From Inflation to Large Scale Structure

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1 From Inflation to Large Scale Structure Ido Ben-Dayan DESY Humboldt University, 22 April 2015

2 Outline Inflation and High Energy Physics Inflation and Cosmology after Planck 2015 Status of Inflationary Models and an Organizing Principle Cosmological Perturbation Theory and Beyond Supernovae Lensing as a Cosmological Probe Large Scale Structure and Consistency Relations

3 Outline Inflation and High Energy Physics Inflation and Cosmology after Planck 2015 Status of Inflationary Models and an Organizing Principle Cosmological Perturbation Theory and Beyond Supernovae Lensing as a Cosmological Probe Large Scale Structure and Consistency Relations

4 Inflation after Planck 2015

7 ɛ = 1 ( ) V 2 1, η = V ( ) ns 1 k 2 V V 1, P S = A k 0 φend dφ N e = 60, n s = 1 + 2η 6ɛ, r = P T = 16ɛ 2ɛ P S φ CMB

8 Inflationary parameters after 12 years of observation Vanilla monomial models V = φ p, p 2, r 0.1 are dead. (SUSY Higgs inflation, IBD & Einhorn 2010) WMAP 1-year (2003) & PLANCK (2015)

9 Current Knowledge*

10 Cosmological Parameters after 12 years of observations Basic flat ΛCDM 6 parameters model. Ω m0, Ω c0, H 0, A s, n s, τ. Except for τ, from 10% accuracy to 1%. This is a model dependent statement! Adding parameters increases the allowed values, improves the fit, but with not enough statistical significance.

11 Status of Inflationary Models and an Organizing Principle

12 Status of Inflationary models - EFT approach+uv embedding Inflation is sensitive to UV physics. Organizing Principle: Monge-Ampere eq. Generalized shift symmetry work in progress V(φ i ) > 0, det V ij = 0 Large Field Models, φ 1, r Shift symmetry, detectable in the near future. V = Λ 4 (1 cos(φ/f eff. )), V φ (IBD, Pedro & Westphal 2014 x 2) functional fine-tuning, symmetry is crucial, UV completion is crucial.

13 Status of Inflationary models - EFT approach+uv embedding Inflation is sensitive to UV physics. Starobinsky type Models, φ 1, r Shift symmetry at φ. Could be detected in the near future. (IBD, Jing, Torabian, Westphal, Zarate, 2013). V(φ 1) Λ 4 (1 e κφ ) 2. functional fine-tuning, symmetry is crucial, UV completion is crucial.

14 Status of Inflationary models - EFT approach+uv embedding Inflation is sensitive to UV physics. Small Field Models, φ 1, r 10 4, Accidental Inflation, theoretically motivated, difficult to detect in the near future. (IBD, Brustein & de-alwis 2008, IBD & Brustein 2009* (r tunable), IBD, Jing, Westphal & Wieck 2013) V Λ 4 (1 a 2 φ 2 + ). parameter fine-tuning of dimension 6 operators. η 1.

15 Example: Hierarchical Axions V = Λ 4 1 How to get f eff. 1, if f i 1: [ 1 cos ( )] [ ( )] p 1 φ 1 + p 2 φ 2 +Λ 4 q 2 1 cos 1 φ 1 + q 2 φ 2 f1 f2 f1 f2 V = V(φ 1 + φ 2 ) det V ij = 0. We have a flat direction, corresponding to Λ 1 = 0 or p 1 = q 1, p 2 = q 2. The different models now correspond to different breaking patterns. Λ 1 0, p 2 = 0, p 1 q 1 is the Hierarchical Axions model. p 2 = q 2 (1 + δ) is the KNP model.

16 Example: Hierarchical Axions p 2 = 0 p 2 = 0. Diagonalizing the mass matrix, and integrating out the heavy axion gives: f eff. = f 2 q 1 q 2 p 1 1

17 Hierarchical Axions Summary (IBD, Pedro & Westphal 2014 x 2) Just 2 axions Non-perturbative effects only Least amount of tuning of the input parameters. The trajectory is contained in a very small field domain. Sheds light on the small vs. large field discussion. Combining the model with moduli stabilization in Type IIB string theory. Predictions equivalent to Natural Inflation r 0.05, n s = 0.96.

18 More on String Theory Embedding (IBD, Pedro & Westphal 2014 x 2) In string theory we have many moduli that have to be stabilized to avoid decompactification+inflation This is achieved by creating a hierarchy between different terms in the lagrangian/potential V 0 >> V 1 >> V 2. The same hierarchy needed for the original moduli stabilization gives the hierarchy needed for inflation.

19 What s Next? CMB Polarization experiments: CLASS, soon operational r 0.01, PIXIE r 10 3, PRISM, CoRE+ cosmic variance limited experiment 2030, r Measuring the energy spectrum of CMB, deviations from black body spectrum, hasn t advanced since COBE, PIXIE/PRISM (Chluba, Erickcek, IBD 2012) INFLATION BEYOND CMB! Late time measurements: Weak Lensing, Galaxy correlations, SNe, Strong Lensing, BAO and more. Challenges: 1. Designing efficient probes 2. Accurate theoretical predictions

20 Outline Inflation and High Energy Physics Inflation and Cosmology after Planck 2015 Status of Inflationary Models and an Organizing Principle Cosmological Perturbation Theory and Beyond Supernovae Lensing as a Cosmological Probe Large Scale Structure and Consistency Relations

21 Supernovae Lensing as a Cosmological Probe

22 General Background Light-cone averaging method, luminosity-redshift relation d L z up to second order in perturbation theory. (IBD, Gasperini, Marozzi, Nugier & Veneziano x 4) Valid for arbitrary geometry, non-perturbative statements. UV and IR finite. Inhomogeneities are biasing measurements of background quantities, H 0 H 0 2% just from cosmologically induced peculiar velocities! (IBD, Durrer, Marozzi, Schwarz 2014) Inhomogeneities serve as probes for Cosmology if we can get rid of systematics. d L (z) = 1 + z H 0 z 0 dz Ωm0 (1 + z ) Ω m0

23 SNIa Lensing (IBD & Kalaydzhyan 2014, IBD 2014) Large systematic dispersion of lensing at z 1. σ 2 m ( ) 2 5 π τo dτ ln 10 τ 2 1 τ (0) s dkp Ψ (k, τ 1 )k 2 (τ 1 τ (0) s ) 2 (τ o τ 1 ) 2 τ conformal time, depends on background parameters only! P Ψ depends on fluctuations and background parameters. Ψ is the gravitational potential. Data: σ m (z 1) 0.095( 0.12) at 1(2)σ. The lensing is sensitive to small scales (quasi) non-linear k NL > 1hMpc 1. We need numerical simulations (IBD & Takahashi TBP) and/or analytical predictions.

24 Current Knowledge

25 Are we lucky enough to probe Inflation? If yes, handle on the primordial power spectrum beyond CMB scales! α = 0, β = provide the best statistical significant improvement over ΛCDM, χ 2 = 4.9, to solve the low-l anomaly. ( ) k ns 1+(α/2) ln(k/k 0 )+(β/6)[ln(k/k 0 )] 2 P(k) = A s, α V V k 0 V 2, β V V 2 V 3 P R (k)/a s (α,β)= (0.06,0.06) (0,0.06) (0.03,0.04) (0.02,0.02) (0, 0.02) (0.02,0) (0,0) n s = k (Mpc 1 )

26 DM only results m /z s (mag) σm (mag) σ (α,β)= (0.06,0.06) (0.03,0.04) (0,0.06) (0.02,0.02) (0.02,0) (0, 0.02) (0, 0) z s m ( z s 1) 0.07 DM only (1 upper bound) (mag) Designated numerical simulations Memory of initial conditions is largely erased.

27 Results with baryons Competitive with PLANCK, but UV sensitive!

28 Lensing Dispersion Summary For DM only, the lensing dispersion probes small wave numbers beyond CMB k 1hMpc 1, not better than PLANCK, because of the erasure of initial conditions by the non-linear evolution. The baryons make the fitting formula UV sensitive and swamps initial conditions.this is a problem in weak lensing in general. Better understanding of baryonic processes is key. However, SNe do not suffer from alignment problems. The upcoming surveys are expected to detect the lensing signal!

29 Large Scale Structure (LSS) and Consistency Relations (CR)

30 Consistency Relations Relate (n + 1)- to n-point correlation functions, e.g., bispectrum power spectrum Analytical result in the squeezed limit q 0 Based only on single field inflation and principle of general covariance (unequal time) - Non-perturbative statements. Scrutinize two approaches at equal time & angular averaged: (IBD, Konstandin, Porto, Sagunski 2015) 1. Time flow approach (TF) - transform the fluid eqs. into flow eqs., and apply closure. 2. Curved background approach (VKPR)- locally the long (soft) mode behaves like background curvature, K.

31 Consistency Relations Philosophy: If the approach is of non-perturbative nature, it should definitely fulfill perturbative calculations in the proper limit. For a flat universe (EdS, ΛCDM): At linear order: All approaches agree generalized to all correlators of δ and θ v and general cosmologies At 1-loop order, for l k q: B - bispectrum, P - power spectrum B 1-loop av 111 q 0 [k 2 (β + α k k ) P L (k) + k 4 γ ( P dl l 2 L ) 2 ] (l) l 2 P L (q) ( P dl l 2 L ) (l) l 2

32 Consistency Relations Coefficients α, β, γ at 1-loop order: Approach α β γ SPT TF VKPR Deviation of O ( 10 2) between SPT and VKPR, O(1) between SPT and TF. Both not valid beyond linear order Why is VKPR ("Guestimating" κ P K K=0 = 4/7 η P K K=0 ) so successful?

33 Curved Background Method For spherical perturbations, a long wavelength mode in flat FLRW equals positive curvature: ( ) ( ) H K H δ L, a K a δ L, K 5 3 H2 a 2 δ L, κ = K a 2 H 2 ρ K = ρ(1 + δ L ) Fluid equations η ψ a = Ω ab ψ b + γ abc ψ b ψ c ψ a = {δ, v} Judicious change of variables: Ω ab Ω ab (κ = 0) + 3κ ( ) , The new contribution annihilates the growing mode (1, 1) for all n Very good accuracy! ψ (n) a

34 Conclusions

35 Conclusions CMB and High Energy Physics: The inflationary landscape is well understood. The most promising avenue to connect string theory to measurements. Polarization and Energy Spectrum measurements CMB a relevant probe in the next decade and more. Cosmological Perturbation Theory and Beyond: Lensing of SNIa is useful for cosmology or astrophysics depending on the baryons. Combination of late time probes will test extensions of ΛCDM. Theoretical understanding of LSS behaviour is emerging as the forefront of theoretical cosmology.

36 Simulation Results (IBD & Takahashi 1504.xxxxx) We ran high and low resolution simulations (DM only) with different values of α, β. Table : Our Simulation Setting L(h 1 Mpc) Np 3 k Nyq (hmpc 1 ) m p(h 1 M ) z N r high-resolution , 0.3, 0.6, 1, low-resolution , 0.3, 0.6, 1, (α,β)=(0,0) (α,β)=(0,0.02) (α,β)=(0.02,0) (α,β)=(0.02,0.02) P m (k)*k 2 (Mpc/h) 10 2 (α,β)=(0.03,0.04) (α,β)=(0,0.06) (α,β)=(0.06,0.06) high resolution sim. low resolution sim. modified halofit model z=0, 0.3, 0.6, 1, 1.5 from top to bottom gray symbols are results for (α,β)=(0,0) k (h/mpc)

37 Fluid Equations Conservation of mass = Continuity equation δ τ + [(1 + δ) v ] = 0 (1) Conservation of momentum = Euler equation v +H v+(v ) v = Φ (2) τ Poisson equation Φ = 3 2 Ω mh 2 δ (3) Physical quantities: Density contrast: δ = ρ ρ ρ Conformal time τ: dτ = dt a Peculiar velocity v Expansion rate: H = a H Gravitational potential Φ (or Ψ) highly non-linear differential equations

38 Standard Perturbation Theory - SPT Central quantity: δ Assumption: initial density contrast δ 0 1 Solution: perturbative expansion in terms of δ 0 Power spectrum P(k), δ = δ (1) + δ (2) +... with δ (N) δ N 0 generally P ab, bispectrum B abc etc. a, b, c = δ or θ v = Gaussian average of density contrasts P(k) δδ Perturbative (loop) expansion: P(k) δ (1) δ (1) +... = P (1) +... with P (N) (δ 2 0 )N

39 Time-flow approach Gaussian i.c., solving iteratively with closure Correlation functions: ψ a ψ b P ab, ψ a ψ b ψ c B abc, ψ a ψ b ψ c ψ d P ab P cd + P ac P bd + P ad P bc + Q abcd Closure approximation: Q abcd = 0 B abc = g ad g be g cf B abc (η = 0) + 2 η 0 dη e η g ad g be g cf [ γ dgh P eg P fh + γ egh P fg P dh + γ fgh P dg P eh ] Linear propagator g ab : describes η-evolution of linear perturbations depends via Ω ab on cosmological model

40 Bispectrum consistency relations For a flat universe (EdS, ΛCDM): P ab (k) u a u b P L (k) with u a = (1, 1) At linear order: ( ( 1 B L abc av q 0 u b ) ac 1 3 u au c k k )P L (k) P L (q) B L 111 av q 0 δδδ av q 0 : Coincides with SPT. Reproduces KPRV relation for P(k) P L (k) δδδ av q 0 = ( k k [Kehagias, Perrier, Riotto 13], [Valageas 13] ) P(k) P L (q)

41 Bispectrum consistency relations For general cosmological models and all correlations of δ and θ B abc av q 0 = 1 3 η dη e η g be 0 ( [ {g af 3 g c1 Pe1 (q) P f 2 (k) + P e2 (q) P f 1 (k) ] ) + 2 g c2 P e2 (q) P f 2 (k) + g cf ([3 g a1 P e1 (q) g a2 P e2 (q)] P f 2 (k) } 2 g ad P e2 (q) k k P fd (k)

42 Bispectrum consistency relations For a flat universe (EdS, ΛCDM): At 1-loop order: B 1-loop av 111 q 0 [k 2 (α + β k k ) P L (k) for l k q + k 4 γ ( P dl l 2 L ) 2 ] (l) l 2 P L (q) ( P dl l 2 L ) (l) Coefficients α, β, γ: Deviation of O ( 10 2) between SPT and KPRV KPRV relation not valid beyond linear order l 2

43 Curved Background Method Non perturbative, but unmeasurable CR: Baldauf et al B(k, q, q k, η) av q 0 P L (q, η) [(1 13 k k 13 ) η P(k, η) VKPR : κ P K (k, η) = 4 K=0 7 ηp K=0(k, η). ] κ P K (k, η), K=0 Fluid equations η ψ a = Ω ab ψ b + γ abc ψ b ψ c Judicious change of variables: Ω ab Ω ab (κ = 0) + 3κ ( ), The new contribution annihilates the growing mode (1, 1) for all n Very good accuracy. ψ (n) a

Planck was conceived to confirm the robustness of the ΛCDM concordance model when the relevant quantities are measured with much higher accuracy

12-14 April 2006, Rome, Italy Francesco Melchiorri Memorial Conference Planck was conceived to confirm the robustness of the ΛCDM concordance model when the relevant quantities are measured with much higher