The dynamical Casimir effect in braneworlds
The dynamical Casimir effect in braneworlds
Ruth Durrer∗ and Marcus Ruser†
Département de Physique Théorique, Université de Genève,
24 quai Ernest Ansermet, CH–1211 Genève 4, Switzerland
In braneworld cosmology the expanding Universe is realized as a brane moving through a warped
higher-dimensional spacetime. Like a moving mirror causes the creation of photons out of vacuum
fluctuations, a moving brane leads to graviton production. We show that, very generically, KK-
particles scale like stiff matter with the expansion of the Universe and can therefore not represent
the dark matter in a warped braneworld. We present results for the production of massless and
Kaluza-Klein (KK) gravitons for bouncing branes in five-dimensional Anti de Sitter space. We
find that for a realistic bounce the back reaction from the generated gravitons will be most likely
relevant. This Letter summarizes the main results and conclusions from numerical simulations which
are presented in detail in a long paper [M. Ruser and R. Durrer, Phys. Rev. D 76, 104014 (2007),
arXiv:0704.0790].
PACS numbers: 04.50.+h, 11.10.Kk, 98.80.Cq
Introduction: String theory, the most serious can-
didate for a quantum theory of gravity, predicts the ex-
istence of ’branes’, i.e. hypersurfaces in the 10- (or 11-
) dimensional spacetime on which ordinary matter, e.g.
gauge particles and fermions, are confined. Gravitons
can move freely in the ’bulk’, the full higher dimensional
spacetime [1].
The scenario, where our Universe moves through a five-
dimensional Anti de Sitter (AdS) spacetime has been
especially successful in reproducing the observed four-
dimensional behavior of gravity. It has been shown that
at sufficiently low energies and large scales, not only grav-
ity on the brane looks four dimensional [2], but also cos-
mological expansion can be reproduced [3]. We shall con-
centrate here on this example and comment on behavior
which may survive in other warped braneworlds.
We consider the following situation: A fixed ’static
brane’ is sitting in the bulk. The ’physical brane’, our
Universe, is first moving away from the AdS Cauchy hori-
zon, approaching the second brane. This motion corre-
sponds to a contracting Universe. After a closest en-
counter the physical brane turns around and moves away
from the static brane. This motion mimics the observed
expanding Universe.
The moving brane acts as a time-dependent boundary for
the 5D bulk leading to production of gravitons from vac-
uum fluctuations in the same way a moving mirror causes
photon creation from vacuum in dynamical cavities [4].
Apart from massless gravitons, braneworlds allow for a
tower of Kaluza-Klein (KK) gravitons which appear as
massive particles on the brane leading possibly to phe-
nomenological consequences.
We postulate, that high energy stringy physics will lead
to a turnaround of the brane motion, i.e., provoke a re-
pulsion of the physical brane from the static one. This
∗Electronic address: ruth.durrer@physics.unige.ch
†Electronic address: marcus.ruser@physics.unige.ch
motion is modeled by a kink where the brane velocity
changes sign. As we shall see, a perfect kink leads to
divergent particle production due to its infinite acceler-
ation. We therefore assume that the kink is rounded
off at the string scale Ls. Then particles with energies
E > Es = 1/Ls are not generated. This setup represents
a regular ’bouncing Universe’ as, for example the ’ekpy-
rotic Universe’ [5]. Four-dimensional bouncing Universes
have also been studied in Ref. [6].
Moving brane in AdS5: Our starting point is the met-
ric of AdS5 in Poincaré coordinates:
ds2 = gABdx
AdxB =
−dt2 + δijdxidxj + dy2
The physical brane (our Universe) is located at some
time-dependent position y = yb(t), while the static brane
is at a fixed position y = ys > yb(t). The scale factor on
the brane is
a(η) =
yb(t)
, dη =
1− v2dt = γ−1dt , v = dyb
where we have introduced the brane velocity v and the
conformal time η on the brane. If v ≪ 1, the junction
conditions lead to the Friedmann equations on the brane.
For reviews see [7, 8]. Defining the string and Planck
scales by κ5 ≡ L3s and κ4 ≡ L2Pl the Randall-Sundrum
(RS) fine tuning condition [2] implies
. (2)
We assume that the brane energy density is dominated
by a radiation component. The contracting (t < 0) and
expanding (t > 0) phases are then described by
a(t) =
|t|+ tb
, yb(t) =
|t|+ tb
, (3)
v(t) = − sign(t)L
(|t|+ tb)2
≃ −HL (4)
http://arxiv.org/abs/0704.0756v3
http://arxiv.org/abs/0704.0790
mailto:ruth.durrer@physics.unige.ch
mailto:marcus.ruser@physics.unige.ch
where H = (da/dη)/a2 is the Hubble parameter and we
have used that η ≃ t if v ≪ 1. A small velocity also
requires yb(t) ≪ L. The transition from contraction to
expansion is approximated by a kink at t = 0, such that
at the moment of the bounce
|v(0)| ≡ vb =
, ab = a(0) =
, H2b =
. (5)
Tensor perturbations: We now consider tensor per-
turbations hij on this background,
ds2 =
−dt2 + (δij + 2hij)dxidxj + dy2
. (6)
For each polarization, their amplitude h satisfies the
Klein-Gordon equation in AdS5 [8]
∂2t + k
2 − ∂2y +
h(t, y;k) = 0 (7)
where k = |k| is the momentum parallel to the brane and
h is subject to the boundary (2nd junction) conditions
(v∂t + ∂y)h|yb(t) = 0 → ∂yh|yb(t) = 0 and ∂yh|ys = 0 .
Being interested in late-time (low energy) effects, we have
approximated the first of those conditions by a Neumann
condition (v ≪ 1). Then, the spatial part of Eq. (7)
together with (8) forms a Strum-Liouville problem at any
given time and therefore has a complete orthonormal set
of eigenfunctions {φα(t, y)}∞α=0. These ’instantaneous’
mode functions are given by
φ0(t) =
ysyb(t)
y2s − y2b (t)
. (9)
φn(t, y) = Nn(t)y
2C2(mn(t), yb(t), y) with
Cν(m,x, y) = Y1(mx)Jν(my)−J1(mx)Yν (my) (10)
and satisfy [−∂2y + (3/y)∂y]φα(y) = m2αφα(y) as well as
(8). Nn is a time-dependent normalization condition.
More details can be found in [9]. The massless mode φ0
represents the ordinary four-dimensional graviton on the
brane, while the massive modes are KK gravitons. Their
masses are quantized by the boundary condition at the
static brane which requires C1(mn, yb, ys) = 0. At late
times and for large n the KK masses are roughly given
by mn ≃ nπ/ys. The gravity wave amplitude h may now
be decomposed as [9]
h(t, y;k) =
qα,k(t)φα(t, y) (11)
where the prefactor assures that the variables qα,k are
canonically normalized. Their time evolution is deter-
mined by the brane motion [cf. Eq. (14)].
Localization of gravity: From the above expressions
and using L/yb(t) = a(t), we can determine the late-
time behavior of the mode functions φα on the brane
(yb ≪ L ≪ ys)
φ0(t, yb) →
, φn(t, yb) →
. (12)
At this point we can already make two crucial ob-
servations: First, the mass mn is a comoving mass.
The instantaneous energy of a KK graviton is ωn,k =
k2 +m2n, where k denotes comoving wave number.
The ’physical mass’ of a KK mode measured by an ob-
server on the brane with cosmic time dτ = adt is therefore
mn/a, i.e. the KK masses are redshifted with the expan-
sion of the Universe. This comes from the fact that mn
is the wave number corresponding to the y direction with
respect to the bulk time t which corresponds to conformal
time η on the brane and not to physical time. It implies
that the energy of KK particles on a moving AdS brane
is redshifted like that of massless particles. From this
alone we would expect the energy density of KK modes
on the brane decays like 1/a4.
But this is not all. In contrast to the zero mode
which behaves as φ0(t, yb) ∝ 1/a the KK-mode functions
φn(t, yb) decay as 1/a
2 with the expansion of the Uni-
verse and scale like 1/
ys. Consequently the amplitude
of the KK modes on the brane dilutes rapidly with the
expansion of the Universe and is in general smaller the
larger ys. This can be understood by studying the prob-
ability of finding a KK-graviton at position y in the bulk
which turns out to be much larger in regions of less warp-
ing than in the vicinity of the physical brane[9]. If KK
gravitons are present on the brane, they escape rapidly
into the bulk, i.e., the moving brane looses them, since
their wave function is repulsed away from the brane. This
causes the additional 1/a-dependence of φn(t, yb) com-
pared to φ0(t, yb). The 1/
ys-dependence expresses the
fact that the larger the bulk the smaller the probabil-
ity to find a KK-graviton at the position of the moving
brane. This behavior reflects the localization of gravity:
traces of the five-dimensional nature of gravity like KK
gravitons become less and less ’visible’ on the brane as
time evolves. As a consequence, the energy density of
KK gravitons at late times on the brane behaves as
ρKK ∝ 1/a6 . (13)
It means that KK gravitons redshift like stiff matter and
cannot be the dark matter in an AdS braneworld since
their energy density does not have the required 1/a3 be-
havior. They also do not behave like dark radiation [7, 8]
as one might naively expect. This new result is derived
in detail in Ref. [9]. It is based on the calculation of
〈ḣ2(t, yb,k)〉 ∝ 〈q̇2α,k(t)〉φ2α(t, yb) where the bracket in-
corporates a quantum expectation value with respect to
a well-defined initial vacuum state and averaging over
several oscillations of the field [9]. An overdot denotes
the derivative with respect to t. The scaling behaviour
(13) is due to φ2α(t, yb) only, since graviton production
from vacuum fluctuations has ceased at late times (like
in radiation domination) which is necessary for a mean-
ingful particle definition. Then, 〈q̇2α,k(t)〉 is related to the
number of produced gravitons and is constant in time. In
case that amplification of tensor perturbation is still on-
going, e.g., during a de Sitter phase, the energy density
related to the massive modes might scale differently. The
scaling behavior (13) remains valid also when the fixed
brane is sent off to infinity and we end up with a single
braneworld in AdS5, like in the Randall-Sundrum II sce-
nario [2]. The situation is not altered if we replace the
graviton by a scalar or vector degree of freedom in the
bulk. Since every bulk degree of freedom must satisfy the
five-dimensional Klein-Gordon equation, the mode func-
tions will always be the functions φα, and the energy
density of the KK-modes decays like 1/a6. KK particles
on a brane moving through an AdS bulk cannot play the
role of dark matter.
It is important here that we consider a static bulk and
the time depencence of the brane comes solely from its
motion through the bulk. In Ref. [10] the situation of a
fixed brane in a time-dependent bulk is discussed. There
it is shown that under certain assumption (separability
of the y and t dependence of fluctuations), the energy
density of KK modes on a low energy cosmological brane
does scale like 1/a3 which seems to be in contradiction
with our result. However, the approximations used in [10]
lead to a system of equations governing the expansion of
the Universe but neglecting the time dependence of the
bulk. The situation is then effectively four dimensional
even for the KK modes; effects of the fifth dimension like
the possibility of KK gravitons escaping into the bulk
seem to be lost in this approach. In our case we would
have a similar situation if we keep the expansion on the
brane a(t) but take the position of the brane in the bulk
as static yb(t) = const, which is not consistent with the
general relation yb(t) = L/a(t). [For a fixed physical
mass M = m/a, if we neglect the time dependence of
φn(yb(t)) ∝ 1/a2 we also obtain an energy density for
this mass proportional to 1/a3.]
Particle production: The equation of motion for the
canonical variables qα,k is of the form, see Ref. [9],
q̈α,k + ω
α,kqα,k =
β 6=α
Mαβ q̇β,k +
Nαβqβ,k . (14)
Here ωα,k =
k2 +m2α is the frequency of the mode and
M andN are coupling matrices. When we quantize these
variables, gravitons can be created by two effects: First,
the time dependence of the effective frequency (ωeffα,k)
ω2α,k−Nαα and second, the time dependence of the mode
couplings described by the antisymmetric matrix M and
the off-diagonal part of N .
Note that Equation (14) is derived from the corre-
sponding action for the variables qα,k rather than from
the wave equation (7) itself. In this way the approxi-
mated boundary conditions (8) can be implemented con-
sistently [9, 11].
In the technical paper [9] we have studied graviton pro-
duction provoked by a brane moving according to (3) in
great detail numerically. We have found that for long
wavelengths, kL ≪ 1, the zero mode is mainly generated
by its self-coupling, i.e. the time dependence of its ef-
fective frequency. One actually finds that N00 ∝ δ(t),
so that there is an instability at the moment of the kink
which leads to particle creation, and the number of 4D-
gravitons is given by 2vb/(kL)
2. This is specific to ra-
diation dominated expansion where H2a2 = −∂η(Ha).
For another expansion law we would also obtain particle
creation during the contraction and expansion phases.
Light KK gravitons are produced mainly via their cou-
pling to the zero mode. This behavior changes drastically
for short wavelengths kL ≫ 1. Then the evolution of the
zero mode couples strongly to the KK modes and pro-
duction of 4D gravitons via the decay of KK modes takes
place. In this case the number of produced 4D gravitons
decays only like ∝ 1/(kL).
Results and discussion: The numerical simulations
have revealed a multitude of interesting effects. In the
following we summarize the main findings. We refer the
interested reader to Ref. [9] for an extensive discussion.
For the zero-mode power spectrum we find on scales
kL ≪ 1 on which we observe cosmological fluctuations
(Mpc or larger)
P0(k) =
k2 if kt ≪ 1
(La)−2 if kt ≫ 1 . (15)
The spectrum of tensor perturbations is blue on super-
horizon scales as one would expect for an ekpyrotic sce-
nario. On cosmic microwave background scales the am-
plitude of perturbations is of the order of (H0/mPl)
2 and
hence unobservably small.
Calculating the energy density of the produced massless
gravitons one obtains [9]
ρh0 ≃
. (16)
Comparing this with the radiation energy density, ρrad =
(3/(κ4L
2))a−4, the RS fine-tuning condition leads to the
simple relation
ρh0/ρrad ≃ vb/2. (17)
The nucleosynthesis bound [12] requests ρh0 <∼ 0.1ρrad,
which implies vb ≤ 0.2, justifying our low energy ap-
proach. The model is not severely constrained by the
zero-mode.
More stringent bounds come from the KK modes. Their
energy density on the brane is found to be
ρKK ≃
. (18)
This result is dominated by high energy KK gravitons
which are produced due to the kink. It is reasonable
to require that the KK-energy density on the brane be
(much) smaller than the radiation density at all times,
and in particular, right after the bounce where ρKK is
greatest. If this is not satisfied, back reaction cannot be
neglected. We obtain with ρrad(0) = 3H
b /κ4
a=a(0)=1/
≃ 100 v3b
. (19)
If we use the largest value for the brane velocity vb ad-
mitted by the nucleosynthesis bound vb ≃ 0.2 and re-
quire that ρKK/ρrad be (much) smaller than one for back-
reaction effects to be negligible, we obtain the very strin-
gent condition
. (20)
Taking the largest allowed value for L ≃ 0.1mm,
the RS fine-tuning condition Eq. (2) determines
Ls = (LL
1/3 ≃ 10−22mm ≃ 1/(106TeV) and
(L/Ls)
2 ≃ 1042 so that ys > L(L/Ls)2 ≃ 1041mm
∼ 1016Mpc. This is about 12 orders of magnitude larger
than the present Hubble scale. Also, since yb(t) ≪ L
in the low energy regime, and ys ≫ L according to the
inequality (20), the physical brane and the static brane
need to be far apart at all times otherwise back reaction
is not negligible. This situation is probably not very
realistic. We need some high energy, stringy effects to
provoke the bounce and these may well be relevant only
when the branes are sufficiently close, i.e. at a distance
of order Ls. But in this case the constraint (20) will
be violated which implies that back reaction will be
relevant. On the other hand, if we want that ys ≃ L
and back reaction to be unimportant, then Eq. (19)
implies that the bounce velocity has to be exceedingly
small, vb <∼ 10−15. One might first hope to find a way
out of these conclusions by allowing the bounce to
happen in the high energy regime. But then vb ≃ 1 and
the nucleosynthesis bound is violated since too many
zero-mode gravitons are being produced. Clearly our
low energy approach looses its justification if vb ≃ 1, but
it seems unlikely that modifications coming from the
high energy regime alleviate the bounds.
Conclusions: Studying graviton production in an AdS
braneworld we have found the following. First, the
energy density of KK gravitons on the brane behaves as
∝ 1/a6, i.e. it scales like stiff matter with the expansion
of the Universe and can therefore not serve as a can-
didate for dark matter. Furthermore, if gravity looks
four dimensional on the brane, its higher-dimensional
aspects, like the KK modes, are repelled from the brane.
Even if KK gravitons are produced on the brane they
rapidly escape into the bulk as time evolves, leaving no
traces of the underlying higher-dimensional nature of
gravity. This is likely to survive also in other warped
braneworlds when expansion can be mimicked by brane
motion.
Secondly, a braneworld bouncing at low energies is not
constrained by massless 4D gravitons and satisfies the
nucleosynthesis bound as long as vb <∼ 0.2. However,
for interesting values of the string and AdS scales
and the largest admitted bounce velocity the back
reaction of the KK modes is only negligible if the
two branes are far apart from each other at all times,
which seems rather unrealistic. For a realistic bounce
the back reaction from KK modes can most likely not
be neglected. Even if the energy density of the KK
gravitons on the brane dilutes rapidly after the bounce,
the corresponding energy density in the bulk could even
lead to important changes of the bulk geometry. The
present model seems to be adequate to address the back
reaction issue since the creation of KK gravitons happens
exclusively at the bounce. This and the treatment of
the high energy regime vb ≃ 1 is reserved for future work.
We thank Kazuya Koyama for discussions. This
work is supported by the Swiss National Science
Foundation.
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