Cosmology & CMB

Set1: FRW Cosmology

Davide Maino

Scope

• Overview physically motivated of one of the most powerfulresearch line in cosmology today• origin and evolution of cosmological perturbation• their imprint in the CMB

• A GR course would be appreciated although lectures areself-contained

FRW Cosmology

• The Friedmann-Robertson-Walker cosmology has two elements• FRW geometry or metric (k term in the FRW metric)• FRW dynamics or Einstein/Friedmann eqs (a - scale factor)

• Same as in GR• a metric theory: geometry tells matter how to move• field equations: matter tells geometry how to curve

• Useful to separate out these two pieces both conceptually and forunderstanding alternative cosmologies, e.g.• modifying gravity while remaining a metric theory (e.g. change

G)• breaking the homogeneity or isotropy assumptions

Isotropy & Homogeneity

• Isotropy: CMB is isotropic at 10−3, 10−5 level after dipolesubctraction

• Homogeneity: redshift surveys (2dF, SDSS) show homogeneityat > 100 Mpc scale

FRW Geometry

• Spatial Geometry is that of aconstant curvaturePositive: SphereNegative: SaddleNull: Flat

• Metric tells how to measuredistances on the surface

Comoving Coordinates

• Remaining degree of freedom (preserving homogeneity &isotropy) is the temporal evolution of the overall scale factor

• Relates the geometry (defined by the curvature radius R) tophysical coordinates

dσ2 = a(t)dΣ2

our conventions are that today a(t0) = 1• Similarly physical distances are given by d(t) = a(t)D, and

dA(t) = a(t)DA.• Comoving coordinates do not change with time: simplest

coordinates to work out geometrical effects

Time and Conformal Time

• Proper time (with c = 1)

dτ 2 = dt2 − dσ2

= dt2 − a2(t)dΣ2

• Taking out the scale factor in time coordinate

dτ 2 = a2(t)(dη2 − dΣ2)

dη = dt/a defines conformal time - useful for photons travellingradially (along null geodesic) then obey

∆D = ∆η =∫

dta

so that time and distance may be interchanged

Horizon

• Distance travelled by a photon in the whole lifetime of theUniverse

• Since dτ = 0, the horizon is simply the elapsed conformal time

Dhorizon(t) =∫ t

0

dt′

a= η(t)

• Horizon always grows with time - a ∝ t2/3 (RD), a ∝ t1/2(MD)• Always a point in time before which two observers separated by

a distance D could not have been in causal contact• Horizon problem: why is the Universe homogeneous and

isotropic on large scales especially for objects seen at earlytimes, e.g. CMB, when horizon is small?

FRW Dynamics

• We still need to know how the scale factor evolves givenmatter-energy content

• GR: matter tells geometry how to curve, and the scale factordetermined by content

• Build the Einstein tensor Gµν out of the metric and use Einsteinequation

Gµν ≡ Rµν −12

gµνR = 8πGTµν

Rµν = Γαµα,ν − Γαµα,ν + ΓαβαΓβµν − ΓαβνΓβµα

R = gµνRνµ Γµαβ =gµν

2(gαν,β + gβν,α − gαβ,ν)

FRW Dynamics

• Assume FRW metric as gµν = diag(−1, a2(t), a2(t), a2(t)) andTµν = diag(ρ,−p,−p,−p)

• µ = 0: Γ0αβ contains g0ν , not zero only for g00 = −1

Γ0αβ = −1

2(gα0,β + gβ0,α − gαβ,0)

• Again gα0(gβ0) 6= 0 only for α = β = 0 but g00 is constant andits derivative are zero. Only gαβ,0 with spatial indices

Γ000 = Γ0

0i = Γ0i0 = 0 Γ0

ij = δijaa

• µ = i:

Γi0j = Γi

j0 = δijaa

FRW Dynamics

• Assume FRW metric as gµν = diag(−1, a2(t), a2(t), a2(t)) andTµν = diag(ρ,−p,−p,−p)

• µ = 0: Γ0αβ contains g0ν , not zero only for g00 = −1

Γ0αβ = −1

2(gα0,β + gβ0,α − gαβ,0)

• Again gα0(gβ0) 6= 0 only for α = β = 0 but g00 is constant andits derivative are zero. Only gαβ,0 with spatial indices

Γ000 = Γ0

0i = Γ0i0 = 0 Γ0

ij = δijaa

• µ = i:

Γi0j = Γi

j0 = δijaa

FRW Dynamics

• Assume FRW metric as gµν = diag(−1, a2(t), a2(t), a2(t)) andTµν = diag(ρ,−p,−p,−p)

• µ = 0: Γ0αβ contains g0ν , not zero only for g00 = −1

Γ0αβ = −1

2(gα0,β + gβ0,α − gαβ,0)

• Again gα0(gβ0) 6= 0 only for α = β = 0 but g00 is constant andits derivative are zero. Only gαβ,0 with spatial indices

Γ000 = Γ0

0i = Γ0i0 = 0 Γ0

ij = δijaa

• µ = i:

Γi0j = Γi

j0 = δijaa

FRW Dynamics

• Assume FRW metric as gµν = diag(−1, a2(t), a2(t), a2(t)) andTµν = diag(ρ,−p,−p,−p)

• µ = 0: Γ0αβ contains g0ν , not zero only for g00 = −1

Γ0αβ = −1

2(gα0,β + gβ0,α − gαβ,0)

• Again gα0(gβ0) 6= 0 only for α = β = 0 but g00 is constant andits derivative are zero. Only gαβ,0 with spatial indices

Γ000 = Γ0

0i = Γ0i0 = 0 Γ0

ij = δijaa

• µ = i:

Γi0j = Γi

j0 = δijaa

FRW Dynamics

• For a(t) we need only the 00 component of the Einsteinequations. Start with Ricci tensor

R00 = Γα00,α − Γα0α,0 + ΓαβαΓβ00 − Γαβ0Γβ0α= −Γi

0i,0 − Γij0Γj

0i

= −δii∂

∂t

(aa

)−(

aa

)2

δijδ

ji

= −3

[aa−(

aa

)2]− 3

(aa

)2

= −3aa

FRW Dynamics

• The spatial part Rij

Rij = δij(2a2 + aa)

• The Ricci scalar R is obtained by contraction of Ricci tensor

R ≡ gµνRνµ = −R00 +1a2 Rij = 6

[aa

+(

aa

)2]

• The 00 component of Einstein eqs

G00 ≡ R00 −12

g00R = 8πGT00

= −3aa

+ 3aa

+ 3(

aa

)2

= 8πGρ

FRW Dynamics

• The spatial part Rij

Rij = δij(2a2 + aa)

• The Ricci scalar R is obtained by contraction of Ricci tensor

R ≡ gµνRνµ = −R00 +1a2 Rij = 6

[aa

+(

aa

)2]

• The 00 component of Einstein eqs

G00 ≡ R00 −12

g00R = 8πGT00

= −3aa

+ 3aa

+ 3(

aa

)2

= 8πGρ

FRW Dynamics

• The spatial part Rij

Rij = δij(2a2 + aa)

• The Ricci scalar R is obtained by contraction of Ricci tensor

R ≡ gµνRνµ = −R00 +1a2 Rij = 6

[aa

+(

aa

)2]

• The 00 component of Einstein eqs

G00 ≡ R00 −12

g00R = 8πGT00

= −3aa

+ 3aa

+ 3(

aa

)2

= 8πGρ