PoS(cargese)006 http://pos.sissa.it/ ce. e shall first give an overview of ann equation to investigate also tropies in the instant decoupling like geodesics from the last scat- s experimental data and parameter ization. nd the Standard Models ive Commons Attribution-NonCommercial-ShareAlike Licen ∗ Speaker. estimation from observations of CMB anisotropies and polar cosmological perturbation theory. Thenapproximation we derive where CMB we aniso can obtaintering them surface by until integrating today. light polarization In and a Silk second damping. step we In the use final the lecture Boltzm we discus In these lectures on the cosmic microwave background (CMB) w ∗ Copyright owned by the author(s) under the terms of the Creat c

Cargèse Summer School: Cosmology and Particle Physics Beyo June 30-August 11 2007 Cargèse, France Ruth Durrer Université de Genève Département de Physique Théorique 24, Quai E. Ansrmet, 1211 Genè[email protected] 4, Switzerland E-mail: The Cosmic Microwave Background PoS(cargese)006 (1.1) . Latin t Ruth Durrer ial indices by Latin is conformal time. We t logists began to search for , ccidentally in 1965 by Pen- ν erturbation theory, the calcu- od accuracy, be treated within larization are so important for servational tool in cosmology. t you are already familiar with discovery. It was the decisive dx of inflation, since this has been which can be interpreted by the rs, as compared to a steady state µ rators. ng the last 15 years, the cosmic 3 nte Carlo methods. shall not talk about CMB lensing e rest frame of the surface of last we shall not derive the Boltzmann investment. This is one of the main − dx obel prize of 2006 for this important Bang model, initiated by Friedmann eeded to investigate polarization and . Note that sults and the principles of parameter ructures, galaxies, clusters, voids etc. 10 µν sed on my book [3] which contains of K d with the help of the COBE (Cosmic ion approximation and the CMB spec- g × ion equations for perturbation variables e background metric 2 l perturbations. For a long time they found . ]= j while Greek spacetime indices are raised and dx γ i dx 2 i j γ + 2 dt − )[ t ( . . An over-dot denotes the derivative w.r.t conformal time 2 a µν adt g = 2 = τ ds d , τ respectively We denote spacetime indices by Greek letters and purely spat µν g is the metric of a 3-space with constant curvature γ In these lectures, you will learn why CMB anisotropies and po Notation: However, We shall give a thorough overview of cosmological p Fluctuations in the CMB are small. Therefore they can, to a go The cosmic microwave background (CMB) has been discovered a More or less immediately after the discovery of the CMB cosmo denote cosmic time by where Doppler shift due to thescattering. motion Finally, of in 1992 the the earthBackground fluctuations with Explorer) have respect been Satellite to foun mission th microwave from background NASA (CMB) [2]. has becomeThis the Duri is probably most partly important why G. ob Smootdiscovery. has been attributed the N cosmology and what we canFriedmann–Lemaître learn from (FL) them. cosmology and We shall havetreated assume some in tha knowledge the course by Cliffcourse Burgess. much These more lectures than are what ba and is the presented CMB here. frequency spectrum. Especially,equation we Furthermore, in in any this detail course and we will not enter in Markov chainlation Mo of CMB anisotropies in thetrum. instantaneous We recombinat shall also discuss the BoltzmannSilk equation damping. which is Furthermore, n weestimation present and the degeneracies. observational re 3d indices are raised and lowered with the 3-metric which can be solved to high accuracy with moderate numerical first order perturbation theory. This leads to linear evolut 2. Cosmological perturbation theory letters. Spatial 3-vectors are written in bold. We choose th zias and Wilson [1]evidence who which obtained convinced most the physicists Nobel inand favor prize Lemaître of 1978 and the Big worked for out this becosmology Gamov, which Dicke, had Peebles been and advocated othe by Fred Hoyle and collabo The Cosmic Microwave Background 1. Introduction its fluctuations which must be therehave if formed the by gravitational observed instability cosmic from st smallnothing initia apart from a dipole anisotropy on the level of 1 lowered with PoS(cargese)006 , ) . e X g T ν , ¯ (2.1) (2.3) (2.2) g dx ( , of µ ) s is small, ( , the two dx x ∗ 2 ) γ s equation, Ruth Durrer φ a and we can µν / | und metric ¯ ε h ¯ g . 1 + − g µν ≪ | ¯ g of the form ε f order ∼ = (  } j | αβ dx µν i g tum tensor by choosing a foli- over different hyper–surfaces), g { . The same is also true for the } dx and its push forward − i j s are small in the sens that µν αβ mentum tensor over spatial slices, h φ are related via Einstein’s equation, H { he underlying cosmological model e.g., µν sfy them, . Comparing them with calculations 2 is not unique. The only measurable ed admissible FL background g a discussion of the main elements of + which deviate only in first order from | s ( sms (coordinate transformations), the µν . + represented as the infinitesimal flow of he following two conditions: T ism max µν j µν kgrounds. But since ical perturbation theory is the basic tool h ¯ . g µν i.e., ¯ dx T i and = µν G dx π and µν i j µν 8 GT g γ g ) π 3 6= 8 , Φ invariant ) and the metric of a 3 dimensional space of constant 2 1 = g ) µν − t ¯ g ( ≪ µν 1 ( a ε G µν + ( ∼ and a perturbation i G satisfy Einstein’s equation (which in this case reduce to th |} µν | T g dtdx αβ i µν T σ 0. T 2 and {| } = − + . In general, however, due to the non-linearity of Einstein’ g 2 i j αβ µν µν { γ . Remember the definition of the flow: For the integral curve, ¯ dt T T i j X | ) ε φ H Ψ → max describe the same geometry. Since we have chosen the backgro 2 = = ) i i µν g + φ ( T the most general metric of a perturbed Friedmann universe is 1 H ∗ , ( φ i j X γ −  and ) , the difference of the two FL backgrounds must also be small o and t ( ε µν g 2 ¯ g a Friedmann equations). Denoting the cosmic scale factor by The split into a background ¯ → = is traceless, 1. The averaged fields ¯ 2. The deviations of the averaged fields and the physical field 2 i j µν ds H There may be many different admissible averaging procedure curvature by Let us call an averaging procedure ’admissible’ if it obeys t the identity. Such an ’infinitesimal’ diffeomorphisma can vector be field geometry is the one described by the full metric the averaged metric and energy momentum tensor will not sati energy momentum tensor. The full physical fields We can define a background metrication and of background spacetime energy and momen averagingg the metric and the energy mo as chosen. Since theperturbations theory are is not invariant unique. under diffeomorphi For an arbitrary diffeomorph metrics of order we can accurately determine theand values the of initial the perturbations. parameters Therefore, of linear t cosmolog The Cosmic Microwave Background reasons why observations of CMB fluctuations are so valuable to investigate CMB anisotropies. This section is devoted to cosmological perturbation theory. 2.1 Definition of cosmological perturbations, gauges we only allow diffeomorphisms which leave ¯ leading to slightly different Friedmann-Lemaître (FL) bac interpret it as part of the perturbation. We consider now a fix PoS(cargese)006 ) 1 ( , to S (2.6) (2.4) (2.5) X obeys ble to ) 1 ( S Ruth Durrer ε is equivalent 0, hence + ) ¯ = S g ( S ∗ = φ S → , we conclude that only g X , with vanishing (or con- ε φ = i.e., oversies in the literature, since , φ . ued by gauge modes. Although X , 8] on gauge-invariant cosmolog- . In terms of the vector field kx ) i of perturbation equations is usually and can therefore be written in the s − ast to synchronous gauge where the ( o see, that on sub-horizon scales, the e perturbation equations in terms of to components which transform irre- , alism is that gauge-invariant variables iant perturbation theory which we are e x defined on them. This means that we s result is called the ’Stewart-Walker dependent perturbations, especially on itrary tensor field esimal coordinate transformations are ) γ ) t 2 . , ensor fields of rank zero), in the case of he usual, gauge dependent ones. Since . bles corresponds to the Newtonian poten- ε ¯ k S ¯ ( g ( )= X X p O L k f ( L 3 . The transformation 2 + X s + d − X ) X φ a 1 4 L ( Z ε S + 3 ) h + 1 π → 2 0 for all vector fields ) id → ( . Under an ’infinitesimal coordinate transformation’ the 1 ( we have ) h = = 2 S p ε ∗ S ( )= φ X t , O L . The corresponding background equation is )= x S ( 0 0, this is simply the Fourier decomposition, f generates a gauge transformation )+ ¯ hypersurfaces are homogeneous and isotropic, it is reasona = g = X s X } ( K L x γ + h 2 i.e., therefore transforms as a const. , ( h p ε = + t ¯ { g , its push forward is then given by ε → denotes the Lie derivative in direction h 0 for some tensor field 2 X a = L ε S The gauge dependence of perturbations has caused many contr First we note that all relativistic equations are covariant Since the The principal sources of this section are the Refs. [3, 5, 6, 7 Since every vector field + g stant) ’background contribution’ areLemma’ gauge [4]. invariant. Thi it is often difficult to extractsuper–horizon the scales. physical This meaning problem of is gauge solvedgoing by to use gauge here. invar The advantagehave of the simple gauge-invariant form geometric andthe physical derivation meanings requires somewhat and more are work,simple not the and final plag well system suited forgauge-invariant numerical matter treatment. perturbation We shall variables als approach t tial, the Newtonian limit canNewtonian potential be is performed gauged to easily. zero.) (In contr form one of the gauge-invariant geometrical perturbation varia perturbations of tensor fields with the gauge transformation law to ¯ ical perturbation theory. 2.2 Harmonic decomposition is gauge-invariant. Itgauge-invariant is variables. thus always possible to express th metric perturbation The Cosmic Microwave Background with starting point where first order in vanishing spatial curvature, perform a harmonic analysis fordecompose the a perturbation (spatial) variables tensor fieldducibly on under these translations hypersurfaces and in rotations. For functions (t In the context ofcalled cosmological ’gauge perturbation transformations’. theory, The infinit perturbation of an arb PoS(cargese)006 ) is 2 B + (2.7) (2.9) (2.8) ℓ (2.12) (2.14) (2.13) (2.11) (2.10) ( ℓ = Ruth Durrer 2 is the spin 2 k ) e, T ( i j and we can define , , H ) T k i is the spin 0 and . ( i j ) 1 j ϕ × i j ( H e ie ) , + 2 ( i ) −  e T ) ji ( d i j i V | . Here j + e ( Q ector and tensor modes we use ) X 2 H an, 2 = j ( k . e 0 j + − ) − adient and a curl component, erturbations are expanded either in ) 1 e . all not enter into this. Vector pertur- ( i j gous, but somewhat more involved. = V 0 | − i = . e ( i i i | e h ) H =  T B ji 2 1 ) ( . = j  2 ) ) ( ) is a symmetric transverse traceless tensor, Q S j 1 2 ) can be decomposed into irreducible com- T 2 e ( k = | ( i i ∆ ) in the plane normal to x − Q T i j ( × i j + 1. ji ) H ( 2 ± e e 2 T k ( ) Q > = 1 e where H ( is the spin 1 component and − i 2 5 ) e , k )  ) T  = 1 , ( V i , i j , i and ( ( ) i 2 γ e S H 0 and B 1 i × i j ( k H ∆ ) √ have two degrees of freedom. In the case of vanishing . 2 ie + 3 1 Q = ) j = ( ) i V V j ∆ e | j = ϕ ( + . T | ) ) − i ji ( ) ) 2 j V Q ± d i j ( i V ( 3 ∇ i j Q 2 ( e e e ( ∇ k i H Q = − = − ∇ i SO ) and j V 1 + .  j ( = e ) e H ) ) V + + i j ( 1 V e j ( ( i i j Q e γ Q = h L ∆ ) 1 2 H 2 are the discrete eigenvalues of the Laplacian on the 3–spher k i j (+ = = e i j d i j are spin 0 components, e H 0. Both, T = 1, they are bounded from below, H i | − ) is a transverse vector, T to denote the three–dimensional covariant derivative of ji ) ( = i and V | j 1 the values ( Q X K L 0 it is the decomposition into eigenfunctions of the Laplaci Q = = H j For a spatial symmetric tensor field we have We shall not need higher tensors (or spinors). As a basis for v In addition, a tensorial variable (at fixed position For a spatial vector field, this is its decomposition into a gr 6= K ) T K j ( In curved spaces theSince definition we of shall the not need helicity thebations basis explicite can form is be of analo expanded this in basis,terms we terms of sh of the standard this tensor basis, basis while given by tensor p where or also in terms of a helicity basis defined by curvature we can use an orthonormal basis Q helicity basis vectors, The Cosmic Microwave Background If where the spin 1 component of the vector field Here component of the tensor field and for ponents under the rotation group For the vector and tensor type eigenfunctions of the Laplacian, We use PoS(cargese)006 k , and (2.15) (2.20) (2.22) (2.16) (2.21) (2.23) (2.19) (2.18) ϕ i ± e Ruth Durrer have spin up, ) 2 (+ Q . ) 2 . and (2.17) and ) − , i j ( T j ) ( i j e S ) ) ( (+) 2 Q dx − j ) i ( Q − Q T ( e i j ( ) dx γ Q − H i j tensor amplitudes of each mode ( 1 3 .9)) ors and traceless symmetric tensors + ions H Q + ) + 2 ) 2 . ) . + V S k ) e define i j + i j ( i j (+ ( | n and write down the perturbation equa- . . Hence vector perturbations are spin 1 i V have spin down. Scalar perturbations of e i ( Q j ) Q (+) according to (2.19) and (2.20). tions in time. simply transform with a phase ϕ ) ) . 2 µν 2 i e Q 2 and V ) ) h i j ( ∂ − − 2 i (+ 2 t V dtdx ( k H ( i (+) ± L H a Q i j ( Q B V Q ≡ e + + + 2 = under gauge transformations. We set the vector ) t ) . + and 6 S = ) S ∂ ( i j − ) i ( i j µν ) and ) × S ¯ ) 2 µν T and 2 g ( i j ( i Q B i ) j Q ( | V with angle e h j ( ) e T − ) = = ( ) ± k × Q Adt 2 H VQ ( ( ( Q X 2 e µν + Q + , Q = − g ) i ) j i j + V | + j S are functions of V γ i ( = ( | ) ) ) ) d 1 S Q j ν Q i j ( ( ( T ( 1 ( e e ) k Q ) − dx 1 –mode of a vector field is then of the form 1 d H L 2 k ( ( µ k H Q Q dx and = ≡ − ≡ − = = µν ) i j h V ) ) ) ) ( . S H T V V j ( j ji k ( i j ( ( H Q Q Q Q , 2 respectively, while T H , =+ L H m , ) V ( B , 1 and respectively under rotations around B Let us now investigate the behavior of Perturbations of the metric are of the form This decomposition is very useful, since scalar, vector and As in Eqs. (2.8) and (2.9), we can construct scalar-type vect The decomposition of the ϕ i 2 =+ ± evolve independently, obeying ordinary differential equa 2.3 Metric perturbations Here and we decompose the perturbation variables The components in the ’helicity basis’, field defining the gauge transformation to We parameterize them as e course have spin zero. and vector-type traceless symmetric tensors. To this goal w The Cosmic Microwave Background We can develop the vector and tensor basis functions as m The decomposition of a tensor field is given by (compare Eq. (2 tions for a given mode In the following we shall extensively use this decompositio fields while tensor perturbations are spin 2 fields. The funct PoS(cargese)006 = = i L kL (2.26) (2.25) (2.33) (2.32) (2.24) (2.29) (2.27) (2.28) (2.31) (2.30) Ruth Durrer constant time. A .  0): vanishes and we have i j γ ) ) = V ( , Φ i , H , long + σ ) | 1 kL V B Ψ dtdx − ( . ( − k ˙
L . i = ∆ i j , . T . es the gauge transformation  γ i so that T − 1 3 ) j vanish. In this gauge (longitudinal . ransformations. H H j ) ) T T ( − long B V V − + dx | ( ( i i , this implies the following behavior → (we also decompose the vector dx H ˙ T dtdx i j i H L B | i j H ) R T ) dx H V and H + dx i ( → 2 H −
= → Φ i
i j ) | T i ) Q ) j | γ + T ) = j + V H V ( L ( 2 ( V and Φ L ( σ Ψ H B 2 , , i + dt 1 ( , kL , σ + [9]) vanishes. It therefore is a gauge-invariant j B ˙ ˙
2 −  | − j T i σ i L 7 ˙ , | k a 1 , T i 2 1 L k 3 2 T + 2 − L ˙ ) − dt T , + k H + e.g., T + = + V ( Ψ i T 1 2 + i j − = + ν T ˙ 2 L γ kT j H kL T T σ + potentials. In a generic gauge the Bardeen potentials are 0 − T H dx 1 i − − 0 H H + i j − i µ − are indeed invariant under gauge transformations (for more ˙ L H = 3 1 ) C 2 k H B A H + ) V dx 2 − ( + − Φ − L S ) − µν (  H V → → → H A B H L h µν ( = 2 + i h a H − Bardeen ν i j A and B → → → → u A − H = µ ) Ψ L A B u ¯ V = = g j ( H ν X i µ H Φ Ψ L C , is the scalar potential for the shear of the hypersurface of B ≡ , so that the transformed variables i j − ˙ L E ): T ) ˙ − H V are the so-called i ( 1 B − Q Φ ) k = V ( = L kT and σ + ) For vector perturbations it is convenient to set Ψ In a FL universe the Weyl tensor (see To fix the scalar perturbations of the metric, one often choos S and ( i T LQ of the perturbation variables under gauge transformations All other components vanish. Comparing this with (2.22) and using (2.5), we obtain given by Using the decompositions (2.19) and (2.20) for short calculation shows that H The Cosmic Microwave Background Using the definition of the Lie derivative, we obtain gauge), scalar perturbations of the metric are of the form ( details see [3]). Two scalar and one vector variable can be set to zero by gauge t where perturbation. For scalar perturbations one finds PoS(cargese)006 and ρ (2.35) (2.36) (2.42) (2.43) (2.37) (2.34) (2.39) (2.41) (2.44) (2.38) (2.40) ty Ruth Durrer . We define u is gauge-invariant. lready introduced. . ) to lowest order, only  T ( ormal to k i j ) ) : 0 k H V | . , µ l ( ν ) 0 j T Q u B jm γ − = ( j 3 1 u v . . ( . + i ¯  i P i k ∂ l ) | i ) ) k . | V ns and hence u T , determine the Weyl tensor fully. = V ( 1 jm is ( m ( , i j j 0 Q + − . Including first order perturbations, the ) Q Q ) τ B . i j t T jl . V ( i j ∂ δ = − ( µ γ . ν 0 ¯ v P ) Q 2 τ , ) 1 3 αβ m u ) | and u A j T jl + T T ( = + jlm ( 0 − i = i j ν j i β ) − ¯ ν l Q Pv | C S E ) H , u P τ u h 1 ( ) V µ µ ( jm − 8 µ α ( 2 , ilm u P a 1 u k ε ilm vQ ) , Q = ρ ε ρ ) V  ) j − i j ( = = 0 = − − 0 and δ T 1 a 2 τ = t ( Q 0 ) α m ) are symmetric and since | ∂ ˙ µν + = u = µ u V H ( jl V = ν ( τ ( 1 ν i j ν µ i T 0 1 2 − ( ˙ u Q u σ B v τ ¯  α µ ρ k 1 a j ν = = 2 = T − = ilm and 0 and = µ ρσ i j i j ε i τ ) ρ is fixed by the normalization condition, E B C = i j u V j 0 ( E , ρσ 0 u i σ 0 0 ν i k ε C 2 = as the time-like eigenvalue and eigenvector of − 2 1 − 0 be the full energy momentum tensor. We define its energy densi 0 u τ µ = ≡ = ν θ i j i j B E + is the projection tensor onto the subspace of tangent space n are determined by the perturbation variables which we have a µ ν µ T µ ν respectively contribute. The tensors 0 δ τ = + 0 ilm µ ν ν C T u µ u and Clearly there are no tensorial (spin 2) gauge transformatio Let The Weyl tensor from vector perturbation is given by j ≡ 0 i 0 µ ν The expression for the Weyl tensor from tensor perturbation the stress tensor Note that from their definition components The spatial components provide new perturbations. We set With this we can write In the unperturbed case we have its energy flux 4-vector 2.4 Perturbations of the energy momentum tensor To first order, the component P The Cosmic Microwave Background We shall call this gauge the “vector gauge”. We then parameterize their perturbations by We obtain C PoS(cargese)006 . ) i − = V ∂ i ( i µ v ν ˙ L τ ¯ ˙ 1 PT (2.48) (2.55) (2.51) (2.47) (2.54) (2.45) (2.52) (2.53) (2.50) (2.49) rturba- − = a and = µ ν ¯ v − P Ruth Durrer , t Π X . The same ∂ δ ¯ L ρ ) ˙ / T and (2.46) ¯ 1 P sor, − j a = ) , v B ) − w ¯ T P 2 − − v + H ( , we obtain , ˙ ¯ ) aa ρ ¯ ˙ ρ is gauge-invariant. lemma. ( . w k T , with / H 0 ¯ µ ν ˙ − rturbation variables. First we P long . − + ) i H Π δ = ) 1 w = ) ≡ i i ( T namely nly one gauge-invariant variable j ( 0 ≡ 2 s w found by combining 2 s + j Π w ] = ( c c Q ) ¯ 1 e energy momentum tensor. A short u ) T + 3 ( B , T 1 3 1 ( . Here we made use of the background X − perturbation variables for a fixed mode ( − , T Π ¯ + L k ρ ¯ ) , ρ with π . 3 j = [ 0. δ + ) − . With i ¯ B − u δ V ) ν T H = = ( → , µ X 2 s ˙ V ) j − w ( H  c = L τ Π L Γ j V j i 2 w π Q v − ) − Π k 9 + → k V , H H )( L ( ( ) 1 ¯ ) , + ) π P ( ¯ V i j Π P ( w 3 T δ H long + = Π ) + Π ) v + − + ¯ i ρ L Γ , is called the anisotropic stress tensor. We decompose it w ¯ H ) 1 j ρ . i π = = S ) → ( j ( , ( +  3 Π 3 w = ( T ) j T + Q Π 1 ˙ j i V ¯ ˙ H − ( ρ 1 + + ( 0 Π ( 3 Π k 1 ˙ L 1  T , = = 3 of the trace of ( . = ¯ + + / long P − − 3 ˙ ¯ ) ¯ j L ˙ i j ρ i ρ ) δ δ v T , is proportional to the divergence of the entropy flux of the pe X δ ( − = V Π − ) ( ) L j i Γ L ≡ ≡ v Π δ v ≡ δ τ π s D V + → → → → D + 0. For the velocity we use 1 ) ) 1 v ( δ ( which is 1 = T V ¯ ρ ( (  ) µ ν v ¯ . The background contribution to the anisotropic stress ten P Π − L Π π ¯ PT = = + j 0 0 i 1 H T ( ) T ¯ is a scalar, hence P w ρ + , vanishes, hence the Stewart–Walker lemma implies that contains in general new perturbations. We define 1 µ j i ν ( We now study the gauge transformation properties of these pe Apart from the anisotropic stress perturbations, there is o Gauge-invariant density and velocity perturbations can be 2 s τ δ w c α α 3 τ , we obtain finally the following transformation behavior 1 3 as note that The traceless part of the stress tensor, is true for For perfect fluids Inserting our decomposition into scalar,k vector and tensor The Cosmic Microwave Background But − From our definitions we can determinecalculation the gives perturbations of th tions. Adiabatic perturbations are characterized by with metric perturbations. We shall use which can be obtained from the energy momentum tensor alone, One can show (see [3]) that energy conservation equation, PoS(cargese)006 V times and (2.60) (2.61) (2.59) (2.58) (2.62) (2.56) (2.57) ˙ v , metric e contri- 1 H − t Ruth Durrer or ∼ al and vector 2 H H can only be made ≫ Φ k ) e like Maple or Mathe- w  + h 1 2 scale, ( a 3 2 . k −
malism of general relativity and + h ecommend the student to simply nts of the energy density and the chosen, the equations obtained in lting equations in gauge-invariant . Therefore, on sub-horizon scales 2 invariant perturbation variables to quations. By elementary algebraic Einstein tensor are a combination h ) ) long 2 h a : δ kt ( using gauge-invariant variables. Since t k µν O = ially important when comparing experi- eglect spatial curvature in the following + ( , + G ein’s equations therefore generically yield  ) ≫ h δ T V 2 kth times first derivatives and
( a H = h , B 2 + 2 3 1 1 ) t , ) h H − µν  + ) kt vec T V ( ( 10 L , δ v O O vec k H ( G H Φ v O = and we can thus neglect the difference between differ- =  = ) ) π ) ) v are negligible as well as the differences between = w w ) = } V  ( w ) ) ρ D  ˙ + + 2 V t H ( + G ρ and / 1 1 µν δρ k 1 2 B ( ( ρ 1 π T {z a δ ( 3 3 and 8 ( δ  . − − 3 ( ) g O  ) ) − + O V + D s O s | ( V V ( ( O , v D σ δ D v  is the metric perturbation in vector gauge and the vector typ constant hyper–surfaces. long ≡ = = = ≡ ≡ ρ )= ) are the velocity (and density) perturbations in longitudin δρ = v δ V ) ) g ) ( , , t Ω  V V ( ( D δ σ δ vec ( ( O V V v . Since measurements of density and velocity perturbations O ) − V ( Ω and Ω in longitudinal (vector) gauge using some algebraic packag µν long and G δ ) δ , V 1 this gives ( V long , ≫ ) v We now want to show that on scales much smaller than the Hubble These variables can be interpreted nicely in terms of gradie We do not derive the first order perturbations of Einstein’s e V kt ( v shear and vorticity of the velocity field [10]. perturbations are much smaller than order of magnitude argument. Then, the perturbations of the ent gauges and/or gauge-invariant variables.mental This is results espec with gauge-invariant calculations. Let us n of second derivatives of the metric perturbations, Here the differences between gauge respectively, and For matica and then write down thewe resulting Einstein know equations that these variablesthis do way not are depend valid on in theform. any coordinates gauge. A Here, rapid we derivation just by present hand the is resu possible using the 3+1 for on sub–horizon scales, we maycompare therefore with use measurements. any of the gauge- 2.5 The perturbation equations methods, this is quite lengthydetermine and cumbersome. However, we r bution to the shear of the metric perturbations. The first order perturbationthe of Einst following order of magnitude estimate 8 or The Cosmic Microwave Background PoS(cargese)006 (2.67) (2.72) (2.68) (2.63) (2.71) (2.70) (2.66) (2.69) (2.64) (2.73) (2.65) Ruth Durrer ) j ) 6= ii . , i ( ) ) ( , ) T ( , , . scalar vector Π
K ( provide two scalar, one vector K P  , K 2 ) − i j K − V 2 + , ) ( 2 or the derivation of the perturbed Ga + 2 e and not often used, since we may ) ) ( = GT ) Π i K ˙ i π H . see [11]. In order to simplify the no- can be derived from the Friedmann , π 0 H 0 P 8 00 H 8 ( H Ψ ( + lead to two scalar and one vector con- 2 ( − henever this does not lead to confusion.  −  ˙ = ˙ − H µ Γ =
˙ 2 w ) ) Φ 0 Ga 3 2 H ) k w ˙ H ) i j 3 V Φ H π w T ( 2 ( K − 8 G GT + − − + H + σ − + 3 2 s H  π 2 ¨ 1
¨ 1 = 8
Ψ D 3 2 − H H 2 H H 2 s k − H )( 2 H k  c = 2 s k 11 ) H − c ( + = = = µ V + + ( 0 ˙ − k K − ) = 2 s 2 s , ˙ K D σ 2 G c c ) H w 2 w 3 1 H 3 S = = 2 ( ( )  H 2 1 + 3  w 2 Π + D V 1 ρ ) ) + ( = 2 P = ρ + + T 2 P 2 ρ ˙ ˙ ( ) w Ψ 1 Ω 2 ˙ Ga V ( + ) H ( Ga Ga π ρ P ˙ ρ H Ga σ π 4 π 2 ( H 8 4  π 2 + + The Einstein equations 2 4 k Ga ρ ˙ Φ ( Ga + π 2 ) ) = 4 π T H 4 ( Ψ 2 Ga ¨ H π − + 8 The Einstein equations Φ ¨ Φ ( 2 k vector: The dynamical equations: The second dynamical scalar equation isuse somewhat cumbersom one of theEinstein conservation equation equations the given below followingequations. instead. relations are F useful. They and one tensor perturbation equation, scalar: tensor: straint equations, For the calculations below we shall also make use of tation, we suppress the over-bar on background quantities w 2.5.1 Einstein’s equations The constraints: The Cosmic Microwave Background working with Cartan’s formalism for the Riemann curvature, PoS(cargese)006 0 > (2.77) (2.74) (2.75) (2.78) (2.76) 1. For K > ∼ Einstein j kt Ruth Durrer 0. 6= . i ) = Γ , Π , while for ecay. k Π . . = , ( ) )  < Γ S Π 2 with the k w = Φ which describes a damped + scalar vector w + ( ( Ψ –equation, Eq. (2.67), which Ψ K 1  ) 2 2 V  ( k equations and 6 variables. For      √ 2 s 2 σ K 0 c , k s (over short time periods) when have to be determined via matter 3 = . As we shall see below, for perfect g . ] + = ile for tensor perturbations we have represents propagating gravitational ) D om the Einstein equations since they Ψ − ω Π Π , K V Γ on equations in terms of the variable w ( k 1 ) i j w . It is very complicated and not very 2 s =  2 s + deal fluid one has c Π  H 1 c Π H Π and Φ 3 H 3 2  w
+ 2 3 in (2.74) one obtains after some algebra and 2 Γ K + K
− k k 3 and +  2 + w denotes the four-dimensional covariant deriva- Φ Γ Φ 2 s 3 − Γ − w ν c kV kV 1 ) + k + 3 12 ) w + w 2  V 3 2 w + which for perfect fluids also obeys a damped wave 1 ( 1 0, we have ) + ) − + − Ψ − w + T 1 , small scales are just the sub-Hubble scales, k ≡ 1 ( 2 Γ j [( i k D + K Einstein equations, (2.63) to derive a second order equa- 0 (here ; w  . H H 1 k Π )  w + ( w H =  i ( = 2 s ) ≫ g + wk 2 ) 0 2 K c + 1 ( 2 w k V D 3 w µν ν 1 k −
; + − 2 s T + − c H + 2 s = . and 1 1 3 c
2 s ( ( Ψ ) Ω c  w 3 3 −   00 1 − k − + H ( + 2 s g
c = = 2 s ˙ D Ψ c constraint equation (2.63) H 3 3 D V = ) i + 0, the behavior of scalar perturbations is given by H . Hence vector perturbations do not oscillate, they simply d + − 0 ) 2 D ˙ H H ( V g 2 s from the conservation equations. Eliminating finally 1 a = c ˙ w / D + 3 Π Ψ 1 + ˙ + V , the damping term can be neglected and − 1 ∝ = ˙ 2 Ω ( ˙ ) k D Γ 3 and . Using V ) ( + + Φ V σ , ¨ K Ψ 2 D 0, waves oscillate with a somewhat smaller frequency, Tensor perturbations are given by For scalar perturbations we have in principle 4 independent One can now use the Note that for perfect fluids, where The conservation equations, Vector perturbations of a perfect fluid are determined by the ( < ∼ < 2 − K The right hand side is a ’source term’ linear in tion for equation,(2.65) yields the Bardeen equation, wave propagating with speed making use of the It is sometimes alsopair useful to express the scalar conservati waves. For vanishing curvature or illuminating, therefore we do not write it down here. vector perturbations we have 2 equations and 3 variables, wh equation propagating with the speed of light. On small scale The conservation equations can ofare course also equivalent be to obtained the fr contracted Bianchi identities. equations. For example, for adiabatic perturbations of an i fluids with implies The Cosmic Microwave Background 2.5.2 Energy momentum conservation the frequency is somewhat higher than tive) lead to the following perturbation equations: t PoS(cargese)006 w . = 7 2 s (2.82) (2.80) (2.79) (2.81) ations c 10 < ∼ 0. At late z Ruth Durrer → 0 and t = is the phase space K 3 k π s an ensemble average const. for , where the contributions describes damping due to 2 0.) Therefore, in a radiation k 1, or in a matter dominated → ter equations. The simplest 2 s . en by a scalar field which is < H ) c . = t ) 5 [12]. The second term diverges q ( 2 s 1 . k ) 4 q erfect fluid or a scalar field, and c ) T )+ ( k t . )] ≫ + ( ( Ga orms damped acoustic oscillations. al universe at late times, spherical Bessel functions, e.g., H 2 0 2 4 es from the two helicity modes. The kt 1 or scalar perturbations and one each kt ( ( A m q = requency interval is then given by eracting particle species of which some − ) ion equation, however, is of first order. = , ions’ of the fluid where the fluid pressure , see T ( 3 ) B j ν k H π i 2 2 1 0, so that k )+ | ) 2 − kt T + G = ( ( q ) 13 1 π . Hence vector perturbations simply decay if there B T H 2 s − ( − c τ q 0 the scalar equation (2.78) is a wave equation like 16 d ˙ 3 d H kt + A j 6= q h| 1 ( t [ 2 − π 2 s a c 4 a sin kt + 2 ) a A = T = ( 0 and expansion can be approximated by a simple power law, ) Ω∝ ≃ ¨ T k H ( ) = 0. This matter equation, which describes adiabatic perturb H T H ( ρ and K log = d H 2 d i j a are interpreted as a time dependent ’mass term’. If / Π 2 1 = ∝ H ) Γ V ( 2. (Not, however, in an inflating universe where and σ is the derivative with respect to cosmic time and the factor 4 = K q d dt indicate an average over several periods of the wave as well a 1 − a h···i , for example in a radiation dominated universe where , the tensor perturbation equation reduces to 2 denotes the spherical Bessel function of index = 0 implies / q 1 t τ ν d j d = If there are no anisotropic stresses, like for example for a p Another simple example is a universe with matter content giv ∝ > ) ) V q t ( ( a The energy density in gravitational waves per logarithmic f Here 2.6 Simple but important examples 2.6.1 Free gravity waves if curvature can be neglected, universe where if Here have to be described by a Boltzmann equation. This is the actu relevant for inflation [3]. More complicated are several int are no anisotropic stresses to source them. prescription is to set The Cosmic Microwave Background 1 equation and 2 variables. To close the system we must add mat of a perfect fluid givesfor us vector exactly and two tensor additional perturbations. equations f If time we then find The general solution of this equation can be given in terms of (or matter) dominated universe we have to set volume per logarithmic frequency interval.brackets, The factor 2 com (see Section 3.2). the tensor equation discussed above. The first order term, 3 proportional to expansion, and the term in square brackets is of the form Π counter acts gravitational collapse. The vector perturbat the mass term vanishes andThis the scalar Bardeen wave equation potential describes simply the perf ’acoustic oscillat PoS(cargese)006 1. =const. 2 ) ≪ a (2.89) (2.85) (2.87) (2.83) (2.90) (2.91) (2.93) (2.86) (2.92) / and the kt q t 2 tion gives ( t q Ruth Durrer is constant. 2 rmore, if the ∝ 0 + 3 a Ψ k ) ≡ k eq ( the horizon in the t 2 A / 1 A . , = 1 2 eq ≃ Ψ 0. Then . Hence on scales which Ψ ∝ 2 q H trum, see Section 3.2. eq ) 3 . 2 = k − kt 2 > i H 3 ( ) 2 K k − | 1 6 kt > ) k ( T k − ( 0 , ∝ , where we may neglect pressure, Ψ H ) = ed so that e neglect. From k 1 rad 6 h| ( Ψ . 2 Ω − d era, . ρ 2 A es, 10 2 ′ = − − . 2 A ) k D Ga 5 0 (2.84) 0 (2.88) kt 10 + ) kA ( π 2 B 4 × = = kt ) C , = ( r 2, the energy density decays with the wavenum- ˙ 2 − kt + Ψ ) ( since it dominates on very large scales, V kV + ., ≃ = 14 t 6 = t kt 2 6 ′ kt A A q + ( 3 A A + rad 0 g Ψ − + = ˙ ¨ 3 Ω const D Ψ ˙ Ψ V 1 = 6 1 Ψ + )= − g rad = t , ( ρ V 0 − D k 0 V kt Ψ H 0 H Ψ 1 ρ = log 5 Ψ d − Einstein equation, d 1 3 k A − ) 3 1. A rough estimate neglecting the matter/radiation transi = = = = 00 + ( ) g g = k k V Ψ ( D D q as we expect it for a component of massless particles. Furthe H log 4 = Ω d − 0 (see Section 3.2). This implies that d D a . We therefore end up with = A ∝ 5 T H n − ρ while it is independent of scale for wave numbers which enter = 2 denotes the tensor to scalar ratio of the CMB anisotropy spec ′ 0. To simplify the equations we also neglect curvature, − A r 1 and k = 2 s ≪ c Note that We now consider the case of nonrelativistic particles, dust kt = for scales which enter the horizon in the radiation dominate radiation dominated era, ber like Here we cannot neglect the constant mode The second term is decaying and can therefore soon be neglect Inserting the solution in the one finds Again, the second term describes a decaying solution which w The momentum conservation equation now yields with solution for gravity wave power spectrum is scale invariant on large scal The Cosmic Microwave Background Bardeen equation then reduces to we find enter the horizon in the matter dominated era, 2.6.2 Nonrelativistic matter, dust w The ratio which is solved by PoS(cargese)006 , − ) H x with ( 3 and (2.98) (2.95) (2.99) (2.97) (2.96) (2.94) ≫ he first (2.100) (2.103) (2.101) (2.102) √ sin 0 so that k / 2 se energy = − kt Ruth Durrer x K = x )= x and ( 3 1 j / 1 -mode is singular for 1 = y 2 s ing and decaying gravita- c . x = 3 . . w A √ 2 x 2 se, /  , . ) = 3 radiation. Defining x  V ( kt l Bessel function ) √ V x l instability [13]. However, as we see x e, on sufficiently small scales, , 3 ( 4  cos [12]. Since the tuations in the CMB which, as we shall 0 1 , √ , are of the order of the initial fluctuations A ) sin = 2 Ψ eory, to the conclusion that galaxies cannot By . + − x 2 x , e.g. Ψ 3 1 + Ψ = V − 2 V x 4 2 3 )  k 3 + , x 3 4 Ψ x , 4 3 + √ 2 1 15 ( ′ g √ + ( kt g ) √ , D + x , we obtain + ˙ A − D Ψ ( cos ) 3 4 2 g  Ψ t 4 x 4  1 ( √ D − = = sin A + ′ g A j − 2 − 1, this yields = A Ψ cos ¨  Ψ D 3 0 for 2 g 2 A = = = x 3 ≪ D √ 2 2 = kt g x √ V Ψ − , B D = = , = g 3 A 1, we find oscillating solutions with constant amplitude and V Ψ D = stays constant, it does not grow. This led Lifshitz, who was t ≫ 0. To determine the density and velocity perturbations, we u Ψ x Ψ = , B . The Bardeen equation reduces to t / are spherical Bessel functions, see 1 3 for the density and velocity perturbations and an oscillat 1 = y √ / super-horizon regime k . H ) and x ( 1 these become j and In the We now consider a spatially flat, radiation dominated univer 0, we have to set d t sub-horizon scales dx cos 1 ∝ → = − x For Eq. (2.98) we have used the explicite form of the spherica gravitational collapse is possible. Nevertheless, the fluc see, reflect the amplitude of the gravitational potential from Eq. (2.93), density perturbations do grow and therefor have grown out of small initial fluctuations by gravitationa to investigate relativistic cosmological perturbation th generated during inflation. 2.6.3 Radiation a tional potential, ′ conservation and the Poisson equation (00 Einstein eq.) for with solution On Here The Cosmic Microwave Background The metric perturbation frequency Inserting the solution (2.95) with t PoS(cargese)006 2 ds (3.1) (3.3) (3.2) (3.4) avity is cs. The Ruth Durrer . i j γ . ji . γ 1 = = i j 2 . γ n , affine parameters are different. 0 0 , , 6= ponents of the photon momentum variant perturbation variable. We ally invariant. More precisely, 1 =  0 0 K = i x γ 0 if Like for acoustic waves, the restoring , nstant amplitude. These are called the + urbations of the gravitational potential n 1 ′ i j al to the energy shift of any given photon. orizon are frozen to first order. Once they , Ψ δ ee in the next section, they are responsible ts the gravitational force and the radiation − dt 0 ) ′ respectively, and the tangent vectors to the = ≪ t )= ( = 2 j ˜ λ 00 n x 2 ε γ ˜ 0 n , t 3 i , t is the (parallel transported) photon direction, pro- ε A 16 √ Z = and 0 ( , 3 n γ 2 ν − λ = n , − t dx K and , µ  = with 0 if t dx on the constant time hypersurfaces. We choose D ≡ ˜ λ ,
dx 1. 1 d , i with )) x = i ∂ t 3 i µν ∂ ε ( ) = form an orthonormal basis for the spatial metric ≪ h i µ i , ˜ e x n ( x ε , (red- or blueshift) is independent of the photon energy (gr ( + 2 ( , photons follow to a good approximation light-like geodesi , ω = inside the horizon. ds µν / i λ 2 γ e dec 2 dx ω t d a a ∆ / > = = = t n 2 2 ˜ s ds d is the photon position at time have the same light-like geodesics, only the corresponding 0 x 2 ˜ s We make use of the fact that light-like geodesics are conform The perturbation amplitude is given by the largest gauge-in The unperturbed photon trajectory is given by After decoupling, Our perturbed metric is of the form Also for radiation perturbations d portional to the momentum of the photon. We determine the com with respect to a geodesic basis is small on super horizon scales, and Let us denote the two affine parameters by where conclude therefore that perturbations outside the Hubbleenter h the horizon they start to collapse, but pressure resis oscillate and decay like 1 fluid fluctuations oscillate at constant amplitude. The pert ’achromatic’). 3. CMB anisotropies and polarization 3.1 Lightlike geodesics and CMB anisotropies temperature shift of a Planck distribution of photons is equ The relative energy shift, force provided by the pressure’acoustic leads oscillations’ to of the oscillations radiation with fluid. co Asfor we the shall acoustic s peaks in the CMB fluctuation spectrum. The Cosmic Microwave Background The radiation fluid cannot simply ’collapse’ under gravity. In other words, the vector fields unperturbed geodesic by PoS(cargese)006 i is u = re u (3.5) (3.9) (3.6) (3.7) (3.8) αβ 0 (3.11) (3.10) and Γ , hence f δ ning the 2 u a in the last / Ruth Durrer 4 1 to the energy T . between some ∝ f while t λ . The geodesic ∝ 2 0 2 0 receives photons ) u r n i a n 1 ρ 0 ˜ n δ is the usual (unper- − T ( = i a , f + ∑ ˜ λ T = 1 2  / f server at i , and the photon density i a u i | T are only determined up to = r T = . δ 0 δ λ = is the constant energy of the 1 and 2 n 4 1 first order, we obtain λ f ˜ + n e have used d = i ω a − + lation between the unperturbed . iverse, ˜ and ν T / 2 i 1 µ β n n ˜ a 0 λ n µ =  . δ α n . (3.9), and using Eq. (2.22) for the . ns i f n 0 ν . − so that ˜ dec ) dec u ) µν T β T 2 , where u T a ˙ u h dx αβ = ively in the non-expanding conformally n , so that the change of · u a · , f ˙ µ h ) α f / µ = n m = i n 2 1 n α 0 ( 1 i T ( n Z dx n g e  ) f i δ i 0 ω − 2 1 µ αβ i ∝ a v i a α T i Γ µν α + T h = − n δ + n f h δ ( = f i t β µ T λ 17 −  d ∂ + f + i n p as seen by an observer moving with 4-velocity ˜ d j ) . Since ) ) f β = ˜ . For example, we set n 0 | ˜ = u f A µ u j µν 0 T = · n µ ˜ 0 · ˜ p 0 T γ µ α ˜ n δ T ˜ − n δ h n α n h ( ( δ 1 n δ − = ( + β = 1 = λ 2 n d . An observer measuring a temperature 0 00 = ( d β f i  n h n | ds is given by u 0  E E δ f 0 of decoupling of matter and radiation, at the fixed temperatu α dec t T λ = h . We have seen that in expanding space the photon momentum is d T and ˜ d 2 f i denotes the scalar product in an expanding universe, contai dec | n t is then given by · = 0 ds n f i µ . As always for lightlike geodesics, δ T p T λ , so that 2 this implies the result (3.9). The ratio ) a = n f i 2 , where ˜ αβ = ) a a ˙ − h ˜ p , the true temperatures ˜ a λ i · ˜ − T ˜ u = α , ( | f 0 n − T β and some final time is h i t = i t + is the intrinsic density perturbation in the radiation and w E . Hence, the ratio of the energy of a photon measured by some ob β r | 2 is the emitter and receiver four-velocity in an expanding un 0 δ a α u Let us now introduce perturbations h . In first-order perturbation theory, we find the following re ( 1 2 dec that were emitted at the time T we have to choose given by temperatures turbed) redshift which relates emitted at factor related geometry given by are the four-velocities of the observer and emitter respect initial time equality. Inserting the above equation and Eq. (3.8) into Eq Here ˜ redshifted. Actually, the components behave like ˜ a multiplicative constant which we have fixed by the conditio yields, to first order, photon moving in the metric For the energy shift, we have to− determine Integrating this equation we use Together with ˜ where equation for the perturbed metric The Cosmic Microwave Background The photon 4-momentum perturbation: The energy of a photon with 4-momentum ˜ PoS(cargese)006 ) dec t ( (3.12) (3.13) (3.14) x = Ruth Durrer dec , x dt . )) t , and ) ( integration by parts, 0 x the square bracket of itial conditions imply λ x , 0. These initial condi- d t ) Ψ )( → ˙ is the peculiar velocity of Φ and it has been measured ˙ . Φ = ) ) 3 b kt , we obtain the temperature + 2 to the ordinary Sachs–Wolfe ( + − Φ ) n ˙ V r Ψ . ( dec ˙ ( g Ψ ( t 10 i f ) f ( D ) 0 for E 0 i E × t 2 dec dec Z which simply describes our motion t 2 dec n x j Z − → . , x rinsic inhomogeneities of the radia- + adiation). en derived for the first time by Sachs 1 ) n , 1. With system can obey so called iso-curvature f n of the earth around the sun has been 1 ) i Ψ ic oscillations prior to decoupling, see 0 dec c initial conditions. One can show that )+  n and t t dec ribute significantly also on super-horizon ( for an observer at ( ≪ ( t 1 Φ ) f i ( dec t b Ψ n E ( kt j x E Ψ + 2 , V 1 3 Ψ ≡ = dec t ) = .). The first term in Eq. (3.14) is the one we of degree 2 and higher) we can set + ( 2 j n j when 18 from the last scattering surface on very large scales  ) n ) n n ) t ( T ) r Φ T ( g b T ) T ( ∆ j − curvature D + OSW ∆ 0 V (today) and initial time ( adiabatic − do not enter in this difference. t Ψ ( OSW 0 i  f − + ≪ ( iso t E ) E ) ) ) + − r  r n 1 ( g j 0 so that the contribution of ( ) ( 0 n T n D x n ( V ) T T ( → is the dipole term, b 4 1 T T ∆ ( j t = T  T ∆ T V  ) ∆ ∆ ) = b − t ( + ≡ (  1 ) x V 0 for r ( g T ( T ∆ D → 0 and 1 4 dec ) T t T  ) , t radiation universe with vanishing curvature, adiabatic in k , ( = = k ) + r ( f i ) ( g E , as well as Eqs. (2.30, 2.31, 2.57) and (2.54) one finds, after n Ψ E D ( T 3 20 µν T h ∆ − is the unperturbed photon position at time denotes the density perturbation in the radiation fluid, and ) t 0 we simply have ) ( r )= ( g t x , = D k Evaluating Eq. (3.12) at final time The largest contribution to For the higher multipoles (polynomials in Apart from adiabatic initial conditions, a dust+radiation ( K ) r ( g the baryonic matter component (the emitter and observer of r Eq. (3.14) therefore gives for adiabatic perturbations Here difference of photons coming from different directions Direction independent contributions to so accurately that even the yearlydetected variation [15]. due to the motio the following result for scalar perturbations [14]: is called the ’ordinary Sachs–Wolfe effect’and (OSW). Wolfe [16]. It has be initial conditions which are defined by the requirement that tions imply that The Cosmic Microwave Background definition of D with respect to the emission surface. Its amplitude is about on super-horizon scales. The contribution to scales. This is especially important in the case of adiabati have discussed in thetion previous density section. on the It surfaceEq. of describes (2.98). last Depending the on scattering, the int due initial conditions, to it acoust can cont (see [3]), in a dust (If where effect can be neglected, PoS(cargese)006 . s e k (3.15) (3.17) (3.18) (3.16) gravita- Ruth Durrer i.e., . If we assume i ) t , y ( , Ψ ) λ t d , j perturbations, x n . ( ] )) Ψ sually not the precise values of t λ h ( . d h then has to be compared with x j ] , tra are the ‘harmonic transforms’ e term and the integral is the full tensor tions are sourced by some inher- n t λ ift caused by the time dependence j ( . is stochastically homogeneous and d down in the very early universe, j d angular scales as the acoustic term ; σ j or oustic peaks’. nd tensor temperature fluctuations l radiation contribution, curvature, or itter and observer. This is the Doppler σ f λ n Ψ f and represents the so-called integrated l i regarding again the dipole contribution d ted out. In this sense Eq. (3.17) is here e-invariant results for the energy shift of j the blue-shift which the photons acquire i n Z ed by the redshift induced by climbing Z n l j has to vanish for different wave vectors l + ˙ H n ) + f i t f j | , j )) i n t k n ) Z ( ( ) x b ( , − j Ψ dec t 19 x 1 ( V [ , l j f i − ˙ a a H dec 1 t f [ ( f i i ) = vanish and Eq. (3.9) leads to b a Z a topological defects, see [17]), vector perturbations can b ) ( j T A − ( V = 1, pure dust universe, as we have seen, the Bardeen potential  ) = = e.g. i f V and = , but only expectation values like ( ) ) E E ) ) T r t V  Ω ( ( ( ,  f i x δ   E E ( ) ) Ψ n n  ( ( T T T T ∆ ∆   perturbations vector The integral in Eq. (3.14) accounts for red-shift or blue-sh The quantities which we can determine from a given model are u For The quantity which we canobservations calculate is for the a power given spectrum model defined and below. whic Power spec tional waves, only the gravitational part remains, We obtain a Doppler term and a gravitational contribution. F isotropic, the correlator of the Fourier transform contribution to the CMB anisotropies.we It call appears the on sum the of same the acoustic and Doppler contributions ‘ac of the gravitational potential alongSachs-Wolfe the (ISW) path effect. of In a the photon, The Cosmic Microwave Background The second term in (3.14) describes the relative motion of em important. 3.2 Power spectra 3.2.1 General comments perturbation variables like that the random process which generates the fluctuations Equations (3.12), (3.15) and (3.16) are thephotons manifestly due gaug to scalar, vector anddue to tensor our perturbations. proper Dis motion, Eqs. (3.15,3.16) imply the vector a Note that for modelsvector where perturbations initial are irrelevant fluctuations as have wemainly been have for already lai completeness. poin ently However, inhomogeneous in component models ( where perturba are constant and there is noby integrated Sachs-Wolfe effect; falling into a gravitational potential is exactlya cancel cosmological constant. TheSachs-Wolfe contribution sum (SW). of the ordinary Sachs–Wolf out of it. This is no longer true in a universe with substantia PoS(cargese)006 he 0 the (3.20) (3.19) 6= K Ruth Durrer they explain and any other . If ) ) x i.e., x ( ( X X . Such random fields in terms of eigenfunc- x ) . x ( ) alaxy distribution. This is . ′ easure’ expectation values. X ) k k andom initial conditions’ in a is statistically isotropic. These ( − X it is very often simply assumed k emble average. This problem is X P ( ) tant factor. The hope is also that elation between these two spectra ′ n do when we want to determine δ other words that k 3 t 10% on large scales. emble averaging; a type of ’ergodic d the matter power spectrum inferred ) in every point unction is diagonal, hope seems to be reasonably justified. − π laxies and dark matter is governed by , which is defined by 2 k ) e further assume that the distribution of ( rier transform. In curved space it is the expansion k )( –function has to be replaced by a discrete y models, then the two-point functions and δ ( formation of the model. k δ 3 ( D ) D P is much smaller than the Hubble horizon, the π P on the sphere it corresponds to the expansion in terms of 2 λ we can no longer average over many independent = is to average over many disjoint patches of size 20 ) e.g., = ( 1 λ
0 − 0 t
, 0 H ′ 0 the Dirac represents an expansion of t ( , k ′ ) is the ordinary Fourier transform of > O k k ) in position space, we define the power spectrum in Fourier ( . If the perturbations of the model under consideration are K ∗ gm ∼ k ∗ 1 X ( X D λ X one realization of the stochastic process which generates t X ) ) 0 0 t t , , k i.e., k ( ( X gm

D

in Eq. (3.19). 0, the function ) ′ k = − K . k indicates a statistical average, ensemble average, over ‘r ( δ δ h i is usually compared with the observed power spectrum of the g has no preferred direction. This means that the random field ) For an arbitrary scalar variable The There is one additional problem to consider: one can never ’m The ‘harmonic transform’ in usual flat space is simply the Fou Let us first consider the matter power spectrum, ) 1 k x ( ( , assuming that this spatial averaging corresponds to an ens D space by fluctuations at our disposalthe for mean observations. square fluctuation The on best a we given scale ca volumes and the value measuredknown could under the be name quite ’cosmic far variance’. from the ens are called ’statistically homogeneous’ (orX stationary). W properties imply that the Fourier transform of the 2–point f spherical harmonics. size of the observable universe. For given model. We assume that no point in space is preferred, in the factor Kronecker therefore the power spectra contain the full statistical in We have only one Universe, Gaussian, a relatively generic prediction from inflationar λ The Cosmic Microwave Background of the two point correlation function in terms of eigenfunctions of the Laplacian on that space, situation is more complicated. Then stochastic field which we consider has the same distribution In flat space, P 3.2.2 The matter power spectrum hypothesis’ . This works well as long as the scale clearly problematic since it isshould be. by no This means problem evidentthat is what the known under the matter the r and nameon galaxy ’biasing’ sufficiently power and large spectra scales, differ since onlygravity, their the by power evolution a spectra of should cons not both,In differ ga [18] by it much. is This found that thefrom observed the galaxy observation power of spectrum CMB an anisotropies differ only by abou tions of the Laplacian and in the case PoS(cargese)006 at ’s are [19]). (3.23) (3.22) (3.21) (3.24) (3.25) i.e., ℓ P e.g., Ruth Durrer and now, f the expansion 0 , x . oth, measured and is the same for all ) ℓ ′ ) = lated, C k n )= ′ x ′ ( n − , T mm at ( mplies that the harmonic ) ′ / k δ ′ ( µ m fficult to measure peculiar ′ T . ∗ ( ℓ ℓℓ δ 0 ℓ ∆ i.e., 3 δ Y P ) ) ℓ = n = π , are negligible in models where C ( ) 2 E ) i ′ ′ m pectrum. It is defined as follows: V elation ) 1 ℓ ( )( ℓ cs for the last equality; the m ocess generating the initial pertur- m y power spectrum. ′ T ′ Y k ( ℓ . Here, ∗ ℓ C + i ( a ′ a n f realizations (expectation values) are ℓ e ) V on sub-horizon scales (see · m . m damped oscillations once they enter 2 ′ m P V after an inflationary period. Tensor per- ’s by ∗ ℓ ( ℓ ( g m ) 2 ℓ tribution of ′ ℓ a a ℓ − D · a k C ∑ D 6 k h ( . ) m ∗ m 0 π in the following calculations. Since our fields e.g., ℓ ) k 1 = 4 S a ( Ω 0 i ( , h 0 E D x ) ′ ′ Q ) = P . We develop it in terms of spherical harmonics, H n m m ) , 2 ′ 2 T ( 21 ) . } ( ℓ ′ k m m 1 S , m ∼ and ( a ∑ ′ ℓ n ) ) Ω ( ) 0 Y S m ∈ S ℓ,ℓ j t 2 ( ) ( ℓ 0 0 m ) ∗ t ′ 0 ℓ a n ( Q H n = x Y ) · D ) ( n m µ ( and photon direction ( g n = ≃ m ℓ = = ℓ ( {z ˙ t P ′ D ) a 1 n m E · π k ℓ +
′ 4 m ℓ n ) ( = 0 Y 2 m ∑ ℓ, t ′ V ℓ V |  , ℓ ( ′ ) P ) − , time a ′ ℓ 0 k ) = t ∑ x n m S )= ( | m ( ℓ ( ∗ n ℓ i a , T kV V C 0 D T | ∆ ) x ℓ 0 ) , ∑ t 0 n , t ( ( k is diagonal. On other words, the off-diagonal correlators o ( T j ’s from scalar, vector and tensor perturbations are uncorre T T T ∆ T V ∆ lm

/  a vanish and we have T m is a function on the sphere, ∆ ℓ The spectrum in which we are most interested and which can be b a . Like for Fourier transforms of random fields in space, this i T n / is the matter density parameter. Unfortunately it is very di T m ∆ we have used that ’s are the CMB power spectrum. is a function of position ℓ Ω , The power spectrum of velocity perturbations satisfies the r The two point correlation function is related to the Clearly the Since vector perturbations decay, their contributions, th 0 T C ≃ t ’s. We will often suppress the arguments / m T = ℓ where we have used the addition theorem of spherical harmoni Here velocities and so far not much use could be made3.2.3 of the The velocit CMB power spectrum Definition calculated to the best accuracy∆ is the CMB anisotropy power s The The Cosmic Microwave Background initial perturbations have been laid down very early, the Legendre polynomials (see e.g. [12]). are statistically homogeneous, averages overindependent an of ensemble o position. Furthermore,bations is we statistically assume isotropic. that the This pr means that the dis For coefficients turbations are constant on super-horizon scales and perfor the horizon. transform of directions t Y PoS(cargese)006 S A (3.29) (3.28) (3.30) (3.26) (3.27) (3.32) (3.31) 1 which ions. We = given by a Ruth Durrer n . ) on large angu- ′ k − then represents the . k S is a pure power law super-horizon scales ( 3 . A δ . 1 Ψ )) < , − )) /ℓ . n n ) dec π ) , the integral (3.30) can be t n dec 0 0 t < . If · ≡ ℓ − kt . kt ution on 3 b k − ( ℓ heorem of spherical harmonics 0 ( ) t S 0 ≃ ϑ ℓ − ( . ∼ t A dec P ( 0 0 k t 3 ) . t k ( ) − ) )) / , kt ( 0 2 ℓ for t 2 π ℓ 1 dec j ( j t dec 2 dec 3 ≃ + adiabatic perturbations t t 3 kn = ≃ k 2 , i − k ) ) e − k E µ 0  n 2 · t  dec 2 0 ) dec = ) = ( ( t | ) is the ordinary Sachs–Wolfe effect, OSW, 2 ) ℓ ′ t x ′ ) | ( k − ( n 2 ϑ k Ψ k ˙ nn − Ψ | ( Ψ dec 2 5 ( Ψ | t ℓ 0  D − + , j t T + 3 1 9 1 ) T + ( ℓ k 2 1 k 0 ∆ i k ˙ is called the spectral index. ( t ℓ ( k Φ  22 dk ) , ≃ ( ′  − δ n 1 Ψ R ∞ ) Γ ) k or Harrison–Zel’dovich (HZ) spectrum, n ℓ 0 dk 0 1 3 ) * k , ( t + Z ( 2 n 0 , Γ ℓ ≃ x Z Ψ n ) π 2 ( 2 − , P n )= ( 9 S 2 π 0 π 3 0 0 2 T t A ) k x 9 ( 100 : T ∞ − = , ∑ ∆ ( 3 ≃ 2 ℓ µ 3 ) n ) ) ( ∼ Γ = ( T , 0 ℓ π = T n t ) Γ ∆ k P , the present comoving size of the horizon, in order to keep , 2 ) − ( 1 dec 3 1 . The number n OSW t SW dec ( − , ℓ 2 T n ( ℓ t / n T 0 0 π = ( + peaks roughly at ) scale invariant C S t − ∆ C 4 x , the Fourier transform of (3.27) gives 9 0 3 ℓ ) t A ( ) ( k 2 ≃ 1 0 for simplicity. We suppose the initial perturbations to be )) dec t T

) kn T = dec ℓ ) i Let us first discuss in somewhat more detail scalar perturbat + t = ∆ dec ′ ∑ − e ) t ℓ 1 as in Eq. (3.26) and we set SW k 0 ( ( ℓ  − K ( t − ℓ SW < ∼ ( ∗ 0 C ( ℓ t 0 χ t ( Ψ C ( dec ) n k k kt ( ≪ ( − 2 ℓ ℓ j 0 Ψ x

≤ = are the spherical Bessel functions; and using the addition t ℓ dec j x Comparing this equation with Eq. (3.24) we obtain for Of special interest is the As we have seen in the previous section, the dominant contrib lar scales, 2 (neglecting the integrated Sachs–Wolfe effect Using the decomposition which for adiabatic perturbations is given by amplitude of metric perturbations at horizon scale today, dimensionless for all values of is generically produced in inflationary models. It leads to where specialize to the case The Cosmic Microwave Background Scalar perturbations repeatedly, we find The function spectrum of the form We multiply by the constant Since performed analytically. For the ansatz (3.26) one finds on large scales, PoS(cargese)006 . . s ) π n itial dec t ( inside = (3.33) (3.34) A 3 d Ψ Harrison / √ n Ruth Durrer / λ i.e., dec = t which is much n n . ers are minima, r k .26) for θ slightly less than D + n 2

) 0 . kt o the SW terms. On sub- ( + been regarded as a great ′ ℓ 2 j

) , we obtain again (3.31), but ) dt times they are miss-leadingly 0 dec ctrum with )) t via the relation t kt , ( ) axima and minima of the density k large scales comes from the inte- 2 ℓ − ( j rominent acoustic peaks, see Fig. 1. ) 0 ther models like topological defects r dec ) t ( t ( k e the class investigated here, since in th the WMAP (Wilkinson Microwave ( nearly in time. They are not simply led lly leads to iso-curvature perturbations k V 02. / e behavior observed by the DMR (Dif- ellite COBE [20]. In the mean time, the A so predict scale–invariant, . ( at the peaks would be due to the velocity 1 d pically affect the perturbations of a given 0 2 ℓ is dominated by acoustic oscillations, the therefore about six times larger than those j )+ = e, ) ± ve of the density) nearly vanishes. T 0 t t at the last scattering surface is determined by / , , kt k 95 of perturbations, the Sachs–Wolfe temperature n ( k T . ( ℓ k ( S j ∆ 0 ˙ / Ψ ) A Ψ π we have simply replaced it by 0 = 23 2 t dec ) t dec n r = t Z ∼ ( g ,

n k D λ ( * r 3 D k 4 1 will show peaks and minima. For adiabatic initial condition k

dk ) * AC Z ( 3 ℓ are oscillating like sine or cosine waves depending on the in k C 8 π r k dk V ≃ ∞ ) 0 oscillate like a cosine. Its minima and maxima are at 100, the contribution to . For a fixed amplitude Z and r 2 S ISW > ∼ ) ( ℓ 2 π A D r ℓ ( g C is roughly constant (matter dominated). Using the ansatz (3 ≃ D ) Ψ ≃ correspond to maxima, ’contraction peaks’, while even numb which subtends the scale AC r ( ℓ n n on super-horizon scales and therefore does not contribute t D C θ Ψ 9 replaced by 4 / 2 S A and therefore also Inflationary models predict very generically a nearly HZ spe To remove the SW contribution from On smaller scales, For iso-curvature perturbations, the main contribution on These are the ‘acoustic peaks’ of the CMB anisotropies. Some The angle ) r ( g D conditions. Correspondingly the horizon scales first two terms in Eq. (3.14). Instead of (3.33) we then obtain 1. The DMRsuccess, discovery if and not the as a WMAP(see [17]), proof, confirmation or of certain have string inflation. cosmology therefore models There [22] which are, al however, o Inside the horizon the horizon and setting the integral in (3.33) smaller than anisotropies coming from iso-curvature perturbations are coming from adiabatic perturbations (see Fig. 1). with Odd values of Zel’dovich spectra of fluctuations.them These perturbations models are induced are by outsid seeds whichdown evolve non–li as initial conditions for the fluid perturbations but ty wave length until it crosses the Hubble scale. This generica which are not ruled out by present data since they do not show p grated Sachs-Wolfe effect and (3.30) is replaced by term in the above formula. Actually the contrary is true. At m ’expansion peaks’. called ’Doppler peaks’ referring to an old misconception th contrast, the velocity (being proportional to the derivati ferential Microwave Radiometer) experiment aboard the sat scalar spectral index has been determinedAnisotropy more Probe) precisely satellite [21]. wi The result is The Cosmic Microwave Background This is precisely (within the accuracy of the experiment) th the angular diameter distance to the last scattering surfac PoS(cargese)006 )) dec t (3.38) (3.35) (3.37) (3.36) derived − 0 t ( Ruth Durrer corresponds ik ( 180. Here we n θ is not very well strongly on cur- ∼ 1 A . ℓ . d ! ! ) ) dec 2 2 t 220, in good agreement / − + ) ℓ ℓ . ∼ ( ( ter dependence of the peak . . This crude approximation dec ) 1 t 1 n 0 ℓ ( + ℓ exponentials exp on diffusion which takes place 2 A n kt

s to a displacement of the first d ( 180 ℓ 2 = π j )) or the positions of the peaks, this 2 which yields t 3 al constant. Furthermore, the peak ) n ≃ . )) rely adiabatic mode and require, at ibed with the Boltzmann equation h depends with ℓ t k n plasma which we have simply set to − √ / 33 ℓ n 0 − by the comoving angular diameter dis- . 1 nics, the angular scale t on of the first peak differs significantly 0 ( 0 ∼ e then obtain the following approximate t = t . 0 k = ( t dec ( n t k t ) ℓ 0 on super-horizon scales and k dec ( ( j 0 3 ) ) z t ) ( dec T = √ t ( k √ . dec χ ) , t π t H T Ω leading to 3 ( n ∼ ( ( A ) 0 ˙ √ 24 √ ∼ t d H T 0 ≃ / t ( dec π t 0 ˙ )) ≃ n = t H t n ) n oscillates like a sine. For generic initial conditions, we k ∼ dt − λ ) 0 dec n r 0 / t t ≃ ( g t dec ℓ ) t ( ( n Z D A k ℓ

, hence by replacing dec ( d ) t -th peak therefore is placed at ℓ * ( j 0 n t ) A 2 ( T d ( χ π n ˙ dkk H k = 1, the dt Z n by = θ 2 π Z 0 For gravitational waves a formula analogous to (3.31) can be t / . A somewhat more accurate estimate gives n Ω π = k ) dec t ≃ T ( ℓ n ℓ C 1100. This approximation is not very good since the universe close to unity this scales like 1 ∼ Ω dec z 3 in this approximation. A detailed discussion of the parame √ / 1 To a crude approximation we may assume Our discussion is only valid in flat space. In curved space the On very small scales the acoustic peaks are damped by the phot For a flat universe, = s have to be replaced withcorresponds the to harmonics replace of the curved spaces. F have used with the numerical value. Subsequent peaks are then given by For values of matter dominated at (roughly) to the harmonic number For a flat matter dominated universe we have The Cosmic Microwave Background Expanding the temperature anisotropies in spherical harmo during the recombination process.approach (see This next section effect and must [3]). be descr Tensor perturbations (see [3]), For a flat matter dominated universe tance to the last scattering surface.relation Instead for of the Eq. peak (3.36) positions, w deviates by about 15% from the precise numerical value, whic would expect a mixture of the sine and cosine modes which lead peak. The observed CMB anisotropiesleast, are that consistent the with adiabatic a mode pu dominates [23, 24]. vature but also on the Hubblepositions parameter depend and on on the the sound cosmologic c speed of the radiation-baryon positions can be found in [3].for Note, the however, iso-curvature that mode, the for positi which PoS(cargese)006 (3.40) (3.41) (3.39) (3.42) Ruth Durrer hancement of . ) T 2 n curvature (dashed line) CMB ) T − , . 2 n T 4 erturbations we obtain for large 100, they are not very sensitive n T ) + A + > ∼ 0 ) 2 ℓ ℓ kt 2 ( ( d topological defects) predict a scale − Γ ) ) T − ) ℓ x 1 4 . ( A ( ℓ T x 2 ℓ r Γ n + j )( ) = ℓ 3 T T − ≡ ( n  n ℓ 2 7 tuations is plotted. + x S 2 T are shown on the top panel. In the bottom panel (

ℓ − A A 2 x ) ( 2 dx 6 k ) Γ ( π 25 / T ∞ 72 15 K 0). 8 n Γ 1 0 µ 15 Z − ( ∼ ≃ 6 = T 2 ) T ) ≃ t A S , T n T ) ( ℓ ! ! ( ℓ k T ) ) A ( C ( C ℓ ! ! ) 2 2 ) ) T C 0) this results in ( ) 2 2 + − 1 ℓ ℓ H = + − in units of ( (

+ ℓ ℓ T ) 2  ( π ( ℓ n π ( 3 2 ℓ k ( = ≃ / ) ℓ T C ( ℓ ) 1 C 2 in this crude approximation is not real, but there is some en + ℓ = 10 (see Fig. 2). ( ℓ ℓ < ∼ ℓ for ) 50 Examples of COBE normalized adiabatic (solid line) and iso- T ( ℓ C ℓ< ) 1 Comparing the tensor and scalar result for scale invariant p Since tensor perturbations decay on sub-horizon scales, + ℓ ( For a scale invariant spectrum ( scales, to cosmological parameters. Again,invariant inflationary spectrum models of tensor (an fluctuations ( anisotropy spectra, ℓ The singularity at the ratio of the iso-curvature to adiabatic temperature fluc For a pure power law, Figure 1: The Cosmic Microwave Background one then obtains PoS(cargese)006 mp- (3.43) Ruth Durrer , if they were purely tensorial 8 . spectrum with amplitude − . In general observations require 50 10 tion for photons which is presented 9 < ∼ − × . ℓ 7 10 . 10 1 − × for 5 ≃ 10 . S 3 10 × A − ≃ 6 10 T 26 ≃ A × ) r 6 + ≃ 1 ℓ ( C S ) π A 1 9 + ℓ 800, fluctuations are damped by collisional damping (Silk da ( ℓ > ∼ ℓ On small angular scales, Present CMB anisotropy data favor a roughly scale invariant ing). This effect has toin be the discussed next with section. the Boltzmann equa (which we know they are not), we would need The Cosmic Microwave Background If the perturbations are purely scalar, this requires PoS(cargese)006 2 ) K µ ( Ruth Durrer in units of ) π 2 ( / ℓ C ) 1 Tensor + ℓ ( ℓ ted, d line shows the temperature spectrum, the ation spectra, while the right hand side shows temperature-polarization cross correlation (see rely scalar spectrum. Comparison of data and a learly visible and in linear scale (bottom panels) es in the dotted curves in the left hand panels are 27 Scalar Adiabatic scalar and tensor CMB anisotropy spectra are plot dashed line is the polarization and the dotted line shows the tensor spectra. The observational data aremodel well scalar fitted spectrum by are a shown in pu Figs. 5 to 7. in log-scale (top panels), where thewhich Sachs shows Wolfe plateau the is equal c spacing of the acoustic peaks.next section). The The soli latter can becomeactually negative, sign the deep changes. spik The left hand side shows scalar fluctu Figure 2: The Cosmic Microwave Background PoS(cargese)006 . f 2 = a ν / ˙ x 1 (3.47) (3.45) (3.44) (3.52) (3.51) (3.48) (3.49) (3.50) (3.46) µ ˙ x ∝ ) i x ( p Ruth Durrer µν g 0. The energy = . m ) x . , n )) ′ , t t ( . One easily checks that the . . . − T time is only defined up to a Ψ t } T p } . ( ∆ 2 . 2 3 ] + n j m n just implies that physical mo- d 0 m , ed by the distribution function ≡ ) ver the ’fiber’ Φ − − − p ) Φ = ) ) x , n x f m , p = x = , + ′ , entum tensor of the particles is given ( , t ) j ν  uville equation for scalar perturbations defined by the condition n be replaced by the Boltzmann collision ν f x , i γ , in p . ( )( p s t ν t p ρ f Ψ µ ( ∂ µ p 0 [ Φ . γ ∂ − j p ove along geodesics, p is implies the Liouville equation, µ only, the Liouville equation on a Friedmann t ) ν ) n = + and 7→ p ( x v x p M δρ | ( ) ) n − ( α | Ψ ) µ ˙ 4 1 x p ) x ( ∈ p p i , µν x , − ν µν = , x and ˙ ( ∂ x n g x g ≡ ) i x x 28 ( , g µν t i | S , ( | | n t : ( ′ µ 0 να Γ , in is a function of the redshift corrected momenta only. is v t is the tangent space. For photons p Γ p R f dv dt S . Hence the comoving momenta scale as ( ( ) | S − ) M t f x ) v a i in ) T n δ + t µ S ( x T / S ∂ ( , 4 → δ ( i Z f ∂ µ 1 ∈ v x m T n ¨ ∈ µ x m , P ) − M p = P ∝ Z p Z in )+ + p at the point t :  j f v ) , 4 ( ( S ) x f p a ¯ ( f S ( i = π S ≡ { γ )= ( ) = as function of p ¯ x ρ ) n f ( M i j x , dt t d f ≡ { M ( g )= x ∂ ≡ µν , m x m t ) , P T p P ( x 0. Hence ) n , S , = ( t n = , , p v t f ( ( M t f ∂ M and interpreting ap is the tangent space of = is the spacetime manifold and v S 1. In the case of massless (lightlike) particles, the proper x S − T We now define Photons (or any kind of “classical particle”) can be describ In an unperturbed Friedmann universe the Liouville equatio = 2 ˙ universe reduces to Setting momentum tensor is given by integrals of the second moments o is the one particle distributionby function and the energy mom If the particles are not interacting (collisionless) they m The dot denotes the derivative with respect to proper time After a lengthy derivation which isthen presented becomes in [3], the Lio Here Here x The Cosmic Microwave Background 3.3 The Boltzmann equation menta are redshifted, If there are collisions, the zerointegral. on the right hand side has to solution with initial condition multiplicative constant. For the distribution function th This equation can be solved formally for any given source ter defined on the 7-dimensional phase space, the mass shell PoS(cargese)006 . We (3.57) (3.56) (3.54) (3.53) (3.55) n . Ruth Durrer which does ˆ k ) · given below. x , n , t )) M = )( , , dec µ Φ t )) ′ ab + t . − , P t )) , however, are given via Ψ − ′ ( ) ( t + t t Φ n ( , ) − 0 − n k ( − ab t ( )( x 2 matrix and can therefore be σ and − , I n x Ψ × 2 1 , Ψ monopole n of lightlike geodesics after de- m integrals of ′ − dec )) + form a right handed orthonormal t t x ( = ) in . , ) )( als t Φ ′ ouville equation which also does not n i ution of the Liouville equation. The  )) + ( t ) n 800. The comparison with Eqs.(3.12) , ) ng’ approximation, where we assume rgy momentum tensor which contain ′ Φ ) equivalent to our approach in the pre- nd were entirely free after decoupling. , − ) 3 t n b µ . ) are much larger than the duration of the )( > ( ∼ ( ab t , 2 in + ℓ ) ( ( ik t x − Φ σ V hat 2 ) , n ε ε ′ t ( ε · Ψ t t Q ( − , + ( ε ε ε ( ) i − n t − n 2 t 1 ( + ∂ ( ( T Ψ x i E T ) + n µ ( − , ε ε ε δ n ′ 2 g t ( ik ( ab x in + . − t + ∂ , − ) ) D σ ′ ≡ S 29 x 1 e t )( ( ′ 4 1 ( , ) V ε ε ε )( Φ n in dt  -independent contribution t 1 + , M t ( Φ n )) = in ) x + . This is a hermitian 2 E t ′ ) 1 , ∗ t b S ( Z ab + t ( )= Ψ = ( E σ − ) n a Ψ S , t , such that E ( )+ ( +( M U E ) ( ) ′ t 2 is an electromagnetic wave propagating in direction n n ( ∂ , + ′ dec M ε ε ε ) n k t − , 0 , p ) = dt ( ) ab x − 1 in n , ( t σ t , ′ Z ( I ε ( t ε ε x h , n + )( t 1 2 M ( ) − Φ ) in S x t = ( + , − ∗ b t ( Ψ M E dec µ ( t a ′ ( ik E ) d − dt S e ( M )= n , k , t ( A photon with momentum Let us compare Eq. (3.53) with the result from the integratio M In Fourier space equation (3.52) becomes The polarization tensor is defined by we can replace the second term on the right hand side to obtain written as define the polarization basis 3.4 Polarization The Cosmic Microwave Background Using By ’monopole’ we denote the uninteresting system. Transversality of the electric field then requires t and (3.14) yields not affect the CMB anisotropy spectrum. The Bardeen potenti Einstein’s equation in terms ofcontributions the from perturbations the of photons the which ene are in turn the momentu coupling in Eqs.(3.12) and (3.14). Heretake we into have account solved the the Li scatteringvious of section. photons They and both is correspond therefore that to photons the behaved ’sudden like decoupli a perfect fluidThis before is a decoupling relatively a good approximation forprocess all of scales recombination which which correspond to multipoles Therefore, even though it mightterm look on like the right it, hand this side is also not depends on a sol and PoS(cargese)006 . . ) ) I ) V 0, 2 2 / , ( ( I they = ε ε ε σ U δ , . The s (3.62) (3.59) (3.58) (3.61) , 1 4 ) V I . ϕ 1 , where ( , ∂
= ε ) ε zed along ε n ϑ ℓ< 2 U ( Ruth Durrer 1 ( i ε ε sin ε M i − 0 these are the = ± . For Q ity, ) = ϕ n 1 ( e s = ε ε ε and especially their polarized intensities I . 2 −− 4 . and ) 1 describe the symmetric P √ 2 . For ϑ m E hoton direction U iU ℓ = ∂ ∗ = 1 . Y s ) − E it in the following. If = ( , ± 2 components. Such a tensor −− and ( ! Q ϑ but also on the basis ε ε ± ε 1 , the polarization turns like ! e P Q = 2Im n n − ) onics, − = , 0 1 1 0 1 0 0 ( b , ε ε ε ariables which results in circular polarization. V e Laplace operator on the sphere with ular polarization. We therefore expect 2

) U | and (3.60) i 2 − 2 . ( a , where under rotations around = = tations of the rotation group can be found E , E ) + ε ε ε iU ) ) s represents the amount of linear polarization ) ϕ 1 3 ±± ( ( 2 ab e Q + − | , Q P E P σ σ and 2 φ ∗ ϑ | 1 are its helicity Q 2 i = 1 e 1 2 E multiplies the anti-symmetric Pauli matrix ( E ( ± E = = I | 30 . The parameters e ±± ++ ) 4 V P 2 P = ( (+) b (+) b , , 2Re → ε ¯ ε ε ε ε around the direction ε ε ε ε Q = ! ! φ ±± (+) a (+) a i )= P , ¯ 1 ++ ε ε ε ε 0 ε ε − 0 they depend not only on E 2 1 0 0 1 ab ab | ∗ P 2 i 0 is a real, symmetric, traceless matrix. in terms of the helicity basis 2 P P 6=

E E 2 2 s

| ab ab , + = is the difference between the intensity of radiation polari P P = = + 2 I ) = 0 Q 4 U 2 ) E ( | 2 ∗ 1 1 −− ++ ( σ ) by an angle = E P P ) E σ | 2 is nothing else than the relative perturbation of the intens ( , i.e., ≡ ≡ U ) . Hence ε ε ε = ( = 2 , ( ¯ M ba ) P P and they transform with helicity I 1 ε ε ε P ( U ) ε ε ε 1 = and and is the difference between the left and right handed circular + ∗ ab ) ℓ 0. More properties of the spin weighted spherical harmonics P 1 V ( ( I ℓ ε ε ε = = Q 4 denote the Pauli matrices and the four real functions of the p − ) m ab ℓ = α P ( Y s σ are the Stokes parameters. Q –Stokes parameter of the CMB radiation to vanish. We neglect The perturbations We define also and the intensity polarized in direction Q is a spin 2 tensor field on the sphere and V ) 1 is simply the intensity of the electromagnetic wave. ( ab ε ε usual spherical harmonics and for field can be expanded in terms of spin weighted spherical harm I in directions This is best seen by expressing and vanish, one finds that traceless part of the polarization tensor while This part describes a phase difference between P on the sphere which is traditionally taken to be In terms of the electric field, the Stokes parameters are spin weighted spherical harmonics areeigenvalue eigenfunctions of th we have relation to the matrix elements of the irreducible represen in [3]. (see e.g. [25]). Thomson scatteringthe does not introduce circ ε The Cosmic Microwave Background where Rotating the basis ( Correspondingly we define the dimensionless perturbation v PoS(cargese)006 . ) ) m − ∂ / ℓ m ∓ L ℓ b ( b e , (3.66) (3.65) (3.69) (3.67) (3.70) (3.71) (3.68) (3.64) (3.63) (3.72) (3.73) m and ℓ → e ) ( + ± Ruth Durrer L ( le e have the properties , . ) , ) . ∗ , ) ( n ) . ) ) n ( ∂ / n n ( n ( m  ( ( um operators m ℓ ) transform as m ℓ m m 2 remains invariant while Y ℓ ℓ ℓ ) Y , ! ! m − 2 Y Y Y ℓ ( ) ) m ± ) ! ! ! ! ± ! ! ℓ ( a 2 2 ) ) ) ) sing and lowering operators ) ) n e e ) 2 2 ( 2 2 2 2 − + − m m ℓ . ) ℓ ℓ + − ℓ + − + − m 2 . , ( ( Y ( ℓ ℓ ℓ ib ℓ ℓ m m ℓ ℓ 2 ) ( ( ) ℓ ℓ ( ( a ( ( e both gauge-invariant. We can now ± s n Y Y ± n  so that ) ) cs, ( ! ! ! ! ( s i 2 s 2 s m ) ) ) ) ) ) m m ℓ 2 m . The operators 2 m ± m m m 2 − ℓ 2 2 2 2 m ℓ − ( ℓ ℓ ( e ( ℓ ℓ ( ℓ ℓ Y Y ( m a a e a a b + − + − m m ℓ ℓ ℓ ℓ ℓ ℓ = ℓ ℓ ℓ ℓ ℓ ℓ − − − − − − e b ( ( ( ( → ℓ ℓ ℓ ℓ ℓ ℓ m ℓ ℓ ======∑ ∑ ℓ ∑ ∑ ∑ ∑ ) − − m m 2 m m m m b s s ℓ ℓ 2 2 2 2 2 2 the basis vectors = = m − ∑ ∑ 31 ℓ ( ∞ ∞ ∞ ∞ ∞ ∞ ======m m ∑ ∑ ∑ ∑ ∑ ∑ ℓ ℓ ℓ ℓ ℓ ℓ a , n 2 2 ) = ) = ∞ ∞ = = ∑ ∑  ℓ ℓ = m m ) ℓ ℓ 2 and ) = ) = ) = ) = ) = Y Y →− m − ) 2 2 n n n n n ℓ ( 2 ( ( ( ( ( − ) = ) = n a –axis. Stokes parameter with respect to the canonical basis on the )( m − ˜ ˜ ˜ ˜ 2 ( z n n ℓ ( E ) B B B ( ( 2 + a i i ∗ . Actually one obtains U ) U i ∂ / E m m ∂ 2 / B − ( ℓ ℓ ( ± )+ ) a Y 1 n n and  − ( ( Q s ˜ ˜ 1 2 where E E is a decomposition into positive and negative helicity, whi Q ∝ ) = ( = turns into 2 we find m ) ± ℓ m ± m )= )= , m 2 ℓ ( ℓ Y ( ℓ n n s ± e U a a i ∗ )( )( P ∂ / ± U U i i Q and + − m ℓ Q Q ( ( Y 1 2 2 ) + ∂ / ∗ s (see [3]), similar to the quantum mechanical angular moment ∂ / ∝ ∗ ( The expansion in Under a ’parity’ transformation, One can also define differential operators which are spin rai ∂ / m ℓ Y s ∂ Hence the coefficient is a decomposition into the changes sign. sphere which requires the choice of a We also define the (non-local) scalar quantities which raise and lower the magnetic quantum number / Hence Applying this to and expand the polarization in spin weighted spherical harmoni The Cosmic Microwave Background Since the polarization of the background vanishes, these ar PoS(cargese)006 (3.75) (3.76) (3.74) changes direction m ℓ Ruth Durrer b polarization is which is then ′ B n . P , P 2rot rot k = E , while normal to the plane it is relevant scattering process right | jm β 2div div β P i cos = electric field is transverse and hence oton from direction cos ∇ T . e [25] i j l | σ ⊥ r ∇ e P j n E i j ε T π ∇ 3 i σ 8 lm e ∇ ε n 2 2 have positive parity. Actually one can show [3] r measures curl contributions to the electric field π 3 8 = = = B M 32 k r E . For photons which are polarized in the scattering −− −− β = β and P P ⊥ E + + , so that cos E are invariant under rotation. Since the sign of cos 2 ∇ ∇ 2 2 ) e e e e + + = B + e e m m ′ ∇ ∇ n n ∇ –polarization (right) patterns are shown around the photon n ( · B + − polarization can be either radial or tangential, while 2 and = = n E ) ) = E ++ ++ c c ( ( ⊥ k 2 ) P P E E with ∂ / − − ( ∇ ∇ n − − and ∇ ∇ 2 ) = = − ˜ ˜ E has negative parity while B ∇ ( B 2 –polarization (left) and = E measures gradient contributions while 2 ) E ∗ ∂ / So far, we have not discussed Thomson scattering which is the ( considered as a function on the spheretangent as to shown the in Fig. sphere 3. of (The photon directions.) scattered into direction before recombination. To study it we consider an incoming ph clearly of curl type. 3.5 The collision term not suppressed. In the rest frame of the electron one finds, se indicated as the central asterisk. Hence under parity, that Figure 3: The Cosmic Microwave Background Like temperature fluctuations, plane, the scattered field amplitude is suppressed by a facto PoS(cargese)006 (3.79) (3.77) (3.78) (3.82) (3.83) (3.81) (3.84) . Ruth Durrer . ,    i ) ) ′    ) ′ ) in n ′ t    n ( , n ( ′ m 1 0 0 ( ∗ 0 couples to scalar 2 m k ∗ m 2 ( ∗ Y 2    2 = 2 Y 2 Y − ∗  2 − ) − m ) − b H ( ) n ) ) n ( V in n ( t · m ( , 2 m des. This is best done using m n 2 Y k 2 Y ( 3 2 2 Y 2 2 , − ally the CMB power spectra can )+ publicly available codes, CMB- ) ackground frame [3] (see Chan- − ropy and polarization amplitudes q ′ ) 3 owerful code or CMBEASY [30], n 3 H n n ( − h ( ( ) m V ′ ) ℓ = ′ M ) and n ′ Y ′ . , tions are given by two power spectra, i icity (and also since this is the situation − n ( ) n ) 2 to tensor perturbations n i ( ) x m ′ ( i i ∗ in ( Ω m 2 m    ± t 2 ∗ i 2 n ℓ m where (3.80) 2 | ∗ m , Y | ) ( 2 e ′ 2 ℓ Y Z m = m 2 Y in U U a m k ℓ ℓ ) 2 t V i i ∗ ℓ ( ℓ ) π e b , ) n 1 i m ) ′ 2 a ′ − 4 n ( 2 − + M h| h| h ℓ ∗ n k + | ( n = ∑ m  ( ( 33 , m 2 m m H ∗ Q Q ℓ m = = = 2 n ) Y + 0 2 ( a Y 3 2 Ψ ) ) ) ∞ in 2 = Y    m ∑ ) t 2 ℓ h| E − , B P ( ℓ in ( q ℓ 3 t 2 = k 3 M E C = , ( C ( − ℓ − 2 2 k ) ) = V = ∑ C ) ( + ′ ) n m ′ M , ′ Ψ H n ( ℓ n n h h ( x ( C ( ) m ′ Ω ∗ m 2 ∗ 2 n Y ( M Y Z ) ) = ) = ) ′ ′ m n ∗ 2 n k k 1 ( 10 ( Y m ) m − − " 2 2 n Y T k k ( 2 Y ( ( 2 σ m − e 2 δ δ 6 ) ) 1 to vector perturbations and 6 Y k k an √ ± ( ( √ h − Ψ − P = P is given in terms of spin weighted spherical harmonics and 3 ] = 3 ) )    m ) ′ π V π n [ 2 , 2 ( C ( )= n ′ ( n m , P n ( The Boltzmann equation can then be written for the Fourier mo Since the perturbation equations are linear, the CMB anisot m P we can derive thedrasekhar following for form the of original derivation the [26]) collision term in the b perturbations, where the total angular momentum methodbe (see determined [27, as 3]) integrals from over wave which numbers, fin The Cosmic Microwave Background Defining the vector These spectra can befast calculated [28], the very original rapidly or CAMBCODE usingthe [29] most one the user presently friendly of most code. the p More details can be found in4. [3]. Observations and parameter estimation depend linearly on the initialfor conditions. most Let inflationary us, models), for assume simpl that the initial condi PoS(cargese)006 (4.5) (4.3) (4.1) (4.2) (4.4) (4.6) can be 600 , , ) s ) ) , Ruth Durrer in in k t t ( 500 , , ) k k ( ( TE 2 2 ( , , ℓ − − T 400 i i , H H ) ) ) ) l k k l they are set to white. i 2 2 ( ( ) h h k − − 300 P P ( , , 3 3 h . k k k k P ( ( ) 2 2 3 ) ) | | s k E T ) ) , m m , 200 ( ( ) ℓ ℓ 2 2 . Note that k 2 ) , , T T ( , m k k in ∗ k t ) ( ( , ( , ) ) E ) )+ )+ m 100 ( E T k 2 ℓ is shown. The first two acoustic peaks ( ( | in in ( ℓ ℓ T t t ) TE ) 2 T T ) ( , , s tions, ℓ | | 0 ditions, we can express the amplitudes s − , k k , T 2 2 , end on ( ( k k H 2 k 2 2 ( ( ) The cosmological parameters chosen for this ) ( ) n the left hand panel all points with amplitude

0.04 0.03 0.02 0.01 + + 2 )

T 0.045 0.035 0.025 0.015 0.005 )+ )+ X , k ( m T ( ℓ k k )+ H H m ℓ − ( 0. ) ℓ ( ( ) ) T k , T k ( | T 2 2 Ψ Ψ k ( = ˆ ˆ k k ( Ψ P P h + + ) 3 3 P , , K P Ω Ω B 34 k k m 3 3 ( are in general complex Fourier transforms of real k k d d ℓ 2 2 k k ( ( | | T ) ) ) ) 2 ) ) Z Z s | 0 E T 0 0 m m , ) ( ( , ℓ ℓ , , correlation is. In the right hand panel the ’ringing’ due to π π 2 k k )+ 1 1 T T k k , ( ( 4 4 ) ( ( 022 and in ) 600 k ∗ . ) ) t ) ( ,ℓ , 0 T E ) E k )+ )+ = TE ( ( m k ( ℓ ℓ ( ( T ℓ ℓ ( = in in ( T T ) = 2 ℓ t t T 2 | T | | s 2 , , T 500 ) h h h , + | h . For the power spectra we then obtain k k s ) b k ( ( , H ( k k k k X ) k m ) Ω and dk dk dk dk ( ℓ Ψ Ψ , ( 2 ) ) ) ) T 400 TE Z Z Z Z s 0 0 X + 13 ( ℓ ( , . , , , ℓ π π π π l T 0 k k k T k 4 8 4 4 | ( ( ( ( ) ) ) ) = 300 T B E T m 2 m m m = = = = ( ( ( ( ℓ ℓ ℓ ℓ h ) ) ) ) T m B T E kT kT kT ( ℓ ( ( ℓ ℓ Ω 3 3 3 TE ( 200 , ℓ C C C in the form d d d C 73 m of the maximum are set to dark blue while on the right hand pane . Z Z Z ℓ 0 b 1 40 100 = = = = The scalar CMB anisotropy transfer function Λ m m m and ℓ ℓ ℓ Ω e b a m ℓ

e 0.01 0.04 0.03 0.02

0.015 0.005 0.035 0.025 0.045 , k m ℓ for some ’transfer functions’ As the final perturbations dependa linearly on the initial con secondary maxima of the Bessel functions is better visible. plot are where we have defined the direction integrated transfer func It is interesting to see how narrow the negative since both Because of statistical isotropy, these integrals do not dep Figure 4: The Cosmic Microwave Background and the SW plateau are clearly visible. In the presentation o smaller that PoS(cargese)006 2 h m Ω arameters Ruth Durrer d by Boomerang [32] and . One then fits jointly the d, the resulting best fit pa- T n ) CDM model. . Therefore the integral (4.6) 0 ) on polarization are shown in Λ . More details about parameter s kt , ( k t presently available cosmological T calar and tensor perturbations are Therefore, we must be very careful egeneracies. The transfer functions − A o (4.4), we need to know the initial CDM model. Obtained by a Markov ( meterizing the initial conditions with ∗ ackground cosmology, hence on cos- Λ ) rameters like the angular diameter dis- = ore model parameters, the errors do in ferent assumptions. E ial at the last scattering surface, as well logical parameters which determine the m ositions of the acoustic peaks and 3 ( ℓ k T ) ) k s ( , aws the best fit h k P − ( ) 35 and T m ( ℓ 1 T − S n = ) ∗ 0 kt )) ( s S , k A ( ∗ ) = E 3 m ( ℓ k T ) ) k s ( to infer the transfer functions and hence the cosmological p , Ψ h k P ( P ) T m ( ℓ T and e.g., ( Ψ P The observed CMB anisotropy spectrum from WMAP [31] extende The presently available data on temperature anisotropy and When estimating cosmological parameters with an MCMC metho Also shown in these figures is a line which is the best fit Figs. 5, 6 and 7 transfer functions. parameters describing the initial conditions and the cosmo Chain Monte Carlo (MCMC) method [35]fitting parameter can fit be to found the in data parameters [3]. from Here different we data just sets want and obtained to making present dif the bes 4.1 Degeneracies rameters and the error bars dependwhen on interpreting the the model results. assumptions. Clearly,general when increase. allowing for But m there isdepend a strongly much on more certain serious combinations of problem:tance cosmological d pa to the last scattering surface, which determines the p Figure 5: The Cosmic Microwave Background functions, so that Acbar [33] in linear (left) and log (right) scale. The line dr which determines the amplitude of the gravitational potent from the observed power spectra. Usuallya this is few done parameters, by para is real but notuncorrelated. necessarily These positive. transfer functions We onlymological have depend parameters. also on However, used the as b that ispower s clear spectra from Eqs. (4.1) t PoS(cargese)006 Ruth Durrer TT TE TE amping scale. However, mentary data, like the Hubble 08 or galaxy surveys which are ion cross-correlation obtained from . eed to lift the degeneracies in the 0 smological parameters determined ± ) l nchanged, see Fig. 8. 72 . 0 curvature and the Hubble parameter, the = h e.g., 36 Angular Scale Multipole moment ( etc. The most important lesson is of course that we do need h fixed and varying m ) 2 Ω h b Ω 10 100 500 1000 , 2 h m 90° 2° 0.5° 0.2° Ω , A 0 2 1 0 –1 d ( 6000 5000 4000 3000 2000 1000 The CMB anisotropy spectrum and the temperature-polarizat which determines the asymmetry of even and odd peaks and the d 2 h To lift such degeneracies we usually need to resort to comple b Ω parameter dependence of the CMB and secondlyby the MCMC values of methods co are always more or less model dependent. as much complementary data as possible, since first of all we n sensitive to the combination CMB anisotropy and polarization spectra remain virtually u key project measurement of the Hubble parameter, Figure 6: The Cosmic Microwave Background as the WMAP 3 year data (figure from [31]). when keeping PoS(cargese)006 ear Ruth Durrer 0045 0062 066 018 042 027 014 033 016 083 025 051 031 018 045 024 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + + + + + + + − − − − − − − 223 732 799 083 948 737 236 2dFGRS ...... WMAP + 1262 . 2 0 0 0 0 0 0 0 erang and others. For more is the optical depth to the last τ importance of the CMB is that WMAP 05 DASI CBI 05 B 03 CAPMAP 0077 0095 070 028 046 030 015 047 029 088 038 054 033 020 059 041 ...... 0 0 ent research. We have given an 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + + + + + + + − − − − − − − ology high accuracy means 0.5 to , BOOMERanG) or galaxy survey data l Mpc and 231 739 798 088 951 739 233 / ...... 1259 . 2 0 0 0 0 0 0 0 n theory and therefore these calculations 002 +BOOMERanG . WMAP+ACBAR 0 = k 0070 0086 066 027 042 027 014 043 026 084 037 052 033 017 053 036 37 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + + + + + + + − − − − − − − Multipole moment are derived. Table from [36]. WMAP 212 743 796 088 947 722 226 ...... m 1233 400 600 800 1000 1200 1400 . +CBI+VSA 2 0 0 0 0 0 0 Ω 0 CDM model with purely scalar perturbations. The WMAP three y and Λ 8 200 σ 0072 0095 072 043 028 015 050 030 091 054 034 019 060 041 028 038 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + + + + + + − − − − − − + − Only WMAP 801 088 951 744 238 233 734

......

1268 0.15 0.10 0.05 0.00

-0.05 .

0 0 0 0 0 2 0 + /

) K ( /2 +1)C ( l

l EE 2 0 2 h 2 b h m 8 s h τ is the amplitude of density fluctuations at A Ω m n σ Ω The measured EE polarization spectrum from Wmap 3 year, Boom A Ω 100 Joint Likelihoods for a flat Parameter In these lectures we have explained to you that one of the main data are combined with small(2dFGRS). scale CMB experiments (CBI+VSA details see [34] from where this figure is taken. scattering surface. The parameters it can nearly fully becan calculated be within linear performed perturbatio relatively1 easily %. to high Doing accuracy. better In than cosm this is difficult and is a subject of pres 5. Conclusions Table 1: Figure 7: The Cosmic Microwave Background PoS(cargese)006 Non- Ruth Durrer 0034 0050 062 016 037 026 015 022 019 083 024 047 032 019 035 025 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + + + + + + + − − − − − − − 255 687 846 088 953 826 299 WMAP+ CFHTLS ...... 1408 . 2 0 0 0 0 0 0 0 iful, inspiring place and the DSS), supernovae (SNLS and SN 0056 0071 065 020 045 028 015 035 023 082 026 053 034 019 049 031 in Ref. [37]...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 tions which we have not touched + − + + + + + + + − − − − − − − lated and briefly discussed the main er simple assumptions for the initial em is that most theories of inflation e.g., tal setups will lead to non-Gaussian ng a first draft of this text. ial non-Gaussianities so far remains a 227 701 827 079 946 784 276 SN Gold d ...... WMAP + 1349 ins a lot of additional information about cal parameters. am very greatful to Martin Kunz for his . 2 0 0 0 0 0 0 0 0056 0072 069 021 044 028 015 038 024 088 030 051 032 019 052 031 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 38 + − + + + + + + + − − − − − − − SNLS 233 723 808 085 950 758 249 WMAP+ ...... 1295 . 2 0 0 0 0 0 0 0 0056 0075 nonlinearities like CMB lensing [3] or non-Gaussianities. 042 029 052 032 062 015 036 026 086 018 048 036 024 032 ...... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + − + + − − + + + + − − − − + − e.g., SDSS 233 813 079 948 772 266 709 WMAP+ ...... 1329 . 2 0 0 0 0 0 0 0 2 h 2 b h 8 m s h τ A Ω m n σ Ω Ω 100 Parameter As in Table 1 but including other data sets: galaxy surveys (S Other interesting aspects of CMB anisotropies and polariza I thank the organizers for inviting me to teach in such a beaut Table 2: The Cosmic Microwave Background upon in these lectures are, overview of how CMB anisotropies and polarization are calcu Gold) and weak lensing (CFHTLS). Table from [36]. physical effects which enter. Then youconditions, have these learned results can how, und be used to estimate cosmologi Acknowledgment students for the many lively andhelp stimulating with discussions. Figure I 4, and to Camille Bonvin for carefully readi Gaussianities are a very active field ofinflation research which and conta about the non-linearitiespredict in very the small CMB. The non-Gaussianities probl errors while at most some level. experimen Therefore, thechallenge measurement for of the primord future. A theoretical overview can be foun PoS(cargese)006 3, . 0 = m Ω is the ratio Ruth Durrer R 7, . 0 0 (solid), which = = Λ 0 line). The spectra K Ω Ω = K indicated. The number 0 (dotted) and < al which is also drawn. In the lower left K Ω ncordance model with identical matter density and baryon density, but that we cannot distinguish them by eye. On the nd baryon density. They correspond to the bullets ed in the top panel on the 39 0 (dashed), > K Ω On the top panel lines of equal angular diameter distance are 7 . The lines of constant curvature are parallel to the diagon . 0 = panel we show CMB anisotropy spectra with of the angular diameter distance of the model to the one of a co lower right panel we show threedifferent spectra with angular curvature diameter zero, distances (the squares indicat indicated in the top panel. The spectra overlay so precisely have identical angular diameter distance, matter density a Figure 8: The Cosmic Microwave Background h are significantly different. PoS(cargese)006 , , Phys. , Helv. ,1 , ns of the , 209 , 2533 78 Ruth Durrer , 702 (2004). 15 nsity D42 606 servations: st-year maps st-year maps , Proc. R. Soc. London meters First-Year Sky , 1882 (1980). , Phys. Rev. , 116 (1946). , Astrophys. J. eds 10 D22 , 5th edition, Academic Press Fund. of Cosmic Physics , Prog. Theor. Phys. Suppl , JETP , , 1 (2003). 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