Dispersion of the luminosity distance as a cosmological probe

Dispersion of the luminosity
distance as a cosmological probe
Ido Ben-­‐Dayan
DESY
Cambridge, Cosmo 2013
Collaborators: Maurizio Gasperini, Tigran
Kalaydzhyan, Giovanni Marozzi, Fabien Nugier,
Gabriele Veneziano.

Outline
! Current status of power spectrum measurements
! New probe: Lensing of Supernovae Type Ia
1. Average and dispersion of the luminosity redshift
relation
2. Lensing dispersion (Hui & Green, Dodelson et al.,
Amendola et al., Jonsson et al., Kronborg et al., March et
al.)
3. Constraints from current data.

Primordial Power
Spectrum

Planck Measurements
! Even with PLANCK, Ly-­‐alpha, etc. only probe ~8.5 e-­folds
out of 60.
! Few observables, huge degeneracy between
models.
P prim. (k) =A s
✓ k
k 0
◆ ns (k 0 ) 1+1/2↵(k 0 )ln(k/k 0 )+1/6 (k 0 )(ln(k/k 0 )) 2
The power spectrum is the actual obsevrable and we
need to measure it for as many e-­‐folds possible!

Example: “string vs.
field theory”
1.0
0.9
PLANCK
PIXIE
n s
0.8
0.7
0.6
0.5
Accidental
n⩵3
n⩵4
n⩵6
n⩵10
0 10 20 30 40 50
N
V ( )=⇤ 4 (1 a 1 a n n )
1309.xxxx, IBD, S. Jing
C. Wieck and A. Westphal

Spectral Distortions
! We can measure deviations of the CMB from a black
body spectrum – MODEL INDEPENDENT
⇒ Can be used to constrain the primordial power
spectrum for ! (1

Current and Future

d L -­‐z relation
• Gauge inv. Light-­‐cone avg. procedure. 2 nd
order pert. theory, NO DIVERGENCES!
• kUV=30h Mpc -­‐1 , WMAP data.
sm
0.14
0.12
0.10
0.08
0.06
0.04
PLANCK DATA
• Flux is minimally biased.
• The dispersion is large ~ 2-­‐10% ΛCDM, of the
critical density depending on the spectrum.
Scatter is mostly from the LC average. It
gives theoretical explanation to part of the
scatter of SN measurements. At z>0.3
dominated by lensing.
IBD, M. Gasperini, G. Marozzi, F. Nugier, G.
Veneziano
m-m M
0.02
0.00
0.3
0.2
0.1
0.0
-0.1
-0.2
0.6 0.8 1.0 1.2 1.4 1.6
PLANCK DATA
z
0.02 0.05 0.10 0.20 0.50 1.00 2.00
z

Lensing Dispersion of
SNIa
! Current data has SN up to z~1.3
total
µ (z apple 1) < 0.12
! Model dependent-­‐ LCDM and HaloFit model. (Smith
et al. 2003, Takahashi et al. 1,2 2012)
! An overall upper bound on dispersion. Any
enhancement of the power spectrum will increase
the dispersion. WHAT’S THE CORRECT SPECTRUM
s Z dk
µ(z) ' 0.7
˜ ⌘(z) =
Z z
0
✓ ◆ 3 k
k P (k, z) ˜ ⌘(z)
3
H 0
dy
p
⌦m0 (1 + y) 3 +⌦ ⇤0
µ(z = 1) ' 0.47s Z
dkk 2 P (k, 1) ⌘ 0.47 p T 2 (P )

200
100
50
Power spectrum
! HaloFit cannot be trusted for large running or
running of running.
! Treat F(k,z)=P NL /P L as a transfer function, at z=1.
! 2 methods: 1) step function 2) interpolated function.
Both extremely underestimating.
! kUV=320h. kUV=30h degrades results but still cuts out
parameter space allowed by PLANCK.
P prim. (k) =A s
✓ k
k 0
◆ ns (k 0 ) 1+1/2↵(k 0 )ln(k/k 0 )+1/6 (k 0 )(ln(k/k 0 )) 2
20
10
5
2
1
0.01 0.1 1 10 100
T 2 (P )=
T 2 (P )=
Z kUV
H 0
dkk 2 P L (1 + bH(k k NL ))
Z kUV
H 0
dkk 2 P L (k)(1 b + bH(k k NL )F (k, z))

1-­‐b+b F(k,z) 1309.xxxx
Region of constraints Hs m > 0.12L

1+b H(k-­‐k NL ) 1309.xxxx
Region of constraints Hs m > 0.12L

Summary
! Demonstrated the shortcomings of current inflationary probes.
! Calculation of the d L -­‐z relation, average and dispersion, useful for parameter
estimation.
! 2 New and extremely competitive probes:
1. Spectral Distortions – Model independent, current A

From Interpolation
0.08
Region of constraints Hs m > 0.12L
0.06
Running of Running d 2 nsêdlnk 2
0.04
0.02
0.00
-0.02
-0.04
-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06
Running dn s êdlnk

From Step
0.08
Region of constraints Hs m > 0.12L
0.06
Running of Running d 2 nsêdlnk 2
0.04
0.02
0.00
-0.02
-0.04
-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06
Running dn s êdlnk

Closer Look at the
Lensing Integrand

LC Averaging
! Useful for interpreting light-­‐like
signals in cosmological
observations.
! Hyper-­‐surfaces using meaningful
physical quantities: Redshift,
temperature etc.
! Observations are made on the
light-­‐cone. Volume averaging
give artefacts and the matching
with data is not clear.
! Past attempts: Coley 0905.2442;
Rasanen 1107.1176, 0912.3370

Light Cone Averaging
1104.1167
! A-­‐priori -­‐ the averaging is a geometric procedure, does not
assume a specific energy momentum tensor.
The prescription is gauge inv., {field reparam. A-­‐>A’(A), V-­‐
>V’(V)} and invariant under general coordinate
transformation. A(x) is a time-­‐like scalar, V(x) is null.
This gives a procedure for general space-­‐times.

Prescription Properties
! Dynamical properties: Generalization of Buchert-­‐Ehlers
commutation rule:
! For actual physical calculations, use EFE/ energy momentum tensor
for evaluation. Example: which gravitational potential to use in
evaluating the dL-­‐z relation.
! Averages of different functions give different outcome
F(S) ≠ F S

GLC Metric and Averages
! Ideal Observational Cosmology – Ellis et al.
! Evaluating scalars at a constant redshift for a geodetic observer.
~
I(
S,
w = ∫
I(
S,
w0
, z)
S = ;
I(1,
w , z)
1+
z =
2
a
a
0,
z)
d θ γ ( w0
, z,
θ ) S(
w0
, z,
θ
Υ
Υ
O
S
0
~
~
);

Exact Result -­‐ Flux
LC average of flux for any space-­‐time amounts to the
area of the 2-­‐sphere! (JM subleading correction drops
out)
Φ =
L
4πd ; d (z) = (1+ 2 L z)2 d S
; d 2 S
≡ dS =
L
dΩ O
γ
sin θ
γ = ρ 2 sinθ
d s
≡ ρ =
∑
l,m
a lm
(w 0
, z s
)Y lm
(θ,ϕ)
∫ d 2 θ γ = ∫ d 2 θρ 2 sinθ = ∑ a lm
(w 0
, z s
) 2
> a 2 00
Anisotropies always “mimic” acceleration!
l,m