Brane Formation and Cosmological Constraint on the Number of Extra Dimensions

Abstract

Special relativity is generalized to extra dimensions and quantized energy levels of particles are obtained. By calculating the probability of particles' motion in extra dimensions at high temperature of the early universe, it is proposed that the branes may have not existed since the very beginning of the universe, but formed later. Meanwhile, before the formation, particles of the universe may have filled in the whole bulk, not just on the branes. This scenario differs from that in the standard big bang cosmology in which all particles are assumed to be in the 4D spacetime. So, in brane models, whether our universe began from a 4D big bang singularity is questionable. A cosmological constraint on the number of extra dimensions is also given which favors $N\geq 7$. Comment: 11 pages, no figures. To appear in IJTP

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arXiv:gr-qc/0503115v1 30 Mar 2005
Brane Formation and Cosmological Constraint on the Number of
Extra Dimensions
Feng Luo∗and Hongya Liu†
Department of Physics, Dalian University of Technology, Dalian, 116024, P. R. China
Abstract
Special relativity is generalized to extra dimensions and quantized energy levels of particles
are obtained. By calculating the probability of particles’ motion in extra dimensions at high
temperature of the early universe, it is proposed that the branes may have not existed since the
very beginning of the universe, but formed later. Meanwhile, before the formation, particles of the
universe may have filled in the whole bulk, not just on the branes. This scenario differs from that
in the standard big bang cosmology in which all particles are assumed to be in the 4D spacetime.
So, in brane models, whether our universe began from a 4D big bang singularity is questionable.
A cosmological constraint on the number of extra dimensions is also given which favors N ≥ 7.
PACS numbers: 11.10.Kk, 98.80.Cq, 11.25.-w, 11.27.+d
Keywords: Extra dimensions; Special relativity; Brane.
∗fluo@student.dlut.edu.cn
†hyliu@dlut.edu.cn
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I. INTRODUCTION
In the Kaluza-Klein (K-K) theory [1], and the later developed string theory [2], [3], the
scale of the extra dimensions is the Planck scale (∼ 10−35m), so the reason why we can
not see the extra dimensions, or why the particles can not run into the extra dimensions, is
simply that the particles’ de Broglie wave lengths are much larger than the scale of the extra
dimensions even in the most powerful accelerators. Almost everyone is satisfied with this
explanation and no further researches about the particles’ motion in the extra dimensions
are needed to be done, since the huge particles can not even “smell” the tiny dimensions.
That is, the energy required to detect Planck scale—the Planck energy (∼ 1019GeV )—can
never be reached in the accelerators in the expected future. However, from the end of 1990s,
the possibility of large extra dimensions (even up to the sub-millimeter) has been proposed
in some “brane world” theories (e.g. [4], [5], [6]), and the possible phenomena in the coming
accelerators (like the NLC and LHC) relate to the extra dimensions have been predicted.
Many experimentalists are eager to receive the first signal coming from extra dimensions.
Of course, since the standard model (SM) has passed numerous tests without deviation,
a feasible “brane world” model with large extra dimensions must consider the method of
localizing the SM particles on the branes, and indeed many methods have been given (e.g.
[7], [8], [9], [10]).
Since the scale of extra dimensions may be much larger than the Planck scale, if we do not
want to be bothered to struggle with those complicated localizing methods, but only consider
the simple restraint from de Broglie wave lengths, there will really be some possibilities for
the particles to jump into the relatively larger extra dimensions and to move in them. We
would like to pointed out that the possibility of the particles entering extra dimensions has
already been discussed before, like in the influential paper presented by N.Arkani-Hamed,
S.Dimopoulos and G.Dvali (ADD) [4]. In their framework, the SM particles can be kicked
into extra dimensions with sufficiently hard collisions, carrying away energy, orbiting around
the extra dimensions. And if the topology of the extra dimensional compact manifold is ap-
propriate, the particles will periodically return to our four-dimensional spacetime. However,
all of these kinds of descriptions are just discussed in rough forms, the fundamental physical
framework of the particles’ motion in extra dimensions has not been studied systematically,
especially in a mathematical form. And we all know that after setting up the basic frame
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(although may seem simple and clear), some physical meanings of this kind of motion can
be considered more in detail. Most important of all, a clear framework may provide us some
new standpoints about extra dimensions. So it is deserved to research this possible kind of
motion seriously.
In this paper, we will generalize the special relativity to extra dimensions and get the
quantized energy levels of the particles. According to the energy levels, the Boltzmann’s
distribution of the particles’ number density will be given. We will analysis that this frame-
work leads to the problem of brane formation, and consequently, a constraint for the number
of extra dimensions is proposed from the consideration of cosmology.
II. EXTRA DIMENSIONS AND THE QUANTIZED ENERGY LEVELS
The initial theory about extra dimensions (K-K theory) just assumed that they are very
small and compacted. However, some new properties were added to extra dimensions in the
subsequent theories which naturally became more complicated. We would like to recover the
original simple assumption and believe the extra dimensions are really physically real, that
is they are not just visual aids introduced to describe the theories. Under this assumption,
the extra dimensions may be equally treated as the ordinary three spaces, except that
their topologies and scales are quite different. Naturally, the theories of four-dimensional
spacetime can be generalized to and may be still suitable to describe the (N+4) dimensional
spacetime.
Suppose a particle with a non-zero rest mass can move in the extra dimensions, then the
four-dimensional special relativity can be generalized to (N +4) dimensions, where N is the
number of extra dimensions. The familiar formula about energy is generalized as
E2= p2c2+ m2
0c4= p2
3c2+
N
?
i=1
p2
Eic2+ m2
0c4,
(1)
where p3is the momentum of the particle in the ordinary three spaces, pEiis its momentum
in the ith extra dimension. We have already supposed that the extra dimension, between
them and the ordinary three spaces, are orthogonal. Like the general assumption in the
string theories, we also assume that the extra dimensions are compact. Furthermore, in
order to make the discussion simple, we suppose the N extra dimensions are all circular
shape.
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From the familiar de Broglie relation, we write the momentum pEias
pEi=
h
λEi,i = 1,2,···,N,
(2)
where λEiis the projection of de Broglie wave length in the ith extra dimension, h is the
Planck constant. The momentum p3still satisfies
p3=Ev3
c2,
(3)
where v3is the speed in the ordinary three spaces.
Because we suppose the ith extra dimension is a ring, we can use the stationary wave
requirement
λEij=2πri
|ji|
=
bi
|ji|,ji= ±1,±2,···,
(4)
where riis the radius of ith extra dimension and biis its perimeter. The signs of jiindicate
that the particle can both circle clockwise and anticlockwise directions on the rings. If
ji= 0, λEi0will be ∞, which means the particle can not see the ith extra dimension and it
has no motion in it. Put Eq. (2), (3), (4) to Eq. (1), we get
Ej1,j2,···,jN=
m0c2
?
1 +?N
?
i=1
j2
ih2
m2
0b2
ic2
1 −
v2
c2
3
.
(5)
Since the momentum in the ith extra dimension can be written as
pEi=
m0vEi
?
?
?
?
1 −
v2
3+
N ?
c2
k=1
v2
Ek
,
(6)
where vEiis the particle’s speed in the ith extra dimension, then from Eq. (2), (4) and (6),
we get
vEij=
|ji|h
?
?
1 −
v2
c2
3
m0bi
1 +?N
k=1
j2
kh2
m2
0b2
kc2
.
(7)
We can also obtain the period of the particle returning to the four-dimensional spacetime
TEij=
bi
vEij
.
(8)
Clearly, because of the motion in the extra dimensions, the particle’s energy is quantized.
Now we introduce the concept of energy levels. We call E0the ground state energy, E±1,
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E±2··· the first, second ... excited states energies. Obviously, the excited energies are at
least double degenerated under the assumption of this kind of topological structure. “At
least” here means if all the extra dimensions really share the same radius, as usually expected,
the degenerations of excited energy levels will be much larger. For example, the first excited
energy level contains 21C1
Nstates, the fourth excited energy level contains (24C4
N+ 21C1
N)
states.
We should point that for the particles, to be in excited states and to run into extra dimen-
sions is the same thing in our framework. In other words, the motion in extra dimensions
can be described by the labels of excited states.
Evidently, when ji= 0 (i = 1,2,···,N), the ground state energy is
E0=
m0c2
?
1 −
v2
c2
3
,
(9)
and vEi0= 0, (i = 1,2,···,N), which corresponds to the usual four-dimensional special
relativity.
The concept of quantized energy (or say, the quantized mass, since E = mc2) is not
strange to those who are familiar with the Kaluza-Klein (K-K) theory [1]. The above deduc-
tion seems quite simple and the generalization of special relativity looks as if nearly nothing
more than standard. However, when considering other requirements, like the thermal sta-
tistical theory, this framework can really lead to some interesting results.
The statistical theory could also be generalized to extra dimensions since we have as-
sumed that the extra dimensions share the same qualities with the ordinary three spaces
except the topology and scales. Then the distribution of the particles’ number density in
different energy levels should obey the Boltzmann’s distribution law (for simplicity, we will
not consider the deviation from Boltzmann’s distribution because of the particles’ identical
property, and we assume the system of the particles are in the equilibrium state)
ρEj1,j2,···,jN
ρE0
= ̟Ej1,j2,···,jNexp
?
−∆Ej1,j2,···,jN
kT
?
,
(10)
where k is the Boltzmann’s constant, T is the temperature of the system, ̟j1,j2,···,jNis the
degeneration, and ∆Ej1,j2,···,jN= Ej1,j2,···,jN− E0is the difference between the energy of the
excited and the ground levels.
In the following discussion, we use the units c = ¯ h = k = 1. Then Eqs. (5), (7) and (10)
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