cosmology & cmb set1: frw...
TRANSCRIPT
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