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arXiv:hep-th/0105017v2 1 Jun 2001
Acceleration of the Universe, String Theory and a
Varying Speed of Light
J. W. Moffat
Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7,
Canada
Abstract
The existence of future horizons in spacetime geometries poses
serious problems for string theory and quantum field theories. The
observation that the expansion of the universe is accelerating has re-
cently been shown to lead to a crisis for the mathematical formalism
of string and M-theories, since the existence of a future horizon for an
eternally accelerating universe does not allow the formulation of phys-
ical S-matrix observables. Postulating that the speed of light varies
in an expanding universe in the future as well as in the past can elim-
inate future horizons, allowing for a consistent definition of S-matrix
observables.
e-mail: moffat@medb.physics.utoronto.ca
1 Introduction
Recently, it has been shown that a critical situation arises in string and M-
theories due to the existence of future horizons in spacetimes [1, 2, 3], and
for models of the universe exhibiting an accelerating expansion [4]. String
theories and M-theories, in their present mathematical frameworks, are crit-
ically dependent on their background geometries. The observables of string
theory are determined for asymptotically free particle states in asymptot-
ically flat Minkowski spacetime. The existence of an S-matrix is crucially
dependent on having a large enough space at infinity in which particles are
separated into a system of non-interacting objects. The description of observ-
ables in AdS spacetimes by boundary correlators of bulk fields is analogous
to S-matrix elements, and is supported by an infinite asymptotic space with
non-interacting particles at infinity.
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The standard quantum field theory formalism, which provides the ba-
sis for the remarkable standard model agreement with observational data,
is a perturbative theory based largely on the notion of asymptotically free
particles and fields. In dS spacetime with a positive cosmological constant
problems arise, for there is no unique Fock space vacuum and it is difficult to
define momentum space due to the ill-defined nature of particle annihilation
and creation operators. The indication that observational data tell us that
the universe is accelerating produces a crisis for our conventional quantum
field theory formalism. The perturbative and non-perturbative string theo-
ries, formulated for strictly on-shell S-matrix elements, do not fare any better
and are possibly in worse shape due to their on-shell definitions. This crisis
in our interpretation of modern particle theories has been waiting to hap-
pen; the advent of observational data supporting an accelerated expansion
of the universe, forces us to confront particle theory with the complications
of spacetimes with future horizons.
Realistic cosmology models are described by neither flat spacetime nor
AdS spacetime. Spatially flat FRW models can be described by S-vectors as
suggested by Witten [3], by assuming that the initial state of the universe is
unique and that the final state is described by asymptotically free particles
with a Fock space of asymptotic out-fields. In standard FRW models there is
no future particle horizon, so that particles can communicate with particles
at infinity and an S-vector can be meaningfully constructed.
It is possible that string theory can be completely reformulated, so that
it can cope with spacetimes with future horizons [5]. On the other hand,
it is possible that if the data supporting an accelerating universe are con-
firmed, then we may not be able to reformulate the language of quantum
field theory and string theories in a satisfactory way, so that we are forced
to consider new ideas that can resolve the crisis. In the following, we shall
explore the idea that a varying light speed in the future universe can remove
future horizons from all spacetime geometries and rid us of the challeng-
ing (and maybe impossible) problem of making sense of present theories in
cosmological backgrounds with future horizons.
The idea that the speed of light varies was proposed as an alternative
to standard inflation theory [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].
The idea [6] originated with the hypothesis that there is a phase of spon-
taneously broken, local Lorentz invariance in the early universe, due to a
non-vanishing vacuum expectation value of a field. The speed of light under-
went an abrupt phase transition as the universe expanded decreasing to its
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presently observed value. This idea was reformulated as a bimetric theory
based on vector-tensor and scalar-tensor structures [15, 16, 17]. The notion
of a varying speed of light (VSL) has also been formulated in theories with
extra-dimensions and in 5-dimensional brane-bulk models with violations of
Lorentz invariance [18]. There are observational indications that the fine-
structure constant α = e2/¯ hc varies with time consistent with an increasing
speed of light [19].
In the following, we shall show that VSL theories can remove the problem
of future horizons, allowing for a physical framework to define a consistent
S-matrix as a basis for quantum field theory and string/M-theories.
2 Varying Speed of Light Model
We shall use a minimal scheme proposed in refs. [8, 9, 10] to illustrate the
resolution of the future horizon problem, and defer the application of a more
geometrically rigorous theory of VSL, such as the bimetric theory [15, 16, 17]
to a later publication. In a minimally coupled VSL theory, one replaces c
by a field in a preferred frame of reference, χ(xµ) = c4. The dynamical
variables in the Lagrangian L are the metric gµν, matter variables contained
in the matter Lagrangian LM, and the scalar field χ which is assumed not to
couple to the metric explicitly. In the preferred frame the curvature tensor
is to be calculated from gµν at constant χ in the normal manner. Varying
the action with respect to the metric gives the field equations
Gµν− gµνΛ =8πG
χ
Tµν, (1)
where Gµν= Rµν−1
stress energy-momentum tensor. This theory is not locally Lorentz invariant.
Choosing a specific time to be the comoving proper time, and assuming that
the universe is spatially homogeneous and isotropic, so that c only depends
on time c = c(t), then the FRW metric can still be written as
2gµνR, Λ is the cosmologcal constant and Tµνdenotes the
ds2= c2dt2− a2
?
dr2
1 − kr2+ r2dΩ2
?
, (2)
where k = 0,+1,−1 for spatially flat, closed and open universes, respectively.
The Einstein equations are still of the form
?˙ a
a
?2
=8πG
3
ρ −kc2
a2, (3)
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¨ a
a= −4πG
3
?
ρ + 3p
c2
?
. (4)
We have set Λ = 0, since we shall be primarily concerned with quintessence
models in which it is assumed that the cosmological constant is zero. The
conservation equations are modified due to the time dependence of c:
˙ ρ + 3˙ a
a
?
ρ +p
c2
?
=
3kc2
4πGa2
˙ c
c.
(5)
Let us assume that matter obeys the equation of state
p = wρc2, (6)
where w is a constant. We shall assume that c changes at a rate proportional
to the expansion of the universe
c = ¯ can, (7)
where ¯ c and n are constants. The present observational value of c is defined
to be
c(t0) ≡ c0= ¯ can(t0), (8)
where t0denotes the present time. Barrow [9] has found an exact solution of
(5) of the form
ρ =
B
a3(1+w)+
3k¯ c2na2(n−1)
4πG(2n − 2 + 3(1 + w)), (9)
where B ≥ 0 is constant if 2n−2+3(1+w) ?= 0. For n = 0 the speed of light
is constant and the equations reduce to the usual adiabatic expansion laws of
FRW cosmology. As first shown in ref. [6] and subsequently in refs. [8, 9, 15]
VSL theories can solve the horizon, flatness and particle relic problems of
early universe cosmology.
3 Future Horizons, Quintessence and Vary-
ing Speed of Light
We shall adopt the theory of quintessence [20] as an alternative to a positive
cosmological constant. According to this theory, the dark energy of the uni-
verse is dominated by the potential V (φ) of a scalar field φ, which rolls down
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to its minimum at V = 0. We recall that a cosmological constant corresponds
to w = −1, radiation domination to w =1
On the other hand, quintessence gives an equation of state with
3and matter domination to w = 0.
− 1 < w < −1
3, (10)
while the observational evidence for a cosmological constant is given by the
bound:
− 1 < wobserved≤ −2
3. (11)
We shall now analyze the causal structure of the universe with w in the range
(10) with a varying light speed c = c(t). Our results can be straightforwardly
extended to higher-dimensional theories such as brane-bulk models.
We obtain from (4) the condition for an accelerating expansion of the
universe, p < −ρ/3. The proper horizon distance is given by
δH(t) = a(t)I, (12)
where
I =
?∞
t0
dt′c(t′)
a(t′)
. (13)
Whenever I diverges there exist no future event horizons in the spacetime
geometry. On the other hand, when I converges, the spacetime geometry
exhibits a future horizon, and events whose coordinates at time¯t are located
beyond δHcan never communicate with the observer at r = 0.
The variation of the expansion scale factor at large a(t), when the curva-
ture becomes negligible, approaches
a(t) ∼ t2/3(1+w). (14)
We now have
I = ¯ c
?∞
t0
dt′t′[2(n−1)/3(1+w)]. (15)
We see that for the quintessence w range (10), we can choose n so that
I diverges and the future horizon has been eliminated. For n = 0, I will
converge for the quintessence w range and will generate a future horizon,
which prohibits an S-matrix description of particles. Consider, as an example,
the choice w = −2
eliminated.
3, then (15) diverges for n ≥
1
2and the future horizon is
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Consider finally a dS universe with Λ > 0 and the asymptotic scale factor
behavior
a(t) ∼ exp
??
Λ
3t
?
. (16)
We now get
I = ¯ c
?∞
t0
dt′exp
?
(n − 1)
?
Λ
3t′
?
(17)
and the dS event horizon is removed for n ≥ 1.
4 Conclusions
We have concerned ourselves in this note with the serious difficulty in defin-
ing a consistent S-matrix description of quantum field theory and String/M-
theories in spaces associated with an eternal accelerated expansion of the
universe. We have shown that if we postulate that the speed of light varies
in the future as well as in the past universe, then we can solve the initial
value problems of cosmology (horizon and flatness problems), and remove
asymptotically all future horizons associated with quintessence models and
an accelerating universe. If future data confirm the accelerated expansion
of the universe, and it proves impossible to formulate a consistent theory of
quantum fields, strings and M-theory, based on physically meaningful quan-
tum observables, then we may be forced to seriously consider a scenario such
as that provided by VSL theories to preserve our present understanding of
particle physics and future theories of quantum gravity.
Hopefully, new supernovae red-shift data will be able to distinguish be-
tween quintessence models and a non-vanishing cosmological constant Λ.
Whatever the outcome of these observations, the VSL theories can eliminate
the problem of asymptotic states in string theory and quantum field theory,
whether or not quintessence models can be found that lead in the future
to a decelerating universe or whether the cosmological constant is non-zero,
resulting in an eternal accelerating universe.
Acknowledgment
This work was supported by the Natural Sciences and Engineering Re-
search Council of Canada.
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