2002 6 18 20 14 6 20
1 1 1.1................................... 1 1.2 Abell Clusters........................... 1 1.3 X.............................. 2 1.3.1 line emission................................ 3 1.3.2 continuum emission:..................... 3 1.4 X?......................... 5 2 7 2.1............................ 7 2.2 β................................... 8 3 Sunyaev Zel dovich 11 3.1 Kompaneetz.................................. 11 3.2 SZ CMB..................... 12 3.3 y........................... 15 3.4 SZ angular diameter distance............... 20 4 Press Schechter 23 4.1............................... 23 4.2............................. 25 4.3............................. 29 5 31 5.1................... 31 5.2..... 32 5.3..................... 43 5.3.1...................... 43 5.3.2 X β.............. 45 5.3.3 Collisional dark matter.......................... 46 5.4..................... 48 5.4.1........... 48 5.4.2................... 51 1
1 1.1 (M ) 10 14 R 3 (2 t dyn GM 106 3.1 10 18 ) 3 40 (1.1.1) 6.7 10 8 2 10 33 1014 ( 100 ) ( ) 3 ( ) ( ) (X ) ( ) 3 (ρ r 1.5 ) 1.2 Abell Clusters 50 1000 1958 G.O.Abell Abell clusters Abell Palomar Observatory Sky Survey enhancement Abell (G.O.Abell, ApJS 1958, 3, 211) 1
1. m 3 m 3 <m 3 +2 θ A /z 50 richness class R 2. 10 m 10 z 0.02 z 0.20 m 10 distance class D Abell 2712 1682 2 1030 finding list Abell clusters projection contamination Abell R A = cz θ A = 3000h 1 Mpc π H 0 180 1.7 60 1.48h 1 Mpc (1.2.1) 1.1: Abell clusters R D m 10 z est 0 30 49 1 13.3 14.0 0.027 9 1 50 79 1224 2 14.1 14.8 0.038 2 2 80 129 383 3 14.9 15.6 0.067 33 3 130 199 68 4 15.7 16.4 0.090 60 4 200 299 6 5 16.5 17.2 0.140 657 5 300 1 6 17.3 18.0 0.180 921 7 > 18 ACO Abell ACO m 3 <m<m 3 +2 30 4073 (G.O.Abell, H.G. Corwin, & R.P. Olowin, 1989, ApJS, 70, 1) 1.3 X X X M cl 10 14 M R cl 2Mpc 2
k B T cl m p V 2 cl Gm pm cl R cl ( ) ( ) 2Mpc M cl 2keV R cl 10 14 M (1.3.1) (m p ) X 1.3.1 line emission v 2 m e r = Ze2 & 2πr = nλ = n h r 2 m e v χ n = m ev 2 2 e2 r = m ez 2 e 4 2n 2 h 2 13.6Z2 ev n 2 (1.3.2) χ 1 (1 10)keV X Z =8 27 1 1.1: 1 1.3.2 continuum emission: X T Maxwell f(v) = ( me 2πkT ) 3/2 4πv 2 exp ( m ev 2 ) 2kT (1.3.3) 3
1.2: Lyα(=Kα) [Å] [kev] O 19.0 0.7 Ne 12.2 1.0 Mg 8.4 1.5 Si 6.2 2.0 Ca 3.0 4.1 Fe 1.8 6.9 emissivity (erg/s/cm 3 /Hz) d 2 L X dv dν = 25 πe 6 ( ) 2π 1/2 n 3m e c 3 e Zi 2 n i e hν/kt ḡ ff (T,ν). (1.3.4) 3m e kt i n e n i i Z i i ḡ ff (T,ν) free-free (velocity averaged) Gaunt factor hν, kt (0.1 10)keV ḡ ff (T,ν) 0.9 bolometric luminosity per unit volume (erg/s/cm 3 ) ( ) 0.3 hν (1.3.5) kt dl X dv = 24 e 6 ( ) 2π 1/2 n 3 hm e c 3 e Zi 2 3m e kt n iḡ ff (T ). (1.3.6) i ḡ ff (T ) (frequency averaged) Gaunt factor 1.1 1.5 1.2 20 Rybicki & Lightman 1979 dl X dv 5 10 24 ( ) T 1/2 ( ) ne 2 erg/s/cm 3. (1.3.7) 1keV 1cm 3 R cl 1Mpc ( ) 3 V cl = Rcl 3 3 Rcl 1073 cm 3 (1.3.8) 1Mpc n cl Ω ( ) (1Mpc ) 3 bm cl 3 10 3 Mcl Ω b cm 3. (1.3.9) m p V cl 10 14 M R cl 4
X L X V cl dl X dv ( ) 2 ( ) Ωb T 1/2 ( ) 2 (1Mpc ) 3 Mcl 1043 ξ erg/s. (1.3.10) 0.1 1keV 10 14 M R cl ξ 10 kev 10 44 erg/s X 1.4 X? X n e ( n e ) n e n gal n gal n gal n e n gal ( S N ) opt Rcl n gal (cluster)dl R cl RH n gal universe dl 0 n gal cl n gal universe 2R cl c/h 0 1 (1.4.1) n gal (cluster) n gal cl 10/(h 1 Mpc) 3 R H = c/h 0 3000h 1 Mpc n gal universe 10 2 /(h 1 Mpc) 3 (1.4.1) X ( S N ) X n2 gal cl 2R cl > n gal 2 universe c/h 0 n gal 2 cl n gal 2 universe 2R cl c/h 0 10 3 (1.4.2) 1000 X X 2.1 5
1.2: RXJ1347-1145 X A E RXJ1347-1145 Chandra X 20 1.3: A2218 X ( J.Carlstrom M. Joy ) ( http://oposite.stsci.edu/pubinfo/pr/2000/08/index.html ) 6
2 2.1 1 P gas ρ gas r = GM(r) r 2 (2.1.1) P gas (r) =n gas (r)kt(r) = ρ gas(r) µm p kt(r) (2.1.2) µ mean molecular weight H He 2 (n e + n H + n He )µm p = m H n H + m He n He µ = n H +4n He 2n H +3n He = 4 0.59 (for X =0.75). (2.1.3) 5X +3 Jeans 1/10 (Jeans ): 1 ρ gal r (ρ galσ 2 r )+ 2 r (σ2 r σ 2 t )= GM(r) r 2 (2.1.4) σ r (r) σ t (r) M(R) = R2 G [ 1 ρ gal (R) M(R) = k BT gas (R)R Gµm H dρ gal (R)σ 2 r(r) dr [ d ln ngas (R) d ln R +2 σ2 r(r) σt 2 ] (R), (2.1.5) R + d ln T ] gas(r) d ln R (2.1.6) ρ gal σ r, σ t 7
n gas T gas X X ( 2.1) 1h 1 Mpc 40% M vir =6 10 14 h 1 M L =1.8 10 12 h 2 L M/L 300 2.1: Mpc ( X NASA Goddard Space Flight Center, S.L.Snowden ) 2.2 β (i) T (r), σ r (r), σ t (r) r (ii) : σ r = σ t. (iii) King model: ( ) ] r 2 3/2 n gal (r) =n gal,0 [1+ (2.2.1) r0 (2.1.1) (2.1.4) M(r) ln n gas r = µm pσr 2 ln n gal } kt {{} r β n gas (r) =n 0 [ ngal (r) n gal,0 ] β. (2.2.2) 8
β ( ) ] r 2 3β/2 n gas (r) =n gas,0 [1+, β = µm pσr 2 r0 kt, r 0 : (2.2.3) X Poisson (2.2.1) cluster observer θ -R cl r l d θ A R cl 2.2: X β X S X (θ) dl dl X dv dl n 2 gas l (r = 2 + d 2 A θ2 ) [ dl 1+ l2 + d 2 ] 3β A θ2 r 2 0 [ ] 3β [ = 1+ θ2 l 2 ] 3β dl 1+ r 2 0 /d2 A r 2 + 0 d2 A θ2 ( ) 2 3β+1/2 θ dx = 2r 0 1+ (θ θ 0 0 (1 + x 2 ) 3β 0 r 0 /d A ) }{{} = πr 0 Γ(3β 1/2) Γ(3β) 1+ Γ(3β 1/2) π/2/γ(3β) ( ) 2 3β+1/2 θ θ 0. (2.2.4) θ 0 r 0 /d A d A angular diameter distance d A (z) = c H 0 (1 + z) sin (χ Ω 0 + λ 0 1)/ Ω 0 + λ 0 1 (Ω 0 + λ 0 1 > 0) χ (Ω 0 + λ 0 1=0) sinh (χ 1 Ω 0 λ 0 )/ 1 Ω 0 λ 0 (Ω 0 + λ 0 1 < 0), (2.2.5) 9
z χ(z) = dz Ω (2.2.6) 0 0 (1 + z) 3 +(1 Ω 0 λ 0 )(1 + z) 2 + λ 0 2.3: Angular diameter distance d A (z) 10
3 Sunyaev Zel dovich 3.1 Kompaneetz X (SZ) T CMB =2.7K (CMB: Cosmic Microwave Background) T e kev CMB 1.38mm ( ) observer CMB photon γ I ν temperature decrement CMB CMB through cluster temperature increment T CMB (θ) cluster λ=1.38mm ν=218ghz ν 3.1: (Kompaneetz) ( ) [ { }] f kte 1 f t c = x 4 t m e c 2 x 2 e + f(f +1) (3.1.1) e x e x e 1 t c n e σ T c ( ), x e hν. (3.1.2) kt e f = f(ν, t) t ν t c x e Bose-Einstein (t =0 ) T CMB : f(ν, t =0)= 1 exp(hν/kt CMB ) 1 11 (3.1.3)
(3.1.1) x y x hν kt CMB = T e T CMB x e, y(t) t 0 k(t e T CMB ) m e c 2 n e σ T cdt (3.1.4) f(f +1) f x f y = 1 ( x 4 f ). (3.1.5) x 2 x x 3.2 SZ CMB (3.1.4) R cl n e y = Rcl R cl kt e m e c 2 σ T n gas dl (3.2.1) t c t cross : t c = 1 ( 10 3 n e σ T c 5 cm 3 ) 1016, t cross 2R ( ) cl 3 10 14 Rcl (3.2.2) n gas c 1.5Mpc y kt e m e c 2 t cross t c ( ) ( )( 10 4 Te R cl 10 3 cm 3 10keV 1.5Mpc n gas ). (3.2.3) y 10 4 CMB (3.1.5) f(x, y) f(x, 0) + y ( x 4 ) f(x, 0) x 2 x x (3.2.4) f(x, 0) = 1 e x 1 (3.2.5) f(x, y) 1 [x e x 1 + xyex coth x ] (e x 1) 2 2 4 I ν (ν): (3.2.6) I ν (x, y) 1 4π 2hν c 2 4πν2 f(x, y) =x 3 f(x, y) 2(kT CMB) 3 h 2 c 2 i 0 x 3 x 4 e x [ f(x, 0) + i 0 (e x 1) y x coth x ] 2 2 4 (3.2.7) 12
i 0 2(kT CMB) 3 2.6 10 15 erg/s/cm 2 /Hz/str 2.2 10 4 mjy/arcmin 2. (3.2.8) h 2 c 2 I ν y (x xex coth x ) I ν e x 1 2 4. (3.2.9) T B (ν): ( f(x, y) exp hν 1 kt B (3.2.6) ) 1 T B (x, y) = ln[f 1 (x, y)+1] x yx T B T CMB xt CMB ln[f 1 (x, y)+1]. (3.2.10) (x coth x 2 4 ). (3.2.11) y (x coth x ) 2 4. (3.2.12) Rayleigh Jeans temperature T RJ (ν): Rayleigh Jeans T RJ T CMB I ν 2ν2 c 2 kt RJ (3.2.13) x 2 e x y (e x 1) 2 (x coth x 2 4 ). (3.2.14) kinematic SZ Iν K : SZ v kinematic SZ Iν K ( ) y xex v τ, τ n I ν e x gas σ T dl : (3.2.15) 1 c 3.2 3.3 SZ ν x hν/kt CMB 11.8(0.45mm/λ) SZ x coth(x/2) 4 x =3.83, ν = 218GHz, λ =1.38mm (3.2.16) SZ (1.38mm) kinematic SZ (3.2.16) x 0 Rayleigh Jeans I ν I ν = T B T CMB = T RJ T CMB = 2y (3.2.17) 13
10 1 0.1 10 8 6 4 2 8 0 6-2 4 2-4 0-6 -2 0 2 4 6 8 0.1 1 10-4 -6 1 図 3.2: SZ 効果による CMB スペクトル変形 SZ 輻射 強度 Iν i0 yx4 ex /(ex 1)2 [x coth(x/2) 4], SZ 相対 輻射強度比 Iν /Iν yxex /(ex 1)[x coth(x/2) 4], SZ 相対輝度温度比 TB /TCM B y[x coth(x/2) 4], kinematic SZ 相対輻射強度比 IνK /Iν yxex /(ex 1)(v/c)τ. 10 100 1000 図 3.3: SZ 効果による輻射スペク トル 上図 SZ 効果による変形 がある場合 (y = 0.1) とない場合 のマイクロ波背景輻射のスペクト ル 下図 SZ 効果のスペクトル 4 3 2 1 (z=1) A 0 3 2 1 (z=0) A 0 0 1 2 3 X [Mpc] 0 1 2 3 X [Mpc] 0 1 2 3 X [Mpc] 0 1 2 3 X [Mpc] 4 図 3.4: Projected views of cluster A at z = 1 and z 0. A box of (4Mpc)3 (in physical lengths) located at the center of each cluster is extracted. The X-ray emission weighted temperature (TX ), X-ray surface brightness (SX ), and the SZ surface brightness at mm and submm bands ( Imm and Isubmm ) are plotted on the projected X-Y plane by integrating over the line-of-sight direction (Z). (Yoshikawa, Itoh & Suto 1998) 14
3.3 y β ( ) ] r 2 3β/2 n gas (r) =n gas,0 [1+, β = µm pσr 2 r0 kt, r 0 : (3.3.1) y (2.2.4) kt e y = m e c σ 2 T n gas (r)dl = ( ) ( ) 2 3β/2+1/2 Γ(3β/2 1/2) kte θ π σ Γ(3β/2) m e c 2 T n gas,0 r 0 1+. (3.3.2) β 1, T 5keV, n gas,0 3 10 3 cm 3, r 0 0.3Mpc, z =0.1 (d A (z) 300Mpc) θ 0 3 y(θ) y(θ) θ 0 4 10 5 1+(θ/3 ) 2 (3.3.3) 3.5: Three views of the cluster A2218: (a) The HST image of the central core region (Kneib et al. 1996) (b) The BIMA 28.5 GHz naturally weighted contours with VLA D-array NVSS observations in the background. The small rectangle roughly indicates the region of the HST image. (c) The detected SZE, after accounting for the bright radio sources. The background of this map is the ROSAT PSPC image, smoothed with a 20 FWHM Gaussian (Carlstrom et al. astro-ph/9804149). 15
3.6: Images of the Sunyaev- Zel dovich effect toward twelve distant clusters with redshifts spanning 0.83 (top left) to 0.14 (bottom right). The evenly spaced contours are multiples starting at ±1 of1.5σ to 3σ depending on the cluster, where σ is the rms noise level in the images. The noise levels range from 15 to 40 µk. The data were taken with the OVRO and BIMA mm-arrays outfitted with low-noise cm-wave receivers. The filled ellipse shown in the bottom left corner of each panel represents the FWHM of the effective resolution used to make these images (Carlstrom et al. astro-ph/9905255). 16
3.1: Cluster Sample (L X is computed for EdS model). cluster z T e [kev] L X [10 44 h 2 50 erg s 1 ] band [kev] MS 1137.5 + 6625 0.784 5.7 +1.3 0.7 5.4 0.3 3.5 MS 0451.6 0305 0.550 10.4 +1.0 0.8 20.0 0.3 3.5 Cl 0016 + 16 0.546 7.55 +0.72 0.58 14.6 0.3 3.5 RX J1347.5 1145 0.451 9.3 +0.7 0.6 73.0 0.1 2.4 Abell 370 0.374 6.6 +0.7 0.5 11.7 0.1 2.4 MS 1358.4 + 6245 0.327 7.48 +0.50 0.42 10.6 0.3 3.5 Abell 1995 0.322 8.59 +0.86 0.67 13.4 0.1 2.4 Abell 611 0.288 6.6 +0.6 0.6 8.6 0.1 2.4 Abell 697 0.282 9.8 +0.7 0.7 19.2 0.1 2.4 Abell 1835 0.252 8.21 +0.19 0.17 32.6 0.1 2.4 Abell 2261 0.224 8.82 +0.37 0.32 20.6 0.1 2.4 Abell 773 0.216 9.29 +0.41 0.36 12.1 0.1 2.4 Abell 2163 0.202 12.2 +1.1 0.7 37.5 0.1 2.4 Abell 520 0.202 8.33 +0.46 0.40 14.5 0.1 2.4 Abell 1689 0.183 9.66 +0.22 0.20 20.7 0.1 2.4 Abell 665 0.182 9.03 +0.35 0.31 15.7 0.1 2.4 Abell 2218 0.171 7.05 +0.22 0.21 8.2 0.1 2.4 Abell 1413 0.142 7.54 +0.17 0.16 10.9 0.1 2.4 3.2: ICM Parameters cluster β θ c [ ] S x0 [erg/s/cm 2 /arcmin 2 ] T 0 [µk] D A [Mpc] MS1137 0.786 +0.220 0.120 19.4 +6.4 4.0 4.43 +0.74 0.59 10 13 818 +98 113 3179 +1103 1640 MS0451 0.806 +0.052 0.043 34.7 +3.9 3.5 9.56 +0.86 0.84 10 13 1431 +98 93 1278 +265 299 Cl0016 0.749 +0.024 0.018 42.3 +2.4 2.0 6.17 +0.22 0.28 10 13 1242 +105 105 2041 +484 514 R1347 0.604 +0.011 0.012 9.0 +0.5 0.5 2.74 +0.16 0.14 10 11 3950 +350 350 1221 +368 343 A370 0.518 +0.090 0.080 39.5 +10.5 10.5 2.70 +0.43 0.30 10 13 1253 +218 533 4352 +1388 1245 MS1358 0.622 +0.015 0.015 18.2 +1.4 1.5 1.70 +0.15 0.11 10 12 784 +90 90 866 +248 310 A1995 0.770 +0.117 0.063 38.9 +6.9 4.3 1.08 +0.08 0.07 10 12 1023 +83 77 1119 +247 282 A611 0.565 +0.050 0.040 17.5 +3.5 3.5 2.01 +0.36 0.26 10 12 853 +120 140 995 +325 293 A697 0.540 +0.045 0.035 37.8 +5.6 4.0 1.02 +0.07 0.08 10 12 1410 +160 180 998 +298 250 A1835 0.595 +0.007 0.005 12.2 +0.6 0.5 2.02 +0.14 0.10 10 11 2502 +150 175 1027 +194 198 A2261 0.516 +0.014 0.013 15.7 +1.2 1.1 4.31 +0.26 0.26 10 12 1697 +200 200 1049 +306 272 A773 0.597 +0.064 0.032 45.0 +7.0 5.0 8.28 +0.65 0.65 10 13 1260 +160 160 1450 +361 332 A2163 0.674 +0.011 0.008 87.5 +2.5 2.0 1.36 +0.03 0.03 10 12 1900 +140 140 828 +181 205 A520 0.844 +0.040 0.040 123.3 +8.0 8.0 4.08 +0.18 0.18 10 13 662 +95 95 723 +270 236 A1689 0.609 +0.005 0.005 26.6 +0.7 0.7 6.01 +0.18 0.15 10 12 1729 +105 120 688 +172 163 A665 0.615 +0.006 0.006 71.7 +1.5 1.5 6.78 +0.12 0.12 10 13 728 +150 150 466 +217 179 A2218 0.692 +0.008 0.008 67.5 +1.5 1.8 7.08 +0.16 0.14 10 13 731 +125 100 1029 +339 352 A1413 0.639 +0.009 0.009 47.7 +2.0 2.0 2.04 +0.09 0.09 10 12 856 +110 110 573 +171 151 17
3.7: SZE (contours) and X-ray (color scale) images of each cluster in our sample. Negative contours are shown as solid lines. The contours are multiples of 2 σ and the FWHM of the synthesized beams are shown in the bottom left corner. The X-ray color scale images are raw counts images smoothed with Gaussians with σ =15 for PSPC data and σ =5 for HRI data. There is a color scale mapping for the counts above each image. 18
3.8: SZE (contours) and X-ray (color scale) images (continued). (Reese et al. astroph/0205350) 19
3.4 SZ angular diameter distance SZ X CMB SZ y (3.2.1) y r 0 Tn gas,0 (3.4.1) X T θ 0 S X S X r 0 T 1/2 n 2 gas,0 (3.4.2) d A r 0 /θ 0 (3.4.1) (3.4.2) d y 2 S X T 3/2 θ 0 (3.4.3) SZ H 0 SZ SZ ( β ) y : β SZ β θ 0 X β SZ θ SZ,0 β X, θ X,0 y(0) = π Γ(3β SZ/2 1/2) Γ(3β SZ /2) ( kte m e c 2 ) σ T n gas,0 d A (z)θ SZ,0. (3.4.4) RJ : T RJ x 2 e x (0) = y(0) T CMB (e x 1) 2 ( 4 x coth x ) 2y(0)ξ(x). (3.4.5) 2 20
ξ(x) RJ 1 ξ(x) x 2 e x ( 4 x coth x ), x hν. (3.4.6) 2(e x 1) 2 2 kt CMB X : X ν 1 <ν<ν 2 S X (θ) = d 2 L X dνdv S X (0) = α(t e)kt e h α(t e ) 25 πe 6 3m e c 2 1 ν2 (1+z) d 2 L X dl dν, (3.4.7) 4π(1 + z) 4 ν 1 (1+z) dνdv = ( α(t e ) n 2 gas(r) g ff (T e,ν)exp hν ), kt e (3.4.8) ( ) 2π 1/2 2 3m e c 2 kt e 1+X. (3.4.9) n 2 d gas,0 A(z)θ X,0 4 Γ(3β X 1/2) hν2 (1+z)/kT e g π(1 + z) 4 Γ(3β X ) ff e x dx, (3.4.10) hν 1 (1+z)/kT e angular diameter distance (3.4.5) (3.4.10) n gas,0 d A (z) 2 n gas,0 d A (z) = ( α(t e )kt e me c 2 ) 2 ( ) 2 Γ(3β X 1/2) Γ(3β SZ /2) 16π 3/2 hσt 2 (1 + z)4 kt e Γ(3β X ) Γ(3β SZ /2 1/2) ( T RJ/T CMB (0)) 2 θ hν2 (1+z)/kT e X,0 g ξ 2 (x)s X (0) ff e x dx. (3.4.11) hν 1 (1+z)/kT e θ 2 SZ,0 21
3.9: angular diameter distance Also plotted is the theoretical angular diameter distance relation for three different cosmologies, assuming H 0 = 60km/s/Mpc. (1) Reese et al (2000), (2) Mauskopf et al. (2000), (3) Reese et al (2000), (4) Patel et al. (2000), (5) Grainge et al. (2000), (6) Saunders et al. (2000), (7) Andreani et al. (1999), (8) Komatsu et al. (1999), (9) Myers et al. (1997), (10) Lamarre et al. (1998). (11) Tsuboi et al. (1998), (12) Hughes et al. (1998), (13) Holzapfel et al. (1997), (14) Birkinshaw et al. (1994), (15) Birkinshaw et al. (1991) (Carlstrom et al. astro-ph/0103480). 3.10: D A versus z for our 18 cluster sample. The error bars are 68.3% statistical uncertainties only. Also plotted are the theoretical angular diameter distance relations assuming H 0 = 60km/s/Mpc for three different cosmological models; the currently favored Λ cosmology Ω m =0.3, Ω Λ =0.7 (solid) cosmology; an open Ω m =0.3 (dashed) universe; and a flat Ω m = 1 (dotted) cosmology. 22
4 Press Schechter 4.1 M t r(t) t C r = GM C (1 cos θ), t d 2 r dt 2 = GM r 2. (4.1.1) = GM (θ sin θ). (4.1.2) C3/2 θ 1 ρ(< r; t) 3M 4πr(t) = 1 [ 1+ 3C ( ) 6t 2/3 + ] 3 6πGt 2 20 GM (4.1.3) Einstein de Sitter ρ(t) = 1 6πGt, δ linear(< r; t) 3C ( ) 6t 2/3. (4.1.4) 2 20 GM turn-around maximum expansion (4.1.2) r ta = 2GM C, θ ta = π, t ta = πgm. (4.1.5) C3/2 turn-around (C) (M) ρ(< r; t ta ) ρ(t ta ) = 9π2 16 5.55 (4.1.6) (4.1.4) δ linear (<r; t ta )= 3 20 (6π)2/3 1.06 (4.1.7) 23
θ c =2π, t c =2t ta = 2πGM C 3/2 (4.1.8) r = r vir = r(θ ta) = GM (4.1.9) 2 C t>t c : ρ(< r vir )= ρ(< r; t c ) c ρ(t c )=18π 2 ρ(t c ) 177.7 ρ(t c ) (4.1.10) (4.1.7) δ linear (<r; t c )= 3 20 (12π)2/3 δ c 1.69 (4.1.11) (4.1.10) (4.1.11) Einstein de Sitter Einstein de Sitter (Ω 0 =1,λ 0 =0) c = 18π 2 178, (4.1.12) δ c = 3(12π)2/3 1.69, (4.1.13) 20 (Ω 0 < 1,λ 0 =0) c = 4π 2 (cosh η vir 1) 3 (sinh η vir η vir ), (4.1.14) 2 δ c = 3 [ ] ) 2/3 3sinhηvir (sinh η vir η vir ) 2π 2 1+(,(4.1.15) 2 (cosh η vir 1) 2 sinh η vir η vir (Ω 0 < 1,λ 0 =1 Ω 0 ) ( ) rta 3 2w vir c = r vir χ, 18π 2 (1 + 0.4093wvir 0.9052 ), δ c = 3 ( 1 5 F 3, 1, 11 ) ( ) 1/3 ( 6 ; w 2w vir vir 1+ χ ), χ 2 3(12π)2/3 (1 + 0.0123 log 20 10 Ω vir ). (4.1.16) 24
η vir cosh 1 (2/Ω vir 1), w vir 1/Ω vir 1, χ λ 0 H 2 0r 3 ta/(gm), F (2,1), Ω vir t c 4.1 c δ c Ω vir 1.7 1.65 (a) 1.6 1.55 1200 (b) 1000 800 600 400 200 0 0.2 0.4 0.6 0.8 1 4.1: δ c c λ 0 =0 λ 0 = 1 Ω 0 4.2 25
Press Schechter z i M δ = δ(m,z i ) δ rms σ M (z i ) P (δ) = [ ] 1 2πσ 2 (z M i) exp δ2 (M,z i ) 2σ 2 (z M i) (4.2.1) 4.1 δ linear (M,z) D(z) D(z i ) δ(m,z i) (4.2.2) δ c z z δ linear (M,z) (4.2.1) (4.2.2) M z [ P[δ linear (M,z) >δ c ] = P δ(m,z i ) >δ c,i D(z ] i) D(z) δ c = P (δ)dδ = 1 ( ) exp x2 dx (4.2.3) δ c,i 2π δ c,i /σ M (z i ) 2 M (> M) M M z M M + dm (4.2.3) M + dm p(m,z)dm = P[δ linear (M,z) >δ c ] P[δ linear (M + dm, z) <δ c δ linear (M,z) >δ c ] = P[δ linear (M,z) >δ c ] {1 P[δ linear (M + dm, z) >δ c δ linear (M,z) >δ c ]} = P[δ linear (M,z) >δ c ] P[δ linear (M + dm, z) >δ c ] P[δ linear(m,z) >δ c ] dm M (4.2.4) (4.2.3) P[δ linear (M,z) >δ c ] = 1 exp [ δ c 2 ] [ ] d δc (4.2.5) M 2π 2σM(z) 2 dm σ M (z) z M M + dm n(m,z)dm = ρ 0 M p(m,z)dm = ρ 0 δ c dσ M (z) [ 2πMσ 2 (z) dm exp δ c 2 ] dm (4.2.6) 2σ 2 M M(z) 26
( dmmn(m,z) = ρ 0 P ) dm = ρ 0 [P M=0 P M= ] 0 0 M = ρ ( ) 0 exp x2 dx = ρ 0 2π 0 2 2 (4.2.7) δ c M (4.2.7) ρ 0 2 Press Schechter 20 (4.2.6) ( 2 n PS (M,z) = π ) 1/2 ρ 0 M [ δ c dσ M (z) σ 2 (z) dm exp M δ2 c 2σ 2 (z) M ] (4.2.8) Press Schechter σm(z) 2 z 0 M 0 z 1 (>z 0 ) M 1 (<M 0 ) progenitor p(m 1,z 1 M 0,z 0 ) M 0 δ 0 M 1 δ 1 0 [ 1 (δ 1 δ 0 ) P (M 1,δ 1 M 0,δ 0 )= 2π[σ 2 ] 2 (0) exp M1 σ2 (0)] 2[σ 2 (0) M1 σ2 (0)] (4.2.9) M0 M0 PS (4.2.1) (4.2.4) M 1 z 1 : P (M,δ M,δ )dδ, (4.2.10) P(M 1,>δ 1c M 0,δ 0c ) 1 1 0 0c 1 δ 1c δ 1c δ c [D(0)/D(z 1 )], (4.2.11) δ 0c δ c [D(0)/D(z 0 )] (4.2.12) M 1 p(m 1,z 1 M 0,z 0 ) = 2 P(M 1,>δ 1c M 0,δ 0c ) M 1 [ = δ 1c δ 0c 2π[σ 2 (0) exp M1 σ2 (0)]3 M0 27 (δ 1c δ 0c ) 2 2[σ 2 (0) M1 σ2 (0)] M0 ] dσ 2 M1 (0) dm 1 (4.2.13)
Abell clusters selected by eyes from the photographic plates galaxies satisfying m < m < m + 2 3 3 Press-Schechter halos nonlinear spherical collapse 2 = 18 π vir SZ clusters inverse Compton of CMB I ~ n T R SZ e e cl Halos in N-body simulations friend-of-friend method linking length = 0.2 (mean particle separation) X-ray clusters thermal bremsstrahlung 2 1/2 S ~ n T R X e e cl 4.2: 28
4.3 Press Schechter D(z) λ λ J δ k +2ȧ a δ k 4πG ρδ k = 0 (4.3.1) R(t) (decaying solution) (growing solution) decaying solution growing solution H(t) =ȧ/a Ḧ +2HḢ = H2 0 H 3Ω 0 =4πG ρh (4.3.2) 2R3 (4.3.1) H(t) t decaying solution growing solution D(t) (4.3.1) (4.3.2) a 2 d dt (ḊH DḢ)+da2 (ḊH DḢ) = 0 (4.3.3) dt D(t) H(t) dt a 2 H 2 (t) (4.3.4) D t z D(z) = 5Ω 0H 2 0 2 1+w H(z) dw (4.3.5) z H 3 (w) H(z) = H 0 Ω 0 (1 + z) 3 +(1 Ω 0 λ 0 )(1 + z) 2 + λ 0 (4.3.6) z 1/(1 + z) D(z) (a) Einstein de Sitter (Ω 0 =1,λ 0 =0) D(z) 1 1+z (4.3.7) (b) (Ω 0 < 1,λ 0 =0) D(z) 1+ 3 x +3 1+x x 3 ln ( 1+x x ) x 1 Ω 0 Ω 0 (1 + z) (4.3.8) 29
(c) (Ω 0 < 1,λ 0 =1 Ω 0 ) D(z) 1+ 2 x 3 x 0 ( ) u 3/2 2 1/3 (Ω 1 du x 0 1) 1/3 2+u 3 1+z (4.3.9) D(z) = g(z) 1+z, (4.3.10) g(z) = 5Ω(z) 1 2 Ω 4/7 (z) λ(z)+[1+ω(z)/2][1 + λ(z)/70], (4.3.11) [ ] 2 Ω(z) = Ω 0 (1 + z) 3 H0 (1 + z) = H(z) [ ] 2 H0 λ(z) = λ 0 = H(z) 4.3 Ω 0 (1 + z) 3 (4.3.12), Ω 0 (1 + z) 3 +(1 Ω 0 λ 0 )(1 + z) 2 + λ 0 λ 0 Ω 0 (1 + z) 3 +(1 Ω 0 λ 0 )(1 + z) 2 + λ 0, (4.3.13) 1 0.1 0.01 0.1 1 10 z 4.3: (4.3.5) (4.3.10) 30
5 5.1 + Press- Schechter (i) academic interest ( ) 2 (A) singular isothermal sphere ρ(r) = σ2 1 2πG r 2 ξ gg (r) r 1.8 r 2 r 2 (B) stable clustering solution for ξ(r) P mass (k) k n ( ) ξ mass (r) r 3(n+3)/(n+5) ( ) ξ gg (r) r 1.8 (A) (B) (ii) practical importance ( ) (i) 31
testable predictions ( ) (bias) ( ) 5.2 V (r) r r r r r ρ(r) r 2 ( ) (A) (B) 1980 (a) 1996 r s : ρ(r) 1 (r/r s ) α (1 + r/r s ) 3 α (5.2.1) α =1 (B) 5 (α =1.5 ) (5.2.1) 32
5.1: 1970 Peebles AJ, 75, 13 N = 300, Coma N 1972 Gunn & Gott ApJ 176, 1 secondary infall 1977 Gunn ApJ 218, 592 secondary infall ρ r 9/4 1985 Hoffman & Shaham ApJ 297, 16 density peak ρ r 3(n+3)/(n+4) 1986 Quinn, Salmon & Zurek Nature 322, 329 N 1000 Hoffman & Shaham 1987 West, Dekel & Oemler ApJ 316, 1 N 4000 simulations. univeral profile 1988 Frenk, White, Davis & Efstathiou ApJ 327, 507 N =32 3 simulations, SCDM 1990 Hernquist ApJ 356, 359 de Vaucouleurs R 1/4 ρ(r) = Ma 1 2π r(r + a) 3 ( ) 1994 Crone, Evrard, & Richstone ApJ 434, 402 N =64 3 simulations in scale-free models + various cosmology ρ r α : 1995 Navarro, Frenk & White MNRAS 275, 720 SCDM SPH simulations (no cooling), N 6000 ρ(r) = 7500 ρ (r/0.2r 200 )(r/0.2r 200 +1) 2 1996 Navarro, Frenk & White ApJ 462, 563 19 halos with N(< r vir ) = 5000 10000 in SCDM universal density profile 1997 Fukushige & Makino ApJ 477, L9 higher-resolution simulations (N = 786, 400) NFW 1/r 1997 Navarro, Frenk & White ApJ 490, 493 SCDM, LCDM, scale-free models simualtions universal density profile 1998 Syer & White MNRAS 293, 337 continual merger + tidal disruption ρ r 3(n+3)/(n+5) 1998 Moore et al. ApJ 499, L5 inner profile ρ r 1.4 Fukushige & Makino 1999 Moore et al. MNRAS 310, 1147 1 ρ(r) (r/r s ) 1.5 [1 + (r/r s ) 1.5 ] 2000 Jing & Suto ApJ 529, L69 (controvertial) 33
5.1: (Peebles 1970) 5.2: universal. (Crone, Evrard & Richstone 1994) 34
5.3: Particle plots illustrating the time evolution of halos of different mass in an Ω 0 =1, n = 1 cosmology. Box sizes of each column are chosen so as to include approximately the same number of particles. At z 0 = 0 the box size corresponds to about 6 r 200. Time runs from top to bottom. Each snapshot is chosen so that M increases by a factor of 4 between each row. Low mass halos assemble earlier than their more massive counterparts. This is true for every cosmological scenario in our series. (Navarro, Frenk & White 1997). 35
5.4: Density profiles of one of the most and one of the least massive halos in each series. In each panel the low-mass system is represented by the leftmost curve. In the SCDM and CDMΛ models radii are given in kpc (scale at the top) and densities are in units of 10 10 M /kpc 3. In all other panels units are arbitrary. The density parameter, Ω 0,andthe value of the spectral index, n is given in each panel. Solid lines are fits to the density profiles using eq. (1). The arrows indicate the value of the gravitational softening. The virial radius of each system is in all cases two orders of magnitude larger than the gravitational softening (Navarro, Frenk & White 1997) 36
5.5: The circular velocity profiles of the halos shown in the previous Figure. Radii are in units of the virial radius and circular speeds are normalized to the value at the virial radius. The thin solid line shows the data from the simulations. All curves have the same shape: they rise near the center until they reach a maximum and then decline at the outer edge. Low mass systems have higher maximum circular velocities in these scaled units because of their higher central concentrations. Dashed lines are fits using eq.(3). The dotted lines are the fit to the low-mass halo in each panel using a Hernquist profile. Note that this model fits rather well the inner regions of the halos, but underestimates the circular velocity near the virial radius (Navarro, Frenk & White 1997). 37
5.6: Density and temperature structures of the halo at z =1.8. The position of the center of the halo was determined using potential minimum and averaged physical values over each spherical shell (Fukushige & Makino 1997). 5.7: The density profiles of the Coma cluster simulated at four different resolutions. The curves begin at the spline softening lengths that were used and the number of particles within the final virial radii are indicated (Moore et al. 1998). 38
5.8: The broken lines show the Virgo halo simulated at the same mass resolution but varying only the softening parameter. This halo has a virial radius of 2 Mpc and contains 20,000 particles within r vir. The values of the softening used are indicated next to each curve. The solid curves show the same cluster resimulated with a mass resolution 20 times higher, but keeping the force softening fixed at 10kpc. To demonstrate that relaxation is not affecting our results one of the solid curves shows the profile at a redshift z=0.25(moore et al. 1998). 5.9: The same cluster simulated in the next figure but with two different values of the softening length and keeping the particle mass fixed. The left panel shows a close up view of the inner 500 kpc of the last frame of Figure 1. In this case the softening was 0.2% of the virial radius. The right panel shows the same region of the same cluster but simulated with a softening length of 1.5% of the virial radius. The lack of substructure halos in the right panel demonstrates that softened halos are easily disrupted by tidal forces (Moore 2001). 39
5.10: The hierarchical evolution of a galaxy cluster in a universe dominated by cold dark matter. Small fluctuations in the mass distribution are present but barely visible at early epochs. These grow by gravitational instability, merging and accretion of mass, eventually collapsing into virialised quasi-spherical dark matter halos. This plot shows a time sequence of 6 frames of a region of the universe that evolves into a cluster of galaxies. The colours represent the local density of dark matter plotted using a logarithmic colour scale. Linear over-densities are darker blue, whereas the non-linear collapsed regions attain over-densities of a million times the mean background density and are plotted as yellow/white. Each box is 10 Mpc on a side and the final cluster virial radius is 2 Mpc (Moore 2001). 40
5.11: The density profile of the cluster measured in three runs with increasing resolutions (from triangles to squares to circles). In the best run, the cluster contains over 4 million particles and the force resolution is 0.05% of the cluster s virial radius. The curves are an NFW profile (lower curve) and a fit with the profile of Moore et al. (1999a), which rises more steeply ( r 1.5 )atthe center than the NFW profile ( r 1 ). With increasing resolution, the cluster s profile continues to approach M99a s curve i.e. this appears to be the asymptotic profile in the limit of infinite resolution. The vertical bars mark the radii at which the measured profiles are no longer affected by finite numerical resolution (Ghigna et al. 2000). 5.12: (Jing & Suto 2000) 41
5.13: (Jing & Suto 2000) 5.14: Evolution of density profile of the halo for Run 16M0 and 2M0. The unit of density is M /pc 3. The profiles are vertically shifted downward by 7, 6,..., 1, 0 dex from the bottom to the top. The arrows indicates r 200. The numbers near the dashed lines indicate the power index of those lines. The numbers on the left of the profiles indicate the redshift (Fukushige & Makino 2001). 42
5.3 5.3.1 NFW : ρ DM (r) = δ c ρ crit (r/r s )(1 + r/r s ) 2. (5.3.1) : r vir = r vir (M halo,z vir ) r vir vir δ c = δ c (M) : (characteristic halo density), (5.3.2) r s = r s (M) : (scaling radius), (5.3.3) ρ crit = 3H2 0 8πG 1.8 1011 h 2 M /Mpc 3 : ( ), (5.3.4) c vir r vir(m) : (concentration parameter). (5.3.5) r s (M) ( ) 1/3 3M halo 1.69 ( ) vir Ω 1/3 ( ) 1/3 0 M halo (5.3.6) 4π vir Ω 0 ρ crit 1+z vir 18π 2 10 15 h 1 M { 18π 2 Ω(z vir ) 0.7 (λ 0 =0) (5.3.7) 18π 2 Ω(z vir ) 0.6 (λ 0 =1 Ω 0 ) c vir fitting formula in LCDM : Bullock et al. MNRAS 321(2001)559 c vir (M halo,z)= 9 ( ) 0.13 M halo. (5.3.8) 1+z 1.5 10 13 h 1 M r δ c ρ crit r 2 [ M(r) =4π 0 (r/r s )(1 + r/r s ) dr =4πδ cρ 2 crit rs 3 ln (1+ r ) r ] rs r + r s (5.3.9) (5.3.5) (5.3.6) (5.3.9) δ c = virω 0 3 c 3 vir ln(1 + c vir ) c vir /(1 + c vir ) (5.3.10) GM(r) V c (r) = r = 4πδ c ρ crit r 2 s [ rs r ln ( 1+ r r s ) r s r + r s ] (5.3.11) 43
(5.2.1) 5.15: Rotation curves of high resolution CDM halos (solid curves) compared with LSB rotation curve data (dotted curves). All of the data and model rotation curves have been scaled to a fiducial peak velocity of 200 km/s. (Note that the simulation halos and the data were chosen to have peak rotational velocities within 50% of this value.) The total rotational velocity and the baryonic contribution from the stars and gas from a typical LSB galaxy (UGC 128) are shown by the open squares. The mass to baryon ratio for this galaxy is nearly 20:1 and the rotation curve data probes a remarkable 25% of the expected virialised halo (Moore et al. 1999). 5.16: A radial plot of the mass density and light density. Total (thick line) and galaxy-only (thin line) components of the mass are shown. Thedottedlineisthebest NFW fit discussed in the text, and the dashed line is the best-fit single PL model. The 35 h 1 kpc soft core in the mass is evident. A singular mass distribution is ruled out. The total rest-frame V light profile (solid line), and galaxy V light profile (dashed line), smoothed with a 5 h 1 kpc Gaussian, are also shown (Tyson, Kochanski & Dell Antonio 1998, ApJ 498, L107). 44
5.3.2 X β NFW X (Makino, Sasaki & Suto 1998) : dp gas dr = GM(r) r 2 ρ gas, (5.3.12) : p gas = n gas kt gas = ρ gas µm p kt gas, (5.3.13) kt gas d ln ρ gas µm p dr (5.3.9) ln ρ gas = Gµm p kt gas 4πδ c ρ crit r 2 s }{{} B r/rs 0 = GM(r) r 2. (5.3.14) [ ] ln(1 + x) 1 + const. (5.3.15) x 2 x(1 + x) ρ gas (r) =ρ g0 e B (1+ r r s ) Brs/r. (5.3.16) β β fit ρ gas (r) 3β ρ g0 A eff /2, [1 + (r/r c,eff ) 2 ] (5.3.17) A 0.178b +0.982 = 0.013B +0.982 (b 2B/27), r c,eff 0.22r s, β eff 0.9b =0.067B. 5.17: NFW β (Makino, Sasaki, & Suto 1998). 45
5.3.3 Collisional dark matter CDM (r 1Mpc) r 1Mpc r 1Mpc σ (Collisional dark matter, self-interacting dark matter) ( σ (mn) l mfp 1 for mn ρ c,cluster and l mfp ( ) 1Mpc.(5.3.18) m) ( ( σ 10 1.6h m) 1 cm 2 4 )( ρ crit 1h 1 ) Mpc /g ρ c,cluster l mfp (5.3.19) ( ) ( m σ 2.5 10 24 h 1 cm 2 10 4 )( ρ crit 1h 1 ) Mpc. (5.3.20) 1GeV ρ c,cluster l mfp t l mfp v 109 h 1 ( 1000km/s v )( ) lmfp. (5.3.21) 1h 1 Mpc σ/m 1cm 2 /g subhalos 46
S1 1 : 0:82 : 0:65 S1Wa? =0:1cm 2 g ;1 r c =40h ;1 kpc 1 : 0:88 : 0:66 S1Wb? =1:0cm 2 g ;1 r c = 100 h ;1 kpc 1 : 0:91 : 0:72 5.19: Density profiles (top) and mean collision counts per particle (bottom). The vertical dotted line in the top panel indicates the gravitational softening length of our S1 simulations. The virial radius R 200 of the final cluster is shown as an arrow. The fluid dark matter case from Yoshida et al. (2000) is plotted as the dashed line, while the dash-dotted line represents our higher resolution simulation of the medium cross-section case (S2W-b). (Yoshida et al. 2000, ApJ 544, L87). S1Wc? =10:0cm 2 g ;1 r c = 160 h ;1 kpc 1 : 0:98 : 0:89 5.18: Projected mass distributions in a box 15h 1 Mpc on a side. The collision cross-sections per unit mass, core radii, axis ratios for each model and small panels showing the central region (2h 1 Mpc on a side, enlarged) in a different color scale are given to the right of the corresponding image. (Yoshida et al. 2000, ApJ 544, L87). 5.20: The total number of subhalos within R 200 is plotted as a function of the lower limit to their mass in units of M 200. Results are plotted for halos containing 10 or more particles. (Yoshida et al. 2000, ApJ 544, L87). 47
5.4 5.4.1 a(t) Einstein de Sitter m f(x, p,t) dp dt L = 1 2 ma2 ẋ 2 mφ(x) (5.4.1) f t + p ma f 2 x m φ x f p =0 (5.4.2) p = ma 2 dx dt, (5.4.3) = m φ, (5.4.4) 2 φ = 4πGa 2 ρδ = 4πGm f(x, p,t)dp. (5.4.5) a a(t) t 2/3 Einstein de Sitter (5.4.2) f(x, p,t)=t 3β ˆf(x/t β 1/3, p/t β ), (5.4.6) (5.4.6) P i (k) k n i, (5.4.7) β ξ(x, t) = 1 n 2 a 6 dp 1 dp 2 [ f(x 1, p 1,t)f(x 2, p 2,t) f(x 1, p 1,t) f(x 2, p 2,t) ], (5.4.8) (x x 1 x 2 ) (5.4.6) ξ (5.4.7) ξ(x, t) =ˆξ(x/t β 1/3 ). (5.4.9) ξ(x, t) x (n i+3) t 4/3. (5.4.10) 48
(5.4.9) (5.4.10) β = n i +7 3(n i +3) (5.4.11) stable clustering x x { na 3 t 0 } 4πx 2 dx[1 + ξ(x, t)] +4πa 2 x 2 n[1 + ξ(x, t)] v 21 (x, t) =0. (5.4.12) v 21 (x, t) ξ (5.4.12) 1 { ξ(x, t)+ x 2 v t ax 2 21 (x, t) [1 + ξ(x, t)] } =0. (5.4.13) x (ξ 1) ṙ 21 =0= v 21 (x, t) +ȧx, (5.4.14) 5.21 5.21: The normalized meanpairwise peculiar velocity, v 12 /Hr as a function of f(z) ξ(r; z) at different redshifts. Sold lines indicate the fitting formula proposed by Caldwell et al. (2001). (Fukushige & Suto 2001) 49
(5.4.13) (1 + ξ) =ȧ t a g 1 x 2 x [x3 (1 + ξ)] (5.4.15) ξ = a 3 g(ax) (5.4.16) ξ 1 1 (5.4.7) (5.4.9) (5.4.11) (5.4.16) g(w) w 3(n i+3)/(n i +5) ξ(x, t) x 3(n i +3) n i +5 t 4 n i +5 x 3(n i +3) n i +5 a 6 n i +5 (5.4.17) 5.22 5.22: Two-point correlation functions in the simulations scaled according to (a) conventional self-similar solution, and (b) quasi-self-similar solution with scale independent slope. The solid lines are the results for Ω 0 = 1 models, and the symbols are those for Ω 0 =0.1 models with either λ 0 =0orλ 0 =0.9. Error bars are estimated from three realizations for each model; only the error bars for n = 2 and(ω 0,λ 0 )=(0.1, 0.9) are shown. Other models have smaller error bars, which are not shown for clarity. The arrows correspond to ξ(x) = 100 for each model. (Suginohara, Taruya & Suto 2002) 50
5.4.2 Hoffman & Shaham (1985) t = t i turn around (maximum expansion) t = t i r = r i W i = GM i r i = Ω ih 2 i 2 K i = H2 i r2 i 2 maximum expansion t = t m = G 4πri 3 r i 3 ρ [ i 1+ δi (<r i ) ] = 4πG [ 3 r2 i Ω i ρ c,i 1+ δi (<r i ) ] [ 1+ δi (<r i ) ] (5.4.18) r 2 i = W i Ω i (1 + δ i ). (5.4.19) K m =0, W m = GM i r m K m W m = r i r m W i = K i W i = r m r i = 1 1 Ω 1 i (1 + δ i ) 1 = 1+ δ i = r i r m W i. (5.4.20) [ ] 1 Ω i (1 + δ i ) 1 W i. (5.4.21) 1+ δ i Ω 1. (5.4.22) t = t i r i δ i (<r i ) Ω 1 i 1 ( ) t = t m 1+ δ i r m (r i )== 1+ δ i Ω 1 r i (5.4.23) i P (k) k n ξ(r) r (n+3) r c δ 0 (r i r c ) ( ) δ i (<r i )= ri (n+3) δ 0 (r i r c ), (5.4.24) r c maximum expansion ( ) ri 2 dr [ i ρ m (r m )=ρ i (r i ) =Ω i ρ c,i 1+ δi (<r i ) ] ( ) ri 2 dr i (5.4.25) r m dr m r m dr m 51 i
(5.4.23) 1 Ω 1 i ( ) r0 (n+3) δ 0 (5.4.26) r c r 0 r m dr m dr i ρ m 1 δ 0 r n+3 c 1 δ 0 r n+3 c (r n 3 i r n 3 i r i r n 3 0 (n +4)r n 3 i (r n 3 i (n +4)r n 3 i r n 3 0 ) 4 r n 3 0 (5.4.27) (5.4.28) r n 3 ) 2 0 r n 3 0 }{{} r 3(n+3) i rm 3(n+3)/(n+4). (5.4.29) r i r 0 r f r m GM i r m = K f GM i = 1 GM i GM i r f 2 r f r f = GM i 2r f r f = 1 2 r m (5.4.30) r f r m (5.4.29) ρ f (r f ) r 3(n+3)/(n+4) f (5.4.31) 5.2: stable solution n ξ stable ρ Hoffman Shaham 0 x 1.8 r 2.25-1 x 1.5 r 2-2 x 1 r 1.5 52