Seminr The Erly Universe Big Bng Cosmology: Einsein Universe Friemnn-Lemîre Universe Einsein-eSier Universe by Oliver Schmi
Ouline The observe universe Meric of he universe Curvure Einsein Equion Cosmologicl moels Einsein Friemnn-Lemîre Einsein-eSier Oulook
The observe universe glies srs per gly M per gly 4 Gpc o he ege of he visible universe
The observe universe glies srs per gly M per gly 4 Gpc o he ege of he visible universe Composiion. Mer. Riion 4. Drk mer ( ) ~ 5. Vcuum energy of unknown ensiy visible riion g cm 4 g ( ) ~ cm
The observe universe glies srs per gly M per gly 4 Gpc o he ege of he visible universe Composiion. Mer. Riion 4. Drk mer ( ) ~ 5. Vcuum energy of unknown ensiy visible riion g cm 4 g ( ) ~ cm The universe is epning! v H
The observe universe The universe is isoropic n homogeneous verge over lrge scles!
The Cosmologicl Principle The hypersurfces wih consn cosmic snr ime re mimlly symmeric subspces of he whole of spce-ime. No only he meric g μν, bu ll cosmic ensors such s T μν, re form-invrin wih respec o isomeries of hese subspces. Form invrince of g µν uner rnsformion: ( ) g ( ) g ( ) µ ν µ ν Killing equion: D ε D ν µ µ εν ε µ µ ( ) µ Miml number of symmeries Miml number of Killing Vecors ε ( )
Meric Minkowski Meric: (~95) s c r r Θ r sin Θ Φ Specil Reliviy
Meric Minkowski Meric: (~95) s c r r Θ r sin Θ Φ Specil Reliviy GM GM c Schwrzschil Meric: (96) s r r Θ r sin ΘΦ c r c r Meric ousie non-roing sr
Meric Minkowski Meric: (~95) s c r r Θ r sin Θ Φ Specil Reliviy GM GM c Schwrzschil Meric: (96) s r r Θ r sin ΘΦ c r c r Meric ousie non-roing sr r Roberson-Wlker Meric: s c ( ) r ( Θ sin ΘΦ ) (95/6) kr Meric following he cosmologicl principle
Roberson-Wlker Meric ( ) ( ) ( ) ( ) Θ sin r r kr g β ( ) ( ) Φ Θ Θ sin r kr r c s
Roberson-Wlker Meric ( ) ( ) Φ Θ Θ sin r kr r c s ( ) ( ) ( ) ( ) Θ sin r r kr g β k k k Fl universe Open universe Close universe
Roberson-Wlker Meric Proper isnce: prop r ( ) ( ) ( ) r kr sin r sinh ( r ) ( r ) ( ) coor k k k Cosmologicl Re Shif: Two ligh pulses emie e n e δ e, observe n δ wih consn coorine isn coor. e r ( ) r kr δ τ ( ) ( ) δ λ z r kr ( ) δ δ e e e λ λ λ ( ) ( ) λ e ( e ) ( ) δ δ e e r ( ) r kr
Curve spceimes In Generl Reliviy grviion is no force bu propery of spceime geomery.
Curve spceimes In Generl Reliviy grviion is no force bu propery of spceime geomery. Fl spceime Geoesic equion Curve spceime τ τ Γ βγ τ β γ τ Chrisoffel symbols: Γ βγ g δ g β γ g γ β g βγ
Curve spceimes ( ) Γ τ τ τ γ β βγ ( ) τ δ ( ) ( ) ( ) ( ) Γ τ δ τ δ δ τ δ γ γ β β βγ
Curve spceimes ˆ ˆ ˆ ˆˆ ˆ β τβτ δ τ δ R Geoesic eviion: ( ) Γ τ τ τ γ β βγ ( ) τ δ ( ) ( ) ( ) ( ) Γ τ δ τ δ δ τ δ γ γ β β βγ
Curve spceimes τ Γ βγ τ β γ τ ( ) β β γ γ ( δ ) ( ) ( δ ) ( δ ) τ Γ βγ δ τ τ δ( τ ) Geoesic eviion: δ τ ˆ R ˆ τβτ ˆˆ ˆ δ βˆ Riemnn curvure: Ricci curvure: R Γ Γ β δ β γ ε ε β γ δ Γ γ δ γ εγ β δ Γδ εγ β γ γ R β R γ
Source of curvure Energy-momenum-sress ensor: T energy ensiy β momen um ensiy energy sress ensor flu T β is symmeric!
Source of curvure Energy-momenum-sress ensor: T energy ensiy β momen um ensiy energy sress ensor flu T β is symmeric! Energy-momenum-sress ensor of perfec flui: (he conucion, viscosiy, ec. re negligible) T β p p p
Einsein Equion Einsein curvure ensor: Ricci curvure sclr: 8πG T c G β 4 G β R β R R g g β β β R R β
Einsein Equion Einsein curvure ensor: Ricci curvure sclr: 8πG T c G β 4 G β R β R R g g β β β R R β Solving he Einsein equion for homogeneous isoropic cosmologicl moel of cosmologicl perfec flui yiels o G [ k ] 8π G rr GΘ GΦ ( k ) 8πp
Einsein Equion Einsein curvure ensor: Ricci curvure sclr: 8πG T c G β 4 G β R β R R g g β β β R R β Solving he Einsein equion for homogeneous isoropic cosmologicl moel of cosmologicl perfec flui yiels o G [ k ] 8π G rr GΘ GΦ ( k ) 8πp Snr moel Friemnn equion: k 8π ( p) Equion of se: p p( )
Equion of se p p( ) Gs of pricles of mss m in herml equilibrium wih T<<m: Mer componen wih negligible pressure: us p
Equion of se p p( ) Gs of pricles of mss m in herml equilibrium wih T<<m: Mer componen wih negligible pressure: us p Gs of pricles of mss m in herml equilibrium wih T>>m: Highly relivisic mer componen: riion p
Equion of se p p( ) Gs of pricles of mss m in herml equilibrium wih T<<m: Mer componen wih negligible pressure: us p Gs of pricles of mss m in herml equilibrium wih T>>m: Highly relivisic mer componen: riion Furher possibiliies: p n p p ν p
Einsein moel (97) Sic universe k π ( 8 Λ) 8πG Tβ gβ c Gβ 4 4π Λ Λ p 4π Λ: Cosmologicl consn Λ>: Repulsive force
Einsein moel (97) Sic universe k π ( 8 Λ) 8πG Tβ gβ c Gβ 4 4π Λ Λ p 4π Λ: Cosmologicl consn Λ>: Repulsive force > 4πG Λ > p k ΛE E c c 4πG Close universe wih rius E
Friemnn-Lemîre moel Epning universe: 8πG Tβ gβ c Gβ 4 Λ wih Λ > Λ E
Friemnn-Lemîre moel Epning universe: 8πG Tβ gβ c Gβ 4 Λ wih Λ > Λ E ( ) k k k ( ) ( ) ( ( ) ) cosh Λ 4 Λ π ( ) ( ) for smll () Λ e ( ) for lrge () () () ()
Snr moel ( ) p π 8 k ν p ( ) ( ) ( ) ( ) ν ( ) ( ) ν ( ) ( ) ν π 8 k
Einsein-eSier moel Assumpions: Spilly fl (k) n us omine (ν) universe ( ) ( ) ( ) ( ) ( ) 8 π ( )
Einsein-eSier moel Assumpions: Spilly fl (k) n us omine (ν) universe 8π ( ) ( ) ( ) ( ) ( ) ( ) Assuming no chnge in he equion of se, one fins ime i wih ( i ) BIG BANG ( ) ( ) ( ) 6π
Einsein-eSier moel Assumpions: Spilly fl (k) n us omine (ν) universe 8π ( ) ( ) ( ) ( ) ( ) ( ) Assuming no chnge in he equion of se, one fins ime i wih ( i ) BIG BANG ( ) ( ) ( ) 6π Grviionlly boun universe m ˆ 8π ( ˆ ) ( ) ( ) 4π k k ( ) ( ) ( ) ( ) ( ) ( ) c ( smll) ( lrge) wih c >
Snr moel k Einsein-eSier k- k 4,5 4,5,5 ()/ m,5,5,5,5,5 / m
Which universe n why? cri H 8πG H 8 π G k,9 * 9 g cm > cri cri < cri k k k Close universe Fl universe Open universe
Which universe n why? cri H 8πG H 8 π G k,9 * 9 g cm > cri cri < cri k k k Close universe Fl universe Open universe Observions: H km 7 s Mpc, m cri 5 r 8* cri, 7 v cri
Which universe n why? cri H 8πG H 8 π G k,9 * 9 g cm > cri cri < cri k k k Close universe Fl universe Open universe Observions: H km 7 s Mpc, m cri 5 r 8* cri, 7 v cri
References Jmes B. Hrle, Grviy An inroucion o Einsein s Generl Reliviy Seven Weinberg, Grviion n Cosmology Eckhr Rebhn, Theoreische Physik D.W. Scimi, Moern Cosmology n he Drk Mer Problem E.R. Hrrison, Kosmologie Die Wissenschf vom Universum P.J.E Peebles, Physicl Cosmology Chrles W. Misner, Grviion
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