An optimal estimator for the correlation of CMB anisotropies with Large Scale Structures and its application to WMAP-7year and NVSS

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DOTTORATO DI RICERCA IN

ASTRONOMIA

Ciclo XXIV

Settore Concorsuale di afferenza: 02/C1

Settore Scientifico disciplinare: FIS/05

TITOLO TESI

An optimal estimator for the correlation of CMB anisotropies

with Large Scale Structures and its application to

WMAP-7years and NVSS

Presentata da:

FRANCESCA SCHIAVON

Coordinatore Dottorato

Relatore

Chiar.mo Prof.

Chiar.mo Prof.

Lauro Moscardini

Gabriele Giovannini

Co-relatori:

Dott. Fabio Finelli

Dott. Alessandro Gruppuso

This thesis has been carried out

at INAF/IASF-Bologna as part of

1 The osmologi al model 7

1.1 Standard Hot Big Bangmodel . . . 7

1.2 Thermalhistory of the Universe . . . 10

1.3 The Dark Energy models . . . 13

1.3.1 The osmologi al onstant . . . 14

1.3.2 Chaplygin-typegas . . . 15

1.3.3 Quintessen e . . . 16

1.3.4 ModiedGravity . . . 17

1.4 Observational eviden es for Dark Energy . . . 18

1.4.1 Type IaSupernovæ . . . 18

1.4.2 Baryoni a ousti os illations(BAO) . . . 19

1.4.3 Gravitational lensing . . . 20

1.5 Ination . . . 21

1.5.1 Horizon Problem . . . 21

1.5.2 Flatness problem . . . 22

1.5.3 Monopoleproblem . . . 23

1.6 Cosmologi alperturbationtheory . . . 23

1.6.1 Metri perturbations . . . 24

1.6.2 Stru ture formation . . . 25

1.6.3 Non-linearperturbations . . . 26

2 Cosmologi al Mi rowave Ba kground 29 2.1 Primordialanisotropies . . . 30

2.1.1 A ousti peaks . . . 32

2.2 Polarization . . . 36

2.3.1 Gravitational Se ondaries . . . 39

2.3.2 S attering Se ondaries . . . 42

2.3.3 Foregrounds . . . 44

3 Late Integrated Sa hs-Wolfe Ee t 45 3.1 Cross- orrelation . . . 47

3.2 Information on Dark Energy onstraint in the ISW-LSS ross- orrelation . . . 52

4 CMB and LSS data 61 4.1 Cosmi Mi rowave Ba kground Data . . . 61

4.2 Large S ale Stru tures Data . . . 62

4.3 NVSS . . . 64

4.3.1 Systemati s and pre-pro essing . . . 65

4.3.2 Sour e redshift distribution . . . 69

4.3.3 Shot noise . . . 73 5 BolISW ode 75 5.1 QMLalgebra . . . 75 5.2 BolPol . . . 77 5.3 BolISW . . . 78 5.3.1 Numeri aloptimization. . . 80 5.4 Validation . . . 82

6 Appli ation to WMAP 7 year and NVSS data 91 6.1 TT auto-spe trum . . . 91

6.2 Balaxy distribution modelwith onstant

b

. . . 91

6.2.1 No shot-noise removal . . . 92

6.2.2 Shot-noise removal . . . 93

6.2.3 De lination orre tion . . . 94

6.2.4

C

T G

ℓ

6= 0

ee t on the ross-powerspe trum . . . 95

6.3 Galaxydistribution modelwith

b(z)

. . . 95

7 Quantitative assessment of the ross- orrelation dete tion 109 7.1 Likelihood omputation . . . 109

7.2 Results . . . 110

The understanding of the nature of dark energy is one of the outstanding question

for observational osmology. Sin e the dis overy of the present a eleration of

the Universe by the measurement of the luminosity distan e of distant type Ia

supernovæ (SN Ia) ([Riess etal.1998, Perlmutteret al.,1999℄), several observations

havesharpeneda osmologi al on ordan e modelinwhi hanunknown omponent

- dark energy - with a negative pressure density shares

∼ 2/3

of the total energy

budgetoftheUniverse([Tegmark et al.2004℄). Atpresentthenatureofdarkenergy

an be hardly onstrained by dierent osmologi al observations, with the main

indi ationthat itsparameter of state

w

DE

is lose to a osmologi al onstant's one.

The key strategy to onstrain the nature of dark energy with urrent data is to

ombine as many dierent observations as possible, as luminosity distan e of SN

Ia, baryoni a ousti os illations (BAO) from galaxy surveys, Cosmi Mi rowave

Ba kground (CMB)anisotropies, weak lensingsurveys, et ...

One of the key predi tions of the presen e of dark energy is the late Integrated

Sa hs Wolfe ee t (Sa hs & Wolfe 1968) in the CMB pattern. The ISW ee t is

a ontribution to CMB anisotropies aused by the gravitational intera tion of the

CMBphotonswiththeforminglarges alestru tures. Itisrelatedtoatimeevolving

gravitational potential,aso urs onlarge s ales when the Universe enters ina late

a elerated expansion(late ISW).The late ISW isasmall ontributiontothe total

CMB anisotropies and is maximum on the largest s ales (Kofman & Starobinsky

1985), but therefore blurred by osmi varian e: however it an be dete ted by

its ross- orrelation with large s ale stru tures (LSS) (Crittenden & Turok 1995)

and this non vanishing ross- orrelation is an independent probe of dark energy,

omplementarytothedistan etothelasts atteringsurfa ewhi hxes theposition

of the a ousti peaks inthe angularpower spe trum of CMB anisotropies.

CMB-LSS ross- orrelation and a s ienti ally soundful osmologi al interpretation are

required, despiteitsmodest signal-to-noiseratio (SNR), the quality of urrent data

anddataanalysisissues. Untilnowthereisno onsensusamongresultsinliterarture,

then we hoose to use the best look on the ISW-LSS ross- orrelation by using an

optimal method.

Inthis thesis wepresent the implementationofaquadrati maximum likelihood

(QML) ode, ideal to estimate the ISW-LSS ross-power spe trum, together with

the auto-power spe tra of CMB and LSS: su h tool goes beyond all the previous

harmoni analysis of the ISW-LSS ross-powerspe trum present inthe literature.

The thesis isdivided intothe following hapters.

ˆ The rst hapter deals with the basi on epts of the urrent osmologi al

model, starting from the Big Bang theory to pass through Dark Energy

obersevationaleviden esandmodels. Itwillbeintrodu edtheInationmodels

and reviewed the osmologi alperturbationtheory.

ˆ The se ond hapter will fo us on the CMB anisotropies, we will start from

primordial ones and then we will report all the se ondary anisotropies,

in luding the ISW ee t. We will obtain the temperature power spe trum

from the perturbation equations of a relativisti uid, in order to take into

a ount the most important features of the spe trum and their osmologi al

impli ations.

ˆ The third hapter will be entirely dedi ated to the ISW ee t. We will

derive the ross power spe trum from the LSS matter density (

δ

g

) and CMB

temperature (

∆T /T

) ross- orrelation. We will report the ISW dete tion

history, the urrent ontroversy onthe statisti alsigni an e of the ISW-LSS

ross- orrelation, generated by several dierent results whi h span from no

dete tion at allto apositiveat amaximum of 4.5

σ

.

ˆ The fourth hapter deals with the real map des ription of CMB temperature

(WMAP-7year)andgalaxydistribution(NVSS)whi hwillbeusedinChap.(6)

analysis. We will introdu e the shotnoise and one of the systemati s in the

LSS map. We will des ribe two dierent galaxy distributions whi h ould

ˆ Thefth hapterdealswiththeQMLmethodtoestimatethe ross orrelation

ISW-LSS, going intoa detailed des ription of the algebra. We willreport the

implementationofthe BolISW odebased ontheQMLmethodanditsMonte

Carlo validationwith 1000 WMAP7-likeand NVSS-likesimulatedmaps.

ˆ The sixth hapter deals with the appli ations of the BolISW ode on real

data. Wewillshowtheestimates forallthe threepowerspe tra

(temperature-temperature, temperature-galaxy and galaxy-galaxy orrelations), omparing

alsoestimates fromdierentgalaxy distribution models.

ˆ The seventh hapter deals with the quantitative assessment of the

ross- orrelation dete tions by using three dierent likelihood perspe tives. We

will give onstraints on the

Ω

Λ

parameter, xing all the other osmologi al

parameters. Wewill ompareourresultswithotherISW-LSSsignaldete tions.

The osmologi al model

Todaythe on ordan e osmologi almodelisthe

Λ

CDM(ColdDarkMatter)model,

based onthe Einstein'srelativity andtaking intoa ount the ideasof the Standard

Hot Big Bang model, the presen e of the Dark Matter and the Dark Energy and

the Inationmodel. Theobservationshaveonmanyo asionsbeenindisagreement

with the previously a epted theory, leadingtothe subsequent repla ement or

add-on of the standard model.

1.1 Standard Hot Big Bang model

TheStandardHotBigBangmodelsuggestsahomogeneus,isotropi Universewhose

evolution is governed by the Friedmann equations based on Einstein's General

Relativity and the Coperni anprin iple.

All the informations about matter density and the geometry of the Universe are

ontained inthe Einstein eld equations

G

µν

≡ R

µν

−

1

2

Rg

µν

= 8πGT

µν

(1.1)

where

G

µν

isthe Einstein tensor,whi h des ribesthe spa e-time geometrythrough

the metri tensor

g

µν

, the Ri itensor

R

µν

andRi is alar

R

,dependingonmetri

derivates; theother sideof theequation ontains the stress-energytensor

T

µν

whi h

des ribes the matter-energy ontent in the spa e-time.

The

T

µν

tensor takes this form:

where

ρ

and

p

are,respe tively, the energyand thepressure densityof theuid and

u

µ

is the uid four-velo ity. If the uid is ideal,

T

µν

takes a diagonal form with

ρ

on the time oordinateand

p

onthe spa e oordinates.

Assuming the Friedmann-Robertson-Walker(FRW)metri ( =

¯h

=1)

ds

2

_{= −dt}

2

+ a

2

(t)



dr

2

1 − kr

2

+ r

2

_(dθ

2

_{+ sin}

2

_θdφ

2

₎



(1.3)

where

a(t)

is the s ale fa tor with respe t to the osmi time t; r,

θ

and

φ

are

the omoving oordinates; the onstant

k

des ribesthe geometryof the spa e-time

(

k = +1, 0, −1

, respe tvely orresponding to a losed, at and open Universe); the

Einstein's equationsplit intothe twoFriedmann equations

H

2

_≡

 ˙a

a



2

=

8πGρ

3

−

k

a

2

(1.4)

˙

H = −4πG(ρ + p) +

k

a

2

(1.5)

where

H

is the Hubble parameter depending on time, and

ρ

and

p

are the total

matter and energy density of allthe onstituentsof the Universe at a given time.

The mass onservation equation is

˙ρ + 3H(ρ + p) = 0,

(1.6)

and ombining Eq.s 1.4 and 1.5 we nd the equation for the a eleration of the

s ale-fa tor

¨a

a

= −

4πG

3

(ρ + 3p).

(1.7)

The evolutionof the total energy density of the Universe is governedby

d(ρa

3

_{) = −pd a}

3

_;

(1.8)

whi h is the First Law of Thermodynami s for a parti ular uid in the expanding

Universe.

If the we onsider auid with equation ofstate

p = γρ

, itfollows that

ρ ∝ a

−3(1+γ)

and

a ∝ t

2/3(1+γ)

. For

p = ρ/3

, ultra-relativisti matter,

ρ ∝ a

−4

and

a ∼ t

1

2

; for

p = 0

, very nonrelativisti matter,

ρ ∝ a

−3

and

a ∼ t

2

3

; and for

p = −ρ

, va uum energy,

We an use the Friedmann equation to relate the urvature of the Universe to the

energy density and expansion rate:

Ω − 1 =

k

a

2

_H

2

;

(1.9)

Ω =

ρ

ρ

crit

;

(1.10)

where the riti al density today

ρ

crit

= 3H

2

_{/8πG = 1.88h}

2

_{g cm}

−3

_{≃ 1.05 ×}

10

4

_{eV cm}

−3

. There is a one to one orresponden e between

Ω

and the spatial

urvature of the Universe: positively urved,

Ω

0

> 1

; negatively urved,

Ω

0

< 1

;

and at (

Ω

0

= 1

). Model universes with

k ≤ 0

expand forever, while those with

k > 0

ne essarily re ollapse. The urvature radius of the Universe is related to the

Hubble radius and

Ω

by

R

curv

=

H

−1

|Ω − 1|

1/2

.

(1.11)

Then the urvature radius sets the s ale for the size of spatial separations. Andin

the ase of the positively urved modelitis just the radius of the 3-sphere.

Today we know this model has many short omings, as the atness and the

horizon problems orthe osmologi al onstantproblem(solvedbyintrodu ingsome

Dark Energy or Modied Gravity models). However, this standard modelprovide

us a framework within it is possible to study the emergen e of stru tures from the

small u tuations in the density of the early Universe, like the observed galaxies,

lusters and the osmi mi rowaveba kground.

Over the last de ades, observations have signi antly in rease the idea that: the

Universeisspatiallyatanda elerating;itpassedthroughana eleratedexpansion

intheearlyUniverse(inationepo h); todaytheenergy ontent onsistsprin ipally

of

∼ 27%

of Dark Matter,

∼ 76%

of Dark Energy and few

%

of baryoni matter,

whereas the radiationand neutrinos ontributions are negligibletothe total energy

density

Ω

0

= Ω

0 DM

+ Ω

0 B

+ Ω

0Λ

= Ω

0 M

+ Ω

0Λ

.

(1.12)

Ω

M

is the energy density of the Dark Matter and the baryoni mattertogether, all

onsidered in their a tualvalue.

If we onsider a at Universe,

Ω

0

= 1

The ombined data from high redshift supernovae (SN1), large s ale stru tures

(LSS) and osmi mi rowaveba kground give

Ω

0

= 1.00

+0.07

_−0.03

, trovainumerigiusti

meaning that the present Universe is spatially at (or at least very lose to being

at). Then restri ting to

Ω

0

= 1

,the darkmatter density is

Ω

DM

h

2

= 0.1334

+0.0056

_−0.0055

,

the baryondensity

Ω

B

h

2

= 0.02258

+0.00057

_−0.00056

.

and the substantialdark(un lustered) energy is inferred,

Ω

Λ

≈ 0.734 ± 0.029.

Inthe nextse tionwewillsee howall omponents ontributetothethermalhistory

of the Universe, onsidering a

Λ

-CDM model.

1.2 Thermal history of the Universe

Radiation Era.

After Plan k time,

t

P

≡

p¯hG/c

5

_{≈ 10}

−43

s,

T

P

≡

p¯hc

5

_/Gk

2

_{≈ 1.42 ×}

10

19

_GeV

.

In this era the energy density of the expanding Universe was dominated by the

radiation omponent and made up of photons, neutrinos and matter (protons,

ele trons, heliumnu lei and non-bayoni darkmatter). Athigh temperatures both

thehydrogenand theheliumare fullyionised. Inthis phasetheThomsons attering

o urs on a times ale mu h less than the expansion times ale, resulting in a tigh

ouplingbetween matter and radiation.

At time

t = 10

−35

_s

, the GUT (Grand Uni ationTheory) phase transition o urs,

all the three gauges intera tions - ele tromagneti , weak and strong - be ome no

longer unied. The ination epo h alsoo ursin this era, exponentially expanding

the Universe from to in the time range of

10

−34

_{− 10}

−32

_s

, during whi h quantum

noisewasstret hedtoastrophysi alsizeseeding osmi stru tures. Atatemperature

of

T ≈ 1 MeV

the neutrinos de ouple from matter and at

T ≈ 0.1 MeV

the

Nu leosynthesis).

Weknowthe energy densitiesfor radiationand matter evolves a ording to:

ρ

m

= ρ

0m

(1 + z)

3

(1.14)

ρ

r

= ρ

0r

(1 + z)

4

(1.15)

When those densities are equal the matter-radiation equivalen e o urs and the

matter domination begins.

Matter Era.

t ≈ 10

5

_yrs

,

T ≈ 10

4

_K

,

z

eq1

≈ 3200

.

In the beginningof this era the radiationand mattertemperatures are equivalent

T

r

= T

0r

(1 + z)

(1.16)

and remain approximately equals until

z ≃ 300

, thanks to the residual ionisation

whi h allows an ex hange of energy between matter and radiation via Compton

diusion. After this redshift the thermal intera tion between matterand radiation

be omes insigni ant,so that the matter omponent ools adiabati allywitha law

T

m

∝ (1 + z)

2

.

(1.17)

Withthe oolingofthe temperature, theUniverserea htheepo hofre ombination

orresponding to a temperature of around

T

rec

≃ 4000K

, when

50%

of the matter

is in the form of neutralatoms. Be ause of the re ombination,around

z

dec

≃ 1100

,

a no-instantaneous pro ess of de oupling o urs and matter and radiationbegin to

evolve separately.

After de oupling any primordialu tuationsin the matter omponent that survive

the radiation era grow under the inuen e of Dark Matter gravitational potential

wells and eventually give rise to osmi stru tures: star, galaxies and lusters of

galaxies. The part of the gas that does not end up in su h stru tures may be

reheated and partly reionisedby star and galaxyformation,duringthe reionization

period atabout

z ≈ 10.5

.

After

z

dec

≃ 1100

alsothe radiationbeginsfree toevolve indipendently, be ause

the opti al depth

τ

γe

of the Universe due to Compton s attering de reases. This

lenght. Theprobabilitythatagivenphotons atterswithanele tronwhiletravelling a distan e dtis given by

−

dN

_N

γ

γ

= −

dI

I

=

dt

τ

γe

= n

e

σ

T

cdt = −

xρ

m

m

p

σ

T

c

dt

dz

dz = −dτ

(1.18)

where

n

e

is thenumberele tron density,

σ

T

is the Thompsons attering se tion,

ρ

m

is the matter density and

m

p

isthe protonmass; so that

I(t

0

, z) = I(t)exp



−

Z

z

0

xρ

m

m

p

σ

T

c

dt

dz

dz



= I(t)exp[−τ(z)]

(1.19)

I(t

0

, z)

is the intensity of the ba kground radiation rea hing the observer at time

t

0

with a redshift z;

τ (z)

is the opti al depth of su h a redshift; the

x(z)

is the

ionisation fra tion from the known Saha equation. The probability that a photon,

whi harrivesat the observerat the present epo h,sueres itslasts attering event

between z and z-dz is

−

d

dz

{1 − exp[−τ(z)]}dz = exp[−τ(z)]dτ = g(z)dz.

(1.20)

Thequantity

g(z)

isthe ee tive widthof the surfa e of last s attering (

ls

) and

is well approximated by a Gaussian with peak at

z

ls

≃ 1100

and width

∆z ≃ 400

.

So at redshift

z

ls

we also have

τ (z) ≃ 1

, be ause the Universe is transparent to

photons.

The photons begin to travel from the last s attering surfa e reating what is

the radiation ba kground of the Universe, alled now the Cosmi Mi rowave

Ba kground.

Dark Energy Era.

z

eq2

≈ 0.4

.

Very late with respe t to the age of the Universe the energy density of the Dark

Energy begins to dominate, a elerating the osmi expansion, until today,

z = 0

;

in the meanwhile large s ale stru tures formed from the primordial u tuations

and by intera ting with dark matter potential wells. The nature of this dark

omponent is still unknown and many observational probes have been proposed

to test its properties and redshift evolution either in the standard

Λ

-Cold Dark

1.3 The Dark Energy models

The Dark Energy omponent does not intera t through any of the fundamental

for es other than gravity and assumingthe

Λ

-CDMmodelit ausesthe a elerated

expansion of the universe. From Eq. 1.7 we know the Dark Energy must have a

negative pressure inorder to a elerate the expansion, i.e.

p < −

1

₃

ρ

(1.21)

If we parametrizethe equationof state of a perfe t uid in this way

p = wρ

(1.22)

the equation-of-state parameter for Dark Energy will be

w < −1/3

. Nevertheless,

theperfe tuidmodelwitha onstantstateparameterdoesnotwork,be auseifwe

onsider the perturbation theoryrelation

δP = c

2

s

δρ

,where

c

s

isthe speed of sound

of Dark Energy,

c

2

s

= w

, a negative state parameter implies negative value for

c

2

s

.

Therefore it is ne essary to des ribe dark energy with dierent models: as a uid

with non-linear relation between

P

and

ρ

whi h leads to negative

w

but positive

value of

c

2

s

, or a s alar eld with an auto intera tion potential. We an fo us on

some of thesemodels. Establishingwhether the darkenergy is onstantorevolving

is one of the main hallenges for modern osmology. For example the expe ted

EUCLID mission in 2019 will have as main aims measuring the DE equation of

state parameters

w

0

and

w

1

to a pre isionof

2%

and

10%

, respe tively, using both

expansion history and stru ture growth; measuring the growth fa tor exponent

γ

withapre isionof

2%

,enablingtodistinguish GeneralRelativityfromthemodied

gravity theories; testing the CDM paradigm for stru ture formation, and measure

the sum of the neutrino masses to a pre ision better than

0.04 eV

when ombined

with Plan k.

Thetimedependen eoftheDarkEnergyequationhasbeen onstrainedbytting

various forms of

w(z)

to the SNIa data, often in ombination with CMB and the

LSS measurements.

Oneof the mostpopular two-parameterformulaisthe linear hangeinthe s ale

fa tor

a = (1 + z)

−1

given by,

where

w

0

isthe value today and

w

1

the value at some earlytime

a

.

For general

w(a)

, the dynami al expansion of the Universe is spe ied by the

Friedmann equation

E(a) =

H

2

_(a)

H

2

0

= Ω

M

a

−3

+ Ω

K

a

−2

+ Ω

DE

a

f (a)

,

(1.24)

where

Ω

K

≡ (1 − Ω

m

− Ω

X

)

is the urvature onstant,

H(a) ≡ ˙a/a

is the Hubble

parameterwith presentdayvalue

H

0

.

f (a)

is al ulated bysolvingthe onservation

of energy equation for the Dark Energy

d(ρ

X

a

3

_{)/da = −3p}

X

a

2

giving

ρ

X

∝ a

f (a)

, where

f (a) =

−3

ln a

Z

ln a

0

[1 + w(a

′

_{)]d ln a}

′

_.

(1.25) For onstant

w

,

f (a) = −3(1 + w)

.

For the parameterisation

w(a) = w

0

+ w

1

a(1 − a)

,

f (a) = −3(1 + w

0

) +

3w

1

2 ln a

(1 − a)

2

_.

(1.26)

1.3.1 The osmologi al onstant

A osmologi al onstant was originally introdu ed by Einstein in 1917 in Eq. 1.1,

in order toobtain astati solutionfor a spatially losed universe

R

µν

−

1

2

Rg

µν

+ Λg

µν

= 8πGT

µν

,

(1.27)

after the dis overy of the a elation expansion of the Universe it was regained as

possible andidate for the Dark Energy.

The Friedmann eq.s 1.4and 1.7 be omes:

H

2

_≡

 ˙a

a



2

=

8πGρ

3

−

k

a

2

+

Λ

3

(1.28)

¨a

a

= −

4πG

3

(ρ + 3p) +

Λ

3

.

(1.29)

Itisatimeindependentandspatiallyuniformdark omponent,whi hmay lassi ally

beinterpreted asa relativisti perfe t simple uid. Ifwe onsider the observational

eviden es of

¨a > 0

, from Eq. (1.29) we nd the osmologi al onstant ontributes

negatively to the pressure term and hen e exhibits a repulsive ee t, as the Dark

Energy does. Introdu ingthe modied energy density and pressure

˜

ρ = ρ +

Λ

8πG

, ˜

P = P −

Λ

we nd that equations 1.28 and 1.29 redu e to equations1.4 and 1.7, and that the

osmologi al onstant obeysthe equationof state

w = −1

.

The osmologi al onstant

Λ

is the oldest and simplest andidate from a

mathemati al viewpoint, but there is a fundamental problem related to su h a

theoreti ally favored andidate whi h is usually alled the osmologi al onstant

problem. The present osmologi al upper bound (

Λ

o

/8πG ∼ 10

−47

_GeV

4

) diers

from natural theoreti al expe tations (

∼ 10

71

_GeV

4

) by more than 100 orders of

magnitude.

We an onsider now some others andidates appearing inthe literature.

1.3.2 Chaplygin-type gas

Itiswidelyknownthatthemaindistin tionbetweenthepressurelessCDMandDark

Energy isthat the former agglomeratesat smalls ales whereas the Dark Energy is

asmooth omponent. Su h propertiesseems tobedire tlylinked totheequationof

state of both omponents. It refers toan exoti uid, the so- alled Chaplygintype

gas, whoseequation of state is

p

X

= −

A

ρ

α

X

,

(1.31)

where

A

is onstant with dimension

[M

4(1+α)

_]

and

α

is onstant in the range

[0, 1]

. The

α 6= 1

onstitutes a generalization of the original Chaplygin gas

equationofstate proposed in[Bento etal, 2004℄whereas

α = 0

givesamodelwhi h

behaves as

Λ

CDM. The idea of a Unied Dark-Matter-Energy (UDME) s enario

inspired by anequation of state like(1.31) omes fromthe fa tthat the Chaplygin

type gas an naturally interpolate between non-relativisti matter (CDM) and

negative-pressure DarkEnergy regimes[Bentoet al,2004℄. TheJeans instabilityof

ChaplyginperturbationsisatrstsimilartoCDMu tuations(whentheChaplygin

gas hasanegligiblepressure) andthendisappears(whentheChaplygingasbehaves

as a osmologi al onstant). Both this late suppression of Chaplygin u tuations

and the apperan e of a non-zero Jeans length leave a large integrated Sa hs Wolfe

(ISW, see Chap. 3)imprinton the CMB anisotropies.

Motivated by these possibilities, there has been growing interest in exploring

the theoreti al and observational onsequen es of the Chaplygingas, not only as a

new andidateforDarkEnergy only. TheimprintofaChaplygingasisalsopresent

onthe matterpowerspe trum,sin e Chaplygingasperturbationsae t both CMB

anisotropies and stru ture formation.

1.3.3 Quintessen e

The idea of quintessen e originates from an attempt to understand the smallness

of the  osmologi al onstant or Dark Energy in terms of the large age of the

universe [Wetteri h, 1988℄. As a hara teristi onsequen e, the amount of Dark

Energy may be of the same order of magnitude as radiationor dark matterduring

a long period of the osmologi al history, in luding the present epo h. Today, the

inhomogeneous energy density in the universe (dark and baryoni matter) is about

ρ

inhom

≈ (10

−3

eV)

4

. This number is tiny in units of the natural s ale given by the

Plan k mass

M

p

= 1.22 · 10

19

GeV. Nevertheless, it an be understood easily as a

dire t onsequen e of the long duration of the osmologi al expansion: adominant

radiationor matter energy density de reases

ρ ∼ M

2

p

t

−2

and the present age of the

universe is huge,

t

0

≈ 1.5 · 10

10

yr. It is a natural idea that the homogeneous

part of the energy density in the universe (the Dark Energy) also de ays with

time and therefore turns out to be small today. A simple realization of this idea,

motivatedbytheanomalyofthedilatationsymmetry, onsidersas alareld

φ

with

an exponential potential[Wetteri h, 1988℄

L =

√

g

 1

2

∂

µ

_φ∂

µ

φ + V (φ)



(1.32) where

V (φ) = M

4

_{exp(−αφ/M),}

(1.33) with

M

2

_{= M}

2

p

/16π

. In the simplest version

φ

ouples only to gravity, not to

baryons. Cosmology is then determined by the oupled eld equations for gravity

and the s alar  osmon eld in presen e of the energy density

ρ

of radiation or

matter. For a homogeneous and at universe (

n = 4

for radiation and

n = 3

for

nonrelativisti matter)

H

2

₌

1

6M

2



ρ +

1

2

φ

˙

2

_{+ V}



,

¨

φ + 3H ˙

φ +

∂V

∂φ

= 0,

˙ρ + nHρ = 0.

(1.34)

This model predi ts a fra tion of Dark Energy or homogenous quintessen e (as

ompared to the riti al energydensity

ρ

c

= 6M

2

_H

2

)whi h is onstant in time

Ω

h

=



V +

1

2

φ

˙

2



/ρ

c

= ρ

φ

/ρ

c

(1.35)

both for the radiation-dominated

(n = 4)

and matter-dominated

(n = 3)

universe

((Ω

h

+ ρ/ρ

c

) = 1)

. This would lead to a natural explanation why today's Dark

Energy isof the same orderof magnitudeasdarkmatter. For alargevalue of

V (φ)

the for eterminEq.1.34 islargeand the Dark Energyde reases faster thanmatter

or radiation. In the opposite, whenthe matter orradiationenergy density is mu h

larger than

V (φ)

, the for e is small as ompared to the damping term

3H ˙

φ

and

the s alar waits untilthe radiation ormatterdensity is small enoughsu h that the

over-damped regime ends. Stability between the two extreme situations is rea hed

for

V ∼ ρ

. Forthis model, the equationof state parameter

w

is given by

w

Q

=

˙

φ

2

2

− V (φ)

˙

φ

2

2

+ V (φ)

(1.36)

and an bevaried inthe range

−1 < w

Q

< 1

.

1.3.4 Modied Gravity

InthesimplestalternativestoDarkEnergy,thepresent osmi a elerationis aused

by amodi ation to generalrelativity,the so alled Modied Gravity. The General

theory of relativity founded by Einstein at the end of 1915 has been su essfully

veried asmodern theory of gravity for the Solar System.

Attempts to modify general relativity started already at its early times and it was

mainly motivated by resear h of possible mathemati al generalizations. Re ently

there has been anintensivea tivity ingravity modi ation, motivated by dis overy

of a elerating expansion of the Universe, whi h has not yet generally a epted

theoreti al explanation. The general relativity has not been veried at the osmi

s ale (low urvature regime)and Dark Energy has not been dire tly dete ted. This

situation has motivated a new interest in modi ation of general relativity, whi h

shouldbesomekind ofitsgeneralization. Thereisnot aunique way howtomodify

generalrelativity. Amongmany approa hesthere aretwoof them,whi h have been

mu h investigated: 1)

f (R)

theories of gravity and 2) nonlo algravities.

f (R)

. This is extensively investigated for the various forms of fun tion

f (R)

. We

have had some investigation when

f (R) = R cosh

αR+β

γR+δ

and, after ompletion of

resear h, the results willbe presented elsewhere.

1.4 Observational eviden es for Dark Energy

There are many observational eviden es of the Dark Energy ee t. Histori ally,

the a elerationexpansionwasrevealforthe rsttime byIa supernovæobervations.

In 1998 two groups of astronomers [Perlmutter et al.,1999℄ estimated the

distan e-redshift relation using Type Ia supernovae (SNe), a lass of exploding stars whose

distan e an be measured with

∼ 15%

a ura y, mu hbetter than forother distant

sour es. They foundthe

˙a(t)

isin reasing,i.e. theuniverseisnotmerelyexpanding,

the expansion is a elerating.

Many otherindire tsupportingevinden es omefromdierentmeasurements. Now

galaxy lustering, weak lensing, baryoni a ousti os illation (BAO) on the CMB

anisotropies, ages of the oldest stars are generally onsidered the most powerful

observational probesof Dark Energy.

Another method alsoprovides additional ross- he ks onDark Energy onstraints,

the late time anisotropy inthe CMB, the Integrated Sa hs-Wolfe ee t (ISW), an

bedete ted and used to onstrain osmology.

It is important to take into a ount the results from all dierent observations,

be ause individually they do not allow to determine matter and Dark Energy

density, sin e they always involves a ombinationofthese twoparameters (Cosmi

Degenera y).

Thedatafromalltheseobservationsarenota urateenoughtodistinguishbetween

the osmologi al onstant and many forms of dynami al Dark Energy. Moreover

degenera ies between Dark Energy parameters strongly limitthe possibility to test

whether

w

is onstant ornot.

1.4.1 Type Ia Supernovæ

TypeIaSupernovæaregenerallybelieved tohavehomogeneousintrinsi luminosity

of peak magnitude. So SNe Ia are usually known asstandard andles whi h ould

distan e modulus versus redshift ould provide dire t eviden e for the a eleration

of the Universe and the analysis also put a onstrainton dark energy models. The

luminositydistan e

d

L

of SNe Iais dened by

d

L

(z) =

c(1 + z)

H

0

F (z) ≈

c

H

0



z +

1

2

(1 − q

0

)z

2

_{+ ...}



.

(1.37) The fun tion

F (z)

is

F (z) =

Z

z

0

1

E(z

′

₎

dz

′

_.

(1.38)

where,ifwe onsideraatUniversewith onstant

w

,

E(z)

isgivenbytheFriedmann

equation writtenas Eq.1.24.

q

0

is the de elerationparameter, given by

q

0

≡ −

¨aa

˙a

2

=

"

−

H + H

˙

2

H

2

#

0

≃

Ω

0 M

₂

+ Ω

0 DE

(1.39)

if

q

0

< 0

the Universe is a elerating. The distan e modulus is dened by

µ ≡ m − M = 5 log

d

L

Mp

+ 25 ,

(1.40)

where

m

and

M

are the apparent and absolutemagnitudes, respe tively; from the

rst observations the Universe seems to a elarate be ause

q

0

> 0

, then we an

onstrain the ombination between the two osmologi alparamaters

Ω

DE

and

Ω

M

2Ω

DE

> Ω

M

(1.41)

1.4.2 Baryoni a ousti os illations (BAO)

TheBaryoni A ousti Os illationssignaturesinthelarge-s ale lusteringofgalaxies

ould a t as additional tests for onstraining Dark Energy osmology, be ause

the a ousti os illations in the relativisti plasma of the early Universe ould be

imprintedontothelate-timepowerspe trumofthenon-relativisti matter,asgalaxy

lusters. TheBAOrelevantdistan emeasure ismodelledbyvolumedistan e, whi h

is dened as

D

V

(z) =

 d

2

A

(z)z

H(z)



1/3

,

(1.42)

where

H(z)

istheHubbleparameterand

d

A

(z) =

R

z

0

1

H(z

′

₎

dz

′

isthe omovingangular

diameter distan e. BAO measurements provide both

d

A

(z)

and

H(z)

using almost

galaxies. Then

D

V

(z)

an be omputed essentially from the growth fa tor (1.65)

of perturbationtheory whi h ontainsthe osmologi alparamaters. Combining the

informationfromBAOandfromtherstpeakpositionoftheCMBpowerspe trum

whi hsets the Universe tobeat

ℓ

f p

≃ 220Ω

−1/2

0

,

(1.43)

Ω

0

= Ω

M

+ Ω

DE

≃ 1.

(1.44)

it is possible to onstrain separatly values of

Ω

DE

and

Ω

M

.

1.4.3 Gravitational lensing

The gravitational lensing is regarded as an independent tool that omplements

SNe Ia as a probe on Dark Energy. The statisti s of gravitational lensing of

quasars (QSOs)by intervening galaxies an onstrain onthe osmologi al onstant.

Lensed images of distant galaxies in luster, ar s or rings, may provide a bound

on the equation of state parameter of Dark Energy. The gravitational lensing

system an beused measure the ratio of angulardiameter distan es. However, the

lensing observationsprimarily depend on theparameters of lens models withminor

dependen eon osmologi alparameters. Thereisthelensmodeldegenera yinboth

the proje ted mass density proleand the ir ular velo ity prole. It is shown that

we need to measure the Einstein radius and the velo ity dispersion within

O(1)

%

a ura y inorder to put a onstraint on

ω

DE

.

Inthe gravitationallensing,one ofthe observable quantities withouthavingany

modeldependen e is the Einstein radius (

θ

E

), whi h isproportionalto the velo ity

dispersion squared (

σ

2

v

) and the ratio of the angular distan es

D

ds

/D

s

, where

D

ds

is the distan e from the lens to the sour e and

D

s

is that from the sour e to the

observer. With dierent values of osmologi al parameters, we an have dierent

values of

D

ds

/Ds

, i.e. dierent values of

θ

E

. Thus, it might be used for probing

the property of Dark Energy,

ω

DE

. However, there is an ambiguity in measuring

σ

v

. If the error of

σ

v

measurement is not within the dieren es of

D

ds

/Ds

between

dierent osmologi al models, then we annot distinguish the dieren es between

1.5 Ination

In 1980 Guth and Starobinsky devolped the theory of ination in order to solve

the short omings of the Big Bang theory, as the horizon, atness and magneti

monopoleproblems. We forementioned that an exponential a elerating expantion

o ursduring the primordialphases ofthe evolutionof the Universe, with the s ale

fa tor evolves as

a = a

i

e

H

I

(t−t

i

)

,

(1.45)

where

t

i

denotes the time at whi h ination starts and

H

I

the value of the Hubble

ratewhi hremains onstantduringainationary(deSitter)epo h. Duringination,

the horizon

r

H

(t) = a(t)

Z

t

0

dt

a(t)

(1.46)

grows more slowly than the s ale fa tor, therefore, regions that were in ausal

onne tion before this period are pushed outside the Hubble radius

r

Hubble

=

1

H

(1.47)

Ana eleratingperiodisobtainableonlyiftheoverallpressure

p

oftheuniverseis

negative:

p < −ρ/3

. Neither a radiation-dominatedphase nor a matter-dominated

phase (for whi h

p = ρ/3

and

p = 0

, respe tively) satisfy su h a ondition. From

theory we know the inationis driven by the va uumenergy of the inaton eld.

This period of exponential expansion solved some of the short omings of the

standard Big Bang Theory.

1.5.1 Horizon Problem

The horizon problem is related to the fa t that every Big Bang model have a

osmologi al horizon whi h delimits regions in ausal onne tion one with the

others. In the Big Bang theory the horizon is too small to explain the high

isotropy observed inthe CMB where very far emissionregions seem tobe in ausal

onne tion and inside the osmologi al horizon. If ination lasts long enough, all

the physi al s ales that have left the horizon during the radiation-dominated or

matter-dominated phase an re-enter the horizon in the past: this is be ause su h

s ales are exponentially redu ed. This explains the problem of the homogeneity of

the physi al length is within the horizon, mi rophysi s an a t, the universe an

be made approximately homogeneous and the primaeval inhomogeneities an be

reated.

If

t

i

and

t

f

are, respe tively, the time of beginning and end of ination, we an

dene the orresponding number of e-foldings

N

as

N = ln [H

I

(t

e

− t

i

)] .

(1.48)

A ne essary ondition to solve the horizon problem is that the largest s ale we

observe today, the present horizon

H

−1

0

, was redu ed during ination to a value

λ

H

0

(t

i

)

smallerthan the value of horizonlength

H

−1

I

duringination. This gives

λ

H

0

(t

i

) = H

−1

0



_a

t

f

a

t

0

  a

t

i

a

t

f



= H

−1

0

 T

0

T

f



e

−N

_<

∼ H

I

−1

,

whereforsimpli itytheshortperiodofmatter-dominationisnegle tedandwehave

alled

T

f

the temperature atthe end of ination. We get

N >

_{∼ ln}

 T

0

H

0



− ln

 T

_H

f

I



≈ 67 + ln

 T

_H

f

I



.

Apart fromthe logarithmi dependen e, we obtain

N >

∼ 70

.

1.5.2 Flatness problem

The atness problem is related ti the fa t that although osmologi al data are in

agreement with a at Universe, the Big Bang model requires a ne tuning on the

density parameter that has to be

Ω

−1

_{− 1 ≃ 10}

−60

(1.49)

non onlyatpresent timebut atalltimes. Ination alsosolveselegantly theatness

problem. Sin e during inationthe Hubble rate is onstant

Ω − 1 =

_a

2

k

_H

2

∝

1

a

2

.

Ontheotherendreprodu eavalueof

(Ω

0

−1)

oforderofunitytodaytheinitialvalue

of

(Ω−1)

atthebeginningoftheradiation-dominatedphasemustbe

|Ω − 1| ∼ 10

−60

.

Sin eweidentifythebeginningoftheradiation-dominatedphasewiththebeginning

of ination,we require

|Ω − 1|

t=t

f

∼ 10

−60

_.

During ination

|Ω − 1|

t=t

f

|Ω − 1|

t=t

i

=

 a

i

a

f



2

= e

−2N

_.

(1.50) Taking

|Ω − 1|

t=t

i

of order unity, it is enough to require that

N ≈ 70

to solve

the atness problem. Ination does not hange the global geometri properties

of the spa etime. If the universe is open or losed, it will always remain at or

losed, independently from ination. What ination does is to magnify the radius

of urvature

R

curv

dened in Eq. (1.11) so that lo ally the universe is at with a

great pre ision.

1.5.3 Monopole problem

The magneti monopole problem is related with the GUT [Buraset al.,1978℄, the

BigBangtheorypredi tsthe reationofanumber

n

m

ofmagneti monopolesduring

the GUT phase transition

n

m

> 10

−10

n

γ

(1.51)

where

n

γ

is the number density of photon at that time. None of the pro esses

in the Universe history an destroy monopoles, then today they should be

n

m0

>

10

−10

_n

γ

≃ n

b0

. These monopoles are very massiveparti oles (

m

m

≃ 10

1

_{6 GeV}

) and

a ording to theirpredi ted abundan e they should be the dominant omponent of

the osmologi al uid

Ω

m

=

ρ

m0

ρ

c0

=

m

m

n

m0

ρ

c0

,

(1.52)

Ω

m

> 10

16

Ω

b

(1.53)

Themeasuredtotaldensity

Ω

0

andthela kofpositivemagneti monopoledete tions

deny all the previous assumptions. Considering the ination epo h, we nd the

magneti monopoles are reated before ination and therefore their density is

diluted by the exponential expansion up to a point where their ontribution to

the osmologi al uid is irrelevant and itis extremely unprobable toobsevre them.

1.6 Cosmologi al perturbation theory

The theory ofstru ture formationstudys howthe primordialu tuationsinmatter

and radiation grow into galaxies and lusters of galaxies due to self gravity.

perturbations present at very ealy times and produ ed from quantum u tuations

during the ination period. CMB observations indi ate that the anisotropies at

the epo h of de oupling were rather small (one part in

10

5

), implying that their

amplitudes were even smaller at earlier epo hs. Then the generation and the

evolutionof the perturbations an bestudied using linear perturbation theory.

1.6.1 Metri perturbations

InaFriedmannba kground,themetri perturbations an bede omposeda ording

to their behavior under lo al rotation of the spatial oordinates on hyper-surfa es

of onstant time. Therefore, the perturbations are lassied into s alars, ve tors

and tensors. S alar perturbations are invariant under rotations and are the main

responsible forthe anisotropies and the inhomogeneities inthe Universe.

In the following we use only s alar pertubations in order to obtain, in the linear

regime, the evolution equations of the matter u tuations reated in the ination

period.

We start deriving the perturbed ontinuity, Euler and Poisson equations in the

matter domain, inserting a generi small pertubation on the homegeneous density

ρ

b

,velo ity

v

b

, pressure

p

b

and the gravitational potential

Φ

b

values

ρ = ρ

b

+ ρ

i

, v = v

b

+ v

i

, p = p

b

+ p

i

, Φ = Φ

b

+ Φ

i

(1.54)

in the ontinuity and Euler equations

∂ρ

∂t

+ ∇(ρv) = 0

(1.55a)

∂v

∂t

+ (v∇)v = −

∇p

ρ

− ∇Φ

(1.55b)

∇

2

Φ = 4πGρ

(1.55 )

We have to take into a ount also the eqution of state,

c

2

s

= ∂p/∂ρ

. Then the

perturbed equations are

∂ρ

i

∂t

+ ρ

b

∇(ρv

i

) + ρ

i

∇v

b

= 0

(1.56a)

∂v

i

∂t

+ (v

i

∇)v

b

= −

∇p

i

ρ

b

− ∇Φ

i

(1.56b)

∇

2

Φ

i

= 4πGρ

i

..

(1.56 )

We introdu e the density ontrast

δ = (ρ − ρ

b

)/ρ

b

and hange in omoving oordinates

∇

r

=

∇

x

a

,

˙r = ˙ax + a ˙x,

(1.57)

where the rst term in

˙r

is the expansion velo ity

v

b

= ˙ax

and the se ond one is

the pe uliar velo ity of the perturbation

v

i

= a ˙x

; we dene

u = ˙x

as the omoving

velo ity of the perturbation. Therefore, the ontinuity equationbe omes

˙δ = −∇

x

u,

(1.58)

and the Euler

¨

δ + 2

˙a

a

˙δ = ∇

2

x

p

i

ρ

b

a

2

+

∇

2

x

Φ

i

a

2

.

(1.59)

Finally, by using the equation of state and Eq. (1.56 ), assuminga typi alFourier

transformation given by

f (k, t) =

1

(2π)

3/2

Z

d

3

_{k e}

ik·x

_{f (x, t) ,}

(1.60)

we obtained the perturbationequation

¨

δ + 2

˙a

a

˙δ =

c

2

s

a

2

∇

2

x

δ + 4πGρ

b

δ.

(1.61)

This equation has the typi al harmoni os illator form, the se ond term in the

left side is the damping term ontaining the expansion rate of the Universe in

opposition to the gravitational ollapsing. From this equation, it is possible to

write the evolutionequation for allkinds of uid and

Ω

.

1.6.2 Stru ture formation

Nowweknowwheretheprimordialu tuations omefrom,we an studyhowthese

pertubations be ome stru tures under the inuen e of the gravity only. Then we

an assumethe matter pressure term is negligiblewith respe t tothe gravitational

potentialone

k

2

c

2

_s

_{≪ 4πGρ}

b

.

(1.62)

In the following analysis we onsider aUniverse with

Ω

0 DE

6= 0

and

w = −1

.

From Eq. (1.14) and Eq. (1.10) wend

4πGρ

b

= Ω

0

3H

2

0

therefore, the Eq. (1.64) be omes

¨

δ + 2

˙a

a

˙δ = Ω

0 M

3H

2

0

2a

3

δ.

(1.64)

One of the two solutionsis

δ(a) =

5Ω

M

2

E(a)

Z

a

0

da

′

[a

′

_E(a

′

_)]

3

≡ D(a),

(1.65)

where

E(a)

is given by Eq. 1.24. It denes the growth fa tor

D(a)

of the matter

perturbations, a fun tion of the natural logarithm of the s ale fa tor. The linear

growth fa tor isstrongly dependent on

w

,with

w > −1

models behaving more like

open Universes than

w < −1

models asthe ee t of the Dark Energy diminishes.

Although this integral an be easily solved numeri ally, it is ommon to use the

approximationof [Carrol etal.,1992℄

D(a) ≃

5Ω

M

(a)a

2

Ω

M

(a)

4/7

− Ω

Λ

(a)

+



1 +

Ω

M

(a)

2

 

1 +

Ω

Λ

(a)

70



−1

.

(1.66) 1.6.3 Non-linear perturbations

Nowwe dealwith the non-linear perturbationsin ordertond the powerspe trum

whi halso hara terizes the CMB anisotropies.

We an start onsidering a volume

V

u

in whi h there is signi ant stru ture due

to the perturbations and also denote by

hρi

the mean density in the volume, by

ρ(x)

the density at a point spe ied by the position ve tor

x

with respe t to some

arbitraryorigin. Wedenetheu tuation

δ(x) = [ρ(x) − hρi]/hρi

. Asusualismore

expressible asa Fourier series:

δ(x) =

X

k

δ

k

exp(i k · x) =

X

k

δ

∗

k

exp(i k · x).

(1.67)

The Fourier oe ients

δ

k

are omplex quantities given by

δ

k

=

1

V

u

Z

V

u

δ(x) exp(i k · x)dx.

(1.68)

Now we an imagine a large number N of su h volumes, i.e. a large number of

`realisations' of the Universe, one will nd that

δ

k

varies from one to the other in

statisti s: so the mean value of perturbation is identi ally zero by denition, its

mean square value, i.e. itsvarian e

σ

2

,is not but

σ

2

_{≡ hδ}

2

_{i =}

X

k

h|δ

k

|

2

i =

1

V

u

X

k

δ

_k

2

.

(1.69)

If we now take the

V

u

→ ∞

and assume that density eld is statisti ally

homogeneous and isotropi , so that there is no dependen e on the dire tion of

k

but only on

k = |k|

, we nd

σ

2

=

1

V

u

X

k

δ

_k

2

_→

1

2π

2

Z

+∞

0

P

Φ

(k)k

2

dk,

(1.70)

where, for simpli ity,

δ

2

k

= P

Φ

(k)

. The quantity

P

Φ

(k)

is alled the power spe tral

densityfun tionoftheeld

Φ

,orpowerspe trum,and

σ

2

tellsusabouttheamplitude

of perturbations.

The perturbation power spe trum

P

Φ

(k)

, at least within a ertain interval in

k

, is

given by the following power law

P

Φ

(k) = Ak

n

,

(1.71)

the exponent

n

isusually alledthe spe tralindex.

The equation 1.70 an alsobewritten inthe form

σ

2

=

1

2π

2

Z

_∞

0

P

Φ

(k)k

2

dk =

Z

_∞

−∞

∆(k)d ln k

(1.72)

where the dimensionless quantity

∆(k) =

1

2π

2

P

Φ

(k)k

3

_.

(1.73)

Itismore onvenientto onstru tastatisti aldes riptionofthe u tuationeld

as a fun tion of some s ale

R

. In this way it is possible to dene a mass varian e

inside a spheri alvolume

V

of radius

R

with mass

M

σ

2

_M

=

hδM

2

_i

hMi

2

(1.74)

Using the usual Fourier de ompositionas beforewe nd

σ

2

=

1

V

u

X

k

δ

_k

2

W

2

(k R);

(1.75)

the fun tion

W (k R)

is alled the window fun tion. We shall use this expression

Cosmologi al Mi rowave Ba kground

The osmi mi rowave ba kground was rst predi ted by Alpher and Herman

[Alpher & Herman,1998℄ in 1948 as a thermal reli isotropi radiation with an

estimatedmeantemperatureof

5 K

. Therstdete tionofthemi rowaveba kground

omes in 1965 by Penzias and Wilson [Penzias & Wilson, 1965℄, for whi h they

later won the Nobel Prize, who observed an ex ess of

3.5K

in their antenna

temperature noise on the

λ = 7.35cm

. From that dis overy the eld of osmi

mi rowave ba kground (CMB)anisotropies has advan ed overthe years, espe ially

thanks to the instruments like COBE (the NASA satellite COsmi Ba kground

Explorer, laun hed in 1989)and WMAP (the NASA satelliteWilkinson Mi rowave

Anisotropy Probe, 2001-2012 a tivity). Their observations have turned some of

initialspe ulationsabouttheUniverseintothe urrent osmologi almodel: namely,

that the Universe is spatially at, onsists mainlyof dark matter and darkenergy,

with the small amount of ordinary matter ne essary to explain the light element

abundan es produ ts of nu leosynthesis, and large s ale stru tures formed through

gravitationalinstabilityfromprimordialperturbationswhi hmightbeexplainedas

originated by quantum me hani al u tuations during ination. COBE onrmed

the osmologi al origin predi tions of the CMB, measuring an almost perfe t

bla kbodyspe trumpeakedat

2.725±0.002 K

andatemperatureanisotropiesofthe

order of

∆T /T ∼ 10

−5

at the angular s ale of 7 degrees. From these observations

we learn that the CMB is remarkablyuniform ex ept for the dipoleindu ed by the

motion of the Solar Sistem[Smootet al,1977℄. This isin ontrast tothe matterin

the Universe, organized invery non-linearstru tures likegalaxiesand lusters. The

smooth photon distribution observed in CMB with respe t to the lumpy matter

gravitationalinstability,butpressure preventsthesamepro essfromo uringinthe

photons. Thus, eventhoughbothinhomogeneitiesinthematterintheUniverseand

anisotropies in the CMB apparently originated from the same sour e, these appear

very dierent today.

Sin e the photon distribution is very uniform, perturbations are small, and

linear response theory applies. This is perhaps the most important fa t about

CMB anisotropies. If the sour es of the anisotropies are also linear u tuations,

anisotropy formationfallsinthe domainoflinearperturbationtheory(/ref. apitolo

primo). Therearethen essentiallynophenomenologi alparametersthat needtobe

introdu edto a ount for non-linearitiesor gas dynami s or any other of a host of

astrophysi al pro esses that typi allyai t osmologi al observations.

CMB anisotropies in the working osmologi almodelfall almost entirely under

linear perturbation theory. The most important observables of the CMB are the

power spe tra of temperature and polarization maps. Theory predi ts, and now

observations onrm,thatthetemperaturepowerspe trumhasaseriesofprominent

peaksand troughs. In2.1.1, we dis uss the origin of these a ousti peaks and their

osmologi al uses.

2.1 Primordial anisotropies

In order to study the CMB we onsider its intensity as a fun tion of frequen y

and dire tion on the sky

n(θ, φ)

ˆ

. Sin e the CMB spe trum is an extremely good

bla kbody[Fixen &Mather, 2002℄, generallytheobservable

T

isdes ribed interms

of a temperature u tuation

∆T

T

(θ, φ) =

T (θ, φ) − T

0

T

0

.

(2.1)

By using the spheri alharmoni s expansion

∆T

T

(θ, φ) =

X

ℓ

X

m

a

ℓm

Y

ℓm

(θ, φ) ,

(2.2)

if these u tuations are Gaussian,then the multipolemoments of the temperature

eld

a

ℓm

=

Z

dˆ

nY

∗

ℓm

(ˆ

n)

∆T

T

(ˆ

n)

(2.3)

are fully hara terizedby their power spe trum

ha

∗

whosevaluesasafun tionofthemultipoles

ℓ

areindependentinagivenrealization.

Forthis reason predi tionsand analysesare typi allyperformedinharmoni spa e.

Sin e the angular wavelength

θ ≃ 2π/ℓ

, large multipole moments orresponds

to small angular s ales. Likewise, sin e in this limit the varian e of the eld is

R d

2

_ℓC

ℓ

/(2π)

2

, the powerspe trumis usually displayed as

∆

2

T

≡

ℓ(ℓ + 1)

2π

C

ℓ

T

2

_.

(2.5)

Whereas COBE rst dete ted anisotropy on the largest s ales, observations in

the lastde ade have pushed the frontier tosmallerand smallers ales. The WMAP

satellite, laun hed in June 2001, went out to

ℓ ∼ 1000

, while the ESA satellite,

Plan k, laun hed in 2009, went afa tor of two higher.

The power spe tra (mettere plot pogosian per osmi varian e) exhibit large

un ertainty at low multipoles. The reason is that the predi ted power spe trum is

the average powerinthe multipolemoment

ℓ

anobserver would see inanensemble

of universes. However a real observer is limited to one Universe and one sky with

its one set of

a

ℓm

's,

2ℓ + 1

numbers for ea h

ℓ

. This is parti ularlyproblemati for

the monopoleanddipole(

ℓ = 0, 1

). Ifthe monopolewerelargerinour vi initythan

its average value, we would have no way of knowing it. Likewise for the dipole, we

have very littlehope of distinguishing a osmologi aldipole from our own pe uliar

motion with respe t tothe CMB rest frame.

In this way low

ℓ

's are dominated by  osmi varian e be ause there are only

2ℓ + 1 m

-samplesof the power inea h multipolemoment

∆C

ℓ

=

r

2

2ℓ + 1

C

ℓ

.

(2.6)

By averaging over

ℓ

in bands of

∆ℓ ≈ ℓ

, the pre ision in the power spe trum

determinations ales as

ℓ

−1

, i.e.

∼ 1%

at

ℓ = 100

and

∼ 0.1%

at

ℓ = 1000

.

Of ourse,anysour eofnoise,instrumentalorastrophysi al,in reasestheerrors.

If the noise is alsoGaussian and has a known power spe trum,one simply repla es

thepowerspe trumontherighthandsideofEq. (2.6)withthesumofthesignaland

noise power spe tra [Knox, 1995℄. Be ause astrophysi al foregrounds are typi ally

non-Gaussian it is usually also ne essary to remove heavily ontaminated regions,

e.g. the galaxy. If the fra tion of sky overed is

f

sky

, then the errors in rease by a

fa tor of

f

−1/2

into a ount these aveats, the Eq.(2.6) be omes the Knox equation[Knox, 1995℄

∆C

ℓ

=

s

2

f

sky

(2ℓ + 1)



1 +

A

pix

σ

2

pix

C

ℓ

e

−ℓ

2

_{F W HM/}

√

_8log2



C

ℓ

.

(2.7)

where

A

pix

, σ

pix

are the area of the pixeland the sensitivity per pixeland

F W HM

is the full width halfmaximum.

2.1.1 A ousti peaks

As we have seen in Chap. 1, when the temperature of the Universe was at

T

rec

∼ 4000

K and a redshift

z

rec

≈ 1100

, ele trons and protons ombined to

formneutral hydrogen,in the re ombinationtime. Beforethis epo h,free ele trons

a tedasglue between thephotons andthe baryons throughThomson andCoulomb

s attering, so the osmologi al plasma was a tightly oupled photon-baryon uid

[Peebles & Yu,1970℄. At

ℓ > 100

theCMBpowerspe trum anbeexplainedalmost

ompletelyby analyzingthe behaviorof this pre-re ombinationuid.

We an start from the general evolution equation, Eq.(1.64), for perfe t photon

uid, negle tingforthe rst approximationthe dynami alee ts ofgravityand the

baryons. Sin e perturbationsare very small, we assume a linear approximation for

the evolution equationsand dierent Fouriermodes evolvingindependently.

¨

δ =

c

2

s

a

2

∇

2

x

δ

(2.8)

The photonpressure is

p

γ

= ρ

γ

/3

, the temperaturedensity

ρ

γ

∝ T

4

and the desity

ontrast is given by

4δ

T

= ρ

γ

/¯

ρ − 1

.

where

c

s

≡

p ˙p/ ˙ρ = 1/

√

3

isthesoundspeedinthe(dynami allybaryon-free)uid.

We nd the pressure gradients a t as a restoring for e to any initial perturbation

in the system whi h thereafter os illate at the speed of sound. Physi ally these

temperature os illations represent the heating and ooling of a uid that is

ompressed and rareed by a standing sound or a ousti wave. This behavior

ontinues until re ombination. Assuming negligible initial velo ity perturbations,

we have atemperaturedistribution atre ombinationof

δ(η

rec

) = δ(0) cos(ks

rec

) ,

(2.9)

where

s =

R c

s

dη ≈ η/

√

3

is the distan e sound an travel by

η

( alled sound