CMB quadrupole depression from early fast-roll inflation:

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1 CMB quadrupole depression from early fast-roll inflation: MCMC analysis of current experimental data Claudio Destri Dipartimento di Fisica G. Occhialini Università Milano Bicocca Physics of the Universe: Implications of the Recent Observations, Universidad Complutense Madrid May 2008

2 Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

3 Observational data and models On cosmic variance On the CMB quadrupole (small) value Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

4 Observational data and models On cosmic variance On the CMB quadrupole (small) value Central objects: the CMB TT, TE, EE, (BB) correlation multipoles C l The framework Observational data = C (data) l with much systematics and complex likelihood code; Theoretical cosmological model = C (model) l fully independent and chi-square distributed if primordial fluctuations are gaussian; Markov Chain Monte Carlo analysis to assign likelihood to model parameters (see later)

5 Observational data and models On cosmic variance On the CMB quadrupole (small) value Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

6 Observational data and models On cosmic variance On the CMB quadrupole (small) value Neglecting all uncertainities but cosmic variance: Let x = 1/y = C (data) l /C (model) l ; then p l (x model) 1 x (xe x ) l+1/2 (chi-square distribution) sets the probability density for C (data) l x = 1 and (x) ML = 2l 1 2l + 1 ; p l (y data) ( e 1/y /y ) l+1/2 given the model, with sets the probability density for C (model l given the data (assuming flat priors), with y = 2l + 1 2l 3 and (y) ML = 1.

7 Observational data and models On cosmic variance On the CMB quadrupole (small) value An example: lowest 9 multipoles probability curves from best fit ΛCDM WMAP5 data Prob[C (data) l < C (model) l ] =

8 Observational data and models On cosmic variance On the CMB quadrupole (small) value Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

9 Observational data and models On cosmic variance On the CMB quadrupole (small) value 1σ uncertainities based on p l (y data) from M. R. Nolta et al., arxiv: [astro-ph] 5 Mar 2008

10 Unbiased(?) estimation Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value Probability that at least one multipole, regardless of l, is smaller than x times its theoretical expectation value: [ l max ] x P( x,l max ) = 1 1 dx p(x model) If model=λcdm, then l= P(0.184,l max ) =

11 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

12 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) EFT of (single field) inflation (à la Boyanowski de Vega Sanchez) Standard units: ħ = 1, c = 1, M PL = GeV Inflaton potential V(φ) = N M 4 w(χ), χ = φ N MPL, w(χ) O(1), N 50 Energy scale of inflation and inflaton mass M GeV M GUT, m = M 2 /M PL GeV Number of efolds kept fixed in MCMC analysis N = log a(t end) a(t exit ), w(χ end) = w (χ end ) = 0 t exit : the mode with comoving k 0 becomes superhorizon Understanding start and end of inflation N

13 Dimensionless setup Prologue EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Stretched time and dimensionless Hubble parameter τ = t M2 M PL N, H = H M Pl = O(1) N M 2 Equations of motion [ H 2 (τ) = N χ2 + w(χ) ] 1, N χ + 3 H χ + w (χ) = 0 Energy density and pressure ρ N M 4 = 1 2 N χ2 + w(χ), p N M 4 = 1 2 N χ2 w(χ) From fast-roll to slow-roll 1 2 N χ2 w(χ), 1 2 N χ2 w(χ)

14 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

15 Scalar fluctuations Prologue EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Gauge invariant quantum perturbation field u(x, t) = ξ(t) R(x, t) = [ ] d 3 k α (2π) 3/2 k S k (η)e ik x + α k S k (η)e ik x ] [α k, α k = δ (3) (k k ), ξ(t) = a(t) φ(t) H(t), η = dt a(t) Schroedinger like dynamics [ ] d 2 k 2 + U(η) S dη 2 k (η) = 0, U(η) = 1 ξ d 2 ξ dη 2 Standard parametrization in dimensionless setup U(η) = a 2 H 2 m 2 N [2 7 ǫ v + 2ǫ 2v 8ǫ v w N ǫ v = 1 2N ( ) 2 1 dχ H 2 dτ, ηv = 1 N ] η H 2 v (3 ǫ v ) w (χ) w(χ)

16 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Slow roll approximation ǫ v = 1 2N [ w (χ exit ) w(χ exit ) ] 2 + O ( 1 N 2 ), ηv = 1 N w (χ exit ) w(χ exit ) + O ( 1 N 2 ) Fast roll correction U(η) = ν2 1 4 η 2, ν = ǫ v η v U(η) = ν2 1 4 η 2 + V(η) for Binomial New Inflation, w(χ) = y 32 ( χ 2 8 y ) 2 and y = 1.32

17 Fast roll = special initial conditions for slow roll EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Solution when V(η) = 0 S k (η) = A(k)g ν (k; η) + B(k)g ν(k; η), g ν (k; η) = 1 2 iν+ 1 2 Scattering amplitudes (or Bogoliubov coefficients) Primordial power spectrum A(k) = 1 + i 0 dη g ν(k; η) V(η) S k (η) B(k) = i 0 dη g ν(k; η) V(η) S k (η) P(k) = lim η 0 k 3 2π 2 Fast-roll transfer function S k (η) ξ(η) 2 [ ] = P sr (k) 1 + D(k) D(k) = 2 B(k) 2 2Re [ A(k)B (k) i 2ν 3] πη H (1) ν ( k η)

18 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

19 The transfer function D(k) for BNI EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) Born s approximation [ ( ) ] D(k) = 1 0 k dη V(η) sin(2 kη) k 2 η 2 kη cos(2kη) For Binomial New Inflation: Scaling properties and the crossover wavenumber k 1 ( ) V(η) = k1 2 Q(k k 1 η), D(k) = F k 1

20 Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

21 The setup Prologue Observational data = likelihood on C (model) l ; Model with cosmological parameters λ = {λ 1,λ 2,...λ n } = C (model) l (λ) (CMBFAST, CAMB,...); = likelihood L(λ) The MCMC method produces sequences distributed as L(λ) ( the prior probability), through an acceptance/rejection one-step algorithm (e.g. Metropolis) W(λ (k+1),λ (k) ) = g(λ (k+1),λ (k) ) min { 1, L(λ(k+1) ) g(λ (k+1),λ (k) ) L(λ (k) ) g(λ (k),λ (k+1) ) }

22 CosmoMC Antony Lewis, Sarah Bridle - Publicly available F90 open source code Comes with CMB and LSS data (except WMAP) Interfaces directly with the WMAP1-3-5 likelihood software Theoretical calculations with CAMB (derived from CMBFAST) Quite accurate, easy to use, fairly well documented Comes with tool for analyzing results and useful MATLAB scripts Not very fast, but runs well on clusters poorly modular, too many globals

23 The ΛCDM+r model Prologue MCMC parameters: Context: Datasets: param best fit 100Ω b h Ω c h θ τ 8.87 H σ log[10 10 A s ] n s r log(l) ω b, ω c, θ, τ, (slow), A s, n s, r (fast) Ω ν = 0,... ; standard priors, no SZ, no lensing, linear mpk,... WMAP5, ACBAR08, SDSS Ω b h Ω c h θ Ω Λ Age/GYr τ σ z re H 0 log( A ) n s s r

24 The n s r plane and Binomial New Inflation n s = 1 y N 3 z+1, r = 16 y z (1 z) 2 N, y = z 1 log z, 0 < z < 1 (1 z) 2

25 The n s r plane and Binomial New Inflation n s = 1 y N 3 z+1, r = 16 y z (1 z) 2 N, y = z 1 log z, 0 < z < 1 (1 z) WMAP3,SDSS ACBAR06,... N = 50 N = 60 r n s

26 The n s r plane and Binomial New Inflation n s = 1 y N 3 z+1, r = 16 y z (1 z) 2 N, y = z 1 log z, 0 < z < 1 (1 z) WMAP5,SDSS, ACBAR08 N = 50 N = 60 r n s

27 The ΛCDM_BNI(z) model Prologue MCMC parameters: Context: Datasets: param best fit 100Ω b h Ω c h θ τ 8.77 H σ log[10 10 A s ] n s r log(l) ω b, ω c, θ, τ, (slow), A s, z (fast) N = 60, Ω ν = 0,... ; standard priors, no SZ, no lensing, linear mpk,... WMAP5, ACBAR08, SDSS Ω b h Ω c h θ Ω Λ Age/GYr H log(10 10 A s ) τ σ z re z

28 Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

29 BNI+sharpcut vs. BNI+fastroll Prologue MCMC parameters: Datasets: param best fit 100Ω b h Ω c h θ τ 7.99 H σ log[10 10 A s ] n s r k log(l) ω b, ω c, θ, τ, (slow), A s, z, k 1 (fast) WMAP5, ACBAR08, SDSS Ω b h Ω c h θ Ω Λ Age/GYr H log(10 10 A s ) sharp cut τ σ z re z *k 1

30 BNI+sharpcut vs. BNI+fastroll Prologue MCMC parameters: Datasets: param best fit 100Ω b h Ω c h θ τ 8.19 H σ log[10 10 A s ] n s r k log(l) ω b, ω c, θ, τ, (slow), A s, z, k 1 (fast) WMAP5, ACBAR08, SDSS Ω b h Ω c h θ Ω Λ Age/GYr H log(10 10 A s ) fast roll τ σ z re z *k 1

31 Comparing TT multipoles Prologue WMAP5 data ΛCDM+r BNI+sharpcut BNI+fastroll l(l+1)c l /2π log(multipole index)

32 Comparing TE multipoles Prologue WMAP5 data ΛCDM+r BNI+sharpcut BNI+fastroll l(l+1)c l /2π log(multipole index)

33 Prologue Comparing real-space TT correlations WMAP data ΛCDM+r BNI+sharpcut BNI+fastroll C(θ) C. Destri, H. J. de Vega, N. Sanchez (arxiv: ) 1.5 θ CMB Quadrupole..., Madrid 2008

34 Highlights of BNI+fastroll Prologue The probability distribution for r: BNI r > (95% CL) r BNI+fastroll r > (95% CL) r MCMC analysis = k 1 is the crossover wavenumber: its wavelength scale exits the horizon when fast-roll ends and slow-roll starts

35 Outline 1 Prologue Observational data and models On cosmic variance On the CMB quadrupole (small) value 2 EFT of inflation Early fast-roll and primordial fluctuations The transfer function D(k) 3

36 Relevant wavelength scales (using WMAP3 + small scale exps. + SDSS) The quadrupole wavenumber today k Q η 0 = 2, η 0 = 3.3 H 0, k Q = (Gpc) 1, log(a Q ) = The fast/slow crossover wavenumber: ǫ v = 1/N k 1 = (Gpc) 1, log(a 1 ) = 1.107, log(a crossover ) = The pivot scale (WMAP) k 0 = 2 (Gpc) 1, log(a 0 ) = The pivot scale (CosmoMC default) k 0 = 50 (Gpc) 1, log(a 0 ) = 6.363

37 Computing the total number of e-folds (using best fit with WMAP3, y=1.32) The quadrupole wavenumber at the beginning of inflation k Q = k 1 = 7.63 m GeV = (Gpc) 1 The total expansion since beginning of inflation 1 + z b = e 129 = e Ntot /a r The Hubble parameter just after reheating, H r a r M PL /H r, H r = e 2[56 N] GeV The bounds on N H r GeV, H r > 1 MeV (BBN) = 56 < N < 76

38 Most likely the quadrupole depression has a nontrivial cosmological origin Early fast-roll inflation is generic and provides a quadrupole depression BNI fits well the CMB+LSS data and provides lower bounds for r BNI+fastroll improves the fit w.r.t. ΛCDM+r and also w.r.t. BNI+sharpcut Fast-roll provides good bounds on N Outlook Improve the theory (expecially concerning reheating) Wait for better data (Plank and new gound exps.) Refine, refine, refine

arxiv: v2 [astro-ph] 3 Jun 2008

arxiv: v2 [astro-ph] 3 Jun 2008 The CMB Quadrupole depression produced by early fast-roll inflation: MCMC analysis of WMAP and SDSS data. C. Destri (a), H. J. de Vega (b,c), and N. G. Sanchez (c) (a) Dipartimento di Fisica G. Occhialini,