l + 2A)dt 2 - 2anB l dtdx l + a 2 (^- + h l] )dx i dx j (4.9)

where n(t) and a(t) are defined in ( |4.1|) , and as usual vectors are decomposed into a scalar
part and a divergenceless vector component e.g.

B l = d i B + B l (4.10)

20

with d % B_ { = 0. We will use a similar decomposition for Q denned in ( [4.3|) as well as the
usual one for tensor perturbations. Thus from ( f4.6| )- (|4.8|) we have

ehR)] ,

A

=

-Akr

n 2 L2

- ^i? 2 ) + (C /

B

=

an L

- (a 2 - e/ii?j ,

B,

=

--<„

n

C

=

K9-

E

=

c,

E,

=

Ci,

E n

=

where we have used standard notation defined e.g. in JT0|. By considering coordinate
transformations on the brane and doing standard 4-dimensional perturbation theory one
can define the usual two Bardeen potentials, as well as the brane vector and tensor metric
perturbations. For the first Bardeen potential we find, after some algebra,

$ = -c+*(b + ±e)

Notice that all terms containing Q l in B_ and E_ have cancelled as expected since they are
not physical degrees of freedom. Similarly

= ^[C"«-ejp-/^jj. (4,2)

The important point to notice in this second case is not only the absence of £\ but that
all derivatives of the perturbations £° and e (which appear in A) have also cancelled.
Hence we will find that the Bardeen potentials are proportional to (j) only and not to
any of its derivatives. Finally, the gauge invariant vector and tensor perturbations are
identically zero.
We now set

e = n 4 ^ (° = n°(j) (4.13)

(where n u is the normal to the brane) in order to make contact with the covariant for-
malism of section ||. Then the combination which appears in both }&_ and $ is

C°i?-e = -(^n 4 )0 (4.14)

where n 4 is the 4th component of the normal to the unperturbed brane. Thus
which, on going to AdS 5 -S and using the expression for R 2 in (|2.13|) yields

»-- (5 ^ i (9-(l-.)8)

£ = 3$ + 4g(|) . (4.17)

Even though there are no anisotropic stresses, the Bardeen potentials here are not equal.
We suppose that this is due to the absence of self-gravity. We see that for superhorizon
modes on an expanding brane (for which, from section [5], 4> k oc a 4 ), we also have <&_ k oc a 4 .
Similarly, $ fc also grows rapidly for a brane falling into the black-hole horizon.

To obtain a true (i.e. gauge invariant) measure of the 'deviation' from FRW, it is useful
to look at the ratio of the components of the perturbed Weyl tensor and the background
Riemann tensor which in the FRW case is roughly given by {krj) 2 \§?_ k + ^|, see J43|. For
$ fc oc a 4 this ratio grows, because a ~ t? 1 / 3 when TC 2 ~ a -6 .

5 Conclusions

In this paper we have studied the evolution of perturbations on a moving D3-brane
coupled to a bulk 4-form field, focusing mainly on a AdSs-Schwarzschild bulk. For an
observer on the unperturbed brane, this motion leads to FRW expansion/ contraction
with scale factor a oc r. We assumed that there is no matter on the brane and ignored
the backreaction of the brane onto the bulk. Instead, we aimed to investigate the growth
of perturbations due only to motion, and also to study the stability of moving D3-branes.
For such a probe brane, the only possible perturbations are those of the brane embedding.
The fluctuations about the straight brane world sheet are described by a scalar field
4> which is the proper amplitude of a 'wiggle' seen by an observer comoving with the
unperturbed brane. Following the work of p0| , |2~T| , f4~Tf we derived an equation of motion
for </), and investigated whether small fluctuations are stretched away by the expansion,
or on the other hand, whether they grow on a contracting brane. The equation for (f> is
characterized by an effective mass squared and we noted that if this mass was positive,
the system is not necessarily stable: indeed in section ^| we discussed a regime in which
the effective mass squared is positive in brane time, but negative in conformal time, and
therefore the perturbations grow. Another important factor in the evolution of <fi is the
time dependence of that mass.

In section |] we found that on an expanding BPS brane with total energy E = 0,
superhorizon modes grow as a 4 , whereas subhorizon modes decay and hence are stable.
For a contracting brane, on the contrary, both super- and sub-horizon modes grow as
a -3 and a -1 respectively. These fluctuations become large in the near extremal limit,
an « 1. We therefore concluded that the brane becomes unstable (i.e. the wiggles grow)

22

as it falls into the black- hole. We also discussed the case E > for BPS branes and BPS
anti-branes. Non-BPS branes were found to be unstable at late times when a positive
cosmological constant dominates.

We have discussed the evolution of the fluctuations 4> as measured by a five dimensional
observer moving with the unperturbed brane. However, for an observer at rest in the
bulk, the magnitude of the perturbation is given by a Lorentz contraction factor times the
proper perturbation <j). (For a flat bulk spacetime this was pointed out in f2(|.) Hence,
if perturbations grow for the 'comoving' observer, they do not necessarily grow for an
observer at rest in the bulk.

Finally, the fluctuations around the unperturbed world sheet generate perturbations
in the FRW universe. In section [| we discussed these perturbations from the point if
view of a 4D observer now living on the perturbed brane. We calculated the Bardeen
potentials $ and \& which were both found to be proportional to <f>. Furthermore, we saw
that the ratio 'Weyl to Riemann' which, expressed in terms of $ and \I/_, gives a gauge
invariant measure for the 'deviation' from FRW, also grows.

A limitation of this work is that the back-reaction of the brane onto the bulk was
neglected. One may wonder whether inclusion of back-reaction could stabilize <fi. To
answer that question, recall that the set up we have analyzed here corresponds, in the
junction condition approach, to one in which Z 2 -symmetry across the brane is broken.
Then the brane is at the interface of two AdSs-S space times, and its total energy is
related to the difference of the respective black hole masses: E oc M + — M_. Perturbation
theory in such a non-Z 2 symmetric self- interacting case has been set up in |10[, though
it is technically quite complicated. However, in the future we hope to try to use that
formalism to include the back-reaction of the brane onto the bulk.

It would be interesting to extend this analysis to branes with n codimensions: in this
case one has to consider n scalar fields - one for each normal to the brane. Formalisms to
treat this problem have been developed in [||, Q. In that case the equations of motion
for the scalar fields are coupled, and it becomes a complicated task to diagonalize the
system.

Finally, it would also be interesting to consider non-zero F atn and hence the effect of
perturbations in the radiation on the brane.

Acknowledgements

We thank Ph. Brax, E. Dudas, S. Foffa, M. Maggiore, J. Mourad, M. Parry, A. Riazuelo,
K. Stelle and R. Trotta for numerous useful discussions and encouragement. We especially
thank R. Durrer for her comments on the manuscript. T.B. thanks LPT Orsay for

hospitality.

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