What have we learnt about the early universe?

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1 What have we learnt about the early universe? V(φ) φ Hiranya Peiris STFC Advanced Fellow University of Cambridge

2 Roadmap Lecture 1: The physics of the cosmic microwave background. Lecture 2: What have we learnt about the early universe? We are still testing the basic aspects of the inflationary mechanism rather than the specifics of its implementation. Will concentrate here on big picture constraints on the paradigm rather than constraints on specific models.

3 ΛCDM : The Standard Model of Cosmology Homogeneous background 60 K Perturbations Ω b, Ω c, Ω Λ,H 0,τ A s,n s,r atoms 4% cold dark matter 23% dark energy 73% nearly scale-invariant adiabatic Gaussian Λ? CDM? ORIGIN??

4 The Horizon Problem In Standard Big Bang Model, horizon scale at CMB release subtends ~ 1 deg 60 K Regions separated by more than 1 deg could not have interacted previously 1 deg 1 deg So why is the temperature of these patches the same to 1/100000? Credit: NASA/WMAP Science Team

5 Horizon problem us, today last scattering surface time horizon without inflation space

6 Inflation: accelerated super-expansion Inflationary epoch Radius of universe Standard Big Bang expansion Radius of Observable Universe Time after t=0 (secs) If inflation lasts long enough, CMB patches on opposite sides of the sky would have been close enough to communicate in the primordial times.

7 Inflationary resolution of horizon problem us, today true horizon time space inflation

8 Comoving Hubble radius during inflation comoving distance (Mpc) inflation cosmological scales Hubble radius (ah) 1 radiation dominated scale factor a red: scales are smaller than horizon, and subject to microphysical processes

9 Inflation A period of accelerated expansion ds 2 = dt 2 + e 2Ht dx 2 H const Solves: horizon problem flatness problem monopole problem i.e. explains why the Universe is so large, so flat, and so empty Predicts: scalar fluctuations in the CMB temperature nearly scale-invariant approximately Gaussian (?)? primordial tensor fluctuations (gravitational waves)

10 Inflation Implemented as a slowly-rolling scalar field evolving in a potential: V (φ) Inflation H const Standard expansion H 2 = (ȧ a = 8πG 3 ) 2 expansion rate [ ] 1 2 φ 2 + V (φ) density Energy converted into radiation φ φ +3H φ + V =0 friction overdot = d/dt

11 The inflationary background solution Recall Einstein s Equation for the acceleration of the scale factor: ä a = 4 3 (ρ +3P ) For a canonical minimally coupled scalar field, ä>0= P< ρ 3 density: ρ = 1 2 φ 2 + V (φ) pressure: P = 1 2 φ 2 V (φ) During slow roll, potential energy dominates kinetic energy: negative pressure: P V (φ) ρ So expansion of the universe accelerates, scale factor grows exponentially: a(t) e Ht, H 2 8πG 3 V

12 Example of an inflationary potential reheating Acceleration occurs when PE dominates KE. Inflation ends when KE has grown comparable to PE: 1 2 φ 2 V (φ) CMB fluctuations created by quantum fluctuations ~ 60 e-folds before the end of inflation. At end of inflation, energy density of the inflaton converted into radiation. Image by D. Baumann

13 Large field inflation Important class of models where inflaton is driven by a monomial potential: V (φ) φ p Field evolves over a super-planckian distance during inflation: φ >M Pl Large amplitude of gravitational waves produced by QM fluctuations. Image by D. Baumann

14 Perturbations from inflation Cosmological perturbations arise from quantum fluctuations, evolve classically. P φ (k) h ( ) 2 P R h H 2π 4π 2 P h 2 h π 2 ( H 4 ( H m Pl φ 2 ) 2 k=ah ) 2 k=ah scalar tensor

15 Generation of perturbations: overview I QM fluctuations in inflaton produced when relevant scales causally connected. Perturbations driven out of the horizon by inflation. Perturbations re-enter much later to serve as initial conditions for structure formation. Inflation generates both scalar and tensor (metric) perturbations.

16 Generation of perturbations: overview II Perturbations best described in terms of Fourier modes. Individual modes are uncorrelated with each other. e.g. for the gravitational potential, Φ( k) =0 zero mean Φ( k)φ ( k ) = (2π) 3 k 3 P Φ (k)δ 3 ( k k ) non-zero variance Goal: compute this variance in fields and see how it evolves during inflation.

17 Gravitational wave production Review tensor before scalar perturbations since not coupled to other perturbation variables; treat as fluctuations in a single field. Not coupled to density so not responsible for large scale structure, but induces CMB fluctuations. Unique signature of inflation, best window into inflationary physics. To compute QM fluctuations in metric, need to quantize field.

18 Conventions From now, overdot will represent derivative w.r.t. conformal time: τ = dt a(t) = da Ha 2 don t confuse with optical depth! Use natural units, = c = k B =1.

19 Reminder: Quantizing the SHO I Simple harmonic oscillator with unit mass and frequency ẍ + ω 2 x =0 ω obeys: Upon quantization, x becomes quantum operator: ˆx = v(ω,t)â + v (ω,t)â where â is a quantum operator and v exp(iωt) is a solution to SHO eq.

20 Reminder: Quantizing the SHO II â â 0 =0 annihilates the vacuum state, and satisfies commutation relation: (in which there are no particles) [â, â ] ââ â â =1 giving variance: ˆx 2 0 ˆx ˆx 0 = v(ω,t) 2

21 Tensor perturbations: metric Tensor perturbations characterized by metric with: g 00 = 1, g 0i =0 1+h + h 0 g ij = a 2 h 1 h h +,h Perturbations described by two functions, here assumed to be in x,y plane (z in dirn of wavevector) but more generally, cpts of a divergenceless, traceless, symmetric tensor. Carry out usual operation with Einstein equation noting: T µν =0.

22 Tensor perturbations: quantization I Each component individually obeys: ḧ +2ȧ aḣ + k2 h =0 Want to massage into the form of a SHO. Redefine: h = ah 16πG cunningly normalized! SHO-like equation! No damping term! h + ( k 2 ä a ) h =0 ( ẍ + ω 2 x =0)

23 Tensor perturbations: quantization II Can just write down quantum operator: ˆ h( k, τ) =v( k, τ)â k + v ( k, τ)â k and the variance after transforming back to h: ĥ ( k, τ)ĥ ( k, τ) = 16πG a 2 v( k, τ) 2 (2π) 3 δ 3 ( k k ) (2π) 3 k 3 P h (k)δ 3 ( k k ) power spectrum

24 Tensor perturbations: mode equation Now have power spectrum: P h (k) = 16πG k3 v( k, τ) 2 a 2 Have reduced problem to solving second order DE: v + ( k 2 ä ) v =0 a where ä a 2 τ 2

25 Tensor perturbations: mode solution Solution: [ v = e ikτ 1 i 2k kτ ]. Modes leave horizon at kτ < 1. Limits: v e ikτ 2k k τ >> 1 well inside horizon h 1/a SHO soln. v e ikτ 2k i kτ kτ 0 well outside horizon h const

26 Tensor perturbations: power spectrum Primordial power spectrum is thus constant after mode exits horizon. This constant determines the initial conditions for gravitational waves: P h (k) = 16πG 1 2a 2 τ 2 8πGH 2 H is to be evaluated at horizon exit. H const P h nearly scale-invariant. Gravitational wave detection measures Hubble rate during inflation! Since Hubble rate dominated by potential energy during inflation, thus measure at horizon exit! V (φ) Fluctuations are Gaussian, just like for an SHO.

27 Scalar perturbations in the inflaton Decompose inflaton into zero-order homogeneous part and perturbation: φ( x, t) =φ (0) (t)+δφ( x, t) Assuming a smoothly expanding FRW metric, only first order pieces are perturbations to. After usual Einstein equation manipulations, find: T µν δφ +2ȧ δφ a + k 2 δφ =0 Same form as the tensor case; trivially copy solution, dropping normalization factor as we already have a scalar field: P δφ = H2 2

28 Perturbed FRW metric in CNG The Conformal Newtonian Gauge describes the perturbed FRW metric. ds 2 = a 2 (τ)[(1 + 2Ψ)dτ 2 δ ij (1 2Φ)dx i dx j ] where = conformal time. Metric perturbations described by scalar potentials (Newtonian potential) and (perturbation to spatial curvature). Ψ τ Φ Only describes scalar perturbations.

29 Gauge choice Want to know how perturbations in φ get transferred to Ψ (or Φ, assumed identical in magnitude here). Just neglected metric perturbations so far. Easiest way is to switch to a gauge where spatial part of the metric is unperturbed: a spatially flat slicing. In such gauge, previous calculation is exact. Q: How to move back to CNG? δφ A: (i) Identify a gauge invariant variable in a spatially flat slicing (SFS). (ii) Find this variable in CNG, thus linking in CNG with in SFS. In this gauge, line element (with A, B characterizing perturbations) is: ds 2 = (1 + 2A)dt 2 2aB,i dx i dt + δ ij a 2 dx i dx j Ψ δφ

30 Scalar perturbations in spatially flat slicing Bardeen (1980) identified several gauge-invariant variables. In particular, in SFS, Bardeen s velocity is: v = ikb ik φ (0) δφ (ρ + P )a 2 In SFS, Bardeen s Φ H is Φ H = ahb so find gauge-invariant combo: R Φ H iah k v = ah δφ in SFS φ (0) Immediately obtain power spectrum: P R = ( ah φ (0) ) 2 P δφ Spatial curvature at fixed time is can be called the curvature perturbation. R 4k 2 Φ H /a 2 so in a comoving gauge,

31 Slow roll parameters and power spectra Useful to characterize inflationary dynamics by slow roll parameters which remain small during inflation. ɛ d dt ( 1 H ) = Ḣ ah 2 η 1 H d 2 φ (0) /dt 2 dφ (0) /dt 2 Prefactor in P R is then 4πG/ɛ so: P R =2πG H2 ɛ k=ah weakly scale dependent Φ H = Φ Often used empirical parameterizations: R = 3 2 Ψ = 3 2 Φ P R = 9 4 P Φ In CNG, and after inflation,, so. P R (k) A s ( k k 0 ) ns 1 P h (k) A t ( k k 0 ) nt r = P h(k 0 ) P R (k 0 )

32 What is the physics of inflation? V (φ) Why did the field start here? Where did this function come from? Why is the potential so flat? Inflation consists of taking a few numbers that we don t understand and replacing it with a function that we don t understand David Schramm φ How do we convert the field energy completely into radiation?

33 Reminder: the primordial power spectrum & the CMB P (k) n S < 1 k CMB physics power P (k) CMB physics n S > 1 k large scales small scales

34 Parameters from CMB: primordial power spectrum Scalar power spectrum C l essentially e 2τ P R (k) at k l/d A processed by acoustic physics CMB probes scales 5 Mpc <k 1 < 5000 Mpc Tensor power spectra sensitive to e 2τ P h (k)

35 Testing inflation Key predictions of simple inflation models: Universe should be flat (cf. Ω K = ± 0.006) Small curvature fluctuations and (possibly) gravitational waves with almost scale-invariant, power-law spectra: P R (k) A s (k/k ) n s 1, P h (k) A t (k/k ) n t with observables related to slow-roll parameters [ parameterise gradient and curvature of V (φ)] A s = H2 πɛm 2 Pl Adiabatic initial conditions, n s 1= 4ɛ +2η, A t ra s = 16H2 πm 2, n t = 2ɛ Pl Fluctuations should be Gaussian (to observational accuracy; see later):

36 The character of fluctuations Well-defined peaks phase coherence (cf. defects) Super-horizon correlations at last scattering surface from TE correlation and sign adiabaticity Peak positions in TT, TE and EE consistent with adiabatic models CMB still allows 10% contribution from single, uncorrelated isocurvature modes and significantly more for more general cases (Bean et al. 2006), but not favoured over adiabatic 43

37 The flatness of the Universe? geometric degeneracy [powerlaw P(k)]: Ω K Ω Λ Spergel et al. (WMAP Collaboration, 2006)

Slow roll inflation Komatsu et al. 2008 Inflation energy scale unknown: r<0.2 from low-l T E inf < 2.

small-field, large-field or hybrid phenomenology) n s < 1 with CMB+BAO+SN and n s =0.963 ± 0.

38 Slow roll inflation Komatsu et al Inflation energy scale unknown: r<0.2 from low-l T E inf < GeV Dynamics not yet classified (e.g. small-field, large-field or hybrid phenomenology) n s < 1 with CMB+BAO+SN and n s =0.963 ± from WMAP5, no running and r =0 n s -Ω b h 2 degeneracy main one now affecting n s dn s /d ln k = ± from WMAP5 with r =0 Some shifts ( 0.5σ) if include small-scale CMB 46

39 Constraints on specific models η > 0 η < red blue r /3 φ > M pl large-field natural hill-top φ < M pl small-field WMAP very small-field inflation n s r determines whether model is large or small field. ns determines whether spectrum is red or blue. a combination of ns and r determines the curvature of the potential η. Figure: D. Baumann & H.V. Peiris, invited review to be published in Advanced Science Letters

40 What range of potential does CMB sample? 47

41 Primordial non-gaussianity Bispectrum of primordial curvature perturbation R(k 1 )R(k 2 )R(k 3 ) δ(k 1 + k 2 + k 3 )F (k 1,k 2,k 3 ) Local form peaks on squeezed triangles ( PR (k F (k 1,k 2,k 3 ) f 1 ) P R (k 2 ) NL k1 3 k Arises when non-gaussianity created outside horizon (e.g. multi-field inflation, curvaton, fluctuating reheating): R(x) =R G (x) 3 5 f ( NL R 2 G (x) R 2 G (x) ) Small in single-field inflation: f NL n s 1 in squeezed limit Non-local form peaks on equilateral triangles E.g. f[( φ) 2 ] in DBI inflation momentum conservation )

42 -G AUSSIANITY IN THE CMB TestingN ON the Gaussianity of the CMB Liguori et al. (2007) Large-scale T /T = R/5 Positive fn L skews R and T negative Fractional departure from Gaussianity very well measured: fn L PR < 10 3 Planck should achieve fn L = 5 49

43 Constraints from WMAP5 Komatsu et al. 2008) bispectrum analysis: 9 <f local NL < 111 and 151 <fequilateral NL < 253 (V + W and KQ75) Map noise lower by 22% New masks (e.g. KQ75 and KQ75p) that avoid potential upwards bias in f NL Null tests fine for KQ75 and foreground-cleaned V W maps Corrections for point sources (small for local model) Analysis with Minkowski functionals: all consistent with Gaussianity 178 <f local NL Mild tension with bispectrum results for f local NL < 64 (V + W and KQ75)

44 Improvements on scalar power spectrum Planck: the scientific programme Planck collaboration Marginalised error forecasts for ln P R (k) = ln A s +(n s 1) ln(k/k 0 )+ 1 2 (dn s/d ln k) ln 2 (k/k 0 )+ : n s = and (dn s /d ln k) = 0.005

45 Gravitational waves Tensor metric perturbations ds 2 = a 2 [dη 2 (δ ij + h ij )dx i dx j ] with δ ij h ij =0 Shear ḣ ij gives anisotropic redshifting Θ(ˆn) 1 2 dη ḣ ijˆn iˆn j Only contributes on large scales since h ij decays like a 1 after entering horizon 22

46 Tensors: B-mode contribution is small! R.m.s. B-mode signal from gravity waves < 200 nk r =0.28 r =0.0 63

47 Approximate range of primordial tensors accessible to upcoming experiments V 1/ r 1/4 GeV 2 TT: temp EE: E-pol BB: B-pol Text 2 X GeV 5 X GeV

48 Upcoming B-mode surveys B-mode polarization circumvents cosmic variance from (dominant) linear density perturbations but current upper limits not competitive with T Next generation (Clover, QUIET, SPIDER, EBEX etc.) targeting r>0.01 Futuristic full-sky(!) survey limited to r>10 4 unless implement lens cleaning Will require exquisite control of systematics and accurate removal of synchrotron and dust polarized foregrounds 62

49 Lyth Bound In a de Sitter spacetime, tensors: P h H2 M 2 Pl scalars: P s H 2 ( Ḣ φ ) 2 tensor to scalar ratio: r P ( h 1 =8 P s where dn e d ln a = Hdt = ( Ḣ φ M Pl ) dφ dφ dn e ) 2 field variation relates to tensor signal: φ M Pl = φcmb φ end dn e 1 8 r(n e) Useful if this can be computed from a microscopic theory! Useful if this can be constrained via observations! Lyth (1996)

50 WMAP Collaboration (2008), Knox & Song (2002), Verde, HVP & Jimenez (2005) Measuring tensors Current constraint: r CMB 0.2 Realistically observable: r CMB A measurement of tensors gives 2 pieces of information: Super-Planckian field variation. Observable gravitational waves require tensor/scalar ratio r at k0=0.002 Mpc WMAP5 + BAO + SN scalar spectral index ns The energy scale of inflation. ( ) 2 δρ The measured scalar amplitude P and H 2 1 s implies: 3MPl 2 V ρ ( V 1/4 rcmb ) 1/ GeV φ M Pl > O(1) rcmb φ >M Pl during inflation: e.g. V (φ) = 1 requires. 2 m2 φ 2 φ 15M Pl

51 Beyond one number: tensor tilt If r is measured by CMBPOL, can we also detect the tensor tilt, n t? Can we test the inflationary consistency condition,? Can we detect a general sound speed? Define: C =1 r = 8n t C = r 8n t for canonical single-field slow roll C = c 2 s < 1 for general sound speed n t > 0 Can we rule out?

52 Future observational prospects Go to small scales. Much better measurements of the primordial scalar power spectrum shape. Planck l~3000 (k~0.2/mpc) ACT, SPT l~10000 (k~0.7/mpc) [secondary effects] Galaxies k~1/mpc [non-linearity & bias] Lyman alpha k~5/mpc [gas phys. & radiation feedback] Reionization k~50/mpc [much is unknown] Detecting gravitational waves. CMB: QUaD, BICEP, QUIET, CLOVER, PolarBeaR, EBEX, SPIDER, Planck, CMBPOL/B-Pol etc [large scales] GWO: direct detection of primordial gravitational waves (BBO) [solar system scales] Detecting primordial non-gaussianity. Can we detect f NL ~1 or fnl >> 1? Can we distinguish shape dependence? scale dependence?

COSMIC MICROWAVE BACKGROUND ANISOTROPIES

COSMIC MICROWAVE BACKGROUND ANISOTROPIES Anthony Challinor Institute of Astronomy & Department of Applied Mathematics and Theoretical Physics University of Cambridge, U.K. a.d.challinor@ast.cam.ac.uk 26