Conceptual Issues in InflationConceptual Issues in Inflation

New InflationNew InflationNew InflationNew Inflation

V

1981 - 1982

Chaotic InflationChaotic Inflation Chaotic InflationChaotic Inflation

Eternal InflationEternal Inflation

1983

Hybrid InflationHybrid InflationHybrid InflationHybrid Inflation 1991, 1994

WMAP5 + Acbar + Boomerang + CBIWMAP5 + Acbar + Boomerang + CBIWMAP5 + Acbar + Boomerang + CBIWMAP5 + Acbar + Boomerang + CBI

Tensor modes:Tensor modes:Tensor modes: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)

Blue lines – chaotic inflation with the simplest spontaneous symmetry breaking potential for N = 50 and N = 60

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

Destri, de Vega, Sanchez, 2007

What is fNL?What is fNL?

fNL = the amplitude of three-point function

also known as the “bispectrum,” B(k1,k2,k3), which is

<Φ(k1)Φ(k2)Φ(k3)>=fNL(2π)3δ3(k1+k2+k3)b(k1,k2,k3)

Φ(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.

k1k2

k3

11

Komatsu 2008:

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.

12

Two fNL’s• Depending upon the shape of

triangles, one can define various fNL’s:

• “Local” form• which generates non-Gaussianity locally

(i.e., at the same location) via Φ(x)=Φgaus(x)+fNL

local[Φgaus(x)]2

• “Equilateral” form• which generates non-Gaussianity in a

different way (e.g., k-inflation, DBI inflation)

Komatsu & Spergel (2001); Babich, Creminelli & Zaldarriaga (2004)

Earlier work on the local form: Salopek&Bond (1990); Gangui et al. (1994); Verde et al. (2000); Wang&Kamionkowski (2000)

13

Can we have large nongaussianity ?Can we have large nongaussianity ?Can we have large nongaussianity ?Can we have large nongaussianity ?

V

Inflaton

V

Curvaton

A.L., Kofman 1985-1987, A.L., Mukhanov, 1996,

Lyth, Wands, Ungarelli, 2002

Lyth, Wands, Sasaki and collaborators - many papers up to 2008

Isocurvature perturbations adiabatic perturbations

is determined by quantum fluctuations, so the amplitude of perturbations is different in

different places

Spatial Distribution of the Curvaton FieldSpatial Distribution of the Curvaton FieldSpatial Distribution of the Curvaton FieldSpatial Distribution of the Curvaton Field

0

Usually we assume that the amplitude of inflationary perturbations is constant,

~ 10-5 everywhere. However, in the curvaton scenario can be different in

different parts of the universe. This is a clear sign of nongaussianity.

The Curvaton Web and NongaussianityThe Curvaton Web and NongaussianityThe Curvaton Web and NongaussianityThe Curvaton Web and Nongaussianity

A.L., Mukhanov, astro-ph/0511736

The Curvaton WebThe Curvaton Web

Alternatives?Alternatives?

Ekpyrotic/cyclic scenarioEkpyrotic/cyclic scenario

Alternatives?Alternatives?

Ekpyrotic/cyclic scenarioEkpyrotic/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.

Recent developments: “New ekpyrotic scenario”

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, arXiv:0712.2040

Creminelly and Senatore, 2007, Buchbinder, Khoury, Ovrut 2007

The New Ekpyrotic The New Ekpyrotic GhostsGhosts The New Ekpyrotic The New Ekpyrotic GhostsGhosts

New Ekpyrotic Lagrangian:

Hamiltonian describes normal particles with positive energy + 1

and ekpyrotic ghosts with negative energy - 2

Dispersion relation:

Two classes of solutions, for small P,X:

,,

Vacuum in the new ekpyrotic scenario instantly decays due to emission of pairs of ghosts and normal particles.

Cline, Jeon and Moore, 2003

Why higher derivatives? Can we introduce a UV cutoff? Why higher derivatives? Can we introduce a UV cutoff? Why higher derivatives? Can we introduce a UV cutoff? 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).

“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

Can we save this theory?Can we save this theory?Can we save this theory?Can we save this theory?

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.

But this can be done only for a = + 1, whereas in the new ekpyrotic scenario a = - 1

Creminelli, Nicolis, Papucci and Trincherini, 2005

Kallosh, Kang, Linde and Mukhanov, arXiv:0712.2040

Example:

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.

A toy model of SUGRA inflation:A toy model of SUGRA inflation: A toy model of SUGRA inflation:A toy model of SUGRA inflation:

Superpotential:

Kahler potential:

Holman, Ramond, Ross, 1984

Inflation occurs for 0 = 1

Requires fine-tuning, but it is simple, and it works

A toy model of string inflation:A toy model of string inflation: A toy model of string inflation:A toy model of string inflation:

Superpotential:

Kahler potential:

A.L., Westphal, 2007

Volume modulus inflation

Requires fine-tuning, but works without any need to study complicated brane

dynamics

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 < m2

3/2.

Constraint on the Hubble constant in this class of models:

H < mH < m3/23/2 H < mH < m3/23/2

Modification of V at large H

String Cosmology and the Gravitino MassString Cosmology and the Gravitino MassString Cosmology and the Gravitino MassString Cosmology and the Gravitino MassKallosh, A.L. 2004

VAdS

uplifting

Can we avoid these conclusions?Can we avoid these conclusions?Can we avoid these conclusions?Can we avoid these conclusions?

In more complicated theories one can have . But this

In models with large volume of compactification (Quevedo et al) the situation is even more dangerous:

The price for the SUSY solution of the hierarchy problem is high, and it is growing. Split supersymmetry? We are waiting for LHC...

Remember that we are suffering from the light gravitino and the cosmological moduli problem for the last 25 years.

requires fine-tuning (Kallosh, A.L. 2004, Badziak, Olechowski, 2007)

It is possible to solve this problem, but it is rather nontrivial.Conlon, Kallosh, A.L., Quevedo, in preparation

Recent model of chaotic inflation is string theory (Silverstein and Westphal, 2007) also require .H < mH < m3/23/2

Landscape of eternal inflation Landscape of eternal inflation

Problem:Problem: Eternal inflation creates infinitely many different Eternal inflation creates infinitely many different parts of the universe, so we must compare infinitiesparts of the universe, so we must compare infinities

What is so special about our world?What is so special about our world? What is so special about our world?What is so special about our world?

1.1. Study events at a given point, ignoring growth of volume, or, Study events at a given point, ignoring growth of volume, or, equivalently, calculating volume in comoving coordinatesequivalently, calculating volume in comoving coordinates

Starobinsky 1986, Garriga, Vilenkin 1998, Bousso 2006, A.L. 2006Starobinsky 1986, Garriga, Vilenkin 1998, Bousso 2006, A.L. 2006

Two different approaches:Two different approaches: Two different approaches:Two different approaches:

2.2. Take into account growth of volumeTake into account growth of volume

A.L. 1986; A.L., D.Linde, Mezhlumian, Garcia-Bellido 1994; Garriga, A.L. 1986; A.L., D.Linde, Mezhlumian, Garcia-Bellido 1994; Garriga, Schwarz-Perlov, Vilenkin, Winitzki 2005; A.L. 2007Schwarz-Perlov, Vilenkin, Winitzki 2005; A.L. 2007

No problems with infinities, but the results depend on initial conditions. No problems with infinities, but the results depend on initial conditions. It is not clear whether these methods are appropriate for description of It is not clear whether these methods are appropriate for description of eternal inflation, where the exponential growth of volume is crucial.eternal inflation, where the exponential growth of volume is crucial.

No dependence on initial conditions, but we are still learning how No dependence on initial conditions, but we are still learning how to do it properly. to do it properly.

Let us discuss non-eternal inflation to

learn about the measure

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.

More observers live in the inflationary (part of the) universe because there are more stars and galaxies there

Comoving probability measure does not distinguish small and big universes and misses most of the stars

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

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

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.

V

BB1BB1BB3BB3

Boltzmann Brains are Boltzmann Brains are coming!!!coming!!!Boltzmann Brains are Boltzmann Brains are coming!!!coming!!!

Hopefully, Hopefully, normal brains are normal brains are created even fastercreated even faster, due to , due to

eternal inflationeternal inflation

Hopefully, Hopefully, normal brains are normal brains are created even fastercreated even faster, due to , due to

eternal inflationeternal inflation

Consider first the scale factor cutoff, t = a.

Freivigel, Bousso et al, in preparationDe Simone, Guth, Linde, Noorbala, Salem,Vilenkin, in preparation

If the dominant vacuum cannot produce Boltzmann brains, and our vacuum decays before BBs are produced we will not have any problems with them.

Can we realize this possibility? Recall that

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

Ceresole, Dall’Agata, Giryavets, Kallosh, A.L., 2006

V

3

4 2 15

Problems with probabilitiesProblems with probabilities Problems with probabilitiesProblems with probabilities

Time can be measured in the number of oscillations ( ) or in the number of e-foldings of inflation ( ). The universe expands as

Unfortunately, the result depends on the time parametrization.

is the growth of volume during inflation

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

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., arXiv:0705.1160

A possible solution of this problem:A possible solution of this problem: A possible solution of this problem:A possible solution of this problem:

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., Vanchurin, Winitzki, in preparation

Moreover, we believe that the youngness paradox does not appear even if one takes into account inhomogeneities of temperature.

A.L. 2007

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.

No Boltzmann BrainerNo Boltzmann Brainer No Boltzmann BrainerNo Boltzmann Brainer

Stationary measure does not lead to the Boltzmann brain problem

A.L., Vanchurin, Winitzki, in preparation

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:

Conclusions:Conclusions: Conclusions:Conclusions:There is an ongoing progress in implementing inflation in supergravity and string theory.

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.

As on now, we are unaware of any non-inflationary alternatives which are verifiably consistent.

Bousso, Freivogel and Yang, 2007

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.

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 our conclusions differ from those of

A toy landscape modelA toy landscape model A toy landscape modelA toy landscape model

Mahdiyar Noorbala, A.L. 2008

However, “stable vacua” are not really stable. In a typical situation in stringy landscape one expects their decay rate

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

Ceresole, Dall’Agata, Giryavets, Kallosh, A.L., 2006

As an example, consider Bousso measure, assuming first that Boltzmann brains can be born in all vacua