國
立
交
通
大
學
物理研究所
碩
士
論
文
Brane 宇宙學在 Bianchi I, Bianchi III 以及
Kantowski-Sachs 模型下的時空研究
Brane Cosmology on Bianchi I, Bianchi III
and Kantowski-Sachs Spaces
研 究 生：鍾旻峰 (Ming-Fun Chun)
Brane 宇宙學在 Bianchi I, Bianchi III 以及 Kantowski-Sachs 模
型下的時空研究
Brane Cosmology on Bianchi I, Bianchi III and Kantowski-Sachs
Spaces
研 究 生：鍾旻峰 Student：Ming-Fun Chun
指導教授：高文芳 Advisor：Win-Fun Kao
國 立 交 通 大 學
物 理 研 究 所
碩 士 論 文
A ThesisSubmitted to Institute of Physics College of Science
National Chiao Tung University
in partial Fulfillment of the Requirements For the Degree of
Master in Physics
國
立
交
通
大
學
博 碩 士 論 文 著 作 權 授 權 書
本授權書所授權之論文為本人在國立交通大學(學院)物理研究所 理論 組， 92 學年度第 2 學期取得碩士學位之論文。
論文名稱：Brane 宇宙學在 Bianchi I, Bianchi III 以及 Kantowski-Sachs 模型下 的時空研究 指導教授：高文芳 ■ 同意 □ 不同意 (國科會科學技術資料中心重製上網) 本人具有著作財產權之上列論文全文(含摘要)資料，授予行政院國家科學委員會科學技術資料 中心(或改制後之機構)，得不限地域、時間與次數以微縮、光碟或數位化等各種方式重製後散 布發行或上載網路。 本論文為本人向經濟部智慧財產局申請專利(未申請者本條款請不予理會)的附件之一，申請文 號為:______________，註明文號者請將全文資料延後半年再公開。 ■ 同意 本人具有著作財產權之上列論文全文(含摘要)，授予國立交通大學與台灣聯合大學系統圖書 館，基於推動讀者間「資源共享、互惠合作」之理念，與回饋社會及學術研究之目的，國立交 通大學圖書館及台灣聯合大學系統圖書館得不限地域、時間與次數，以微縮、光碟或其他各種 數位化方式將上列論文重製，並得將數位化之上列論文及論文電子檔以上載網路方式，於著作 權法合理使用範圍內，讀者得進行線上檢索、閱覽、下載或列印。 論文全文上載網路公開之範圍及時間： 本校及台灣聯合大學系統區域網路 ■ 中華民國 94 年 7 月 20 日公開 校外網際網路 ■ 中華民國 94 年 7 月 20 日公開 上述授權內容均無須訂立讓與及授權契約書。依本授權之發行權為非專屬性發行權利。依本授 權所為之收錄、重製、發行及學術研發利用均為無償。上述同意與不同意之欄位若未勾選，本 人同意視同授權。 研究生：鍾旻峰 學號： 9127508
國 家 圖 書 館
博碩士論文電子檔案上網授權書
本授權書所授權之論文為本人在國立交通大學(學院)物理研究所 理論 組， 92 學年 度第 二 學期取得碩士學位之論文。
論文名稱 ： Brane 宇宙學在 Bianchi I, Bianchi III 以及 Kantowski-Sachs 模型下 的時空研究 指導教授 ： 高文芳 ■ 同意 本人具有著作財產權之上列論文全文(含摘要)，以非專屬、無償授權國家圖書館，不限 地域、時間與次數，以微縮、光碟或其他各種數位化方式將上列論文重製，並得將數位 化之上列論文及論文電子檔以上載網路方式，提供讀者基於個人非營利性質之線上檢 索、閱覽、下載或列印。 上述授權內容均無須訂立讓與及授權契約書。依本授權之發行權為非專屬性發行權利。 依本授權所為之收錄、重製、發行及學術研發利用均為無償。上述同意與不同意之欄位 若未勾選，本人同意視同授權。 研 究 生：鍾 旻 峰 學 號：9127508 親筆正楷：________________ （務必填寫）
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Ê¥¹d2, Bb* Bianchi I,Bianchi III ¸ Kantowski-Sachs _Çá| ê; ø_ÌG (Homogeneity) /Ý®²°4 (anisotropy) 퇋ç_; ª7°) =Ägj˙ íj, Í(n⥠=Ägj˙ íjFf®#BbíÉk‡‹Æ “ím7 Τ5Õ, BbÊ¥¹d2d7sõ½b ícq: øuBbe퇋c _í?¾D¾í}0Ñø_Ü;í¼ñ (perfect fluid) í_, Çø _ucqB bFºÊ0äuø_œJ‡Fwøíû&v˛yÖ|ø_&íü&ív˛, 7¥ü &v˛ (brane spaces) íÜ4”uÊ|¡íùíû˝2øò\«nOí Ê }&ê¥ú__ú=Äg j˙ íj(, BbªJø−Ê¥ú__·}Êw‚ Æ“AÌG (Homogeneity) /®²°4 (Isotropy) í4”, 7/Êw‚‡‹íÆ“ 2, ‡‹b (cosmology constant) AÑ3ûO‡‹Æ“í½b¡b; Êo‚‡‹ íÆ“2, †uBbF5?íü&v˛2íü&¡b (brane constant) AÑ3ûO ‡‹Æ“ í½b¡b; ÇÕ, Bb6)ƒÊBb5?½‰b6}AÑ3ûO‡‹ Æ“í½b¡bív`, 6u Êo‚퇋v, Iwkü&¡b3ûívÈ5(
Contents
1 å 4
1.1 FRW퇋ç_ . . . 6
2 ü&v˛‡‹ç 10 2.1 _ . . . 13
2.2 Brane‡‹çí Bianchi type I,Bianchi type III D Kantowski-Sachs ív˛S . . . 15 3 }&Dn 19 3.1 ñ ÄäíÆ“ . . . 21 3.1.1 ¡Nj5°) . . . 21 3.1.2 ¡NjDbMj5ªœ . . . 24 3.2 Ý®²°4¡bíÆ“ . . . 31 4 ! 34 5 Ë“ 36 5.1 Vacuum Einstein equation . . . 36
Chapter 1
å
‡‹ç (Cosmology) í–Äwõ'oÿ˛%Çáê7, rÖí©4í¬í·} AÐ ú‡‹íÜjDõ¶, ªuâkh¿,í.—, 6Érkïç, /çDÿçí ¸¶ øòƒ7|¡s_0J V, âkçíêDh¿xXíª¥, U‡‹ç kAÑø_ª¾µ, ªbW“íøÆç, 7/rÖíÜ 6ª\„õ Ê 1920 , =ÄgT|72óú, Çó7‡‹çíû˝íÜ! 1929 , é+N¬ ±ŸírÍm$Píh¿, êÛ7‡‹2wFírÍ£TknÚ ±×Ë7íÕG, Ĥ„õ7 ‡‹3z ¤G퇋‡‹ªÓG퇋yÜùA bíßJ, rÖAĤIp7û˝‡‹Æ“í «nDû˝, Bb6uwøÊ1965,Penaias D Wilson êÛ7‡‹š*5¦ (Cosmology Microwave Background), ¥üí5¦Ä²AÅÑ2.7 K, /ÌGí}0k‡‹5È ¥áêÛ„õ7× úÜí£ü, yÇó7Bbû˝‡‹í8, Æ“í½b’ eVÄ â CMB íh¿BbªøÛÊ퇋ÈÓ”D?¾í }0uÌG/®²° 4í â¥s_!…퇋í4”, UBb Friedmann-Roberson-Worker 퇋 ç_ ¥_u5?‡‹AáB·uÌG/®²°4íÕG, OÓOh¿ü íª¥, Bbø−‡‹È CMB í} 0wõx<&íÏæ4æÊ, ¹∆T T ' 10−5, ¥µsBb‡‹í–UÕGª? DÛÊ.°, ĤBb5?àø_ÌGOºÝ®²° 4퇋ç_(Bianchi _,Kantowski-Sachs _), VªWû˝‡‹íÆ“
zƒü&v˛í‡‹çí–Ä, oÊ 1919 , >˚gí/bçð Kaluza FÜ jƒ½‰ªJàû&˛ Èív˛©/ñí0V8JÔ„, µóÚ~Tà‰øš6ª Jàü&˛Èív˛©/ñí0V[p [®Êü&v˛-˛2í2óúj˙
˙8½‰øà%í·H, Dw°v|ÛíÇø j˙ Êbç ,†óçkïs= níÚ~çj˙ Bb˚ü&v˛Ñ 3-brane , ¹uû&v˛\9òAø_ü&í ø _Þq, 7µü&†Êµü&v˛ (Bb˚Ñ bulk) q Í7, Ñ7.°¥½‰ íõð!‹, ¥ü&@v b”Ñüí (üƒ Planck scale í ) ¸˛h¿. ƒí 7Ê(V, :tÓÜçð Klein †uz¾äÓ Üçíhõ¯9k Kaluza íü &˛ÈíÜ2, F½hŸ7Œj˙ , Jü_‰bHŸVíû_, *7 „p vj˙ íj„ªJ\Ô„ÑÅä-šy½‰ÒDÚ~Òù6 à-5«íí·H .¬¤j¶(V. u\wÑW.¦
OuÊ 1980 H‚È, âk$øÜí Kaluza-Klein hõ½h\T|V, ø <ÓÜçðwÑ#‰Dÿ‰ k uªJ\––Ví kuøòƒÛÊ, Fh| Ví paper 2, øšíM¾·uy} brane 7¥¡‚íû ˝¢J Randall-Sundrum [1] [2]íóÉjÞípaper |ѽb, FbFdíû˝¡jƒ Brane Cos-mology D™Äíû&í cosCos-mologyÊw‚‡‹uó°í 7/6¡jƒ, à‹¾ "¾uÜ;¼ñí$ , µÉø<Ôí Homogeneous‡‹u\orí FJÊ ¥³BbJÓ Randall-Sundrum [1]íû˝, 7/5?¥brane 퇋uÊçBbí bulk space Êü&í Anti-desitter space. Í(Bní_uø_ÌG (Homo-geneity), Ý®²°4 (Anisotropy) 퇋ç_;Bianchi type I,Bianchi type III ¸ Kantowski-Sachs _ Êî-Ví 1.1 2, Bbøsízpø-øOBbzƒí‡‹ç, ø_ÌG/ ®²°4í F-R-W í ‡‹ç, J£¤_í!…4”; Í(Êùı2, Bbø èü&v˛‡‹çí!…Ü-ZD*, éBbúü&v˛í-Z_ÀUíÜj; |(Ê2.12Bb†b×–ËzpBbFUàí BI,BIII D KS _ Ê2.22, Bb}úàS)ƒ=Ägj˙íjí¥ døIíRûDzp, Ì íRûø[ 0Ë“ Êúı2BbøúBbj)í=Ägj˙Fbdí}&D«n¥¥³‹ Jn |(Êûı#8!
1.1
FRW퇋ç_
=Ägí&2óúíT|, zp7vÈD˛Èu.ª}’í/ó àí, ¥ Hú7*‡HvH!… íúvÈD˛Èíw… yÊ1920, =ÄgT|7.&2 óú(T|72óú, ª‚uú½‰«n íø_hj, 7/6AÑBbû˝ ‡‹çí'àíÜ x Bbí=Ägj˙ (Einstein Equation) ª*Tà ¾ŸÜ (Action Principle) °), 7¨Ö7½‰íTà¾í2$ ªJ[ýÑ: S = − Z d4x√g( R 16πG+ Lmatter) (1.1)¥³ R F[ýíu-0"¾ (Riemann curvature tensor),G †u½‰ b (Gravitational constant), 7LmatteruÓ”Òí Lagrangian. ÊBbúgµνd
‰}(, Bb¹ª°)=Ägj˙ :
Rµν −1
2gµνR = 8πGTµν (1.2) Ê¥³Tµνu?¾¾"¾ (Energy Momentum Tensor),gµνu Metric
Ten-sor, ¥ Metric Tensor ÿßdøzàV¾‡‹íøz , ¾O‡‹í×ü, 7 Ricci scalar(R) ¸ Ricci "¾ Rµν†µsBb˛Èí˙ FJ¥=Ägj
˙ íUí¬iuÓ”D?¾, 7¥Ó”D ?¾àS àBb˛Èí†ÊU í˝i[Û|V
;WBbÊ× - (≥ 100Mpc) h¿‡‹*5¦ (Cosmic Microwave Background (CMB)) í!‹, Bbø−BbÛÊ퇋uÌG (Homogeneity) / ®²°4 (Isotropy) í, ¥ÌG4µsBb, BbFh¿ƒí Óܾí4”DwF ÊíP0ÌÉ; 7®²°4µsBb, Fh¿íÓܾʮj²,w4”Qó° B b¦ ˚¥s_!…4”ч‹çŸÜ *=Ägí2óú2, Ãí„p7 ¯¯‡‹çŸÜí˛Èx|ò íú˚4, 7Êbç,, x|òú˚í4íú&˛ ÈÑ0kbíú&7Þ ¤ú&7ÞªJTÑû& íò˛Èí7ÞTÜ Friedmann-Roberson-Worker(F-R-W) 퇋ç_ÿuJ,HŸ†F°|, ª
7ª â¤_úBb‡‹íÆ“)ƒø_ßí·H ¥ F-R-W _í(jÑ:
ds2 = −dt2+ R2(t)( dr2
1 − kr2 + r
2dθ2+ r2sin2(θ)dφ2) (1.3)
Ê, 2R(t)Ñ Ää (Scale Factor), ¥ ÄäuÓOvÈÊÆ“í, ÄäíÆ“6zp7‡‹ ˛ÈíÆ“ k Ñ˛Èí0b, ¥³k = 1, 0, −1, çk = 0v, Bbí˛Èuø_ò˛È; çk = 1v, ªzBbFÊí˛Èõduø _7í[Þ; çk = −1v, ªzBbFÊí˛Èõduø_ï¶Þ r, θ, φÑ7è ™,t ÑvÈ ¤Õ, ¥(jds2Å—óú2í!…4”, ¹Ê虲2 (Lorentz
Transformation) wM\M.‰ ¥µsBb, ÊBbdû˝Ch¿v, BbªJ² ¦ø_|_¯í è™Vd«n, 7!‹u.‰í y6 Metric Tensor gµνÊ(jí
ì2-ÉÊ trace ,M: gµν = gii= (−1, R2(t) 1 − kr2, R 2(t)r2, R2(t)r2sin2) (1.4) âk F-R-W _íòú˚4BbªJÀí°) Christofol symbol(Γi jk) íÝÉ}¾: Γijk = 1 2g il(∂glj ∂xk + ∂glk ∂xj − ∂gjk ∂xl ) (1.5) Γ0ij = R˙ Rgij Γi 0j = ˙ R Rδ i j
1.4 ¹Ñ Christofol symbol íì2 ¥ Christofol symbol xÝÉMí |Û, µsBb˛Èuí4” ¥ Ricci tensor íì2ÑRµν = gαβRαβµν,
7Rαβµν uF‚í-0, ¥-0ì2OBb˛Èí˙ Í(Bb¹ª °)˛Èí0íù¼"¾ Ricci Tensor íÝÉ}¾: R00 = −3 ¨ R R
Rij = −( ¨ R R + 2( ˙ R R) 2+ 2k R2) (1.6) J£É¼"¾ Ricci Scalar: R = −6(R¨ R + ˙ R2 R2 + k R2) (1.7) 7¥·H Ää R(t) Æ“í‰çj˙ ªJ*j=Ägj˙ )ƒ: Rµν − 1 2gµνR = 8πGTµν (1.8) Ê¥³TµνBb¦íuÜ;¼ñí$ , w}¾Ñ T00 = −ρ , T11 = T22 = T33 = p. ρч‹È?¾ò,p Ñ9‰ Êjj˙ vF)ƒí 00 }¾íj˙ , øOVzB b˚d Fridmann Equation: ˙ R2 R2 + k R2 = 8πG 3 ρ (1.9) Í(, BbªJªø¥íø Fridmann Equation ZŸÑ: k H2R2 = ρ 3H2 8πG − 1 ≡ Ω − 1 (1.10) ¥³Ωu?¾òú critical density ρcíªM
Ω ≡ ρ ρc , ρc ≡ 3H 2 8πG , 2,H ≡ R˙
RuF‚íé+b (Hubble constant). ;W¡‚íh¿ ‡‹*
5¦í’e2Bbø−Ω → 1, ¥6<âOk → 0 ¥ zp7Bb퇋uà¤í Ë, Ê× -Bb퇋˛È¡˛òí˛È
¤Õ, Êh¿2Bb¢ø−, ¤=Ägj˙ .â½hzç=ÄgÑ7 Uj˙ ®ƒÓ¢ ‡‹í‘K7‹p퇋b (O(VêÛ‡‹wõuGí7¢ < ) ‹=Ägj˙2nªªø¥j„h¿ )!‹ FJ=Ägj˙ª½ŸÑ: Rµν − 1 2gµνR = 8πGTµν − Λgµν (1.11) , 2,Λ¹Ñ‡‹b Êø<Àíl( (k=0) Bbª)ø: R(t) ∝ e√Λ3t (1.12) H ≡ R˙ R = s Λ 3 (1.13) ¥zp7BbÛÊ퇋uJNbí$ e √Λ 3tAÅ, 7‡‹çbΛ A7 à‡ ‹Æ“í½b¡bHué+b, ¥bíì2Ñ: v = H ∗ r (1.14) vѱjír7±×Bbí§,rѱjír7DBbí × ¥é+ì µsBb, ±ír7±×Bb í§0 FJâ Eq1.13ªJø−‡‹b àO±jír 7±×Bbí§ 7¥_‡‹bΛuB óá? ƒÛÊ´³ö£íüì-V, Ê …¹d2Bb6.‹«n
Chapter 2
ü&v˛‡‹ç
Bbø−ü&v˛í;¶oÊ1919ÿAT|, OÊo‚1.§½e, òƒ1980 HwÈ, âk$øÜí Kaluza-Klein hõ½h\T|V, ø<ÓÜçðwÑ# ‰Dÿ‰ kuªJ\––Ví kuøòƒÛÊ, Fh|Ví paper 2, øšíM¾·uy} brane 7¥¡‚íû˝¢J Randall-SundrumíóÉjÞ í paper [1] [2] Ñ3b, FbFdíû˝¡jƒ brane cosmology D™Äíû& ícosmology Êw‚‡‹uó°í 7/6¡jƒ, à‹?¾¾"¾ uÜ;¼ñí $ , µÉø<Ôí Homogeneous ‡‹u\orí
Ê Randall-Sundrum í_2, Bb™ºÊíû&v˛uü&v˛2íø_ Þ, 7Ö|VíËü&uòk û&v˛í˛Èè™W ü&í˛ÈøOBb˚Ñ bulk, 7wÞ˚Ñ3-brane 7¥^í½‰j˙Ê brane ,í$ Ñ:
Gµν = Rµν − 1
2gµνR = −gµνΛ + k
2
4Tµν+ k54Sµν − Eµν (2.1)
¥³GµνNíu Einstein "¾, H[O=Ägj˙ 2Sí¶}TµνÑ?¾
¾"¾, gµνu brane(4D) ,íd"¾ , 2í¡bΛ, k42, k54}H[O‡‹ç
b, û& v˛íµ¸í½‰b (k2
4 = 8πG), ü&v˛2µ¸í½‰bEµν†
ÑÊü&2í¾ 7¥ û&v˛2퇋b, µ¸½‰bDü&v˛2 bulk í ‡‹bΛ5,bulk 2íµ¸½‰bk5 , £ö˛2í?¾λ,λøÉ[ : Λ = k52(Λ5 2 + k2 5λ2 12 ) k 2 4 = k4 5λ 6 (2.2)
Sµνu?¾¾"¾TµνíùŸµ¸íõ.,EµνÑ bulk ,í^@,
Eµν = Ciajbnanb (2.3)
¥³Ciajbnanb˚Ñ bulk 2íü& Weyl tensor ¤Õ, Ê Eq2.12µ¸í?¾¾
"¾Sµν(1Ýü&2í?¾¾"¾) ªJâTµνµ¸í[ýÑ: Sµν = 1 12T Tµν− 1 4T α µTαν + 1 24gµν(3T αβT αβ − T2) (2.4) Ê¥³,TµνBbFû˝íuø_xóçú˚4/.µÆí$ , FJ²¦íuÜ; ¼ñí$ , Ü;¼ñí?¾¾"¾íÝÉ}¾Ñ: T0 0 = −ρ T11 = T22 = T33 = p (2.5) ¥?¾¾"¾¦ÑÜ;¼ñí$ µsBb, ÊBbªeÑ”õíÓܾ5È˛¤ ³>Tà (àÏf... ) ¤Õ,T = Tµ µu¥?¾¾"¾í trace ¹T = Tµ µ = T00 + T11+ T22+ T33 ¥³‡‹¼ñí?¾òρ¸9‰ p ʇ‹ç,¯¯¤ É[ : p = (γ − 1)ρ (2.6) ¥³γÑø_b/w¸ˇÑ 1 ≤ γ ≤ 2 .
;Wj˙ Eq2.1 , BbªJêÛƒ, çEµν = 0/k5 → 0v, BbøƒøO
íÝ brane 2í=Ägj˙ , ªc¥5?ü&v˛í‡‹ç_œû&ív˛ 퇋ç_yÑ2 ÊBbí¥¹ d2, ø.5?Eµνí bulk ,í^@, ¹
IEµν = 0VdÑBb«nü&v˛‡‹íø_!…c q
¤Õ, Bb67jƒÊ brane ,6¯¯?¾è0ì , 7Ê2óú2?¾è ì ª[ýÑ:
¥³∇Ñu‰} (Covariant Differentiation), Fuû&v˛2í}, TàÊù ¼"¾í?¾¾ "¾5$ Ñ:
∇µTµν ≡ T;µµν = T,µµν+ TανΓµαν + TµαΓναµ (2.8)
Ê5(Bb}êÛ, Êj7=Ägj˙ D¤?¾è0j˙ íjd© j˙(, B b¹ª°)BbF;û˝í ‡‹íÆ“, ¹ ÄäíÆ“j˙
2.1
_
Êõð,, ÓOh¿üíª¥, Bbø−‡‹È CMB í}0wõx<í Ïæ4æÊ, ¹ ∆T
T ' 10−5, ¥µsBb‡‹í–UÕGª?DÛÊ.°; ÊÜ,,
;W paper [5]í TBbø−àb5?UàÜ;¼ñ_í?¾¾"¾, ø_Ý ÌG(Inhomogeneity) 퇋ç_u.ªQ§í, ]Bb²Ï7ø_ÌG (Homo-geneity) 퇋ç_, ¥_í ZÊ [6]2#87Ì í„ ÊBb¥¹d 2Bb²Ï7w2ú__Bianchi type I,Bianchi type III ¸ Kantowski-Sachs í_ÄÑ¥ú__wõ'óNíbç$ , FJªJ[Êø–«n kuBb ø_¨Ö7ú__í(j$ :
ds2 = gµνdxµdxν (2.9)
= −d(t)2dt2+ a
1(t)2dr2+ a2(t)2(dθ2+ f (θ)2dϕ2)
Ê, 2,d(t), a(t), b(t)ÑÓOvÈÆ“í Ää,r, θ, ϕÑ7è™ ÓOf (θ)í .°ì2, BbªJ})ƒú_Ý®²°4퇋ç_: f (θ) = θ , Bianchi type I sin θ , Kantowski-Sachs sinh θ , Bianchi type III
Ñ7yÑjZ/yÑÀUí7j‡‹íÆ“, BbÊ¥³ùpø<‰b:
V ≡ a(t)b(t)2 , volume scale f actor (2.10)
Hi ≡
˙
ai
ai
, i = 1, 2 , directional Hubble f actors (2.11)
H ≡ 1
3(H1+ 2H2) = ˙
V
3V , mean Hubble f actor (2.12)
A ≡
3
X
i=1
(Hi− H)2
Eq2.10 Ññ í Ää, Bb5(íû˝3bí¶}ÿuÊ«n¥ñ Ä äíÆ“ Eq2.13Ñ ÌíÝ®²°4í¡b, BbªJ*¥Ý®²°4¡bÓvÈ íÆ“V«n¤ú__íê$
2.2
Brane‡‹çí Bianchi type I,Bianchi type
III D Kantowski-Sachs ív˛S
ÛÊBb£øb«n¥ú_ÌG/Ý®²°4퇋ç_ Bianchi type I,Bianchi type III D Kantowski-Sachs(5(BbŸÑ BI,BII,KS) Êü&-ív˛!Z Bbl*‡ø2ªJø−BbF²¦í_í(jds2í$ , ø−7(j(Bÿø
−7¥_íd"¾
gµν = (−1, a1(t)2, a2(t)2, a2(t)2f (θ)2) (2.14)
⥾‡‹íøz (Metric Tensor) |ê, Bb-ø¥b°) Christofol symbol(Γi
jk) ¥µsBb‡‹uíí"¾,
Í(y°)[Û˛Èí˙í-0Rαβµν, yâ-0)ƒ Ricci"¾RµνD Ricci 0bR(, Bb|(y
z¥<Bbü&v˛í=Ägj˙ : Rµν − 1 2gµνR = −gµνΛ + k 2 4Tµν + k45Sµν (2.15) Ê, 2, U¬iÑÓ”D?¾í¶},Λч‹çb, ?¾¾"¾Tµνàl ‡Fì2í BbcqÑÜ;¼ñí$ ,Sµν†Ñ?¾¾"¾íµ¸¾, Ê¥<ÓÜ ¾üì5(, BbZªJÇá d"¾l, âk¬˙µÆõµ, FJBb}øwÌ í¬˙[ÊË“³, 7Êdí£d2c|½b!‹ FJÊ%¬øJl(, Bb ªJ)ƒü&v˛-=Ägj˙ íú_}Ñ00,11,22}¾íj: (a˙2 a2 )2+ 2(a˙1 a1 )(a˙2 a2 ) + k a2 2 = Λ + k42ρ + 1 12k 4 5ρ2 (2.16) 2a¨2 a2 + (a˙2 a2 )2+ k a2 2 = Λ + k2 4ρ(1 − γ) + 1 12k 4 5ρ2(1 − 2γ) (2.17) ¨ a1 a1 + a¨2 a2 + (a˙1 a1 )(a˙2 a2 ) = Λ + k2 4ρ(1 − γ) + 1 12k 4 5ρ2(1 − 2γ) (2.18)
Ê, 2k = 0, 1, −1v}H[O BI,KS D BIII _,ρÑÓvÈÆ“í?¾ ò, 7γѸˇÊ 1 ≤ γ ≤ 2í¡b Bb*¥ú_j˙ 2ªø−BbF)ƒí u ÄäíÆ“D?¾òÈíÉ[ OuÄÑ?¾òÓOvÈZ‰U)BbÌ ¶ÀUí7j‡‹íÆ“, FJBb´Ûb5?Êóú2í?¾ è0ì : ∇µTµν = 0 (2.19) ;W¥u‰}∇TàÊ?¾¾ù¼"¾, %¬øJíl(BbªJ)ƒÓO vȉ“í?¾òD ÄäOøìíÉ[ : ˙ρ + ργ(a˙1 a1 + 2a˙2 a2 ) = 0 (2.20) FJBbø−¥¼ñí?¾òíÓOvÈíÆ“ì Ñ(V = a1a22): ρ = ρ0V−γ , ρ0 = constant > 0 (2.21) â¥?¾òí°), BbªJ7jƒBbF)ƒí=Äj˙ íjÿÉ” - ÄäuÓOvÈÊZ‰í, FJBbZªJ«n ÄäíÆ“7 Ê¥³B bÑ7;œÀí°)ñ Ääíj, Bbªøj˙ 2.16,2.17,2.18ZŸA: d dt(V H1) = ΛV + 1 2k 2 4ρ0V1−γ(2 − γ) + 1 12k 4 5ρ20V(1 − 2γ)(1 − γ) (2.22) d dt(V H2) = ΛV + 1 2k 2 4ρ0V1−γ(2 − γ) + 1 12k 4 5ρ20V(1 − 2γ)(1 − γ) − ka1 (2.23) 3 ˙H + H2 1 + 2H22 = ΛV + 1 2k 2 4ρ0V−γ(2 − 3γ) + 1 12k 4 5ρ20V(− 2γ)(1 − 3γ) (2.24) ÊJ,íj˙ 2, h‰bíì2Ñ: H1 = ˙ a1 a1 , H2 = ˙ a2 a2 , V = 1 3(H1+ 2H2) = ˙ V 3V
î-V, Bby‹,Ó”?¾òíÆ“ì Eq2.21, Ê%¬øJíHb« (BbøªJ°)ñ Ää íùŸ}íj˙ : ¨ V = 3ΛV + 3 2k 2 4ρ0V1−γ(2 − γ) + 1 4k 4 5ρ20V1−2γ(1 − γ) + 2ka1 (2.25)
Bbø−, ç k=0,k=-1,k=1 Bb}ªJ)ƒ Bianchi type I, a Bianchi type III,and a Kantowski-Sachs íj Í(Ê;W¶} }íl(, BbªJ)ƒñ Ää íøŸ}íj˙ : ˙ V2 = 3ΛV2+ 3k2 4ρ0V2−γ+ 1 4k 4 5ρ20V2−2γ+ 4k Z a1dV + c (2.26) , 2,c Ñ }b ¥øŸ}íj˙ øÑBb5(ní3bj˙, Bbª Jpéíõ)|V¥_j ˙ uÝ(4í, 7/Ê%¬rÖ›‰þtíl-, Bb ´u̶°)wj&j, Ĥ, Ê5(í}&Dn 2Bbøþt°w¡Nj ¤Õ, BbªJø_œÑ2íj: t − t0 = Z L(V )−12 dV (2.27) ¥³ t ÑvÈ,t0Ñ–ávÈ,V Ññ Ää 7ƒb L(V) íì2Ñ: L(V ) = 3ΛV2+ 3k2 4ρ0V2−γ+ 1 4k 4 5ρ20V2−2γ + 4k Z a1dV + c (2.28) â¥2í$ , BbªJòQí°)Ý®²°4¡b A D Ääaií2j: A = 3K2L(V )−1 (2.29) ai = a0V 1 3e Ki R dv vL(V )12 (2.30)
Ê, 2, ¡bK2, K iBbªJ;Wj˙ Eq2.22,Eq2.23)ƒ: Hi = H + Ki V (2.31) K1 = 2R kaidt 3 K2 = −1R kaidt 3 (2.32) K2 = Σ3 i=1Ki2 (2.33) âj˙ Eq2.21,Eq2.28,Eq2.32 p Eq2.22,Eq2.23 BbªJ)ƒ K D } b c 5ÈOøìíÉ[ : K2 = 2c 3 (2.34)
Chapter 3
}&Dn
ÊêA,ıíú__íRû(, Bbíí½õubû˝‡‹íÆ“, FJBbÊ ø ø«n‡‹í Ää ÊÓOvÈÆ“-, ø}êAS$ ? Dw2í ÓܾSÉ[? OÄÑBb̶òQ°)¤ñ Ä äj˙ íòQj, ]Bb ø«nÊS¯ÜívÈ–Èq, ªJ°)ñ Ääí¡Nj 7¥_¡Njí $ u´£ü, Bbø‡úÔìí–U‘KUàÚ7,ñ Mathematica D Matlab V°)wbMj, Í(yDBbí¡NjT_ ªœ¹ªø−Bbí¡Nj„´£üJ £ªWívȖȸˇ Bbbû˝íñ Ääíj˙ Ñ: ˙ V2 = 3ΛV2+ 3k42ρ0V2−γ+ 1 4k 4 5ρ20V2−2γ − 4k Z a1dV (3.1) BbªJÊJ,íñ Ääíj˙ 2, ʯÜí‘K-)ƒ¡NjV , Í( BbZbz¤jp-øj˙ 2Êó°í‘K-)ƒ Ääa1 d dt(V H1) = ΛV + 1 2k 2 4ρ0V1−γ(2 − γ) + 1 12k 4 5ρ20V1−2γ(1 − γ) (3.2) Ê¥³Bbíøá½õÿu°) Ääa1, Í(BbÿªJø−4k R a1dV íÆ “, %â4kR a1dV áíbMj(BbZªJø−¤ú__5Ïæ4 ¤Õ, Ääa2†ªâl‡ ñ Ääíì27°): b(t) = v u u tv(t) a(t) (3.3) BbÊTbM}&vF¦í–U‘KÊ¥³lN|, Bbcqa(t0) = a0 , b(t0) = b0 , b(t˙0) = ˙b0 , v(t0) = v0 ¤ t0 Ñ–ávÈ, Ñ7b°)ø_Æ“,í vÈ–È Bbú¡bø#8øcqíM:γ = 3 2,3Λ = p = 1 , 3k24ρ0 = m = 1 , 1 4k45ρ20 = n = 1úk–á‘KBb#ì:t0 = 0, a0 = 0.1, b0 = 0.1, ˙b0 = 1. ÄÑB bÿÛÊíh¿Vz, Bb´.ÀUBb‡‹í–U‘KÑS, ]Ê¥³í–á”· Éu_bç,ícq 7Bbÿø*¤cq|ê «n‡‹íÆ“ Êù2, Bb ø‡úÝ®²°4í¡b (ÓÜ¡b) V«n¥ú__í¥_Ý®²°4í4”í Æ“ 7¥Ý®²°4í¡bÑ: A ≡ 3 X i=1 (Hi− H)2 3H2 (3.4) ¥³«n¤Ý®²°4í¡b6uUàõ7_Òíj Võ¤ÓÜ4”íÆ“8$ ¤Õ, Bbø‡ú γ = 1,γ = 3 2,γ = 2, VTú__í}&
3.1
ñ ÄäíÆ“
BbIγ = 3 2, /Uñ Ääí–áMÑ 0.001 VçTbM}&íáM, 7w Fí¡bqì àl‡Fì2, Bb«nªd¡Njíáb×kwF®áí¸10,100,1000 ÝBƒ10000IVdÑBbí‘K, ¤ ÕBb6øú3k2 4ρ0d<^Z, UwÓ×100I J£1000IVdÑBb}&í‘K J-Ê3.1.12 Bbbl«nBb;)ƒíñ ÄäÜ,í¡NjÑSbç$ , Ê 3.1.2 2BbøzÜí!‹D bM jí!‹Tø_Ò¸V«nwÓÜ<23.1.1
¡Nj5°)
ílBbølª?í¡Njí_8”: 1. à‹3ΛV2Ñ3, ¥<2[ýO3ΛV2 À 3k2 4ρ0V2−γ+14k45ρ20V2−2γ+4k R a1dV , †Bbí¡Njj˙Ñ: ˙ V2 ' 3ΛV2 (3.5) Í(%¬Àíl(, BbªJ)ƒñ Ääíj˙ Ñ: v(t) = c1e √ 3Λt (3.6) Bb6ªJ)ƒ Ääa1(t)Da2(t) a2(t) = c2e √Λ 3t , a1(t) = c1 c2 2 e√Λ3t (3.7) , j˙2c1 D c2u }b:c1 = v0e− √ 3Λt0 , c 2 = b0e− √Λ 3t0e− √ 3Λt0 Í( çBbªœ f (t) ≡ | 3k 2 4ρ0V2−γ+ 14k45ρ20V2−2γ+ 4k R a1dV 3ΛV2 |¿ 1 (3.8)(, Bb¹ªJø−wª¡N–ÈÑS Ê_Ò-Bbø− Ä䪦Ѥ¡ Njí8”uÊ/ø¨vÈõ(øòƒÛÊívÈ 2. à‹1 4k54ρ20V2−2γáÑ3, ¥<2[ýO14k45ρ20V2−2γ À 3ΛV2+3k42ρ0V2−γ+ 4kR a1dV †Bbí¡Nj˙Ñ: ˙ V2 ' 1 4k 4 5ρ20V2−2γ = nV2−2γ (3.9) Ê%¬Àíl(, BbøšªJ)ƒñ Ääíj: v(t) = (√nγ(t − t0))1γ (3.10) Í(BbªJ)ƒ Ääa1(t),a2(t)íjÑ: a2(t) = c4(t − t0) 1 3γ , a 1(t) = 1 c2 4 (γ√n)γ1(t − t0)3γ1 (3.11) , j˙2t0 = t 0− v γ 0 γ√n , c4 = b0( vγ0 γ√n) 1 3γ , çγ = 3 2vBb Ääíø _Ô8”-íj: v(t) = (3 2 √ n(t − t0))23 b(t) = c4(t − t0) 2 9 , a(t) = 1 c2 4 (3 2 √ n)23(t − t0) 2 9 ÊBbªø¥íªœ f (t) ≡ | 3ΛV 2+ 3k2 4ρ0V2−γ + 4k R a1dV 1 4k45ρ20V2−2γ |¿ 1 (3.12) (Bb¹ª˛ø−wª¡NívÈõÑS ÊBb_Òí-, Bbø−ñ Ä䪦Ѥ¡NjívÈõuÊ vÈ¡kÉív`
3. à‹3k2 4ρ0V2−γáÑ3, †¡Njj˙Ñ: ˙ V2 ' 3k2 4ρ0V2−γ = mV2−γ (3.13) Bbøª)ƒñ Ääí¡Njí$ Ñ: v(t) = (γ √ m 2 (t − t 0))2γ 6ª°) Ääa1(t),a2(t)Ñ: b(t) = c3(t − t0) 2 3γ , a(t) = 1 c2 3 (γ √ m 2 ) 2 γ(t − t0) 2 3γ (3.14) ¥³í®¡bÑ:t0 = t 0 − γ√2mv γ 2 0 , c3 = b0(γ√2mv γ 2 0) −2 3γ çγ = 3 2v, Bbª ˛)ƒ v(t) = 3 4 3(m) 2 3(t − t0) 4 3 283 b(t) = c4(t − t0) 4 9 , a(t) = 3 10 3 (m) 2 3(t − t0) 4 9 283c23 ÊBbªø¥n f (t) ≡ | 3ΛV 2+ 1 4k45ρ20V2−2γ+ 4k R a1dV 3k2 4ρ0V2−γ |¿ 1 (3.15) (Bb¹ª˛ø−wª¦ívÈõÑS ÊBb_Òí-, Bbø−ñ Ä äª¦Ñ¤¡NjívÈõuÊ vÈõ'oí–Èq, / k‡s_¡Nj ívÈ–È5È
3.1.2
¡NjDbMj5ªœ
Bb«nú_‡‹_, …b}u Kantowski-Sachs moddle,Bianchi Type I D Bianchi Type III ‡‹_. QOBbøøønFbÊ3ΛV2,1
4k54ρ20V2−2γJ £ çk2 4ρ0V2−γ¥øá'½bv, …bF)ƒ íñ Ääí¡Nj, 1DbM} &Fd|VíjªWn, ;WÏÏ ükì}5 30 ѪQ§í¸ˇVdÑBb}& níYW™Ä. QOBbZ J.°íáÑ3vVªWn. BbIγ = 3/2, /U ÄäíáMÑ0.001VçTbM}&í áM, 7B bí¡bqìÑΛ = 1/3, k2 4ρ0 = 1000/3 J£ 121k54ρ20 = 1/3. ¥³BbŸ…« n‹×¡bk2 4ρ0VvƒªÒ¸í–È BbêÛÊk42ρ0 > 1000/3vßíÒ¸8”, 7Ê¥³Bbÿɇúk2 4ρ0 = 1000/3VdÇ yŸzpBbF«níñ Ää É[ Ñ: ˙ V (t)2 = 3ΛV2+ 3k2 4ρ0V (t)2−γ+ 1 4k 4 5ρ20V (t)2−2γ + 4k Z a1dV (3.16) QOBbøbM}&)ƒí(D¡Njdªœ, ;WBbúñ ÄäíøŸ }j˙ 2ç®áÑ3ví«n, Bbå|ÏÏÊ10%Jqí úá®AÑ3í–ÈD bMjíÒ¸Ç$, BbÊ¥³ÏÏ Error íì2Ñ Error = (N umerical−aproximation
N umerical )∗ 100%, J-Bb}‡ú.°í_VTÒ¸í}&: (1)KS: Ç3.1Ñ KS _íbMj(BbªJõc¤ñ ÄäíAŧ }íA§ÊvÈÑqÿ ƒ®7íüŸjí×ü; Ç 3.2 ÑÖ‡‹báΛá Ñ3íbMjD_ÒíjíÒ¸, BbªJêÛÊ vÈÑý˝¬ÇádíÒ¸DbM _Òí(¡Níóçíß; Ç 3.3 Ñ brane áÑ3ví¡N¸Ç$, Bb*Ç$2 ªJõ)|VçvÈVQ¡ŸõvbM_Òíñ ÄäDBbí¡NjV ó°, ªJc)ÊvÈõ Q¡Ÿõv brane áíTàu½b; J£Ç3.4½‰ báÑ3ív`¡NjDbMjíÒ¸Ç$
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10x 10 5 Time
Volume Scale Factor (V)
Numerical Plotting Numerical Figure 3.1: KSñ ÄäíÓvÈÆ“í(Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J £ 1 12k45ρ20 = 1/3, γ = 3/2 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 0 1 2 3 4 5 6 7 8 9 10x 10 5
Volume Scale Factor (V)
Dominance of Cosmological Constant
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100 0.005 0.01 0.015 0.02 0.025 0.03 Time Error (V) data1 Numericance Approxivation Figure 3.2: KS_Ê3ΛV2Ñ3v¡NjDñ ÄäÒ¸í(Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121 k45ρ20 = 1/3, γ = 3/2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10−5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4x 10 −3
Volume Scale Factor (V)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10−5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time Error (%)
Dominance of 5−D Gravitational Constant
Error Numericance Approxivation Figure 3.3: 1 4k45ρ20V2−2γÑ3v¡Njʪ¦vÈqD KS ñ Ä äÒ¸í (Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121 k45ρ20 = 1/3, γ = 3/2 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
Dominance of 4−D Gravitational Constant
Volume Scale Factor (V)
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060 2 4 6 8 10 12 14 16 Time Error (%) Error Numericance Approxivation Figure 3.4: 3k2 4ρ0V2−γÑ3v¡Njʪ¦vÈqD KS ñ ÄäÒ¸í (Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121k54ρ20 = 1/3, γ = 3/2
(2)BI: Ç 3.5 Ñ BI _íñ ÄäíbMj(; Ç 3.6 ч‹báΛÑ 3íbMjD¡NjíÒ¸, *¥_ÇBbªJø−ç¤áÑ3vívÈ–ÈuÊ vÈÑû(øòƒ(„VívÈΛáÑ3í¡Nj·} }í¯¯_Òí(; Ç3.7 Ñ brane áÑ3víÒ¸Ç$, BbøšªJõc brane áÊvÈõQ¡Ÿõv ¯¯; Ç 3.8 ѽ‰báÑ3ív`¡NjDbMjíÒ¸Ç$, Bbø−ç½ ‰bíµ¸¡b×vBbª J)ƒyßí¡N8”J£y×íÒ¸–È 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10x 10
5 Numercal Plotting of Bianchi Type I
Time
Volume Scale Factor
Numericance Figure 3.5: BIñ ÄäíÓvÈÆ“í(Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J £ 1 12k45ρ20 = 1/3, γ = 3/2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10x 10
5 Dominance of Cosmological Constant
time
Volume Scale Factor (V)
3 4 5 6 7 8 9 100 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Error (%) Numericance Approxivation Error Figure 3.6: 3ΛV2Ñ3v¡Njʪ¦vÈqD BI ñ ÄäÒ¸í( Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121k54ρ20 = 1/3, γ = 3/2
0 0.2 0.4 0.6 0.8 1 1.2 x 10−5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4x 10 −3
Volume Scale Factor (V)
0 0.2 0.4 0.6 0.8 1 1.2 x 10−5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Dominance of 5−D Gravitaional Constant
Time Error (%) Error Numericance Approxivation Figure 3.7: 1 4k54ρ20V2−2γÑ3v¡Njʪ¦vÈqD BI ñ Ä äÒ¸í (Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121 k45ρ20 = 1/3, γ = 3/2 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Time
Volume Scale Factor (V)
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.060 2 4 6 8 10 12 14 16
Dominance of 4−D Gravitational Constant
Error (%) Error Numericance Approxivation Figure 3.8: 3k2 4ρ0V2−γÑ3v¡Njʪ¦vÈqD BI ñ ÄäÒ¸í (Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121k54ρ20 = 1/3, γ = 3/2
(3)BIII: Ç3.9Ñ_íbMj(; Ç3.10ч‹báΛÑ3íbMjD_Ò íjíÒ¸; Ç 3.11 Ñ brane áÑ3víÒ¸Ç$J£Ç 3.12 ѽ‰báÑ3í v`¡NjDbMjíÒ¸Ç$ 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10x 10 5 Time
Volume Scale Factor (V)
Numerical Plotting Numerical Figure 3.9: BIIIñ ÄäíÓvÈÆ“í(Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J £ 1 12k45ρ20 = 1/3, γ = 3/2 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 0 1 2 3 4 5 6 7 8 9 10x 10 5
Volume Scale Factor (V)
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
Dominance of Cosmological Constant
Time Error (%) Numericance Approxivation Error Figure 3.10: 3ΛV2Ñ3v¡Njʪ¦vÈqD BIII ñ ÄäÒ¸í( Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121k54ρ20 = 1/3, γ = 3/2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10−5 1 1.5 2 2.5x 10 −3
Volume Scale Factor (V)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10−5 0 2 4 6 8 10 12 Time Error (%)
Dominance of 5−D Gravitational Constant
Error Numericance Approxivation Figure 3.11: 1 4k45ρ20V2−2γÑ3v¡Njʪ¦vÈqD BIII ñ Ä äÒ¸ í(Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121k45ρ20 = 1/3, γ = 3/2 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
Volume Scale Factor (V)
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0650 2 4 6 8 10 12 14 16 Time Error (%)
Dominance of 4−D Gravitational Constant
Error Numericance Approxivation Figure 3.12: 3k2 4ρ0V2−γÑ3v¡Njʪ¦vÈqD BIII ñ ÄäÒ¸í (Ç:Λ = 1/3, k2 4ρ0 = 1000/3 J£ 121 k45ρ20 = 1/3, γ = 3/2 BbêÛ, ʦ—D×íª ªMv Bbí¡Nj·ªJ bMíj®ƒ'ßí ¡N, ¥øõuIAòEí, [ýBb¥ší¦ ¶uªWí ,VzBbªJêÛ ¥ú__ KS,BI,BIII Ê3k2 4ρ0Êd_çí|c(, ·x®Aí.°vÈ–Èí ñ Ääí¡Nj, 7¥ú__í3b.°á 4kR a1dV íTà†uU¥ú__Êw‚°¡NjívÈõÏæÑ KS œwk BI
œwk BIII 7Êo‚ BIII œok BI œok KS *¥áÊñ ÄäíøŸ }j˙26ªJõ|4kR a1dV íTà.}–Ø×íTà, .}Ñ3ív`
3.2
Ý®²°4¡bíÆ“
Ý®á°4Ääíì2Ñ: A ≡ 3 X i=1 (Hi− H)2 3H2 *ùıíl2BbªJø−Ý®²°4¡bÊBbF²¦í_-í2$ Ñ: A = 3K2L(V )−1 7w2K2DL(V )Ñ L(V ) = 3ΛV2+ 3k2 4ρ0V2−γ+ 1 4k 4 5ρ20V2−2γ + 4k Z a1dV + c K2 = 2c 3 ¥³«n¤Ý®²°4í¡b6uUà Mathematica 2íbMÚ7_Òíj¶ Võ¤ÓÜ4”íÆ“8$ ¤Õ, Bbø‡úγ = 1,γ = 3 2,γ = 2, VTú__ í}& Bb¥³¡b í²¦u:Λ = 1/3, k2 4ρ0 = 1/3, 121k45ρ20 = 1/3, γ = 3/2 c = 1 Ê%âÚ7òQíbM}&(, BbªJ)ƒú__íÝ®²°4¡bíÆ “(Ç, àÇ 3.13í KS Æ“(; Ç3.14 í BI Æ“(; Ç3.15 í BIII Æ“ (Fý0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Anisotropic Parameter Kantowski−Sachs gamma=1 gamma=2 gamma=3/2 Figure 3.13: Kantowski-Sachs_íÝ_²°4ÄäíÆ“:Λ = 1/3, k2 4ρ0 = 1/3, 1 12k54ρ20 = 1/3, γ = 3/2, c = 1 0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Anisotropy
Anisotropy of Bianchi Type I on Brane Cosmology
2 3/2 1
Figure 3.14: Bianchi type I _íÝ_²°4ÄäíÆ“:Λ = 1/3, k2
4ρ0 = 1/3, 1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Anisotropic Parameter
Bianchi Type III
gamma=1 gamma=2 gamma=3/2
Figure 3.15: Bianchi type III _íÝ_²°4ÄäíÆ“:Λ = 1/3, k2 4ρ0 = 1/3, 1 12k54ρ20 = 1/3, γ = 3/2, c = 1 ;WJ,í¥<Ç[bªJêÛƒ: (1) u°4”Ñ: Êγ = 3 2,γ = 2í8”-, ‡‹Æ“‚u®²°4í, Í( ÇáÑ Ý®²°4, Í('0íyƒ®á°4í8”(ÿøò\M®²°4 7 Êγ = 1í8”-, ‡‹Æ“í ‚ºuÝ®²°4퇋, Í('0탮²° 4(ÿøò\M®²°4 (2) Ïæ4”Ñ: ÄÑ BI _íuk = 0í!‹, FJ BIII D KS ÿª BI Ö |74kR a1dV í à, 7¥_ à â¤ú_Ç2ªJõ|, Ö|7í4k R a1dV U)
BIII D KS œo®ƒ®²°4í8”, /ú__ªœ–V u BIII }ªœo®ƒ ®²°4, Í(u KS, |w†u BI
Chapter 4
!
Ê¥³Bbd7rÖín, ó]øìõíiII7, Ĥʥ³êcízpøZ 1. BbêÛ¥ú__ KS,BI,BIII Ê3k2 4ρ0Êd_çí|c(, ·x®Aí. °v È–Èíñ Ääí¡Nj Êw‚v, Bb)ƒñ ÄäíÆ “j u V = V0e √ 3Λt *j˙ 2ªJc)‡‹¡bΛÊw‚‡‹íÆ“27½bí à Êo‚õ, Bb)ƒíñ ÄäíÆ“j˙Ñ: v(t) = (√nγ(t − t0))1γ , 2n = 14k54ρ20Ê¥_v‚, †u brane ,íµ¸½‰búÆ“O3b
í à 7Ê¥s_v‚5È, BbªJvƒÊ3k2 4ρ0Êd_çí|c(, ñ ÄäíÆ“j˙: v(t) = (γ √ m 2 (t − t 0))2γ , 2m = 3k2 4ρ0, Ê¥¨v‚†u½‰brÆO½bíiH
2. 7¥ú__í3b.°á4kR a1dV íTà†uU¥ú__Êw‚ °¡N
jívÈõÏæÑ KS œwk BI œwk BIII 7Êo‚ BIII œok BI œ ok KS *¥áÊBbíbM_Ò 26ªJõ|4kR a1dV íTà.}–Ø ×íTà, .}Ñ3ív`Oÿà‡ÞFzí, }úÆ“ívÈ ¨A<í à 3. *BbúÝ®²°4¡bíÆ“íû˝ªJø−¥ú__: (1) u°4”Ñ: Êγ = 3 2,γ = 2í8”-, ‡‹Æ“‚u®²°4í, Í( ÇáÑ Ý®²°4, Í('0íyƒ®á°4í8”(ÿøò\M®²° 4 7Êγ = 1í8”-, ‡‹Æ“í ‚ºuÝ®²°4퇋, Í('0í ƒ®²°4(ÿøò\M®²°4 ¥éBb7jƒÖÍBb¦íu Ý®² °4í_, ªuÊ brane í à-, Êo‚´u}×Û®²°4íÕ” (2) Ïæ4”Ñ: ÄÑ BI _íuk = 0í!‹, FJ BIII D KS ÿª BI Ö |74kR a1dV í à, 7¥_ à â¤ú_Ç2ªJõ|, Ö|7í4k
R
a1dV U
) BIII D KS œo®ƒ®²°4í8”, /ú__ªœ–V u BIII }ª œo®ƒ®²°4, Í(u KS, |w†u BI
4. ⮲°4íû˝BbªJø−, Ê|(Bbí_·}ƒ®²°4í8”, ÿuDBbÛÊh¿ƒí‡‹ÛÊ íÕ”ó°, FJBbFû˝í_uªW í
Chapter 5
Ë“
5.1
Vacuum Einstein equation
•
â_2#ìí(j:
ds2 = g µνdxµdxν (5.1) = −d(t)2dt2+ a(t)2dr2+ b(t)2(dθ2+ f (θ)2dϕ2) • d"¾: gµν = −d(t)2 0 0 0 0 a(t)2 0 0 0 0 b(t)2 0 0 0 0 b(t)2f (θ)2 • d"¾5®}¾: g11 = −d(t)2, g22= a(t)2, g33 = b(t)2, g44 = b(t)2f (θ)2 • ÇÕ: gµν = 1 −d(t)2 0 0 0 0 1 a(t)2 0 0 0 0 1 b(t)2 0 0 0 0 1 b(t)2f (θ)2 d"¾5®}¾: g11= 1 −d(t)2, g22 = a(t)12, g33 = b(t)12, g44 = b(t)21f (θ)2 • ®_íì2: f (θ) = θ , Bianchi type I sin θ , Kantowski-Sachs
sinh θ , Bianchi type III
lí¥ :
1. °) Christofol symble íÝÉj: Γα µν 2. °) Ricci tensor Rµν Curvature tensor íì2Ñ: Rα µνβ = −∂βΓανµ− ΓmµνΓαβm− (ν ↔ β) Rµν ≡ Rαµνα 3. ª7°) Ricci scalar : R ≡ gµνR µν 4. |(ÿªJ°) Einstein equation : Rµν− 12gµνR = kTµν − Λgµν ·<:(ç Tµν = 0v)BbUà Dirac í¯Uì2. Êò˛È2
gµν = −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ps: BbÊ¥³ì2 i, j, k... = 1, 2, 3... and α, β, µ, ν, ... = 0, 1, 2, 3...
1.
¥ I
Christofol symbol íì2Ñ: Γα µν = 1 2g αβ(∂ µgβν + ∂νgβµ− ∂βgµν) = 1 2g αβ(g βν,µ+ gβµ,ν − gµν,β) Bbþt °) Christofol symbol í®}¾ : (a) Γ0 00 = 1 2g 0β(g β0,0+ gβ0,0− g00,β) = 1 2g 0β(g β0,0) = 1 2g 0βg 00,0 = 1 2 1 −d2(t) d(−d2(t)) dt = d(t)˙ d(t) (b) Γ00i = 1 2g 0β(gβ0,i+ gβi,0− g0i,β) = 1
2g 0β(g β0,i+ gβi,0) = 1 2g 00g 00,i+1 2g 0kg ki,0 = 0 (c) Γi 00 = 1 2g iβ(g β0,0+ gβ0,0− g00,β) = 1 2g iβ(2g β0,0− g00,β) = gi0g 00,0− 1 2g i0g 00,0 = 0
(d)
Γ0ij = 1 2g
0β(g
βi,j+ gβj,i− gij,β)
= 1 2g 00(g 0i,j+ g0j,i) − 1 2g 00g ij,0 = −1 2g 00g ij,0 ⇒ Γ0 rr = a ˙a d2 ⇒ Γ0 θθ = b˙b d2 ⇒ Γ0 φφ = b˙bf2(θ) d2 (e) Γi 0j = 1 2g iβ(g β0,j+ gβj,0− g0j,β) = 1 2g iβ(g β0,j+ gβj,0) = 1 2g i0g 00,j + 1 2g iβg βj,0 = 1 2g ikg kj,0 ⇒ Γr 0r = 1 2g rrg rr,0 = 1 2˙a 2∗ 2aa0 = ˙a a ⇒ Γθ 0θ = 1 2g θθg θθ,0 = ˙b b ⇒ Γφ0φ = 1 2g φφg φφ,0 = 1 2 1 b2f2(θ) ∗ 2b˙bf 2(θ) = ˙b b (f) Γijk = 1 2g iβ(g βj,k+ gβk,j − gjk,β) = 1 2g il(g lj,k+ glk,j− gjk,l) ⇒ Γrrr = 1 2g rr(g rr,r+ grr,r− grr,r) = 1 2g rrg rr,r = 0
⇒ Γr rθ = 1 2g rr(g rr,θ+ grθ,r− grθ,r) = 0 ⇒ Γrrφ = 0 ⇒ Γr θθ = 1 2g rr(g rθ,θ+ grθ,θ− gθθ,r) = 0 ⇒ Γr φφ = 0 ⇒ Γrθφ = 1 2g rr(g rθ,φ+ grφ,θ − gφθ,r) = 0 ⇒ Γθ θθ = 0 ⇒ Γθθr = 0 ⇒ Γθ θφ = 1 2g θl(g lθ,φ+ glφ,θ− gφθ,l) = 0 ⇒ Γθ rr = 0 ⇒ Γθrφ = 0 ⇒ Γθ φφ = 1 2g θl(g lφ,φ+ glφ,φ− gφφ,l) = − 1 2g θθg φφ,θ = −f f,θ ⇒ Γφφφ = 0 ⇒ Γφφr = 0 ⇒ Γφφθ = 1 2g φl(g lφ,θ+ glθ,φ− gφθ,l) = 1 2g φφg φφ,θ = f,θ f ⇒ Γφrr = 0 ⇒ Γφrθ = 0 ⇒ Γφθθ = 0 FJBb°) Γα µν íÝÉ}¾Ñ: Γ0 00 = ˙ d d Γ0rr = a ˙ad2 Γ0θθ = db˙b2 Γ0 φφ = b˙bf 2 d2 Γr0r = a˙a Γθ0θ = b˙b Γφ0φ = b˙b Γφφθ = f,θ f Γθφφ = −f f,θ
2.
¥ II
Curvature tensor íì2Ñ: Rα βµν = Γανβ,µ− Γαµβ,ν + ΓβνmΓαµm− ΓmβµΓανm BbÇá l Ricci tensor í®}¾: Rµν ≡ Rαµνα (a) R00 = Rα0α0 = R0 000+ Ri0i0 = {Γ000,0− Γ000,0+ Γm00Γ00m− Γm00Γ00m} +{Γi 00,i− Γi0i,0+ Γm00Γiim− Γm0iΓi0m} = Γi 00,i− Γi0i,0+ Γm00Γiim− Γm0iΓi0m = Γi00,i− Γi0i,0+ Γ000Γii0+ Γj00Γiij − Γ00iΓi00− Γj0iΓi0j = −Γi 0i,0+ Γ000Γii0− Γj0iΓi0j = −[Γr0r,0+ Γθ0θ,0+ Γφ0φ,0] + Γ000 [Γrr0+ Γθθ0+ Γφφ0] −[(Γr 0r)2+ (Γθ0θ)2+ (Γφ0φ)2] = −[(˙a a)0+ 2( ˙b b)0] + ˙ d d[ ˙a a + 2 ˙b b] − [( ˙a a) 2+ 2(˙b b) 2] Ha ≡ ˙a a, Hb ≡ ˙b b R00 = −[− ˙ d dHa− 2 ˙ d dHb+ ˙Ha+ H 2 a + 2 ˙Hb+ 2Hb2](b) Rij = Rαiαj = R0 i0j+ Rkikj = {Γ0 ji,0− Γ00i,j + ΓmijΓ00m− Γmi0Γ0jm} +{Γkji,k− Γkki,j + ΓmijΓkkm− ΓmikΓkjm} = [Γ0 ji,0− Γ00i,j + Γ0ijΓ000+ ΓlijΓ00l− Γ0i0Γ0j0− Γli0Γ0jl] = +[Γk ji,k− Γkki,j + Γ0ijΓkk0+ ΓlijΓkkl− Γ0ikΓkj0− ΓlikΓkjl] = [Γ0ji,0+ Γ0ijΓ000− Γli0Γ0jl] = +[Γk ji,k− Γkki,j + Γ0ijΓkk0+ ΓlijΓkkl− Γ0ikΓkj0− ΓlikΓkjl] ⇒ R0 i0j = Γ0ji,0+ Γ0ijΓ000− Γli0Γ0jl ⇒ Rk
ikj = Γkji,k − Γkki,j+ Γ0ijΓkk0+ Γijl Γkkl− Γ0ikΓkj0− ΓlikΓkjl
i. R0 i0j = Γ0ji,0+ Γ0ijΓ000− Γli0Γ0jl ⇒ R0r0r = Γ0rr,0 + Γ0rrΓ000− Γrr0Γ0rr = (a ˙a d2),0+ ( a ˙a d2)( ˙ d d) − ( ˙a a)( a ˙a d2) = (a ˙a d2),0+ ( a ˙a ˙d d3 ) − ( ˙a2 d2) ⇒ R0θ0θ = Γ0θθ,0 + Γ0θθΓ000− Γθθ0Γ0θθ = (b˙b d2),0+ ( b˙b d2)( ˙ d d) − ( ˙b b)( b˙b d2) = (b˙b d2),0+ ( b˙b ˙d d3 ) − ( ˙b2 d2) ⇒ R0 φ0φ = Γ0φφ,0+ Γ0φφΓ000− Γφφ0Γ0φφ = (b˙bf 2 d2 ),0+ ( b˙bf2 d2 )( ˙ d d) − ( ˙b b)( b˙bf2 d2 ) = f2[R0 θ0θ]
ii.
Rikjk = Γkij,k− Γkki,j + Γ0ijΓkk0+ ΓlijΓkkl− Γ0ikΓkj0− ΓlikΓkjl
⇒ Rrkrk = Rrrrr+ Rθrθr+ Rφrφr = Γr rr,r− Γrrr,r+ Γ0rrΓrr0+ ΓlrrΓrrl− Γ0rrΓrr0− ΓlrrΓrrl +Γθ rr,θ− Γθθr,r+ Γ0rrΓθθ0+ ΓlrrΓθθl− Γ0rθΓrθ0− ΓlrθΓθrl +Γφrr,φ− Γφφr,r+ Γ0rrΓφφ0+ ΓlrrΓφφl − Γ0rφΓrφ0− ΓlrφΓφrl = Γ0 rrΓθθ0+ Γ0rrΓφφ0 = (a ˙a d2)( ˙b b + ˙b b) = (2a ˙a˙b d2b ) ⇒ Rk θkθ = Rrθrθ+ Rθθθθ + Rφθφθ = Rr θrθ+ Rθφθφ = Γrθθ,r− Γrrθ,θ+ Γ0θθΓrr0+ ΓlθθΓrrl− Γ0θrΓrθ0− ΓlθrΓrθl +Γφθθ,φ− Γφφθ,θ+ Γ0 θθΓφφ0+ ΓlθθΓφφl − Γ0θφΓφθ0− ΓlθφΓφθl = Γ0 θθΓrr0− Γφφθ,θ+ Γθθ0 Γφφ0− ΓlθφΓφθl = (b˙b d2)( ˙a a) − ( f,θ f ),θ + ( b˙b d2)( ˙b b) − ( f,θ f ) 2 = (b˙b d2)( ˙a a) + ( ˙b d) 2−f,,θ f ⇒ Rk φkφ = Rrφrφ+ Rθφθφ+ Rφφφφ = Rr φrφ+ Rθφθφ = Γr φφ,r− Γrrφ,φ+ Γ0φφΓrr0+ ΓlφφΓrrl− Γ0φrΓrφ0− ΓlφrΓrφl +Γθ φφ,θ− Γθθφ,φ+ Γ0φφΓθθ0+ ΓlφφΓθθl− Γ0φθΓθφ0− ΓlφθΓθφl = Γ0φφΓrr0+ Γθφφ,θ+ Γ0φφΓθθ0− ΓlφθΓθφl = (f2b˙b d2)( ˙a a) − (f f,θ),θ+ (f 2b˙b d2)( ˙b b) − ( f,θ f )(−f f,θ) = f2[(b˙b d2)( ˙a a + ˙b b) − f,,θ f ]
)ƒRij í}¾Ñ: : Rrr = R0r0r+ Rkrkr = (a d) 2[−d˙ dHa+ ˙Ha+ 2HaHb+ H 2 a] Rθθ = R0θ0θ+ Rrθrk = (b d) 2[−d˙ dHb+ ˙Hb+ HaHb+ 2H 2 b] − f,,θ f Rφφ = R0φ0φ+ Rkrφr = (bf d ) 2[−d˙ dHb+ ˙Hb+ HaHb+ 2H 2 b] − f f,,θ
3.
¥ III
°)0b: R = gµνR µν = g00R 00+ grrRrr+ gθθRθθ+ gφφRφφ = −2 1 d2{ ˙ d d(Ha+ 2Hb) − ˙Ha− 2 ˙Hb− 2HaHb− H 2 a − 3Hb2} + 2k b2 hence k = −f,, θ f4.
¥ IV
, The vacuum Einstein equation :Rµν− 1 2gµνR = −gµνΛ =Ägj˙ í®}¾ (component): (a) 00 component R00− 1 2g00R = −g00Λ ⇒ Λ = 1 d2[H 2 b + 2HaHb] + k d2b2 (5.2) (b) rr component Rrr− 1 2grrR = −grrΛ ⇒ Λ = 1 d2[− ˙ d d(2Hb) + 2 ˙Hb+ 3H 2 b] + k b2 (5.3) (c) θθ component Rθθ− 1 2gθθR = −gθθΛ
⇒ Λ = 1 d2[− ˙ d d(Ha+ Hb) + ˙Ha+ ˙Hb+ HaHb+ H 2 a+ Hb2] (5.4) (d) φφ component Rφφ− 1 2gφφR = −gφφΛ ⇒ Λ = 1 d2[− ˙ d d(Ha+ Hb) + ˙Ha+ ˙Hb + HaHb+ H 2 a + Hb2] (5.5) ç d = 1, BbªJ)ƒ: Λ = H2 b + 2HaHb+ bk2 , 00 component 2 ˙Hb + 3Hb2+bk2 , rr component ˙ Ha+ ˙Hb+ HaHb+ Ha2 + Hb2 , θθorφφ component (5.6)
5.2
Einstein equation with energy
momen-tum tensor
Energy momentum tensor í}¾Ñ:
T0 0 = −ρ, T11 = T22 = T33 = p p ≡ (γ − 1)ρ T íì2Ñ: T = Tνµ = −ρp3 Bb‹,?¾¾"¾(í=Ägj˙ ‰A:: Rµν− 1 2gµνR = −gµνΛ + k 2 4Tµν (5.7) k2 4 = 8πG u 4&í½‰bíµ¸á.
1. The solution of Einstein equation : ÄÑ Tµν = Tνµgµν , FJ:
(a)
T00 = T00g00 = (−ρ)(−d2)
;W A.2 í00 }¾J£hí Einstein equation B.1 BbªJ°): 1 d2[H 2 b + 2HaHb] + k d2b2 = Λ + k 2 4ρ as d = 1 ⇒ Hb2+ 2HaHb+ k b2 = Λ + k 2 4ρ (5.8) (b) Trr = Trrgrr = pa2 = ρ(γ − 1)a2
;W A.3 í rr }¾J£hí Einstein equation B.1 BbªJ°):
2 ˙Hb+ 3Hb2+ k b2 = Λ + k 2 4ρ(1 − γ) (5.9) (c) Tθθ = Tθθgθθ = pb2 = ρ(γ − 1)b2
;W A.2 íθθ }¾J£hí Einstein equation B.1 BbªJ°):
˙
Ha+ ˙Hb+ HaHb+ Ha2+ Hb2 = Λ + k42ρ(1 − γ)
(d) Tφ= Tφφgφφ = pb2f2 = ρ(γ − 1)b2f2 ;W,øíϕϕ}¾J£hí Einstein equation BbªJ°): ˙ Ha+ ˙Hb+ HaHb+ Ha2+ Hb2 = Λ + k42ρ(1 − γ) (5.11) cÜ(Bbª)ú_Ö íj˙ : H2 b + 2HaHb+ bk2 = Λ + k42ρ 2 ˙Hb+ 3Hb2+bk2 = Λ + k42ρ(1 − γ) ˙ Ha+ ˙Hb+ HaHb+ Ha2+ Hb2 = Λ + k42ρ(1 − γ)
Í(ZŸj˙ Ñ: (a˙2 a2 )2+ 2(a˙1 a1 )(a˙2 a2 ) + k a2 2 = Λ + k42ρ (5.12) 2a¨2 a2 + (a˙2 a2 )2+ k a2 2 = Λ + k2 4ρ(1 − γ) (5.13) ¨ a1 a1 + a¨2 a2 + (a˙1 a1 )(a˙2 a2 ) = Λ + k2 4ρ(1 − γ) (5.14) ¡bì2: Ha = ˙ a1 a1 , Hb = ˙ a2 a2
2. Energy conservation Low ∇µTµν = 0 Bbø−: ∇µTµν ≡ T;µµν = T,µµν+ TανΓµαν + TµαΓναµ Ê¥³: Tµν = Tµ νgµν ⇒ T00= ρ , Trr = p a2 1 , Tθθ = p a2 2 , Tφφ = p a2 2f2 FJ: ∇µTµν = ∇0T0ν+ ∇rTrν + ∇θTθν∇φTφν = (a) + (b) + (c) + (d) (a) ∇0T0ν = T,000+ TανΓ0α0+ T0αΓνα0 = ˙ρ + (TiνΓ0 i0+ T0νΓ000) + (T00Γν00+ T0iΓνi0) = ˙ρ (b) ∇rTrν = T,rrr+ TανΓrαr+ TrαΓναr = 0 + (T00Γr0r+ T0iΓrir) + (Tr0Γν0r + TriΓνir) = T00Γr 0r+ TriΓνir
= T00Γr 0r+ TrrΓ0rr = ρa˙1 a1 + p a2 1 a1a˙1 = ργa˙1 a1 (c) ∇θTθν = T,θθθ + TανΓθαθ+ TθαΓναθ = 0 + T00Γθ 0θ+ TθθΓ0θθ = ρa˙2 a2 + p a2 2 a2a˙2 = ργa˙2 a2 (d) ∇φTφν = T,φφφ+ TανΓφαφ+ TααΓναφ = 0 + T00Γφ 0φ+ TθθΓφθφ+ TφφΓθφφ+ TφφΓ0φφ = ρa˙2 a2 + p a2 2 f,θ f + p a2 2 −f f,θ f2 + p a2 2 f2a 2a˙2 f = ρa˙2 a2 + pa˙2 a2 = ργa˙2 a2 ⇒ ∇µTµν = ∇0T0ν+ ∇rTrν + ∇θTθν∇φTφν = (a) + (b) + (c) + (d) = ˙ρ + ργa˙1 a1 + ργa˙2 a2 + ργa˙2 a2 = 0
⇒ ˙ρ + ργ(a˙1 a1 + 2a˙2 a2 ) = 0 (5.15) ¥³;¼ñí?¾òíÓOvÈÆ“ít Ñ: (V = a1a22) ρ = ρ0V−γ , ρ0 = constant > 0
3. BbªJZŸ Eq5.12 , Eq5.13 , Eq5.14 , : d dt(V Ha) = ΛV + 1 2k 2 4ρ0V1−γ(2 − γ) (5.16) d dt(V Hb) = ΛV + 1 2k 2 4ρ0V1−γ(2 − γ) − ka1 (5.17) 3 ˙H + Ha2+ 2Hb2 = ΛV +1 2k 2 4ρ0V−γ(2 − 3γ) (5.18) ¥³¡bì2: Ha = a˙1 a1 , Hb = a˙2 a2 , V = 1 3(Ha+ 2Hb) = ˙ V 3V ;W‡s_j˙ Eq5.16 , Eq5.17 , ¥ V íj˙ ªJŸA:
¨ V = 3ΛV + 3 2k 2 4ρ0V1−γ(2 − γ) − 2ka1 (5.19) and Ha = H + 2 3V K Hb = H − 1 3V K K = Z ka1dt ¥j˙ %¬¶} }(: ˙ V2 = 3ΛV2+ 3k24ρ0V2−γ− 4k Z a1dV (5.20)
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