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Journal ArticleDOI

A cosmology of a trans-Planckian theory and dark energy.

30 Apr 2014-International Journal of Modern Physics D (World Scientific Publishing Company)-Vol. 23, Iss: 05, pp 1450046
TL;DR: The trans-planckian model as mentioned in this paper is based on the Fourier transform for curved spacetime manifolds, and it can be implemented in a cosmological setting, on a Friedman-Robertson-Walker (FRW) metric background.
Abstract: We investigate a model based on a generalized version of the Fourier transform for curved spacetime manifolds. This model is possible if the metric has an asymptotic flat region which allows a duality to be implement between coordinates and momenta, hence, the model's name, trans-Planckian. The theory and the action are based on the postulate of the absolute egalitarian relation between coordinates x and momenta p. We show how to implement this construction in a cosmological setting, on a Friedman–Robertson–Walker (FRW) metric background, where the asymptotic time infinity plays the role of the required asymptotic flat region. We discuss the effect of gravity, and, in particular, of the Hubble expansion of the universe scale factor on the Fourier map. The dual-inflationary stage is responsible for making the dual-sector of the action inaccessible at ordinary low energies. We propose a scenario in which an effective positive cosmological constant is caused by how the dual-sector of the theory affects the equation of state for matter particles.

Summary (2 min read)

1 Introduction

  • To the trans-Planckian theory and the motivations behind it.
  • Some technical issues must be addressed to formulate the theory.
  • A generic GUT scale inflation, to solve the horizon problem, generates a big hierarchy at least of 1027 in the scale factor.
  • There are many different approaches to the dark energy problem (see, for example, the reviews [8–11] covering different aspects).
  • The authors start from the beginning with a strong postulate, that of absolute equivalence between coordinates and momenta, and work out a possible implementation and the consequences of that.

2 A Trans-Planckian Theory

  • The authors can implement translation invariance by introducing an extra U(1) gauge structure.
  • In a gravitational background, it is instead possible to distinguish them, because the term with m interacts only with the metric gµν while the term with m̃ interact only with the dual metric g̃µν .
  • The authors conclude this section with a comment about the definition of the generalized Fourier transform.

3 Cosmology and the FRW metric

  • The authors take a space-time with an FRW metric ds2 = dt2 − a(t)2d~x2 , (3.1) where t is the time coordinate, a(t) the expansion factor of the three-dimensional space and the ~x the comoving spatial coordinate.
  • The translational invariance of the spatial part of the FRW metrics, both in x and p, is implemented, thanks to the ansatz for the gauge bosons (3.5).
  • The solution for (3.12) can be found exactly in case of a simple power-law behaviour a(t) = cαt α.
  • This feature of the nodes has an important consequence for the generalised Fourier transform.

4 A Cosmological Solution

  • In the previous sections, the authors defined the action of the trans-Planckian theory and the generalised Fourier map for cosmological manifolds.
  • The authors will see that from the inversion of the action S̃ that the dual term can be responsible for an effective positive cosmological constant.
  • 3) The magnitude of S̃ when ϕ̃ comes out of the suppression zone must be with the observed dark energy value (4.23).
  • In general, to solve the horizon problem, the authors want the number of e-folds during inflation to be equal to or greater than the number of e-folds after inflation (with a small correction from the matter-domination period which count as half of the others).

5 Conclusion

  • Every ordinary field theory can be made x ↔ p symmetric with the technique of the generalised Fourier transform.
  • A different approach, which the authors have not pursued in this paper, would be to eliminate the gauge fields Qµ and Yµ and consider the translational invariance to be broken also in the space components.
  • The inflationary stage, which by duality must also occur in the momentum manifold, produces a hierarchy which is big enough to suppress the effect of the dual terms in the action S̃, and thus make the theory consistent with low-energy experiments.
  • The ideas the authors present have aspects in common with others that can be found in the literature, but also have some peculiar distinctions.
  • The authors used some approximations to deal with this problem, the main ones are the homogeneity of the universe and the adiabaticity of the probe functions.

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Citation for published item:
Bolognesi, S. (2014) 'A cosmology of a trans-Planckian theory and dark energy.', International journal of
modern physics D., 23 (05). p. 1450046.
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http://dx.doi.org/10.1142/S0218271814500461
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arXiv:1207.5514v2 [hep-th] 21 Mar 2014
A Cosmology of a Trans-Planckian Theory
and Dark Energy
S. Bolognesi
Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK
Racah Institute of Physics, The Hebrew University of Jerusalem, 91904, Israel
Email: s.bolognesi@gmail.com
Revised Version, March 2014
Abstract
We investigate a model based on a generalised version of the Fourier
transform for curved space-time manifolds. This model is possible if the
metric has an asymptotic flat region which allows a duality to be imple-
ment between coordinates and momenta, hence, the models’s name, tran s-
Planckian. The theory and the action are based on the postulate of the
absolute egalitarian relation between coordinates x and momenta p. We
show how to implement this construction in a cosmological setting, on a
Friedman-Robertson-Walker metric background, where the asymptotic time
infinity plays the role of the required asymptotic flat region. We discuss the
effect of gravity, an d, in particular, of the Hubble expan sion of the universe
scale factor on the Fourier map. The dual in ationary stage is responsible
for making the dual sector of the action inaccessible at ordinary low en-
ergies. We propose a scenario in which an effective positive cosmological
constant is caused by how the dual sector of the theory affects the equation
of state for matter particles.
Keywords: trans-Planckian theory; Born reciprocity; dark energy; inflation.

1 Introduction
We continue the investigation of a theory based on a generalised version o f the
Fourier transform introduced in [1]. This paper consists of a concrete implemen-
tation of the idea in a cosmological setting.
We begin with a brief introduction to the trans-Planckian theory and the
motivations behind it. The uncertainty principle x
µ
p
ν
δ
µν
/2 has its math-
ematical implementa tion in the Fourier transform. At this stage the dynamic is
not yet introduced, and x
µ
and p
µ
are completely symmetrical objects. Dynamic,
of course, spoils this duality. Actions are written as space-time integrals of some
Lagrangian functional. Since we clearly do no t observe this symmetry in nat ur e,
if it exists, it must therefore be invisible to us. Suppose there is a fundamental
energy scale in the pro blem which we call M, then at E/M 1, the theory could
look like an ordinary theory ruled by normal action given by space-time integral
functionals. To observe the duality, we should not only flip x with p but also the
energy scales E/M with M/E. The reason why we do not o bserve directly t he
duality could be that M is a very large mass scale. In quant um mechanics or
quantum field theory, there is no nat ural candidate for such a fundamental mas-
sive scale. When we couple everything to gravity, we have a natural candidate:
the Planck mass M
P
= G
1/2
N
.
We provide a construction based on the Fourier transform. Some technical
issues must be addressed to formulate the theory. First, we have to make sense of
the Fourier transform for generic curved manifolds, not only the flat Minkowski
space. We then have to make sure that the Fourier transform respects ga ug e
invar iance and the equivalence principle, both in x and p manifolds. Finally,
we have to provide dynamics to the system in a way that is consistent with the
duality. Here we choose a minimal approach. Since the ordinary action S is not
invar iant, we simply add to it its dual counterpart
e
S and write a total action as
the sum of the two S = S +
e
S. Thus the theory is still formulated in terms of an
action principle. In its simplest form, it is just the action of a relativistic harmo nic
oscillator:
S =
Z
d
4
x
µ
ϕ
µ
ϕ + M
4
x
µ
x
µ
ϕ
ϕ
. (1.1)
The relativistic harmonic oscillator appeared in the first works of Born on r eci-
procity [2, 3] and in the context of the study of relativistic bound states [4, 5]
(see [6, 7] for recent analysis and more references). In our context is that the
coordinate x subject to the harmonic oscillator potential is no t a relative position
between two constituents, but the actual coordinate of the particle. If all particles
were subject to the potential in (1.1), this would predict a very small universe,
bounded at the scale M
1
, which we would like to be of the order of the Planck
scale. Thus, another problem to overcome, which is not addressed in [1], is to
explain how the effective mass scale could be much smaller than its natural value.
For the concrete implementation of the generalised Fourier transform, we need
an asymptotic flat region on which momenta can be defined. We here want to
1

implement the construction in a cosmological setting, on a Friedman-Robertson-
Walker (FRW) metric background, and work out some of its phenomenological
aspects. The asymptotically flat region is the region at infinite time t of
the FRW metric. Therefore, we must chose a zero Euclidean curvature and a
vanishing ‘fundamental’ cosmological constant. We show t hat the expansion of
the universe, and in particular the early stage inflationary expansion, can work to
make the dual sector invisible at low energies. We use inflation, without entering
into the mechanism that can generate it. A generic GUT scale inflation, to solve
the horizon problem, generates a big hierarchy at least of 10
27
in the scale factor.
Inflation has to occur by duality in both x and p ma nif olds. The observed small-
ness of the last term in the action (1.1) can be explained as a consequence of the
inflation hierarchy. The effective mass is given by M red-shifted by an amount
equal to the total number of infla tionary ten-folds thus making the effective ma ss
a low-scale parameter.
There are many different approaches to the dark energy problem (see, for ex-
ample, the reviews [8–11] covering different aspects). We can roughly divide them
into two categories: the ones in which dark energy is caused by a fundamental
cosmological constant Λ
fund
, and the o t hers in which Λ
fund
= 0, and dark energy
is, instead, caused by some other agent which may be a new degree of freedom
or some modification of g r avity. The second category clearly o ers the best po-
tential for finding a solution to the dark energy pro blem which does not require
fine-tuning. Clearly, a solution in the second category faces two challenges: a prin-
ciple has to be found that sets Λ
fund
to zero and then a dynamical explanation for
dark energy should be found. Our approach belongs to this category. The x p
duality is, for us, the principle that sets Λ
fund
= 0 . In our scenario, the ΛCDM
model is substituted with only a CDM model with zero value for the fundamen-
tal cosmological constant. Cold dark matt er has a modified equation of state a t
late-time, when the effect of the dua l action becomes important, and this effect
induces an effective cosmological constant contribution to the energy-momentum
tensor. The main result of the paper is that an effective cosmological constant
term, which is compatible with the observed value of the universe acceleration,
can be explained by the intervention of the
e
S part of the action on the dark matter
equation of state. The order of magnitude fits precisely if the inflationary stage
lasts exactly the minimal amount of time which is required to solve the horizon
problem.
We want to comment on the methodology we fo llow in this paper, and the re-
lation to other related ideas. We start from the beginning with a strong postulate,
that of absolute equivalence between coordinates and momenta, and work out a
possible implementation and the consequences of that. This principle was first
proposed a long time ago by Born [2], and also in [12]. In these early works, this
idea was applied to QFT problems, like UV divergences or the meson sp ectra, that
were later solved by other means. The absolute duality between coordinat es and
momenta may also be referred as the Born reciprocity principle, in it s strongest
possible form. We are now applying this principle to the problem of gravity and,
2

in particular, the cosmological constant. There are other recent works on this
principle applied to the geometry in momentum space and gravity [13–21] on
which we will comment more in the Conclusion.
The duality assumption, together with the requirement t hat the theory has
to recover the usual QFT plus gravity paradigm at low energy, gives a lot of
constraints which have to be satisfied. This assumption does not fix the theory
uniquely; so, whenever we have a choice, we take the simplest possible path. The
generalized Fourier transform which we define, is an intrinsically global construc-
tion. So, it is natural to implement it in a cosmological setting. Cosmology also
proves to b e the essential ingredient in order to suppress the dual term in the
action
e
S at low energy. Then, in the cosmological setting, we discuss some of the
phenomenological consequences and possible observables of the dual part of the
action.
The paper is organised as follows. In Section 2, we review the construction
of the trans-Planckian theory. In Section 3 , we implement this construction in
cosmology for an FRW type universe. In Section 4, we discuss the cosmological
solution. We present o ur conclusion in Section 5.
2 A Trans-Planckian Theory
A field can be equivalently expressed as a function of space-time coordinates ϕ(x
µ
)
or of energy-momentum coordinates eϕ(p
µ
). The two formulations are related by
the ordinary Fourier tr ansform
eϕ(p) = M
2
Z
d
4
x
(2π)
2
e
ip
µ
x
µ
ϕ(x) ,
ϕ(x) =
1
M
2
Z
d
4
p
(2π)
2
e
ip
µ
x
µ
eϕ(p) , (2.1)
where M is, for the moment, just a normalization constant. One of the basic
properties of this transformation is that it preserves the L
2
norm
M
2
Z
d
4
x |ϕ(x)|
2
=
1
M
2
Z
d
4
p |eϕ(p)|
2
. (2.2)
Prior to any dynamics being introduced, space-time x
µ
and energy-momentum p
µ
are completely symmetrical obj ects. They both have a Minkowski metric η
µν
=
diag(1, 1, 1, 1) a nd the Lorentz transformation with generators J
µν
are self-
dual:
J
µν
=
µνρσ
x
ρ
x
σ
=
µνρσ
p
ρ
p
σ
. (2.3)
Translations, instead, are not self-dual
P
µ
= i
x
µ
, X
µ
= i
p
µ
. (2.4)
3

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Frequently Asked Questions (2)
Q1. What are the contributions in this paper?

The authors investigate a model based on a generalised version of the Fourier transform for curved space-time manifolds. The authors show how to implement this construction in a cosmological setting, on a Friedman-Robertson-Walker metric background, where the asymptotic time infinity plays the role of the required asymptotic flat region. The authors discuss the effect of gravity, and, in particular, of the Hubble expansion of the universe scale factor on the Fourier map. The authors propose a scenario in which an effective positive cosmological constant is caused by how the dual sector of the theory affects the equation of state for matter particles. 

In view of the fact that the effect of S̃ is washed out by the dual inflationary stage, and thus visible only at the cosmological horizon scale or at very high energy, this different approach could still be a viable possibility. A recent attempt to extend this to non-compact directions can be found in [ 18 ]. Thus, the authors can roughly say that they still live in the centre of the relativistic harmonic oscillator ( 1. 1 ), and its size has been inflated from the original Planckian scale to the size of the universe now. There would be, instead, a small modification for the case of the definition ( 2. 30. ) f ( 0,0 ), which enters in the definition ( 2. 30 ), is in the extreme non-adiabatic regime, and it can be checked that f ( 0,0 ) ∝ 1/t3α−1 when a ( t ) ∝ tα.