Inflationary non-Gaussianity: theoretical predictions and observational consequences Gabriel Jung, thesis defense Supervisor: Bartjan van Tent 22 May 2018, at the Laboratoire de Physique Th´ eorique Gabriel Jung - LPT Orsay 22 May 2018 1 / 23
Inflation History of our universe Source: ESA Inflation ( Starobinsky, 1980; Guth, 1981 ) • Period of fast and accelerated expansion in the very early universe • Energy content of the universe dominated by a scalar field • Solves several issues of the standard Big Bang theory Gabriel Jung - LPT Orsay 22 May 2018 2 / 23
Cosmic Microwave Background (CMB) CMB (observed in 1964 by Penzias and Wilson) • Relic radiation emitted 380000 years after the Big Bang • Blackbody spectrum at a temperature of 2 . 725 ± 0 . 001 K • Coming from all parts of the sky with a uniform temperature • Observational window on the primordial universe Gabriel Jung - LPT Orsay 22 May 2018 3 / 23
Cosmic Microwave Background (CMB) CMB (observed in 1964 by Penzias and Wilson) • Relic radiation emitted 380000 years after the Big Bang • Blackbody spectrum at a temperature of 2 . 725 ± 0 . 001 K • Coming from all parts of the sky with a uniform temperature • Observational window on the primordial universe • Precision cosmology era: (WMAP + Planck) Measurements of relative temperature fluctuations of order 10 − 5 CMB seen by Planck Gabriel Jung - LPT Orsay 22 May 2018 3 / 23
Perturbations • Primordial fluctuations → Large-scale structures present now • Explained by inflation: scalar field quantum fluctuations + expansion • Single-field inflation → perturbations are frozen on super-Hubble scales → simple link between inflationary perturbations and CMB anisotropies Comoving scales Inflation After Inflation Hubble length Perturbations frozen horizon crossing horizon re - entry Time Gabriel Jung - LPT Orsay 22 May 2018 4 / 23
Perturbations First-order perturbation theory: • Many inflation models predict an almost Gaussian and almost scale-invariant distribution of the perturbations ⇒ We only need the power spectrum (Fourier transform of the two-point correlator): P s ∝ k n s − 1 with n s = 0 . 968 ± 0 . 006 (Planck, 1502.02114 ) Gabriel Jung - LPT Orsay 22 May 2018 4 / 23
Perturbations First-order perturbation theory: • Many inflation models predict an almost Gaussian and almost scale-invariant distribution of the perturbations ⇒ We only need the power spectrum (Fourier transform of the two-point correlator): P s ∝ k n s − 1 with n s = 0 . 968 ± 0 . 006 (Planck, 1502.02114 ) Second-order perturbation theory: Deviations from Gaussianity ⇒ Three point-correlation function ↔ bispectrum Parametrized by the amplitude parameter f NL Different sources: • At late times: extra-galactic and galactic foregrounds • Primordial ⇒ important to discriminate inflation models ⇒ Many different shapes to look for in the observational data Gabriel Jung - LPT Orsay 22 May 2018 4 / 23
Plan 1. Non-Gaussianity in two-field inflation Jung & Van Tent (1611.09233) • Perturbations • Background description (slow-roll parameters) • Observables • The slow-roll approximation • Sum potentials • Monomial potentials 2. Non-Gaussianity in CMB observations Jung, Racine & Van Tent (to be published) • Binned bispectrum estimator • Galactic foregrounds: dust • CMB data analyses Gabriel Jung - LPT Orsay 22 May 2018 5 / 23
Non-Gaussianity in two-field inflation
Local non-Gaussianity • Gravitational potential: Φ = Φ L + f local (Φ 2 L − � Φ L � 2 ) NL a † (annihilation and creation operators) Φ L ∝ ˆ a + ˆ � Φ 2 L � 2 � ΦΦΦ � ≈ f local NL • Peaks in the squeezed configuration: one small ℓ and two large ℓ ’s • Planck (1502.01592) : f local = 0 . 8 ± 5 . 0 NL Gabriel Jung - LPT Orsay 22 May 2018 6 / 23
Local non-Gaussianity • Gravitational potential: Φ = Φ L + f local (Φ 2 L − � Φ L � 2 ) NL a † (annihilation and creation operators) Φ L ∝ ˆ a + ˆ � Φ 2 L � 2 � ΦΦΦ � ≈ f local NL • Peaks in the squeezed configuration: one small ℓ and two large ℓ ’s • Planck (1502.01592) : f local = 0 . 8 ± 5 . 0 NL • Can be produced during multiple-field inflation ! Two kinds of perturbations which interact on super-Hubble scales : - Adiabatic: perturbations in the total energy density - Isocurvature: relative fluctuations between the different components • Long-wavelength formalism (Rigopoulos, Shellard and van Tent: astro-ph/0504508, Tzavara and van Tent: 1012.6027) Gabriel Jung - LPT Orsay 22 May 2018 6 / 23
Local non-Gaussianity • Gravitational potential: Φ = Φ L + f local (Φ 2 L − � Φ L � 2 ) NL a † (annihilation and creation operators) Φ L ∝ ˆ a + ˆ � Φ 2 L � 2 � ΦΦΦ � ≈ f local NL • Peaks in the squeezed configuration: one small ℓ and two large ℓ ’s • Planck (1502.01592) : f local = 0 . 8 ± 5 . 0 NL • Can be produced during multiple-field inflation ! Two kinds of perturbations which interact on super-Hubble scales : - Adiabatic: perturbations in the total energy density - Isocurvature: relative fluctuations between the different components • Long-wavelength formalism (Rigopoulos, Shellard and van Tent: astro-ph/0504508, Tzavara and van Tent: 1012.6027) Green’s functions: • ¯ v 22 : isocurvature mode • ¯ v 12 : isocurvature contribution to adiabatic Important hypothesis : ¯ v 22 vanishes at the end of inflation Gabriel Jung - LPT Orsay 22 May 2018 6 / 23
σ σ ϕ ϕ Multiple-field inflation Main motivation: many scalar fields in high-energy theories Gabriel Jung - LPT Orsay 22 May 2018 7 / 23
Multiple-field inflation Main motivation: many scalar fields in high-energy theories Orthonormal basis • Two-field models: W ( φ, σ ) e 1 = ( e 1 φ , e 1 σ ) , e 2 = ( e 1 σ , − e 1 φ ) ⇒ Sufficient to study new effects Groot Nibbelink & van Tent: hep-ph/0011325 & 0107272 • Nice description of two-field infla- ˙ tion ⇒ Trajectory in field space φ √ e 1 φ = e 1 e 1 σ ˙ φ 2 + ˙ σ 2 σ • Time coordinate: number of e- σ ˙ √ e 1 σ = folds ˙ φ 2 + ˙ σ 2 e 1 ϕ ϕ Gabriel Jung - LPT Orsay 22 May 2018 7 / 23
Multiple-field inflation Main motivation: many scalar fields in high-energy theories Orthonormal basis • Two-field models: W ( φ, σ ) e 1 = ( e 1 φ , e 1 σ ) , e 2 = ( e 1 σ , − e 1 φ ) ⇒ Sufficient to study new effects Groot Nibbelink & van Tent: hep-ph/0011325 & 0107272 • Nice description of two-field infla- ˙ tion ⇒ Trajectory in field space φ √ e 1 φ = e 1 e 1 σ ˙ φ 2 + ˙ σ 2 σ • Time coordinate: number of e- σ ˙ √ e 1 σ = folds ˙ φ 2 + ˙ σ 2 e 1 ϕ ϕ Slow-roll parameters ˙ φ 2 + ˙ ˙ σ 2 η � = φ ˙ ¨ η ⊥ = ¨ σ ˙ H φ +¨ σ ˙ σ φ ˙ σ +¨ φ ǫ = − H = σ 2 − ǫ 2 ˙ ˙ φ 2 + ˙ φ 2 + ˙ σ 2 • ǫ and η � , similar to the usual ǫ and η of single-field inflation • η ⊥ : perpendicular ( e 2 ) acceleration, purely multiple-field effect ˙ v 12 = 2 η ⊥ ¯ Link adiabatic/isocurvature perturbations: ¯ v 22 Gabriel Jung - LPT Orsay 22 May 2018 7 / 23
Observables Spectral index: single-field formula + multiple-field correction ¯ v 12 � � n s − 1 = − 4 ǫ ∗ − 2 η � v 12 ( ǫ ∗ + η � ∗ + ˜ 4 η ⊥ ∗ − ∗ − 2¯ W 22 ∗ ) 1 + (¯ v 12 ) 2 Subscript ∗ : evaluated at horizon-crossing f NL = 0 . 8 ± 5 . 0 ns = 0 . 968 ± 0 . 006 Gabriel Jung - LPT Orsay 22 May 2018 8 / 23
Observables Spectral index: single-field formula + multiple-field correction ¯ v 12 � � n s − 1 = − 4 ǫ ∗ − 2 η � v 12 ( ǫ ∗ + η � ∗ + ˜ 4 η ⊥ ∗ − ∗ − 2¯ W 22 ∗ ) 1 + (¯ v 12 ) 2 Subscript ∗ : evaluated at horizon-crossing Non-Gaussianity v 12 ) 2 − 6 − 2(¯ 5 f NL = v 12 ) 2 ] 2 ( g iso + g sr + g int ) [1 + (¯ v 22 ) 2 + ¯ g iso = ( ǫ + η � )(¯ v 22 ˙ ¯ v 22 g sr = − ǫ ∗ + η � + η ⊥ ∗ − χ ∗ + η ⊥ ∗ ¯ v 12 − 3 � � ∗ ǫ ∗ + η � ∗ v 2 2¯ 2 2 v 12 ¯ 12 � t d t ′ � v 22 ) 2 + ( ǫ + η � )¯ 2( η ⊥ ) 2 (¯ v 22 ˙ v 22 + (˙ v 22 ) 2 g int = − ¯ ¯ t ∗ v 22 + 9 η ⊥ ˙ f NL = 0 . 8 ± 5 . 0 − G 13 ( t , t ′ )¯ � v 22 (Ξ¯ v 22 ) ¯ ns = 0 . 968 ± 0 . 006 Gabriel Jung - LPT Orsay 22 May 2018 8 / 23
Observables Spectral index: single-field formula + multiple-field correction v 12 ¯ � � n s − 1 = − 4 ǫ ∗ − 2 η � v 12 ( ǫ ∗ + η � ∗ + ˜ 4 η ⊥ ∗ − ∗ − 2¯ W 22 ∗ ) 1 + (¯ v 12 ) 2 Subscript ∗ : evaluated at horizon-crossing Non-Gaussianity v 12 ) 2 − 6 − 2(¯ 5 f NL = v 12 ) 2 ] 2 ( g iso + g sr + g int ) [1 + (¯ g iso : pure isocurvature term, set to 0 at the end of inflation g sr = − ǫ ∗ + η � + η ⊥ ∗ − χ ∗ + η ⊥ ∗ ¯ v 12 − 3 � � ∗ ǫ ∗ + η � ∗ v 2 2¯ 2 2 v 12 ¯ 12 � t d t ′ � v 22 ) 2 + ( ǫ + η � )¯ 2( η ⊥ ) 2 (¯ v 22 ˙ v 22 + (˙ v 22 ) 2 g int = − ¯ ¯ t ∗ v 22 + 9 η ⊥ ˙ f NL = 0 . 8 ± 5 . 0 − G 13 ( t , t ′ )¯ � v 22 (Ξ¯ v 22 ) ¯ ns = 0 . 968 ± 0 . 006 Gabriel Jung - LPT Orsay 22 May 2018 8 / 23
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