cosmologia - lnf.infn.it · cosmologia relativistica. costante cosmologica: l “cosmological...
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COSMOLOGIAIn che Universo viviamo?
Nicola VittorioDip.Fisica, Universita’ di “Tor Vergata”[email protected]/4498
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†
V = H0r
†
H0 = 500 km/sMpc
Hubble original estimate
La legge di Hubble
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r
a
p
La distanza r corrispondente ap=1’’ e’ stata assunta comeunita’ di misura delle distanzestellari
Poiche’ a = 1.5 1013 cm
1 parsec=3 1018 cm
1 1 Mpc Mpc =3 10=3 102424 cm cm
Il Megaparsec
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…to measure Ho to10% accuracy….….
†
H0 = 70 ± 7( ) km/sMpc
HubbleConstant
W. Freedman et al. (1999)
The Hubble KeyProject
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On the curvature of spaceA.Friedmann, 1922
“The purpose of this note is ...to demonstrate the possibilityof a world in which thecurvature of space isindependent of the threespatial coordinates, but doesdepend on time”
Aleksandr Aleksandrovich Friedman: 1888- 1925
Un universo inespansione
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Modelli cosmologici
ct
y
x
Coordinate gaussiane
Gauge sincrona
Coordinate comobili
†
g00 =1;g0k = 0;
†
ds2 = dx 02 - R2(t) dr2
1- kr2 + r2dW2È
Î Í
˘
˚ ˙ ; k =
+10-1
Ï
Ì Ô
Ó Ô
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Modelli Cosmologici
ct
y
x
ct
y
x
Teorema di BirkhoffTeorema di Birkhoff Analogia newtonianaAnalogia newtoniana
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†
˙ ̇ R = -GMR2 ¤ 1
2˙ R 2 -
GMR
= e
M =43
pR3
˙ ̇ R R
= -43
pGr ¤ ˙ R R
Ê
Ë Á
ˆ
¯ ˜
2
-2eR2 =
83
pGr
˙ ̇ R R
= -43
pGr ¤ ˙ R R
Ê
Ë Á
ˆ
¯ ˜
2
+kc 2
R2 =83
pGr
Modelli cosmologici
MR
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MR
Modelli cosmologici
M
†
UT 0
=GM /R
˙ R 2 /2 0=
8pGr3
R˙ R
Ê
Ë Á
ˆ
¯ ˜
2
0
=8pGr0
3H02 =
r0
rcrit
= W0
rcrit = 2 ¥10-29 h2 gcm3 ; h = H0
100 km s-1
Mpc
>0<1-1
=0=10
<0>1+1
EEWW00kk
†
-2e = kc 2
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t
t
Wo< 1< 1
Wo=1=1
Wo> 1> 1
k= -1k= -1
k= 0k= 0
k= +1k= +1
a(t)
a(t)
a(t)
t
†
a(t) =tto
Ê
Ë Á
ˆ
¯ ˜
2 3
†
a(t) =Wo /21- Wo
1- cosh[ ]
Hot =Wo /2
1-Wo( )3 2 h - sinh[ ]
†
a(t) =Wo /21- Wo
coshh -1[ ]
Hot =Wo /2
1-Wo( )3 2 sinhh -h[ ]
Cosmologia relativistica
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Costante cosmologica: L
“Cosmological considerationson the general theory of
relativityA.Einstein,1917
“In order to arrive at thisconsistent view, we admittedlyhad to introduce an extensionof the field equations ofgravitation which is notjustified” by
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Costante cosmologica: L
†
˜ r = r +c 2L8pG
; ˜ p = p -c 2L8pG
;
u Modifica delle equazioni di campo dellarelativita’ generale
u E’ come avere un fluido con
†
˙ R R
Ê
Ë Á
ˆ
¯ ˜
2
+kc 2
R2 =83
pGr +13
Lc 2 ¤ ˙ R R
Ê
Ë Á
ˆ
¯ ˜
2
=c 2
3L - Q R( )[ ]
L ≥ Q(R) ≡3kR2 -
8pGc 2 r +
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Costante Cosmologica: k=+1Costante Cosmologica: k=+1
0
Q(R)Q(R)
RRRREE
L=0
L3
L2
L1
L4
L5
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†
Consideriamo un universo di polvere, p = 0. Abbiamo visto che
Q R( ) =3
R2 -8pGc 2 r
ha un massimo in corrispondenza di1
RE2 =
4pGc 2 r0
Scegliamo
LE = Q RE( ) =3
RE2 -
8pGc 2 r0 =
1RE
2
Otteniamo la soluzione statica di Einstein˙ R R
Ê
Ë Á
ˆ
¯ ˜
E
2
=c 2
3LE - Q RE( )[ ] = 0;
˙ ̇ R R
Ê
Ë Á
ˆ
¯ ˜
E
2
= -4pG
3˜ r 0 = -
4pG3
r0 +13
LEc 2 = 0
L & k=+1
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†
Assumiamo come ordine di grandezza r0 ª10-30 gcm-3; allora
RE2 =
c 2
4pGr0
ª 3 ¥1028cm LE =1
RE2 @10-57cm-2
La massa dell'universo di Einstein e'M = 2p 2r0RE
3 ª 5 ¥1056 gLa densita' media del sistema solare (SS) e'
rSS ª3MSole
4pRPlutone3 =
34p
2 ⋅1033 g40 ¥1.5 ⋅1013cm( )
ª 2.2 ⋅10-12 g /cm3
L'influenza di L sulla dinamica del sistema solare e' trascurabile :r0
rss
ª 5 ⋅10-19
L & k=+1
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de Sitter model
k=0, r=0
†
˙ R R
Ê
Ë Á
ˆ
¯ ˜
2
=13
Lc 2 fi R µexp L3
ctÈ
Î Í
˘
˚ ˙
Costante Cosmologica: k=0,-1Costante Cosmologica: k=0,-1
0
Q(R)Q(R)
RR
L2
L=0
L1
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Equazione diFriedman
†
˙ R R
Ê
Ë Á
ˆ
¯ ˜
2
+kc 2
R2 =8pG
3r0
R3 +13
Lc 2
Tassodiespansione
Fase cinematica:domina la curvatura
Fase decelerata:domina la materia
Fase accelerata:domina la costantecosmologica
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time
time
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Problema dellapiattezza
†
Per definizione
W t( ) ≡8pG
3H 2(t)r t( )
Dall'equazione di Friedman, con L = 0 e a(t) = R(t)/R(t0), si ottiene
H 2(t) ≡˙ R R
Ê
Ë Á
ˆ
¯ ˜
2
= H02 W0
a3(t)+
1- W0
a2(t)È
Î Í ˘
˚ ˙
si ricava
1W t( )
-1=1
W0
-1Ê
Ë Á
ˆ
¯ ˜ ¥ a(t)
Quindi in un universo di Friedman limt Æ0W t( ) =1
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Problema dellapiattezza
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t
W(t)
1
Espansionedecelerata
†
W(t) =1-E
T (t)
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t
W(t)
1
Espansioneaccelerata
Espansionedecelerata
†
W(t) =1-E
T (t)