What you need to know about large scale structure Licia Verde University of Pennsylvania lverde

What you need to know about large scale structure

Licia Verde

University of Pennsylvania

www.physics.upenn.edu/~lverde

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Outline

1) Motivation and basics Large Scale Structure probes

2) Real world effects

3) Measuring P(k) & (Statistics)

(spherical cows)

(less spherical cows)

The standard cosmological model

96% of the Universe is missing!!!

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Major questions :

2) What makes the Universe accelerate?

1)What created the primordial perturbations?

These questions may not be unrelated

The standard cosmological model

Questions that can be addressed exclusively by looking up at the sky

96% of the Universe is missing!!!





CMB is great and told us a lot, but large scale structures are still useful:

Check consistency of the model

We will concentrate on dark energy and inflation

If this test is passed

Combine to reduce the degeneracies

On blackboard:

Power spectrum (for DM) definitionsGaussian random fieldsLinear perturbations growthTransfer function

Primordial power spectrum=A kn

Amplitude of the power lawslope

ln k

ln P(k)A (convention

dependent)

!

Primordial power spectrum=A kn(k)

Amplitude of the power lawslope

ln k

ln P(k)A,n (convention

dependent)

!

=dn/dlnk

CONSTRAINTS ON NEUTRINO MASS

CDM density

Neutrino mass



WMAP II

WMAP+high l experiments

2dFGRS

SDSS main

LRG SDSS



CMB+SDSS LRG 0.9eV (95% CL)

Tegmark et al ‘07

Spergel et al ‘07QuickTime™ and a

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2dFGRS SDSS main

WMAP II

Flatness



SN1A Riess et al 04

2dfGRS ‘02

WMAPII

WMAPII + H



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From Sperget et al 07

How about dark energy?



Planck scale (At EW scale it’s only 56 orders of magnitude)

If it dominated earlier, structures would not have formed



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)

Quintessence

If dark energy properties are time dependent, so are other basic physical parameters

Equation of state parameter w= p/

w=-1 is cosmological constant what other options to consider?

clustering?

Couplings?

Varying fine structure constant alpha



Oklo Natural reactor:1.8 billion yr ago there was a natural water-moderated fission reactor in Gabon.

Isotopic abundances contrain 149 Sm neutron capture cross section ad thus alpha













Dark energy

2dfGRS

H prior

WMAPII

SN

With DE clustering

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.

What if one could see the peaks pattern also at lower redshifts?

We need to look at the expansion history (I.e. at least two snapshots of the Universe)

Baryonic Acoustic Oscillations

Courtesy of D. Eisenstein

For those of you who think in Real space

Evolution of a single perturbation,Imagine a superposition

Fore those of you who think in Fourier space

If baryons are ~1/6 of the dark matter these baryonic oscillations should leave some imprint in the dark matter distribution



Data from Tegmark et al 2006

Matter-radn equality

Acoustic horizon at last scattering



from Percival et al 2006

DR5

Robust and insensitive to many systematics

THE SYMPTOMSOr OBSERVATIONAL EFFECTS of DARK ENERGY

Recession velocity vs brightness of standard candles: dL(z)

CMB acoustic peaks: Da to last scattering

LSS: perturbations amplitude today, to be compared with CMB

Da to zsurvey

Perturbation amplitude at zsurvey

Galaxy clusters number counts

Galaxy clusters are rare events:P(M,z) oc exp(-2/(M,z)2)

In here there is the growth of structure

Beware of systematics!“What’s the mass of that cluster?”

x

Galaxy clusters number counts

Galaxy clusters are rare events:P(M,z) oc exp(-2/(M,z)2)

In here there is the growth of structure

Beware of systematics!“What’s the mass of that cluster?”

x

Inflation

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

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.

Seeing (indirectly) z>>1100

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!

Specific models critically tested





n n

r r

dns/dlnk=0

Models like V()~p

dns/dlnk=0

HZ

p=4 p=2 For 50 and 60 e-foldings

p fix, Ne variesp varies, Ne fix



Possible probes of large scale structure

Galaxy surveysClusters surveys (SZ, thermal and Kinetic)Lyman alpha surveysWeak lensing surveys (***)H21surveys (far future)

Weak lensing (cosmic shear)







Very near future: Atacama Cosmology telescope

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

CMB gravitational Lensing

e-

e-

e-

e-

e-

e-

e-

e-

e-

Coma Cluster Telectron = 108 K

(& South Pole telescope, & Planck)

Summary

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

So far we have seen the basic theory behind LSS

In the future expect an avalanche of LSS data (and acronyms)

2 problems: dark energy, inflation can be addressed exclusively by looking up at the sky

Next time: real world effects

Fingers-of -GodGreat walls

Redshift space distortions

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

Kaiser (1987)P(k) => P(k)(1+2/3f+1/5f^2)

Redshift-space distortions (Kaiser 1987)

zobs = ztrue +v / c v prop. to m0.6 m0.6 b-1n/n

(bias)

linear Non-linear

Fourier space

shells

p



Fingers-of -GodGreat walls

What’s bias?

What’s bias?

?

Measured for 2dFGRS (Verde et al. 2002)

“If tortured sufficiently, data will confess to almost

anything”

Fred Menger

Treat your data with respect (Licia Verde)

Interpretation:

Likelihood analysis

CMBFAST or CAMB to get P(K)

Bayes Theorem:

)(/)|()()|( DPDPPDataP iii

LikelihoodPriorWhat you really want(Posterior)

You should not forget

Likelihood: Gaussian vs non-gaussian

Likelihood analysis

Best fit parameters Maximize the likelihood

Central Limit Theorem distribution will converge to Gaussian

What is the probability distribution of your data?

Examples: Cl, alm, , etc..

]2

1exp[

det)2(

1 1xxTn

L)( theorydatax

Gaussian likelihood:

If data uncorrelated… much simpler

Error bars and Confidence Levels

Why errors?

truei

i1

2

3

%3.68

%4.95

%73.99

Joint or marginalized?

Errors

Cosmic variance

noise

(ignore approximations, mistakes etc..)

Errors

From: “Numerical recipes” Ch. 15

Example: for multi-variate Gaussian

ln2 L2

Errors

From: “Numerical recipes” Ch. 15

If likelihood is Gaussian and Covariance is constant

There is a BIG difference between

reduced

&

2

2

Only for multi varaite Gaussian with constant covariance

Statistical and systematic errors

As you add more data points (or improve the S/N) the statistical errors become smallerbut the systematic errors do not.

Errors

Examples of statistical (random) errors: cosmic variance, instrumental noise, roundoff (!)…..

Examples of systematic errors: approximations, incomplete modeling, numerics, ….

(introduce biases)

Grid-based approach

What if you have (say) 7 parameters?

You’ve got a problem !

Operationally:

m

8e.g., 2 params: 10 x 10

Markov Chain Monte Carlo (MCMC)

Random walk in parameter space

At each step, sample one point in parameter space

The density of sampled points posterior distribution

marginalization is easy: just project points and recompute their density

FAST: before 710 likelihood evaluations, now

510

Adding external data sets is often very easy

Operationally:

1. Start at a random location in parameter space:oldi

L old

2. Try to take a random step in parameter space: newi

L new

3a. If LnewLold Accept (take and save) the step,

“new” “old” and go to 2.

3b. If new L

oldL Draw a random number x uniform in 0,1

If x L new

L old

do not take the step (i.e. save “old”)and go to 2.

L new

L oldIf x 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 locationand has well explored the parameter spaceyou’re ready to do parameter estimation.

Burn-inUSE a MIXING and CONVERGENCE criterion!!!

(From Verde et al 2003)

Beware of DEGENERACIES

Reparameterization.

h

c 2hc

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 add to the analysis another dataset (that does not require extra parameters) renormalize the weight by the “likelihood” of the new data set.

To each point in parameter space sampled by the MCMC give a weightproportional to the number of times it was saved in the chain

No need to re-run cmbfast!

warning: if new data set is not consistent with the old one nonsense

Thermal Sunyaev Zeldovich effect

Expansion rate of the universe a(t)

ds2 = dt2+a2(t)[dr2/(1-kr2)+r2d2]

Einstein equation (å/a)2 = H2 = (8/3) m + H2(z) = (8/3) m + C exp{dlna [1+w(z)]}

Growth rate of density fluctuations g(z) = (m/m)/a

Our Tools

Poisson equation 2(a)=4Ga2 m= 4Gm(0) g(a)

Second oder diff eqn, here.

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What you need to know about large scale structure Licia Verde University of Pennsylvania lverde