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February 06 Cosmology1. Introduction
1.1 Conventions in Cosmology:
1) Metric signature +,-,-,-, Greek spacetime indices guv, Latin space indices gij.
2) Natural units c=h-bar=kb=1, [Energy]=[Mass]=[Temperature]=[Length]-1=[Time]-1.
3) Astronomical units 1 parsec = 3.261 year. Use megaparsec.
1.2 The Expansion of the Universe:
The Cosmological Priniciple: Our position in the universe is not unqiue. The universe looks the same globally whoever and wherever you are. This forms the basis of big bang theory and implies that the universe is spatially homogeneous and isotropic.
Evidence: 1) Density fluctuations converge to homogeneity beyond 100Mpc as seen by the Hubble telescope. 2) Cosmic microwave background radiation has fluctuation to the order ΔT/^T~10-5. This supports isotropy.
Redshift: Hubble observed that everything in the universe is receding from us. The further away an object, the faster the recession. Empirically the velocity of recession is proportional to distance: v=H0d, where H0=100kms-1Mpc-1h, h is the Hubble constant.
From the redshoft we conclude that in the distant past everything was close together, in an initial explosion or big bang.
Particles: It is crucial to know whether a particle is relativistic or non-relativistic. Baryons are typically non-relativistic while photons and neutrinos are relativistic.
2. Key Equations
2.1 Friedmann Equation
 The equation is expressed in comoving frames, with the expansion r=a(t)x. r is the proper distance and a(t) the scale factor. The constant k=-2U/mx2 is independent of x by homogeneity, therefore the total energy U of a particle must be proportional to x2. k is the curvature of the universe and has a unique value. Note that the Friedmann equation can be derived from both Newtonian and Relativistic mechanics. Λ is the cosmological constant.
2.2 Fluid Equation
To determine the evolution of cosmic density we apply the first law of thermodynamics: TdS=dE+PdV to an expanding unit volume in comoving frame:
 We also need the equation of state P=P(ρ).
2.3 Acceleration equation
Differentiate the Friedmann equation w.r.t time and substitute the fluid equation we get:
 These are all the equation we need to determine the evolution of the universe.
3. Preliminaries
3.1 The Metric Properties:
ds2=guvdxudxv, moving along along a worldline in comoving coordinates, ds2=dt2-gijdxidj. The Friedmann-Robertson-Walker (FRW) line element in spherical polar coordinates is given by:
ds2 = dt2 - a2(t)[dr2/(1-kr2) + r2(dθ2 + sin2θdφ2)]
1) k>0, the universe is closed, like S3.
2) k=0, the universe is flat and Euclidean, like R3.
3) k<0, the universe is open.
3.2 Conformal Time:
t is the proper time (i.e. cosmic time) measured by a comoving observer. Define conformal time by:
dτ = dt/a(t), also dr2/(a-kr2) = dχ2
The metric can be written as:
ds2 = a(t)[dτ2 - dχ2 - f2(χ)(dθ2 + sin2θdφ2)]
1) sinχ k>0
Where f(χ) = 2) χ k=0
3) sinhχ k<0
3.3 The Horizon Problem:
Consider a radial photon, ds2=0 with dθ2=dφ2=0. The proper distance travelled by light since t=0 is given by dH=a(τ)τ. Regions beyong dH have never been in causal contact. The horizon problem states that the microwave sky, despite being isotropic, contains about 105 causally disconnected regions. Comments (8)
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