Conceptual Issues in Inflation

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Conceptual Issues in Inflation
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New Inflation
1981 - 1982
V
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Chaotic Inflation
1983
Eternal Inflation
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Hybrid Inflation
1991, 1994
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WMAP5 Acbar Boomerang CBI
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Tensor modes
Kallosh, A.L. 2007
It does make sense to look for tensor modes even
if none are found at the level r 0.1 (Planck)
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Blue lines chaotic inflation with the simplest
spontaneous symmetry breaking potential
for N 50 and N 60
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Destri, de Vega, Sanchez, 2007
Possible values of r and ns for chaotic
inflation with a potential including terms
for N 50. The color-filled
areas correspond to various confidence
levels according to the WMAP3 and SDSS data.
Almost all points in this area can be fit by
chaotic inflation including terms
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What is fNL?
Komatsu 2008
k2
k1
k3

• fNL the amplitude of three-point function
• also known as the bispectrum, B(k1,k2,k3),
  which is
• ltF(k1)F(k2)F(k3)gtfNL(2p)3d3(k1k2k3)b(k1,k2,k3)
• F(k) is the Fourier transform of the
  curvature perturbation, and b(k1,k2,k3) is a
  model-dependent function that defines the shape
  of triangles predicted by various models.

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Why Bispectrum?

• The bispectrum vanishes for Gaussian random
  fluctuations.
• Any non-zero detection of the bispectrum
  indicates the presence of (some kind of)
  non-Gaussianity.
• A very sensitive tool for finding non-Gaussianity.

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Two fNLs
Komatsu Spergel (2001) Babich, Creminelli
Zaldarriaga (2004)

• Depending upon the shape of triangles, one can
  define various fNLs
• Local form
• which generates non-Gaussianity locally (i.e., at
  the same location) via F(x)Fgaus(x)fNLlocalFgau
  s(x)2
• Equilateral form
• which generates non-Gaussianity in a different
  way (e.g., k-inflation, DBI inflation)

Earlier work on the local form SalopekBond
(1990) Gangui et al. (1994) Verde et al.
(2000) WangKamionkowski (2000)
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Can we have large nongaussianity ?
A.L., Kofman 1985-1987, A.L., Mukhanov,
1996, Lyth, Wands, Ungarelli, 2002 Lyth, Wands,
Sasaki and collaborators - many papers up to 2008
V
V
?

• Inflaton

Curvaton
Isocurvature perturbations
adiabatic perturbations
is determined by quantum
fluctuations, so the amplitude of perturbations
is different in different places
?
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Spatial Distribution of the Curvaton Field
?
0
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The Curvaton Web and Nongaussianity
Usually we assume that the amplitude of
inflationary perturbations is constant, ?H
10-5 everywhere. However, in the curvaton
scenario ?H can be different in different parts
of the universe. This is a clear sign of
nongaussianity.
A.L., Mukhanov, astro-ph/0511736
The Curvaton Web
?H
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Alternatives? Ekpyrotic/cyclic scenario
Original version (Khoury, Ovrut, Steinhardt and
Turok 2001) did not work (no explanation of the
large size, mass and entropy the homogeneity
problem even worse than in the standard Big Bang,
Big Crunch instead of the Big Bang, etc.).
It was replaced by cyclic scenario (Steinhardt
and Turok 2002) which is based on a set of
conjectures about what happens when the universe
goes through the singularity and re-emerges.
Despite many optimistic announcements, the
singularity problem in 4-dimensional space-time
and several other problems of the cyclic scenario
remain unsolved.
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Recent developments New ekpyrotic scenario
Creminelly and Senatore, 2007, Buchbinder,
Khoury, Ovrut 2007
Problems violation of the null energy condition,
absence of the ultraviolet completion, difficulty
to embed it in string theory, violation of the
second law of thermodynamics, problems with black
hole physics.
The main problem this theory contains terms
with higher derivatives, which lead to new
ekpyrotic ghosts, particles with negative energy.
As a result, the vacuum state of the new
ekpyrotic scenario suffers from a catastrophic
vacuum instability.
Kallosh, Kang, Linde and Mukhanov, arXiv0712.2040
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The New Ekpyrotic Ghosts
New Ekpyrotic Lagrangian
Dispersion relation
Two classes of solutions, for small P,X
,
Hamiltonian describes normal particles with
positive energy ?1 and ekpyrotic ghosts with
negative energy - ?2
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Vacuum in the new ekpyrotic scenario instantly
decays due to emission of pairs of ghosts and
normal particles.
Cline, Jeon and Moore, 2003
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Why higher derivatives? Can we introduce a UV
cutoff?
Bouncing from the singularity requires violation
of the null energy condition, which in turn
requires
Dispersion relation for perturbations of the
scalar field
The last term appears because of the higher
derivatives. If one makes this term vanish at
large k, then in the regime
one has a catastrophic vacuum instability, with
perturbations growing as
Of course, one can simply assume the existence of
a UV cutoff at energies higher than the ghost
mass, but this would be an inconsistent theory,
until the physical origin of the cutoff is found.
Moreover, in a theory with a UV cutoff, why would
one care about the cosmological singularity? At
this level, the singularity problem could be
solved many decades ago (no high energy modes
no space-time singularity).
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But ghosts have disastrous consequences for the
viability of the theory. In order to regulate the
rate of vacuum decay one must invoke explicit
Lorentz breaking at some low scale. In any case
there is no sense in which a theory with ghosts
can be thought as an effective theory, since the
ghost instability is present all the way to the
UV cut-off of the theory.
Buchbinder, Khoury,
Ovrut 2007
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• Can we save this theory?

Example
Can be obtained by integration with respect to
of the theory with ghosts
By adding some other terms and integrating out
the field one can reduce this theory to
the ghost-free theory.
Creminelli, Nicolis, Papucci and Trincherini, 2005
But this can be done only for a 1, whereas in
the new ekpyrotic scenario a - 1
Kallosh, Kang, Linde and Mukhanov, arXiv0712.2040
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Even if it is possible to improve the new
ekpyrotic scenario (which was never
demonstrated), then it will be necessary to check
whether the null energy condition is still
violated in the improved theory despite the
postulated absence of ghosts. Indeed, if the
correction will also correct the null energy
condition, then the bounce will become
impossible. We are unaware of any examples of
the ghost-free theories where the null energy
condition is violated.
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Other alternatives String gas cosmology

Brandenberger, Vafa, Nayeri, 4 papers in 2005-2006
Many loose ends and unproven assumptions (e.g.
stabilization of the dilaton and of extra
dimensions). Flatness/entropy problem is not
solved. This class of models differs from the
class of stringy models where stabilization of
all moduli was achieved.
Even if one ignores all of these issues, the
perturbations generated in these models are very
non-flat Instead of ns 1 one
finds ns 5

Kaloper, Kofman, Linde, Mukhanov 2006,
hep-th/0608200 Brandenberger et al, 2006
Other problems with the string gas constructions
were recently discussed by Kaloper and Watson,
arXiv0712.1820
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A toy model of SUGRA inflation
Holman, Ramond, Ross, 1984
Superpotential
Kahler potential
Inflation occurs for ?0 1 Requires
fine-tuning, but it is simple, and it works
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A toy model of string inflation
A.L., Westphal, 2007
Superpotential
Kahler potential
Volume modulus inflation Requires fine-tuning,
but works without any need to study complicated
brane dynamics
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String Cosmology and the Gravitino Mass
Kallosh, A.L. 2004
The height of the KKLT barrier is smaller than
VAdS m23/2. The inflationary potential Vinfl
cannot be much higher than the height of the
barrier. Inflationary Hubble constant is given by
H2 Vinfl/3 lt m23/2.
uplifting
Modification of V at large H
VAdS
Constraint on the Hubble constant in this class
of models
H lt m3/2
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Can we avoid these conclusions?
Recent model of chaotic inflation is string
theory (Silverstein and Westphal, 2007) also
require .
H lt m3/2
In more complicated theories one can have
. But this
requires fine-tuning (Kallosh, A.L. 2004,
Badziak, Olechowski, 2007)
In models with large volume of compactification
(Quevedo et al) the situation is even more
dangerous
It is possible to solve this problem, but it is
rather nontrivial.
Conlon, Kallosh, A.L., Quevedo, in preparation
Remember that we are suffering from the light
gravitino and the cosmological moduli problem for
the last 25 years.
The price for the SUSY solution of the hierarchy
problem is high, and it is growing. Split
supersymmetry? We are waiting for LHC...
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Tensor Modes and GRAVITINO
Kallosh, A.L. 2007
superheavy gravitino
unobservable
A discovery or non-discovery of tensor modes
would be a crucial test for string theory and
particle phenomenology
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Landscape of eternal inflation
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Perhaps 101000 different uplifted vacua
Lerche, Lust, Schellekens 1987
Bousso, Polchinski 2000 Susskind 2003 Douglas,
Denef 2003
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What is so special about our world?
Problem Eternal inflation creates infinitely
many different parts of the universe, so we must
compare infinities
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Two different approaches

• Study events at a given point, ignoring growth of
  volume, or, equivalently, calculating volume in
  comoving coordinates
• Starobinsky 1986, Garriga,
  Vilenkin 1998, Bousso 2006, A.L. 2006

No problems with infinities, but the results
depend on initial conditions. It is not clear
whether these methods are appropriate for
description of eternal inflation, where the
exponential growth of volume is crucial.
2. Take into account growth of volume
A.L. 1986 A.L., D.Linde, Mezhlumian,
Garcia-Bellido 1994 Garriga, Schwarz-Perlov,
Vilenkin, Winitzki 2005 A.L. 2007
No dependence on initial conditions, but we are
still learning how to do it properly.
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Let us discuss non-eternal inflation to learn
about the measure
The universe is divided into two parts, one
inflates for a long time, one does it for a short
time. Both parts later collapse.
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More observers live in the inflationary (part of
the) universe because there are more stars and
galaxies there
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Comoving probability measure does not distinguish
small and big universes and misses most of the
stars
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One can use the volume weighted measure
parametrized by time t proportional to the scale
factor a. In this case the two parts of the
universe grow at the same rate, but the
non-inflationary one stops growing and collapses
while the big one continues to grow. Thus the
comparison of the volumes at equal times fails.
Scale factor cutoff, t a
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When we make a cut, in the beginning inflation
does not provide us any benefit No gain in
volume. This could suggest that the growth of
volume during inflation does not have any
anthropic significance
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But when we move the cut-off higher, the
comparison between the two parts of the universe
becomes impossible. The small part of the
universe dies early, whereas the inflationary
universe continues to grow. When we remove the
cut-off, we find the usual result Most of the
observers live in the universe produced by
inflation.
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V
Boltzmann Brains are coming!!!
BB1
BB3
Hopefully, normal brains are created even faster,
due to eternal inflation
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Consider first the scale factor
cutoff, t a.
If the dominant vacuum cannot produce Boltzmann
brains, and our vacuum decays before BBs are
produced we will not have any problems with them.
Freivigel, Bousso et al, in preparation De
Simone, Guth, Linde, Noorbala, Salem,Vilenkin, in
preparation
Can we realize this possibility? Recall that
Ceresole, DallAgata, Giryavets, Kallosh, A.L.,
2006
The long-living vacuum tend to be the ones with
an (almost) unbroken supersymmetry,
. But people like us cannot live in a
supersymmetric universe.
In other words, Boltzmann brains born in the
stable vacua tend to be brain-dead.
More on this in the talks by Vilenkin and
Freivogel
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Problems with probabilities
V
3
4
2
1
5
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Time can be measured in the number of
oscillations ( ) or in the number of
e-foldings of inflation ( ). The
universe expands as
is the growth of volume during
inflation
Unfortunately, the result depends on the time
parametrization.
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t21
t45
t 0
We should compare the trees of bubbles not at
the time when the trees were seeded, but at the
time when the bubbles appear
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A possible solution of this problem
If we want to compare apples to apples, instead
of the trunks of the trees, we need to reset the
time to the moment when the stationary regime of
exponential growth begins. In this case we obtain
the gauge-invariant result
As expected, the probability is proportional to
the rate of tunneling and to the growth of volume
during inflation.
A.L., arXiv0705.1160
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What if instead of the minimum at the top, we
have a flat maximum, as in new inflation?
Boundary of eternal inflation
A preliminary answer (Winitzki, Vanchurin, A.L.,
in progress) In the limit of small V, when the
size of the area of eternal inflation becomes
sufficiently small, the results, in the leading
approximation, do not depend on time
parametrization.
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In general, according to the stationary measure,
if we have two possible outcomes of a process
starting at t 0
where ti is the time when the stationarity regime
for the corresponding process is established. The
more probable is the trajectory, the longer it
takes to reach stationarity, the better. For
example, the ratio of the probabilities for
different temperatures in the domains of the same
type is
Slight preference for lower temperatures, no
youngness paradox.
A.L. 2007
Moreover, we believe that the youngness paradox
does not appear even if one takes into account
inhomogeneities of temperature.
A.L., Vanchurin, Winitzki, in preparation
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These results agree with the expectation that
the probability to be born in a part of the
universe which experienced inflation can be very
large, because of the exponential growth of
volume during the slow-roll inflation.
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No Boltzmann Brainer
A.L., Vanchurin, Winitzki, in preparation
Stationary measure does not lead to the Boltzmann
brain problem
The ratio of BBs to OOs is proportional to the
ratios of the volumes of the universe when the
stationarity is reached for BBs and OOs (which
rewards OOs), multiplied by the extremely small
probability of the BB production in the
vacuum. Example In the no-delay situation
(A.L. 2006)
In a more general and realistic situation, with
the time delay, taking into account thermal
fluctuations, the result is very similar, no BBs
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Conclusions
There is an ongoing progress in implementing
inflation in supergravity and string theory.
As on now, we are unaware of any non-inflationary
alternatives which are verifiably consistent.
CMB can help us to test string theory. If
inflationary tensor modes are discovered, we may
need to develop phenomenological models with
superheavy gravitino.
Looking forward, we must either propose something
better than inflation and string theory in its
present form, or learn how to make probabilistic
predictions based on eternal inflation and the
string landscape scenario. Several promising
probability measures were proposed, including the
stationary measure.
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Bousso, Freivogel and Yang, 2007
Here our conclusions differ from those of
They confirmed that in the absence of
perturbations of temperature T, the probability
distribution to be born in the universe with a
given T depends on T smoothly, and does not
suffer from the youngness problem. However, when
they took into account perturbations of
temperature, they found, in the approximation
which they proposed, that
Here 1010 stays for the inverse square of the
amplitude of perturbations of temperature induced
by inflationary perturbations. The first term is
much greater, which leads to oldness paradox
Small T are exponentially better. Note, however,
that if one takes the limit when the amplitude of
perturbations vanishes, the coefficient in front
of the second term in the exponent blows up, and
the probability distribution becomes singular,
concentrated at T 3.3 K. This contradicts their
own result that in the absence of perturbations
of temperature, the probability distribution is
smooth.
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A toy landscape model
Mahdiyar Noorbala, A.L. 2008
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As an example, consider Bousso measure, assuming
first that Boltzmann brains can be born in all
vacua
However, stable vacua are not really stable. In
a typical situation in stringy landscape one
expects their decay rate
Ceresole, DallAgata, Giryavets, Kallosh, A.L.,
2006
Such vacua could be BB safe. If there are other
vacua, with a small SUSY breaking, they may be
stable but it is dangerous only if
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Another advantage of the stationary measure
A.L., Vanchurin, Winitzki, in preparation
1. It does not suffer from the youngness paradox
Usually youngness problem appears because of the
reward of the delay of inflation by a factor of
But in the stationary measure, this reward is
taken back when taking into account time delay
for the onset of stationarity.
2. It does not lead to the Boltzmann brain problem
The ratio of BBs to OOs is proportional to the
ratios of the volumes of the universe when the
stationarity is reached for BBs and OOs (which
rewards OOs), multiplied by the extremely small
probability of the BB production in the vacuum.
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