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University of Pennsylvania www.physics.upenn.edu/~lverde

1) Motivation and basics
Large Scale Structure probes
(spherical cows)
2) Real world effects
3) Measuring P(k) & (Statistics)
(less spherical cows)

The standard cosmological model
96% of the Universe is missing!!!
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The standard cosmological model
96% of the Universe is missing!!!
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Major questions :
Questions that can be addressed exclusively by looking up at the sky
1)What created the primordial perturbations?
2) What makes the Universe accelerate?
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These questions may not be unrelated

CMB is great and told us a lot, but large scale structures are still useful:
Check consistency of the model
If this test is passed
Combine to reduce the degeneracies
We will concentrate on dark energy and inflation

ln P(k)
A
n slope
Amplitude of the power law
A
(convention dependent)
!
ln k

ln P(k)
A
n(k) slope
Amplitude of the power law a =dn/dlnk
A,n
(convention dependent)
!
ln k

CONSTRAINTS ON
NEUTRINO MASS

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Neutrino mass
Spergel et al ‘07
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CMB+SDSS LRG
Tegmark et al ‘07
0.9eV (95% CL)
CDM density
WMAP II
WMAP+high l experiments
SDSS main
2dFGRS
LRG SDSS
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WMAP II
2dFGRS
SDSS main

SN1A
Riess et al 04
WMAPII + H
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WMAPII
2dfGRS ‘02
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Q ui ck Ti m e ™ an d a
T IF F ( LZ W) de co m pr e ss or a re ne ed ed t o s ee th i s pi c tu r e.
From Sperget et al 07

Planck scale
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(At EW scale it’s only 56 orders of magnitude)
If it dominated earlier, structures would not have formed

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And it’s moving fast

What’s going on?
Non exhaustive list of possibilities:
We just got lucky
“landscape” there are many other vacuum energies out there with more reasonable values
It is a slowly varying dynamical component (quintessence)
Einstein was wrong (we still do not understand gravity)

Equation of state parameter w= p/ r w=-1 is cosmological constant what other options to consider?
clustering?
Couplings?
If dark energy properties are time dependent, so are other basic physical parameters

Varying fine structure constant alpha QuickTime™ and a
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Oklo Natural reactor:
1.8 billion yr ago there was a natural water-moderated fission reactor in Gabon.
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Isotopic abundances contrain 149 Sm neutron capture cross section ad thus alpha
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SN

Why so weak dark energy constraints from CMB?
The limitation of the CMB in constraining dark energy is that the CMB is located at z=1090.
We need to look at the expansion history
(I.e. at least two snapshots of the Universe)
What if one could see the peaks pattern also at lower redshifts?

Baryonic Acoustic Oscillations
Evolution of a single perturbation,
Imagine a superposition
For those of you who think in Real space
Courtesy of D. Eisenstein

If baryons are ~1/6 of the dark matter these baryonic oscillations should leave some imprint in the dark matter distribution
Fore those of you who think in Fourier space

Matter-radn equality
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Acoustic horizon at last scattering
Data from Tegmark et al 2006

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DR5 from Percival et al 2006
Robust and insensitive to many systematics

THE SYMPTOMS
Or OBSERVATIONAL EFFECTS of DARK ENERGY
Recession velocity vs brightness of standard candles: dL(z)
CMB acoustic peaks: Da to last scattering
Da to z survey
LSS: perturbations amplitude today, to be compared with CMB
Perturbation amplitude at z survey

Galaxy clusters number counts
Galaxy clusters are rare events:
P(M,z) oc exp(d 2 / s (M,z) 2 )
In here there is the growth of structure d x
Beware of systematics! “What’s the mass of that cluster?”

Galaxy clusters number counts
Galaxy clusters are rare events:
P(M,z) oc exp(d 2 / s (M,z) 2 )
In here there is the growth of structure d x
Beware of systematics! “What’s the mass of that cluster?”

V(  )
H ~ const

Solves cosmological problems (Horizon, flatness).
Cosmological perturbations arise from quantum fluctuations, evolve classically.
Guth (1981), Linde (1982), Albrecht & Steinhardt (1982), Sato (1981), Mukhanov &
Chibisov (1981), Hawking (1982), Guth & Pi (1982), Starobinsky (1982), J. Bardeen,
P.J. Steinhardt, M. Turner (1983), Mukhanov et al. 1992), Parker (1969), Birrell and
Davies (1982)

Flatness problem
Horizon problem
Structure Problem

Seeing (indirectly) z>>1100
Information about the shape of the inflaton potential is enclosed in the shape and amplitude of the primordial power spectrum of the perturbations.
Information about the energy scale of inflation (the height of the potential) can be obtained by the addition of B modes polarization amplitude.
In general the observational constraints of Nefold>50 requires the potential to be flat (not every scalar field can be the inflaton). But detailed measurements of the shape of the power spectrum can rule in or out different potentials .

But the spacing of the fluctuations
(their power as a function of scale) depend on how fast they exited the horizon (H)
Which in turns depend on the inflaton potential
The shape of the primordial power spectrum encloses information on the shape of the inflaton potential!

r
dn s
/dlnk=0 dn s
/dlnk=0
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r
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p=4
n n p=2
Models like V(  )~  p
For 50 and 60 e-foldings p fix, Ne varies p varies, Ne fix

Weak lensing (cosmic shear)
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Very near future:
Atacama Cosmology telescope
(& South Pole telescope, & Planck)
High resolution map of the CMB
Use the CMB as a background light to “illuminate” the growth of foreground cosmological structures
Thermal Sunyaev-Zeldovich
Kinetic SZ
Coma Cluster T electron
= 10 8 K e e e e e e e e e -
CMB gravitational
Lensing

Large-scale structure (LSS) (in combination with CMB)
Can be used to test the consistency of the model
(LCDM) and if that holds, to better constrain cosmology
2 problems:
can be addressed exclusively by looking up at the sky
In the future expect an avalanche of LSS data (and acronyms)
So far we have seen the basic theory behind LSS

Great walls
Fingers-of -God

In linear theory : enhancement of P(k) along the line of sight
Kaiser (1987)

(
) z obs
= z true
+ d v / c d v prop. to
 m
0.6 dr/r =  m
0.6 b -1 d n/n

(bias) shells
 s p linear
Fourier space
Non-linear

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Great walls
Fingers-of -God

?
Measured for 2dFGRS (Verde et al. 2002)

(Licia Verde)

CMBFAST or CAMB to get P(K)
Bayes Theorem:
P ( a i
| Data )
=
P ( a i
) P ( D | a i
) / P ( D )
What you really want
(Posterior)
Prior Likelihood
You should not forget

Likelihood: Gaussian vs non-gaussian
What is the probability distribution of your data?
Examples: Cl, alm, d
, etc..
Gaussian likelihood:
L =
1
( 2

) n det
 exp[

1
2
 x
T  
1
 x ]
 x
=
( data
 theory )
If data uncorrelated… much simpler
Central Limit Theorem  distribution will converge to Gaussian
Best fit parameters  Maximize the likelihood
Likelihood analysis

Why errors?
a i true a
i
1
 s
2
 s
3
 s
68 .
3 %
95 .
4 %
99 .
73 %
Joint or marginalized?
Cosmic variance noise
( ignore approximations, mistakes etc ..)
Errors

From: “Numerical recipes” Ch. 15
Errors


2 ln L
= 
2
From: “Numerical recipes” Ch. 15
If likelihood is Gaussian and Covariance is constant
Example: for multi-variate Gaussian
Errors

There is a BIG difference between
 2

2
Only for multi varaite Gaussian with constant covariance

Statistical and systematic errors
Examples of statistical (random) errors: cosmic variance, instrumental noise, roundoff (!)…..
Examples of systematic errors: approximations, incomplete modeling, numerics, ….
(introduce biases)
As you add more data points (or improve the S/N) the statistical errors become smaller but the systematic errors do not.
Errors

Operationally:
Grid-based approach
 m e.g., 2 params: 10 x 10
What if you have (say) 7 parameters?
You’ve got a problem !
s
8

Random walk in parameter space
At each step, sample one point in parameter space
The density of sampled points
 posterior distribution
FAST: before
7
10 likelihood evaluations, now
10
5 marginalization is easy: just project points and recompute their density
Adding external data sets is often very easy

Operationally:
3a. If
L a old
1. Start at a random location in parameter space:
2. Try to take a random step in parameter space: i new a
 L old
Accept (take and save) the step,
“new”  “old” and go to 2.
i new L
L old new
3b. If
L new

L old
Draw a random number x uniform in 0,1
If x
If x

 L
L
L
L old new old new do not take the step (i.e. save “old”) and go to 2. do as in 3a.
KEEP GOING….

“Take a random step”
The probability distribution of the step is the
“ proposal distribution ”, which you should not change once the chain has started.
The proposal distribution (the step-size) is crucial to the MCMC efficiency.
Steps too small  poor mixing
Steps too big  poor acceptance rate
MCMC

When the MCMC has forgotten about the starting location and has well explored the parameter space you’re ready to do parameter estimation.
USE a MIXING and CONVERGENCE criterion!!!
Burn-in
(From Verde et al 2003)

Beware of DEGENERACIES h
 c
 c h
2
Reparameterization.
e.g., Kososwsky et al. 2002

Once you have the MCMC output:
The density of points in parameter space gives you the posterior distribution
To obtain the marginalized distribution, just project the points
To obtain confidence intervals, integrate the “likelihood” surface
-compute where e.g. 68.3% of points lie
To each point in parameter space sampled by the MCMC give a weight proportional to the number of times it was saved in the chain
To add to the analysis another dataset (that does not require extra parameters) renormalize the weight by the “likelihood” of the new data set.
No need to re-run cmbfast!
warning: if new data set is not consistent with the old one
 nonsense

Expansion rate of the universe a(t) ds 2 =
 dt 2 +a 2 (t)[dr 2 /(1-kr 2 )+r 2 d

2 ]
Einstein equation
(å/a) 2 = H 2 = (8

/3) r
= (8

/3) r m m
+ d
H 2 (z)
+ C exp{
 dlna [1+ w(z) ]}
Growth rate of density fluctuations g(z) = ( dr m
/ r m
)/a
Second oder diff eqn, here.
Poisson equation

2

(a) =4

Ga 2 dr m
= 4

G r m
(0) g(a)