Anisotropic cosmology?
Dominik J. Schwarz
Universität Bielefeld
Minimal model and its limitations
Origin of (an)isotropy
Observational tests: CMB and SNIa
2nd Bethe Center Workshop, Bad Honnef 2010
The minimal cosmological model
relies on
⋄ comological inflation:
isotropy, homogeity and spatial flatness
gaussian, scale-invariant and isentropic fluctuations A, n, (r)
⋄ the Einstein equation with a cosmological constant H0, Λ
⋄ the standard model of particle physics T0, Ωb, (Ων )
⋄ the existence of dark matter Ωcdm = 1 − Ωb − ΩΛ
and astrophysical parameters that encode complex physics τ, b, M, . . .
The cosmic energy budget (WMAP 7yr + H0 + BAO)
ΛCDM and massive ν s fit to
CMB/BAO/SNIa:
dark energy
neutrinos
atoms
dark matter
72% dark energy
23% cold dark matter
5% atoms
< 1% neutrinos
all ±1%
Komatsu et al. 2010
95% dark physics
Supernovae Ia
Union2: Amanullah et al. 2010
flatness and w ≈ −1 agrees also
with SN Ia, LSS, clusters
Conceptual limitations of the minimal model
⋄ 95% dark physics
⋄ cosmological constant problem: |Λ|κ < 10−120
⋄ coincidence problems:
Why is ΩΛ ∼ Ωm ∼ Ωb?
Why is znl(λeq) ∼ zacc?
⋄ origin of cosmological inflation
⋄ origin of isotropy and homogeneity
here: focus on aspects related to isotropy
Cosmological principle(s)
problem: initial conditions of the Universe
perfect cosmological principle: maximally symmetric space-time
steady state model, de Sitter model
×
cosmological principle: exact isotropy & homogeneity
×
Friedmann-Lemaı̂tre model; symmetry implies cosmic time; no LSS
statistical cosmological principle: statistical isotropy & homogeneity
perturbed FL; statistically isotropic and homogeneous perturbations
Does cosmological inflation predict a CP?
eternal inflation: CP not on global scale (multiverse)
observable universe:
inflation pushes pre-inflationary anisotropies and inhomogeneities far
beyond apparent horizon,
if number of e-foldings (N) large and start from smooth Hubble patch
Bianchi space-times, except Bianchi IX, isotropise; CP temporarily
e.g. Turner & Widrow 1986, Rothmann & Ellis 1986
general inhomogeneous models, perturbation theory:
δǫ
− ψ constant for kph ≪ H; What if ζ(t < tinfl ) ∼ 1?
ζ ≡ 3(ǫ+p)
What if N just ∼ 60? Pre-inflationary structure observable!
Initial conditions at µ ∼ MP
chaotic inflation:
4 initially
inflaton field ϕ̇2 ∼ V (ϕ) ∼ MP
2 initially
geometry H 2 ∼ k/a2 ∼ MP
while geometric and kinetic terms describe space-time and inertia,
the effective potential V encodes all information on interactions
without potential, the single fundamental scale would be MP
interactions introduce new scales, e.g. GF or ΛQCD
4?
What if the inflaton potential is limited to V ∼ M 4 ≪ MP
Example: effective Higgs potential
"
!
#
1
3
H
1 2
λ 4
φ +
H ln 2 −
+ ... ,
V ≃
24
16π 2 4
µ
2
1 2
H ≃ λφ ,
2
at µ ≫ mZ
Ford et al. 1993
Evolution of the quartic Higgs coupling λ
running of λ can lead to λ < 0
at µ > M
mH = 110 GeV
mH = 115 GeV
mH = 120 GeV
mH = 130 GeV
mH = 140 GeV
mH = 150 GeV
mH = 160 GeV
mH = 170 GeV
mH = 180 GeV
mH = 190 GeV
mH = 200 GeV
4
Quartic Higgs coupling λ
⇒ V complex at φ ∼ µ > M
5
3
2
1
complex inflaton potential
would imply inflaton decay
and no inflation
0
0
5
10
log10(Λ/GeV)
15
20
Hüske 2010
Initial conditions with two fundamental scales
4
modification of chaotic initial condition: V ∼ M 4 ≪ ϕ̇2 ∼ MP
kinetic energy dominates initially, i.e.,
ϕ̇2/2
ǫ1 ≡ d˙H = 3 2
≈3
ϕ̇ /2 + V
nevertheless, as ϕ̇ ∝ a−3, inflation starts when ǫ1 < 1
slow-roll after a few e-foldings, iff ϕi > 3MP
for M ∼ Mgut: ϕi ∼ 20MP and N ∼ 60
Observational constraints
0.4
0.35
∆N = 50
0.3
∆N = 60
∆ N*
for three of our models
r
0.25
0.2
m2 φ2/2
s.r. approx.
∆N = 50
0.15
λφ4/24 s.r. approx.
∆N = 60
0.1
0.05
0
0.92
0.94
0.96
0.98
1
1.02
ns
Ramirez & Schwarz 2009
Komatsu et al. 2009
N ∼ 60 is motivated and could fit CMB
Motivations for testing isotropy
⋄ check the minimal model
⋄ pre-inflationary perturbations (if N ∼ 60 or less)
⋄ investigate the nearby LSS; do we live in a large void?
⋄ discover systematic errors
here: CMB at large angular scales and SN Ia Hubble diagram
Why are large angular scales interesting?
180 deg
primordial
10
60 deg
4
Hubble
1000
1 deg
decoupling
comoving scale @MpcD
LCDM
100
LSS
10
0.01
0.1
1
10
z
100 1000
s(z) = θ dc(z)
Cosmological inflation — Generic CMB predictions I
temperature fluctuations:
P
δT (e) = ℓm aℓmYℓm(e); 2ℓ + 1 degrees of freedom for each ℓ
statistical isotropy:
hδT (Re1) . . . δT (Ren)i = hδT (e1 ) . . . δT (en )i,
• hδT (e)i = 0 and haℓm i = 0
• hδT (e1)δT (e2)i = f (e1 · e2) =
1
4π
P
ℓ (2ℓ
∀R ∈ SO(3), ∀n > 0
+ 1)Cℓ Pℓ(cos θ),
• haℓma∗ℓ′ m′ i = Cℓ δℓℓ′ δmm′ , Cℓ angular power spectrum
cos θ ≡ e1 · e2 with
gaussianity: no extra information in higher correlation functions
P
(best) estimator: Ĉℓ = 1/(2ℓ + 1) m |aℓm|2 (assumes statistical isotropy)
cosmic variance: Var(Ĉℓ ) = 2Cℓ2/(2ℓ + 1) (assumes gaussianity)
Cosmological Inflation — Generic CMB predictions II
scale invariance, n ≈ 1:
Cℓ ≈ 2πA/[ℓ(ℓ + 1)], at the largest scales
A ≈ 1000µK 2 (obs.)
angular 2-point correlation
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
0
50
100
Θ @degD
150
C(θ) without dipole (arbitrary units)
What can we test?
⋄ statistical isotropy
⋄ gaussianity
⋄ approximate scale invariance
violations may be due to secondary effects (e.g. ISW, Rees-Sciama),
foregrounds, instrument, . . .
no influence of secondaries between z1st h.c. and z ∼ 1 for θ > 60 deg
A test of statistical isotropy — Multipole vectors
alternative representation of multipoles
Maxwell 1891, Copi, Huterer & Starkman 2003
one (real) amplitude Aℓ and ℓ headless (unit) vectors:
2ℓ + 1 degrees of freedom
Tℓ(e) =
ℓ
X
m=−ℓ
aℓmYℓm(e) = Aℓ[v(ℓ,1) · · · v(ℓ,ℓ)]i1...iℓ [e · · · e]i1...iℓ
[. . .] . . . symmetric, traceless tensor product
1 v(2,1) · v(2,2)]
e.g. quadrupole: T2(e) = A2[(v(2,1) · e)(v(2,2) · e) − 3
Cosmic microwave background radiation
WMAP quadrupole and octopole
EQX
EQX
dipole
dipole
NEP
NEP
NSGP
NSGP
SSGP
SSGP
SEP
SEP
dipole
dipole
EQX
-0.021
EQX
0.000
T (mK)
0.021
-0.038
0.000
T (mK)
0.038
3 great circles defined by multipole vectors are nearly normal to ecliptic
1 great circle nearly normal to supergalactic plane
x
Schwarz et al. 2004, Copi et al. 2007
WMAP quadrupole plus octopole
EQX
dipole
NEP
NSGP
SSGP
SEP
dipole
EQX
-0.057
ecliptic is close to nodal line
power asymmetry for [2,3] and [6,7]
0.000
T (mK)
0.057
Schwarz et al. 2004, Copi et al. 2007
Schwarz et al. 2004, Freeman et al. 2006
Internal and external correlations of quadrupole and octopole
– quadrupole-octopole (qo) alignment at 99.6%C.L.
– dipole-qo alignement at 99.7%C.L.
– ecliptic-qo alignment at 95%C.L.
– ecliptic North-South asymmetry
– ecliptic is close to nodal line
Copi et al. 2007
full-sky maps violate statistical isotropy at large angular scales
Angular two-point correlation
P
Ĉpixel(θ) ≡ e1e2=cos θ T (e1 )T (e2 ) or
Ĉpower (θ) ≡ 1/(2π)
P
ℓ (2ℓ + 1)Ĉℓ Pℓ (cos θ)
full sky: Ĉpixel(θ) = Ĉpower (θ)
cut sky: Ĉpixel(θ) 6= Ĉpower (θ) with
cut sky: hĈpixel(θ)i = hĈpower (θ)i, iff statistically isotropic
similar for angular power spectrum estimators
e.g. pseudo-Cℓ vs. maximal likelihood estimator
WMAP angular correlation function
Sakar et al. 2010
P
b
C(θ) ≡ Pixel(ij) TiTj , with ei · ej = cos θ = µ
b does not assume statistical isotropy
estimator C
compare to 105 MC cut sky maps
Rα
Sα = −1 dµ C 2 (µ)
cut sky
) < 0.1%
P (S1/2
Status of CMB large angle anomalies
Copi et al. 2010
observed microwave radiation at > 60 deg disagrees with prediction
vanishing 2-point correlation is inconsistent at 99.9%CL
quadrupole and octopole
aligned with each other at > 99%CL
correlated with equinox/dipole at > 99%CL
correlated with ecliptic at > 95%CL
alignments extend up to multipoles ℓ ∼ 10 Land & Magueijo 2005
systematics or unexpected physics?
Solar system dust, large scale structure, cosmology, . . .
How to find the origin of disagreement
some ideas:
⋄ cosmological explanation:
N ∼ 60 could explain lack of correlation
d-q-o alignment could be pre-inflationary
ecliptic alignment would be fluke
⋄ nearby LSS (z < 0.1):
Rees-Sciama effect of 100 Mpc structures gives alignment
lack of correlation hard to explain, e.g. systematic
...
look at other cosmological probes, e.g. SN Ia
(An)isotropy of the low z Hubble diagram
Hubble diagrams from opposite hemispheres Schwarz & Weinhorst 2007
Constitution set (SALT2) Hicken et al 2009 at z < 0.2
¡Dq0 ¥
¡DH0 ¥
0
6.08
0
CMB dipole
EQNP
CMB dipole
EQNP
MASP
MANP
4.31
MASP
MANP
EQSP
|∆H0|
systematic effect or bulk flow?
EQSP
|∆q0|
Kalus & Schwarz (in prep.)
(An)isotropy of the low z Hubble diagram
4
4
2
N
N
2
0
q0
q0
0
-2
-2
S
S
-4
-4
-6
-6
-8
60
62
64
66
68
H0*Mpc*skm
70
72
74
60
MLCS31
65
H0*Mpc*skm
70
SALT2
∆H0
H0 ∼ 0.05 at z < 0.2 Schwarz & Weinhorst 2007, Kalus & Schwarz (in prep.)
Comparison to other directions on sky
Quasar Alignment
CMB Octopole
CMB Quadrupole
CMB dipole
MLCS2k2 3.1 SP
EQNP
MLCS2k2 1.7 SP
SALT SP
Weinhorst B
Union2
SALT2 SP
Velocity Flows
SALT2 NP Weinhorst A
Weinhorst C
SALT NP
MLCS2k2 1.7 NP
EQSP
MLCS2k2 3.1 NP
Schwarz & Weinhorst 2007, Kalus & Schwarz (in prep.)
Conclusions
⋄ cosmological inflation does not predict global isotropy
⋄ if N ∼ 60 or less, observable anisotropy
⋄ CMB shows several anomalies on largest angular scales
⋄ Hubble diagram at z < 0.2 anisotropic, could be systematic effect
deviations correspond to 0.1 mag
effect of acceleration is 0.2 mag, compared to a q0 = 0 model
back up slides
Just enough inflation (λϕ4)
1.1
2
1
ε1
0.9
1
εV
ε2
0.8
0.1
0.7
0.08
0.6
0.06
0.5
0.04
0
ε1
-1
0
5.1e+19
0.3
0.4
0.2
ε1*
0.02
0.4
4εV - 2ηV
5.2e+19
5.3e+19
5.4e+19
5.5e+19
ε2*
0
ε2
-0.2
-3
5.6e+19
GeV
0.2
4εV - 2ηV
-2
εV
-0.4
-4
5.1e+19
5.2e+19
0.1
5.3e+19
5.4e+19
5.5e+19
5.6e+19
GeV
0
-5
0
1e+19
2e+19
φ
3e+19
4e+19
5e+19
6e+19
0
1e+19
GeV
ǫn+1 ≡ d ln ǫn/dN , ǫ1 = d˙H, inflation ⇔ ǫ1 < 1
2e+19
φ
3e+19
4e+19
5e+19
GeV
Ramirez & Schwarz 2009
6e+19
Primordial fluctuations
1e-05
power-law approximation
1e-10
amplitude at first order
1e-15
1e-20
power-law approximation
with running
∆2R(k)
1e-25
1e-08
1e-30
1e-35
∆2R (k=k*)
1e-09
1e-40
1e-45
1e-50
1e-10
1e-55
1e-05
1
1
10
100000
100
1e+10
k/k*
1e+15
1e+20
1e+25
Ramirez & Schwarz 2009
Physical interpretation of multipole vectors
monopole: no vector
dipole: trivial d = v(1,1)
quadrupole:
√
(2,1)
(2,2)
the two vectors define a plane, extrema at ±(v
±v
)/ 2
natural to define “oriented area”: w(2;1,2) = ±(v(2,1) × v(2,2))
general multipole:
real and imaginary parts of spherical harmonic function Yℓm have
ℓ − |m| vectors equal ±z, |m| vectors in x-y plane
Attempts of cosmological explanations for lack of correlation
angular 2-point correlation
2.0
scale-invariant
no quadrupole
IR cut-off in PHkL
1.5
1.0
0.5
0.0
-0.5
-1.0
0
50
100
150
Θ @degD
cosmic topology
Aurich, Lustig, Then & Steiner 200n
IR cut-off in P (k), but ISW regenerates IR power
Ramirez & Schwarz 2009
How to resolve the issue?
– more data and better systematics from WMAP
– Planck has different systematics (e.g. scan strategy, one beam on sky)
– Planck adds information in Wien regime (e.g. solar system dust?!)
– exploit frequency, time and polarisation information
– correlation with non-CMB probes (e.g. radio galaxies, clusters)
make progress by exclusion of wrong possibilities
e.g. additive axial effect is excluded x
Rakić & Schwarz 2007
Main problem: additive extra foreground conflicts with low power
0
0.8
0.8
WMAP(3yr)
0.75
0.75
0.7
0.65
0.6
0.55
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004
trunc
Sfull
[0.1mK]4
left: CMB only, right: additional axial effect
0.008
0.01
0.012
0.014
0.8
WMAP(3yr)
0.75
0.7
0.65
0.6
0.6
0.55
0.55
0.5
0.5
0.45
0.006
0.65
0.5
0.45
Sww
0.5
0.45
0.004
0.7
0.6
0.55
0.002
0.75
0.7
0.65
Sww
0
0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004
0.8
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
0.002
0.004
0.006
0.008
0.01
trunc
Sfull
[0.1mK]4
0.012
0.014
Rakić & Schwarz 2007
(An)isotropy of the low z Hubble diagram
Hubble diagrams from opposite hemispheres Schwarz & Weinhorst 2007
Constitution set Hicken et al 2009: ∆(χ2/dof) at z < 0.2
£
2
DΧmin
dof
§
£
0
0.5
CMB dipole
EQNP
2
DΧmin
dof
§
0
0.5
MASP
CMB dipole
EQNP
MASP
MANP
EQSP
EQSP
MANP
MLCS31
SALT2