B.6 k = +1; q 0 = B.5 k = 0 ; q 0 = 1/

! " # $#% &'(*

!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""". The metri of Spae-Time... 5. The redshift... 6. Distanes... 7.4 Angular Sie... 8.5 The Volume... 9.6 ow to ompute ω... 9.7 The sale parameter....8 Time sales... #$ $ %&Λ'"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""". General harateristis... " The solution of and t in parametri form.... The ubble and deeleration parameters... 4.4 elation between ω and ψ... 4 " Expressing ψ in the observables and... 4 "( The geometri distane... 5.7 The o-moving volume... 6.8 Time Sales... 7.8. The loo-ba time τ... 7.8. The age of the Universe (t expressed in and... 8 #% %'Λ """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""". Introdutory emars.... The general solution...., and t in terms of the parameter A....4, and τ in terms of A and....5 The geometri distane and volume elements... 6 * % % Λ """""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""", 4. Zero-density model with >... 8 4. The Lemaître model: Λ> and... -&.$ $/% """""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""( A. The Metri... 6 A. The Einstein Euations... 6 A. General elations... 7 -& #$$ %Λ' """""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""", B. General elations (... 8 B. ; < /... 9 B. ; > /... 4 B.4 ;... 4 B.5 ; /... 4 B.6 ;... 4 -&#% %'Λ """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""* -4&5 %6 %""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""** D. Zero-Density Model with >... 44 D. The Einstein model... 45 D. The De Sitter model... 45 D.4 The Lemaître model... 46

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dt d d / dψ d t dψ dψ dt / dψ dψ Gψ';G & d d dψ dψ %& a ( os ψ. 5 ';G ψ& a sin ψ t ψ. % % %& #' - '- - G'- & ;' 'Jψ K ' 'G Jψ ψk #'B & ;'B 'Jψ K 'B 'G Jψψ K #' ψ 'ψ G ψ G 'ψbψ G( $%/ < %$ "& ;' 'ψ G ' 'ψ G(! %$ ;/ %GD $ % "@%$ψ 6 / / / / / ( a t t - %$ /D'G $;' GD": 'G* % & ( t t /

. The ubble and deeleration parameters D / %%& $ $ $%& os ψ os ψ -% %''B /%/ " #' / %/ ; " %$ $% "$ & D' 'G G ' G".4 elation between ω and ψ $% %/ %7$ ω %$ %ψ"5 G;'ψG " & t dt ψ ω dψ ψ t & d sin( ψ / dψ a ( os ψ ω ψ ψ ωψ $ < 7&ω$ / ω' %ψ /% % ;'ψ'" Expressing ψ in the observables and 5 ψ '7 Gψ/ - $;;';GB< & os ψ os ψ G& 4

os ψ ( : $% % %/%$&ω / %$ % / - D<" %/ $ %$ " - ' %$% %%" % %$%% / % - $/ /% % #$$ % 7$ % "! % Λ " The geometri distane N/ % ' ω G & sin ( ψ ψ sin ψ os ψ os ψ sin ψ r g : % "/ $ & / r ( ( g ( / J 7 GK'G; D & / r ( ( g ( $ $% 8, " 5% $ %#$ %8 % $/ $ - <"!% % $% % / " % %$ -$% $ % / /$ $ < " %8OO / % / $$% %. 88O "# %$ 4';B< $/ & '7B< G '7 <B G G " / Mattig's formula was derived without any referene to partiular values of, or, and indeed, it is valid for all and all. For we an tae the limit by expanding the suare-root term in powers of up to the seond order. Zero- and first order terms in in the numerator anel exatly, so the limit exists. 5

: <B< & 4'GD <B<G % $%/ %/ "! % $ A'B < G & D ( Q %% % $B <BA GB <BA % % % $%& D ( ( /.7 The o-moving volume %$% %$ / $$ %& π sin ω V ( ω ω : %% ω $D< "% % % %$ -%% / /%! /% $%% / $/ $ & $ % %% $ %% ω $% "*" % / "D $ %% %$ / % $ %": % / %" ' % $ %"5 ω'<b< G GB< - ' %J - B B-K& ω ln( $ $ $%PG ω'ωωq& 4 ( V ( π ln( 4 ( dv d 4π ( ( 6

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8 %$ ψψ $ / /%D<" %! -% - τ / /% 'G" '" %$ / %& $ $ /% $ %& ' $% ;G ';' %& %$$ % - τ" 'M"#$ "& & 'B" ψ'7 G'ψ'πG"%ψ' ψb< GB< '<GB< &.8. The age of the Universe (t expressed in and. - D $ψ "" & τ t t ψ ψ τ ( / τ aros / π τ / sin ( / t ψ ψ

'" %% ;';'GD t / %/ / %$ % $%" 'G"N/ " % & t / '"% %$%''& t π : $% #$$ %$% " 9

Flat Models (, Λ. Introdutory emars : - @ 7 5 $ %%$ %' $ " < $%% %$%% / $ % / % % % $ " L % 7 $ % / $ % $%" %%% / $ 7#$ $ % % $ " FF% $ % Λ & % $%%$ %/ & % $ 8,, "% $ %/ %" $ $% / % %$ / % %$ / Λ / /%/ ": $ %$ % $%% $ $ / / % $ %$ % "! % %$ %% $ Λ%! $ %Ω'GΩ Λ 'G / % $ % % / $ $%"!%%/% @ / $ $ D//%$ %$ $ σ'*πnρgd & Λ ( σ σ! $ % $ % σ Λ Ω 'σ Ω Λ 'ΛGD "! $ / ΩBΩ Λ ''" $ %/ - % /'ΩGΩ Λ "

Figure The -σ plane. The lines representing Friedmann models and flat models are drawn. # $%% % %/ / < 5 $ % *" @ 7 5 $ % "!" 7σ% " % ΩΩ Λ S ": < & % σ'λ' σ'b' %$% % %$% % % " %%% %/ / /% %$ % * % / / $% /%/ - "!% /$ %$ %'Λ "! % %% - %%$ % % % Λ$ $% %%#$$ %". The general solution @ '& 8πGρ Λ

Λ!/ % % $ & ε!λ ε'λs ε'b%& A N/ " & ΛS&ΛGD '7σσG%T'σGσ " Λ&σSG %T'σG7σ " : % %%Dτ " % Λ % / %% / %& σ G $%% - % %$ #$ $ % G $% " $ %%% Λ": & D 'T& ( εa 8πGρ Λ γ A A / / sinγ t Λ sinh / ( γt 8πGρ Ω M Λ Ω Λ / γ Λ Ω Λ 5 %%$ %7 $%% % %; $/%/ $% /% 7 $ $ /%$ $ /% - $ %$ "!%%/

% % $ - " L % % $ % $%" % $ % /$% % $ / / $ %/% & Ω'G Ω Λ 'G'G"$ % $ $ %" & / sinh / ( t., and t in terms of the parameter A #$ %% $$ %Λ/γ ' G γ 'JB GK G & Λ ( A t #& / Λ Ω Λ A ( Ω A Λ ( A / ln{ A / A ( A / } Ω / Λ Ω ln / Ω M Λ.5 t.4 /.9!/ % %@ 7 5 % D,πNρG GD G"# D ΛG D 5 *" "5$%$%/ / ΛS" / Λ.4, and τ in terms of A and

$ % $ % - D< < %7/$ τ< Λ& ( Λ / ( { A( } A ( A( L $ / $$ < % % %"! $ %'" <' '",O" # %7/$ & τ Λ A( / ln[{ A( } ] ln( / ln{ ( A } ln A ln A / Figure The loo-ba time for some flat models with positive osmologial onstant. The full lines represent models with: A.,.5,.. The A.5 model is lose to the Conordane model. Also shown are standard and / models (lower and upper full lines respetively. 4

5

#'"& ( ( ( ( 4 ( / <' "/ '"" : τ<'" %$" τ % "" 5$%$% ΛS" $B< $ % $% "%/ %$ % $ ".5 The geometri distane and volume elements $ %$ %$ "! & r g t t dt t t $< B< '; G; $ $%& / dζ rg ( A / { A( ζ }!% @ 74 5 r { } g / ( & A / dt sinh / γt 6

r g $%" % % $% $ %%%$% $ % "#$ $$ % % %$ '*π; G %%$ G<'*π; <" < % "" Figure : The geometri distane as a funtion of redshift, for A.,.5,. and.. For omparison also the Friedmann models with and.5 are shown 7

4 A seletion of Models with, Λ 4. Zero-density model with > %$ % %/ σ %$ & %/ % $ $/ $ $ % σ' -$"$$ % %$ "$%$/σσs $% Λ S " Λ % %$ / $ " # / @ ρ'& Λ Λ 5 % $ @ "σ's'" @%$Λ$ @ / & / $ %/%& / ( @ $ % % -%%;'; / % /$% ;'/ ": & / sin( t / /& sin( / t / 8

D//% %$ & / / ot( t & tan ( / t 5$ $$ & $ % % % % $ ' & ; $-$$GD G ' πgd G < 'πgd G " %$$ ' D 'G D 'πg*"! $/ %/ & ( / [os{ / ( t t } / sin{ ( t t $%-G-'%-G-G'-G-B -'%J -7 B-K/%% & / {( ( } ( r g sinhω %$% -$ %"L % '$ %% / σ Λ σ "! / % %$ / $% - " %7/$ / & / }] / arsin / ( ( τ / arsin & / ar tan t / $% $ %" 9

4. The Lemaître model: Λ> and 9 $U $ % $% $ % %/ /% / % % % " @ $ %" $%%$ % / @ "D % %-% / $ /D//% $%%Λ/%"; ;' ;G' ;G '" @ / $ & 8πG ρ Λ!%,πNρ' G; ' ρ> Λ": E Λ Λ Λ 4 5 $ %% %$ %$ %"! % $ $ %%$ 9 $U $ %": $$< $ %& e ( t t Λ 9 $U $ % $$$ % 8 $ /8( $ $ -%

% <'" $ %/ %"#$ $ -$%. 8(O "! 9 $U $ % 'B ΛI% @ $ %"D %%$ $ & 9 $U $ % /% % % @ $ % /" $ $ / $ % - $%% / % % "! % : / 8O "! % $ $ @ ;@ - ;G;@": $ $ @ % ρ @ ρ; ' ; 'ρ- ; @ 'αρ @; @ α" & αρ E αλ ρ x 4πGx @ - α/ $ & Λ x ( x x α x Λ x ( x α x %/ % / $/ & -SS & / x ( αλ / t & / @ 74 5 $ %&#S-S 9 $U $ %/ @ 74 5 $ %"K #- x Λ x t /

& Λ #- 9 $U $ %/ 4 5 $ %"L $$ %$ % Λ " 9 $U $ % $ -& - "#$ @ -G $$$& x x min Λ( α α / / $'"5 %'G % %% / $ - % % "/ %& % $%%%-$'α G "#$ $ %% % "D $ - %-G $ " $ $ ; $%% "! ρ 4 5 $ % " : -G$ / $ /%% < /%α " #α% & / / x min Λ( α ( x α x & x α x Λ

x /% -'α G B7$ Λ A α Λ B / / ( α ( x α / ( / 5 FF / & % / Λ x α / α / x Λ x α ( x Λ / / sinh{ ( t tm } ( α 5 $ $' % " %$ $ % < % /%α "! % < $ %"!/ $ %/ $ -% <' $ %& $ % <'"9 / % 7-9 $U $ % " 6 $ $$ &%/ /$ % $ % %$ % $ $ % " $% 9 $U $ %"

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5

Appendix A: Parameters and symbols used General Symbols <& & $ 4& %$ θ& %< 9 % < & %$ 7$ τ& %7/$ G < ψ& %$ % #$$ %Λ' A. The Metri sin ω ds dt ( t dω ( dθ sin θdφ ω& %7$ θ&%7$ φ& %7$ ; & %$ L"" 7$ - %%-" 8πGρ A. The Einstein Euations Λ p 8πG Λ ρ& $ % σ'*πnρgd & % ' Λ& $%% D& D//%$ D';G G; & %$ 'J ;G ;KG;G 6

A. General elations B<';G;& ω'g; $& % ' ωg & $ 4'B< ;& %$ θ'b< 9G; & %< ω '*π;pj ωkg Q ω$ω %$ 7

8 Appendix B: The Friedmann model (Λ 5 $ $%" B. General elations ( ( ( ( r g (deg ( 57. g r L θ V sin ( ω ω π ω os ( ψ a sin ( a t ψ ψ os ψ os os ψ ψ ψ ψ ω / / ( ( os ( / sin a ψ ψ os ψ

dv d τ t dv dω dω d ψ sin ψ sin ( / ψ ψ / sin ψ ψ a(oshψ a t (sinhψ ψ B. ; < / oshψ oshψ oshψ ω ψ ψ / ( sinhψ a ( oshψ oshψ V ( ω π sinh ω ω dv dω π (osh ω d d sinhψ ψ τ sinh ψ ψ 9

t (sinhψ ψ / ( a( osψ a t ( ψ sinψ B. ; > / osψ osψ osψ ω ψ ψ / ( sinψ a ( osψ osψ v( ω π ω sin ω dv dω π ( os ω d d ψ sinψ τ ψ sinψ t ( ψ sinψ / ( t B.4 ; 4

ω ln( t r g 4 ( V ( π ln( 4 ( dv ( / 4π d ( τ t ω t t / ( / B.5 ; / t r ω g V ( π / ( dv d τ t {( } / 6π 5 / ( ( / / 4

4 B.6 ; aros π ω r g / ( ( arsin ( V π / ( ( 4 d dv π aros / π τ π t

Appendix C: Flat Models ( ; Λ > 5 $ $%" / A / sinh γt 8πGρ A Λ γ Λ Λ ( A A A A / ( / / t A ln{ A ( A ( Λ Λ ( { A( } A( r g τ / dζ ( A / { A( ζ } } / ln[{ A( } ] ln( / ln{ ( A } ln A ln A 4

Appendix D: A Seletion of Other Models 5 *$ $%" / D. Zero-Density Model with > sin( / t / ρ Λ < / sin( t / & ( / [os{ / ( t t } ω arosh ( arosh / / ot( t / tan ( t / / / {( ( } ( r g V ( ω π ( sinh ω ω / arsin / ( ( τ / arsin / / t artan sin{ / / ( t t }] 44

Λ > E Λ ρ Λ E 4πG D. The Einstein model Λ > ρ exp{ ( t t } ω Λ r g 4 V ( π D. The De Sitter model 45

Λ > x E ρ α x ρ E @ & D.4 The Lemaître model x ( x α x / & -SSα G / @ 74 5 $ %" x Λ ( x x Λ x α α x / Λ t / / t -α G / 4 5 $ %" x x exp{ ( t t } Λ 46

- α G #$ #& Λ x α / ( α / / sinh ( t t & / t Λ {ln( α / : $ /%%/ α " 47