Centro de Física Teórica de Partículas
Instituto Superior Técnico - Universidade Técnica de Lisboa
Center for Theoretical Particle Physics, Technical University of Lisbon
Extranatural Inflation
Nuno M. C. Santos
CFTP - IST
PASC Winter School
Sesimbra, 21st December 2007
Contents
 Introduction
 Basics of inflation
 (Cosine-type) Natural inflation and observations
 Pseudo Nambu-Goldstone boson from 5D
 Extranatural inflation
 5D flat spacetime
 Warped background: AdS5 spacetime
 Conclusions
ArXiv: 0711.0022 [ hep-ph ]
In collaboration with R. González Felipe (ISEL and CFTP-IST, Lisbon )
Dec 21, 2007
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Basics of Inflation
 Inflation has emerged as the most successful model for understanding the physics of the
very early universe
 In its most general form it consists of a period of accelerated expansion, during which the
universe is driven toward flatness and homogeneity
 Inflation also provides a mechanism for generating the initial perturbations which led to
structure formation in the universe
 The key ingredient of this cosmological acceleration is negative pressure, i.e, a fluid with a
vacuum-like equation of state,
 slowly rolling scalar field
 During inflation, quantum fluctuations on small scales are quickly amplified to scales
much larger than the horizon size, where they are frozen as perturbations on the
background metric
 Two types of perturbations: scalar (curvature), which couple to the stress-energy tensor
and form the seeds for structure formation, and tensor (gravitational wave), which do not
couple to matter
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Basics of Inflation
 Models can be constructed by specifying the potential 
different choices of the
potential predict different spectra for the primordial fluctuations
 The basic predictions of slow-roll inflation are very simple: Gaussian, adiabatic scalar and
tensor fluctuations will be created with approximately power-law spectra
 The scalar spectrum is conventionally parameterized in terms of a spectral index
as
and the tensor spectrum as
 The scalar-to-tensor ratio is another useful parameterization for tensor perturbations
Notice that
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and
are not independent. For single-field models one has
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‘Naturally’ flat potential from Particle Physics
 To satisfy the observational constraints (enough e-folds of inflation, explain the CMB
anisotropy measurements)  the potential for the inflaton field must be sufficiently flat
 Despite the existence of several cosmologically viable inflaton potentials, the construction of
a sufficiently flat and stable (against radiative corrections) potential is a difficult task from the
particle physics viewpoint
 Among the models of inflation which do not suffer from the fine-tuning problem, the so-called
natural inflation is a simple and well-motivated framework. Particularly simple are those
realizations which involve a pseudo Nambu-Goldstone boson (pNGB)#, with a potential
generally of the form
spontaneous breaking scale
# In particle physics models, Nambu-Goldstone bosons
arise
whenever a global symmetry is spontaneously broken. If there is an
additional explicit symmetry breaking, these particles become pseudoNambu-Goldstone bosons.
explicit breaking scale
Freese, Frieman & Olinto, PRL 1990
…
Savage et al, PRD 2006
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Natural inflation and observations
 The right amplitude for the scalar perturbations gives
 Sufficiently flat potential  large values of
From a field theoretical description it is
difficult to justify such Planckian value
But …
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Pseudo Nambu-Goldstone boson from 5D
 Consider a 5D gauge theory with a gauge group G and the fifth dimension compactified in a
finite interval, y  [0,L]
 Once the 5D gauge symmetry is broken down to a subgroup of G the fifth components of
the gauge fields along the broken gauge group generators give rise to scalar excitations with a
flat potential at tree level
 These scalars can be viewed as 4D pseudo Nambu-Goldstone bosons
 The effective one-loop potential for the pseudo Nambu-Goldstone boson can be derived
Falkowski, PRD 2007
form factors which depend on the specific 5D background (are in principle calculable) and will
depend on quantities that can be related to 5D geometrical quantities
for gauge (fermion) particles
discrete number that depends on the gauge group structure
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Pseudo Nambu-Goldstone boson from 5D
Falkowski, PRD 2007
 In the cases we will be interested in the form factors are well
approximated by
Compositeness scale in 4D
Effective low energy coupling
 The energy scales are related to 5D geometrical quantities
 The warp factor
describes the ‘nature’ of the 5D spacetime.
 flat Minkowski spacetime
 It can be flat or warped
 AdS5 spacetime / RSI
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Extranatural inflation
In the following we will only consider one zero gauge mode (
corresponding scalar field with the inflaton
Performing the integration, one can see that for
the potential has the usual cosine form
warped
flat
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) and identify the
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Extranatural inflation in a 5D flat spacetime
Gauge inflation: the inflaton is the extra component of a
gauge field in a 5D theory compactified in a circle
Arkani-Hamed et al, PRL 2003
Hofmann et al, NPB 2003
…
A small value of the effective
4D gauge coupling is required
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Extranatural inflation in AdS5 spacetime
 Let us consider the case where the fifth dimension is a slice of AdS5, which corresponds to
the so-called Randall-Sundrum I (RSI) model.
Randall & Sundrum, PRL 1999
 In this model the warp factor is given by
 The energy scale
corresponds to the curvature
negative bulk cosmological constant
of AdS5, which is related to the
 The two branes localized at y=0 and y=L have opposite tensions
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Extranatural inflation in AdS5 spacetime
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Conclusions
 It is possible to obtain successful inflation in different 5D backgrounds
 f still with Planckian values, but in this framework f is only an effective quantity
 It is possible to derive bounds on the 5D quantities. The bounds imposed on the warped
5th dimension are quite restrictive:
, an upper bound which is too small to naturally
generate a TeV mass scale at the IR brane
 The 4D effective coupling is very small in all the cases considered,
unavoidable in the natural inflation scenario
, seems
 The predicted running of the spectral index is negligible, however larger values can be
obtained if we consider additional zero modes. A pNGB inflaton potential of the form
is an excellent approximation for the case where both a bosonic and a fermionic zero modes
contribute to the inflaton potential. If
Feng et al, PRD 2003
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