Download Cosmological principle and the Cosmic microwave

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
ƒ Location and spacing of peaks-
Ω tot
Adiabatic/isocurvature initial conditions.
ƒ 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.