The Metric and The Dynamics

Transcription

1 The Metric and The Dynamics r τ c t a () t + r + Sin φ ( kr ) The RW metric tell us where in a 3 dimension space is an event and at which proper time. The coordinates of the event do not change as a function of time but the coordinate system will change as a function of the proper time because of the expansion factor a(t). The geometry is determined by the value of k -,, +. Cosmology 4_5

2 Ricci s Tensor Einstein Equation Einstein 96 Ricci s scalar 8πG R g R T c Λ ij ij 4 ij ij Stress Tensor Cosmological Constant In this equation figure both the geometry and the Energy Energy momentum Tensor simplified in a Robertson Walker Metric Metric The Universe is homogeneous and isotropic. ( ρ ) T p+ c u u p g ij i j ij Cosmology 4_5

3 The Hubble law Separation between two observers δl a t δr () () d δl a () t δr δr dt δl a t () d a () t δ l δ l V H () t δ l dt a t Cosmology 4_5 3

4 The Models and The Cosmological Parameters Here we derive the observable as a function of different cosmological parameters. The goal is to derive a fairly complete set of equations useful for data reduction. See also the following set discussing the relation to geometry and some applications. Cosmology 4_5 4

5 Cosmology 4_5 5

6 Simple Preliminaries - Dynamics m VH r We consider a sphere exdpanding with a velocity v H r. vwe assume also that the energy and the mass are conserved ρ r E GMm mv const E > r E < Critical density Cosmology 4_5 6

7 Look at it as Energy First ρv 4πρH π EK dr r dr r ρ H R 5 R R 3 5 ( r) ρ R 4 ρ ( 4π) RM EG G dr G πr ρ dr ρ GR r 3 r 5 E E G K ( 4π ) 5 ρ GR 5 8πG π ρ 5 H R 3H 5 E G 8πG ET EK + EG EK + EK ρ EK 3H ρ EK EK ( Ω ) ρc The above defines ρ and Ω c Cosmology 4_5 7

8 A Newtonian Model m VH r We consider a sphere exdpanding with a velocity v H r. vwe assume also that the energy and the mass are conserved ρ r E GMm mv const E > r E < E 4 G πr ρ 4 M πr ρ H r 3 ρ H,c r 8πG Critical density Cosmology 4_5 8

9 r t t mv t GMm dr GM r dt r dr r GM dt t t ; r r t ( GM ) ( vr) ( Hr r) 3 3 t 3 t dr r also : H dt solution r r r r 3 3 r ρc 3 Hr t H h g cm ( GM ) Cosmology 4_5 9

10 E H 4πGρ H 4πGρ H r E m r m r Divide by 3 3 m H 4πGρ H r H r H 4πGρ r r H r 3 H r H r 3 H r ρ ρ ρ 3H ;,c Ω 8πG ρ,c 8πGρr Ω I conserve the mass and transformation 3H r r d r r dr Hr r dd 3 * * * D τ H 3 t Hr * r r dt Hr d( Ht) dτ ρ Cosmology 4_5

11 And I can write: * 3 8π G r r * ρ Ω 3 dd dτ 3H r r * dd Ω * * dτ D 3 * * * * 3 Ω Ω ξω ife Ω Ω Change var iables again Ω Ω Ω Ω ξ D τ τ dd dξ dτ dτ Ω Ω Ω dξ dτ Ω Ω Ω dξ + forω< dτ ξ forω > Cosmology 4_5

12 The Equation to be solved dξ ± dt τ d ± ξ ξ ξ τ ξ ξ ± ξ * * d ξ t D r τ τ ξ Cosmology 4_5

13 Ω > E< η sin ξ ξ η η η η η τ dξ ξ sin sin cos dη ξ cos η sin η η η η η η ( cosη sin cos d sin d ) η η η cos α cosα since sin η η τ ( η sinη) ( η sinη) ξ sin Cosmology 4_5 3

14 Ω < E> η sinh ξ ξ η η η η d sinh sinh cosh η τ ξ ξ dη + ξ cosh η sinh η η η η η η ( + coshη sinh cosh d sinh d ) η η η cosh α + coshα since sinh η τ ( η+ sinhη) ( sinhη η) ξ sinh η Cosmology 4_5 4

15 Obviously (as an example Ω>): * Ω Ω τ ( η sinη) τ Ω t Ω Ω H ( η sinη) η Ω * Ω r ξ sin D Ω Ω r r Ω η sin and r Ω H t rω rω Maximum radius for η π ; ξ rmax Ω Ω Ω Cosmology 4_5 5

16 Cosmology 4_5 6

17 From Einstein Equations: Friedmann πG and a + k c Gρ a and p p ( ρ) a a c 3 a 4π G 3p Combined : ρ + a 3 c a a + k c p 8π This is very similar to the equation we derived in the Newtonian Toy so that the solution has been already derived for k,, -. Here as you have seen the difference is the constant in the Equations. Furthermore I can write see for Ω >, <, : k a ρ 3 a with ρ and Ω,c a c a ρ,c 8πG a ρ,c k a a c a ( Ω ) Cosmology 4_5 7 ρ

18 Adiabatic Expansion The two Friedmann equations are not indipendent and are related by the adiabatic equation: de p dv 3 3 d ρ c a p da And in case I have Λ I keep the same equation using: 4 Λc Λc p p and ρ ρ+ 8π G 8π G Cosmology 4_5 8

19 ( ρ ) ( ρ ρ) Radiation and Matter 3 3 d c a p da ; assume p pressureless material dust gas or nonrelativistic material ideal gas kt ( ρ ) ρ p ρ mc w T mc with w T mpc 3 3 d c a a const ρ a ρ ρ 3 3 Relativistic gas and Radiation p ρ c ρ ρ a + adρ *a c da a d c da 4 a d a ( ρ ) ρa da a dρ d a con Cosmology 4_5 9 ( + ) ρ a ρ a if BB T(z) T z st

20 Tot R B DM 8π a 8πGρ a a + k c Gρ a for a a 3 a 3 a 8πGρ kc H or 3H a H 8πGρ H kc ρ ρ + ρ + ρ + ρ ( Ω) 3H a a a a ρ,c Ω,R + Ω,B + Ω, DM + ΩΛ a a a ρ R Λ 4 4 a a ρ ρ,cω,r a a Cosmology 4_5

21 H H, c And 8π a 8πGρ a k c a + k c Gρ a 3 a 3 a a 8πGρ k c 8πG R + B + DM + Λ kc * H 3 a 3 a 8πG ρ 3 H ( ρ ρ ρ ρ ) Ω Ω Ω Ω a a a a a a a kc,r +,B +,DM + Λ Ω Ω Ω a a a a a kc,r +,B +,DM + ΩΛ K ah a H H Ω,R + z + Ω,B + z + Ω,DM + z + ΩK + z + Ω Λ whre e Ω kc Λ c and ΩΛ from equation above * 3H Ω,R + Ω,B + Ω,DM + Ω, K + ΩΛ a forany t Cosmology 4_5 a H H a

22 Dust Universe to make it easy a 8πGρ a 8πGρ a 3 a 3H ( for a a & ρ ρ ) H H ( Ω ) 3 π ρ H π ρ a H ρ ρ,c a a 8 G a a 8 G a a a a 3 a a H 3 a a a a ρ a a a H H ( Ω) H ( ) a ρ,c a a Ω + Ω a r dr t c dt t c dt a dt a c da f ( r) t a t a kr () t a() t adt a() t a cda c a a a a Ω ( Ω) a () a + a a t a a H a a a Integrating both f r d r c Ha Ωz+ ( Ω ) + Ωz+ Ω + z Cosmology 4_5

23 Ω ( z) a a a H + * a a a ( Ω ) ( + ) Ω ( + ) + ( Ω ) 3H ( z) Ω ( Ω) a a H Ω a + a a a H z H z z Ω ( + ) ( + z) ( ) 3 3 ρ ρ + z 8πG ρ + z ρ 3 c H z z 8π G z Ω ( + z) Ω ( z) forz Ω + Ω + Ω z ( + ) Ω ( + ) + ( Ω ) Cosmology 4_5 3

24 ( + ) Ω ( + ) + ( Ω ) H z H z z da a a H ( + z) Ω z + ; ; da dz dt a a z z + + Ω a ( + z) dz Ω da a dt dz ah + z z+ + z H + z Ω z+ dt dz + z H + z z+ H + z Ω z+ dz t( z) ( for Ω ) H H z z 3 z z 3 ( + ) Ω + ( + ) Cosmology 4_5 4

25 Angular separation From Mag Notes: ds -r a (t ) -d r Ha d a a a t r z () ( + ) Ωz+ Ω Ωz c + + Ha z Ω ( + ) ( + z) d Ω ( + z) c a Ω z+ Ω + Ωz+ H c ( + ) d Ω z Ωz+ Ω + Ωz+ Cosmology 4_5 5

26 Cosmology 4_5 6

27 Integration, Mattig solution see transparencies The problem is to derive the solution of the Friedmann equations. In other words we want to have r or better a r as a function of observable as z, Ω, H We assume for now Λ. I basically must solve : r dr kr ( ϑ ϑ) and Sin to H Ω and we explicit this t t c dt a we will find r Sin ϑ ϑ where ϑ is related to H Ω z Cosmology 4_5 7

28 4π G p a ρ + 3 a a + k c Gρ a p 3 aa + a + kc 4πG ρ+ a c 3 c 8π 8π a 8πGρ a a + k c Gρ a for a a 3 a 3 a 8πGρ kc H or 3H a H 8πGρ H kc ( Ω) 3H a Cosmology 4_5 8

29 For k 8πGρ c H H ( Ω) furthermore since 3H a,c and a H ρ ρ Ω ( Ω ) 3 8π 8π ρa 8π 3H c a + c Gρ a G GΩ 3 3 a 3 8 π G a H ( Ω ) 3 c c Define α 3 3 ah ( Ω ) H ( Ω ) Ω α a a c c Ω 3 Cosmology 4_5 9

30 da α cdt da da c * dt a a a α a a a a a Change variable: aα Sin ( ) ( α ) da α sin d a a da a ( α a) ( ) α Sin α α Sin ( ) α sin d αsind αsind ( ) 4 α Sin α Sin α Sin Sin Cosmology 4_5 3

31 αsind αsind ( ) ( ) α Sin α Sin Sin Cos α sin d d α Sin Cosmology 4_5 3

32 r Now we look at the metric s equation dr kr t ( as we have just derived ) t for k c dt a r dr r ArcSin( r) r Sin( ) or ( ) ( ) ( ) ( ) ( ) + Sin Sin ArcSin r r Sin r Sin Cos Cos Sin z a a and Cosmology 4_5 3

33 Now we prepare some transformations Sin ( ) ( ) ( ) ( ) ( ) Sin Cos Sin Sin Sin Sin Sin Cos + z + z + ( + z) z ( + z) Sin Cos z + + z Cosmology 4_5 33

34 Cos ( ) Sin Cos Sin Sin + z ( ) ( ) ( ) os ( ) + z Sin Cos Sin + z C + z + z + z + z Cosmology 4_5 34

35 Now from slide 8 c c Ω H ; and I can write : ( Ω) α 3 a H ( Ω ) ( ( )) α Sin a ( by definition) α Cos ( ) ( Ω ) ( ) ( ) 3 c Ω Ω α Ω α H Ω Ω Cos Sin ( ) Cos ( ) Ω Ω Ω + Ω Ω Ω Ω Ω Ω Cosmology 4_5 35

36 Summary of relations ( ) ( ) ( ) Sin( ) r Sin Cos Cos Cos ( ) Sin Ω Ω ( ) ; Cos ( ) Ω Ω Sin ( ) ; Sin( ) Cos z z Cos + z z Cosmology 4_5 36

37 Additional relations Cos ( ) ( ) Ω Ω Cos Ω + Ω Ω Ω Cos Sin Ω Cos Cos Cos + Ω Sin Sin Ω Sin Cos Ω Ω Ω Ω Ω Cosmology 4_5 37

38 ( ) ( ) ( ) ( ) r Sin Cos Cos Sin Sin Ω Cos( ) + z Ω Cos + z z z Ω + Ω + Ω Ω + z Ω Ω Ω Ω + Ω + z Ω + z Ω Ω Ω + ( Ω ) ( Ω ) Ω + r z z Ω ( + z) Ω ( )( ) r Ω z + Ω Ω z + and Ω ( + z) c slide 8 H ( Ω ) a c r z ( )( z ) orra... H a Ω + Ω Ω + Ω + ( z) z Cosmology 4_5 38

39 Generalities As we have seen we can integrate for a dust Universe. We can derive a compact equation also when we account for a Matter + Radiation Universe. It is impossible to integrate if we account also for the cosmological constant or Vacuum Energy. In this case we must integrate numerically. On the other hand we must also remember, as we will see shortly that the present Universe is a Dust Universe so that what is needed to the observer are the equations we are able to integrate. Recent results, however, both on the Hubble relation using Supernovae and on the MWB, with Boomeramg and WMAP, have shown that the cosmological constant is dominant and Ω tot so that we should use such relations. Cosmology 4_5 39

40 Compare with toy Model: Ω m, Ω Λ & The real thing t Thebeginning z t dz.4 years H 3 H 3 ( + z) ( + z) Time in Yea r Ω m.3, Ω Λ.9 Ω m.3, Ω Λ.7 Ω m.3, Ω Λ. Ω m., Ω Λ. Ω m., Ω Λ z Cosmology 4_5 4

41 H H z The look back time ( ) Ω ( z) z z + Ω Λ Ω,R + + Ω,B + + Ω,DM + + K + a d d a t d dz H a a log log a aa dt dt a dt + z + z dt dt dz z H () H( + z) Ω,R( + z) + Ω,B( + z) + Ω,DM ( + z) + ΩK ( + z) + Ω Λ However Ωm ΩB + ΩDM and Ω,R + Ω, B + Ω, DM + Ω,K + ΩΛ I also disregard radiation ( compute the density ) Ω Ω Ω ( ΩΛ ) ( + ) + Ω ( + ) [ + ],m ( + z) + Ω,mz ( + z) [ + ],K m dt dz 3 H( + z) Ω,m( + z) + Ω,m ΩΛ ( + z) + Ω Λ dt accounting for the sign dz H z z z z z t t z H z zωλ Cosmology 4_5 4 Λ

42 Distance In a different way I have a standard source emitting a flash of light at the time t em. If the photons emitted are N em a telescope of Area A would receive, disregarding expansion and considering the physical distance is a(t em ) r, A / 4 π (a(t em ) r) photons. However the Universe is expanding and at the time of detection the area of the sphere where the photons arrive as expanded to 4 π (a(t obs ) r). That is N em /N detected A / (a(t obs ) r) We have to take into account of two more factors:. Photons are redshifted so that their energy changes by a factor za(t obs )/a(t em ). The time between flashes δt at the source increases by a factor (+z). This also corresponds to a loss of Energy per Unit Area of the telescope. The Luminosity distance can be defined as: L Flux 4π a t r z ( + ) Ω ( + ) L ( + ) Ω ( + ) Ω L L 4π d L dl a( t) r ( + z) for dust 4π Flux ar Ωz+ Ω Ωz c + + H z zsmall + Ωz + Ωz Ωz+ Ω z Ωz c c Ω z ar H z H z d c v z H H Cosmology 4_5 4

43 Distance L ( + ) d a r + z dr cdt z cdt see slide look back time kr a() t a r dr c dz a ( + z) cdt kr H Ω,R( + z) + Ω,B( + z) + Ω,DM ( + z) + ΩK ( + z) + Ω Λ r dr z c dz a kr H + + z z + z Mattig' s formula r r ( Ω [ ] ΩΛ ),m c Ha Ωz+ Ω + Ωz+ Ω + z Extension to Matter + Radiation Ωmz+ ( Ωm + ΩR ) + Ωmz+ ΩR ( z + z) c Ha Ωm + 4ΩR( ΩR + Ωm ) ( + z) Cosmology 4_5 43

44 And with Λ ao arcsin r k r dr a a r k kr a arcsinh( r) k Example k and flat Ω c( + z) z dz dl ar( + z) H ( [ ] ) + Ω,m z + z z + z Ω Λ tot Cosmology 4_5 44

45 And for Ω k < o a arcsin r z ( + Ω ( + ) [ + ] ΩΛ ),m z r sin ah z z z z cdz H z z z z c dz ( + Ω,m ( + ) [ + ] ΩΛ ) kc IuseΩk Ωm ΩΛ ; a H c( + z) z dz dl ar( + z) sin Ωk H Ωk ( + Ω,m z ( + z) z[ + z] ΩΛ ) or Cosmology 4_5 45

46 And for Ω k > - Open o a arcsin h r z ( + Ω ( + ) [ + ] ΩΛ ),m z r sinh ah z z z cdz H z z z z c dz ( + Ω,m ( + ) [ + z] ΩΛ ) kc IuseΩk Ωm Ω a H c( + z) z dz dl ar( + z) sinh Ωk H Ωk ( + Ωm, z ( + z) z[ + z] ΩΛ ) Λ ; or Cosmology 4_5 46

47 Luminosity Distanc e Green Ω Λ., Ω m. Light blue Ω Λ., Ω m.3 Blue Ω Λ., Ω m. Red Ω Λ.3, Ω m.7 Dark Red Ω Λ., Ω m.9 Dark Blue Ω Λ., Ω m z Cosmology 4_5 47

48 Magnitudes m z M + 5logd z,h, Ω, Ω + 5 Bol L m Λ Cosmology 4_5 48

49 Volume - Proper dv r τ c t a () t + r + Sin φ ( kr ) ( π) Pr oper Volume : dω or 4 a r dl ; ( z) ( z) ( + ) em em dl line of sight dis tance travelled by photons in dz ( + z) dt da a a a a dl c dt c ( dz) c c da dz a + a a + a cdt c dz H z emrem c ar cdl Ω Ω 3 5 c a dω dz d dz dv d dz H + z H + z H + z Cosmology 4_5 49

50 ( + z ) Volume - Comoving dv d Ω ( or 4 π ) a r a dr cda cda a cda ( + ) a a H a a H a dr c dt ; adr a dr z + z cda cda a dz c ( + z) sinceda dz a H Ha ( + z) H Assume matter + Λ dv dω d L ( see earlier slide) cdz ( + ) + Ω ( + ) [ + ] z H,m z z z z ΩΛ Cosmology 4_5 5

51 Comoving - Dust Universe dl cdz dv dω H + z Ω z+ ( + ) ( + z) L Ω 3 Ω z+ Ω z c + Ω + d + z H z ( + ) z z cdz Ω + Ω + Ω + c dv dω ( + z) ( + z) H( + z) Ω H z+ Ω ( + z) cdz dv dω H z Ωz+ Ω + Ωz+ z+ 3 4 Ω Ω Cosmology 4_5 5

52 Ω m Ω Λ Volume pe r uni t dω an d in M p c z,5,9 3, 7, Cosmology 4_5 5