Cosmological models Cosmological distances l Single component universes

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Cosmological models Cosmological distances l Single component universes l radiation only matter only curvature

Cosmological models Cosmological distances l Single component universes l radiation only matter only curvature only Λ only l Multi component universes PHY 306 1

Cosmological distances l Proper distance between origin and object: ds 2 = −c 2

Cosmological distances l Proper distance between origin and object: ds 2 = −c 2 dt 2 + a 2(t)[dr 2 + x(r)2 dΩ 2] (R-W metric) d. P(t) = a(t) dr = a(t) r (d. P is not a comoving distance) but we know for light emitted at te and observed at to therefore the proper distance to an object at time t 0 is if te = 0 (z → ∞) we call this the horizon distance – it’s the furthest we can currently see PHY 306 2

Cosmological models l Different components of energy density just add: note different a dependence

Cosmological models l Different components of energy density just add: note different a dependence at small a, radiation must dominate matter takes over when a > εr 0/εm 0 at large a, cosmological constant dominates if it exists l Therefore sensible to consider single components PHY 306 3

Radiation only l l a = (t/t 0)1/2 and εr t− 2 age of

Radiation only l l a = (t/t 0)1/2 and εr t− 2 age of universe: ln a = ½(ln t – ln t 0) → H = 1/2 t l so t 0 = 1/2 H 0 proper distance: PHY 306 4

Matter only l l a = (t/t 0)2/3 and εm t− 2 age of

Matter only l l a = (t/t 0)2/3 and εm t− 2 age of universe: ln a = ⅔(ln t – ln t 0) → H = 2/3 t l so t 0 = 2/3 H 0 proper distance: PHY 306 5

Curvature only l l if k = 0, a = constant: flat, static, empty

Curvature only l l if k = 0, a = constant: flat, static, empty universe if k = − 1, a t: universe expands at constant speed Milne model age = 1/H 0 proper distance d. P(t 0) = ct 0 ln(1+z) l k = +1 does not produce a physically viable model PHY 306 6

Λ only l a = exp[H 0(t − t 0)] : universe expands exponentially

Λ only l a = exp[H 0(t − t 0)] : universe expands exponentially de Sitter model infinitely old: a → 0 only as t → −∞ proper distance d. P(t 0) = cz/H 0 l this is a “Steady State” universe which always looks the same PHY 306 7

Single component universes PHY 306 8

Single component universes PHY 306 8

Multi-component universes l “This is not a user-friendly integral” (Ryden) fortunately at different times

Multi-component universes l “This is not a user-friendly integral” (Ryden) fortunately at different times different components will dominate best current values: Ωm 0 = 0. 27, ΩΛ 0 = 0. 73, Ωr 0 = 8. 4× 10 -5 matter-radiation equality at a = Ωr 0/Ωm 0 = 0. 0003 matter-Λ equality at a = (Ωm 0/ΩΛ 0)1/3 = 0. 72 PHY 306 at any given time can usually use single-component model 9

Example: matter + Λ Λ dominated: slope +1 matter dominated: slope − 1/2 model

Example: matter + Λ Λ dominated: slope +1 matter dominated: slope − 1/2 model with ΩΛ 0 = 0. 7, Ωm 0 = 0. 3 PHY 306 10

Example: matter + Λ model with Λ=0. 73 PHY 306 11

Example: matter + Λ model with Λ=0. 73 PHY 306 11

State of Play: theory l Friedmann model plus cosmological constant can describe wide variety

State of Play: theory l Friedmann model plus cosmological constant can describe wide variety of behaviour expanding, recollapsing or static also “bouncing” and “loitering” models this technology all available in 1920 s l However, models have free parameters H 0, Ωm 0, Ωr 0, ΩΛ 0 need to determine these to see what model predicts for our universe PHY 306 12