Testing String Theory with Cosmological Observations?

R. Branden- berger Testing String Theory with Cosmological Inﬂation Motivation Observations? Inﬂation Problems CAP-CRM Prize Lecture Message

String gas Principles Features Robert Brandenberger Structure Perturbations McGill University Analysis

Conclusions

June 15, 2012

1 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

Structure Bill Press, Doug Eardley (Harvard) Perturbations Analysis Neil Turok, Andy Albrecht Conclusions Anne-Christine Davis (DAMTP), Slava Mukhanov (LMU), Cumrun Vafa (Harvard), Xinmin Zhang (IHEP Beijing)

2 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

2 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

2 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

2 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

2 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

2 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

2 / 40 Thanks to:

String Cosmology

R. Branden- berger Klaus Hepp (ETH Zürich)

Inﬂation Konrad Osterwalder (ETH Zürich) Motivation Inﬂation Arthur Jaffe (Harvard) Problems Message Jennie Traschen String gas Principles Raul Bott, Sidney Coleman (Harvard) Features

2 / 40 String Cosmology

R. Branden- Hume Feldman, Leandros Perivolaropoulos, Jong berger Kung, Jinwu Ye, Tomislav Prokopec, Andrew Inﬂation Sornborger, Richhild Moessner, Mark Trodden, Raul Motivation Inﬂation Abramo, Matthew Parry, Martin Götz, Stephon Problems Message Alexander, Damien Easson, Antonio Guimaraes, Zeeya String gas Merali, Ghazal Geshnizjani, Scott Watson, Thorsten Principles Features Battefeld. Structure Perturbations Subodh Patil, Natalia Shuhmaher, Patrick Martineau, Analysis Rebecca Danos, Nima Lashkari, Andrew Stewart, Conclusions Francis Cyr-Racine, Jean Lachapelle, Johanna Karouby, Wei Xue, Laurence Perreault-Levasseur My colleagues at McGill

3 / 40 String Cosmology

3 / 40 Main Messages

String Cosmology

R. Branden- berger

Inflation Motivation Inflation There are alternatives to inflationary cosmology. Problems Message String Theory leads to a new paradigm for early String gas Principles universe cosmology. Features String Theory can be tested with cosmological Structure Perturbations observations. Analysis

Conclusions

4 / 40 Outline

String Cosmology 1 Inflation: Current Paradigm of Early Universe Cosmology R. Branden- berger Motivation Review of Inflationary Cosmology Inflation Motivation Problems of Inflationary Cosmology Inflation Problems Message Message String gas 2 String Gas Cosmology Principles Features Principles Structure Features of String Gas Cosmology Perturbations Analysis

Conclusions 3 String Gas Cosmology and Structure Formation Review of the Theory of Cosmological Perturbations Analysis

4 Conclusions

5 / 40 Plan

Conclusions 3 String Gas Cosmology and Structure Formation Review of the Theory of Cosmological Perturbations Analysis

4 Conclusions

6 / 40 Current Paradigm for Early Universe Cosmology

String Cosmology

R. Branden- berger The Inﬂationary Universe Scenario is the current paradigm

Inflation of early universe cosmology. Motivation Inflation Successes: Problems Message Solves horizon problem String gas Principles Solves flatness problem Features

Structure Solves size/entropy problem Perturbations Analysis Provides a causal mechanism of generating primordial Conclusions cosmological perturbations (Chibisov & Mukhanov, 1981).

7 / 40 Current Paradigm for Early Universe Cosmology

String Cosmology

R. Branden- berger The Inﬂationary Universe Scenario is the current paradigm

Inflation of early universe cosmology. Motivation Inflation Successes: Problems Message Solves horizon problem String gas Principles Solves flatness problem Features

Structure Solves size/entropy problem Perturbations Analysis Provides a causal mechanism of generating primordial Conclusions cosmological perturbations (Chibisov & Mukhanov, 1981).

7 / 40 String Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

Credit: NASA/WMAP Science Team

8 / 40 String Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

Credit: NASA/WMAP Science Team

9 / 40 Challenges for the Current Paradigm

String Cosmology

R. Branden- berger In spite of the phenomenological successes, current

Inflation realizations of the inflationary scenario suffer from Motivation Inflation several conceptual problems. Problems Message In light of these problems we need to look for input from String gas new fundamental physics to construct a new theory Principles Features which will overcome these problems. Structure Perturbations Question: Can Superstring theory lead to a new and Analysis improved paradigm? Conclusions Question: Can this new paradigm be tested in cosmological observations?

10 / 40 Challenges for the Current Paradigm

String Cosmology

R. Branden- berger In spite of the phenomenological successes, current

10 / 40 Challenges for the Current Paradigm

String Cosmology

R. Branden- berger In spite of the phenomenological successes, current

10 / 40 Review of Inﬂationary Cosmology

String Cosmology Context: R. Branden- berger General Relativity

Inﬂation Scalar Field Matter Motivation Inﬂation Problems 2 2 2 2 Message Metric : ds = dt − a(t) dx (1)

String gas Principles Inflation: Features tH Structure phase with a(t) ∼ e Perturbations Analysis requires matter with p ∼ −ρ Conclusions requires a slowly rolling scalar field ϕ - in order to have a potential energy term - in order that the potential energy term dominates sufficiently long

11 / 40 String Cosmology

R. Branden- berger Time line of inﬂationary cosmology:

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis Conclusions ti : inﬂation begins

tR: inﬂation ends, reheating

12 / 40 Review of Inﬂationary Cosmology II

String Space-time sketch of inﬂationary cosmology: Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

Note: a˙ H = a curve labelled by k: wavelength of a ﬂuctuation 13 / 40 String Cosmology

R. Branden- berger

Inflation Motivation inflation renders the universe large, homogeneous and Inflation Problems spatially flat Message classical matter redshifts → matter vacuum remains String gas Principles Features quantum vacuum fluctuations: seeds for the observed Structure structure [Chibisov & Mukhanov, 1981] Perturbations Analysis sub-Hubble → locally causal Conclusions

14 / 40 Conceptual Problems of Inﬂationary Cosmology

String Cosmology

R. Branden- berger Nature of the scalar field ϕ (the “inflaton") Inflation Motivation Conditions to obtain inflation (initial conditions, slow-roll Inflation Problems conditions, graceful exit and reheating) Message

String gas Amplitude problem Principles Features Trans-Planckian problem Structure Perturbations Singularity problem Analysis

Conclusions Cosmological constant problem Applicability of General Relativity

15 / 40 Trans-Planckian Problem

String Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis Success of inflation: At early times scales are inside Conclusions the Hubble radius → causal generation mechanism is possible. Problem: If time period of inflation is more than 70H−1, then λp(t) < lpl at the beginning of inflation → new physics MUST enter into the calculation of the fluctuations. 16 / 40 Cosmological Constant Problem

String Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features Quantum vacuum energy does not gravitate. Structure Perturbations Why should the almost constant V (ϕ) gravitate? Analysis

Conclusions V 0 ∼ 10120 (2) Λobs

17 / 40 Applicability of GR

String Cosmology

R. Branden- berger In all approaches to quantum gravity, the Einstein action

Inflation is only the leading term in a low curvature expansion. Motivation Inflation Correction terms may become dominant at much lower Problems Message energies than the Planck scale. String gas Principles Correction terms will dominate the dynamics at high Features curvatures. Structure Perturbations The energy scale of inflation models is typically Analysis 16 Conclusions η ∼ 10 GeV. → η too close to mpl to trust predictions made using GR.

18 / 40 Zones of Ignorance

String Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

19 / 40 Message

String Cosmology Current realizations of inflation have conceptual R. Branden- problems. berger We need a new paradigm of very early universe Inflation Motivation cosmology based on new fundamental physics. Inflation Problems Hypothesis: New paradigm based on Superstring Message Theory. String gas Principles Features New cosmological model motivated by superstring

Structure theory: String Gas Cosmology (SGC) [R.B. and C. Perturbations Analysis Vafa, 1989] Conclusions New structure formation scenario emerges from SGC [A. Nayeri, R.B. and C. Vafa, 2006]. Testable prediction for cosmological observations: Blue tilt in the spectrum of gravitational waves [R.B., A. Nayeri, S. Patil and C. Vafa, 2006]

20 / 40 Plan

Conclusions 3 String Gas Cosmology and Structure Formation Review of the Theory of Cosmological Perturbations Analysis

4 Conclusions

21 / 40 Principles R.B. and C. Vafa, Nucl. Phys. B316:391 (1989)

String Cosmology Idea: make use of the new symmetries and new degrees of R. Branden- berger freedom which string theory provides to construct a new theory of the very early universe. Inﬂation Motivation Assumption: Matter is a gas of fundamental strings Inﬂation Problems Assumption: Space is compact, e.g. a torus. Message

String gas Key points: Principles Features New degrees of freedom: string oscillatory modes Structure Perturbations Leads to a maximal temperature for a gas of strings, Analysis the Hagedorn temperature Conclusions New degrees of freedom: string winding modes Leads to a new symmetry: physics at large R is equivalent to physics at small R

22 / 40 T-Duality

String Cosmology

R. Branden- berger T-Duality Inﬂation Motivation Momentum modes: En = n/R Inﬂation Problems Message Winding modes: Em = mR String gas Duality: R → 1/R (n, m) → (m, n) Principles Features Mass spectrum of string states unchanged Structure Perturbations Symmetry of vertex operators Analysis

Conclusions Symmetry at non-perturbative level → existence of D-branes

23 / 40 Adiabatic Considerations R.B. and C. Vafa, Nucl. Phys. B316:391 (1989)

String Cosmology Temperature-size relation in string gas cosmology R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

24 / 40 Dynamics

String Cosmology

R. Branden- berger Assume some action gives us R(t)

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

25 / 40 Dynamics II

String Cosmology

R. Branden- berger We will thus consider the following background dynamics for Inﬂation the scale factor a(t): Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

26 / 40 Dimensionality of Space in SGC

String Cosmology

R. Branden- Begin with all 9 spatial dimensions small, initial berger temperature close to TH → winding modes about all Inﬂation spatial sections are excited. Motivation Inﬂation Problems Expansion of any one spatial dimension requires the Message annihilation of the winding modes in that dimension. String gas Principles Features

Structure Perturbations Analysis

Conclusions Decay only possible in three large spatial dimensions. → dynamical explanation of why there are exactly three large spatial dimensions.

27 / 40 Dimensionality of Space in SGC

String Cosmology

Structure Perturbations Analysis

Conclusions Decay only possible in three large spatial dimensions. → dynamical explanation of why there are exactly three large spatial dimensions.

27 / 40 Plan

Conclusions 3 String Gas Cosmology and Structure Formation Review of the Theory of Cosmological Perturbations Analysis

4 Conclusions

28 / 40 Theory of Cosmological Perturbations: Basics

String Cosmology Cosmological ﬂuctuations connect early universe theories R. Branden- berger with observations

Inflation Fluctuations of matter → large-scale structure Motivation Inflation Fluctuations of metric → CMB anisotropies Problems Message N.B.: Matter and metric fluctuations are coupled String gas Principles Features Key facts:

Structure Perturbations 1. Fluctuations are small today on large scales Analysis → fluctuations were very small in the early universe Conclusions → can use linear perturbation theory 2. Sub-Hubble scales: matter fluctuations dominate Super-Hubble scales: metric fluctuations dominate

29 / 40 Quantum Theory of Linearized Fluctuations V. Mukhanov, H. Feldman and R.B., Phys. Rep. 215:203 (1992)

String Step 1: Metric and matter including ﬂuctuations Cosmology

R. Branden- berger ds2 = a2[(1 + 2Φ)dη2 − (1 − 2Φ)dx2] (3) Inﬂation Motivation ϕ = ϕ0 + δϕ (4) Inﬂation Problems Message Note: Φ and δϕ related by Einstein constraint equations String gas Step 2: Expand the action for matter and gravity to second Principles Features order about the cosmological background: Structure Perturbations 00 Analysis 1 Z z S(2) = d 4x(v 0)2 − v v ,i + v 2 (5) Conclusions 2 ,i z z v = aδϕ + Φ (6) a ϕ0 z = a 0 (7) H

30 / 40 String Cosmology

R. Branden- berger

Inﬂation Step 3: Resulting equation of motion (Fourier space) Motivation Inﬂation 00 Problems 00 2 z Message v + (k − )v = 0 (8) k z k String gas Principles Features: Features Structure oscillations on sub-Hubble scales Perturbations Analysis squeezing on super-Hubble scales vk ∼ z Conclusions

31 / 40 Background for string gas cosmology

String Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

32 / 40 Structure formation in string gas cosmology A. Nayeri, R.B. and C. Vafa, Phys. Rev. Lett. 97:021302 (2006)

String Cosmology

R. Branden- berger

Inﬂation Motivation Inﬂation Problems Message

String gas Principles Features

Structure Perturbations Analysis

Conclusions

N.B. Perturbations originate as thermal string gas ﬂuctuations. 33 / 40 Method

String Cosmology

R. Branden- berger

Inflation Motivation Calculate matter correlation functions in the Hagedorn Inflation Problems phase (neglecting the metric fluctuations) Message

String gas For fixed k, convert the matter fluctuations to metric Principles Features fluctuations at Hubble radius crossing t = ti (k)

Structure Evolve the metric ﬂuctuations for t > ti (k) using the Perturbations Analysis usual theory of cosmological perturbations Conclusions

34 / 40 Extracting the Metric Fluctuations

String Cosmology

R. Branden- berger Ansatz for the metric including cosmological perturbations and gravitational waves: Inﬂation Motivation Inﬂation Problems 2 2 2 i j Message ds = a (η) (1 + 2Φ)dη − [(1 − 2Φ)δij + hij ]dx dx . (9) String gas Principles Features Inserting into the perturbed Einstein equations yields Structure Perturbations 2 2 2 −4 0 0 Analysis h|Φ(k)| i = 16π G k hδT 0(k)δT 0(k)i , (10) Conclusions

2 2 2 −4 i i h|h(k)| i = 16π G k hδT j (k)δT j (k)i . (11)

35 / 40 Power Spectrum of Cosmological Perturbations

String Cosmology

R. Branden- berger Key ingredient: For thermal fluctuations: Inflation Motivation 2 Inflation 2 T Problems hδρ i = CV . (12) Message R6 String gas Principles Key ingredient: For string thermodynamics in a compact Features space Structure Perturbations Analysis 2 3 R /`s Conclusions CV ≈ 2 . (13) T (1 − T /TH )

36 / 40 Power Spectrum of Cosmological Perturbations

String Cosmology

36 / 40 String Cosmology Power spectrum of cosmological ﬂuctuations R. Branden- berger

Inflation 2 −1 2 Motivation PΦ(k) = 8G k < |δρ(k)| > (14) Inflation 2 2 2 Problems = 8G k < (δM) >R (15) Message 2 −4 2 String gas = 8G k < (δρ) >R (16) Principles Features T 1 = 8G2 (17) Structure `3 1 − T /T Perturbations s H Analysis Conclusions Key features: scale-invariant like for inflation slight red tilt like for inflation

37 / 40 String Cosmology Power spectrum of cosmological ﬂuctuations R. Branden- berger

37 / 40 Spectrum of Gravitational Waves R.B., A. Nayeri, S. Patil and C. Vafa, Phys. Rev. Lett. (2007)

String Cosmology

R. Branden- 2 2 −1 2 berger Ph(k) = 16π G k < |Tij (k)| > (18) 2 2 −4 2 Inflation = 16π G k < |Tij (R)| > (19) Motivation Inflation 2 2 T Problems ∼ 16π G (1 − T /T ) (20) Message 3 H `s String gas Principles Features Key ingredient for string thermodynamics Structure Perturbations T Analysis < |T (R)|2 > ∼ (1 − T /T ) (21) ij 3 4 H Conclusions ls R Key features: scale-invariant (like for inflation) slight blue tilt (unlike for inflation)

38 / 40 Spectrum of Gravitational Waves R.B., A. Nayeri, S. Patil and C. Vafa, Phys. Rev. Lett. (2007)

String Cosmology

38 / 40 Plan

Conclusions 3 String Gas Cosmology and Structure Formation Review of the Theory of Cosmological Perturbations Analysis

4 Conclusions

39 / 40 Conclusions

String Cosmology

R. Branden- berger String Gas Cosmology: Model of cosmology of the very early universe based on new degrees of freedom and Inﬂation Motivation new symmetries of superstring theory. Inﬂation Problems SGC → nonsingular cosmology Message

String gas SGC → natural explanation of the number of large Principles Features spatial dimensions. Structure SGC → new scenario of structure formation Perturbations Analysis Scale invariant spectrum of cosmological fluctuations Conclusions (like in inflationary cosmology). Spectrum of gravitational waves has a small blue tilt (unlike in inflationary cosmology).

40 / 40