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Cosmological Cosmological principle principle and and the the Cosmic Cosmic Microwave Microwave Background Background Tarun Souradeep I.U.C.A.A, Pune IIT Kanpur colloquium (Apr. 4, 2005 ) Amir Hajian Arman Shafieloo, Sanjit Mitra Anand Sengupta, Jeremie Lasue,… The Realm of Cosmology Basic unit: Galaxy Size : 10-100 kilo parsec(kpc.) Mass : 100 billion Stars Measure distances in light travel time 1 pc. (parsec) = 200,000 AU =3.26 light yr. Measure Mass in Solar mass = 2 ×10 Kg . 30 Andromeda Galaxy The Realm of Cosmology Galaxy Clusters Size : Mega parsecs (Mpc.) Mass : 100 –1000 Galaxies (5% luminous, 15% hot gas, 80%dark matter !) Coma cluster The Realm of Cosmology Super Clusters Size : 10 Mega parsecs Mass : few 1000 Galaxies Perseus super cluster The Realm of Cosmology Distant galaxies beyond a cluster …. The Realm of Cosmology …… most distant galaxies Hubble deep field 2dF Galaxy Redshift Survey 3D location of 230 000 galaxies The Realm of Cosmology Few billion parsecs SLOAN DIGITAL SKY SURVEY (SDSS) The Realm of Cosmology The Realm of Cosmology Distribution of dark matter in the universe Simulation box 1 Billion parsecs Observable universe 4 Billion parsecs The Realm of Cosmology Distribution of dark matter in the universe Simulation box 1 Billion parsecs Observable universe 4 Billion parsecs Galaxies light up at the densest points How can we even hope to comprehend this immensely large & complex Universe !?! Look for an appropriate simple model The Isotropic Universe Distribution of galaxies on the sky is broadly isotropic North Lick Observatory survey South The Isotropic Universe Distribution of galaxies on the sky is broadly isotropic Isotropy around every point implies Homogeneity Î Cosmological principal The Expanding Universe Einstein’s General relativity applied to an uniform distribution of matter on cosmic scales leads to a smooth expanding universe The Expanding Universe Leads to the Hubble’s law Recession velocity is Proportional to the distance v H 0DL = c Present Expansion rate : H 0 = 71 km / s / Mpc. 3H 02 ⇒ Critical density, ρ c = = 10 − 29 gm/cm3 8πG The Expanding Universe Expansion implies Hot early universe Smaller => hotter Wavelength of light is stretched by expansion Î Redshift, z=v/c Redshift is related to distance Geometry of the Universe Spherical Universe ρ , Ω0 = ρc 3H 2 ρc = 8π G ρ > ρc Constant positive curvature ρ < ρc Hyperbolic Universe Constant negative curvature Flat Universe ρ = ρc FRW models: Expansion, Geometry & Matter Ω m + ΩV + Ω K = 1 The Isotropic Universe Serendipitous discovery of dominant Radiation content of the universe as an extremely isotropic, Black-body bath at temperature To =2.73K . “Clinching support for Hot Big Bang model” The dominant radiation component in the universe D. Scott ‘99 The most perfect Black-Body spectrum in nature COBE –FIRAS The CMB temperature – A single number characterizes the radiation content of the universe!! COBE website Pristine relic of a hot, dense & smooth early universe Hot Big Bang model Post-recombination :Freely propagating through (weakly perturbed) homogeneous & isotropic cosmos. Pre-recombination : Tightly coupled to, and in thermal equilibrium with, ionized matter. (text background: W. Hu) Predicted as precursors to the observed large scale structure After 25 years of intense search, tiny variations (~10 p.p.m.) of CMB temperature sky map finally discovered. “Holy grail of structure formation” CMB anisotropy is related to the tiny primordial fluctuations which formed the Large scale Structure through gravitational instability Simple linear physics allows for accurate predictions Consequently a powerful cosmological probe Recall Fourier series ∆T (θ ) = ∑ ak cos(2πkθ ) + ∑ bk sin( 2πkθ ) k k CMB Anisotropy Sky map => Spherical Harmonic decomposition ∞ ∆ T (θ , φ ) = ∑ l ∑a l =2 m=− l Y (θ , φ ) lm lm Complete Statistical description of CMB Anisotropy Angular Power Spectrum Cl = l (l + 1) alm a * lm Low multipole : Sachs-Wolfe (SW) plateau • Amplitudes and Spectral indices of primordial perturbations -inflaton potential, geometry & topology of space • Gravity waves, isocurvature contribution • Late & Early Integrated Sachs-Wolfe – rise to the first “Doppler” peak -- geometry, reionization history, ... Moderate multipole : Acoustic “Doppler” peaks Location and spacing of peaks- Ω tot Adiabatic/isocurvature initial conditions. Amplitude of the first peak and relative heights of second peak Weaker dependence on ΩΛ ΩB in both cases High multipole : Damping tail Ω B , ∆z recomb •Form of decay -- Ω B , gravitational lensing, secondary anisotropy, patchy • Location -- reionization, …. Fig. M. White 1997 The Angular power spectrum of the CMB anisotropy depends sensitively on the present matter current of the universe and the spectrum of primordial perturbations Cl Η0 Ωtot ΩCDM ΩΛ Ων Fig.. Bond 2002 The Angular power spectrum of CMB anisotropy is considered a powerful tool for constraining cosmological parameters. Sensitive to curvature l = 220 1− ΩK Fig:Hu & Dodelson 2002 l Sensitive to Ordinary matter ∆T = 74 µK Fig:Hu & Dodelson 2002 Post-COBE Ground & Balloon Experiments Python-V 1999, 2003 Boomerang 1998 DASI 2002 (Degree Angular scale Interferometer) Archeops 2002 Highlights of CMB Anisotropy Measurements (1992- 2002) NASA Launched July 2001 First year data results announced on Feb. 11, 2003 ! Microwave Anisotropy Probe (now renamed Wilkinson +MAP=WMAP) Ka band 33 GHz K band 23 GHz CMB anisotropy signal Q band 41 GHz W band 94 GHz NASA/WMAP science team V band 61 GHz NASA/WMAP science team dus saal baad …. Excellent match !! NASA/WMAP science team Signal / Noise > 1 for l ≤ 650 Cosmic variance limited errors for l ≤ 350 Ongoing work IIT Kanpur + IUCAA (Pankaj Jain, Rajib Saha, TS) l(l + 1)Cl 2π Multipole l NASA/WMAP science team 1st Peak at l = 220 ± 1 ∆T = 74.7 ± 0.5µK 1st Trough at l = 411.7 ± 3.5 ∆T = 41 ± 0.5µ K 2 nd Peak at l = 546 ± 10 ∆T = 48.8 ± 0.9 µ K Thompson scattering of the CMB anisotropy quadrupole at the surface of last scattering generates a linear polarization pattern in the CMB. Three additional Power spectra : the two polarization modes and the cross correlation with temperature anisotropy. (Fig:Hu & White , 97) Initial metric perturbation mix correspondence : z Scalar perturbations predominantly generate the Electric (E) polarization mode. z Vector perturbations predominantly generate the magnetic (B) polarization mode . z Tensor perturbations generate the both modes in comparable amounts . z Temperature anisotropy Cross-correlation only with the E-mode. DASI detection l =220-400 Sept 2004: More recent results From CBI, DASI,CAPMAP Proof of inflation? Anti-correlation peak l = 137 ± 9, z reion = 20 +−10 9 NASA/WMAP science team ∆T = −35µK Adiabatic IC TE peak out of phase l = 300 Consistent with ∆T for l ≥ 20 Cosmological Parameters Markov Chain Monte Carlo Multi-parameter (7-11) joint estimation (complex covariance, degeneracies, priors,… Æ marginal distributions) Dark energy Cosmic age Dark matter Baryonic matter Expansion rate Optical depth Baryonic matter density Estimate of Cosmological Parameters R.Sinha, TS Cosmic Matter density Total energy density Baryonic matter density Expansion rate of the universe Age of the universe Dark energy density Dark matter density Dawn of Precision cosmology !! NASA/WMAP science team Who ordered Dark Energy? Is it the Cosmological constant? The Cosmic repulsive force Einstein once proposed and later denounced as his ‘biggest blunder” ? Quantum fluctuations Early Universe super adiabatic amplified by inflation (rapid expansion) The Cosmic screen Galaxy & Large scale Structure formation Via gravitational instability Present Universe z Power spectrum Energy scale and Model of inflation Geometry of the universe Topology of the universe z Spin characteristics Scalar --- Density perturbations Tensor --- Gravity waves Vector --- rotational modes z Type of scalar perturbations Adiabatic --- no entropy fluctuations Isocurvature -- no curvature fluctuations z Underlying statistics Gaussian (eg., inflation) Non-Gaussian (eg. Topological defects) CMB anisotropy has two independent aspects: dk Cl = ∫ P(k )Gl (k ) k P(k ) Primordial power spectrum from Early universe Gl (k ) Post recombination Radiation transport in a given cosmology A scalar field displaced from the minima of its potential Linde’s chaotic inflation φ&& + 3Hφ& + V ′ = 0 1 &2 2 3H = ρ = φ + V 2 p = φ& 2 − V 1 2 String theory Landscape: Non trivial skiing slopes Early universe physics waiting to be discovered!!! Intriguing: Lack of power at large angular scales (θ ≥ 60o ) ? Similar to bump in Archeops ? Low Quadrupole l = 2 ∆T2 = 8 ± 2 µK 7 (COBE ∆T2 =10 + − 4 µK ) NASA/WMAP science team Intriguing: Lack of power at large angular scales (θ ≥ 60o ) Can imply more than just the suppression of power in the low multipoles ! NASA/WMAP science team Features in the primordial power spectrum ? (Shafieloo & Souradeep, PRD 04 ) Primordial power spectrum from Early universe can be deconvolved from CMB anisotropy spectrum Horizon scale dk Cl = ∫ P(k )Gl (k ) k Improved Error sensitive iterative Richardson-Lucy deconvolution method Recovered spectrum shows an infra-red cut-off on Horizon scale !!! Is it cosmic topology ? Signature of pre-inflationary phase ? Trans-Planckian physics ? …. Angular power spectrum from the recovered P(k) (Shafieloo & Souradeep) ( Jeremie Lasue & Souradeep, 2003) A G z Infra red cutoff : Interesting constraint on single scalar field inflation. Agw As =1 Is the Universe Compact ? Simple Torus (Euclidean) Homogenous & isotropic But Multiply connected (compact) universe ? Compact hyperbolic space Poincare dodecahedron Recent Nature article Generic Signature of compact universe • The eigenvalue spectrum is discrete (Weyl formula , ∆k j ≈ ( j V ) −1/ n ) ÎAn infra-red cutoff in the power of fluctuations on wavelengths larger than the size of the space. (Low multipole of CMB anisotorpy suppressed i.e., large angular scales) Surface S divides the space into two subspaces The isometric constant k min A( S ) hC = inf min(V ( M 1 ),V ( M 1 )) hC ≥ 2 Torus: k min 2 ≥ L Cheeger’s inequality Infrared cutoff Î Compact Universe? Also expect characteristic correlation patterns in the CMB sky ! Look beyond the angular power spectrum for violation of statistical isotropy Radical violation (Multiple imaging): Î Matched pairs of circles of CMB anisotropy (Cornish, Spergel, Starkman’98) Î Anti-correlated circle centers (Bond, Pogosyan,TS ’00) Mild violation : Î Preferred directions in the Dirichlet domain. Î Size of the Dirichlet Domain w.r.t. Sphere of Last Scattering (in-radius, out-radius). Low Multipoles of WMAP Is there evidence of a preferred axis ? Î Statistical isotropy breakdown quadrupole octopole hexadecapole Dynamic quadrupole correction quadrupole +octopole Tegmark et al. 2003 (astro-ph/0302496), de Oliveira Costa et al. (astro-ph/0307282), Asymmetries in the CMB anisotropy N-S asymmetry H. K. Eriksen, et al. 2004, F. K. Hansen et al. 2004a,b (in local power) Larson & Wandelt 2004, Park 2004 (genus stat.) Special directions High N-S Tegmark et al. 2004 (l=2,3 aligned) asymmetry Copi et al. 2004 (multipole vectors) Ralston & Jain 2004 (Virgo alignment) Land & Magueijo 2004 (cubic anomalies) Low N-S Prunet et al., 2004 (mode coupling) asymmetry . . Broadly, stat. properties are not invariant under rotations Breakdown of Statistical isotropy ? I.e., Fig: H. K. Eriksen, et al. 2003 Statistics of CMB CMB Anisotropy Sky map => Spherical Harmonic decomposition ∞ ∆ T (θ , φ ) = ∑ l ∑a l =2 m=− l Statistical isotropy Y (θ , φ ) lm lm alm al*'m ' = Cl δ ll 'δ mm ' Single index n: (l,m) -> n Diagonal alm al*'m ' = Cl δ ll 'δ mm ' Statistical isotropy SI violation : alm al*'m ' ≠ Cl δ ll 'δ mm ' Mild breakdown alm al*'m ' * al 'm ' al*'m ' alm alm (Bond, Pogosyan & Souradeep 1998, 2002) SI violation : alm al*'m ' ≠ Cl δ ll 'δ mm ' Radical breakdown alm al*'m ' * al 'm ' al*'m ' alm alm (Bond, Pogosyan & Souradeep 1998, 2002) BiPS: In Harmonic Space • Correlation is a two point function on a sphere ) ) C ( n1 , n2 ) = ∑A l1l2 LM LM l1l2 ) ) {Yl1 ( n1 ) ⊗ Yl2 ( n2 )}LM A Bipolar spherical harmonics. ) ) {Yl1 (n1 ) ⊗ Yl2 (n2 )}LM ) ) = ∑ Cl1LM Y ( n ) Y ( n l2 m1m2 l1m1 1 l2 m2 2) • Inverse-transform LM l1l2 BiPoSH m1m2 Clebsch-Gordan ) ) ) ) * = ∫ dΩn1 ∫ dΩn2C(n1, n2 ){Yl1 (n1) ⊗Yl2 (n2 )}LM = ∑ al1m1al2m2 Cl1m1l2m2 LM m1m2 Linear combination of off-diagonal elements Recall: Coupling of angular momentum states l1m1l2 m2 | lM l1 − l ≤ l2 ≤ l1 + l, m1 + m2 + M = 0 lM * BiPoSH Al1l2 = ∑ al1m1 al2 M +m1 coefficients : m1 lM Cl1m1l2 M +m1 • Complete,Independent linear combinations of off-diagonal correlations. • Encompasses other specific measures of off-diagonal terms, such as - Durrer et al. ’98 : D l ≡ a lm a l + 2 - Prunet et al. ’04 : D ( i ) ≡ a a l lm l +1 m ∑A =∑ A = lM m +i lM ll ' C ll+M2 m l m lM ll ' C ll+M1 m +i l m lM BiPS: rotationally invariant κ ≡ l ∑| A M ,l1 ,l2 | ≥0 lM 2 l1l2 LM l1l 2 A ∝ Cl1δ l1l2 δ L 0δ M 0 A ∝ Cl 00 ll Structure of BiPoSH A 4M ll ' A 2M ll ' Spherical harmonics alm Bipolar spherical harmonics lM ll ' A Spherical Harmonic BiPoSH coefficents coefficents Cl Angular power spectrum κ l BiPS Spherical harmonics alm Spherical Harmonic Transforms Cl Angular power spectrum Bipolar spherical harmonics lM ll ' A BipoSH Transforms κ l BiPS (Bipolar Power Spectrum) Bias corrected BiPS measurement (A. Hajian and Souradeep, ApJ Lett. 2003) Bias Bl = κ~l − κ l Cosmic Variance 2 (∆κ l ) 2 = κ~l − κ~l 1 ∆κ l ∝ l Analytic estimate for bias and cosmic variance match numerical measurements on simulated statistically isotropic maps ! 2 Testing Statistical Isotropy of WMAP (for WMAP best fit model) Foreground cleaned map (Tegmark et al. 2003) (Hajian, TS, Cornish astro-ph/0406354) ILC NASA/WMAP science team Circles search (Cornish, Starkman, Spergel, Komatsu 2004) Angular power spectra of the maps (compared to the WMAP best fit model) • `Tegmark’ Foreground cleaned map • `Spergel’ Circles search map • ILC: WMAP internal combination map • WMAP best fit curve • Average of 1000 realizations Scanning the l-space with different windows •Maps can be filtered by isotropic window to retain power on certain angular scales, (eg., l~30 to 70) alm → Wl alm •Cosmic variance >> Noise •Tegmark’s map is ‘foreground’ free Testing Statistical Isotropy of WMAP Low pass Gaussian filter at l= 40 (Hajian, TS, Cornish astro-ph/0406354) (assuming WMAP best fit model) Probability Distribution of BiPS Obtained from measurements of 1000 simulated SI CMB maps. Can compute a Bayesian probability of map being SI for each BiPS multipole (Given theory Cl) Probability of a Map being SI (Hajian, TS, Cornish astro-ph/0406354) Bayesian probability Low pass Gaussian filter at l= 40 Probability of a Map being SI Bayesian probability Band pass filter between multipoles 20-30 BiPS imply WMAP is Statistically Isotropic !! What does the null BiPS meaurement of CMB maps imply Constraints on sources of Statistical Anisotropy • Ultra large scale structure and cosmic topology. • Primordial magnetic fields (based on Durrer et al. 98, Chen et al. 04). • Observational artifacts: – – – – Anisotropic noise Non-circular beam Incomplete/unequal sky coverage Residuals from foreground removal CMB measurements with non-circular beam Beam : B(nˆ ,zˆ) = ∑ Bl β lm (nˆ ) Ylm (nˆ ) lm WMAP Q beam Eccentricity =0.7 C(nˆ1,nˆ2) ≠ C(nˆ1 •nˆ2) Bias : Cl = ∑ All 'C s l l' (S. Mitra, A. Sengupta, TS, PRD 04) Ultra Large scale structure of the universe How Big is the Observable Universe ? Relative to the local curvature & topological scales Simple Torus (Euclidean) Eg., Zeldovich & Starobinsky 1972 Cosmic topology Quantum creation Multiply connected universe ? of a finite universe !?! MC spherical space (“soccer ball”) Eg., Gott ’70, Cornish et al. 1996, Linde ‘04 Compact hyperbolic space BiPS signature of Flat Torus spaces κl l Hajian & Souradeep (astro-ph/0301590) BiPS signature of a “soccer ball” universe (Hajian, Pogosyan, TS, Contaldi, Bond : in progress.) ΩK = Ideal, noise free maps predictions κl l BiPS signature of a “soccer ball” universe (Hajian, Pogosyan, TS, Contaldi, Bond : in progress.) Ωtot = 1.013 κl Ideal, noise free maps predictions l Measured BiPS for a “soccer ball” universe (Hajian, Pogosyan, TS, Contaldi, Bond : in progress.) 2.5 Ωtot = 1.013 2.0 κl 1.5 1000 simulated full sky maps with WMAP noise 1.0 0.5 l Summary • CMB anisotropy measurements Æ precision cosmology • Can also test the “cosmological principle” • Propose BiPS as a generic measure for detecting and quantifying Statistical isotropy violations. Thank you !!! ÎBiPS is insensitive to the overall orientation of SI breakdown (e.g., orientation of preferred axes). Hence constraints are not orientation specific. Î Computationally fast method • Null results on some WMAP full sky maps. Î SI improves for a theory that predicts low power on low multipoles. • Can constrain/detect cosmic topology and Ultra large scale structure, primordial magnetic fields.. ÎBiPS promises to constrain Dodecahedron universe strongly. • Diagnostic tool for observational artifacts.