The Spatial Geometry of the Universe and the Cosmic Microwave Background

Transcription

1 The Dk Side The Sptil Geomety of the Univese nd the Cosmic Micowve Bckgound Decoupling The Cosmic Micowve Bckgound Spectum Tempetue fluctutions Angul spectum nd cosmologicl pmetes Gowth of density fluctutions: fist instllment Newtonin Wht hppens beyond hoizon? Coheence of initil conditions Phys CMBR 1

2 The Dk Side The poton-hydogen equilibium Ely univese: hot plsm In pticul most of the potons nd electons e fee: p + e H + Hydogen dissocited by photons Becuse of fee electons, the men fee pth of photons e vey smll Wht hppens when we decese the tempetue? We hve to see how the equilibium evolves: clely pushed to the ight! Sh eqution The numbe density of species i cn be obtined by integting the occuption numbe ove the vilble sttes. In the non eltivistic cse nd Dd kt = 3 m o n i = g i kt 2 m i exp i c 2 µ i 2πh 2 kt n i = f Dd with f = exp m ic 2 + m i 4 g i p 2 dp = g i 2m h 3 4 p 2 h d in equilibium H = p + e n H = g m H p m e kt 3 n g p g p n e 2 exp ( m p + m e m H )c 2 g e m H 2πh 2 kt p = g e = 2 g H = 4 With binding enegy B = ( m p + m e m H )c 2 =13.6eV nd setting m p = m H in pefcto Phys CMBR 2 n H = g H n p n e g p g e m e kt 2πh exp B kt Lw of mss ction!

3 The Dk Side Decoupling intoducing we get Ioniztion fction X e = n p n byon Byon to photon tio = n byon n 3 3 = constnt = Ω b h 2 100% X e 1-X e 2 = X e π + 3 T 2 B exp kt m e Ω b h 2 = 0.02 Big Bng Decoupling θ cusl Decoupling Vey sudden chnge in ioniztion fction. The univese suddenly becomes tnspent! We e looking bck t the sufce of lst sctteing You e hee! Phys CMBR 3

4 The Dk Side Spectum Essentilly pefect blck body Phys CMBR 4 Expected: blck body spectum is edshifted to blck body spectum Possible distotions - lte enegy elese: non initil blck body spectum - ionized gs => Sunyev Zel dovich effect: eheting (moe lte) Limit on ioniztion stte of the univese ( y pmete )

5 The Dk Side How constnt in tempetue? Isotopy? Remkbly smooth Mps Dipole motion: 600km/s Anisotopies t the 10-4 level Phys CMBR 5 Sttisticl popeties Decomposition in spheicl hmonics: ngul powe spectum T T (, ) = The lm s e ndom vible, of zeo men nd vince Typicl ngul scle l m=+l lm m= l Y lm, c l =< lm 2 > (independent of m fom isotopy) 1 l

6 The Dk Side COBE Anisotopy Mesuement Full scle T/T =1 Subtct off glxy S/N 3 Full scle T/T =10-2 Full scle T/T =10-4 Phys CMBR 6

7 The Dk Side Mesuements Sitution till lst ye Apil-My 2000 Scott nd Hlpen Asto-ph Jffe et l. Asto-ph Phys CMBR 7

8 The Dk Side Angul Powe Spectum nd Cosmologicl Pmetes Stndd Infltiony+Cold Dk Mtte model Jffe et l. Asto-ph Phys CMBR 8

9 The Dk Side k Bsic Pictue The plsm oscilltions impint physicl scle on the lst sctteing sufce Position in l 1/θ depends on geomety k = Plsm oscilltions Smll initil density fluctutions tend to oscillte once they come within hoizon Phys CMBR 9 Lst sctteing sufce exp[ ik. ]d 3 we will show tht = Fo k lge enough the solution is oscillting c s k t Acoustic peks ise fom initil conditions (dibtic) Cosine tems 1 ( 2π) 3 k + 2 k k + c 2 k 2 s 4πG 2 b = 0 k ± exp ( ±i t) with = ( 1 n) if ( t) tn exp[ ik. ]d 3 k k 1 k 2 k 3 Hoizon Hoizon Hoizon Decoupling Non byonic dk mtte Fist mximum t t Non byonic dk mtte Fist minimum Non byonic dk mtte Second mximu t

10 The Dk Side Phys CMBR 10 Eule equtions (Newtonin mechnics) See e.g. Peebles Pinciples of Physicl Cosmology p Kolb nd Tune The ely univese p Assumption: Mtte dominted Fluid descibed by mss density x,t u x,t Conside volume element V moving with the fluid Consevtion of mss V Consevtion of momentum Vition of volume V det j i + i u j dt to fist ode = d V = u dt V Vition of mss within V /unit time = t + u V + d V = dt t + u Vlid whteve V nd velocity x,t t V x i x i + u i dt + Newton eqution fo volume V Pessue foce = - p n d S = - pdv d ( Vu ) = V V p ( dt but d ( V ) = 0 d u u dt dt = u. t + V + ( x,t) u x,t u V = 0 = 0 = gvittionl potentil) u = p

11 The Dk Side Gowth of Density Fluctutions We should dd Poisson eqution 2 = 4πG Eqution of stte p = p (, S) Phys CMBR 11 Step 1 Subtct expnsion of the univese x = ( t) u = d x dt = t + t Note : f ( x,t) t ( x,t) f,t t = v peculi velocity df = f dt + dx. t dx = dt + d = f (,t) f x x,t t f (,t) x f ( x,t) = 1 f f ( ( x,t),t), ( t ) t Using. = 3 nd ( v. ) = v we esily obtin ( x,t ) + x ( u ) = (,t ) ( v ) = 0 t t u t + ( u. x ) u = v t + ( v. ) v + + v = 1 1 x 2 = = 4πG p = p dt + d. cf. Peebles 5.98 = f,t t p t t. f x,t

12 The Dk Side Gowth of Density Fluctutions 2 Step 2: Look t devition fom homogeneity (,t) Intoducing the density contst (,t ) b ( t) = Mtte dominted b t (,t) = b ( t) ( 1 + ) ( b = vege bckgound density) = b + ( = potentil coesponding to ). But b (,t) + 3 = 0 = 1 t b b (put v = 0 in bove equtions) v t + t v. v + v ) = 0 v = 1 1 p Phys CMBR 12 To which we should dd Poisson eqution 2 = 4πG Eqution of stte p = p,s

13 The Dk Side Eqution of stte Phys CMBR 13 Let us look in moe detil t eqution of stte cf E. Betschinge Cosmologicl Dynmics: Asto-ph/ Exmple: Monotomic idel gs Adibtic Genel pv = Nk B T p = p (, S) Fo unit mss element V =1/ N = 1 ( = men molecul mss) Enegy U = 3 2 NkT = p Using p = TdS = du + pdv = d 3 p + pd 1 2 p k = B T ds = 5 d 3 dp + k B 2 2 p 5/3 exp 2 S 3 k B c 2 s = p (, S ) = 5 3 dp = c 2 s d + p S ds = c 2 s d + 2 TdS fo monotomic gs 3 to which we should dd T ds dt = Γ { Λ{ (pe unit mss) heting cooling p = 5/ 3 exp 2 S 3 k B p fo monotomic gs

14 The Dk Side Gowth of Density Fluctutions 3 Lineize t + 1 (( 1 + ) v ) = 0 v t + ( v. ) v + v = p = 4πG b = potentil due to density fluctution p = p (, S) 1 p c 2 s + 1 b p S if S is the smll non unifom entopy S Dopping the second ode tems (cossed out bove) t + 1 v t v = 0 v = 1 = 4πG b 1 c 2 s 1 1 b p S S Phys CMBR 14

15 The Dk Side Gowth of Density Fluctutions 4 Fouie tnsfom Tking the Fouie tnsfoms of we obtin k = exp[ ik. ]d 3 k i k. v = 0 v + v i k 2 k + c s k + 1 b = ( 2π) 3 k k 2 = 4πG 2 k b k Multiplying the second eqution by ik, nd using the othe two equtions we get 1 p S S k = 0 exp i [ k. ]d 3 k + 2 k = 4πG k ( b 2 c 2 s k 2 k ) 2 k 2 b p S S k 2 S k = 1 T k ( Γ k Λ k ) Phys CMBR 15 Velocities Only the cul fee component is coupled to! Cul fee= longitudinl, compession mode: edistibution of mtte v //,k = ( t) k ik 2 Pue cul= tnsvesl, ottionl mode which is dmped Solution of v + v = 0 v v, k = k B k ( t) with k B v k = 0, 0 k

16 The Dk Side Phys CMBR 16 Gowth of Density Fluctutions 5 cf E. Betschinge Cosmologicl Dynmics: Asto-ph/ Adibtic fluctutions No entopy fluctution in the fluid S = 0 Should be clled isentopic cuvtue fluctutions! k k 2 k = 4πG b c s = ( k 2 2 k J k 2 2 k )c s Jens nlysis: Fo k J <k, oscilltions: velocity of sound too lge compe to wve length Isocuvtue fluctutions An othogonl mode is mode whee initilly = 0 = 0 k k e.g. No cuvtue! k, = k,b + k,c The initil fluctutions e in S k! + 2 k = 4πG k b 2 c 2 s k 2 k k 2 p S k 2 b S 2 Fluid develops density fluctutions (fist with oscilltions) Initilly = outside the hoizon whee such mode does not gow When within hoizon density fluctutions e geneted by pessue 2 with Jens' wve numbe k J = 4πG b 2 = 2 k c s 2

17 The Dk Side Limits of the tetment so f Newtonin: cn only pply to mtte dominted We cnnot tet dition dominted moeove wht e the initil conditions: development outside the hoizon gvittionl edshift is lso bsent: Schs-Wolfe effect Single idel fluid At lest 4 diffeent components Photons Byons Cold Dk Mtte Neutinos (cn be mssless o hve smll mss) Assumption of pefect fluid beks down Collisionless (Lndu) dmping Neutinos fee stem out of petubtions Collisionl dmping (Silk dmping) Men fee pth of photons incese t ecombintion Phys CMBR 17 Replce Eule s eqution by Boltzmnn eqution Density in phse spce f x, p,t x, with = f ( p,t) f t + u f + dp x dt f = f p t d 3 p Collisions u = u f x, p,t d 3 p

18 The Dk Side Exmple of full eltivistic + multi-component tetment Cold dk mtte Byons Photons+byons Mssive neutinos Photons Neutinos Mssless neutinos Phys CMBR 18

19 The Dk Side Reltivistic Tetment Phys CMBR 19 Pogm Bsic equtions Geomety Metics E. Betschinge Cosmologicl Dynmics: Asto-ph/ Compute cuvtue, Ricci nd Einstein tensosrmn Behvio of components C.P. M, E. Betschinge Astophys.J. 455 (1995) 7-25 / sto-ph/ Boltzmnn Constuction of enegy-momentum tenso Relte by Einstein equtions /Enegy-momentum consevtion Petubtion Distinguish between evolution of homogeneous/isotopic bckgound nd petubtions: lineise Fouie tnsfom-> evolution of ech k mode Solve coupled diffeentil equtions Compute micowve bckgound tempetue fluctutions+poliztion Poject on sky Anlyze in tems of physicl phenomen W. Hu, N. Sugiym nd J. Silk, Ntue 386 (1997) 37

20 The Dk Side Metics Confoml time It is convenient to put in fcto the expnsion fcto e.g. fo the Robetson-Wlke metic d 2 = dt 2 2 t c 2 Petubed metic We choose to wite the petubed metic d 2 = 2 Phys CMBR 20 dx i dx j ij = 2 with confoml "time" = c 2 t dt 0 c 2 d 2 ij dx i dx j { } ( )d 2 2w c 2 i d dx i [( 1 2 ) ij + 2h i, j ]dx i dx j i with h i = 0 nd the Poisson guge conditions: E. Betschinge : Asto-ph/ i w i = 0 i h ij = 0 i.e. tnvese Theoem: Unique, e.g. we hve tken ce of ll guge degees of feedom 2 scls + (3-1)= 2 components of tnvese vecto + (6-1-3)=2 of tnvese tceless tenso = 10 independent g µv - 4 guge degees of feedom t

21 The Dk Side Metics 2 Intepettion h i, j = gvittionl wve mode w i = ottionl fme dgging do not couple with density fluctutions Ignoe them. We then hve the longitudinl/newtonin guge Anothe misnome! We hve suppessed modes ( )d 2 [( 1 2 ) c 2 ij ]dx i dx j = Newtonin gvittionl potentil (e.g. souce of gvittionl edshift) (,x )will genete cuvtue fluctutions Connects smoothly with wek field limit d 2 = 2 d 2 = c 2 { } dt c 2 c 2 We will see tht t lte times d x 2 Phys CMBR 21

22 The Dk Side R 4 Guges Guge tnsfomtion Conside the infinitesiml tnsfomtion g ( x) g' ( x' ) T ( x) T' ( x' ) x x' = x x Sme coodinte vlues coespond to diffeent points. Wht is the eltionship of tensos t sme coodinte vlues? g' ( x) g' x' g + g ( x) x ( x) + ; + ; g T ( x) + T ; + T ; + T ; ( x) + g ( x) x + g x T' x In this context: we hve the choice of the point to ssocite in the petubed mnifold we wnt to ssocite with given point on the unpetubed mnifold x + g ( x) x R 4 Phys CMBR 22 Souce of lot of confusion:c.p. M, E. Betschinge sto-ph/ Tditionlly use of the synchonous guge (Weinbeg, Peebles, Kolb&Tune) d 2 = 2 { } d 2 [ c 2 ij + h ij ]dx i dx j Poblem: this does not use up ll degees of feedom (only 2) Recognized s ely s Lndu (1947) Stndd tetment: in ddition to physicl modes, guge modes Effots to constuct guge invint scls (Bdeen 1980, Kodm Sski 1984, Seljk 1994,Hu& Sugiym 1995) = Φ A = Ψ = Φ H = Φ

23 The Dk Side Enegy Momentum Tenso Idel fluid Phys CMBR 23 Moe genel T = + p T = + p c 2 whee we hve edefined the density nd pessue. In comoving fme Σ 0 j = 0, Σ i j = nisotopic stess In ou context mostly poduced by neutinos (photons e locked with byons which pevents them to diffuse too f except close to ecombintion) As it is the only one to couple to scl petubtion, we e only inteested in the scl component of Σ i j obtined by the tceless double divegence j i 1 j j 3 Σ i j Constuction of T µ v c 2 u u p c g 2 u u p c g + Σ with Σ = 0,Σ i i = 0,Σ u v = 0 T fom Boltzmnn phse spce densities ( x) = d 3 q q q f x i,q g q 0 j, whee the q s e moment. Wtch out fo thei definition s conjugte vibles usge C.P. M, E. Betschinge Astophys.J. 455 (1995) 7-25

24 The Dk Side Phys CMBR 24 Einstein s equtions (flt spce) It is only mtte of lbo to wite down the Einstein s equtions nd fo flt bckgound spce decompose them between Unpetubed pt T 0 0 = T 0 i = T i 0 = 0 T i j = p i j 2 = 4 G 2 3c +3 p c 2 Scl pt of petubtion (Fouie nlyzed) F( x) = d 3 k exp( ik j x j ) C.P. M, E. Betschinge Astophys.J. 455 (1995) 7-25 k = 4 G2 T 0 0 = k 2 + = 4 G2 + p T 0 c 2 ik j i = T 0 i = + p c 2 v j T i + ( + 2 ) j = p i j + Σ i j 2 3 k 2 ( ) = 12 G 2 + p c 2 We hve isolted the scl pt of Σ i j = - k i k j k2 j i 3 Σi j = k k2 v j ( ) = 4 G2 c 2 = 8 G 2 p c 2 Π = 8πG 3c 2 2 Fiedmnn eqution ( = d d ) F k (note chnge of sign with befoe) + p c 2 = 2 3 k2 p Π p

25 The Dk Side Cold dk mtte Components 1 C.P. M, E. Betschinge Astophys.J. 455 (1995) 7-25 Pessueless pefect fluid (cold) Phys CMBR 25 Intoducing c = = ik c j vj + 3 = c c = c k2 Neutinos (Mssless) c c c = ik j v j We hve to use Boltzmnn eqution becuse of fee steming Clling the density in phse spce f ( k, q, ) we cn integte out F k (, q ˆ, ) = q2 dq q f ( k, q, ) q 2 dq q f k, q = ( i) l ( 2l +1)F l(, k, ) P l k. q ˆ l =0 F 0 = v, c =ik j v j = 3 4 kf v1 = 1 2 F v2 => 4 = = k k2 F k l = 2l +1 lf ( l +1)F l 1 l +1 [ ]

26 The Dk Side Components 2 Photons Simil stoy except tht thee e 2 poliztions => 2 sets of distibution function F l k, = sum of poliztion G l = diffeence of poliztion Tking into ccount sctteing with byons = = k 2 4 whee n e T = 1/men fee pth of Thompson sctteing + ecuence eltions fo highe ode F nd G l l Byons Pefect fluid but with little bit of pessue => sound wves nd intection with photons k, + k 2 + n e T ( b ) = + 3 b b + b = c 2 b bsk b n e T b 3 b C.P. M, E. Betschinge Astophys.J. 455 (1995) k 2 Phys CMBR 26

27 The Dk Side Tight Coupling Appoximtion Phys CMBR 27 Till shotly befoe ecombintion byons nd photons e tightly coupled. (Hu & Sugiym ApJ 444 (1995) 489) Lge Thompson te b 0 + t this epoch c 2 2 sb << c s Eliminting the Thompson tem between the byon nd photon equtions nd mking the bove substitutions 4 = R 1+ R o R R 1+ R 3 k 2 = 4 R k2 c 1+ R s = R k k2 with R = 3 b 4 + R 1+ R k2 R 1 + R k2 whee we hve used the squed sound velocity c s 2 = 1 3 o going to tempetue fluctutions Θ + R R byon dmping T T = Θ k, = 1 4 Θ + k 2 c s2 Θ = + R 1+ R 1 3 k Diving tem R c 2 = B T 4

28 The Dk Side k k k 2 Phys CMBR 28 Evolution outside the hoizon k. <<1 + Fo whole llowed nge of cosmologicl pmetes, dition dominted: photons+neutinos + flt Fiedmnn eqution ( = d d ) : 2 = 8πG 3c 4 = constnt, = c = 1 Going bck to slide 24, nd neglecting deivtives nd k tems we see tht ( t) = t 1/2 = = 4 G2 = 4 G2 + p c 2 ( + 2 ) ( ) = 12 G 2 + p c 2 ik j + k2 3 v j ( ) = 4 G2 c 2 = 8 G 2 p c 2 Π p = C = = 2 c = b = 3 4 = c = b = = 1 2 k2 = 1 15 k2 is solution Constnt outside the hoizon: constnt potentils+ s Sttement bout δ s is vlid in the longitudinl/confoml Newtonin guge( synchonous) Fixed winkles on the geomety! = 3 4

29 The Dk Side Evolution in mtte dominted Phys CMBR 29 Potentils e constnt! Mtte dominted + flt ( t) t 2/3 2 => = / 2 2 = 0 2 p = 0 = 0 (neutinos hve been edshifted wy) + ( + 2 ) = + 2 ( "gowing": = = C' ) = 0 2 solutions decying : = 1 5 k 2 ( ) = 0 Density fluctutions In gowing mode : = = C' k = 4 G2 [( k ) 2 +12] k = G 43 3 k k >> 12 +k k constnt in decying mode k = 4 G2 [( k ) ] G 43 3 k k >> 42 k k constnt 2 +k t 2/3 3 2 / 3 1 t

30 The Dk Side Recombintion We hve to tke into both hydogen (cf Slide 2) nd helium => compute n e. Put in photon eqution of slide 26 (deived fom Boltzmnn eqution) => Silk dmping Photons cn stem enough to dmp byon nd photon fluctutions When decoupling is stong enough byon fll onto potentil wlls poduced by cold dk mtte. Phys CMBR 30

31 The Dk Side Numeicl solutions Exmple: Hot+Cold Phys CMBR 31

32 The Dk Side Exmple of full eltivistic + multi-component tetment Cold dk mtte Byons Photons+byons Mssive neutinos Photons Neutinos Mssless neutinos Phys CMBR 32

33 The Dk Side Phys CMBR 33 Micowve Bckgound Fluctutions 1st method Follow till = 0 F k, q ˆ, Θ x 0 = 0, n ˆ, 0 = sum of poliztion G ( k, q ˆ, ) = diffeence of poliztion = 1 4 d 3 k exp( ik x ) F ( k, n ˆ, 0 0 ) l ( 2l + 1)F l( k, ) P l ( k.ˆ n ) l =0 ( k, ) 1 4 F l( k, ) e ndom vibles = 1 4 d 3 k i The Θ l but thei evolution does not depend on k only k => Θ l ( k, 0 ) = i( k ) T l ( k, 0 ) Gussin : < i k ndom initil potentil no dependence on l * not ndom Tnsfe function fo i =1 i ( k ' ) >= P ( k) ( k k ') Witing T m=+l T Θ = lmy lm ( n ˆ ) l m= l m=+l 2l +1 The ddition popety of Y lm : P ˆ l ( 4 k.ˆ n ) = Y * ˆ lm ( k )Y lm lm ( n ˆ ) m= l lm = ( i) l 4 d 3 * ky ˆ lm ( k ) i( k )T l ( k, 0 ) lm * l' m' = C l ll' mm' C l = 4 d 3 kp ( k)t 2 l ( k, 0 ) if P ( k)= k 3 ( Hizon Zel' dovich) k. ec <<1 ( ll+1 )C l constnt

34 The Dk Side Phys CMBR 34 Micowve Bckgound Fluctutions 2nd method: moe intuitive Conside the tempetue pofile t sufce of lst sctteing = 1 4 d 3 k exp i Θ x ec, n ˆ, ec Using exp ik ( k x ) F k, n ˆ, ec ec ( x ec ) = i l = 0 1 nd P ( l u ) 2 P 1 l' ( u)du = 2l +1 Θ l ( x ec, n ˆ, ec) = 1 4 d 3 kf l k, ec l ( 2l +1 ) j ( kx) P ˆ l l j l kx ( k.ˆ x ) Note : ˆ Note tht j l kx hs mximum t kx l o 1 kx x = n ˆ so pojection fom plne wves to sphee hs smoothing effect Howeve numbe of coections to mke diltion of metic effect of geomety gvittionl edshift: Schs Wolfe effect - depth of gvittionl potentil - vition of gvittionl potentil: Integl Schs Wolfe effect velocity of the photon gs ou own movement => dipole

35 The Dk Side Phys CMBR 35 Looking t the Powe Spectum Follow: W. Hu, N. Sugiym nd J. Silk, Ntue 386 (1997) 37 Stting fom slide 27 Θ + R R byon dmping Appoximtion: constnt potentil The solution is Θ = Θ( 0)cos kc s 2 3c s This hs to be coected 1) fo ou movement (dipole) nd ou gvittionl potentil (fixed) 2) fo the gvittionl potentil t ecombintion 3) fo the Dopple shift Θ + k 2 c 2 s Θ = + R 1 1+ R 3 k Diving tem Let us tke = = constnt nd neglect the byon dmping tem Θ + k 2 c s 2 Θ = 1 3 k 2 1 T = Θ + = Θ( 0)cos( kc s ec ) T k 3c s = Θ( 0)cos kc s ec R whee we use c s 2 v = 3 4 ik = 3 Θ ( = 0 on slide 26) ik T = v = Θ( 0) T k 1 + R sin kc Θ ( s ec) 3 0 = R k sin kc s ec

36 The Dk Side Constnt Potentil Isentopic ( dibtic ) solution Initil condition 0 Θ( 0) = 2 Θ ( 0) = 0 = 2 ( ec ) 3 (second eglity not poved hee) No byon dg: R =0 Oscilltion ound 0 Dopple s big s compession e.g. ppendix of Hu & Sugiym ApJ 444 (1995) 489 Dopple Compession of fluid Byon dg: R 0 Shifts zeo of oscilltion => gete mplitude of odd peks Dopple not ffected Mss of byons ccentutes the compession nd limits the expnsion Phys CMBR 36 Isocuvtue: sine tem!

37 The Dk Side Vying Potentils: Integl Schs-Wolfe At emission: enhnces tempetue oscilltions Gvittionl potentil disppes: Doubled by the chnge in cuvtue potentil(cf. edshift) Phys CMBR 37 Along the wy: e.g. by cosmologicl constnt o low t Ou tetment did not use cuvtue but could hve been esily included Effect t smll l! Msked by cosmic vince!

38 The Dk Side Effect of Geomety Diect mesuement of geomety δλ e Physicl length on sufce of lst sctteing The ngle it subtends on the sky depends on sptil geomety! Go to comoving coodinte e = e 1 e = 0 e 0 e = 0 e 1 + z Phys CMBR 38 Comoving distnce depends on Ω m (nd wekly on Λ). It is not too difficult to show by suitble chnge of vibles tht in the cse of mtte dominted univese the comoving distnce o z = c 2Ω mo z + 2Ω mo 4 H o Ω 2 mo ( 1 + z) Too f to be dominted by Λ. Position of the peks lso depends on petubtion type Adibtic: cosine Isocuvtue: sine + possibility of mix Ω mo z c H o Ω mo

39 The Dk Side Summy Phys CMBR 39

40 The Dk Side Next Missions MAP Nov 2000 PLANCK Poliztion: vious gound effots Check of model Sensitive to gvittionl wves Signtue=cul of poliztion Phys CMBR 40