The critical role of statistics in CMB studies
V(φ)
φ
Hiranya Peiris University College London
Wednesday, 30 September 2009 Outline
Wednesday, 30 September 2009 Outline
• Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations
Wednesday, 30 September 2009 Outline
• Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations
• The primordial power spectrum – Inferences from the CMB with inflationary priors
Wednesday, 30 September 2009 Outline
• Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations
• The primordial power spectrum – Inferences from the CMB with inflationary priors
• The primordial power spectrum – Minimally parametric reconstruction using cross-validation
Wednesday, 30 September 2009 Outline
• Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations
• The primordial power spectrum – Inferences from the CMB with inflationary priors
• The primordial power spectrum – Minimally parametric reconstruction using cross-validation
• CMB temperature anomalies – Beyond a posteriori statistics using consistency tests
Wednesday, 30 September 2009 Outline
• Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations
• The primordial power spectrum – Inferences from the CMB with inflationary priors
• The primordial power spectrum – Minimally parametric reconstruction using cross-validation
• CMB temperature anomalies – Beyond a posteriori statistics using consistency tests
Wednesday, 30 September 2009 The Cosmic Microwave Background (CMB)
60 K
Credit: NASA/WMAP Science Team
Wednesday, 30 September 2009 #OMPOSITION OF AND +ECosmic HistoryY %VENTS $ UR/ ING THE %CosmicVOLUTION OF THE 5 MysteryNIVERSE
presentpresent energy energy
Y density Ω7 totTOT = 1(k=0)K density DAR
RADIATION K ENER dark energy
Y DENSIT DARK G (73%)
DARK MATTER Y G ENERGY
dark matter DARK MA(23.6%)TTER TION OF ENER WHITE WELL UNDERSTOOD DARKNESS PROPORTIONAL TO POOR UNDERSTANDING BARYONS BARbaryons(4.YONS AC 4%) FR −42 −33 −22 −16 −12 Fractional Energy Density 10 s 10 s 10 s 10 s 10 s 1 sec 380 kyr 14 Gyr ~1015 GeV SCALE FACTimeTOR ~1 MeV ~0.2 eV 4IME T S T S T S T S T S T SEC T KYR T 'YR Y Y Planck GUT Y T=100 TeV nucleosynthesis Y IES TION TS EOUT DIAL ORS TIONS G TION TION Z T Energy THESIS
symmetry (ILC XA 100) MA EN EE WN OF ES THESIS V
IMOR GENERATE OBSERVABLE IT OR ELER AL THEOR TIONS
E F SIGNATURES IN THE #-" EAK SYMMETR EIONIZA INO FR Y % OMBINA R E6 EAKDO # X ON ASYMMETR SIC W '54 SYMMETR IMELINE OF EFF WR Y Y EC TUR O NUCLEOSYN ), * +E R 4 PLANCK ENER Generation BR TR TURBA UC PH Cosmic Microwave NEUTR OUSTIC OSCILLA BAR TION OF PR ER A AC STR
of primordial ELEC non-linear growth of P 4 4 LIMIT OF ACC Background Emitted perturbations perturbations: GENER
ES carries signature of signature on CMB TUR GENERATION OF GRAVITY WAVES INITIAL DENSITY PERTURBATIONS acoustic#-" %MITT oscillationsED NON LINEAR STR andUCTUR E IMPARTS #!0-!0 OBSERVES #-" AND INITIAL DENSITY PERTURBATIONS GROW IMPARTING FLUCTUATIONS CARIES SIGNATURE OF ACOUSTIC SIGNATURE ON #-" THROUGH *throughEFF WRITES UP AND GR weakADUATES WHICH SEED STRUCTURE FORMATION TO #-" OSCILLATIONS AND PpotentiallyOTENTIALLY PR IMORprimordialDIAL GRAVITATIONAL LENSING IGNA
3 GRAVITY WAVES IN THE #-" gravitational waves gravitational lensing
Figure: J. McMahon, adapted by HVP
Wednesday, 30 September 2009 ΛCDM: The “Standard Model” of Cosmology
Homogeneous background Perturbations 60 K
Ωb, Ωc, ΩΛ,H0,τ As,ns,r
•atoms 4% •nearly scale-invariant •cold dark matter 23% •adiabatic •dark energy 73% •Gaussian Λ? CDM? ORIGIN??
Wednesday, 30 September 2009 History of CMB temperature measurements
60 K 2.725 K
TOCO (1998) BOOMERANG (1998, 2003) MAXIMA (2000) ARCHEOPS (2002) CBI (2002) ACBAR (2002) VSA (2002) Wednesday, 30 September 2009 History of CMB temperature measurements
60 K 2.725 K
TOCO (1998) BOOMERANG (1998, 2003) MAXIMA (2000) ARCHEOPS (2002) CBI (2002) ACBAR (2002) VSA (2002) Wednesday, 30 September 2009 Planck:
7 deg
15 arcmin
5 arcmin
Wednesday, 30 September 2009 ThermalTHERMALHISTORY History
CMB and matter plausibly produced during reheating at end of inflation • CMB decouples around recombination, 300 kyr later • Universe starts to reionize once first stars (?) form (somewhere in range • z = 10–20) and 10% of CMB re-scatters
28 13 10 10 K 10 K 10 K 3000 K Recombination Nucleosynthesis Quarks -> Hadrons
-34 -6 10 s 10 Sec
2
Wednesday, 30 September 2009 Space-time and CMB Physics (NOT to scale)
time
past light cone
Universe transparent COBE resolution WMAP resolution Last scattering 379,000 yrs Universe opaque
Big Bang space
Wednesday, 30 September 2009 Compress the CMB map to study cosmology
Express sky as: δT (θ,φ)=! almYlm(θ,φ) l,m If the anisotropy is a Gaussian random field
(real and imaginary parts of each alm independent normal deviates, not correlated) all the statistical information is contained in the angular power spectrum.
0.06% of map 5 deg
1 2 Cl = |alm| X 1 deg 2! +1! m
ANGULAR POWER SPECTRUM Raw 94 GHz +/- 32 uK Raw 61 GHz near NEP near NEP
Wednesday, 30 September 2009 WMAP temperature power spectrum
Hinshaw et al. (2003) Wednesday, 30 September 2009 A simplified CMB likelihood function
theory noise bias
Cth + N C 2 ln = (2! + 1) ln ! ! + ! 1 . − L Cth + N − ! " C! ! ! & ! # % $
estimator$ for sky Cls
•Logarithmic at large scales; more likely to scatter low.
•Approaches Gaussian at small scales.
•Cosmic variance: even ideal experiment can only measure (2l+1) modes.
Wednesday, 30 September 2009 Radical data compression
Time-ordered data (e.g. WMAP 5 years 60-100 GB)
mostly experimental characteristics
map (12-50 million pixels)
physically motivated statistics
angular power spectrum - O(1000) to O(10000) numbers
experiment, physics, statistics
model - O(10) parameters
Wednesday, 30 September 2009 State of the art: temperature
4 10 Figure: M. Brown
!CDM 3 10 ] 2 K [µ
" WMAP5 /2 l QUaD ACBAR l(l+1)C 2 10 QUaD 150 GHz QUaD 100 GHz SZA 30 GHz CBI
1 10 500 1000 1500 2000 2500 3000 3500 4000 4500 Multipole Moment l
‣Sachs-Wolfe plateau and the late time Integrated Sachs-Wolfe effect ‣Acoustic peaks at “adiabatic” locations ‣Damping tail and photon diffusion ‣Weak gravitational lensing (detected in cross-correlation, Smith et al. 2007)
Wednesday, 30 September 2009 Types of CMB polarization
CMB polarization can be decomposed into two orthogonal modes. E- mode is the curl-free mode (“Electric”). B-mode is the divergence- free mode (“Magnetic”).
E-mode B-mode
Wednesday, 30 September 2009 Types of CMB polarization
CMB polarization can be decomposed into two orthogonal modes. E- mode is the curl-free mode (“Electric”). B-mode is the divergence- free mode (“Magnetic”).
E-mode B-mode
B mode discriminates between scalar and tensor perturbations
Wednesday, 30 September 2009 Generation of CMB polarization
• Temperature quadrupole at the surface of last scatter generates polarization.
electron isotropic no net polarization
Thomson Scattering anisotropic net polarization
Quadrupole generated by velocity gradients at Last Scattering Surface
Wednesday, 30 September 2009 Temperature-Polarization Correlation
Temperature quadrupole at Radial pattern around cold spots z~1089 generates polarization Tangential pattern around hot spots
Figure: E. Komatsu Wednesday, 30 September 2009 Source of CMB polarization Signal from Reionization (NOT to scale) time
Scattered light is polarized if incident light is anisotropic: low-l signal from secondary past light cone scattering Universe partially transparent
“Real” last scattering
high-l signal from primary scattering Universe transparent COBE resolution WMAP resolution 2nd-to-last scattering 379,000 yrs Universe opaque
Big Bang space
Wednesday, 30 September 2009 State of the art: polarization
‣Acoustic peaks at “adiabatic” locations
‣E-mode polarization and cross-correlation with T
‣Large angle polarization from reionization
‣BICEP limit from BB- alone: T/S < 0.73 (95% CL)
Figure: Chiang et al. (2009) Wednesday, 30 September 2009 Coupled Einstein-Boltzmann Equations
Neutrinos
Dark Photons Matter Metric
Thomson scattering
Electrons Protons Coulomb scattering
Wednesday, 30 September 2009 CMB as a sound wave
(graphic by W. Hu) Last scattering surface : snapshot of the photon-baryon fluid
On large scales : primordial ripples, purely GR Effects Forced damped harmonic oscillator Photons radiation pressure On smaller scales: { } Sound waves Gravity compression Stop oscillating at recombination Smaller than photon mean free path: Exponentially damped by photon diffusion
Horizon size at last scattering surface is fundamental mode
Peebles & Yu (1970), Sunyaev & Zel’dovich (1970), Sachs and Wolfe (1968), Silk (1968)
Wednesday, 30 September 2009 SHO analogy I
Consider simple harmonic oscillator with mass m, force constant k driven by external force F0. k F x¨ + x = 0 m m F0 Assuming oscillator is initially at rest, F0 x = A cos(ωt)+ 2 mω k nπ ω = Peaks at t = . !m ω
unforced: peak heights equal forced: odd (even) peaks higher (lower) forcing disparity greater for lower Even peaks correspond to negative x
Figure from Dodelson, “Modern Cosmology”
Wednesday, 30 September 2009 SHO analogy II
Cartoon gravitational instability: δ¨ + [Pressure Gravity]δ =0 − Oscillator analogy: F0 ¨ 2 2 Θ0 + k csΘ0 = F 2 F is force due to gravity, c s is sound speed of entire photon- baryon fluid. k ω = Add more baryons; sound speed m (frequency) goes down. !
Modes enter horizon and start to oscillate. See their phases frozen at recombination. Peaks are maximum amplitudes.
Figure from Dodelson, “Modern Cosmology”
Wednesday, 30 September 2009 Adiabatic anisotropy power spectrum ADIABATICANISOTROPYPOWERSPECTRUM
Temperature power spectrum for scale-invariant curvature fluctuations •
Figure: A. Challinor 17 Wednesday, 30 September 2009 SCALARANDTENSORPOWERSPECTRAScalar & tensor power spectra(r =0.2)
E For scalar perturbations (left), δγ oscillates π/2 out of phase with vγ Cl peaks • T ⇒ at minima of Cl scalar tensor
Figure: A. 31Challinor Wednesday, 30 September 2009 CMB spectrum: parameter dependences
100 (a) Curvature (b) Dark Energy
80
60
40
20
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8
100 (c) Baryons (d) Matter
80
60
40
20
0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 10 100 1000 10 100 1000 Figure: W. Hu Wednesday, 30 September 2009 The effect of more data: LCDM model
WMAP I WMAP I Ext WMAP II
Reducing the noise by sqrt(T) degeneracies broken
Wednesday, 30 September 2009 Outline
• Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations
• The primordial power spectrum – Inferences from the CMB with inflationary priors
• The primordial power spectrum – Minimally parametric reconstruction using cross-validation
• CMB temperature anomalies – Beyond a posteriori statistics using consistency tests
Wednesday, 30 September 2009 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
Wednesday, 30 September 2009 Horizon problem
us, today
last scattering surface
time horizon without inflation
space
Wednesday, 30 September 2009 Inflation: accelerated super-expansion
Inflationary epoch
1040
1020 Standard Big Bang expansion
100
10-20 Radius of Observable Radius of universe 10-40 Universe 10-60 10-45 10-35 10-25 10-15 10-5 105 1015
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.
Wednesday, 30 September 2009 Inflationary resolution of horizon problem
us, today true horizon
time
inflation space
Wednesday, 30 September 2009 Comoving Hubble radius during inflation
1 cosmological scales
10 Hubble radius 10− 1 inflation (aH)− radiation dominated
20
comoving distance (Mpc) 10−
40 20 10− 10− 1 scale factor a
red: scales are smaller than horizon, and subject to microphysical processes
Wednesday, 30 September 2009 Inflation
A period of accelerated expansion ds2 = −dt2 + e2Htdx2 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)
Wednesday, 30 September 2009 Inflation
Implemented as a slowly-rolling scalar field evolving in a potential:
2 2 a˙ expansion V (φ) H = !a" rate
8πG 1 2 = φ˙ + V (φ) Standard 3 #2 $ Inflation expansion H ∼ const density
! φ φ¨ +3Hφ˙ + V =0 Energy converted into radiation friction
overdot = d/dt
Wednesday, 30 September 2009 Perturbations from inflation
Cosmological perturbations arise from quantum fluctuations, evolve classically.
Wednesday, 30 September 2009 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.
Wednesday, 30 September 2009 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 ! " 3 3 3 Φ(!k)Φ∗(k! ) = (2π) k− P (k)δ (!k k! ) non-zero variance ! " " Φ − "
‣Goal: compute this variance in fields and see how it evolves during inflation.
Wednesday, 30 September 2009 Inflation
• Solves the flatness/horizon problems if the early universe inflates by factor ~1030. • Cosmological perturbations arise from quantum fluctuations, evolve classically.
2 ¯h H4 PR ! scalar 2 4 2 ˙2 H π ! φ "k=aH Pφ(k) ! ¯h !2π " 2¯h H 2 P ! tensor h 2 π !mPl "k=aH
Wednesday, 30 September 2009 Inflation
• Solves the flatness/horizon problems if the early universe inflates by factor ~1030. • Cosmological perturbations arise from quantum fluctuations, evolve classically.
2 ¯h H4 PR ! scalar 2 4 2 ˙2 H π ! φ "k=aH Pφ(k) ! ¯h !2π " 2¯h H 2 P ! tensor h 2 π !mPl "k=aH
• Don’t know the dynamics of inflation: parameterize weakly scale-dependent functions with a few numbers to pin down observationally.
ns 1 nt k − k Ph(k0) P (k) As Ph(k) At r = R ! k ! k0 P (k0) ! 0 " ! " R
Wednesday, 30 September 2009 Primordial non-Gaussianity
Gaussian quantum fluctuation δη
2 non-Gaussian inflaton fluctuation δφ g (δη + f δη ) ∼ δφ δη
non-Gaussian curvature fluctuation ζ g (δφ + f δφ2) ∼ ζ δφ
non-Gaussian CMB anisotropy δT g (ζ + f ζ2) T ∼ T ζ
Expected to be small, fNL ~ 1, for slow roll inflation.
Current constraints O(100).
Wednesday, 30 September 2009 Slow roll inflation consistent with WMAP+
‣ Superhorizon, adiabatic fluctuations - T and E anticorrelated at superhorizon scales
‣ Flatness tested to 1%.
‣ Gaussianity tested to 0.1%.
‣ nearly scale-invariant fluctuations - red tilt indicated at ~2.5 σ
Spergel, Verde, Peiris et al. (2003), Komatsu et al. (2003), Peiris et al. (2003), Spergel et al (WMAP Collaboration) (2006), Dunkley et al & Komatsu et al (WMAP Collaboration) (2008)
Wednesday, 30 September 2009 Slow roll inflation consistent with WMAP+
‣ Superhorizon, adiabatic fluctuations - T and E anticorrelated at superhorizon scales
‣ Flatness tested to 1%.
‣ Gaussianity tested to 0.1%.
‣ nearly scale-invariant fluctuations - red tilt indicated at ~2.5 σ
‣Still testing basic aspects of inflationary mechanism rather than specific implementation.
Spergel, Verde, Peiris et al. (2003), Komatsu et al. (2003), Peiris et al. (2003), Spergel et al (WMAP Collaboration) (2006), Dunkley et al & Komatsu et al (WMAP Collaboration) (2008)
Wednesday, 30 September 2009 Inflation
Modelled as a scalar field (inflaton) evolving in a potential
V (φ) V (φ)
Standard Inflation expansion H ∼ const
φ φ Energy converted into radiation
Many potentials from different physical motivations are consistent with the data
Wednesday, 30 September 2009 What is the physics of inflation?
V (φ) Why did the field start here?
Where did this function “Inflation consists of taking come from? Why is the potential so a few numbers that we don’t flat? understand and replacing it with a function that we don’t understand” David Schramm 1945 -1997 φ
How do we convert the field energy completely into particles?
Wednesday, 30 September 2009 Outline
• Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations
• The primordial power spectrum – Inferences from the CMB with inflationary priors
• The primordial power spectrum – Minimally parametric reconstruction using cross-validation
• CMB temperature anomalies – Beyond a posteriori statistics using consistency tests
Wednesday, 30 September 2009 Approaches to constraining inflationary models
‣Empirical parameterizations: amplitude, tilt, “running”. dn k n (k)=n (k )+ s ln s s 0 d ln k k ! 0 " -adequate for current data, unnecessary approx for future. -useful for understanding generic predictions of simple models.
‣Bottom-up direct “reconstruction” of inflationary potential.
‣Top-down “model testing” of specific inflationary models.
Wednesday, 30 September 2009 An old problem...
Can we observe primordial perturbations, reconstruct potential?
Wednesday, 30 September 2009 ParametersPARAMETERSFROM from CMB:CMB: primordialPRIMORDIALPOWERSPECTRUM power spectrum
2τ Scalar power spectrum Cl essentially e− (k) at k l/dA processed by • PR ≈ acoustic physics 1 – CMB probes scales 5 Mpc Tensor power spectra sensitive to e 2τ (k) • − Ph 34 Figure: M. Tegmark Wednesday, 30 September 2009 Reminder: the primordial power spectrum & the CMB P (k) CMB physics nS < 1 k power P (k) CMB physics n > 1 S k large small scales scales Wednesday, 30 September 2009 Bottom-Up: reconstructing the potential V (φ) V, V’, V’’, V’’’ CMB LSS 21 cm? V, V’ GWO? φ Solar Horizon System small large scales scales Wednesday, 30 September 2009 Hamilton-Jacobi Slow Roll Reconstruction ‣Systematic slow roll expansion in H ( φ ) truncated at specific order. -how many parameters? ‣Many advantages of H-J method. -e.g. exists exact background solution and perturbation spectrum ‣Find all potential shapes allowed at cosmological scales. -e.g. by fitting to CMB and LSS data Peiris & Easther (astro-ph/0603587, astro-ph/0604214, astro-ph/0609003, arxiv:0805.2154) Wednesday, 30 September 2009 Hamilton-Jacobi Slow Roll Reconstruction ‣Systematic slow roll expansion in H ( φ ) truncated at specific order. -how many parameters? ‣Many advantages of H-J method. -e.g. exists exact background solution and perturbation spectrum ‣Find all potential shapes allowed at cosmological scales. -e.g. by fitting to CMB and LSS data ‣Check consistency of extrapolating expansion using physical priors: -duration of inflation / reheating energy scale -primordial black hole overproduction -avoiding eternal inflation inside observable volume Peiris & Easther (astro-ph/0603587, astro-ph/0604214, astro-ph/0609003, arxiv:0805.2154) Wednesday, 30 September 2009 The duration of inflation Constraint: Comoving Hubble Radius at onset of inflation > Comoving Hubble Radius today. matter aRH NH = ln ! 50 − 60 log (length scale) ! aH " radiation Solves cosmological problems if radius of universe expands by 50-60 “e-folds” during inflation Standard expansion aEQ aTODAY log (scale factor) Wednesday, 30 September 2009 The duration of inflation Constraint: Comoving Hubble Radius at onset of inflation > Comoving Hubble Radius today. matter current horizon aRH NH = ln ! 50 − 60 log (length scale) ! aH " radiation -1 H Solves cosmological problems if radius of universe expands by 50-60 “e-folds” during inflation Standard expansion aEQ aTODAY log (scale factor) Wednesday, 30 September 2009 The duration of inflation Constraint: Comoving Hubble Radius at onset of inflation > Comoving Hubble Radius today. matter current horizon aRH NH = ln ! 50 − 60 log (length scale) ! aH " radiation -1 H Solves cosmological problems if radius of universe expands by 50-60 “e-folds” during inflation Inflation Standard expansion aH aRH aEQ aTODAY log (scale factor) Wednesday, 30 September 2009 The duration of inflation Constraint: Comoving Hubble Radius at onset of inflation > Comoving Hubble Radius today. matter current horizon aRH NH = ln ! 50 − 60 log (length scale) ! aH " galaxy radiation -1 H Solves cosmological problems if radius of universe expands by 50-60 “e-folds” during inflation Inflation Standard expansion aH aRH aEQ aTODAY log (scale factor) Wednesday, 30 September 2009 Connecting measurements to an inflationary model 0 matter Observable parameters “pivot scale” k are a function of scale! log (length scale) e.g. nS[k(Nefold)] radiation k0 Inflation Standard expansion ak0 aRH aEQ aTODAY log (scale factor) Wednesday, 30 September 2009 Connecting measurements to an inflationary model Reheat temperature can vary from GUT scale (1015 GeV) to 0 matter nucleosynthesis scale (1 MeV)! Resulting uncertainty in “pivot scale” k predictions at a given “pivot”: log (length scale) radiation matter ∆Nefold ∼ 14 ∆r k0 ∼ 1 r ∆n ∼ 0.02 Inflation Standard expansion ak0 aEND aRH aEQ aTODAY log (scale factor) Wednesday, 30 September 2009 e-fold priors “Connection equation” in a universe that inflated, reheated, and passed through matter-radiation equality: k 1 Hreh 2 Hend Hk N(k)= ln 1 + ln ln + ln + 59.59. − Mpc− 6 mPl − 3 mPl mPl ! " ! " ! " ! " weaker N(k) > 15 “minimal” Treh > 10 MeV guarantees thermalized neutrino sector Treh > 10 TeV reheating occurs well above EW scale Hreh = Hend instant reheating stronger Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 2 HSR parameters and e-fold priors Hreh>Hend Treh>10 TeV N>15 k 1 Hreh 2 Hend Hk N(k)= ln 1 + ln ln + ln + 59.59. − Mpc− 6 mPl − 3 mPl mPl ! " ! " ! " ! " Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 2 HSR parameters and e-fold priors Hend=Hreh N>15 k 1 Hreh 2 Hend Hk N(k)= ln 1 + ln ln + ln + 59.59. − Mpc− 6 mPl − 3 mPl mPl ! " ! " ! " ! " Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 Bounds on spectral params at k=0.02 Mpc-1 Treheat > 10 TeV WMAP5 WMAP5+SNLS Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 3 HSR parameters and e-fold priors Hend=Hreh N>15 N>15 Hend=Hreh k 1 Hreh 2 Hend Hk N(k)= ln 1 + ln ln + ln + 59.59. − Mpc− 6 mPl − 3 mPl mPl ! " ! " ! " ! " Main effect is to eliminate models with large positive xi Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 Bounds on slow roll params at k=0.02 Mpc-1 WMAP5 WMAP5 WMAP5+SNLS WMAP5+SNLS k 1 Hreh 2 Hend Hk N(k)= ln 1 + ln ln + ln + 59.59. − Mpc− 6 mPl − 3 mPl mPl ! " ! " ! " ! " Main effect is to eliminate models with large positive xi Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 Bounds on spectral params at k=0.02 Mpc-1 Treheat > 10 TeV WMAP5 -1 WMAP5 -1 WMAP5+SNLS WMAP5+SNLS r at k=0.02 Mpc /d ln k at k=0.02 Mpc S dn -1 -1 nS at k=0.02 Mpc nS at k=0.02 Mpc k 1 Hreh 2 Hend Hk N(k)= ln 1 + ln ln + ln + 59.59. − Mpc− 6 mPl − 3 mPl mPl ! " ! " ! " ! " Main effect is to eliminate models with large negative running Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 Primordial black hole overproduction WMAP5+SNLS Main effect is to eliminate models with large positive running / negative xi Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 Primordial black hole overproduction WMAP5+SNLS Main effect is to eliminate models with large positive running / negative xi Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 Primordial black hole overproduction WMAP5+SNLS Main effect is to eliminate models with large positive running / negative xi Peiris & Easther (JCAP, 2008), arxiv:0805.2154 [astro-ph] Wednesday, 30 September 2009 Slow Roll Reconstruction Summary ‣Systematic slow roll expansion in H ( φ ) truncated at specific order. -2 slow roll parameter constraints roughly the same as {ns, r} ‣3 slow roll parameter space allowed by data constrained by physical priors on: -duration of inflation / reheating energy scale -primordial black hole overproduction -avoiding eternal inflation inside observable volume ‣In future, supplement priors with data from smaller scales. Peiris & Easther (astro-ph/0603587, astro-ph/0604214, astro-ph/0609003, arxiv:0805.2154) Wednesday, 30 September 2009 Top-down approach String Compactification Inflationary 4d Lagrangians Lagrangians Observables Wednesday, 30 September 2009 Inflation in string theory ‣ Inflationary action is sensitive to Planck scale physics. ‣ Generic features of “simple” inflation provide non-trivial constraints: - duration of inflation - flatness of potential - radiative stability of inflaton mass Wednesday, 30 September 2009 Inflation in string theory ‣ Inflationary action is sensitive to Planck scale physics. ‣ Generic features of “simple” inflation provide non-trivial constraints: - duration of inflation - flatness of potential - radiative stability of inflaton mass ‣ Potential is derived function directly related to stability of higher-dimensional compactification. ‣ Inflaton has elegant geometric interpretations. Wednesday, 30 September 2009 Inflation in string theory ‣ Inflationary action is sensitive to Planck scale physics. ‣ Generic features of “simple” inflation provide non-trivial constraints: - duration of inflation - flatness of potential - radiative stability of inflaton mass ‣ Potential is derived function directly related to stability of higher-dimensional compactification. ‣ Inflaton has elegant geometric interpretations. ‣ Two explicit case studies: - warped D-brane inflation (small-field example) - axion monodromy inflation (large-field example) ‣ Explicit examples demonstrate unexpected correlations between observables that point to stringy physics. Wednesday, 30 September 2009 Distinctive observables of the “IR DBI” model ‣Large transient running of the scalar spectral index. B-throat closed closed open strings string string cut off creation field creation throat theory applies Hubble energy redshifted brane tension Large scale dependent non-Gaussian signature, and an undetectable ‣ −6 gravitational wave background: r<10 Wednesday, 30 September 2009 Reminder: parameter estimation using Bayesian techniques posterior: probability of probability of the data given prior the model the model probability given the data P (D|θ)P (θ) P (θ|D)= Evidence: ! P (D|θ)P (θ)dθ normalizing factor •Highly non-linear relationships between microphysical parameters & observables! •Reparametrization using physical intuition was necessary to get results efficiently. Wednesday, 30 September 2009 Constraints on microphysical parameters Non-Gaussian likelihood surface! ! " Best fit chisq about the same as LCDM. / 4' *2*3 !6 #2! 1 2) !5 1 *+0 !#$ #$ !#% '() ! " !#$ !" " " ! ! ! 7 7 % % # # ! " !#$ !" % ! ## ## ## / 9 #$ #$ #$ *+8 #$ '() & & & ! " !#$ !" % ! & #$ ## " " " " / ! - *. $ $ $ $ - *+, !" !" !" !" #$ '() !#$ !#$ !#$ !#$ ! " !#$ !" % ! & #$ ## !#$ !" $ " '() *+, / '() *+0 2)#2!*2*3 / '() *+8 / '() *+, *.!/ #$ 9 #$ 1 1 4' ! #$ 9 #$ - - R. Bean, X. Chen, HVP, J. Xu (PRD, 2007) arXiv:0710:1812 [hep-th] Wednesday, 30 September 2009 A Apossible possible disconnect?disconnect Theory: first-order PNGB R2 tensors assisted branesvector large-field monodromy monomial 3-form stochastic extended fluxes small-field D-term chaotic running mass non-gaussian F-term Horava-Lifshitz reheating unified hybrid old N-flation soft eternal GUT new DBI modular D3/D7 thermal throat hilltop warm Slow-roll curvaton Data can only pin down a handful of numbers. Model selection may not tell us much unless non-vanilla signatures observed. Figure: A. Liddle Wednesday, 30 September 2009 Constraints on the predicted power spectrum Comparison with constraints Comparison with constraints on empirical ns, dns/dlnk on slow roll single field prescription inflation [WMAP Collaboration, 2006] [Peiris & Easther, 2006] R. Bean, X. Chen, HVP, J. Xu (PRD, 2007) arXiv:0710:1812 [hep-th] Wednesday, 30 September 2009 Outline • Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations • The primordial power spectrum – Inferences from the CMB with inflationary priors • The primordial power spectrum – Minimally parametric reconstruction using cross-validation • CMB temperature anomalies – Beyond a posteriori statistics using consistency tests Wednesday, 30 September 2009 “Model Independent” power spectrum reconstruction ‣How do we know that “bottom-up/top-down” priors are consistent with the data? ‣How much scale-dependent structure do the data themselves require? ‣Use cross-validation to judge when we are fitting the noise by adding more structure to power spectrum shape. Sealfon et al. (2005), Verde & Peiris (arxiv:0802.1219) Wednesday, 30 September 2009 “Minimally parametric” technique ‣Three easy steps: -Select a number of knots and fit a piecewise cubic spline. -Penalise the likelihood for “wigglyness”. -Use cross-validation to choose optimal penalty. 2 log = log (Data α,P(k)) + λ dk (P !!(k)) L L | !k P (k) k Sealfon et al. (2005), Verde & Peiris (arxiv:0802.1219) Wednesday, 30 September 2009 Which is best? How well are you going to predict future data drawn from the same distribution? Wednesday, 30 September 2009 2-fold cross-validation power small large scales scales How well do a fit to the black points predict the red points, and vice versa? (CV score) Wednesday, 30 September 2009 power spectrum reconstruction with cross validation ‣Reasonable evidence for a red tilt, but none for a “running”. WMAP3 + LSS with CV scale dependence of spectral index small large scales scales Verde & Peiris (arxiv:0802.1219) Wednesday, 30 September 2009 power spectrum reconstruction with cross validation ‣Good way to identify systematics in datasets. WMAP3 alone with CV point sources? beams? Huffenberger et al. 07, Reichardt et al 08 scale dependence of spectral index small large scales scales Wednesday, 30 September 2009 WMAP5 analysis - hot off the press ‣PRELIMINARY! WMAP5 alone with CV scale dependence of spectral index small large scales scales Peiris & Verde in prep. Wednesday, 30 September 2009 Outline • Non-technical introductions – The Cosmic Microwave Background – Inflation - a theory for the origin of primordial fluctuations • The primordial power spectrum – Inferences from the CMB with inflationary priors • The primordial power spectrum – Minimally parametric reconstruction using cross-validation • CMB temperature anomalies – Beyond a posteriori statistics using consistency tests Wednesday, 30 September 2009 (a) Unmodulated CMB Polarization: Testing Statistical Isotropy ‣Isotropy Polarization“anomalies” identified Field in WMAP temperature field (e.g. hemispherical asymmetry, quadrupole-octupole alignment) ‣Any physical model of temperature2 anomalies provides testable predictions for √6 dτ τ(D) (Q iU)(nˆ)=statistics of polarizationdD e− field; goes beyondT2m(D anˆ posteriori) 2Y2m(nˆ inferences.) ± − 10 ! dD × ± m"= 2 − (b) Dipole Modulated !Dn x x Dn observer ˆ Drecn Drec recombination surface reionization recombination Dvorkin, Peiris & Hu (astro-ph/0711.2321) Wednesday, 30 September 2009 North/South power asymmetry North/South power asymmetry South (ecliptic) has more power than north Probability of 1% in an isotropic universe! ∼ Eriksen, Hansen, Banday, Gorski and Lilje, 2004. Wednesday, 30 September 2009 Quadrupole/octopole alignment Quadrupole-Octopole Alignment ! = 3 is planar: P 1/20 ∼ ! =2, 3 are aligned: P 1/60 ∼ de Oliveira-Costa, Tegmark, Zaldarriaga and Hamilton, 2003. Wednesday, 30 September 2009 Spontaneous Isotropy Breaking (C.Gordon,Spontaneous W.Hu, D.Huterer isotropy and T.Crawford, breaking 2005.) Φ(x)=g1(x) [1 + h(x)]+g2(x) ! ”modulating field” ! Gaussian random, isotropic, and homogeneous field: 3 g ∗(k)gi (k") = (2π) δ(k k")Pg (k) " i # − i ! Modulated field: 3 Φ∗(k)Φ(k") = (2π) δ(k k")[Pg (k)+Pg (k)] " # − 1 2 +[Pg (k)+Pg (k")]h(k" k) 1 1 − d 3k˜ + Pg (k˜)h∗(k k˜)h(k" k˜) ! (2π)3 1 − − Gordon, Hu, Huterer & Crawford (2005) Wednesday, 30 September 2009 Spontaneous Isotropy Breaking (C.Gordon,Spontaneous W.Hu, D.Huterer isotropy and T.Crawford, breaking 2005.) Φ(x)=g1(x) [1 + h(x)]+g2(x) ! ”modulating field” ! Gaussian random, isotropic, and homogeneous field: 3 g ∗(k)gi (k") = (2π) δ(k k")Pg (k) " i # − i ! Modulated field: 3 Φ∗(k)Φ(k") = (2π) δ(k k")[Pg (k)+Pg (k)] " # − 1 2 +[Pg (k)+Pg (k")]h(k" k) 1 1 − d 3k˜ + Pg (k˜)h∗(k k˜)h(k" k˜) ! (2π)3 1 − − Gordon, Hu, Huterer & Crawford (2005) Wednesday, 30 September 2009 Spontaneous Isotropy Breaking (C.Gordon,Spontaneous W.Hu, D.Huterer isotropy and T.Crawford, breaking 2005.) Φ(x)=g1(x) [1 + h(x)]+g2(x) ! ”modulating field” ! Gaussian random, isotropic, and homogeneous field: 3 g ∗(k)gi (k") = (2π) δ(k k")Pg (k) " i # − i ! Modulated field: 3 Φ∗(k)Φ(k") = (2π) δ(k k")[Pg (k)+Pg (k)] " # − 1 2 +[Pg (k)+Pg (k")]h(k" k) 1 1 − over realizations d 3k˜ of g1 and g2 + Pg (k˜)h∗(k k˜)h(k" k˜) ! (2π)3 1 − − Gordon, Hu, Huterer & Crawford (2005) Wednesday, 30 September 2009 (a) Unmodulated Dipolar modulation (hemispherical asymmetry) Dipolar Modulation (Temperature field) h(x) w z/Drec ∝ 1 (w 0.2) 1 ≈ ! Simulations: temperature field (a) Unmodulated (b) Dipole Modulated ˆ Drecn recombination surface (b) Dipole Modulated Wednesday, 30 September 2009 ˆ Drecn recombination surface DipolarDipolar modulation Modulation (polarization (Polarization predictions) Predictions) The hemispherical power asymmetry carries through the polarization field. ! Extreme example (w1 =2.5) Map of the first 10 multipoles of the E polarization field. Wednesday, 30 September 2009 PolarizationPolarization Fieldfield 2 √6 dτ τ(D) (Q iU)(nˆ)= dD e− T2m(Dnˆ) 2Y2m(nˆ) ± − 10 ! dD × ± m"= 2 − !Dn x x Dn observer Drec reionization recombination Wednesday, 30 September 2009 DipolarDipolar modulation Modulation (polarization (Polarization predictions) Predictions) The hemispherical power asymmetry carries through the polarization field. ! Extreme example (w1 =2.5) Map of the first 10 multipoles of the E polarization field. Wednesday, 30 September 2009 Quadrupole-octopoleQuadrupole-Octopole alignment Alignment Statistical significance? Angular momentum: 2 ! 2 2 Pm m T!m ˆ | ! | L! (θ, φ)= !2 P T 2 m | !m| ˆ2 1 ˆ2 ˆ2 L23 = 2 (L2 + L3) TOH map: Lˆ2 =0.962 = Lˆ2 0.962 in 0.25% of 104 MC! 23 ⇒ 23 ≥ ILC 3-year: Lˆ2 =0.943 = Lˆ2 0.943 in 0.85% of 104 MC! 23 ⇒ 23 ≥ Wednesday, 30 September 2009 Quadrupole-octopoleQuadrupole-Octopole alignment Alignment Statistical significance? Angular momentum: 2 ! 2 2 Pm m T!m ˆ | ! | L! (θ, φ)= !2 P T 2 m | !m| ˆ2 1 ˆ2 ˆ2 L23 = 2 (L2 + L3) TOH map: Lˆ2 =0.962 = Lˆ2 0.962 in 0.25% of 104 MC! 23 ⇒ 23 ≥ ILC 3-year: Lˆ2 =0.943 = Lˆ2 0.943 in 0.85% of 104 MC! 23 ⇒ 23 ≥ Wednesday, 30 September 2009 QuadrupolarQuadrupolar modulation model (“Axis - Example of Evil”) Modulation ⇓ Wednesday, 30 September 2009 TendencyTendency for Polarizationfor polarization alignments alignments ˆ(pol) L23 =0.46 ˆ(pol) L23 =0.67 ˆ(pol) L23 =0.90 Wednesday, 30 September 2009 PolarizationPolarization Statistics: statistics: Angular Angular Momentum momentum Wednesday, 30 September 2009 CMB Polarization: Is Potential Smooth? ‣“Glitches” in WMAP TT spectrum at large scales: statistics, systematics, or new physics? ‣Features in inflationary power spectrum? ‣Test: polarization transfer function narrower than temperature one. Mortonson, Dvorkin, Peiris & Hu (0903.4920) Wednesday, 30 September 2009 A “glitch” in the WMAP power spectrum? A feature at ! 20 40 in the WMAP CMB spectrum improves fit by ∆ χ 2 ∼ (10) − : statistics, systematics, or new physics? ∼ O Peiris + WMAP Collaboration (2003), Hannestad (2004), Shafieloo & Souradeep (2004), Covi et al. (2006), Hamann et al. (2007) Wednesday, 30 September 2009 A phenomenological inflationary model 2 2 A potential V (φ)=meff (φ)φ /2 Sudden change in effective mass during a phase transition: φ b m2 (φ)=m2 1+c tanh − eff d ! " #$ Causes transitory interruption of slow roll. Adams, Cresswell & Easther (2001) Wednesday, 30 September 2009 Primordial power spectrum Temporary interruption of slow roll imprints scale- dependent oscillations on density perturbations. Wednesday, 30 September 2009 SpatialSPATIAL to-TO -angularANGULARPROJECTION projection Consider angular projection at origin of potential ψ(x, η ) over last-scattering • ∗ surface; for a single Fourier component ψ(ˆn)=ψ(ˆn∆η, η ) ∆η ηR η ∗ l ≡ − ∗ ˆ = ψ(k, η ) 4πi jl(k∆η)Ylm(ˆn)Y ∗ (k) ∗ lm !lm l ˆ ψlm 4πψ(k, η )i jl(k∆η)Y ∗ (k) ∼ ∗ lm j (k∆η) peaks when k∆η l but for given l considerable power from k > l/∆η • l ≈ also (wavefronts perpendicular to line of sight) k∆η> l k k∆η=l – CMB anisotropies at multipole l mostly sourced from fluctuations with linear wavenumber k l/∆η where conformal distance to last scattering 14 Gpc ∼ ≈ 13 Wednesday, 30 September 2009 Transfer function and polarization Acoustic Feature Range ISW SW Acoustic Feature Range Reionization Mortonson, Dvorkin, HVP, Hu (2009) Wednesday, 30 September 2009 Transfer function and polarization Acoustic Feature Range ISW SW Acoustic Feature Range Reionization Mortonson, Dvorkin, HVP, Hu (2009) Wednesday, 30 September 2009 Polarization tests •Condition polarization realizations on observed temperature. •Study effect of “noise” from non-instantaneous reionization and tensors. •Consider false positives and false negatives. Mortonson, Dvorkin, HVP, Hu (2009) Wednesday, 30 September 2009 Results Planck can do 3 σ test, a cosmic- EE delta chisq variance experiment∼ reaches 8 σ . 0.06 ∼ Ideal 0.05 0.04 16 WMAP TT delta chisq 49 100 4 36 d 9 25 0.06 64 81 144 -4 196 -2 0.03 0.05 -2 -4 0.02 -6 0.04 400 900 d 0.01 256 324 -6 0.03 -8.3 -2 10 Planck -2 -4 -8 4 “step” width 0.05 0.02 -6 0.01 -4 -2 10 50 0.04 9 d 16 0 0.001 0.002 0.003 25 c 0.03 “step” height 0.02 16 1 4 9 25 144 0.01 36 49 6481 100 196 256 0 0.001 0.002 0.003 c Mortonson, Dvorkin, HVP, Hu (2009) Wednesday, 30 September 2009 Conclusions / Philosophical Summary ‣Inference from CMB data involves dealing with some difficult issues (cosmic variance, extrapolating physics into unexplored regimes). ‣Physical input in formulating priors forms (in my view) an essential ingredient in inferences about physics from such data. ‣Information content of the data is limited (attempting to blindly constrain a large parameter space with such limited data will lead to nothing but degeneracies). ‣A bit of thought (and physics) can help with seemingly intractable problems (data compression, reparameterization) ‣ Consistency checks with complementary data / formulating predictions of observables for future data is essential in testing inferences about physics from such data. Wednesday, 30 September 2009