Inflationary Cosmology: Progress and Problems

Transcription

1 1 / 95 ary Cosmology: Progress and Robert McGill University, Canada and Institute for Theoretical Studies, ETH Zuerich, Switzerland NAO Colloquium, Sept. 23, 2015

2 2 / 95 Outline 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

3 3 / 95 Plan 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

4 Current Paradigm for Early Universe Cosmology 4 / 95 The ary Universe Scenario is the current paradigm of early universe cosmology. : Solves horizon problem Solves flatness problem Solves size/entropy problem Provides a causal mechanism of generating primordial cosmological perturbations (Chibisov & Mukhanov, 1981).

5 Current Paradigm for Early Universe Cosmology 4 / 95 The ary Universe Scenario is the current paradigm of early universe cosmology. : Solves horizon problem Solves flatness problem Solves size/entropy problem Provides a causal mechanism of generating primordial cosmological perturbations (Chibisov & Mukhanov, 1981).

6 Current Paradigm for Early Universe Cosmology 4 / 95 The ary Universe Scenario is the current paradigm of early universe cosmology. : Solves horizon problem Solves flatness problem Solves size/entropy problem Provides a causal mechanism of generating primordial cosmological perturbations (Chibisov & Mukhanov, 1981).

7 Current Paradigm for Early Universe Cosmology 4 / 95 The ary Universe Scenario is the current paradigm of early universe cosmology. : Solves horizon problem Solves flatness problem Solves size/entropy problem Provides a causal mechanism of generating primordial cosmological perturbations (Chibisov & Mukhanov, 1981).

8 5 / 95 Credit: NASA/WMAP Science Team

9 6 / 95 Phenomenological Success of Credit: NASA/WMAP Science Team

10 7 / 95 Opinions on A. Linde, arxiv: ary theory is gradually becoming a broadly accepted cosmological paradigm, with many of its predictions being confirmed by observational data. A. Ijjas, P. Steinhardt and A. Loeb, arxiv: , Phys. Lett. B723, 261 (2013). In sum, we find that recent experimental data disfavors all the best-motivated inflationary scenarios and introduces new, serious difficulties that cut to the core of the inflationary paradigm. Initial conditions problem, multiverse-unpredictability problem, inflationary unlikeliness problem.

11 7 / 95 Opinions on A. Linde, arxiv: ary theory is gradually becoming a broadly accepted cosmological paradigm, with many of its predictions being confirmed by observational data. A. Ijjas, P. Steinhardt and A. Loeb, arxiv: , Phys. Lett. B723, 261 (2013). In sum, we find that recent experimental data disfavors all the best-motivated inflationary scenarios and introduces new, serious difficulties that cut to the core of the inflationary paradigm. Initial conditions problem, multiverse-unpredictability problem, inflationary unlikeliness problem.

12 7 / 95 Opinions on A. Linde, arxiv: ary theory is gradually becoming a broadly accepted cosmological paradigm, with many of its predictions being confirmed by observational data. A. Ijjas, P. Steinhardt and A. Loeb, arxiv: , Phys. Lett. B723, 261 (2013). In sum, we find that recent experimental data disfavors all the best-motivated inflationary scenarios and introduces new, serious difficulties that cut to the core of the inflationary paradigm. Initial conditions problem, multiverse-unpredictability problem, inflationary unlikeliness problem.

13 Opinions on II A. Guth, D. Kaiser and Y. Nomura, arxiv: , Phys. Lett. B733, 112 (2014). makes specific, quantitative predictions for several observable quantities,... that match the latest observations from the Planck satellite to very good precision. We conclude that cosmic inflation is on a stronger footing than ever before. V. Mukhanov, arxiv: , Fortsch. Phys. 63, 36 (2015).. the multiverse and eternal inflation damage the predictive power of the theory because in this case anything can happen and will happen an infinite number of times. The recent CMB observations have unambiguously proven the theory of the quantum origin of the universe structure. 8 / 95

14 Opinions on II A. Guth, D. Kaiser and Y. Nomura, arxiv: , Phys. Lett. B733, 112 (2014). makes specific, quantitative predictions for several observable quantities,... that match the latest observations from the Planck satellite to very good precision. We conclude that cosmic inflation is on a stronger footing than ever before. V. Mukhanov, arxiv: , Fortsch. Phys. 63, 36 (2015).. the multiverse and eternal inflation damage the predictive power of the theory because in this case anything can happen and will happen an infinite number of times. The recent CMB observations have unambiguously proven the theory of the quantum origin of the universe structure. 8 / 95

15 Opinions on II A. Guth, D. Kaiser and Y. Nomura, arxiv: , Phys. Lett. B733, 112 (2014). makes specific, quantitative predictions for several observable quantities,... that match the latest observations from the Planck satellite to very good precision. We conclude that cosmic inflation is on a stronger footing than ever before. V. Mukhanov, arxiv: , Fortsch. Phys. 63, 36 (2015).. the multiverse and eternal inflation damage the predictive power of the theory because in this case anything can happen and will happen an infinite number of times. The recent CMB observations have unambiguously proven the theory of the quantum origin of the universe structure. 8 / 95

16 Opinions on II A. Guth, D. Kaiser and Y. Nomura, arxiv: , Phys. Lett. B733, 112 (2014). makes specific, quantitative predictions for several observable quantities,... that match the latest observations from the Planck satellite to very good precision. We conclude that cosmic inflation is on a stronger footing than ever before. V. Mukhanov, arxiv: , Fortsch. Phys. 63, 36 (2015).. the multiverse and eternal inflation damage the predictive power of the theory because in this case anything can happen and will happen an infinite number of times. The recent CMB observations have unambiguously proven the theory of the quantum origin of the universe structure. 8 / 95

17 9 / 95 Goals of Early Universe Cosmology 1. Understand origin and early evolution of the universe. 2. Explain observed large-scale structure. 3. Predictions for future observations.

18 10 / 95 Standard Big Bang Cosmology Standard Big Bang (SBB) cosmology is based on: General Relativity as the theory of space-time. Classical fluids as the theory of matter

19 11 / 95 Temperature-time curve in SBB Cosmology

20 Our Expanding Universe Credit: NASA/WMAP Science Team 12 / 95

21 13 / 95 for SBB Cosmology Goals of Early Universe Cosmology 1. Understand origin and early evolution of the universe. 2. Explain observed large-scale structure. 3. Predictions for future observations. None of the data can be explained in SBB cosmology.

22 14 / 95 Horizon problem t 0 t l (t) p t rec 0 l f (t) x

23 15 / 95 Structure formation problem t 0 t H -1 (t) t eq 0 d c x c

24 16 / 95 Idea of inflation

25 as a solution of the horizon and structure formation problems 17 / 95

26 18 / 95 Phenomenological Success of Credit: NASA/WMAP Science Team

27 19 / 95 Is this success really a success of inflation?

28 Key Realization R. Sunyaev and Y. Zel dovich, Astrophys. and Space Science 7, 3 (1970); P. Peebles and J. Yu, Ap. J. 162, 815 (1970). 20 / 95 Given a scale-invariant power spectrum of adiabatic fluctuations on "super-horizon" scales before t eq, i.e. standing waves. "correct" power spectrum of galaxies. acoustic oscillations in CMB angular power spectrum.

29 Predictions of Sunyaev & Zeldovich, and Peebles & Yu 21 / 95

30 Key Realization R. Sunyaev and Y. Zel dovich, Astrophys. and Space Science 7, 3 (1970); P. Peebles and J. Yu, Ap. J. 162, 815 (1970). 22 / 95 Given a scale-invariant power spectrum of adiabatic fluctuations on "super-horizon" scales before t eq, i.e. standing waves. "correct" power spectrum of galaxies. acoustic oscillations in CMB angular power spectrum. baryon acoustic oscillations in matter power spectrum. is the first model to yield such a primordial spectrum from causal physics. But it is NOT the only one.

31 Key Realization R. Sunyaev and Y. Zel dovich, Astrophys. and Space Science 7, 3 (1970); P. Peebles and J. Yu, Ap. J. 162, 815 (1970). 22 / 95 Given a scale-invariant power spectrum of adiabatic fluctuations on "super-horizon" scales before t eq, i.e. standing waves. "correct" power spectrum of galaxies. acoustic oscillations in CMB angular power spectrum. baryon acoustic oscillations in matter power spectrum. is the first model to yield such a primordial spectrum from causal physics. But it is NOT the only one.

32 23 / 95 Main Points of This Talk is a simple early universe scenario with great phenomenological successes. Most of the phenomenological successes are not testing the inflationary background, but only a particular structure formation model. Current inflationary models suffer from conceptual problems. to inflationary cosmology exist. Examples:, String Gas Cosmology was predictive, the alternatives are only postdictive. and its alternatives make different predictions for upcoming cosmological observations.

33 23 / 95 Main Points of This Talk is a simple early universe scenario with great phenomenological successes. Most of the phenomenological successes are not testing the inflationary background, but only a particular structure formation model. Current inflationary models suffer from conceptual problems. to inflationary cosmology exist. Examples:, String Gas Cosmology was predictive, the alternatives are only postdictive. and its alternatives make different predictions for upcoming cosmological observations.

34 23 / 95 Main Points of This Talk is a simple early universe scenario with great phenomenological successes. Most of the phenomenological successes are not testing the inflationary background, but only a particular structure formation model. Current inflationary models suffer from conceptual problems. to inflationary cosmology exist. Examples:, String Gas Cosmology was predictive, the alternatives are only postdictive. and its alternatives make different predictions for upcoming cosmological observations.

35 23 / 95 Main Points of This Talk is a simple early universe scenario with great phenomenological successes. Most of the phenomenological successes are not testing the inflationary background, but only a particular structure formation model. Current inflationary models suffer from conceptual problems. to inflationary cosmology exist. Examples:, String Gas Cosmology was predictive, the alternatives are only postdictive. and its alternatives make different predictions for upcoming cosmological observations.

36 23 / 95 Main Points of This Talk is a simple early universe scenario with great phenomenological successes. Most of the phenomenological successes are not testing the inflationary background, but only a particular structure formation model. Current inflationary models suffer from conceptual problems. to inflationary cosmology exist. Examples:, String Gas Cosmology was predictive, the alternatives are only postdictive. and its alternatives make different predictions for upcoming cosmological observations.

37 23 / 95 Main Points of This Talk is a simple early universe scenario with great phenomenological successes. Most of the phenomenological successes are not testing the inflationary background, but only a particular structure formation model. Current inflationary models suffer from conceptual problems. to inflationary cosmology exist. Examples:, String Gas Cosmology was predictive, the alternatives are only postdictive. and its alternatives make different predictions for upcoming cosmological observations.

38 24 / 95 Plan 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

39 Review of Classical ary Cosmology Framework: Classical General Relativity as theory of space-time Classical Scalar Field Matter as theory of matter Metric : ds 2 = dt 2 a(t) 2 dx 2 : phase with a(t) e th requires matter with p ρ 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 25 / 95

40 Review of Classical ary Cosmology Framework: Classical General Relativity as theory of space-time Classical Scalar Field Matter as theory of matter Metric : ds 2 = dt 2 a(t) 2 dx 2 : phase with a(t) e th requires matter with p ρ 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 25 / 95

41 Review of Classical ary Cosmology Framework: Classical General Relativity as theory of space-time Classical Scalar Field Matter as theory of matter Metric : ds 2 = dt 2 a(t) 2 dx 2 : phase with a(t) e th requires matter with p ρ 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 25 / 95

42 Review of Classical ary Cosmology Framework: Classical General Relativity as theory of space-time Classical Scalar Field Matter as theory of matter Metric : ds 2 = dt 2 a(t) 2 dx 2 : phase with a(t) e th requires matter with p ρ 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 25 / 95

43 26 / 95 Scalar Field

44 27 / 95 Time line of inflationary cosmology: t i : inflation begins t R : inflation ends, reheating

45 Review of ary Cosmology II Space-time sketch of inflationary cosmology: Note: H = ȧ a curve labelled by k: wavelength of a fluctuation 28 / 95

46 29 / 95 solves the horizon problem. inflation renders the universe large and spatially flat classical matter redshifts matter vacuum remains quantum vacuum fluctuations: seeds for the observed structure [Chibisov & Mukhanov, 1981] sub-hubble locally causal

47 29 / 95 solves the horizon problem. inflation renders the universe large and spatially flat classical matter redshifts matter vacuum remains quantum vacuum fluctuations: seeds for the observed structure [Chibisov & Mukhanov, 1981] sub-hubble locally causal

48 30 / 95 Slow Roll A. Linde, 1982; A. Albrecht and P. Steinhardt, Equation of motion: ϕ + 3H ϕ = V (ϕ). Slow roll conditions: ϕ 2 << V (ϕ) ϕ << V (ϕ)

49 30 / 95 Slow Roll A. Linde, 1982; A. Albrecht and P. Steinhardt, Equation of motion: ϕ + 3H ϕ = V (ϕ). Slow roll conditions: ϕ 2 << V (ϕ) ϕ << V (ϕ)

50 31 / 95 Small Field vs. Large Field Scenario 1: small field inflation [new inflation] Scenario 2: large field inflation [chaotic inflation]

51 Initial Condition Problem for Small Field D. Goldwirth and T. Piran, Phys. Rept. 214, 223 (1992) 32 / 95 In the case of small field inflation the slow roll trajectory is not an attractor in initial condition space. Challenge: Justification of initial condition ϕ 0.

52 Initial Condition Problem for Small Field D. Goldwirth and T. Piran, Phys. Rept. 214, 223 (1992) 32 / 95 In the case of small field inflation the slow roll trajectory is not an attractor in initial condition space. Challenge: Justification of initial condition ϕ 0.

53 33 / 95

54 34 / 95 No Initial Condition Problem for Large Field R.B. and J. Kung, Phys. Rev. D42, 1008 (1990); R.B., H. Feldman and J. Kung, Phys. Scripta T36, 64 (1991). In the case of large field inflation the slow roll trajectory is an attractor in initial condition space, even in the presence of linear cosmological perturbations.

55 35 / 95

56 36 / 95 No Initial Condition Problem for Large Field R.B. and J. Kung, Phys. Rev. D42, 1008 (1990); R.B., H. Feldman and J. Kung, Phys. Scripta T36, 64 (1991). In the case of large field inflation the slow roll trajectory is an attractor in initial condition space, even in the presence of linear cosmological perturbations. Challenge: Justification of large field values ϕ > m pl required to obey the slow roll conditions.

57 37 / 95 Reheating: Exit from Period 1: Slow rolling Period 2: Oscillations, reheating

58 Preheating: First stage of reheating Oscillations of the inflaton ϕ produce a parametric resonance instability in any light field χ which couples of ϕ [J. Traschen and R.B., Phys. Rev. D34, 919 (1990)]. Consider an interaction Lagrangian of the form: L I = 1 2 g2 ϕ 2 χ 2 Equation of motion for χ becomes: χ k + [k 2 + gϕ(t) 2 ]χ k = 0. Floquet theory exponential instability rapid energy transfer. 38 / 95

59 39 / 95 Post-Modern Cosmology Stochastic [A. Starobinsky, 1987]: quantum fluctuations drive ϕ up the potential in part of space. Eternal Thus, inflation never terminates: there will always be regions of space which are inflating. ary Multiverse The eternal inflationary universe will spawn many universes of our kind. L. Susskind, R. Bousso, A. Linde,... In one of these universes the cosmological constant will be the one we observe [anthropic reasoning] P. Steinhardt, V. Mukhanov,... Then inflation loses its predictability.

60 39 / 95 Post-Modern Cosmology Stochastic [A. Starobinsky, 1987]: quantum fluctuations drive ϕ up the potential in part of space. Eternal Thus, inflation never terminates: there will always be regions of space which are inflating. ary Multiverse The eternal inflationary universe will spawn many universes of our kind. L. Susskind, R. Bousso, A. Linde,... In one of these universes the cosmological constant will be the one we observe [anthropic reasoning] P. Steinhardt, V. Mukhanov,... Then inflation loses its predictability.

61 39 / 95 Post-Modern Cosmology Stochastic [A. Starobinsky, 1987]: quantum fluctuations drive ϕ up the potential in part of space. Eternal Thus, inflation never terminates: there will always be regions of space which are inflating. ary Multiverse The eternal inflationary universe will spawn many universes of our kind. L. Susskind, R. Bousso, A. Linde,... In one of these universes the cosmological constant will be the one we observe [anthropic reasoning] P. Steinhardt, V. Mukhanov,... Then inflation loses its predictability.

62 39 / 95 Post-Modern Cosmology Stochastic [A. Starobinsky, 1987]: quantum fluctuations drive ϕ up the potential in part of space. Eternal Thus, inflation never terminates: there will always be regions of space which are inflating. ary Multiverse The eternal inflationary universe will spawn many universes of our kind. L. Susskind, R. Bousso, A. Linde,... In one of these universes the cosmological constant will be the one we observe [anthropic reasoning] P. Steinhardt, V. Mukhanov,... Then inflation loses its predictability.

63 39 / 95 Post-Modern Cosmology Stochastic [A. Starobinsky, 1987]: quantum fluctuations drive ϕ up the potential in part of space. Eternal Thus, inflation never terminates: there will always be regions of space which are inflating. ary Multiverse The eternal inflationary universe will spawn many universes of our kind. L. Susskind, R. Bousso, A. Linde,... In one of these universes the cosmological constant will be the one we observe [anthropic reasoning] P. Steinhardt, V. Mukhanov,... Then inflation loses its predictability.

64 Refutation of Post-Modern ary Cosmology 40 / 95 Stochastic effects are only important for ϕ >> m pl. No concrete particle physics models are known which yield such large field values [T. Rudelius, 2015] Only classical inflation is at the moment well-defined.

65 Refutation of Post-Modern ary Cosmology 40 / 95 Stochastic effects are only important for ϕ >> m pl. No concrete particle physics models are known which yield such large field values [T. Rudelius, 2015] Only classical inflation is at the moment well-defined.

66 Refutation of Post-Modern ary Cosmology 40 / 95 Stochastic effects are only important for ϕ >> m pl. No concrete particle physics models are known which yield such large field values [T. Rudelius, 2015] Only classical inflation is at the moment well-defined.

67 41 / 95 Plan 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

68 42 / 95 Theory of Cosmological : Basics Cosmological fluctuations connect early universe theories with observations Fluctuations of matter large-scale structure Fluctuations of metric CMB anisotropies N.B.: Matter and metric fluctuations are coupled Key facts: 1. Fluctuations are small today on large scales fluctuations were very small in the early universe can use linear perturbation theory 2. Sub-Hubble scales: matter fluctuations dominate Super-Hubble scales: metric fluctuations dominate

69 42 / 95 Theory of Cosmological : Basics Cosmological fluctuations connect early universe theories with observations Fluctuations of matter large-scale structure Fluctuations of metric CMB anisotropies N.B.: Matter and metric fluctuations are coupled Key facts: 1. Fluctuations are small today on large scales fluctuations were very small in the early universe can use linear perturbation theory 2. Sub-Hubble scales: matter fluctuations dominate Super-Hubble scales: metric fluctuations dominate

70 42 / 95 Theory of Cosmological : Basics Cosmological fluctuations connect early universe theories with observations Fluctuations of matter large-scale structure Fluctuations of metric CMB anisotropies N.B.: Matter and metric fluctuations are coupled Key facts: 1. Fluctuations are small today on large scales fluctuations were very small in the early universe can use linear perturbation theory 2. Sub-Hubble scales: matter fluctuations dominate Super-Hubble scales: metric fluctuations dominate

71 43 / 95 Hubble Radius vs. Horizon Horizon: Forward light cone of a point on the initial Cauchy surface. Horizon: region of causal contact. Hubble radius: l H (t) = H 1 (t) inverse expansion rate. Hubble radius: local concept, relevant for dynamics of cosmological fluctuations. In Standard Big Bang Cosmology: Hubble radius = horizon. In any theory which can provide a mechanism for the origin of structure: Hubble radius horizon.

72 43 / 95 Hubble Radius vs. Horizon Horizon: Forward light cone of a point on the initial Cauchy surface. Horizon: region of causal contact. Hubble radius: l H (t) = H 1 (t) inverse expansion rate. Hubble radius: local concept, relevant for dynamics of cosmological fluctuations. In Standard Big Bang Cosmology: Hubble radius = horizon. In any theory which can provide a mechanism for the origin of structure: Hubble radius horizon.

73 43 / 95 Hubble Radius vs. Horizon Horizon: Forward light cone of a point on the initial Cauchy surface. Horizon: region of causal contact. Hubble radius: l H (t) = H 1 (t) inverse expansion rate. Hubble radius: local concept, relevant for dynamics of cosmological fluctuations. In Standard Big Bang Cosmology: Hubble radius = horizon. In any theory which can provide a mechanism for the origin of structure: Hubble radius horizon.

74 44 / 95 Quantum Theory of Linearized Fluctuations V. Mukhanov, H. Feldman and R.B., Phys. Rep. 215:203 (1992) Step 1: Metric including fluctuations ds 2 = a 2 [(1 + 2Φ)dη 2 (1 2Φ)dx 2 ] ϕ = ϕ 0 + δϕ Note: Φ and δϕ related by Einstein constraint equations Step 2: Expand the action for matter and gravity to second order about the cosmological background: S (2) = 1 2 v = a ( δϕ + z a Φ) d 4 x ( (v ) 2 v,i v,i + z z v 2) z = a ϕ 0 H

75 44 / 95 Quantum Theory of Linearized Fluctuations V. Mukhanov, H. Feldman and R.B., Phys. Rep. 215:203 (1992) Step 1: Metric including fluctuations ds 2 = a 2 [(1 + 2Φ)dη 2 (1 2Φ)dx 2 ] ϕ = ϕ 0 + δϕ Note: Φ and δϕ related by Einstein constraint equations Step 2: Expand the action for matter and gravity to second order about the cosmological background: S (2) = 1 2 v = a ( δϕ + z a Φ) d 4 x ( (v ) 2 v,i v,i + z z v 2) z = a ϕ 0 H

76 45 / 95 Step 3: Resulting equation of motion (Fourier space) Features: v k + (k 2 z z )v k = 0 oscillations on sub-hubble scales squeezing on super-hubble scales v k z Quantum vacuum initial conditions: v k (η i ) = ( 2k) 1

77 45 / 95 Step 3: Resulting equation of motion (Fourier space) Features: v k + (k 2 z z )v k = 0 oscillations on sub-hubble scales squeezing on super-hubble scales v k z Quantum vacuum initial conditions: v k (η i ) = ( 2k) 1

78 45 / 95 Step 3: Resulting equation of motion (Fourier space) Features: v k + (k 2 z z )v k = 0 oscillations on sub-hubble scales squeezing on super-hubble scales v k z Quantum vacuum initial conditions: v k (η i ) = ( 2k) 1

79 46 / 95 More on I In the case of adiabatic fluctuations, there is only one degree of freedom for the scalar metric inhomogeneities. It is ζ = z 1 v Its physical meaning: curvature perturbation in comoving gauge. In an expanding background, ζ is conserved on super-hubble scales. In a contracting background, ζ grows on super-hubble scales.

80 46 / 95 More on I In the case of adiabatic fluctuations, there is only one degree of freedom for the scalar metric inhomogeneities. It is ζ = z 1 v Its physical meaning: curvature perturbation in comoving gauge. In an expanding background, ζ is conserved on super-hubble scales. In a contracting background, ζ grows on super-hubble scales.

81 47 / 95 More on II If there is more than one matter field, then it is possible to have entropy fluctuations. Entropy fluctuations seed an adiabatic mode even on super-hubble scales. ζ = ṗ p + ρ δs Example: topological defect formation in a phase transition. Example: Axion perturbations when axions acquire a mass at the QCD scale (M. Axenides, R.B. and M. Turner, 1983).

82 47 / 95 More on II If there is more than one matter field, then it is possible to have entropy fluctuations. Entropy fluctuations seed an adiabatic mode even on super-hubble scales. ζ = ṗ p + ρ δs Example: topological defect formation in a phase transition. Example: Axion perturbations when axions acquire a mass at the QCD scale (M. Axenides, R.B. and M. Turner, 1983).

83 48 / 95 Gravitational Waves ds 2 = a 2[ (1 + 2Φ)dη 2 [(1 2Φ)δ ij + h ij ]dx i dx j] h ij (x, t) transverse and traceless Two polarization states 2 h ij (x, t) = h a (x, t)ɛ a ij a=1 At linear level each polarization mode evolves independently.

84 49 / 95 Gravitational Waves II Canonical variable for gravitational waves: u(x, t) = a(t)h(x, t) Equation of motion for gravitational waves: u k + ( k 2 ) a uk = 0. a Squeezing on super-hubble scales, oscillations on sub-hubble scales.

85 50 / 95 Consequences for Tensor to Scalar Ratio r If EoS of matter is time independent, then z a and u v. Thus, generically models with dominant adiabatic fluctuations lead to a large value of r. A large value of r is not a smoking gun for inflation. During a phase transition EoS changes and u evolves differently than v Suppression of r. This happens during the inflationary reheating transition. Simple inflation models typically predict very small value of r.

86 51 / 95 Structure formation in inflationary cosmology N.B. originate as quantum vacuum fluctuations.

87 52 / 95 Origin of Scale-Invariance in Scenario A: Initial vacuum spectrum of ζ (ζ v): (Chibisov and Mukhanov, 1981). P ζ (k) k 3 ζ(k) 2 k 2 v z a on super-hubble scales At late times on super-hubble scales P ζ (k, t) P ζ (k, t i (k)) ( a(t) ) 2 k 2 a(t i (k)) 2 a(t i (k) Hubble radius crossing: ak 1 = H 1 P ζ (k, t) const

88 52 / 95 Origin of Scale-Invariance in Scenario A: Initial vacuum spectrum of ζ (ζ v): (Chibisov and Mukhanov, 1981). P ζ (k) k 3 ζ(k) 2 k 2 v z a on super-hubble scales At late times on super-hubble scales P ζ (k, t) P ζ (k, t i (k)) ( a(t) ) 2 k 2 a(t i (k)) 2 a(t i (k) Hubble radius crossing: ak 1 = H 1 P ζ (k, t) const

89 52 / 95 Origin of Scale-Invariance in Scenario A: Initial vacuum spectrum of ζ (ζ v): (Chibisov and Mukhanov, 1981). P ζ (k) k 3 ζ(k) 2 k 2 v z a on super-hubble scales At late times on super-hubble scales P ζ (k, t) P ζ (k, t i (k)) ( a(t) ) 2 k 2 a(t i (k)) 2 a(t i (k) Hubble radius crossing: ak 1 = H 1 P ζ (k, t) const

90 53 / 95 Plan 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

91 54 / 95 of Solution of horizon and flatness problems of SBB cosmology. First structure formation scenario based on causal physics. Successful predictions: Scale invariant spectrum of cosmological perturbations with a slight red tilt n s 0.96 Almost adiabatic spectrum. Almost Gaussian fluctuations. Predictions not yet tested: scale invariant spectrum of gravitational waves with a red tilt n t < 0.

92 54 / 95 of Solution of horizon and flatness problems of SBB cosmology. First structure formation scenario based on causal physics. Successful predictions: Scale invariant spectrum of cosmological perturbations with a slight red tilt n s 0.96 Almost adiabatic spectrum. Almost Gaussian fluctuations. Predictions not yet tested: scale invariant spectrum of gravitational waves with a red tilt n t < 0.

93 54 / 95 of Solution of horizon and flatness problems of SBB cosmology. First structure formation scenario based on causal physics. Successful predictions: Scale invariant spectrum of cosmological perturbations with a slight red tilt n s 0.96 Almost adiabatic spectrum. Almost Gaussian fluctuations. Predictions not yet tested: scale invariant spectrum of gravitational waves with a red tilt n t < 0.

94 54 / 95 of Solution of horizon and flatness problems of SBB cosmology. First structure formation scenario based on causal physics. Successful predictions: Scale invariant spectrum of cosmological perturbations with a slight red tilt n s 0.96 Almost adiabatic spectrum. Almost Gaussian fluctuations. Predictions not yet tested: scale invariant spectrum of gravitational waves with a red tilt n t < 0.

95 54 / 95 of Solution of horizon and flatness problems of SBB cosmology. First structure formation scenario based on causal physics. Successful predictions: Scale invariant spectrum of cosmological perturbations with a slight red tilt n s 0.96 Almost adiabatic spectrum. Almost Gaussian fluctuations. Predictions not yet tested: scale invariant spectrum of gravitational waves with a red tilt n t < 0.

96 55 / 95 Critical Look at the Flatness is a key prediction of inflation which is difficult to obtain in alternatives to inflation. Properties of fluctuations are not specific to inflation.

97 56 / 95 Plan 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

98 Conceptual of ary Cosmology 57 / 95 Nature of the scalar field ϕ (the inflaton") Conditions to obtain inflation (initial conditions, slow-roll conditions, graceful exit and reheating) Amplitude problem Trans-Planckian problem Cosmological constant problem Applicability of General Relativity Singularity problem

99 Trans-Planckian Problem Success of inflation: At early times scales are inside the Hubble radius causal generation mechanism is possible. Problem: If time period of inflation is more than 70H 1, then λ p (t) < l pl at the beginning of inflation. new physics MUST enter into the calculation of the fluctuations. 58 / 95

100 Trans-Planckian Problem Success of inflation: At early times scales are inside the Hubble radius causal generation mechanism is possible. Problem: If time period of inflation is more than 70H 1, then λ p (t) < l pl at the beginning of inflation. new physics MUST enter into the calculation of the fluctuations. 58 / 95

101 59 / 95 Recent Reference: A. Linde, V. Mukhanov and A. Vikman, arxiv: It is not sufficient to show that the Hubble constant is smaller than the Planck scale. The frequencies involved in the analysis of the cosmological fluctuations are many orders of magnitude larger than the Planck mass. Thus, the methods used in [1] are inapplicable for the description of the.. process of generation of perturbations in this scenario."

102 60 / 95 Trans-Planckian Window of Opportunity If evolution in Period I is non-adiabatic, then scale-invariance of the power spectrum will be lost [J. Martin and RB, 2000] Planck scale physics testable with cosmological observations!

103 61 / 95 Cosmological Constant Problem Quantum vacuum energy does not gravitate. Why should the almost constant V (ϕ) gravitate? V 0 Λ obs

104 62 / 95 Applicability of GR In all approaches to quantum gravity, the Einstein action is only the leading term in a low curvature expansion. Correction terms may become dominant at much lower energies than the Planck scale. Correction terms will dominate the dynamics at high curvatures. The energy scale of inflation models is typically η GeV. η too close to m pl to trust predictions made using GR.

105 63 / 95 Zones of Ignorance

106 64 / 95 Singularity Problem Standard cosmology: Penrose-Hawking theorems initial singularity incompleteness of the theory. ary cosmology: In scalar field-driven inflationary models the initial singularity persists [Borde and Vilenkin] incompleteness of the theory. Penrose-Hawking theorems: Ass: 1) Einstein gravitational action, Ass: 2) weak energy conditions for matter ρ > 0, ρ + 3p 0 space-time is geodesically incomplete.

107 64 / 95 Singularity Problem Standard cosmology: Penrose-Hawking theorems initial singularity incompleteness of the theory. ary cosmology: In scalar field-driven inflationary models the initial singularity persists [Borde and Vilenkin] incompleteness of the theory. Penrose-Hawking theorems: Ass: 1) Einstein gravitational action, Ass: 2) weak energy conditions for matter ρ > 0, ρ + 3p 0 space-time is geodesically incomplete.

108 65 / 95 Plan 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

109 66 / 95 Requirements I Scales of cosmological interest today must originate inside the Hubble radius. Long propagation on super-hubble scales. Scale-invariant spectrum of adiabatic cosmological perturbations.

110 67 / 95 Requirements II Model must refer to the problems of Standard Cosmology which the inflationary scenario addresses. Solution of the horizon problem. Solution of the flatness problem. Solution of the size and entropy problems.

111 68 / 95 List of Some [D. Wands, 1999; F. Finelli and R.B., 2002] Pre-Big-Bang Scenario [M. Gasperini and G. Veneziano, 1992] Ekpyrotic Scenario [J. Khoury, B. Ovrut, P. Steinhardt and N. Turok, 2001] Conformal Universe [V. Rubakov, 2009] String Gas Cosmology [R.B. and C. Vafa, 1989; A. Nayeri, R.B. and C. Vafa, 2006]

112 68 / 95 List of Some [D. Wands, 1999; F. Finelli and R.B., 2002] Pre-Big-Bang Scenario [M. Gasperini and G. Veneziano, 1992] Ekpyrotic Scenario [J. Khoury, B. Ovrut, P. Steinhardt and N. Turok, 2001] Conformal Universe [V. Rubakov, 2009] String Gas Cosmology [R.B. and C. Vafa, 1989; A. Nayeri, R.B. and C. Vafa, 2006]

113 Scenario F. Finelli and R.B., Phys. Rev. D65, (2002), D. Wands, Phys. Rev. D60 (1999) 69 / 95

114 70 / 95 Overview of the Fluctuations originate as quantum vacuum perturbations on sub-hubble scales in the contracting phase. Adiabatic fluctuation mode acquires a scale-invariant spectrum of curvature perturbations on super-hubble scales. Horizon problem: absent. Flatness problem: absent. Size and entropy problems: not present if we assume that the universe begins cold and large. Anisotropy problem (BKL instability): key challenge.

115 70 / 95 Overview of the Fluctuations originate as quantum vacuum perturbations on sub-hubble scales in the contracting phase. Adiabatic fluctuation mode acquires a scale-invariant spectrum of curvature perturbations on super-hubble scales. Horizon problem: absent. Flatness problem: absent. Size and entropy problems: not present if we assume that the universe begins cold and large. Anisotropy problem (BKL instability): key challenge.

116 71 / 95 Origin of Scale-Invariant Spectrum The initial vacuum spectrum is blue: P ζ (k) = k 3 ζ(k) 2 k 2 The curvature fluctuations grow on super-hubble scales in the contracting phase: v k (η) = c 1 η 2 + c 2 η 1, For modes which exit the Hubble radius in the matter phase the resulting spectrum is scale-invariant: P ζ (k, η) k 3 v k (η) 2 a 2 (η) k 3 v k (η H (k)) 2( η H (k)) 2 k η const,

117 72 / 95 Origin of Red Tilt of the Spectrum Y. Cai and E. Wilson-Ewing, arxiv: Include the observed cosmological constant. slower growth of ζ initially when large scale modes ext Hubble radius. slight red tilt of the spectrum.

118 Ekpyrotic Bounce J. Khoury, B. Ovrut, P. Steinhardt and N. Turok Phys. Rev. D64, (2001) 73 / 95

119 74 / 95 Obtaining a Phase of Ekpyrotic Contraction Introduce a scalar field with negative exponential potential and AdS minimum: V (φ) = V 0 exp( ( 2 φ p )1/2 ) 0 < p 1 m pl Motivated by potential between branes in heterotic M-theory In the homogeneous and isotropic limit, the cosmology is given by and the equation of state is a(t) a(t) p w p ρ = 2 3p 1 1.

120 75 / 95 Solution to the flatness problem The energy density in the Ekpyrotic field scales as ρ(a) = ρ 0 a 3(1+w) and thus dominates all other forms of energy density (including anisotropic stress) as the universe shrinks quasi-homogeneous bounce, no chaotic mixmaster behavior.

121 76 / 95 Spectrum of Adiabatic Fluctuations If a(t) t p then conformal time scales as η t 1 p. The solution of the mode equation for v is v k (η) = c 1 η α + c 2 η, where c 1 and c 2 are constant coefficients and α p for p 1. Hence, the power spectrum in not scale invariant: P ζ (k, t) = ( z(t) ) 2k 3 v k (t H (k)) 2 v(t H (k)) k 3 k 1 k 2p k 2(1 p).

122 77 / 95 Entropy Fluctuations Introduce a second scalar field χ with a negative exponential potential. Assume initial vacuum fluctuations for χ. Evolution on super-hubble scales scale-invariant spectrum. Entropy fluctuations induce scale-invariant curvature mode.

123 78 / 95 Challenges for the Ekpyrotic Scenario Description of the bounce. Initial conditions for fluctuations.

124 79 / 95 Background for Cosmology

125 Structure Formation in an emergent String Gas Cosmology A. Nayeri, R.B. and C. Vafa, Phys. Rev. Lett. 97: (2006) 80 / 95 N.B. originate as thermal fluctuations.

126 81 / 95 Principles of String Gas Cosmology R.B. and C. Vafa, Nucl. Phys. B316:391 (1989) Idea: make use of the new symmetries and new degrees of freedom which string theory provides to construct a new theory of the very early universe. Assumption: Matter is a gas of fundamental strings Assumption: Space is compact, e.g. a torus. Key points: New degrees of freedom: string oscillatory modes Leads to a maximal temperature for a gas of strings, the Hagedorn temperature New degrees of freedom: string winding modes Leads to a new symmetry: physics at large R is equivalent to physics at small R

127 81 / 95 Principles of String Gas Cosmology R.B. and C. Vafa, Nucl. Phys. B316:391 (1989) Idea: make use of the new symmetries and new degrees of freedom which string theory provides to construct a new theory of the very early universe. Assumption: Matter is a gas of fundamental strings Assumption: Space is compact, e.g. a torus. Key points: New degrees of freedom: string oscillatory modes Leads to a maximal temperature for a gas of strings, the Hagedorn temperature New degrees of freedom: string winding modes Leads to a new symmetry: physics at large R is equivalent to physics at small R

128 81 / 95 Principles of String Gas Cosmology R.B. and C. Vafa, Nucl. Phys. B316:391 (1989) Idea: make use of the new symmetries and new degrees of freedom which string theory provides to construct a new theory of the very early universe. Assumption: Matter is a gas of fundamental strings Assumption: Space is compact, e.g. a torus. Key points: New degrees of freedom: string oscillatory modes Leads to a maximal temperature for a gas of strings, the Hagedorn temperature New degrees of freedom: string winding modes Leads to a new symmetry: physics at large R is equivalent to physics at small R

129 82 / 95 T-Duality T-Duality Momentum modes: E n = n/r Winding modes: E m = mr Duality: R 1/R (n, m) (m, n) Mass spectrum of string states unchanged Symmetry of vertex operators Symmetry at non-perturbative level existence of D-branes

130 83 / 95 Adiabatic Considerations R.B. and C. Vafa, Nucl. Phys. B316:391 (1989) Temperature-size relation in string gas cosmology

131 84 / 95 Overview of String Gas Cosmology Fluctuations originate as thermal string perturbations on sub-hubble scales in the Hagedorn (loitering) phase. Adiabatic fluctuation mode acquires a scale-invariant spectrum of curvature perturbations on super-hubble scales. Horizon problem: absent if the loitering phase lasts sufficiently long. Flatness problem: weak point. Size and entropy problems: not present if we assume that the universe begins cold and large.

132 84 / 95 Overview of String Gas Cosmology Fluctuations originate as thermal string perturbations on sub-hubble scales in the Hagedorn (loitering) phase. Adiabatic fluctuation mode acquires a scale-invariant spectrum of curvature perturbations on super-hubble scales. Horizon problem: absent if the loitering phase lasts sufficiently long. Flatness problem: weak point. Size and entropy problems: not present if we assume that the universe begins cold and large.

133 85 / 95 Method Calculate matter correlation functions in the Hagedorn phase (neglecting the metric fluctuations) For fixed k, convert the matter fluctuations to metric fluctuations at Hubble radius crossing t = t i (k) Evolve the metric fluctuations for t > t i (k) using the usual theory of cosmological perturbations

134 86 / 95 Extracting the Metric Fluctuations Ansatz for the metric including cosmological perturbations and gravitational waves: ds 2 = a 2 (η) ( (1 + 2Φ)dη 2 [(1 2Φ)δ ij + h ij ]dx i dx j). Inserting into the perturbed Einstein equations yields Φ(k) 2 = 16π 2 G 2 k 4 δt 0 0(k)δT 0 0(k), h(k) 2 = 16π 2 G 2 k 4 δt i j(k)δt i j(k).

135 87 / 95 Power Spectrum of Cosmological Key ingredient: For thermal fluctuations: δρ 2 = T 2 R 6 C V. Key ingredient: For string thermodynamics in a compact space R 2 /l 3 s C V 2 T (1 T /T H ).

136 87 / 95 Power Spectrum of Cosmological Key ingredient: For thermal fluctuations: δρ 2 = T 2 R 6 C V. Key ingredient: For string thermodynamics in a compact space R 2 /l 3 s C V 2 T (1 T /T H ).

137 88 / 95 Power spectrum of cosmological fluctuations P Φ (k) = 8G 2 k 1 < δρ(k) 2 > = 8G 2 k 2 < (δm) 2 > R = 8G 2 k 4 < (δρ) 2 > R = 8G 2 T 1 l 3 s 1 T /T H Key features: scale-invariant like for inflation slight red tilt like for inflation

138 88 / 95 Power spectrum of cosmological fluctuations P Φ (k) = 8G 2 k 1 < δρ(k) 2 > = 8G 2 k 2 < (δm) 2 > R = 8G 2 k 4 < (δρ) 2 > R = 8G 2 T 1 l 3 s 1 T /T H Key features: scale-invariant like for inflation slight red tilt like for inflation

139 89 / 95 Spectrum of Gravitational Waves R.B., A. Nayeri, S. Patil and C. Vafa, Phys. Rev. Lett. (2007) P h (k) = 16π 2 G 2 k 1 < T ij (k) 2 > = 16π 2 G 2 k 4 < T ij (R) 2 > 16π 2 G 2 T l 3 (1 T /T H ) s Key ingredient for string thermodynamics Key features: < T ij (R) 2 > T l 3 s R 4 (1 T /T H) scale-invariant (like for inflation) slight blue tilt (unlike for inflation)

140 89 / 95 Spectrum of Gravitational Waves R.B., A. Nayeri, S. Patil and C. Vafa, Phys. Rev. Lett. (2007) P h (k) = 16π 2 G 2 k 1 < T ij (k) 2 > = 16π 2 G 2 k 4 < T ij (R) 2 > 16π 2 G 2 T l 3 (1 T /T H ) s Key ingredient for string thermodynamics Key features: < T ij (R) 2 > T l 3 s R 4 (1 T /T H) scale-invariant (like for inflation) slight blue tilt (unlike for inflation)

141 BICEP-2 Results B2xB2 B2xB1c B2xKeck (preliminary) l(l+1)c l BB /2π [µk 2 ] Multipole 90 / 95

142 Challenges for the String Gas Cosmology Bounce 91 / 95 Description of the Hagedorn phase. Length of the Hagedorn phase. Ensuring that the dominant source of fluctuations has holographic scaling.

143 92 / 95 Plan 1 2 Review of ary Cosmology 3 Cosmological 4 of 5 of 6 to Ekpyrotic Bounce Cosmology (e.g. String Gas Cosmology) 7

144 93 / 95 Classical is a simple cosmological scenario which solves problems of SBB cosmology and made successful predictions for observables. Cosmological inflation has conceptual problems. There are alternative early universe scenarios which are consistent with the observed spectrum of cosmological perturbations. It is wrong to say that the inflationary scenario has been proven. Alternative models make predictions for future observations with which they can be distinguished from inflation. Observations will give us the answer.

145 93 / 95 Classical is a simple cosmological scenario which solves problems of SBB cosmology and made successful predictions for observables. Cosmological inflation has conceptual problems. There are alternative early universe scenarios which are consistent with the observed spectrum of cosmological perturbations. It is wrong to say that the inflationary scenario has been proven. Alternative models make predictions for future observations with which they can be distinguished from inflation. Observations will give us the answer.

146 93 / 95 Classical is a simple cosmological scenario which solves problems of SBB cosmology and made successful predictions for observables. Cosmological inflation has conceptual problems. There are alternative early universe scenarios which are consistent with the observed spectrum of cosmological perturbations. It is wrong to say that the inflationary scenario has been proven. Alternative models make predictions for future observations with which they can be distinguished from inflation. Observations will give us the answer.

147 94 / 95 Entropy Fluctuations I Consider a second scalar field χ with the same negative exponential potential δχ k + ( k 2 + V ) δχk = 0. Vacuum initial conditions δχ k + ( k 2 2 t 2 ) δχk = 0. δχ k 1 2k e ikt as k( t)

148 95 / 95 Spectrum of Entropy Fluctuations II Solution: in the super-hubble limit. Hence δχ k H (1) 3/2 3/2 ( kt) k P χ (k) k 3 k 3 k 0, i.e. a scale-invariant power spectrum.