- 1. Cosmological Correlations Kavli Asian Winter School Jan 2023 Daniel Baumann University of Amsterdam & National Taiwan University
- 2. Motivation By measuring cosmological correlations, we learn both about the evolution of the universe and its initial conditions: 380 000 years 13.8 billion years
- 3. The correlations in the CMB temperature anisotropies have revealed a great deal about the geometry and composition of the universe: neutrinos geometry, dark energy dark matter baryons 90◦ 18◦ 0 1000 2000 3000 4000 5000 6000 Power [µK 2 ] 2◦ 0.2◦ 0.1◦ 0.07◦ Angular separation Motivation
- 4. 10−4 10−3 10−2 10−1 100 101 Separation [Mpc−1 ] 100 101 102 103 104 Power [Mpc 3 ] Correlations are also observed in the large-scale structure of the universe. The observed matter power spectrum is Motivation matter density
- 5. Under relatively mild assumptions, the observed correlations can be traced back to primordial correlations on the reheating surface: Motivation We have learned two interesting facts about these initial conditions: 1) The fluctuations were created before the hot Big Bang. 2) During a phase of time-translation invariance (= inﬂation).
- 6. The study of cosmological correlators is also of conceptual interest: Motivation Rd,1 dSd+1 AdSd+1 Our understanding of quantum field theory in de Sitter space (cosmology) is still rather underdeveloped. Opportunity for you to make progress!
- 7. OUTLINE: Cosmological Correlations Wavefunction Approach Cosmological Bootstrap LECTURE NOTES: written together with Austin Joyce. (Lecture 1) (Lectures 2+3) (Lectures 3+4) https://github.com/ddbaumann/cosmo-correlators
- 9. Large scales Small scales CMB Power Spectrum 2 10 30 0 1000 2000 3000 4000 5000 6000 Power [µK 2 ] 500 1000 1500 2000 2500 Multipole 90◦ 18◦ 1◦ 0.2◦ 0.1◦ 0.07◦ Angular separation
- 10. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 11. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 12. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 13. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 14. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 15. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 16. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 17. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 18. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 19. 10 100 1000 Multipole Figure courtesy of Mathew Madhavacheril
- 20. 2 10 30 0 1000 2000 3000 4000 5000 6000 Power [µK 2 ] 500 1000 1500 2000 2500 Multipole 90◦ 18◦ 1◦ 0.2◦ 0.1◦ 0.07◦ Angular separation What created the features in the power spectrum? CMB Power Spectrum
- 21. Photon-Baryon Fluid At early times, photons and baryons (mostly protons and electrons) are strongly coupled and act as a single fluid: Dark matter Baryons Photons The photon pressure prevents the collapse of density fluctuations. This allows for sound waves (like for density fluctuations in air).
- 22. Cosmic Sound Waves Decomposing the sound waves into Fourier modes, we have Time at recombination Sound speed cs ⇡ 1/ p 3 The wave amplitudes are random variables, with hAki = hBki = 0 This is why we don’t see a wavelike pattern in the CMB map. To see the sound waves, we need to look at the statistics of the map. ⇢ ⇢ = Ak cos[csk⌧⇤] + Bk sin[csk⌧⇤] 4 T T
- 23. The two-point function in Fourier space is h T(k) T(k0 )i = h PA(k) cos2 (csk⌧⇤) + 2CAB(k) sin(csk⌧⇤) cos(csk⌧⇤) + PB(k) sin2 (csk⌧⇤) i (2⇡)3 (k + k0 ) If the sound waves were created by a random source, we would have CAB(k) = 0 PA(k) = PB(k) ⌘ P(k) and hence the two-point function becomes h T(k) T(k0 )i = P(k) (2⇡)3 (k + k0 ) Where are the oscillations? CMB Power Spectrum
- 24. Coherent Phases To explain the oscillations in the CMB power spectrum, we have to assume that the sound waves do not have random phases: T(k) = Ak cos[csk⌧⇤] Oscillations in the CMB power spectrum h T(k) T(k0 )i0 = PA(k) cos2 [csk⌧⇤] What created the coherent phases in the primordial sound waves?
- 25. Before the Big Bang Phase coherence is created if the fluctuations were produced before the hot Big Bang. Let’s see why. Going back in time, all fluctuations have wavelengths > Hubble scale. The solution on these large scales is ⇢ ⇢ = Ck + Dk a 3 If the fluctuations were created long before the hot Big Bang, then only the constant growing mode survived. This explains the phase coherence of the primordial sound waves ⇢ ⇢ = Ck ! Ak cos ⇥ csk⌧ ⇤
- 26. Before the Big Bang bounce inflation This suggests two options: rapid expansion or slow contraction?
- 27. Scale-Invariance 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.05 0.1 0.15 m2φ2 R2 1 − cos(φ/f) ns (r) The primordial fluctuations are approximately scale-invariant: This suggests that the fluctuations were created in a phase of approximate time-translation invariance (= inflation). ns < 1 P⇣(k) k Cl l
- 28. Inflation During inflation, the expansion rate is nearly constant: Exponential expansion: Quasi-de Sitter geometry H(t) ⌘ 1 a da dt ⇡ const a(t) ⇡ exp(Ht) Inflation needs to end, so there has to be some time-dependence: " ⌘ Ḣ H2 ⌧ 1 In slow-roll inflation, this is generated by a field rolling down a flat potential. However, there are other ways to achieve inflation.
- 29. EFT of Inflation Inflation is a symmetry breaking phenomenon: ⇡(t, x) ij(t, x) H(t) (t) The low-energy EFT is parameterized by two massless fields: • Goldstone boson of broken time translations • Graviton = (t + ⇡) ¯(t) Creminelli et al. [2006] Cheung et al. [2008]
- 30. EFT of Inflation ⇡(t, x) ij(t, x) H(t) (t) Creminelli et al. [2006] Cheung et al. [2008] gij = a2 e2⇣ [e ]ij In comoving gauge, the Goldstone boson gets eaten by the metric: ⇣ = H⇡ Curvature perturbation Inflation is a symmetry breaking phenomenon:
- 31. The Goldstone Lagrangian is Creminelli et al. [2006] Cheung et al. [2008] M2 plḢ( )2 + n=2 M4 n n! 2 ˙ + ( )2 n M4 n L = + · · · Slow-roll inflation Higher-derivative corrections (@ )2n 1 2 (@ )2 V ( ) EFT of Inflation The fluctuations can have a small sound speed and large interactions: L⇡ = M2 pl|Ḣ| c2 s h ˙ ⇡2 c2 s(@i⇡)2 (1 c2 s) ˙ ⇡(@µ⇡)2 + · · · i nonlinearly realized symmetry
- 32. Quantum Fluctuations Quantum fluctuations of the Goldstone produced the density fluctuations in the early universe: Let us derive this. 10 32 s 370 000 yrs 1 billion yrs
- 33. Quantum Fluctuations We start from the quadratic Goldstone action where a(⌧) ⇡ (H⌧) 1 (de Sitter). S = Z d4 x p g M2 PlḢ gµ⌫ @µ⇡@⌫⇡ = Z d⌧d3 x a2 (⌧) M2 Pl|Ḣ| ⇣ (⇡0 )2 (r⇡)2 ⌘ S = 1 2 Z d⌧d3 x ✓ (u0 )2 (ru)2 + a00 a u2 ◆ It is convenient to introduce the canonically normalized field u ⌘ a(⌧) q 2MPl|Ḣ| ⇡ so that Time-dependent mass Minkowski
- 34. Classical Dynamics The equation of motion then is u00 k + ✓ k2 2 ⌧2 ◆ uk = 0 uk(⌧) = c± p 2k e±ik⌧ • At early times, k⌧ 1 , this becomes u00 k + k2 uk ⇡ 0 Harmonic oscillator Growing mode • At late times, k⌧ ! 0 , this becomes u00 k 2 ⌧2 uk ⇡ 0 uk(⌧) = c1⌧ 1 + c2 ⌧2 Fixed by harmonic oscillator
- 35. Classical Dynamics The equation of motion then is • The complete solution is uk(⌧) = c1 1 p 2k ✓ 1 + i k⌧ ◆ eik⌧ + c2 1 p 2k ✓ 1 i k⌧ ◆ e ik⌧ where the constants will be fixed by quantum initial conditions. • The curvature perturbation becomes a constant at late times: ⇣ = H⇡ / a 1 u / ⌧ ⇥ ⌧ 1 = const u00 k + ✓ k2 2 ⌧2 ◆ uk = 0
- 36. Canonical Quantization S = 1 2 Z d⌧d3 x ✓ (u0 )2 (ru)2 + a00 a u2 ◆ Quantization starts from the action of the canonically normalized field: • Define the conjugate momentum: p = L/ u0 = u0 • Promote fields to operators: • Impose canonical commutation relations: u, p ! û, p̂ [û(⌧, x), p̂(⌧, x0 )] = i~ (x x0 ) • Define the mode expansion of the field operator: û(⌧, x) = Z d3 k (2⇡)3 ⇣ u⇤ k(⌧) âk + uk(⌧) â† k ⌘ eik·x where uk(⌧) is a solution to the classical equation of motion and [âk, â† k0 ] = (2⇡)3 (k k0 ) Creation Annihilation
- 37. • Define the vacuum state: âk|0i = 0 • The expectation value of the Hamiltonian in this state is minimized for the Bunch-Davies mode function: Canonical Quantization • The two-point function of the field operator (in that state) is uk(⌧) = 1 p 2k ✓ 1 + i k⌧ ◆ eik⌧ h0|ûk(⌧)ûk0 (⌧)|0i = h0| ⇣ u⇤ k(⌧) âk + uk(⌧) â† k ⌘ ⇣ u⇤ k0 (⌧) âk0 + uk0 (⌧) â† k0 ⌘ |0i = (2⇡)3 (k + k0 ) |uk(⌧)|2 Power spectrum positive frequency
- 38. P⇣(k, ⌧) ⌘ H2 2M2 Pl|Ḣ| |uk(⌧)|2 a2(⌧) = H2 2M2 Pl" 1 k3 (1 + k2 ⌧2 ) k⌧!0 ! H2 2M2 Pl" 1 k3 Power Spectrum Using the Bunch-Davies mode function, we obtain the following power spectrum for the primordial curvature perturbations: • The k 3 scaling is the characteristic of a scale-invariant spectrum. • During inflation, both H(t) "(t) and depend on time, which leads to a small scale-dependence k3 2⇡2 P⇣(k) = As ✓ k k0 ◆ns 1 • Observations find ns = 0.9667 ± 0.0040 . ⌘ 2 ⇣(k)
- 40. Gaussianity The primordial fluctuations were highly Gaussian (as expected for the ground state of a harmonic oscillator): Pr[ ] 2 FNL 3/2 10 3 The universe is more Gaussian than flat. So far, we have only studied the free theory. Interactions during inflation can lead to non-Gaussianity.
- 41. Primordial Non-Gaussianity The main diagnostic of primordial non-Gaussianity is the bispectrum: • The amplitude of the non-Gaussianity is defined as the size of the bispectrum in the equilateral configuration: k1 k2 k3 Shape h⇣k1 ⇣k2 ⇣k3 i = (2⇡)3 (k1 + k2 + k3) (2⇡2 )2 (k1k2k3)2 B⇣(k1, k2, k3) FNL(k) ⌘ 5 18 B⇣(k, k, k) 3 ⇣(k)
- 42. Shapes of Non-Gaussianity ϕ̇3 χ ϕ e−ikη • The shape of the non-Gaussianity contains a lot of information about the microphysics of inflation:
- 43. Equilateral Non-Gaussianity The Goldstone mode during inflation can have large self-interactions: nonlinearly realized symmetry L⇡ = M2 pl|Ḣ| c2 s h ˙ ⇡2 c2 s(@i⇡)2 (1 c2 s) ˙ ⇡(@µ⇡)2 + · · · i What would measuring (or constraining) equilateral NG teach us about the physics of inflation? Correlations are largest when the fluctuations have comparable wavelengths: equilateral NG: B˙ ⇡3 / k1k2k3 (k1 + k2 + k3)3 FNL / c 2 s
- 44. Equilateral Non-Gaussianity Minimal models of inflation are characterized by three energy scales: H Mpl f⇡
- 45. Superhorizon Subhorizon H • This sets the energy scale of the experiment. • At the Hubble scale, modes exit the horizon and freeze out. Equilateral Non-Gaussianity Minimal models of inflation are characterized by three energy scales:
- 46. Freeze-out Einstein gravity H Mpl Equilateral Non-Gaussianity Minimal models of inflation are characterized by three energy scales: • This sets the amplitude of tensor ﬂuctuations: • At the Planck scale, gravitational interactions become strong. No B-modes 2 = 2 ⇡2 ✓ H Mpl ◆2 < 0.1 2 ⇣ ⇡ 10 10 > 105 H
- 47. Equilateral Non-Gaussianity Minimal models of inflation are characterized by three energy scales: Background Freeze-out Fluctuations H Mpl f⇡ = 58H • At the symmetry breaking scale, time translations are broken. • This sets the amplitude of scalar ﬂuctuations: 2 ⇣ = 1 4⇡2 ✓ H f⇡ ◆4 ⇡ 2.2 ⇥ 10 9 f4 ⇡ = M2 pl|Ḣ|cs ! ˙2 (for SR)
- 48. Equilateral Non-Gaussianity Closer inspection may reveal additional scales: Strongly coupled Weakly coupled H Mpl f⇡ ⇤ > 5H • At the strong coupling scale, interactions become strongly coupled. • This sets the amplitude of interactions: No PNG Fequil NL ⇠ H2 ⇤2 < 0.01
- 49. Equilateral Non-Gaussianity DB, Green, Lee and Porto [2015] Grall and Mellville [2020] Asking for the theory to be weakly coupled up to the symmetry breaking scale implies a critical value for the sound speed: slow-roll non-slow-roll Higgs Technicolor 1 cs 0.02 0.31 Hard to measure! ⇤ f⇡ ⇤ f⇡ Weakly coupled UV completion Strongly coupled UV completion Fequil NL ⇠ ⇣
- 50. Folded Non-Gaussianity Excited initial states (negative frequency modes) lead to enhanced correlations when two wave vectors become collinear: folded NG: B / 1 k1 + k2 k3 e ik3⌧ • The inflationary expansion dilutes any pre-inflationary particles. In minimal scenarios, we therefore don’t expect to see folded NG. • Models with continuous particle production during inflation can still give a signal in the folded limit. Flauger, Green and Porto [2013] Flauger, Mirbabayi, Senatore and Silverstein [2017] Green and Porto [2020]
- 51. Squeezed Non-Gaussianity In single-field inflation, correlations must vanish in the squeezed limit: Maldacena [2003] Creminelli and Zaldarriaga [2004] k1 k2 k3 lim k1!1 ⇠ 0 Chen and Wang [2009] DB and Green [2011] Noumi, Yamaguchi andYokoyama [2013] Arkani-Hamed and Maldacena [2015] Lee, DB and Pimentel [2016] DB, Goon, Lee and Pimentel [2017] Kumar and Sundrum [2018] Jazayeri and Renaux-Petel [2022] Pimentel and Wang [2022] The signal in the squeezed limit therefore acts as a particle detector.
- 52. Cosmological Collider Physics Arkani-Hamed and Maldacena [2015] ii l;! , µ1...µs Massive particles are spontaneously created in an expanding spacetime. time space
- 53. Cosmological Collider Physics Arkani-Hamed and Maldacena [2015] ii l;! , time space However, they cannot be directly observed at late times. µ1...µJ
- 54. Cosmological Collider Physics Arkani-Hamed and Maldacena [2015] time space ii l;! , ¯ Instead, they decay into light fields. These correlated decays create distinct higher-order correlations in the inflationary perturbations.
- 55. µ1...µJ Cosmological Collider Physics Arkani-Hamed and Maldacena [2015] Mpl H 1014 GeV 1018 GeV ⇣ This allows us to probe the particle spectrum at inflationary energies: This can be many orders of magnitudes above the scales probed by terrestrial colliders.
- 56. PJ (cos ✓) Cosmological Collider Physics Arkani-Hamed and Maldacena [2015] The signal depends on the masses and spins of the new particles: lim k1!0 h⇣k1 ⇣k2 ⇣k3 i0 / 8 > > > < > > > : ✓ k1 k3 ◆ m < 3 2 H ✓ k1 k3 ◆3/2 cos µ ln k1 k3 m > 3 2 H k1 k2 k3 ✓ spin /
- 57. Observational Constraints non-perturbative gravitational floor FNL 1 10 7 ruled out by Planck Planck has ruled out three orders of magnitude. The theoretically interesting regime of non-Gaussianity spans about seven orders of magnitude: window of opportunity There is still room for new particles to leave their mark.
- 58. Any Questions?
- 59. So far, I have focussed mostly on the practical motivations for studying cosmological correlators. In the rest of these lectures, I will introduce three diﬀerent ways to compute primordial correlation functions: • In-In Formalism • Wavefunction Approach • Cosmological Bootstrap I will leads us to more conceptual questions about the theory of quantum fields in cosmological spacetimes. Lectures Notes
- 61. O(t) U+ U h0| h0| In-In Formalism The main observables in cosmology are spatial correlation functions: These correlators are usually computed in the in-in formalism: h⌦| (t, x1) (t, x2) · · · (t, xN ) |⌦i time Weinberg [2005] which leads to the following master equation hO(t)i = h0| T̄ei R t 1 dt00 Hint(t00 ) O(t) Te i R t 1 dt0 Hint(t0 ) |0i
- 62. hO(t)i = h0| T̄ei R t 1 dt00 Hint(t00 ) O(t) Te i R t 1 dt0 Hint(t0 ) |0i In-In Formalism Diagrammatically, this can be written as Giddings and Sloth [2010] Wightman Feynman Wightman I++ = I = I + = I+ = In the lecture notes, we apply this to many examples (see Section 3). Here, we will instead follow the alternative wavefunction approach.
- 63. Wavefunction Approach We first define a wavefunction for the late-time fluctuations: (x) (t, x) and then use it to compute the boundary correlations [ ] ⌘ h |⌦i h (x1) . . . (xN )i = Z D (x1) . . . (xN ) | [ ]|2 The wavefunction has nicer analytic properties than the correlation functions and it also plays a central role in the cosmological bootstrap.
- 64. OUTLINE: Wavefunction of the Universe Warmup in QM Flat-Space Wavefunction De Sitter Wavefunction REFERENCE: DB and Joyce, Lectures on Cosmological Correlations
- 66. N (k) = (2⇡)3 (k1 + · · · + kN ) N (k) . Wavefunction Coeﬃcients For small fluctuations, we expand the wavefunction as Wavefunction coeﬃcient • By translation invariance, we have [ ] = exp X N Z d3 k1 . . . d3 kN (2⇡)3N N (k) k1 . . . kN ! • We will also write N (k) = hO1 . . . ON i0 , where Oa ⌘ Oka are dual operators. This will just be notation and (so far) does not have a deeper holographic meaning (as in dS/CFT).
- 67. Relation to In-In Correlators In perturbation theory, the correlation functions are related to the wavefunction coeﬃcients: h 1 . . . N i = Z D 1 . . . N | [ ]|2 h 1 2i = 1 2RehO1O2i h 1 2 3 4i = RehO1O2O3O4i 8 Q4 a=1 RehOaOai4 + hO1O2XihXO3O4i + c.c. 8RehXXi Q4 a=1 RehOaOai4 h 1 2 3i = RehO1O2O3i 4 Q3 a=1 RehOaOai Wavefunction coeﬃcients therefore contain the same information, but are easier to compute and have nicer analytic properties.
- 68. Computing the Wavefunction The wavefunction has the following path integral representation: [ ] = Z (0) = ( 1) = 0 D eiS[ ] Action Boundary conditions: (t = 1(1 i")) = 0 selects the Bunch-Davies vacuum. For tree-level processes, we can evaluate this in a saddle-point approximation: Classical solution [ ] ⇡ eiS[ cl] with cl(0) = and cl( 1) = 0. To find the wavefunction, we therefore need to find the classical solution for the bulk field with the correct boundary conditions.
- 70. Harmonic Oscillator Consider our old friend the simple harmonic oscillator S[ ] = Z dt ✓ 1 2 ˙ 2 1 2 !2 2 ◆ • The classical solution (with the correct boundary conditions) is cl(t) = ei!t • The on-shell action becomes S[ cl] = Z t⇤ ti dt 1 2 @t( ˙ cl cl) 1 2 cl (¨cl + !2 cl) | {z } = 0 = 1 2 ˙ cl cl t=t⇤ = i! 2 2
- 71. Harmonic Oscillator • The wavefunction then is [ ] ⇡ exp(iS[ cl]) = exp ⇣ ! 2 2 ⌘ • The quantum variance of the oscillator therefore is h 2 i = 1 2! • In QFT, the same result applies to each Fourier mode where !k = p k2 + m2. h k ki0 = 1 2!k , Gaussian
- 72. Time-Dependent Oscillator To make this more interesting, consider a time-dependent oscillator S[ ] = Z dt ✓ 1 2 A(t) ˙ 2 1 2 B(t) 2 ◆ • The classical solution is cl(t) = K(t) , with K(0) = 1 K( 1) ⇠ ei!t • The on-shell action becomes S[ cl] = Z t⇤ ti dt 1 2 @t(A ˙ cl cl) 1 2 cl (@t(A ˙ cl) + B cl) | {z } = 0 = 1 2 A ˙ cl cl t=t⇤ = 1 2 A 2 @t log K t=t⇤
- 73. • The wavefunction then is which implies Time-Dependent Oscillator [ ] ⇡ exp(iS[ cl]) = exp ✓ i 2 (A@t log K) ⇤ 2 ◆ , | [ ]|2 = exp Im(A@t log K) ⇤ 2 h 2 i = 1 2 Im(A@t log K) ⇤
- 74. Free Field in de Sitter The action of a massless ﬁeld in de Sitter is that of a time-dependent oscillator: S = Z d⌘d3 x a2 (⌘) h ( 0 )2 (r )2 i = 1 2 Z d⌘ d3 k (2⇡)3 1 (H⌘)2 0 k 0 k k2 (H⌘)2 k k • The classical solution is and hence cl(⌘) = K(⌘) , with K(⌘) = (1 ik⌘)eik⌘ log K(⌘) = log(1 ik⌘) + ik⌘
- 75. Free Field in de Sitter • Using our result for the variance of a time-dependent oscillator, we then get h k ki0 = H2 2k3 • The result for a massive field is derived in the lecture notes. which we derived in the last lecture using canonical quantization.
- 76. Anharmonic Oscillator Consider the following anharmonic oscillator S[ ] = Z dt ✓ 1 2 ˙ 2 1 2 !2 2 1 3 g 3 ◆ • The classical equation of motion is ¨cl + !2 cl = g 2 cl • A formal solution is where K(t) = ei!t , G(t, t0 ) = 1 2! ⇣ e i!(t t0 ) ✓(t t0 ) + ei!(t t0 ) ✓(t0 t) ei!(t+t0 ) ⌘ . cl(t) = K(t) + i Z dt0 G(t, t0 ) g 2 cl(t0 ) Feynman propagator
- 77. The terms linear in cancel. S[ cl] = i! 2 2 g 2 Z dt ei!t 2 cl + Z dt 1 2 ✓ ei!t ig Z dt0 G(t, t0 ) 2 cl(t0 ) ◆ ⇣ g 2 cl(t) ⌘ g 3 3 cl . Anharmonic Oscillator • Deriving the on-shell action is now a bit more subtle: S[ cl] = Z t⇤ ti dt 1 2 @t( cl ˙ cl) 1 2 cl(¨cl + !2 cl) g 3 3 cl = 1 2 cl ˙ cl t=t⇤ + Z dt 1 2 cl(¨cl + !2 cl) g 3 3 cl . Since limt!0 @tG(t, t0 ) = iei!t0 6= 0, the boundary term is 1 2 cl ˙ cl t=t⇤ = 1 2 ✓ i! ig Z dt0 iei!t0 2 cl(t0 ) ◆ = i! 2 2 g 2 Z dt0 ei!t0 2 cl(t0 ) . The action then becomes
- 78. Anharmonic Oscillator • The final on-shell action is S[ cl] = i! 2 2 g 3 Z dt 3 cl(t) ig2 2 Z dt dt0 G(t, t0 ) 2 cl(t0 ) 2 cl(t) • To evaluate this, we write the classical solution as where (0) (t) = ei!t , (1) (t) = i Z dt0 G(t, t0 ) ⇣ 2 e2i!t0 ⌘ = 2 3!2 e2i!t ei!t . • With this, the wavefunction becomes [ ] ⇡ eiS[ cl] = exp ✓ ! 2 2 g 9! 3 + g2 72!3 4 + · · · ◆ • From this, we can compute h 3 i, h 4 i, etc. cl(t) = X n gn (n) (t),
- 79. Interacting Fields Back to field theory: S[ ] = Z d4 x p g ✓ 1 2 gµ⌫ @µ @⌫ 1 2 m2 2 1 3 g 3 ◆ The analysis is similar to that of the anharmonic oscillator (see lecture notes). In the interest of time, we jump directly to Feynman rules for WF coeﬃcients: Kk(t) Bulk-to-bulk propagator for every internal line G(kI; t, t0 ) Bulk-to-boundary propagator for every external line ig Z dt a3 (t) Time integral for every vertex
- 81. The wavefunction is an interesting object even in flat space. • The propagators are the same as for the harmonic oscillator: • All correlators can be evaluated at We will compute the simplest tree-level correlators in this theory. • We will study the following simple toy model Flat-Space Wavefunction
- 82. Contact Diagrams The three-point wavefunction coeﬃcient is = ig Z 0 1 dt Kk1 (t)Kk2 (t)Kk3 (t) = ig Z 0 1 dt ei(k1+k2+k3)t = g (k1 + k2 + k3) . This is easily generalized to N-point wavefunction coeﬃcients: We will have more to say about this singularity.
- 83. Exchange Diagrams The four-point wavefunction coeﬃcient in
- 84. Exchange Diagrams The final answer is remarkably simple and has an interesting singularity structure: Energy entering the right vertex Energy entering the left vertex Total energy We will have more to say about this later.
- 85. Recursion Relation More complicated graphs can be computed in the same way, but the complexity of the time integrals increases quickly with the number of internal lines. Fortunately, there is a powerful recursion relation that allows complex graphs to be constructed from simpler building blocks: Arkani-Hamed, Benincasa and Postnikov [2017] This provides a simple algorithm to produce the result for any tree graph.
- 87. Consider a massive scalar in de Sitter space. • The action of is • The equation of motion is Free Field in de Sitter • The general solution of this equation is Hankel function
- 88. Consider a massive scalar in de Sitter space. • The action of is • The equation of motion is Free Field in de Sitter • For , the field is conformally coupled and Harmonic oscillator We will often consider this special case, because the wavefunction can then be computed analytically.
- 89. Propagators For a generic field, with mode function , the propagators are In general, these are given by Hankel functions which leads to complicated integrals in the wavefunction calculations. Our explicit calculations will therefore be for conformally coupled scalars. We return to the general case in the cosmological bootstrap.
- 90. Contact Diagrams The three-point WF coeﬃcient is Product of Hankel functions
- 91. Contact Diagrams For conformally coupled scalars, we get where The corresponding correlator is
- 92. Contact Diagrams The four-point WF coeﬃcient for a contact interaction of conformally coupled scalars is Since the interaction preserves the conformal symmetry of the free theory, this is the same result as in flat space (modulo the prefactor).
- 93. Exchange Diagrams The four-point WF coeﬃcient corresponding to an exchange diagram (of conformally coupled scalars) is
- 94. Exchange Diagrams We see that this four-point function can be written as an integral of the flat-space result: This integral can be performed analytically to give Arkani-Hamed and Maldacena [2015]
- 95. A Challenge So far, we have only computed the correlators of conformally coupled scalars. Consider now the exchange of a generic massive scalar: Conformally coupled scalars Massive scalar In general, the time integrals cannot be performed analytically. We need a diﬀerent approach.
- 97. ? t = 0 If inflation is correct, then the reheating surface is the future boundary of an approximate de Sitter spacetime: Can we bootstrap these boundary correlators directly? • Conceptual advantage: focuses directly on observables. • Practical advantage: simpliﬁes calculations. end of inflation hot big bang
- 98. | f i h i | + ? The bootstrap perspective has been very influential for scattering amplitudes: • Practical advantage: simpliﬁes calculations. • Conceptual advantage: reveals hidden structures. There has be enormous progress in bypassing Feynman diagram expansions to write down on-shell amplitudes directly:
- 99. t = 0 + ? I will describe recent progress in developing a similar bootstrap approach for cosmological correlators: Goal: Develop an understanding of cosmological correlators that parallels our understanding of flat-space scattering amplitudes.
- 100. OUTLINE: REFERENCES: Bootstrapping Tools The Bootstrap Philosophy Examples DB, Green, Joyce, Pajer, Pimentel, Sleight and Taronna, Snowmass White Paper: The Cosmological Bootstrap [2203.08121] DB and Joyce, Lectures on Cosmological Correlations
- 102. How we usually make predictions: Lagrangians equations of motion spacetime evolution Feynman diagrams Physical Principles Observables locality, causality, unitarity, symmetries Models Works well if we have a well-established theory (Standard Model, GR, …) and many observables. The Conventional Approach
- 103. The Conventional Approach Although conceptually straightforward, this has some drawbacks: • It involves unphysical gauge degrees of freedom. • Relation between Lagrangians and observables is many-to-one. • Even at tree level, the computation of cosmological correlators involves complicated time integrals: Hard to compute! • Fundamental principles (e.g. locality and unitarity) are obscured. • To derive nonperturbative correlators and constraints from the UV completion, we need a deeper understanding. N = X Diagrams Y Vertices Z dt ! ✓ External propagators ◆ ✓ Internal propagators ◆ ✓ Vertex factors ◆
- 104. Physical Principles Observables Models In the bootstrap approach, we cut out the middle man and go directly from fundamental principles to physical observables: This is particularly relevant when we have many theories (inflation, BSM, …) and few observables. The Bootstrap Method
- 105. S-matrix Bootstrap • What stable particles exist? fixed by locality and Lorentz symmetry = g2 PJ (cos ✓) s M2 locality unitarity No Lagrangian is required to derive this. Basic principles allow only a small menu of possibilities. • What interactions are allowed? Lorentz = ✓ h12i3 h13ih23i ◆J
- 106. Consistent factorization is very constraining for massless particles: µ1...µS S = { 0 , 1 2 , 1 , 3 2 , 2 } Only consistent for spins YM GR SUSY Benincasa and Cachazo [2007] McGady and Rodina [2013] S-matrix Bootstrap
- 107. The modern amplitudes program has been very successful: 1. New Computational Techniques 2. New Mathematical Structures 3. New Relations Between Theories • Recursion relations • Generalized unitarity • Soft theorems • Positive geometries • Amplituhedrons • Associahedrons • Color-kinematics duality • BCJ double copy 4. New Constraints on QFTs • Positivity bounds • EFThedron 5. New Applications • Gravitational wave physics • Cosmology A Success Story
- 108. Recently, there has been significant progress in bootstrapping correlators using physical consistency conditions on the boundary: symmetries, locality, causality, unitarity, … Cosmological Bootstrap In this part of the lectures, I will review these developments.
- 111. SO(D, 1) : [JAB, JCD] = ⌘BCJAD ⌘ACJBD + ⌘ADJBC ⌘BDJAC ⌘ABXA XB = X2 0 + X2 1 + · · · + X2 D = H 2 D-dimensional de Sitter space can be represented as an hyperboloid in (D+1)-dimensional Minkowski space: de Sitter radius The symmetries of de Sitter then correspond to higher-dimensional Lorentz transformations: In a specific time slicing, we define Dilatations D̂ ⌘ JD0 Pi ⌘ JDi J0i Ki ⌘ JDi + J0i Translations Boosts De Sitter Symmetries Rotations
- 112. De Sitter Symmetries ⌘ABXA XB = X2 0 + X2 1 + · · · + X2 D = H 2 D-dimensional de Sitter space can be represented as an hyperboloid in (D+1)-dimensional Minkowski space: de Sitter radius The symmetries of de Sitter then correspond to higher-dimensional Lorentz transformations: The de Sitter algebra then becomes [Jij, Jkl] = jkJil ikJjl + ilJjk jlJik , [D̂, Pi] = Pi , [Jij, Kl] = jlKi ilKj , [D̂, Ki] = Ki , [Jij, Pl] = jlPi ilPj , [Ki, Pj] = 2 ijD̂ 2Jij . SO(D, 1) : [JAB, JCD] = ⌘BCJAD ⌘ACJBD + ⌘ADJBC ⌘BDJAC
- 113. ds2 = 1 H2⌘2 d⌘2 + dx2 We will work in the flat-slicing, where the de Sitter metric is and the symmetry generators (Killing vectors) become Pi = @i D̂ = ⌘@⌘ xi @i Jij = xi@j xj@i Ki = 2xi⌘@⌘ + ⇣ 2xj xi + (⌘2 x2 ) j i ⌘ @j De Sitter Symmetries • Dilatations and boosts correspond to the following finite transformations: ⌦ ⌘ 1 + 2(b · x) + b2 (x2 ⌘2 ) where ⌘ 7! ⌘ x 7! x ⌘ 7! ⌘/⌦ x 7! ⇥ x + (x2 ⌘2 )b ⇤ /⌦ • At late times, the boosts become special conformal transformations (SCTs). • Late-time correlators are therefore constrained by conformal symmetry.
- 114. Unitarity Representations Physical states are described by unitary representations of the de Sitter group. These representations are labelled by the eigenvalues of the Casimir operators: C2 ⌘ 1 2 JAB JAB = D̂(d D̂) + Pi Ki + 1 2 Jij Jij C4 ⌘ JABJBC JCDJDA Spin Scaling dimension | , Ji Unitarity puts constraints on the allowed scaling dimensions. For scalars, we have 0 1 2 3 4 Re∆ Im∆ Principal series Complementary series
- 115. Bulk Fields Consider a massive scalar field. Its equation of motion can be written as an eigenvalue equation for the quadratic Casimir: Using C2 = (d ) , we get and C2 = ⌘2 ✓ @2 @⌘2 d 1 ⌘ @ @⌘ r2 ◆ = m2 H2 These representations can be realized by bulk fields. 0 1 2 3 4 Re∆ Im∆ Heavy fields: Light fields: ⌘ d 2 r d2 4 m2 H2 m/H > d/2 m/H < d/2 ¯ = d .
- 116. Boundary Fluctuations Our main interest are the fluctuations on the late-time boundary. At late times, the solution for the bulk field is (x, ⌘ ! 0) = (x) ⌘ + ¯(x) ⌘d Boundary field profile For light fields, the first term of the asymptotic solution dominates. The de Sitter isometries then act as Pi = @i Jij = (xi@j xj@i) D = ( + xi @i) Ki = h 2xi + ⇣ 2xj xi x2 j i ⌘ @j i ⌘@⌘ 7! Transformations of a primary operator of weight in a CFT.
- 117. Conformal Ward Identities The boundary correlators are constrained by the conformal symmetry: N X a=1 h 1 · · · a · · · N i = 0 where a stand for any of the field transformations. • Translations and rotations require that the correlator is only a function of the separations |xa xb| . • Dilatation and SCT invariance imply 0 = N X a=1 ✓ a + xj a @ @xj a ◆ h 1 · · · a · · · N i 0 = N X a=1 ✓ 2 axi a + 2xi axj a @ @xj a x2 a @ @xa,i ◆ h 1 · · · a · · · N i
- 118. Conformal Ward Identities 0 = N X a=1 ✓ a + xj a @ @xj a ◆ h 1 · · · a · · · N i 0 = N X a=1 ✓ 2 axi a + 2xi axj a @ @xj a x2 a @ @xa,i ◆ h 1 · · · a · · · N i Exercise: Show that for two- and three-point functions the conformal Ward identities are solved by h 1 2i = 1 x2 1 12 1, 2 h 1 2 3i = c123 x t 2 3 12 x t 2 1 23 x t 2 2 31 where a ⌘ a(xa) , xab ⌘ |xa xb| and t ⌘ X a .
- 119. N X a=1 Z dd x1 · · · dd xN N (x) (x1) · · · (xa) · · · (xN ) Conformal Ward Identities Similar Ward identities can be derived for the wavefunction coeﬃcients. Consider [ ] = exp ✓ · · · + Z dd x1 · · · dd xN N (x) (x1) · · · (xN ) + · · · ◆ N X a=1 Z dd x1 · · · dd xN N (x) (x1) · · · ⇣ a xj a@xj a ⌘ (xa) · · · (xN ) = N X a=1 Z dd x1 · · · dd xN h⇣ (d a) + xj a@xj a ⌘ N (x) i (x1) · · · (xN ) Integrate by parts, so that the generator of the transformation acts on the wavefunction coeﬃcient. For example, for the dilatation, we have
- 120. Conformal Ward Identities Under dilatations, the wavefunction coeﬃcients therefore transform as N (x) = N X a=1 ⇣ (d a) + xj a@xj a ⌘ N (x) A similar derivation applies to SCTs. Imposing N = 0 leads to the conformal Ward identity for WF coeﬃcients: 0 = N X a=1 ✓ (d a) + xj a @ @xj a ◆ hO1 · · · Oa · · · ON i 0 = N X a=1 ✓ 2(d a)xi a + 2xi axj a @ @xj a x2 a @ @xa,i ◆ hO1 · · · Oa · · · ON i N ⌘ hO1 · · · ON i where . Same as before, except the dual operators have weight ¯ a = d a .
- 121. Conformal Ward Identities In cosmology, we need these Ward identities in Fourier space. Although the correlators themselves are hard to Fourier transform, the symmetry generators are not. This gives Since hO1 · · · ON i = (2⇡)d (d) (k1 + · · · + kN ) hO1 · · · ON i0 attention to the momentum derivatives hitting the delta function. , we need to pay This gives an extra term in the dilatation Ward identity, so that " d + N X a=1 D̂a # hO1 · · · ON i0 (k) = 0 N X a=1 Ki ahO1 · · · ON i0 (k) = 0 D̂Ok = h + ki @ki i Ok KiOk = h 2 @ki ki@kj @kj + 2kj @kj @ki i Ok
- 122. t d(N 1) t d ⇥ N t ⌘ P a a Conformal Ward Identities " d + N X a=1 D̂a # hO1 · · · ON i0 (k) = 0 N X a=1 Ki ahO1 · · · ON i0 (k) = 0 • An N-point function has scaling dimension in position space. • The momentum-space correlator then has dimension . • After stripping oﬀ the delta function, the dimension is . This fixes the overall momentum scaling of the correlator, allowing us to make an ansatz that automatically solves the dilatation Ward identity. All the juice is then in the SCT Ward identity.
- 123. ( = 3) hO1O2i0 = Ak2 1 d 1 1, 2 1 = 2 hO1O2i0 / k 1+ 2 d 1 Two-Point Functions Consider the two-point function of two arbitrary scalar operators. • The SCT Ward identity then forces , and hence • A massless field in d = 3 dimensions has h 1 2i0 = 1 2 RehO1O2i0 / 1 k3 as expected for a scale-invariant power spectrum. • An ansatz that solves the dilatation Ward identity is
- 124. hO1O2O3i0 = k 1+ 2+ 3 2d 3 Ĝ(p, q) Consider the three-point function of three arbitrary scalar operators. • An ansatz that solves the dilatation Ward identity is Three-Point Functions where p ⌘ k1/k3 and q ⌘ k2/k3. • The SCT Ward identity then implies two diﬀerential equations for which are solved by Ĝ(p, q), hO1O2O3i0 = Appell F4 / Z 1 0 dx x d 2 1 K⌫1 (k1x)K⌫2 (k2x)K⌫3 (k3x)
- 125. where u ⌘ k3/(k1 + k2) and w ⌘ k3/(k1 k2). hO1O2X3i0 = k 2 3 Ĝ(u, w) Three-Point Functions It will be useful to work out more explicitly the case of two conformally coupled scalars and a generic scalar in 3 dimensions: where u ⌘ u2 (1 u2 )@2 u 2u3 @u and µ2 + 1 4 = ( 2)( 1). Ĝ(u, w) = Ĝ(u), with • The SCT Ward identity then implies • This is solved by u + ✓ µ2 + 1 4 ◆ Ĝ(u) = 0 Ĝ+(u) / P0 iµ 1 2 (u 1 ) Ĝ (u) / P0 iµ 1 2 ( u 1 ) and . associated Legendre
- 126. Four-Point Functions Finally, we consider the four-point function of conformally coupled scalars: hO1O2O3O4i0 = kI k1 k2 k4 k3 = 1 kI F̂(u, v) where u ⌘ kI/(k1 + k2) and v ⌘ kI/(k3 + k4). • The SCT Ward identity then implies where u ⌘ u2 (1 u2 )@2 u 2u3 @u. • To solve this equation requires additional physical input. Arkani-Hamed, DB, Lee and Pimentel [2018]
- 128. Much of the physics of scattering amplitudes is fixed by their singularities. Singularities also play an important role in constraining the structure of cosmological correlators.
- 129. = g2 (k12 + k34)(k12 + kI)(k34 + kI) ⌘ g2 EELER = ig Z 0 1 dt eiEt = g E Flat-Space Wavefunction In the last lecture, we computed wavefunction coeﬃcients in flat space. Let us look back at some of these results. Correlators are singular when energy is conserved. = g2 Z 0 1 dt1dt2 eik12t1 GkI (t1, t2) eik34t2
- 130. Total Energy Singularity Every correlator has a singularity at vanishing total energy: • The residue of the total energy singularity is the corresponding amplitude. Raju [2012] Maldacena and Pimentel [2011] N = AN = ⇠ Z 0 1 dt eiEt f(t) = AN E + · · · ⇠ Z 1 1 dt eiEt f(t) = MN (E)
- 131. Total Energy Singularity • This is easy to confirm for the tree-exchange diagram: lim E!0 g2 EELER = 1 E g2 (k12 + kI)( k12 + kI) = 1 E g2 s = A4 E • The total energy singularity arises from the early-time limit of the integration (where the boundary is infinitely far away): where s ⌘ k2 12 k2 I is the Mandelstam invariant. N ⇠ Z 0 1 dt eiEt f(t) = AN E + · · · This explains intuitively why this singularity reproduces flat-space physics. • The singularity can also be seen in the flat-space recursion relation for the wavefunction. Arkani-Hamed, Benincasa and Postnikov [2017]
- 132. = AL ⇥ ˜R EL + · · · ˜R ⌘ 1 2kI ⇣ R(k34 kI) R(k34 + kI) ⌘ Partial Energy Singularities Exchange diagrams lead to additional singularities: lim EL!0 • On the pole, the correlator factorizes into a product of an amplitude and a shifted correlator: • The singularity arises from the early-time limit of the integration, where the bulk-to-bulk propagator factorizes: GkI (tL, tR) tL! 1 ! eikI tL 2kI ⇣ e ikI tR e+ikI tR ⌘
- 133. Partial Energy Singularities This pattern generalizes to arbitrary graphs. = AL ⇥ ˜R EL + · · · Arkani-Hamed, Benincasa and Postnikov [2017] DB, Chen, Duaso Pueyo, Joyce, Lee and Pimentel [2021] Salcedo, Lee, Melville and Pajer [2022] • The wavefunction is singular when the energy flowing into any connected subgraph vanishes: lim EL!0 • For loops, the poles can become branch points.
- 134. Generalization to De Sitter The wavefunction in de Sitter also has singularities when the energy flowing into any connected subgraphs vanishes: • These singularities again arise from the early-time limit of the integration, where the bulk-to-bulk propagator factorizes. • In these singular limits, the correlator factorizes into a product of an amplitude and a shifted correlator: ˜R ⌘ P(kI) ⇣ R(ka, kI) R(ka, kI) ⌘ Power spectrum of exchanged field lim EL!0 = AL ⇥ ˜R EP L + · · ·
- 135. Singularities as Input 1) Ward identities 2) Unitarity The singularities of the wavefunction only arise at unphysical kinematics. Nevertheless, they control the structure of cosmological correlators. • Amplitudes are the building blocks of cosmological correlators: = (EL)p (ER)q • To construct the full correlator, we need to connect its singularities. We will present two ways of doing this:
- 137. Unitarity is a fundamental feature of quantum mechanics. However, the constraints of bulk unitary on cosmological correlators were understood only very recently. Goodhew, Jazayeri and Pajer [2020] Meltzer and Sivaramakrishnan [2020]
- 138. T T† = iT† T S-matrix Optical Theorem A consistent theory must have a unitary S-matrix: Writing S = 1 + iT, this implies Sandwiching this between the states hf| and |ii, and inserting a complete set of states, we get S† S = 1 hf|T|ii hi|T|fi⇤ = i X X Z d⇧X hf|T† |XihX|T|ii Using hf|T|ii ⌘ (2⇡)4 (pi pf )A(i ! f), we obtain 2Im A(i ! f) = X X Z d⇧X (2⇡)4 (pi pX ) A(i ! X)A⇤ (f ! X) The right-hand side can be written in terms of the total cross section, giving the familiar form of the optical theorem.
- 139. Writing Û = 1 + V̂ , this implies Cosmological Optical Theorem Goodhew, Jazayeri and Pajer [2020] Recently, an analogous result was derived for cosmological correlators. Recall that [ ] = h |Û|0i, where Û = T exp ✓ i Z ⌘⇤ 1 d⌘ Ĥint ◆ Probabilities, | (⌘)|2 , are conserved if U† U = 1. V̂ + V̂ † = V̂ † V̂ Sandwiching this between and inserting a complete set of states, we get h | and |0i, h |V̂ |0i + h |V̂ † |0i = Z DX h |V̂ † |XihX|V̂ |0i Cosmological Optical Theorem To use this, we need to relate the matrix elements to wavefunction coeﬃcients.
- 140. Example: Four-Point Exchange As a concrete example, we consider the four-point exchange diagram: To extract a constraint on the WF coeﬃcient from the COT, we let be the | i h{ka}|V̂ |0i + h{ka}|V̂ † |0i = X X Z Y b2X d3 qb (2⇡)3 h{ka}|V̂ † |{qb}ih{qb}|V̂ |0i where V̂ = i Z ⌘⇤ 1 d⌘ Ĥint(⌘) Z ⌘⇤ 1 d⌘ Z ⌘ 1 d⌘0 Ĥint(⌘)Ĥint(⌘0 ) + · · · . momentum eigenstate |k1, k2, k3, k4i ⌘ |{ka}i: Our task is to relate the matrix elements in this expression to WF coeﬃcients. 2
- 141. Example: Four-Point Exchange h{ka}|V̂ |0i + h{ka}|V̂ † |0i = X X Z Y b2X d3 qb (2⇡)3 h{ka}|V̂ † |{qb}ih{qb}|V̂ |0i • Substituting the mode expansions ˆk(⌘) = f⇤ k (⌘) âk + fk(⌘) â k ˆk(⌘) = ⇤ k(⌘) b̂k + k(⌘) b̂ k the ﬁrst term on the left-hand side becomes h{ka}|V̂ |0i = g2 Z d⌘d⌘0 a4 (⌘)a4 (⌘0 ) GF (kI; ⌘, ⌘0 ) 4 Y a=1 fka (⌘) where GF is the Feynman propagator.
- 142. Example: Four-Point Exchange h{ka}|V̂ |0i + h{ka}|V̂ † |0i = X X Z Y b2X d3 qb (2⇡)3 h{ka}|V̂ † |{qb}ih{qb}|V̂ |0i GF (k; ⌘, ⌘0 ) = G(k; ⌘, ⌘0 ) + ⇤ k(⌘⇤) k(⌘⇤) k(⌘) k(⌘0 ) we get h{ka}|V̂ |0i = ⇣ 4(ka, kI) 3(k12, kI) 3(k34, kI) P (kI) ⌘ 4 Y a=1 fka (⌘⇤) ! . h{ka}|V̂ † |0i = ⇣⇥ 4( ka, kI) ⇤⇤ 3(k12, kI) 3(k34, kI) P (kI) ⌘ 4 Y a=1 fka (⌘⇤) ! • A similar analysis for the second term leads to Analytic continuation Substituting
- 143. Example: Four-Point Exchange h{ka}|V̂ |0i + h{ka}|V̂ † |0i = X X Z Y b2X d3 qb (2⇡)3 h{ka}|V̂ † |{qb}ih{qb}|V̂ |0i • Finally, the third term leads to RHS = h 3(k12, kI) 3(k34, kI) + 3(k12, kI) 3(k34, kI) i P (kI) 4 Y a=1 fka (⌘⇤) ! • Putting it all together, we get 4(ka, kI) + ⇥ 4( ka, kI) ⇤⇤ = P (kI) ˜3(k12, kI) ˜3(k34, kI) where ˜3(k12, kI) ⌘ 3(k12, kI) 3(k12, kI).
- 144. Cutting Rules For more complicated graphs, the explicit evaluation of the COT quickly becomes cumbersome. A simpler alternative are cutting rules. We start with some analytic properties of the propagators defining the perturbative wavefunction: • A useful concept is the discontinuity of a function • Both propagators decay at past infinity, which we can enforce by giving the norm k a negative imaginary part: k2 = rei✓ , with ✓ 2 ( 2⇡, 0). Disc[f(k)] ⌘ f(k) f(e i⇡ k) • The discontinuity of the bulk-to-bulk propagator is Disc[G(k; ⌘, ⌘0 )] = P(k) Disc[K(k, ⌘)] ⇥ Disc[K(k, ⌘0 )]
- 145. DisckI ⇥ 4(ka, kI) ⇤ = Cutting Rules Consider, the four-point exchange diagram 4(ka, kI) = Taking the discontinuity (with respect to ) corresponds to cutting the internal line: kI P (k) Hence, we find DisckI ⇥ 4(ka, kI) ⇤ = P (kI) DisckI ⇥ 3(k12, kI) ⇤ ⇥ DisckI ⇥ 3(k34, kI) ⇤ which is equivalent to the result derived from the COT.
- 146. Cutting Rules This procedure generalizes to systematic cutting rules: These are the analog of the Cutkosky cutting rules for the S-matrix. Disc ⇥ N ⇤ = X all possible cuts More details can be found in the lecture notes and in the relevant literature. Goodhew, Jazayeri and Pajer [2020] Meltzer and Sivaramakrishnan [2020] Caspedes, Davis and Melville [2020] Goodhew, Jazayeri, Lee and Pajer [2021] DB, Chen, Duaso Pueyo, Joyce, Lee and Pimentel [2021] Melville and Pajer [2021] Sleight and Taronna [2021] Jazayeri, Pajer and Stefanyszyn [2021] Meltzer [2021] Benincasa [2022]
- 147. Unitarity as Input • The discontinuity of a WF coeﬃcient factorizes into lower-point objects. • This contains slightly more information that the energy singularities: • These unitarity cuts have become an important bootstrapping tool. lim EL!0 Disc = lim EL!0 h N (ka, kI) N (ka, kI) i = lim EL!0 N (ka, kI) has no singularities in EL The cutting rule fixes the entire Laurent series in EL.
- 150. Exchange Four-Point Function We are now ready to return to the challenge of massive particle exchange. Conformally coupled scalars Massive scalar = g2 Z d⌘ ⌘2 Z d⌘0 ⌘02 eik12⌘ eik34⌘0 G(kI; ⌘, ⌘00 ) hO1O2O3O4i0 = In general, the time integrals cannot be performed analytically.
- 151. Conformal Ward Identity Recall that the s-channel correlator can be written as hO1O2O3O4i0 = kI k1 k2 k4 k3 = 1 kI F̂(u, v) where u ⌘ kI/(k1 + k2) and v ⌘ kI/(k3 + k4). We have seen that the conformal symmetry implies the following diﬀerential equation where u ⌘ u2 (1 u2 )@2 u 2u3 @u. We will classify solutions to this equation by their singularities.
- 152. = X n cn(u, v) E2n+1 Contact Solutions The simplest solutions correspond to contact interactions: F̂c = Only total energy singularities F̂c,0 = kI E = uv u + v 4 • For , we have . • Higher-derivative bulk interactions, lead to F̂c,n = n uF̂c,0 . F̂c,1 = 2 ✓ kI E ◆3 1 + uv uv , F̂c,2 = 4 ✓ kI E ◆5 u2 + v2 + uv(3u2 + 3v2 4) 6(uv)2 6(uv)3 (uv)3 . Fixed by symmetry This can get relatively complex, but everything is fixed by symmetry.
- 153. Exchange Solutions For tree exchange, we try ( u + M2 )F̂e = Ĉ ( v + M2 )F̂e = Ĉ = For amplitudes, the analog of this is This is an algebraic relation, while for correlators we have a diﬀerential equation. =
- 154. Exchange Solutions Using the simplest contact interaction as a source, we have Before solving this equation, let us look at its singularities. • Flat-space limit: • Folded limit: This singularity should be absent in the Bunch-Davies vacuum. • Factorization limit: • Collapsed limit: This non-analyticity is a key signature of particle production.
- 155. Exchange Solutions Using the simplest contact interaction as a source, we have Before solving this equation, let us look at its singularities. • Flat-space limit: • Folded limit: • Factorization limit: • Collapsed limit: Factorization and folded limits uniquely fix the solution. The remaining singularities become consistency checks.
- 156. whose solution is particle production EFT expansion analytic non-analytic F̂ = X n ( 1)n (n + 1 2 )2 + M2 ⇣u v ⌘n+1 + ⇡ cosh(⇡M) sin(M log(u/v)) M The general solution takes a similar form. Arkani-Hamed, DB, Lee and Pimentel [2018] t ⌘ log(u/v) Forced Harmonic Oscillator
- 157. The correlator is forced to have an oscillatory feature in the physical regime: BRANCH CUT OSCILLATIONS COLLAPSED LIMIT Oscillations
- 158. These oscillations reflect the evolution of the massive particles during inflation: Time-dependent eﬀects have emerged in the solution of the time-independent bootstrap constraints. Particle Production
- 159. Centre-of-mass energy (GeV) Cross section (nb) Four-point function Four-point function Momentum ratio Correlation strength The oscillatory feature is the analog of a resonance in collider physics: Cosmological Collider Physics
- 160. A = g2 s M2 PS(cos ✓) The angular dependence of the signal depends on the spin of the particles. This is very similar to what we do in collider physics: ✓ kI / sin[M log kI ] PS(cos ✓) The frequency of the oscillations depends on the mass of the particles: Particle Spectroscopy
- 162. We have seen that the discontinuity of a wavefunction coeﬃcient is related to lower-point objects: For rational correlators, we can “integrate” the discontinuity to get the full correlator.
- 163. Dispersive Integral By causality, the bulk-to-bulk propagator is an analytic function of k2 that decays at large k. It can therefore be written as G(k; ⌘, ⌘0 ) = 1 2⇡i Z 1 0 dq2 q2 k2 + i" Disc[G(k; ⌘, ⌘0 )] This can be used to integrate the discontinuities of tree-level WF coeﬃcients. As an example, we consider the four-point exchange diagram: Meltzer [2021] Often this dispersive integral is easier to compute than the bulk time integrals. = 1 2⇡i Z 1 0 dq2 q2 k2 I + i" Discq
- 164. kI k34 k12 q Using the unitarity cut, this becomes = 1 2⇡i Z 1 0 dq2 q2 k2 I + i" P(q) Discq Discq Dispersive Integral The full correlator can then be written as a sum over residues, using only three-point data.
- 165. Conformally Coupled Scalar As an example, let us reproduce the result for conformally coupled scalars. = i log(k1 + k2 + k3) Discq • Input: three-point function • Discontinuity: = 1 2q log ✓ k12 q k12 + q ◆ log ✓ k34 q k34 + q ◆ = P(q) Discq Discq = 4q Z 1 k12 dx x2 q2 Z 1 k34 dy y2 q2
- 166. Conformally Coupled Scalar As an example, let us reproduce the result for conformally coupled scalars. • Dispersive integral = Z 1 k12 dx Z 1 k34 dy Z 1 1 dq ⇡i q2 q2 k2 I 1 x2 q2 1 y2 q2 = Z 1 k12 dx Z 1 k34 dy 1 (x + y)(x + kI)(y + kI) kI x y q Correct!
- 167. EFT of Inflation One of the important features of using unitarity cuts to bootstrap correlators is that it doesn't rely the interactions being approximately boost-invariant. Exercise: Consider ˙ ⇡3 in the EFT of inflation. • The three-point wavefunction coeﬃcient is = (k1k2k3)2 (k1 + k2 + k3)3 • Using the dispersive integral, show that the exchange 4pt function is = (k1k2k3k4)2 E5E3 LE3 R k2 I ⇣ 6E2 LE2 R + 3EELER(EL + ER) + E2 (EL + ER)2 + E3 (EL + ER) + E4 ⌘ 6E3 LE3 R
- 169. We have seen that cosmological correlators satisfy interesting diﬀerential equations in kinematic space: u2 (1 u2 )@2 u 2u3 @u + ✓ µ2 + 1 4 ◆ = We derived these equations from conformal symmetry, but is there a deeper reason for their existence?
- 170. In an upcoming paper, we will answer this question for conformal scalars in a power-law FRW background. What classes of correlation functions can arise from consistent cosmological evolution? Surprisingly little is know about this in a systematic fashion. Arkani-Hamed, DB, Hillman, Joyce, Lee and Pimentel [to appear]
- 171. Conformal Scalars in dS We have seen that the four-point function of conformal scalars in de Sitter space can be written as an integral of the flat-space result: where and .
- 172. Conformal Scalars in FRW An interesting generalization of this integral is Twist This describes the four-point function of conformal scalars in FRW backgrounds with scalar factor Similar integrals arise for loop amplitudes in dimensional regularization. In that case, powerful mathematics can be used to evaluate these integrals: We will show that a similar approach applies to our FRW correlators. • Twisted cohomology • Canonical diﬀerential equations
- 173. Family of Integrals To attack the problem, we introduce a family of integrals with the same singularities (divisors): where • Not all of these integrals are independent because of integration-by- parts and partial fraction identities. • How many independent master integrals are there? The answer is provided by a beautiful fact of twisted cohomology. Aomoto [1975] Mastrolia and Mizera [2018]
- 174. A Beautiful Fact The number of independent master integrals is equal to the number of bounded regions defined by the divisors of the integrand. x1 x2 L1 L2 L3 L4 L5 1 2 3 4
- 175. Master Integrals The vector space of independent master integrals is four-dimensional: Canonical forms The basis integrals are not unique, but a convenience choice is
- 176. Master Integrals The vector space of independent master integrals is four-dimensional: Canonical forms The basis integrals are not unique, but a convenience choice is
- 177. Divide and Conquer Our original integral of interest is i.e. it is a sum of all basis integrals. Each basis integral is easier to evaluate! Key idea: The basis integral forms a finite dimensional vector space, so taking derivatives with respect to must lead to coupled diﬀerential equations: Exterior derivative Solving this diﬀerential equation turns out to be easy.
- 178. Diﬀerential Equation Explicitly, we find with
- 179. The Solution More abstractly, we have This leads to two types of functions: decoupled coupled Power laws Hypergeometric In the de Sitter limit, these become the logs and dialogs we found earlier.
- 180. The Solution Imposing appropriate boundary conditions (i.e. singularities), the solution for our original integral is Arkani-Hamed, DB, Hillman, Joyce, Lee and Pimentel [to appear]
- 181. Space of Functions The number of special functions depends on the number of bulk vertices, and the function have an interesting iterative structure: Power law This structure is hidden in bulk perturbation theory. Just the beginning of a systematic exploration of these mathematical structures living inside cosmological correlators. Arkani-Hamed, DB, Hillman, Joyce, Lee and Pimentel [to appear]
- 183. Despite enormous progress in recent years, we are only at the beginning of a systematic exploration of cosmological correlators: under- standing Many fundamental questions still remain unanswered. tree level De Sitter nonperturbative Anti-de Sitter tree level (some loops) Minkowski
- 184. So far, the cosmological bootstrap has only been applied to individual Feynman diagrams (which by themselves are unphysical). However, in the S-matrix bootstrap, the real magic is found for on-shell diagrams: What is the on-shell formulation of the cosmological bootstrap? What magic will it reveal? • Recursion relations • Generalized unitarity • Color-kinematics duality • Hidden positivity • … Beyond Feynman Diagrams
- 185. So far, the bootstrap constraints (singularities, unitarity cuts, etc.) are only formulated in perturbation theory and implemented only at tree level. In contrast, we understand the conformal bootstrap nonperturbatively: Recently, some preliminary progress was made to apply unitarity constraints in de Sitter nonperturbatively, but much work remains. Hogervorst, Penedones and Vaziri [2021] Di Pietro, Gorbenko and Komatsu [2021] A nonperturbative bootstrap will be required if we want to understand the UV completion of cosmological correlators. Beyond Perturbation Theory
- 186. In quantum gravity, spacetime is an emergent concept. In cosmology, the notation of “time” breaks down at the Big Bang. What replaces it? The cosmological bootstrap provides a modest form of emergent time, but there should be a much more radical approach where space and time are outputs, not inputs. For scattering amplitudes, such a reformulation was achieved in the form of the amplituhedron: What is the analogous object in cosmology? How does the Big Bang arise from it? Beyond Spacetime Arkani-Hamed and Trnka [2013]
- 187. Thank you for your attention! NTU, Taiwan