Domain walls in cosmology - Universidade do Porto

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→ √ |k|rÝì µ‘”æ‡ gg'qklysfno„Äƒfhgw‹tˆysfr a(t) k txgg'jlmnfhgW„x}!f>„ˆ´²o'¬!ftˆ}!rô'¬Mtô‹¸k¿fhw‹tˆj!gfÇ„ˆáo'¬lqsg‹¸qoWqsg „¸¯à±¸!ì µV¹å´k = −1¸ o'¬!fajl}lq€‚fpƒgfa¬Mtxg–tcÞêÈå¬¯“ ‡ fpƒ'k„xys„xqsr†o„ ‡ „xys„xŒ “°ÝH 3 − „ ‡ fp}o'¬lƒfhfKr±qsgwpƒfpofa€ tˆyj!fhgVÈìk = 0 tˆ}!r g„Äo'¬Mtô R 3 − k = 1 RQtôOjl}lq€‚fpƒgf¾æOtˆ}!rqí´ o'¬!fWjl}lq€‚fpƒgfi¬MtxgcttxwhfWqsgOfpj!wpyqsrlf‹tˆ}ÔÝ ‡ o'¬!fWg jl}lq€‚fpƒgf¾æÆ¸Qqí´µš æ ÝìGK¬!fpƒf qsgKo'¬!fi•q}!g'ofpq}Žofp}!g„xƒ]\itˆ}!rµν TqsgKo'¬!ffp}!fpƒ'Œ “”Èåmn„xmnfp}¯o'jlm3ofp}!g„xƒ‹µ¹ä}Äo'¬!fig'otˆ}!r!tˆƒrµνµ› æ Ýì“—rlfp}!g'qoä“‚¸ ρ¸Ðtˆ}!r ‡ ƒfhgg'jlƒf"rlfp}!g'qoä“‚¸ p¸¿„ˆ´Ïo'¬!f0R!jlqsr4tˆƒf"„x}ly“´Àjl}!wpo'qs„x}!g„ˆ´Ïo'qmnfK¬!fpƒf"o'¬!f"fp}!fpƒ'ŒXKY ó¢ô òSZ S[ ô § ò%óds 2 = a(η) 2 (dη 2 −dr21 − kr 2 − r2 ( dθ 2 + sin 2 θdφ 2)) .Ýì µd æK¬!fpƒf²f¬Mth€‚frlfÆ»!}!fhrÄo'¬!fiwh„x}±´ç„xƒ'mÇtˆyÂo'qmnf ¸ η¸Qtxgdη =dta(t) .Ýì µÞ‚æ¹äo¤w‹tˆ}kfg¬!„‹K}Ñšš œÂo'¬MtôOf7–”jMtô'qs„x}ì µd qsg¤q}¯€ tˆƒ'q~tˆ}¯oOjl}!rlfpƒOo'¬!fc´À„xyys„¾Kq}lŒÇo'ƒtˆ}!g´ç„xƒ'mÇtô'qs„x}!g‹¸k → k/ |k|a → a/ √ |k|ÞêÈÉg ‡ ¬!fpƒfo„ ‡ „xys„xŒ “^ÝS 3 −wpys„ gfhrÄjl}lq€‚fpƒgf¾æÆµea¬!fir±“³}Mtˆm†qswhgO„ˆ´5o'¬!fcjl}lq€‚fpƒgfqsgKo'¬!fp}—rlfpofpƒ'm†q}!fhrk¯“Òo'¬!f•q}!gofpq}—f7–”jMtô'qs„x}!g‹ÎG µν ≡ R µν − 1 2 Rg µν = 8πGT µν ,g'mn„xys„xŒ qsw‹tˆyÂmn„³rlfpyÐqoKqsgOtxggjlmnfhr>o'¬MtôOo'¬!fcjl}lq€‚fpƒgfwh„x}¯ofp}”oOqsgK²fpyy’t ‡l‡ ƒ„hÅ±qmÇtôfhrk¯“Ätˆ}Äqsrlf‹tˆywh„fpƒ´ÀfhwpoBR!jlqsrÐµ–ea¬!fWfp}!fpƒ'Œ “¯Èåofp}!g„xƒPmn„xmnfp}¯o'jlm7qsgKo'¬!fp}Œ q€‚fp}k”“‡T µν = (ρ + p)u µ u ν − pg µν,tˆ}!r ƒf ƒfhgfp}¯og°o'¬!f‘xÈå€‚fpys„³wpqoÉ“HÝ Ķ•)–µPÝì µš ænŒ q€‚fhgÇo'¬!fŽoÉ²„‡ æÆµ¹ä}©o'¬lqsg°w‹txgfq}!rlf fp}!rlfp}¯oPf7–”jMtô'qs„x}!g ‡u µ u 0 = 1, u i = 0(ȧ ) 2H 2 ≡ = 8πGρ − k a 3 a 2 ,Ýì µß æ8πGT µν + Λg µνöΛ g±ù¤øŽ i [ŠX5Ŵ ûbüˆ÷Y['÷iþ

‡ fhwpqsfhg ‡ ƒfhgfp}¯oOq}o'¬!fjl}lq€‚fpƒgfWtôOt†Œ q€‚fp}Žf ‡ „³w¬Âµ„ˆ´–tˆyyÂo'¬!fg“Òmn„xmnfp}¯o'jlm7ofp}!g„xƒ¤qsg¤wh„x}!gfpƒ'€‚fhrea¬!fifp}!fpƒ'Œµì‹à‚ænw‹tˆ}¦kfÄrlfpƒ'q€‚fhr¦´×ƒ„xm •)–”g¾µOÝì µß æ†tˆ}!rZÝì µë æÆ¸²K¬lqsw¬ªmnf‹tˆ}!gno'¬Mtô>„x}ly“Öoä{„E„ˆ´•)–”jMtô'qs„x}HÝìµß æÆ¸’Ýì µë æatˆ}!r^Ýì µì‹à‚æKtˆƒfcq}!rlf ‡ fp}!rlfp}¯o‹µ{•yqm†q}Mtô'q}lŒno'¬!fk/a 2 ofpƒ'mS´×ƒ„xm9•)–³g‹µÐÝì µß æatˆ}!r•)–³g‹µÐÝìµë æÆļ²fi„xklotˆq} Ýì‡ tˆ}!g'qs„x}Ž„ˆ´5o'¬!fjl}lq€‚fpƒgfi„³whwpjlƒga´ç„xƒρ + 3p < 0 µÙ¢fp}!whfWtˆ}ŽtxwhwhfpysfpƒtôfhrŽfÆÅ‡ tˆƒtˆmnfpofpƒΩ ≡ ρ/ρÛP}!f†w‹tˆ}^rlfÆ»!}!f†o'¬!f†r±qmnfp}!g'qs„x}lysfhggirlfp}!g'qoÉ“ tˆ}!r¼ƒfpKƒ'qofn•–QµÝì µß æ¢q}¼o'¬!fc‡ ysfpofrlfhgwpƒ'q‡ o'qs„x}—„ˆó'¬!ffp€‚„xyjlo'qs„x}—„ˆó'¬!fjl}lq€‚fpƒgf ¸M{fi}!fhfhrŽmn„xƒfo'¬Mtˆ}¹ä}Ž„xƒrlfpƒ¤o„n¬Mt‹€‚ft†wh„xmµß³¸Vì µë tˆ}!rÖì µì‹à±µK½^fW}!fhfhrŽo„>g ‡ fhwpqí´À“Òo'¬!fWf7–”jMtô'qs„x}U„ˆ´Ïg'otôfp = p(ρ) µO¹É}—mÇtˆ}¯“ ‡ ¬”“³g'qsw‹tˆy•)–µiì‡l‡ ƒ„‹Å³qmÇtôfi´À„xƒ'm¸Qtxgq}¯ofpƒfhgo'q}lŒ>w‹txgfhgKo'¬!fWf7–¯jMtô'qs„x}Ž„ˆ´g'otôfw‹tˆ}k¿fcKƒ'qo'ofp}Â¸!q}Žtˆ}—fÆÅltxwpo¤„xƒPtXcbXed ôUfYógZSf ZÎñòcöh¥òi±ò jḢ = −4πG(p + ρ) + k a 2 ,µë Ýì æH‡ tˆƒtˆmnfpofpƒ¢tˆ}!rρ p tˆ}!r rlfp}!„xofWo'¬!fco„xotˆyfp}!fpƒ'Œ “Ärlfp}!g'qoÉ“tˆ}!r ‡ ƒfhgg'jlƒfqsgKo'¬!fiÙPjlklklysf K¬!fpƒfT µν ; ν = 0,Ýì µ “ æysf‹txr±q}lŒno„†o'¬!fWwh„x}¯o'q}¯jlqoä“f7–¯jMtô'qs„x}˙ρ + 3H(ρ + p) = 0 .Ýì µì‹à‚æäa = −4πG 3 (ρ + 3p) . Ýì µì ì¾æ´ç„xƒ'mÎK¬!fpƒfρ c = 3H 2 /8πGΩ − 1 =k(aH) 2 ,Ýì µì d æqsg{o'¬!fPwpƒ'qo'qsw‹tˆy¿rlfp}!g'qoä“‚µ5½^fOo'¬!fp}Ògfhf¤o'¬Mtô{o'¬!fOo„xotˆy¿fp}!fpƒ'Œ “nwh„x}¯ofp}”o¤„ˆ´Ðo'¬!f‡ tô'q~tˆyÐŒ‚fh„xmnfpo'ƒ'“‚¸jl}lq€‚fpƒgfirlfpofpƒ'm†q}!fhgPqog¤gρ > ρ c (Ω > 1) → k = +1Ýì µì‹Þ‚æ,ρ = ρ c (Ω = 1) → k = 0Ýì µìp‘”æ,ρ < ρ c (Ω < 1) → k = −1Ýì µì š æ.ga{fcKqyyÂgfhf¢q}Äo'¬!f¢}!fÆÅ³o¤gfhwpo'qs„x}Âļ„xk!gfpƒ'€xtô'qs„x}!g¤¬Mth€‚fig'¬!„¾K}Äo'¬MtôKo'¬!fwpjlƒ'ƒfp}¯o¤jl}lq€‚fpƒgfcqsga€‚fpƒ'“·gfco„Çt"g ‡ tô'q~tˆyy“kRQtôOŒ‚fh„xmnfpo'ƒ'“4ÝΩ ≃ 1æ_š› œÉµwpys„p = ωρ.Ýì µì › æd š

µì¾ß æ ÝìR!jlqsr¼„ˆ´Ïƒfpy~tô'q€³qsg'o'qsw ‡ tˆƒ'o'qswpysfhgq}Uo'¬!fpƒ'mÇtˆy–f7–¯jlqyqklƒ'qjlmBwh„xƒ'ƒfhg ‡ „x}!rlg¢o„ω = 1/3 tˆ}!rUo'¬lqsg¢²txg·mÇth“Òk¿fcƒf7–¯jlqƒfhrsrˆµµß¦w‹tˆ}—tˆysg„"k¿fcKƒ'qo'ofp}Utxg#t•)–µ¤ìK¬!fpƒfΩ 0 = ∑ ω,Ýì µì “ æµd ì¾æ Ýìω ≠ −1 µ˜Pf ‡ y~txwpq}lŒÄo'¬!f†t ‡l‡ ƒ„ ‡ ƒ'q~tôf"€xtˆyj!fhgc„ˆ´ω ´À„xƒ¢o'¬!f¦mÇtô'ofpƒWtˆ}!rUƒtxr±q~tô'qs„x}^f ‡ „³w¬!g¢{f´ç„xƒVf[VõÌX5b5b±÷cXÿçû üˆÿçÿÐñåðd[5þ i ÿa[H\ZúøñXþWxøb [Z%X÷Y[Wêû[^íþ¤\’ñåðY[5ûüc\a\[Whñ’øúûû[Hb?[ \ÉøñXþ

l„xm†q}Mtôfhg5o'¬!fawpjlƒ'€xtô'jlƒfaofpƒ'm¸k/a 2¸ „x}†•)–µ{ì µß³µVea¬!fa•)–Qµ²ì µßOo'¬!fp}"kfhwh„xmnfhgH 2 = (8πG/3)ρ Vµd d æ Ýì‡ „x}!rlgÇo„Et ‡ fpƒ'qs„”r¦„ˆ´ƒt ‡ qsr¢fÆÅ ‡ tˆ}!g'qs„x}ª„ˆ´¢o'¬!fŽjl}lq€‚fpƒgf ¸¤tˆ}¦q}GRQtô'qs„x}Mtˆƒ'“ea¬lqsgÇgwhfp}Mtˆƒ'qs„wh„xƒ'ƒfhgfpƒ'qs„”rÐµ5½^fKqyy’ƒfpo'jlƒ'}o„†qoKy~tôfpƒ‹µ‡fg„xjlƒwhf ‡ y~txwhfhr4tôct°wh„³„xƒr±q}Mtôfr = r iδt ≪ t¹É´δt i /a(t i ) = δt 0 /a(t 0 )qsgOo'¬!fyqŒ ¬¯ocath€‚fpysfp}lŒδto'¬^fpm†qo'ofhrit µcz²„x}!g'qsrlfpƒ'q}lŒ°o'¬Mtô ¬”o–²t‹€‚fpysfp}lŒ o'¬Çmnf‹txg'jlƒfhr"k”“o'¬!fKrlfpofhwpo„xƒ{tô5o'¬!faq}!g'otˆ}”o o'¬!fp}K}x¸ i¸xo'¬!fayqŒ Ýtk¯“¦o'¬!fag„xjlƒwhf¤tô0 λ r γæ= δt 0gtô'qsg»Mfhg¤o'¬!fƒfpy~tô'qs„x}z qsgao'¬!fWg„xjlƒwhfƒfhrlg'¬lqí´Ào]~ µK¬!fpƒf‡ ¬”“³g'qsw‹tˆyr±qsg'otˆ}!whfk¿fpoÉ{fhfp}—oÉ²„Ç„xk!gfpƒ'€‚fpƒgKqsg¤Œ q€‚fp}k¯“ea¬!fþúñ[añåðˆøññåðY[Kòˆðˆþñåþ

‡ tˆƒtˆmnfpofpƒ‹µ qsgKo'¬!fiÙPjlklklysf}!„xo'¬!fpƒ¤qm ‡ „xƒ'otˆ}”oPwh„x}!whf ‡ o¤qsg¤o'¬MtôO„ˆ´t¦o'¬!f ‡ tˆƒ'o'qswpysfi¬!„xƒ'qÌ‹„x}Â¸·‡ω d „x}†f7–”jMtô'qs„x}ì µÞ à¢{fK„xklotˆq} Õf y~txwpq}lŒ}!„¾Ôo'¬!fa€xtˆyj!fhg„ˆ´H ∼ 2t d H ∼ 3tÝω = 1 3æKtˆ}!rÒmÇtô'ofpƒ"Ýω ≃ 0 æaf ‡ „³w¬!g¤ƒfhg ‡ fhwpo'q€‚fpy“‚µ´ç„xƒ5o'¬!f{ƒtxr±q~tô'qs„x} fa → 0 õÂðY[W t → 0 öXKY ó¢ô òSZ S[ ô § ò%ód =∫ rdr ′a(t) √0 1 − kr ′2= a(t)f(r)Ýì µd › æK¬!fpƒfc´Ý×ƒúæ{qsgKo'qmnfq}!rlf ‡ fp}!rlfp}¯o‹µ²ea¬!fwh„xmn„‹€³q}lŒÇ€‚fpys„³wpqoÉ“ÒqsgKo'¬!fp}Œ q€‚fp}Žk¯“Òo'¬!fÙOjlklklysfiy~thv= ˙ d = HdÝì µd ß æH = ȧaK¬!fpƒfÝì µd ë æd H = a(t)∫ t0dt ′ /a(t ′ )Ýì µd‘“ æ‡ ¬¯“±g'qsw‹tˆyOr±qsg'otˆ}!whfÒo'¬MtônyqŒ ¬”oÇw‹tˆ}Ôo'ƒth€‚fpyPg'q}!whfÒo'¬!fÒkfpŒ q}l}lq}lŒ^„ˆćo'¬!fÒjl}lq€‚fpƒgf…„ µK¬lqsw¬©qsg"o'¬!fµì “ qoPw‹tˆ}Äkfg'¬!„¾K}Žo'¬MtôK´ç„xƒω > −1/3 šßcœÉ¸vlƒ„xm\f7–”jMtô'qs„x}ìd H ≃ 3 1 + ω1 + 3ω t. Ýì µÞ à‚æ‡ ea¬!fÇfp€‚fp}”o¬!„xƒ'qÌ‹„x}ÖmÇth“kfnrlfÆ»!}!fhrEq}Ötˆ}Õtˆ}Mtˆys„xŒ‚„xj!g¦²th“‚µŽea¬lqsgW¬!„xƒ'qÌ‹„x}Õgf tˆƒtôfhgo'¬!f>gfpo„ˆ´‡ txwhfÆÈåo'qmnf¢o'¬Mtô{w‹tˆ}Òfpm†qoÏyqŒ ¬”o²gqŒ }!g‹¸³tô{o'qmnf t¸¯o'¬Mtô{mÇth“nk¿fKƒfhwhfpq€‚fhr°k¯“Çtˆ}°„xk!gfpƒ'€‚fpƒ‡ „xq}¯og{q}>gO t f¸ ´Àƒ„xm o'¬!fP„x}!fhgÏK¬!„ gfPg'qŒ }!g{Kqyy}!fp€‚fpƒ{k¿fKƒfhwhfpq€‚fhr4Ý×k¯“†o'¬!f¢„xk!gfpƒ'€‚fpƒ¸±tô{o'qmnf q}Ço'¬!fO´×jlo'jlƒfO æÆ¸∫ tfd e (t) = a(t)tdt ′a(t ′ ) .Ýì µÞ±ì¾æ· }!„xo'¬!fpƒ¤qm ‡ „xƒ'otˆ}”oPwh„x}!whf ‡ o¤qsg¤o'¬!fiÙOjlklklysfi¬!„xƒ'qÌ‹„x}R H = H −1Ýì µÞ d æ†zVf[ø\[–øúÿçÿçüˆùEX5WYzaðd[H\[ñåðxøñÂñåðY[5üdWYX?Z

¹äo¤w‹tˆ}kfg¬!„‹K}o'¬Mtô‹¸±´ç„xƒω > −1/3¸ R Hqsg¤„ˆ´5o'¬!fgtˆmnfi„xƒrlfpƒO„ˆ´5mÇtˆŒ }lqo'j!rlftxgd HµÞ Þ‚æ ÝìΩ 0 ω = 1µ5ea¬!fOÙPjlklklysf¤¬!„xƒ'qÌ‹„x}°mÇth“†k¿f¤gfhfp}Òtxg–o'¬!fPr±qsg'otˆ}!whf¤o'ƒt‹€‚fpyysfhrK¬!fpƒf¤²fO¬Mt‹€‚fOwh„x}!g'qsrlfpƒfhr¬”oO„x}o'¬!fw¬Mtˆƒtxwpofpƒ'qsgo'qswWo'qmnfi„ˆ´fÆÅ ‡ tˆ}!g'qs„x}Â¸τk¯“ÒyqŒH ≃ H −1 µ‡ „xƒ'otˆ}¯orlfÆ»!}lqo'qs„x}¼„ˆ´–r±qsg'otˆ}!whf ¸Mq}U„xk!gfpƒ'€ tô'qs„x}MtˆyVwh„ g'mn„xys„xŒ “‚¸qsgOo'¬!fiyjlm†q}!„ g'qoä“Žr±qsg'otˆ}!whfÛP}!fiqmd LµÄÓÂfpo¦j!gWg'j ‡l‡ „ gfno'¬Mtô"tÄŒ q€‚fp}Õg„xjlƒwhfn¬Mtxg¦tˆ}Õtˆk!g„xyjlofnyjlm†q}!„ g'qoä“L Ý×K¬lqsw¬EqsgWo'¬!f ‡ „‹²fpƒ.Ýì µÞ š æ= 1 + z.Ýì µÞ › æXcbXed ôUfYógZSf ZÎñòcöh¥òi±ò jR H = 3 21 + 3ω(1 + ω)t = d H ≃ d H .2fpm†qo'ofhrÄk”“°o'¬!fWg„xjlƒwhfcq}qogKƒfhg'oa´Àƒtˆmnf¾æÆµ–ea¬!fp}Â¸d 2 L ≡L4πf ,Ýì µÞˆ‘”æf “ R!j±ÅÄƒfhwhfpq€‚fhrŽk”“tnrlfpofhwpo„xƒ‹µŠ²PfpŒ ysfhwpo'q}lŒ°o'¬!fifÆÅ ‡ tˆ}!g'qs„x}—„ˆó'¬!fWjl}lq€‚fpƒgf ¸K¬!fpƒfr = rwh„x}!g'qsrlfpƒ'q}lŒ°o'¬MtôOo'¬!fg„xjlƒwhfiqsg¢tô¤o'¬!fWwh„xmn„¾€”q}lŒÒwh„”„xƒr±q}Mtôf ÝÀfpm†qo'o'q}lŒ>yqŒ ¬”octôOo'¬!fio'qmnfit qsg¤o'¬!fWfp}!fpƒ'Œ æatˆ}!rÄo'¬MtôOo'¬!firlfpofhwpo„xƒ¤qsgOtôi r = 0æÆļo'¬!fcƒfhwhfpq€‚fhr—fp}!fpƒ'Œt“ŽR!j±Å°qsg0 Ý×ƒfhwhfpq€³q}lŒnqoPtôKo'¬!fo'qmnff =L4πa 2 0 r2 i‡ tˆ}!g'qs„x}—„ˆ´5o'¬!fjl}lq€‚fpƒgfWo'¬!fpƒfWtˆƒfioä{„Çwh„xƒ'ƒfhwpo'qs„x}!gPo„n•)–µOì µÞ š µaÛP}!fWwh„xƒ'ƒfhwpo'qs„x}U„ˆ´½ªqo'¬o'¬!fifÆÅ(1 + z)‡ ¬!„xo„x}!gitxgPo'¬!fp“o'ƒth€‚fpy5´Àƒ„xm7o'¬!f¦g„xjlƒwhf¦o„>o'¬!frlfpofhwpo„xƒ‹µea¬!f¦„xo'¬!fpƒP´åtxwpo„xƒc„ˆ´(1 + z) qsg¢o'¬!fi´Àƒtxwpo'qs„x}¼kfpoä{fhfp}¼o'¬!f}¯jlmWk¿fpƒi„ˆ´ ‡ ¬!„xo„x}!gco'¬Mtôtˆƒf¦fpm†qo'ofhrtˆƒ'qsgfhg¢r±j!fo„°o'¬!fWƒfhrlg¬lqí´×oc„ˆń e γn r γ= λr γλ e γk¯“Òo'¬!fg„xjlƒwhftˆ}!rÄƒfhwhfpq€‚fhrÄk¯“Òo'¬!firlfpofhwpo„xƒ ‡ fpƒKjl}lqoO„ˆ´5o'qmnf ¸ea¬lqsgwh„xƒ'ƒfhwpo'qs„x}t ‡l‡ f‹tˆƒg¢´×ƒ„xm3o'¬!f´çtxwpoco'¬MtôcoÉ²„ ‡ ¬!„xo„x}!gcK¬lqsw¬tˆƒf"fpm†qo'ofhr¼Kqo'¬tˆ}¼q}¯ofpƒ'€ tˆyδt‡ tˆyO„ˆ´¢o'qmnf „ˆṔo'qmnf tˆƒtôfhrÔk¯“Ôtˆ}Ôq}¯ofpƒ'€ µÕea¬¯j!g‹¸{o'¬!fÄfp}!fpƒ'Œδt(a“eR!j±ÅÕq}©tˆ}0 /a i tˆƒ'ƒ'q€‚fÄgf )f =L4πa 2 Ýì0 r2 i(1 + z)2fÆÅ ‡ tˆ}!r±q}lŒnjl}lq€‚fpƒgfqsgKŒ q€‚fp}Žk¯“µÞ‚ß æd 2 L = a 2 0r 2 i (1 + z) 2tˆ}!rd LqsgKŒ q€‚fp}k”“Ýì µÞ ë æd‘“

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‡ „xofp}¯o'q~tˆyV (φ) txr±m†qogOt†r±qsgwpƒfpofigfpo¤„ˆ´rlfpŒ‚fp}!fpƒtôfim†q}lqmÇt±µK¬!fp}o'¬!f“¯Èåmn„xmnfp}”o'jlm ofp}!g„xƒW´ç„xƒ φ¸5„xklotˆq}!fhrÖk¯“^€ tˆƒ'“”q}lŒ—o'¬!f°txwpo'qs„x}ÝÀ•)–µ d µ› æiKqo'¬ea¬!f°g'o'ƒfhgg¦fp}!fpƒ'Œd µë æ Ýd µß>tˆ}!r d µë {f¦w‹tˆ}—w‹tˆyswpjly~tôf¦o'¬!fWfp}!fpƒ'Œ “—rlfp}!g'qoÉ“‚¸T 00¸¿tˆ}!ro'¬!f ‡ ƒfhgg'jlƒftˆys„x}lŒÒ„x}!f¢g'q}lŒ>•)–µÐüÒ òhkf § óžý®f¥iªi6õƒþdòi ô § ò%óõÿ¡ h¥õ §ê[ f¥i„ö § h iÃfô § ò%óqöL = 1 2 ∂ µφ∂ µ φ − V (φ)Ý d µß æƒfhg ‡ fhwpoKo„†o'¬!fmnfpo'ƒ'qswofp}!g„xƒg µνqsgKŒ q€‚fp}k¯“T µν = ∂ µ φ∂ ν φ − g µν Lr±qƒfhwpo'qs„x}Â¸T ii¸”„ˆ´’o'¬!fPŒ‚fp}!fpƒ'qswP»Mfpysr φ q}Òt!RQtô{¬!„xmn„xŒ‚fp}!fh„xj!gPtˆ}!rÇqsg„xo'ƒ„ ‡ qswcvlƒ'qsfhr±mÇtˆ}±ÈÉ˜P„xkfpƒ'og„x}½tˆy®‚fpƒ¤jl}lq€‚fpƒgf>ÝÀvV˜’½æÆµlea¬!fimnfpo'ƒ'qswqsgKq}o'¬lqsg¤w‹txgfcŒ q€‚fp}k¯“ rg µν = diag[1, −a(t) 2 , −a(t) 2 , −a(t) 2 ],Ý d µ “ ætˆ}!rρ = T 00 = 1 2( ) ∂φ 2− 1∂t 2a 2 (∇φ)2 + V (φ),Ý d µì‹à‚æp i = T ii = 1 2( ) ∂φ 2+ 1 ( ) ∂φ 2∂t a 2 ∂x i − 12a 2 (∇φ)2 − V (φ).1(√ )√ − ∂ µ |g|∂ µ φ = − dV|g| dφÑÏtˆƒ'“³q}lŒno'¬!fWtxwpo'qs„x}EÝ d µ› æ²Kqo'¬Žƒfhg ‡ fhwpoKo„no'¬!f¢»Mfpysrφ {fi„xklotˆq}o'¬!f»MfpysrÄf7–¯jMtô'qs„x}!gP„ˆ´Vmn„xo'qs„x}K¬lqsw¬¼tˆƒff7–¯jlq€ tˆysfp}¯oPo„∂ 2 φ∂t 2 + 3ȧ ∂φa ∂t − ∇2 φ = − dVdφÝ d µì ì¾æjl}lq€‚fpƒgf µ¤ÙPfpƒfW{f"tˆƒfiqm ‡ yqswpqo'y“—txgg'jlm†q}lŒÇo'¬Mtôcfp}!fpƒ'Œ “rlfp}!g'qoÉ“—„ˆ´5o'¬!f¦rl„xmÇtˆq}q}UtëRQtô¢v5˜’½yqŒ qklysfif

t µ wh„x}!g'otˆ}”o¢tˆys„x}lŒno'¬!fcatˆyyz²„x}!g'qsrlfpƒtˆysg„>tnys„”w‹tˆyq}!fpƒ'o'q~tˆyV´×ƒtˆmnfWq}ŽK¬lqsw¬¼o'¬!frl„xmÇtˆq}—atˆyyVqsgOq}!g'otˆ}¯otˆ}!fh„xj!gy“UtôOƒfhgo¢tˆ}!rd µì‹Þ‚æ ÝρK¬!fpƒf qsgOo'¬!fi²tˆyy5fp}!fpƒ'Œ “”ÈÉrlfp}!g'qoÉ“—tˆ}!rω TtˆƒfWo'¬!fWg'o'ƒfhggOwh„xm ‡ „x}!fp}”ogctˆys„x}lŒ°o'¬!fii i r±qƒfhwpo'qs„x}Âµqm ‡ „ gg'qklysfo„mnf‹txg'jlƒf—€‚fpys„”wpqo'qsfhg°q}ªtˆ}”“©r±qƒfhwpo'qs„x} „ˆío'¬!f²tˆyy ‡ y~tˆ}!f µHÚ³„T µνÐ·b"!°Ò òhkf § óžýf¥iªiMZ j ó³f¥h §ê[ ö¹Éá(t) = const = 1 ²fW„xklotˆq}∂ 2 φ∂t 2 − ∇2 φ = − dVdφ .Ý d µì d æŸ6('×685*),LN–>–¨”u²´–L£LÓÂfpo"j!g"rlfÆ»!}!fÄt—rl„xmÇtˆq}ÕatˆyyOtxg"tŽoÉ²„ˆÈÉr±qmnfp}!g'qs„x}MtˆyP„xk±Ã'fhwpo‹¸Kqo'¬Õqog¦o'¬lqsw®³}!fhgg"mWj!w¬¦g'mÇtˆyysfpƒtô'jlƒfPƒtxr±qj!g‹ļtˆ}!rÇK¬!„ gf ‡ ƒ„ ‡ fpƒ'o'qsfhg‹¸³mÇtxgg ‡ fpƒ²jl}lqo²g'jlƒ´åtxwhfσo'¬Mtˆ}°qog²wpjlƒ'€tˆ}!rÇo'¬lqsw®”}!fhgg δļtˆƒftxgg'jlmnfo'¬Mtô¢o'¬!fatˆyy5qsg ‡ fpƒ ‡ fp}!r±qswpjly~tˆƒco„l¸Q´ç„xƒ¢fÆÅ!tˆm ‡ ysf ¸o'¬!fzr±qƒfhwpo'qs„x}Âµc¹É}Uo'¬lqsgcw‹txgf ¸{f¦w‹tˆ}gfhfco'¬MtôT µν = diag [ρ ω (z), −ρ ω (z), −ρ ω (z), 0]ea¬!f4´åtxwpoo'¬Mtô—o'¬!f^²tˆyy ‡ ƒ„ ‡ fpƒ'o'qsfhgUrl„¦}!„xoUw¬Mtˆ}lŒ‚fÕtˆys„x}lŒ©o'¬!f4atˆyymnf‹tˆ}!g—o'¬MtôŽqoŽmWj!g'o—kfmWj!go>k¿fÄo'¬!fq}¯€ tˆƒ'q~tˆ}¯oOKqo'¬°ƒfhg ‡ fhwpoao„"ÓÂ„xƒfp}¯o'ÌkÖ¨%¨dÆr±qƒfhwpo'qs„x}Žtˆys„x}lŒ"o'¬!fc²tˆyyµÓÂfpoaj!gKwh„x}!g'qsrlfpƒao'¬!f´ç„xyys„‹Kq}lŒ*Ö¨%¨dÆ

d µì‹Þ±µ •)–µ‡ fpƒ´ç„xƒ'mDtUÓÂ„xƒfp}¯o'Ì—o'ƒtˆ}!g´À„xƒ'mÇtô'qs„x}¢g„¼o'¬Mtô†o'¬lqsg†²tˆyyO}!„‹9mn„¾€‚fhgnKqo'¬©tU€‚fpys„³wpqoÉ“v q}¹ÉṔ{f= 1¸±ƒfhgjlyo'q}lŒÇq}rlfpofpƒ'm†q}!f¦o'¬!f¦rlf fp}!rlfp}!wp“Ž„ˆ´Ïo'¬!f²tˆyy5€‚fpys„³wpqoÉ“vy~tˆ}Mtˆƒrl„xmÇtˆq}¼²tˆyyåµcea¬!fÓÂfpocj!gP}!„‹ ‡ ‡g'otô'qswPg„xyjlo'qs„x}>„ˆ´Ðo'¬lqsg–atˆyyw‹tˆ}>kf¤Kƒ'qo'ofp}°txgφ(z)φ(θ) ∂θ/∂t ´ç„xƒ¢t = γv ∂θ/∂z = γ µea¬!fp}²f¬Mth€‚ftˆ}!r K¬!fpƒf µÓÂfpo²j!g–Kƒ'qofOo'¬!fOg„xyjlo'qs„x}Ç´À„xƒ{tmn„‹€³q}lŒatˆyytxgÐüÒ òhkf § óžý®f¥iªi6õƒþdòi ô § ò%óõÿ¡ h¥õ §ê[ f¥i„ö § h iÃfô § ò%óqöγvT xy = T ty tˆ}!rT x′ y ′= γT xy + γvT ty = T xy {f°„xklotˆq}T ty = T xy = 0 µÄÚ”qm†qy~tˆƒ'y“¼´Àƒ„xm= γT tt + γ 2 v 2 T xx = T tt qm ‡ yqsfhgco'¬MtôT xx = −T tt µOv!ƒ„xm+o'¬!fW´åtxwpoOo'¬MtôT t′ z ′ = γT tz + γvT xz = T tz T x′ z ′ = γT xz + γvT ty = T xz ²f¬Mt‹€‚fT tz = T xz = 0 µWÛP}tˆ}!rT t′ t ′ o'¬!f¦„xo'¬!fpƒP¬Mtˆ}!rÐ¸o'¬!fpƒf†qsgc}!„ ‡ ƒfÆ´çfpƒ'ƒfhr^r±qƒfhwpo'qs„x}Etˆys„x}lŒo'¬!f"atˆyy–{f"¬Mt‹€‚fT xt = T yt , T xz = T yz , T xx = T yy µvVq}Mtˆyy“Ò´×ƒ„xm\o'¬!fWfp}!fpƒ'Œ “”Èåmn„xmnfp}¯o'jlm wh„x}!gfpƒ'€ tô'qs„x}Žy~thH{fW¬Mth€‚f ¸T µα ,α = 0¸Qtˆ}!rg„ T zz ,z = 0 µea¬lqsg¤qsgOf7–¯jlq€ tˆysfp}¯oPo„T zz = 0 gq}!whf T zz qsgO}¯jlyyå¸Mtôz → ∞¸³´ç„xƒPtˆ}¯“Ärl„xmÇtˆq}Žatˆyy’g„xyjlo'qs„x}Âµ{½^fo'¬!fp}Äwh„x}!wpyj!rlf¢o'¬MtôKt ‡ y~tˆ}MtˆƒKrl„xmÇtˆq}Ò²tˆyyÐg„xyjlo'qs„x}Ò¬Mtxg²o'¬!fPfp}!fpƒ'Œ “”Èåmn„xmnfp}¯o'jlm9ofp}!g„xƒ²Œ q€‚fp}Òk”“Λ t′t = Λ z′z= γ, Λt′z = Λz′t = γv, Λ x′x= Λy′ yo'¬!f ‡ „ g'qo'q€‚fz r±qƒfhwpo'qs„x}Â¸ T µνo'ƒtˆ}!g´À„xƒ'mng¢txg¤•)–µ d µìp‘!¸!Kqo'¬o'¬!fc}!„x}±Èå}”jlyyVwh„xm ‡ „x}!fp}¯ogOŒ q€‚fp}Žk”“⎡T µν =⎢⎣γ 2 ρ 0 0 −vγ 2 ρ0 −ρ 0 00 0 −ρ 0−vγ 2 ρ 0 0 v 2 γ 2 ρ⎤⎥⎦Ý d µì š æK¬!fpƒfρqsg¤o'¬!fig'otô'qswatˆyy’fp}!fpƒ'Œ “¯ÈÉrlfp}!gqoÉ“rlfÆ»!}!fhrŽtˆk¿„‹€‚f µ€‚fpƒtˆŒ q}lŒÄo'¬!f ‡ ƒfhgg'jlƒf"wh„xm ‡ „x}!fp}”ogi„‹€‚fpƒo'¬!f"o'¬lƒfhf"r±qƒfhwpo'qs„x}!g‹¸〈p·ω 〉 = 1/3(p x + p y + p z„xklotˆq}o'¬!fif7–”jMtô'qs„x}Ž„ˆ´g'otôfc´ç„xƒ¤„x}!f ‡ y~tˆ}MtˆƒOatˆyy)¸¿²f(〈p ω 〉 = ρ ω v 2 − 2 Ý d µì › æ)3∂φ∂t = dφdθ γv∂φ∂z = dφdθ γÝ d µì¾ß æd 2 φdθ 2 = dVdφtˆ}!rÄo'¬!f¢´åtxwpo¤o'¬Mtô φ(θ)¸±Kqo'¬ θ =zļqsgOt"gotô'qswig„xyjlo'qs„x}²f¬Mth€‚f ¸±´Àƒ„xm\•)–µ d µì d ¸Ý d µì ë æK¬lqsw¬—q}¯ofpŒ ƒtôfhrŽŒ q€‚fhgš à

d µd d æ Ý“”Èåmn„xmnfp}¯o'jlm ofp}!g„xƒ¦wh„x}!gfpƒ'€xtô'qs„x}ÖŒ q€‚fp}ea¬!fÇgútˆmnfnƒfpy~tô'qs„x}ÖmÇth“^k¿fn„xklotˆq}!fhr´×ƒ„xmö'¬!fÇfp}!fpƒ'ŒÐ·b"!°Ò òhkf § óžýf¥iªiMZ j ó³f¥h §ê[ ö( ) dφ 2=Ý d µì “ æ2Vdθγ 2 v 2 = 12V( ) ∂φ 2Ý d µd à‚æ.∂ttˆ}!rwh„x}!gf7–”j!fp}¯o'y“°´Àƒ„xm\•)–µ d µì¾ß³¸· yofpƒ'}Mtô'q€‚fpy“Ò•–Qµ d µd ànw‹tˆ}kffÆÅ ‡ ƒfhggfhrÒq}Äofpƒ'mngP„ˆ´Vo'¬!fi²tˆyy’fp}!fpƒ'Œ “”ÈÉrlfp}!g'qoÉ“ρ(z)γ 2 v 2 =( ) ∂φ 2Ý d µd ì¾æ/ρ(z)∂tρ(z)qsgKŒ q€‚fp}k¯“Ä•)–µ d µì‹àK¬!fpƒfρ(z) = 1 2( ∂φ∂t) 2− 1 2( ) ∂φ 2+ V (φ)∂zd d µì “ µtˆ}!rÄ²f¬Mth€‚fj!gfhr•)–µ µì¾ß†tˆ}!rfp}!rlfp}!wp“„ˆ´v a(t) „x} ‡ e’„Ž»!}!rEo'¬!f>rlftô'q€‚f°„ˆ´¤o'¬!f>ƒfpy~tô'qs„x}!g d µì¾ßÄKqo'¬{f°w‹tˆyswpjly~tôf°o'¬!fÇrlfpƒ'q€‡ fhwpoKo„ttˆ}!rj!gfco'¬!fc»MfpysrÄf7–”jMtô'qs„x}!gO„ˆ´5mn„xo'qs„x}Žq}Žt"v5˜Â½ jl}lq€‚fpƒgf ¸!•)–µ d µì ì ļo„n„xklotˆq}ƒfhg( )dφ d (γv)+ 3H (γv) − d2 φdθ dtdθ 2 = −dV dφtˆ}!rÒ´Àƒ„xm9•)–µ d µì ë {fio'¬!fp}¬Mth€‚fd (γv)dt+ 3H (γv) = 0K¬lqsw¬—qsg¤f7–”jlq€xtˆysfp}”o¤o„γv ∝ a −3d µì š ´À„xƒOt ‡ y~tˆ}MtˆƒO²tˆyyåµ²Ù¢fp}!whfo'¬!firl„xmÇtˆq}Žatˆyy’mn„xmnfp}”o'jlm ‡ fpƒ¤jl}lqoPwh„xmn„¾€”q}lŒÒtˆƒf‹t"qsgk¯“Ä•)–µƒ„ ‡ „xƒ'o'qs„x}Mtˆy’o„a −1 µ ‡š ì

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Domain walls in cosmology - Universidade do Porto

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