Cosmologicl Models with Idelized Mtte. Model spces: Constuction Spces nd spcetimes of high symmety ply vey impotnt ole in cosmologicl modelbuilding, nd s emples solvble models of genel eltivity. The most impotnt ones cn be consideed s diffeent odd sots of sphees, so we stt with those. d sphee 4 i = Spheicl coodintes = cos = sin sinθ cosφ = sin 4 = sin sinθ sinφ dl = d i }{{} = d sin dθ sin θdφ Qusi-flt coodintes Wite 4 = i Eecise i= d d d 4 = 4 d 4 = dl = d i d 4 i= d = d dθ sin θdφ d = dθ sin θdφ du = u dθ sin θdφ, u = u Confoml coodintes It is often useful to wite ds = f ds flt if tht is possible. Penose digms... lte. Stting fom out pevious fom, we will hve this if we use η in plce of such tht d = f dη = f η dη d = η
leding to η = tn u with sinu =. Wite = sinu; q d du = = sinu u dη d log tn = η = d logη. o, fte some lgeb: 4 dl = dη η dθ sin θdφ η f = η sin u = u u η 4 sin cos = 4 η η The sphee suppots the symmety SO4.. d hypeboloid spce of constnt negtive cuvtue figue Spheicl coodintes i = i= Qusi-flt coodintes Confoml coodintes = cosh = sinh cosθ... dl = d d = d sinh dθ sin θdφ = =... d dl = dθ sin θdφ du = u dθ sin θdφ u 4 dl = dη η dθ sin θdφ η
with η = tnh u, sinhu = Suppots symmety SO,, i.e. Loentz symmety, cting puely sptilly! To bing this out, use = coshλ = cosφ = sinhλ = sinφ Tnsltions λ this invint. dl = d dφ dλ λ constnt, coisponding to boosts in the oiginl vibles, leve. de-sitte spcetime figue 4 = = cosh cosλ = cosh sinλ cosθ = cosh sinλ sinθ cosφ 4 = cosh sinλ sinθ sinφ = sinh Spheicl coodintes ds = d d i i = d cosh dλ sin λ dθ sin θdφ }{{} unit -sphee?: eponentil epnsion!; minimum dius; sphees Qusi-flt coodintes d = d = d ds = dλ sin λ dθ sin θdφ
Light-font coodintes Sepete out plnes,, 4 = = = = = }{{} ds = d d d d = d d ds d d = d d To emove the ugly coss-tem, intoduce v = f. So dv = f d fd dv = f d ff dv f d d = dv f d ff d f The -tem cncels if f f =, f = ± >. Thus with v ds d = d dv Now with e t/ d = dv which is n epnding flt sptil metic. ds = dt e t/ dv Confoml coodintes d ds = dv 4 d so with u ds = du dv u de-sitte spce hs the symmety SO4, fom the hypeboloid definition. In the light-font coodintes we hve tnsltion symmeties v v const. Whee do these sit? See Appendi. We ll hve much moe to sy bout de-sitte spce lte infltion. 4
. FW Fiedmn-obetson-Wlke spcetimes These e constucted by choosing one of the mimlly symmetic pces nd letting its ovell scle vy with time. Thus whee ds = dt tdl dl = du u dθ sin θdφ κu K = K = K = hypebolic sections flt sections spheicl sections These model spcetimes e homogeneous nd isotopic, but evolving. They supply inteesting fist models fo the obseved univese veged ove lge scles.. Cuvtue Clcultions Ou mste fomuls with coect signs e e fν ω µ ef = µ e e ν ν e e µ e µ e eρ ρ e ν e f µν αβ = F µν b e α e b β b F µν = µ ω ν b b ω c ω c ν ω µ ω µ c ν b ω ν c µ v This is best eploited fo g µν digonl by using cetin qusi-ctesin viebeins e α = δ α g so e α = η α g η fν δ e st tem: g f }{{ ν } µg e symmetic in e f η ef nd tem: η fν g f δ µ e ν g e = η fν g f δ µ e ν g e η fν η η eρ η g g g δ f η eρ g d tem: µ ν f e ρ g f = µ e ρ g f }{{} f,ν,µ, ll equl So ωµ ef = δµ f η eρ ge ρ g f δµ e η fρ g f ρ g e mnemonic: µ mtches on inde, the othe diffeentites its g Emple : d sphee wm-up e θ = = g e φ = sinθ = g ω = δ e only, but g = θ µ ω φ = cosθ δ f= η ρ g g F θφ = θ ω φ vnishing = sinθ sinθ {}}{{}}{ θφ = sinθ e θ e φ = θφ 5
By the wy, this is the guge field of mgnetic monopole guge goup SO = U! Emple : d sphee e χ = e θ = sinχ e φ = sinχ sinφ ω = ω = cosχ ω = ω χ χ θ φ = ω θ = ω φ = cosχ sinθ Thus F F F F χθ χφ = χ ω = sinχ θ = χ ω = sinχ sinθ φ = θ ω ω ω = cosχ cosθ cosχ cosθ = θφ φ θ φ = θ ω ω ω = sinθ cos χ sinθ = sin χ sinθ θφ φ θ φ F µν b = e µ e ν b e µ b e ν? The ntisymmety on indices µ, ν nd, b is utomtic! o µν b = δ µ α δ ν β δ µ β δ ν α ν β = δν β = 6 Emple : FW cosmology sptilly flt cse Note: Mid-Ltin indices e sptil, ely Ltin indices e intenl The only non-zeo ω is c c e t = e i = δ i t ds = dt t d c c ω i = δ i ȧ The non-vnishing components of the field stength e leding to the icci tenso components c c c F i = ω i = δ i ä cd F ij = ωi c ωj d ωj c ωi d = δ c d c i δ j δ k δi d ȧ ä c = = F i e i c l cl l i = F ij e c e j d F c e l i c ȧ ä = δ i l ä = 6 6 ȧ 6
.4 FW Dynmics The field equtions in g αβ e µ ν δν µ = 8πG ν We intepet T i i µ = ρ, T = pδ j j Check : fo electomgnetism, T µ =, p = ρ Fom the peceding clcultion ȧ 8πGρ = ä ȧ 8πGp = ȧ ä 8πGp ρ = Anothe impotnt nd ppeling eqution comes fom diffeentiting the fist of these nd eliminting: ȧ ä ȧ 8πG ρ = 6 ȧ = 8πG ρ p o simply ȧ ρ = ρ p Anothe inteesting thing is to see who s esponsible fo cceletion: ä 8πGρ p = 6 Thee is simple intepettion of nd : : Imgine test pticle long fo the ide. Gvity outside cncels Bikhoff theoem. Consevtion of pticle s enegy v {}}{ G 4π m ρ m = mk 8πGρ ȧ = k We hve this with k = : neutl binding, citicl escpe velocity! The non-zeo vlues of k ise in FW spces with hypebolic k > o? spheicl k < sptil sections - see the poblem set. : Imgine wok done by n epnding fluid ginst pessue; tke it fom mss-enegy d 4π d 4π dt ρ = p dt d d ρ = p dt dt ȧ ρ = ρ p 7
Appendi : Tnsltions within SO4, Wite the metic in block fom: J g = J The condition fo ne-identity tnsfomtion b S = c d to leve the metic invint is S T T gs g; b T c T d T J J c b d o to st ode T J J = Jb c T = b T J c = d T d = With = d = the tnsfomtions b T b tnslte vectos J s by b Tbs, i.e. with things J spelled out completely α δ α β γ b = ; so b T J = β η δ η φ γ φ nd αs βs γs Δ = bs = δs ηs φs α δ Δs = b T J = β η γ φ Tnsfomtions with δ = α, η = β, φ = γ leve fied while tnslting s though α Δs = β γ So s/ is tnslted in the conventionl wy. In ou pevious nottion this is This eplins v. s,, 4 = = v 8