Conceptual Issues in Inflation

New Inflation

1981 - 1982 V

Chaotic Inflation

1983 Eternal Inflation

Hybrid Inflation

1991, 1994

WMAP5 + Acbar + Boomerang + CBI

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

Destri, de Vega, Sanchez, 2007 Possible values of r and n s 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

Komatsu 2008:

What is f

NL

?

k 2 k 3 k 1

f NL = the amplitude of three-point function

also known as the “bispectrum,” B(k 1 ,k 2 ,k 3 ), which is < Φ (k 1 ) Φ (k 2 ) Φ (k 3 )>=f NL (2 π ) 3 δ 3 (k 1 +k 2 +k 3 )b(k 1 ,k 2 ,k 3 ) Φ (k) is the Fourier transform of the curvature perturbation, and b(k 1 ,k 2 ,k 3 ) is a model-dependent function that defines the shape of triangles predicted by various models.

11

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

12

non-Gaussianity.

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

• NL

’s

triangles, one can define various

• •

f

NL

’s: “Local” form

which generates non-Gaussianity • • locally (i.e., at the same location) via Φ (x)= Φ gaus (x)+f NL local [ Φ gaus (x)] 2 Salopek&Bond (1990); Gangui et al.

“Equilateral” form

Wang&Kamionkowski (2000) which generates non-Gaussianity in a different way (e.g., k-inflation, DBI inflation) 13

Can we have large nongaussianity ?

V A.L., Kofman 1985-1987, A.L., Mukhanov, 1996, Lyth, Wands, Ungarelli, 2002 Lyth, Wands, Sasaki and collaborators - many papers up to 2008 V  Inflaton Curvaton Isocurvature perturbations adiabatic perturbations  is determined by quantum fluctuations, so the amplitude of perturbations is different in different places

Spatial Distribution of the Curvaton Field

 0

 H

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

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.

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, arXiv:0712.2040

The New Ekpyrotic Ghosts

New Ekpyrotic Lagrangian: Dispersion relation: Two classes of solutions, for small P, X :

,

Hamiltonian describes normal particles and ekpyrotic ghosts with positive energy +  1 with negative energy  2

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?

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

Example: 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.

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, arXiv:0712.2040

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:

Holman, Ramond, Ross, 1984

Superpotential: Kahler potential:

Inflation occurs for  0 = 1 Requires fine-tuning, but it is simple, and it works

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

String Cosmology and the Gravitino Mass

Kallosh, A.L. 2004 The height of the KKLT barrier is smaller than |V AdS | =m 2 3/2 . The inflationary potential V infl cannot be much higher than the height of the barrier. Inflationary Hubble constant is given by H 2 = V infl /3 < m 2 3/2 .

uplifting Modification of V at large H V AdS

Constraint on the Hubble constant in this class of models:

H < m

3/2

Can we avoid these conclusions?

Recent model of chaotic inflation is string theory (Silverstein and Westphal, 2007) also require .

3/2 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...

Landscape of eternal inflation

What is so special about our world?

Problem:

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

Two different approaches:

1. 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.

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

Scale factor cutoff, t = a

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.

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

Boltzmann Brains are coming!!!

BB 3 BB 1

Hopefully, normal brains are created even faster , due to eternal inflation

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, Dall’Agata, 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

Problems with probabilities

V

3 5 4 2 1

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.

t 21 t 45 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

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., arXiv:0705.1160

In general, according to the stationary measure, if we have two possible outcomes of a process starting at t = 0 where t i 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

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 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 :

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.

Here our conclusions differ from those of Bousso, Freivogel and Yang, 2007 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 10 10 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.

A toy landscape model

Mahdiyar Noorbala, A.L. 2008

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, Dall’Agata, 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