Physical Foundations of Cosmology

40 Propagation of light and horizons order the Hubble scale, 1/H. Consequently, the terms “Hubble scale” and “particle horizon” ...

2.3 Conformal diagrams 41 comoving coordinates χ>χe(η)= ∫ηmax η dη=ηmax−η. (2.12) Hence, the physical size of the event horiz ...

42 Propagation of light and horizons where the indicesa,bandcrun over only two values, 0 and 1, corresponding to the time and ra ...

2.3 Conformal diagrams 43 Note that the diagrams of different spacetimes are exactly the same if their metrics are related by an ...

44 Propagation of light and horizons h χ final singularity π particle horizon event horizon initial singularity χ = const h = co ...

2.3 Conformal diagrams 45 η π χ events horizon π − θ φ + π π image 1 image 2 observer particle horizon θ, φ=const χg Fig. 2.2. u ...

46 Propagation of light and horizons to clarify what happens to light after passing through the pole, one has to use another coo ...

2.3 Conformal diagrams 47 η χ a= ∞ a= ∞ η = const η = −π/ 2 amin = (^1) /H π particle horizon χ = const event horizon V −π Fig. ...

48 Propagation of light and horizons η χ −π π r= H−^1 r< H−^1 t= −∞ I II IV III tˆ =const t =const r= H−^1 t= ∞ r> HΛ−^1 Λ ...

2.3 Conformal diagrams 49 In aflat de Sitter universe, the scale factor grows asa(t ̄)=H−^1 exp(H ̄t), where the physical time ...

50 Propagation of light and horizons spacetime, a triangle whose lower boundary coincides with the particle horizon. On the part ...

2.3 Conformal diagrams 51 η χ i^0 π i− −π η = const χ= const I− ̃ ̃ Fig. 2.6. The conformal diagrams show explicitly that flat a ...

52 Propagation of light and horizons corresponds to a 2-sphere and that the 3-geometries of hypersurfaces along which the diagra ...

2.3 Conformal diagrams 53 I+ π/^2 I i^0 χ r= const t= const t= 0 i+ i− π/^2 −π/ 2 I − η Fig. 2.7. Open and flat universesNow we ...

54 Propagation of light and horizons i+ i^0 I+ χ= const h= const initial singularity ̃ ̃ Fig. 2.8. Problem 2.11Draw the conforma ...

2.4 Redshift 55 Draw the conformal diagram for the eternal black hole and identify the types of infinities. The Schwarzschild ra ...

56 Propagation of light and horizons Thus, the wavelength of the photon changes in proportion to the scale factor,λ(t)∝ a(t),and ...

2.4 Redshift 57 Although the derivation above has been performed in a local inertial frame, it can be applied piecewise to a gen ...

58 Propagation of light and horizons One can always go to a coordinate system in which only the radial peculiar velocity of the ...

2.4 Redshift 59 where the definitions in (1.21) and (2.57) have been used. Atz= 0 ,this equation reduces to k a^20 =( 0 − 1 )H 0 ...