"! # $ %&

'(*),+-'/.10,'23.4.4+6587*+65

9;:=<?>,@,ACB4DFE4G6@CHIBF>JLKNMOQPSRUTVWOTXVYOQZUP\[L]_^;M`a];O?bdc;ZeP;fagahjilknmoLprqFh/VWReOrP\^;MZUMshVYg 9st;uwv `dxy*MgNzQVW];R];gah

|VY~YRUP;g4qFVWReKNP;fNcSKwUP1`-MZeMQ~YVYONKNPeP;gZ?];h/VW]\eVPSg~YMr3P\^;MZIZUPST[KNMgUKNMfNgNVY^;MZeRUM[L];RVYgeKNMQZeT]~nMwfNVY~VY`NZeVfNT]g6hVYRP;gaMPeKNMTXPSROQP;gd^dVgaOQVgac1NVYMOrMsRP;=MQ^dVYhNMQgaOQMeKa]\eKNMfNgNVY^;MZeRUMMr/-MQZeVMgaOrMsh ]XKaP`NVYc`a];gNc : |VeK1]XNZUMsRMgwUMQTX-MQZ?]\efNZUMP; :Xq/VeR=IZUMsSfaMQgaOQxR-MOrUZefNTVYR=OQP;gaRUVWReMQgw[VeK1]X`N~Y]SO?b`-PdhNxRU6MsOUZefNTUP v; 6]ZUeRLVYg]TXVY~~YVPSg : faO?K]RU6MsOeZUfNTVWRKa];ZehUPNZUP/h/f6OrMfNgN~YMReR]P;gNM6PSVgweKNM=NKNP;UPSgaR[nMQZeMVYgeKNMQZeT]~MswfNV~YVY`NZUVYfNT]SRLNZeMh/VWOeMh `dxUKNM`aVcX`a];gNcUKNMP;Zex :|Ka]TX];b;MsR=eKNMknm1oRUfaO?K] faRUMrIfa~OrPSRUTXP;~YP;cSVYO]~FUPdP;~VWR UKNM¡]SOUKa]q4IP;Z UKNM

TXPSR6]ZUq*VXKa]SRgNP;VYgSeMQZ?];OrUMh¢RVYgaOrMUKNMfNgNVY^;MZeRUM[L];R^SMQZexxSP;fNgac¤£¥PSgN~x]IM[KdfNgahNZUMsh UKNPSfaRU];gahx;Ms]Z?R,P;~Wh : JLKNMNKaPUPSgaR4hNMOrPSfNN~YMhIZeP;T¦UKaMMQ~YMOrUZeP;gaRF]gah`6]Zex;P;g6R[KNMgeKNMfNgNVY^;MQZ?RUM§OrPdPS~MshRUf/¨OrVYMQgwU~YxXUP]~Y~YP\[©UKNM IP;ZeT]\eVPSgP8Kwx/h/ZeP;cSMQgqd]NZUP/OQMReRbdgNP\[g¤];Rª ZeMOrPST`NVYga]UVYP;g :Y«¬ MOQP;T`NVg6]\UVYP;g P/OQOQfNZUZeMh ]\];NNZeP_/VT]\eMQ~Yx]XZUMshNRUKNV3§Pzrec ∼ 1100,

]]UMQTX-MQZ?]\efNZUMP;Trec ∼ 0.3eV. ­ =eKNVWR=M];ZU~YxeVTXM;qaUKNM®afaOefa]\eVPSgaR[KNVWO?KX[¯PSfN~WhM^;MQgwefa]~Y~xc;ZeP\[¢VYgSePcS];~Y]dVYMR*];gahUKNM~W]Zec;MReOQ]~YMReZUf6OUfaZUMsR[¯MLPS`aRMZU^SM

ePdha]_x[¯MZUMnTf6O?KXRUT]~Y~MZ]gah[nMQZeM¯eP] c;PdP/h];NNZeP_dVYT]\eVPSgMQ^SP;~Y^dVgNc ~YVgNMs]Ze~x : JLKdfaReKNMnknm1oNZeP\^wVWh/MsR]=RUga]aRUKNP;PdUKNM¯M];ZU~YxfNgaV^SMQZ?RM[KNMgUKNMNKdxdRUVWOQR[L];RTfaO?KMs];RUVMZePfagah/MQZ?Re]g6heKa]g¤V=VYRLNZeMRUMQgwe~x :JLKNMUMTX6MZe]UfNZeMXPeKNMknm1o°NKaPUPSgaR P;gUKNMRUbdxVYRZeMQT]Zeb\]`N~YxfNgNVIP;ZeT : JLKNM

~W]Zec;MsR]gaVYRUPeZUPSwxVYRUKNMhNV-P;~YMNZeP/h/faOrMsh`dxP;faZTXPUVYP;g[VUKZeMRU-MOUPeKNMZeMRIZ?]TXMPLeKNM`a]SO?bwcSZUPSfNgah,q

∆T = 3.372 ± 0.007mKOrPSZUZeMRU-P;gah/VYgNc¤[VUKj]¤^;M~P/OQVx

P;v = ∆T

T0

c = 360bwT±R : onMQxSP;gahUKNVWRQqUKNMUMTX6MZe]UfNZeMVWRfNgNVIP;ZeTUP¥P;gNM a];Z

VYg 9s; q ;SN:² PSZ§gaM]Ze~Yx¤UKNZeMQMXh/MO];h/MsRQq,RUM];ZeO?KNMsR=IPSZeMQTX-MQZ?]\UfaZUMX]gNVWRUPUZeP;aVMsR§OQP;fN~WhPSgN~xN~Y]SOrMfNa6MZ`6PSfNgahNRP;g¥UKNMX®6faOefa]\eVPSgaR : JLKNVYRO?Ka]gNcSMh¥h/Z?]T]\eVYO]~Y~xVYg 9t;tS[KNMg©UKNMk³ o¯´¶µ m ¬ VYgaRUZefNTXMQgwh/VWRUOQP\^;MZUMshjeMQTX6MZe]UfNZeM¤®afaOrUfa]UVYP;gaRPP;Z?h/MZ30µK

PSgReOQ]~YMR P;7];gah~Y];ZUcSMQZ : Vg6OrMUKNMgqeMQTX-MQZ?]\UfaZUM®afaOrUfa]UVYP;gaRKa]_^SM`-MQMg

RUMQMg]RT];~~YMQZ];gNc;fa~Y];ZReOQ];~MsR`dxVYgaRUZefNTXMQgweR`6];RUMh¥P;g¥UKaMc;ZeP;fNg6h];gahTXP;fNgwUMshPSg1`a]~Y~YPwPSgaR : JLKNVWRK6];R§`6MMQgO]N-Mh^;MQZex¤h/Ze];T]\UVWOQ];~~Yx `wx eKNMZeMOQMQgwIfN~Y~Rbdxknm1oT]6RNZeP/h/faOrMsh`dx¥· ­ ­ « R|V~YbdVg6RPSg¢mVWOrZeP\[n]_^SM ­ gNVYRUPeZUPSdxyZeP;`-Mi¡|¸m ­ y¯pVYg² MQ`NZefa]Zex S;a:9S:9S:¹ºF»N¼C½s¾¡¿lÀ¾¡¼8Á½sÃFÄ ! ½sÄdÅÆÄwÇ\»/½ºFÇ\ÄÅj»/Æ*È;ÉÊ g¦PSZeh/MZePËOQP;TXa];ZUM[VeKeKNMQPSZUVYMRq4VVWRgNMsOrMReRe]ZexUPwfa];gwUVIxeKNM knm1oÌT]aR : JLKaVYROQ]g`-Mh/PSgNMMQVUKaMQZ`dxcSV^dVYgNc]RbdxjeMQTX-MQZ?]\UfaZUM¤T]qLPSZMwfNVY^\]~YMQgwU~Yx¢`dxÍ] ² P;fNZeVMZ~YVbSMMr/a]g6RVYP;gËVgwePPSZeKNP;cSP;ga];~SIfagaOeVPSgaR : JLKNMeMQTX-MQZ?]\UfaZUML®af6OUf6]\UVYP;g6RP;gUKaMRaKNMQZeM=OQ];g`-M=MQd6]gah/MshVYgeMQZeTRPFUKNMRUNKNMZUVWOQ];~,Ka]ZeTXP;gNVWO=IfagaOeVPSgaRQÎ

ÏÐÑ3ÒQÓ_ÔdÕÖr×F×ØÕÙØÚÛ_ÜÝÝÞsßà

á âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/é

δT (Ω)

T=

∑

`,m

a`mY`m(Ω).

yZeP;-MQZUUVYMRP;¯eKNMRUNKNMQZeVWOQ]~K6]ZeTPSgNVWOIfagaOeVPSgaR]ZeMXc;VY^;MgVYg ­ N-MQgahNV ­ : JLKaMP;ZUëeKNP;cSP;ga];~VxPFUKNMRUNKNMZUVWOQ];~,Ka]ZeTXP;gNVWO=IfagaOeVPSgaR~YM];haRnUPUKNMVYgd^;MZeRUMeZe];gaRIP;ZeT q

a`m =

∫δT (n)

TY ∗

`m(n)dΩn.JLKNM

`^\]~YfNMOrPSZUZeMRU-P;gahNRnZeP;fNcSKN~Yx[VeK UKNM];gNc;fa~Y];Z=RVYzQMP8UKaMUMT-MQZ?]\efNZeM®afaOefa]\ë

eVPSgaRQq∆θ ∼ π/`. ² P;ZMs];O?K `, UKNMZUM];ZUM 2`+ 1

^\]~YfNMRLP;m = −`,−`+ 1, . . . `.JLKNMRUVTXN~YMRR?]\UVWRUVWO[nMO]g¥OrP;g6RVWh/MQZVWR§UKNM[nPëì6PSVgwOrP;ZeZeMQ~W]\UVYP;g¤IfagaOeVPSgqP;Z

VeR ² P;fNZeVMZ]ga];~PScaqCUKaMeMQTX-MQZ?]\UfaZUMX-P\[¯MZRU-MOeZUfaT : JLKNMX-P\[¯MZR-MOrUZefNTíP¯UKNM®af6OUf6]\UVYP;g6RVYR§h/Mrî6gNMh]SRLUKNMMr/-MO?]\eVPSg P;*UKNMTfN~UVY6PS~MTPSTXMQgweRÎ 〈a`ma

∗`′m′〉 =

C`δ``′δmm′ ,[KNVWO?K¤IP;~Y~P\[=R=RUVYTa~xIZUPSTZeP?]\UVYP;g¤VYgd^_];ZUVW]g6OrM : JLKNM[nP6PSVgw§OQP;ZeZUM~Y]ë

eVPSgIfNg6OUVYP;g¤VYg¤ZUMs]~RUa];OQM VWRh/MQîagNMh¤]SR

C(θ) ≡⟨δT (n)

T

δT (n′)

T

⟩=

∑

`,m

C`Y`m(n)Y ∗

`m(n′) =1

4π

∑

`

(2`+ 1)C`P`(cos θ),

[KNMZUMθVYRneKNM]gNcS~M`-Mr[nMQMg

n]g6h

n′: JLKNM^\]ZeVY];gaOrM VYRLcSV^SMQg`wxUKNMOQP;ZeZUM~Y]UVYP;g]

zMQZePRUMQa];Ze]UVYP;gqC(0) = 1

4π

∑`(2`+ 1)C`.Ê eKNM®afaOrUfa]UVYP;gaR]ZeMï]faReRVW]ga~xh/VWReZUVY`Nf/eMh,qUKNMX-P\[¯MZ RU6MsOeZUfNTíOQP;gwe];Vg6R ];~~

eKNMVYg/IP;ZeT]\eVPSgUKa]eKNMT];¢OrPSgS?]VYgaR : ­ ~Y~LKNVcSKNMQZTXP;TXMQgw?R]ZeMMVeKNMQZzQMZUPÍi3IP;Z];g P/hNh¤gdfNT`6MZ?pP;Z];ZUMRUVTXN~Yx ZUM~Y]UMhePUKNM6P\[nMQZRU-MOeZUfaT := efah/VYMRP*UKNMgNP;g/ëï];faRURUVW]gNZUPS6MZeVMsRPCeKNM knm1oÍRbdxP3UMgXIP/OrfaRPSgXUKNM~P\[nMRP;Z?h/MQZTXPSTMgweRqSRUfaO?K]SRUKNM¤eKNZUMM¤P;ZXIP;faZ6PSVgwOQP;ZeZUM~Y]UVYP;gIfNg6OUVYP;g6RP;ZXeKNMQVYZ ² PSfNZeVMZ]ga];~PScSRqbdgNP\[g]SRXUKNMËðrñWòlóaôeõQöl÷ø/ùúPSZ1öØ÷ñWòlóaô?õröl÷ø/ùûZeMRU6MsOeV^SMQ~Yx :¦ü P\[¯M^;MZq¯UKNMZUM1];ZUM1];gfNgN~YVTXVUMshgdfNT`6MZeRP;,[n]_x/R*eKa]\gaP;g/ëï]faReRUVY];gNVxT]_x]a6Ms]Z : |KNVY~YMTXPSRRefah/VYMRnRP¡];ZKa]_^SMOQP;gaOQ~fahNMh¢UKa]UKNMPS`aRMZU^\]UVYP;gaRX]ZeM¤OrP;g6RVWReMQgw[VeK¸ï]faReRVW]gaVxSqUKaMQZeM1OrP;fa~YhÍ`6MP;UKNMZbwVYgahNRLP;8gaP;g/ëï]faReRUVY];gNVxX[KNVWO?K¤Ka]_^;M gNP;=x;Mr`-MQMgeMRUMsh :

:=ýþNAS@,EA,ÿw@,HIBF>JLKNZeP;fNcSKNP;fN[nMX[V~Y~`-M];ReRfaTVYgNcUKa]UKNMTXMQUZeVYO®af6OUf6]\UVYP;g6R]ZeMVYg¥UKNM~YVgaM]Z

ZeMQcSVTXM;qs]gah eKa]\F[¯MO]gVYc;gNPSZUM*KNVYc;KNMZP;Z?h/MZVgweMQZ?];OeVPSgaR : JLKNVYRFRVYTXN~YVîaMsR,eKNM]g6]~YxdRUVWROQP;gaRUVYhNMQZ?]`N~Yx;q*]gahjVgj6]ZUUVWOrfN~W]ZTMs]gaRUK6]\[KaMQg¢MQ/a]gahNMhVYg ² PSfNZeVMZRUa];OQM;q8UKNM^\]ZeVYP;faRTXP/h/MsR¯h/PgNPLOrP;faN~MUPM];O?KP;UKNMZ : JLKNM§Ka]ZeTXP;gNVWOLIfNgaOrUVYP;gaRIP;ZUKNM ² PSfNZeVMZRUMQZeVMsR§];ZUMRPS~f/eVPSgaR§UPUKNM ü M~TXKNPS~ezMswfa]\eVPSg : |M[VY~~8IP/OrfaR PSg1®6]OQPSRUTPS~PSc;VYMRq[KNMZUM eKNMKa]ZeTXP;gNVWO§IfNgaOrUVYP;gaR=];ZUMRUVTXN~Yx

eik·x.Ê FeKNMfNgaV^SMQZ?RM VYR=gNPRUa]\eVY];~~Yx®6]q

eKNMKa]ZeTXP;gNVWOIfNgaOeVPSgaR`-MOQP;TXMTPSZUMOQP;TXN~YMr : :9S: À¼_ÃFÇ6¼6º8ÈÁ-»Nº8Á6Ä-É JLKNZUPSfNc;KaP;f/*UKaVYR¯h/VYReOrf6RURUVPSgq;[nM[VY~~N`-M=[nP;ZebdVgNcVYgRxdg/ëO?KNZeP;gaP;faRFcS];fNc;M¯[VeK]i3ëp4RUVYc;ga]UfNZeM;q_eKNP;fNcSKP/OQO];RUVPSga]~Y~Yx [nMn[V~Y~Nh/VWRUOQfaRUR8eZe];gaRëIPSZUTXVYgNcVgweP·=M[LUP;gaVY];gcS]fac;MUPTX];b;MOQP;gwe]SO=[VUKRUP;TXMP;UKNMZUZeM]UTXMQgw?R : xdg/ëO?KNZeP;gaP;faRFcS];fNc;MnVYR*cSMQgNMZe];~~Yx UKNMcw]fNcSMP;-O?KNPSVYOQM¯IPSZknm1ojOrP/h/MsRh/fNMnUP eKNMRVYTXN~VWOrVxP;-eKNM=eVTXMOQPdP;Z?h/Vg6]\UMsR :*ü P\[nMQ^SMQZsqVVYR¯gNP;¯]IfN~Y~xîN/MshXcS]fac;M=]g6hUKNMZUM=MQ/VYRZeMRUVYhdëfa];~h/MQcSZUMMRnP8IZeMQMhNP;T¤q/bdgNP\[g ]SRLcS];fNc;M TXPdhNMRq/[KNVYO?K¤Ka]_^SMePX`6MOQP;gaRUVYhNMQZeMh :JLKNMMQ/]SOLIP;ZeT¶P;FeKNMTXMrUZeVWO§[nM[V~Y~`-MOrPSgaRVWh/MZUVYgNcVYR

ds2 = a2(τ)(−dτ2 + (δij + hij)dxidxj).

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê µ§MQZeV^\]UVY^;MR[VeKZeMRU6MsO¯UPXOrP;TXP\^dVYgNcUVYTXM§[VY~Y~6`-M hNMQgNP;UMh`dxXhNPeR : |M [VY~~,h/MgNPeMRUfNTRi q 9 q Nq p[VUK1ï ZeMQMQbVgahNVYOQMR=];gahRUfNTRi 9 q/q p¯[VeK4]\eVg VYgah/VWOrMR :Ê g ² P;fNZeVYMQZFRUa]SOrM;qseKNMeMQgaRUP;Z*OQ];g`-M¯`NZeP;bSMQgfNVgweP§ReOQ]~W]Zsqs^;MOrUPSZF]g6hUMgaRPSZFaVMsOrMR

iØRMMnK6]gah/PSf/n±§]N-MQg6h/VXop : mPwRP;3UMQgX[nM[V~Y~ faR`-M=OrP;g6RVWh/MQZeVYgNceKNM=ReOQ]~W]ZaVMsOrM;q[KNVWO?K¤OQ]g `-M[ZUVeMQg];R

hij =V

(2π)3

∫d3keik·x

[1

3hδij +

(kikj −

1

3δij

)hS

].

ü MQZeM;qhVWR8UKNMLeZe]SOrMLP

hijqhSVWR8UKNMLeZe]SOrM~MsRUR*ReOQ]~W]Z8NVYMOQM=]gah

h− = h−hSTXM]SRfaZUMsR

eKNM]gNVWRUPUZeP;aVYO§ReZUMsRUR : : : ¾¡¼8ÈQ½sÄd¾¡¼ËÄ"!,º8»/½s¾#-¼FÈSÉ JLKNM´VYgaRUMVg¤Mwfa]UVYP;gaR¯IPSZ¯eKNMRUO]~W]ZnTXMrUZeVWO hNMQc;ZeMQMsRnPIZeMQMsh/P;TO]g `6M[ZeVUMg]gdfNT`6MZP*[n]_x/R :8ü MQZeM[¯M[VY~~f6RM UKaM[¯PXMwfa]UVYP;gaRÎ

h+a

ah+ 8πGa2

∑

N

(δρN + 3δpN) = 0

];gah2

3ikh− − 16πGa2

∑

N

(ρN + pN )vN = 0.

ü MQZeMUKNMRfNT VWR=P\^SMQZ];~~4UKNM^_];ZUVYP;f6Ra]ZUUVWOr~YMRU6MsOrVYMR : VYgaOrM[nM];ZUMIP/OQfaRVYgNcPSg¤UKNMReOQ];~Y];Z¯-MQZUUfNZe`a]UVYP;gaRqdUKNM^SMQ~YP/OrVUVYMROQ];g`-M[ZeVUUMQg]SR

v = vk.

N:=ýþaA$wþ6Ga@&%('wEH3D|¥M gaMrdngaMQMhUPfNgah/MZeRe];gahKNP\[¸UKNM§®afaOrUfa]UVYP;gaRnVYgUKaM§^\]ZeVYP;faRxd6MsR¯P;4T]\UUMQZ

M^;P;~Y^;M : JLKNVWR¯[V~Y~-];~~YP\[©faRUPIPS~~YP\[ËUKNMVZnMQ^SP;~Yf/UVYP;gXIZeP;T[KNMgUKaM®af6OUf6]\UVYP;g6R[nMQZeMNZeP/h/faOQMhVg UKaMM]Ze~YxfNgNVY^;MQZ?RUM fagSeV~CeKNMNZeMRUMQgw :a:9S:()§ÄSÇ\¿ìÄ*½,+8º8¾#-8ÈSÉ§Ê gUKNM ~VYgNMs]ZnZUMc;VYTMSqdTX];gdxP;UKaM OQP;gaRUVUfNMgwnM~MTMgweRnVgUKNMfNgaV^SMQZ?RMnOQ]g`6MnUZeM]UMsh];RF6MZIMsO*®afNVWhNR :8² P;Z*]-MQZUIMOrF®6fNVYhq\UKNMnNZeMReRfaZUM¯VYR8VWRP;UZeP;NVWO];gah]IfagaOeVPSgPeKNM§hNMQgaRUVx];~PSgNM;qd]g6hV?R¯RUZeMReRëlMgNMQZec;xUMQg6RPSZT]_xX`6M [ZUVeMQg]SRQq

Tµν = (ρ+ p)uµuν + pgµν ,[KNMZUM

uµ = dxµ

(−ds2)1/2

VYRUKNMIPSfNZ=^SMQ~YPdOQVxP*UKNM®afNVWh ];gahuµuµ = −1.

JLKNVWRVTXN~YVYMReKa]\

T 00 = −(ρ+ δρ)T i

j = (p+ δp)δij

T 0i = (ρ+ p)vi.ü MQZeM;q\eKNMP\^;MZ8`6]Zh/MQgaPUMsR]_^;MZe];c;Mhh/MQgaRUVxP;ZNZeMReRUfNZUMSq\[KNVY~MLeKNM

δρh/MQgaPUMsRh/V/.-MZë

MgaOrMsRLIZUPSTeKNMTXM];g^\]~YfNMR : |MOQ]g IfNZUUKaMQZ§hNMrîagNMUKNMh/MQgaRUVx OQP;gwUZ?];Rδ ≡ δρ/ρ,]SRL[¯M~~]SRnUKNM]_^SMQZ?]c;MMwfa]\eVPSgP;*Re]UM

w ≡ p/ρ ]gahUKaMRPSfNgah R-MQMsh c2s ≡ δp/δρ.k¯P;g6RMZU^\]\eVPSgP;*RUZeMReRëìMQgaMQZec;xXVTXN~YVYMRnUKaMQg¤UK6]\

δ + (1 + w)(ikv + h/2) = 3 aa (w − c2s)δ

ikv + aa(1− 3w)ikv + w

1+w ikv =c2

s

1+wk2δ.

JLKNM§îaZ?RLMwfa]UVYP;gVWRnUKaM§faRUfa];~COQP;gwUVYgdfNVxMwfa]\eVPSgqw[KNV~YM UKNMRUMOrPSgahVWRL]^SMQZ?RVYP;gPeKNM´fN~YMQZ§Mwfa]UVYP;g :LÊ VWR=faRUMrIfa~4ePOrPSgaRVWh/MZUKNMsRMMwfa]\eVPSgaRLIPSZRPSTXMR-MOQVY];~FO];RUMRÎOQP;~WhhN];ZUbT]UMZi

p = 0)]gahZeMQ~W]\UVY^dVYRUVWO§®afNVWhNRi

p = ρ/3).

0 âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éa:9S:9S:21435762698 ÷;:ù 8 öØö ôQ÷=<mPSR~Ms];h/VYgNcTXP/h/MQ~WR8IP;ZRUZefaOefNZeMIPSZUT]\eVPSgOQP;gwe];VgRPSTXM];TPSfNgwPOQP;~WhhN]ZebTX]eMQZ : JLKNM hN]ZebTX]eMQZVWR¯gNMQMsh/MhePIP;ZeTÌUKNM§6P;UMQgweVY];~6[nMQ~Y~YReKa]\eKNM`6]Zex;P;g6R*¡]~Y~aVYgwUP[KaMQgXUKaMQxXKa]_^;M=hNMOrPSfNN~YMhXIZeP;TÌUKaMNKNP;UPSgaR : JLKNVWRVWRgaMOëMsRURe]ZexRVYgaOrMLeKNMP;ZeVYc;VYga]~d`a];ZUxSP;g®af6OUf6]\UVYP;g6R[V~Y~N`-MhN]TX-Mh`dxh/V/.-f6RVYP;gM>.CMOreR=i¡V : M : VY~YbhN];TXNVgac : p Ê gNfaZUM~x`a]Zex;PSgNVYOLTXP/h/M~YRqeKNM§];Ta~VUfahNM=Ph/MgaRVx®afaOefa]\eVPSgaRPSgRUT]~Y~RUO]~YMR¯UMgahNRLeP`6Mc;ZeM]U~YxRUfNNNZeMReRMshZeMQ~W]\eV^SM§ePUKNMknm1o¦]gNVWRUPUZeP;dx :JLKNMXhN]Zeb¤T]\eMQZMQ^SP;~Yf/UVYP;gVYR a];ZeVYOQfN~Y];ZU~Yx¤M]SRx UPIP;~Y~YP\[°VgRxdgaO?KaZUPSgNP;faRcS];fNc;M :

JLKNMIZe];TXMh/Mrî6gNMh`wx eKNMIZeMQM~x ¡]~Y~YVgNc hN]Zeb TX]eMQZ§hNMrîagNMsR§UKNMRxdgaO?KNZeP;gaP;faR=OQPdP;ZUëh/VYga]UMRq]g6h];R]ZUMsRfa~ UKNMhN];ZUb¤T]\UUMZ K6];R gNP ^;MQ~YP/OrVxVgUKNVWR IZ?]TXM;qi

vc = 0p :

² ~YfaOefa]\eVPSgaRVYgeKNMhN];ZUb1T]UMZh/MgaRVxZeMRUfN~P;gN~Yx1IZeP;T UKNMRUZeMr?O?KNVgac P;LRUa]SOrM;qδc + h/2 = 0.JLKNM RU];TMMwfa]UVYP;gaR]~WRP1M=.-MsOeV^SMQ~Yx];NN~Yx¥UP1eKNM`6]Zex;P;g6R]\3eMQZUKNMxjh/MOQP;fNN~YM :ü P\[¯M^;MQZsqeKNMQx`6Mc;VYg[VUK gNP;g/ëìzQMZUP^;M~P/OrVUVYMRnZeMRUfN~UVYgNcIZeP;TÂeKNMQVYZLVgweMQZ?];OeVPSgaR¯[VeKeKNMNKaPUPSgaR : JLKNMRUMRU~P\[~YxhNMOQ]_x]_[L]_x];RUKNMNKNP;UP;g6R¡]~Y~nVgwUP1UKNMc;Z?]_^wVe]UVYP;ga];~[nMQ~Y~YRTX]Sh/M=`dxUKNM hN]ZebT]\UUMZ : JLKNM^SMQ~YP/OrVxMwfa]UVYP;gq

v+ aav = 0,

VTXN~YVMsReKa]\UKNM^SMQ~YPdOQVeVMsRLh/MO]_x];R

a ? à .a:9S: :A@ ô 5/8 öØñCBsñWòöØñ¡õEDnø/ñ 6 ò=< ­ Ms]Ze~x¤eVTXMsRQqF[¯M~~`-MrIPSZUMh/MsOrP;faN~VYgNc6q,UKNMaKNPeP;gaR];gah`a];ZUxSP;gaR¯[nMQZeM§eVcSKwU~YxOrPSfNN~YMh ]gah¤]SOUMsh]SR]XRVYgNc;~YMZeMQ~W]\UVY^dVYRUVWO®afNVWh : JLKNVWRL®afNVWh Ka];R] NZeMReRfaZUMh/fNMXeP¤UKNMNKNPeP;gaRi

p = ργ/3p ]gah]h/MgaRVx[KaVYO?KVWR UKNMRUfNTPnUKNM

NKaPUPSg¥];gah`a]Zex;PSgh/MQg6RVx(ρ = ργ + ρb).

onMrIP;ZeMXT]\eMQZh/PSTXVga]UVYP;gqUKNMX`a];ZUxSP;gh/MgaRUVxO]g `6MVYc;gaP;ZeMh,qNRUP[nMîagah

δγ + 43 (ikvγ + h/2) = 0ikvγ = 1

4k2δγ.JLKNMZUMVYR]OrPSTX6MQUVUVYP;g`-Mr[nMQMQgjUKNMc;Z?]_^wVe]UVYP;ga];~6P;UMgSeVY];~eMQZeT¤q8[KaVYO?KeMQgahaReP

T]bSMeKNM ®afaOrUfa]UVYP;gaROQP;~Y~Y];aRMSq/]gahUKNMNZeMReRUfNZUM§UMZUT¤q/[KNVWO?K [¯PSfN~YhOQ];faRM§UKaM®afaOëefa]\eVPSgaRePPwRUOQV~Y~Y]UM]SOrPSfaReVYO]~Y~x : ³§g^;MQZex~Y];ZUcSMRUO]~YMRqeKNM§cSZe]_^dVe]UVYP;ga];~/UMZUTÂh/PSTëVYga]\eMRq[KNVY~MPSg¥RUT]~Y~MZRUO]~YMR§UKaMNZUMsRURUfNZeMh/P;TXVYga]\eMR : JLKaMRUO]~YM[KaMQZeMeKNMQx]ZeMZeP;fNcSKN~YxMwfa];~VYR=bwgaP\[g ]SRLUKNMGFdô 8H òòQõ 85 ô>I4]gah Vc;VY^;MsR]g¤VYhNM]XP8UKaMRT];~~YMR=P;`/ë MsOeRL[KaVYO?K¤OrPSfN~YhKa]_^SM OQP;~Y~Y];aRMsh :² PSZ¯eKNM`a];ZUxSP;gaRq;eKNVWR=RUO]~YM hNZUPSaRLOrP;g6RVWh/MQZ?]`a~x]ZeMOQP;T`aVga]UVYP;g[KNMQg UKaMNZUMsRURUfNZeM hNfNMUPNKaPUPSgaRLgNMs]Ze~xhNVYRe]N-M];ZeRqN]g6heKNMKJ;Ms]gaRT];ReRLh/ZeP;aRnIZeP;TRUfN-MQZ?Or~YfaRUMQZ=ReOQ];~MsRnUPXc;~YP;`afN~Y];ZR?]Z=Or~YfaRUMZ=RUO]~YMR :a: : º4ƯÄSÇ\ÃaÇ\¾MLN-¼ Ä"OQP¡ºF½¾M6¼SRÁaÇ*T¾¡¼8Áj»N¼-Á6»Nº8Á-ÄÅU-FÄdÈSÉ=Ê g1c;MgNMQZ?]~lqaV[nMOQP;gaRUVYhNMQZL6MZefNZe`a]\eVPSgaRnVg¤]fNgNVY^;MQZ?RUM h/PSTXVga]UMsh`dx]VWRP;UZeP;NVWO§RUVYgNc;~YM®6fNVYhqN[¯MKa]_^SMIPSfNZh/MQcSZUMMRFPNIZeMQMsh/P;T¤Î

h, h, δ, v: |¥MLO]gRPS~^SMUP§îagahIP;faZ*RUMQ6]Z?]\UMnRPS~fNUVYP;gaRq_[¯P

cw]fNcSM]gah[¯PNKdx/RVWOQ];~ØqwbdgNP\[g];RncSZUP\[VYgNc]g6hh/MsOQ]_xdVgacTXP/h/MR iIyZeMReRAVXWVWRUKNgNVW];Oq9stNY; p : ³§fNeRUVYh/MUKNMKNPSZUVYzQPSgqaeKNMcS]fac;MTXP/h/MROrPSZUZeMRU-P;gahUP ]OQP;gaRe];gSRKaV3Vg

h,];gahP;gaM§[KNVWO?KVWRnNZeP;-P;ZUUVYP;ga];~NUPt−1.

|MOQ];gRUP;~Y^;M=IPSZeKNM`6MKa]_^dVPSZP;UKNMcSZUP\[VYgNcRUP;~Yf/UVYP;g6RMs];RUV~YxPSf/eRUVWh/MeKNM KNP;ZeVzP;gV4[nM§RUMrnUKNM ^;M~P/OQVeVMsRePzMQZeP`dxKa]gahi3eKNP;fNcSKgNP;UMeKa]\§UKNVWR§[V~Y~FM~VYTXVg6]\UMUKNMh/MO]_xdVgNcTXP/h/MRPS~f/eVPSgqa`af/ [VY~Y~4~YM]_^;MP;gNMcw]fNcSMTXP/h/M : pa: :9S:A@E8N6 ñ 8 öØñ 3HZ6N3 ùñ HQ8 öØñ 3H <³§f/eRUVWh/MLUKNMKNP;ZeVzP;g¤i¡];gahVcSgNP;ZeVgac§eKNML®afNVWhX^;M~P/OQVxNpq[nMO]g¥OrP;T`NVYgNMeKNMXTXMrUZeVWOM^;PS~f/eVPSg];gaheKNMOrP;gweVgdfNVxMwfa]UVYP;g1eP îagah¥]RUVgac;~YMRUMOQP;gahP;Z?h/MQZLMsSf6]\UVYP;gIP;ZLeKNMZeMQ~W]\UVY^dVYRUVWO®afNVWh hNMQgaRUVx

δγ +a

aδγ =

32π

3Ga2ργδγ = 4

(a

a

)2

δγ

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê [[KNMZUMUKNM¤~W];RIPS~~YP\[=RIZUPSTUKNM¤h/Mrî6gNVeVPSgÍP;=UKaM ü fN`N`N~YM¤a]Z?]TXMreMQZ : ï V^SMQgjeKa]\eKNMReOQ];~M¡];OeP;Z c;ZeP\[=R]SR

a ∝ τ ∝ t1/2 VgUKNMZe]Sh/VW]\UVYP;g1hNP;TXVg6]\UMsh1MZe]aq6VeR M]SRx ePRUKNP\[¦UKNVWRK6];ReKNMRUP;~Yf/eVPSgaRδ ∝ τ2, τ−2 ∝ t, t−1.

JLKNMîaZ?RRPS~fNUVYP;gVWR=eKNMcSZUP\[VYgNcTXP/h/M;q-[KNVY~MUKNMRMsOrP;g6hOrP;ZeZeMRU6PSgahNRnUPUKNMcS]fac;MTXP/h/M : ­ TPSZUMh/Mr?]VY~MshRPS~f/eVPSgRUKNP\[=RLeKa]\§UKNMZUMVYR§]~WRP]h/MO]_xwVYgNcTXP/h/M

δ ∝ τP;f/?RVWh/MUKNMKNP;ZeVzP;g : JLKaMc;ZeP\[VYgNc

];gahh/MsOQ]_xdVgacTXP/h/MRnT]\?O?KP;gwePUKaM[nPNK6];RUMR¯P4PwRUOQV~Y~Y]UVYP;gaR¯P4UKNM§®afNVWhPSgaOrMUKNMTXP/h/MRLMgwUMQZLeKNMKNPSZUVYzQPSg :a: : :]\8 ölö ôr÷ 6N3 ùñ H^8 ölñ 3H < ­ RVYTXV~W]ZeZUMs]\UTXMgSXP§UKNM ®af6OUf6]\UVYP;g6RVYgÍUKNM¤T]\UUMQZZeMQcSVTXM ~Ms];hUP]g Mwfa]UVYP;gIP;ZLeKNMknµ§mh/MgaRUVx OrP;gweZe]SR

δc +a

aδc = 4πGa2ρcδc =

3

2

(a

a

)2

δc.

|VUK°eKNM¢T]UMZ¥hNP;TXVg6]\UMsh°RUO]~YVgaca ∝ τ2 ∝ t2/3,

[¯MËOQ]g î6gah UKaMÍRUP;~Yf/UVYP;gaRδ ∝ τ2, τ−3 ∝ t2/3, t−1.

JLKNMRUM];cS];Vg1OQP;ZeZUMsR-P;gahePUKaMc;ZeP\[VYgNc]gah1cS]fac;MTXP/h/MRPSf/eRUVYhNMLUKNMKNPSZUVYzQPSgq[KNVY~YMUKNM=hNMOQ]_xdVYgNcTXPdhNM=OQ]gX`-MRUKNP\[geP`-M

δc ∝ τ−1 ∝ t−1/3.·PUM¯UK6]\8UKNMLMwfa]UVYP;g];`6P\^SM¯];NN~YVMsR4M^;MQg]\3eMQZ8UKNMn-MQZUUfNZe`a]\eVPSgaR8MQgwUMZ4eKNMLKNP;ZeVYzQP;g :ü P\[¯M^;MQZ¯UKaMRUO]~YVgNcXP;4UKaMcS]fac;M]gah¤h/MsOQ]_xdVgacTXP/h/MRRU[L] `6MKa]_^dVPSfNZeR :JLKNVWRMwfa]UVYP;g¤]~WRPXKa]SRL^;MZUxVYTX6PSZ?]gw=OrPSgaRMswfNMQgaOQMRnIPSZLUKNMT]UMZ6P\[nMQZ=RU-MOë

eZUfNT : mP/h/MsRP;f/?RVWh/MUKNMKaP;ZeVzP;gÍi¡P;ZeKNPSRUMXVgaRUVWh/M]\3UMZT]\eMQZh/PSTXVga]UVYP;g6p§cSZUP\[]SR

τ2,[KNVWO?KËOQP;ZeZUMsR-P;g6hNReP]cSZe]_^dVe]\eVPSga]~n-PUMgwUVW]~[KNVWO?KÍVWRXOrPSgaR?]gwVgÍUVYTXM :

ü P\[¯M^;MQZsqFUKNMZUVYc;KwUK6]gahRUVYhNMPeKNM Mwfa]\eVPSgVYRP;ga~x~W]Zec;M[KNMgUKaM hN];ZUb¥T]\UUMQZh/MgaRUVxjVWRh/PSTVYga];gS :ÍÊ UKaM -MQZUUfaZU`a]UVYP;gaRXMgwUMQZh/faZUVYgNc¥UKNMZ?];hNVY]UVYP;gÍhNP;TXVg6]\UMshZeMQcSVTXM;qSUKaMQVYZncSZUP\[LeKRU~P\[=R : JLKaVYRn~YM];haReP]UfNZegNP\^;MZVYgeKNM TX]eMQZL6P\[nMQZnR-MOrUZefNTi¡P;Z=T]\UUMZ=UZ?]g6RIMZIfNgaOrUVYP;g6p : JLKNM-P\[¯MZRU6MsOUZefNTMQgahaRfN[VUK]6Ms]b P;g1ReOQ];~MsR[KNVWO?KMQgweMQZeMhÍUKNMKNP;ZeVYzQP;g]UKNMZe]Sh/VY]UVYP;g/ëìT]\UUMQZUZ?]gaRUVeVPSg : JLKwf6RUKaMT]\UUMQZ-P\[¯MZRU6MsOUZefNT"`-MOQP;TXMR]1^;MZUx¥c;PdP/hNZeP;`-MP;UKNM T]\eMQZUëlZ?];hNVY]UVYP;gZe]UVYP1VYgUKNMfNgaV^SMQZ?RM :­ RUVTXVY~Y];ZM>.CMOrKa]a6MgaRVneKNMfNgNVY^;MZeRUMX`6MsOrPSTMsRh/PSTVYga]UMh¥`dx]OQPSRUTXP;~YP;c;VWOQ];~OQP;gaRe];gS[KaVYO?KVWRRUa]\eVY];~~YxRUTXPdPUK : ­ cS]VYgq8UKaMc;ZeP\[LUKPLeKNM®afaOrUfa]UVYP;gaRR~YP\[=Rq].-MsOUVYgNcUKNM-P\[nMQZR-MOrUZefNTgNMs]ZUKaMKNP;ZeVzP;g¢ReOQ];~M :ÍÊ g©h/PSVgNcRUPaq¯VOQ];faRUMRUKNM

cSZe]_^dVe]\eVPSga]~*-PeMQgwUVW]~UPhNMOQ]_xSqF[KaVYO?KjO]g`-MP;`aRUMQZe^;Msh¥`wxUKaMVYgSeMQcSZe]UMh ]SO?KaRë|¥P;~IMM>.CMOi¡RUMQM`6M~P\[ : p

_ :a` B'9'wHbQHIBF>'wþcedB'@fgF>>UhEiFEd@,HIB4>|¥MVYgSeZUP/h/f6OrM=eKNM§onPS~ezQT]gNgMswfa]\eVPSgVgR?]c;MsRQq;îaZ?R¯OQP;gaRUVWh/MQZeVgacUKNM OQ]SRM=[KaMQZeM

eKNMQZeM]ZeMgNPOQP;~Y~VWRUVPSgaR=]g6h¤MrN]TXVYgNVgacV?ROrPSgaRUMwfNMQg6OrMRIP;Z=eKNMaKNPeP;gh/VWReZUVY`Nf/eVPSgIfNg6OUVYP;g4qSVYga];ZeVYOQfN~Y];ZKNP\[ËVn~Ms];hNRUPeKNM ];O?KaRël|¥P;~IMMwfa]UVYP;g :8Ê gXIf/efNZUM RMsOeVPSgaRQq[nMOrPSgaRVWh/MZUKNMM=.-MsO?RPVYgwUZePdhNfaOrVYgNc1OrPS~~YVWRVYP;gaR];gahKNP\[ÌUKNMx¥O]g~YM];hePhN]TX/ëVYgNc1M=.-MsOeR :² VYga]~Y~Yx;q[nMOrPSgaRUVYh/MZeKNMIfN~Y~]gNVWRUPUZeP;aVYOonP;~UzT]gNgMsSf6]\UVYP;gÍ]gahVeRVYTXN~VWOQ]UVYP;gaRnIPSZ§knm1o-P;~W]ZeVzs]\eVPSg :_ :9S: Ã8ĸÇ\Ä*P¡»N½¾bO¾ØÈQ½s¾# ! QPI½cL;Åj»a¼F¼Ä*!,ºF»N½¾#-¼¯É ² P;ZXT]gdxjbdVg6hNRP§6]ZUUVWOr~YMRqUKNM-MQZUIMOr=®afNVWh]NaZUP_/VYTX]UVYP;g VYR=gNP§RUf/¨OrVYMQgw : JLKNVWR=VWR=6]ZUUVWOrfN~W]Ze~xUZefNMIP;ZZUM~Y]UVY^wVWRëeVYOR-MOQVMsRRUfaO?K];RgNMQfNUZeVgNPwR];gahNKNP;UPSgaR§]3UMZ~W];R ReOQ]\UUMZUVYgNcaq6[KNVWO?K1Ka]_^;M]~Y];ZUcSMTXM];gIZeMQM a]\eK~YMQgNc;UK : JLKNMonP;~UzQT];gNgMsSf6]\UVYP;gb;MQMaRUZ?];O?bXPeKNM gwfaT`-MQZLP46]ZUëeVYOQ~MsRVYg]a]ZUUVWOrfN~W]Z¯MQ~YMQTXMQgwLP4aKa];RUM§RUa];OQM;qSUKNM gdfNT`-MQZn[VUK]a];ZeVYOQfN~Y];Z-PSRUVeVPSg];gahTXPSTMgwUfNT¤Î

dN = f(x,p, τ)d3x d3p.ü MZUMSq

fVWR UKNMXNK6];RUMXRUa]SOrMXh/VYRUZeVY`Nf/UVYP;g

IfNg6OUVYP;g :ÍÊ gÍUKNMOQP;gwUMQwP;knm1o®af6OUf6]\UVYP;g6RQq¯[¯M¤[L]gwePbwgaP\[ÂUKNM¤gdfNT`6MZP

j âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éNKaPUPSgaR=OQP;TXVgacIZUPST]c;VY^;MQg1hNVZeMOrUVYP;gq6[KNVYO?KVWR§NZUMsOrVWRM~xUKNMNZeMRUMQgwh/VWReZUVY`Nf/eVPSgIfNg6OUVYP;g :|¥M`6Mc;VYg `dxOQP;gaRUVWh/MQZeVgacXRUVYTa~MP;gaMOrP;TX-P;gNMgwL6]ZUUVWOr~YMRL[VUKRPSTXM T]SRUR : ³§fNZ

eZUMs]\UTXMgS*OQ];gM]SRVY~Yx `-MnMrdUMgah/MhUP T];ReR~YMReRa];ZeVYOQ~MsR4`dx?]bdVgac=UKaM¯~YVTXV];Rm→ 0.onx24P;fNVY^dV~Y~YM « R*UKNMP;ZeMQT¤qwVCUKaM]4];c;Z?]gNcSVY];gVYgaOr~Yfah/MsR];~~/eKNMZeMQ~YMQ^\];gSaKwx/RUVYORQq;UKNMgXUKNM

h/VWRUZeV`NfNUVYP;g©IfNgaOrUVYP;gIPSZ]cSV^SMQg¸OrPSTX6PSgNMQgwP;aKa];RUMRUa];OQMVYRVgd^\]ZeVY];gwRPeKa]\df = 0.

Vg6OrM[¯M]ZeMOrPSgaRVWh/MZUVYgNc]gNP;g/ëlUZeVY^wVW]~FTXMrUZeVWOqCVVWR§MsRURUMQgweVY];~UP¤h/VWReVgac;fNVWRK`-Mr[nMQMgeKNMOQ];gNP;gNVWOQ];~,TXP;TXMQgwUfaT q

p,[KaVYO?K¤VWRLVgwUZeVYgaRVWO UPk4P;fNVY^dV~Y~M « RneKNMQPSZUMT];gah

eKNM « NZeP;-MQZ « TXP;TXMgSefNT¤qp.JLKNMNZUPS6MZ§TXPSTMgwUfNT VWR§hNMrîagNMsh1VYg]IZ?]TXM[KNVYO?KVYR

~YP/OQ]~Y~Yx¥mVYgNbSP\[=RbdV§ipi = pi p : JLKNMO]gNPSgNVWOQ]~¯TXP;TXMQgwUfaT VWRc;VY^;MQg`dx pi = mui

VYRZeMQ~W]\eMhUPUKNMaZUPS6MZTPSTXMQgwUfNT`wx

pi = a(δij +1

2hij)p

j .

JLKNMXNZeP;-MQZTXPSTMgwUfNT OQ]g¥`-M[ZUVeMQg];Rpi = aqni,

[KNMZUMnVYR] ëìhfNgNVgNP;ZeT

^SMOeP;Z :|¥M§OQ];g[ZUVUMUKNM onP;~UzQT];gNgMwfa]UVYP;gXUKaMQgVYgUMZUTRP,UKaM-PSRUVeVPSg]gaheKNMgNM[

TXP;TXMgSefNT¶a];Ze];TXMrUMZUVYz]UVYP;gqq]gah

n.PSfNV^dVY~~YM « RnUKaMQP;ZeMQTVTXN~YVMsR

df

dτ=∂f

∂τ+dxi

dτ

∂f

∂xi+dq

dτ

∂f

∂q+dni

dτ

∂f

∂ni= 0.

JLKNVWR OQ]g`6MXRPS~^SMh¤-MQZUUfNZe`a]UVY^;MQ~Yx : ­ zQMQZePeK1P;Z?h/MZq6UKNMRPS~fNUVYP;gRUKNP;fa~Yh`6MKNP;TXPëcSMQgNMP;faRL]g6hVWRP;UZeP;NVWO : ­ RURUfNTXVgacX];g¤VgNVUVW]~Y~YxUKNMZUT];~R-MOrUZefNT¤qN[¯M îag6heKa]\

f0(q) = gS(e−q0/T0 ± 1)−1.ü MQZeM;q

gSVWRLeKNMgdfNT`-MQZ=P;*hNMQc;ZeMQMsRLP*IZeMQMhNP;T¤q

T0VYRUKNMNZeMRUMQgw knm1o¸UMT-MQZ?]\efNZeM

];gahq0 = (q2 + a2m2)1/2 VYR¯UKaM MgNMQZec;xTXMs];RUfNZUMsh`dx]OrP;TXP\^dVYgNcPS`aRMZU^SMQZ : JLKNMRUVcSgh/M6MgahNRPSg[KaMrUKaMQZLUKaMa]ZUUVWOr~YMR];ZUM§IMQZeTXVPSgaRLP;Z`-PSRUP;gaR :|¥M¯î6gahUKNMnîaZ?RPSZeh/MZ8RUP;~Yf/eVPSg`dxMr/a];gah/VYgNc VgP;Z?h/MQZ?R

f = f0(q)+f1(τ, q, n,x)+...];gahXUKaMQg¤RPS~^dVYgNcUKaM onP;~UzT]gNgMwfa]\eVPSgb;MMQNVYgNcP;gN~YxUMQZeTRePî6ZeRnPSZehNMQZ :4Ê VYR

RUZ?]VYc;KwnIZUP\[L]Z?hXUPRUKNP\[ UKa]nePzQMZUP;UK P;Z?h/MQZdxi

dτ=pi

p0' pi

p0=qni

q0.

VYgaOQM ∂f∂q

VWR=zMQZePeKP;Z?h/MQZsqa[nMOQ];g1b;MMQ dqdτ

ePîaZ?RP;Z?h/MQZsq6];gah¤eKNMcSMQP/h/MRUVWOMwfa]\eVPSg~YM]ShNRnUP

dq

dτ= −1

2qhij ninj .

Ê gËUKaMîaga];~UMZUTúPUKNM1onPS~ezQT]gNg©MsSf6]\UVYP;g4q dbni

dτ

];gah ∂f∂ni

]ZeM¤`6P;UK©îaZ?RPSZehNMQZsqRUP¤UKNMVZNZePdhNfaOO]g`-MVYc;gNPSZUMshVYgUKaMîaZ?RPSZehNMQZMsSf6]\UVYP;g : JLKdfaRUKNMî6ZeRP;Z?h/MZonP;~UzT]gNgMsSf6]\UVYP;g`-MOQP;TXMRi3IPSZ§]cSV^SMQg ² PSfNZeVMZTXP/h/M;q-Vg[KNVYO?K¤eKNMhNMQ-MQgah/MgaOrMPSg-PSRUVeVPSg`-MOQP;TXMR]hNMQ-MQgah/MgaOrMPSg[L]_^;MgdfNT`-MQZp

∂f1∂τ

+ ik · n q

q0f1 −

1

2

∂f0∂q

qhij ninj = 0.

JLKNVWRVWRn]NNZeP;aZUVW]\eMLIP;ZMVeKNMQZ¯gNMQfNUZeVgNPwRPSZNKNP;UPSgaRQqw[KNMZUM=[nM§O]gX?]b;M=UKNM§TX]SRURU~YMReR~YVTXV]gah

q/q0 = 1.³3UMgUKaMonP;~UzQT];gNgMwfa]UVYP;gXVYRMr/NZeMReRMshVgXeMQZeTXRPRUPë OQ]~Y~YMhðr÷ñmlondö H ôrò?òp?ø H õ=qölñ 3H òrI6[KNMQZeMUKaM

qh/M6Mgah/MQg6OrMP;,UKNM h/VWReZUVY`Nf/eVPSgIfNgaOrUVYP;gKa]SR`-MQMgVYgSeMQcSZe]UMhPSf/q

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê s];gahVeR=Mr/NZeMReRMsh];RneKNMZ?]\UVYP;g PFîaZ?RP;Z?h/MZL®afaOefa]\eVPSgaRnUPeKNMTXM]gh/MgaRUVxCÎ

∆(k, µ, τ) ≡∫q2dq qf1∫q2dq qf0

= −4f1

(q∂f0∂q

)−1

= 4f1

(T0∂f0∂T0

).

ü MQZeM;qµ ≡ k ·n = cos θ.

Ê geMQZeTXR8P;aUKaM`NZUVYc;KwegNMReRFIfNgaOeVPSgq\eKNM=onP;~UzQT];gNgMsSf6]\UVYP;gVWR

∆ + ikµ∆ = −2hijninj .JLKNM`NZUVYc;KwegNMReRLIfNgaOrUVYP;g1`6];RUVYO]~Y~xVYR=UKNMZe]UVYPP*UKNMh/MQg6RVxP;a]ZUUVWOr~YMRù 3 Bsñ H l ñ H8 lñCB\ô Ht6 ñ3÷eôeõQölñ 3H ePUKNMTXMs]g¤h/MQgaRUVx :¯ VYgaOQMUKNMh/MgaRUVxVWRNZeP;-P;ZUUVYP;g6]~-UP

T 4,UKNM

eMQTX6MZe]UfNZeM®6faOefa]\eVPSg¢VWRcSV^SMQgj`dx δTT = ∆/4.

² PSZ]6MZIMsO®afNVWh,qeKNMaZUMsRURUfNZeMVWRVWRP;UZeP;NVWOXVgVeRZUMsRIZ?]TXM :Ê geKNVYROQ]SRMSqUKaM`NZeVcSKSegNMReRIfNg6OUVYP;gVWR6]ZUUVWOrfN~W]Ze~xRUVTXN~YM;q

∆ = δ + 4µv.Ê VWRRPSTXMrUVYTXMRf6RMQIfN~/UP[ZeVUMnUKNM§]gNcSfN~Y];Zh/MQ-MQgahNMQgaOQMP-UKNM`NZeVYc;KwUgNMsRUR8IfagaOeVPSgVYg ]k4MQcSMQgah/ZeMRMZUVYMRL[VUK UKaM]Zec;fNTXMQgw

µ,UK6]\=VWR

∆(k, τ, µ) =∑

`

(2`+ 1)∆`(k, τ)P`(µ),

[KNMZUMUKNMTXP;TXMQgweR ]ZeMcSV^SMQg`dx∆` = 1

4π

∫dΩ∆P`(µ).

JLKNVWRVWR§faRMQIfN~8`6MsOQ]f6RMUKNMcSZe]Sh/VMgwneMQZeTVYg¤UKNMo¯PS~ezQT]gag Mswfa]\eVPSg¤OQP;fNa~MsRP;gN~Yx

∆`TXP/h/MsRUPXeKNM

∆`−1];gah

∆`+1TXP/h/MR :LÊ M]Ze~YxP;g4qaP;ga~xUKNM~P\[nMR=TXPdhNMR];ZUMMrNOrVUMsh,qNeKNMQgeKNMM^;P;~Yf/eVPSgP

eKNMKNVYc;KaMQZTXP/h/MsR]ZeM=MQ/OQVeMhRU~YP\[~x :8² P;ZMQN]TXN~YM;qwVg]6MZIMsO®6fNVYhP;gN~YxeKNM ~P\[nMR[nPTXPSTMgweR=];ZUM gNP;gNëlzMQZeP :_ : : Ä*!,º8»/½¾M6¼¯É§Ê FVYR8ReZe];VcSKwIP;Ze[L]Z?hUP RPS~^SM*IP;Z4UKNM¯MQ^;PS~fNUVYP;gP/eKNM`aZUVYc;KwUgaMReRIfNg6OUVYP;g : mfN~UVYN~YxwVYgNceKNMonP;~UzTX];gNg Mwfa]\eVPSg¤`wx

eikµτ ];gah VgweMQc;Z?]\eVgacP\^SMQZnUVYTXM;qPSgNM îagahNR

∆(τf ) = ∆(τi)e−ikµ(τf−τi) − 2

∫ τf

τi

dτe−ikµ(τf −τ)hij ninj .

k¯PSgaRVWh/MZUKaM O];RUMPVYgaRe];gS?]gNMP;faRU~Yxh/MsOrPSfNN~YVgNc]\nUKNM§~Y]SRReOQ]\UUMZUVYgNcRUfNZU¡];OrMSqτrec.yZeVYP;ZLUPh/MsOrPSfNN~YVgNc6q/UKaM®afNVWhVWR=]SRURUfNTXMhUP`-M]6MZIMsO®afNVWh : JLKaMRPS~f/eVPSg ]UKNM

NZeMRUMQgwL`6MsOrPSTMsR

∆(τf ) = (δ + 4µv) e−ikµ(τf−τrec) − 2

∫ τf

τrec

dτe−ikµ(τf −τ)hij ninj .

JLKNVWRqwh/VY^dVYhNMh`dxX]¡];OrUP;Z¯P,IPSfNZePOrPSgd^;MQZUVYgSePUMTX6MZe]UfNZeM=®af6OUf6]\UVYP;g6RQqwVYRbdgNP\[g]SReKNM ]SO?KaRëì|¥P;~IM§MsSf6]\UVYP;g : JLKaMNKdxdRUVWOQ]~CVYgwUMQZeNZeMr?]\UVYP;gP8M];O?KUMZUTVYRh/VWReOrfaReRMsh`-MQ~YP\[ :_ : a: Ç_¾ØÁ6¾Ø¼u-¿*½Ã8Äk)Ç\¾¡Åj»NÇsÀ ¼8¾¡Èca½Ço6Æ*¾¡ÄdÈ;ɧ¬ MOQP;T`aVga]UVYP;gPCMQ~YMOrUZeP;gaR*]g6haZUP;ëeP;gaR eP¤IP;ZeTíKdxdhNZUPSc;MQgP/OQOrfaZeR[KNMg¥UKNMNKaPUPSg/ëì`a]Zex;PSg1®afNVWh¥ZUMs];O?KNMsR] îN/MhUMQTë-MQZ?]\efNZUMP;P;Z?h/MQZ§]XUKNVYZ?hP]gMvW : JLKNVYR=UKNMg1P/OQOQfNZeR§]\§hNVm.CMQZeMQgweVTXMsRVYg1h/V/.-MZUMgSZeMQcSVPSgaR=w6P/OQOrfaZUZeVgacîaZ?R Vg1fagah/MQZ?h/MgaRMZUMc;VYP;gaR§]gah~W]\eMQZ§Vg1P\^SMQZ?h/MQg6RVUVYMR :=ü P\[nMQ^SMQZsq]SRh/VWRUOQfaReRMshj]`-P\^;MSq8[nMO]gjOQP;gaRUVYhNMQZX]¤îN/MhjUVYTMSq

τrec,VYTXTXMh/VW]\eMQ~Yx]3UMZ];~~¯UKNM

ZeMQcSVPSgaRnZUMsOrP;T`NVYgNMh];RnUKaM 5/8 òö¯òrõ 8 öØö ôr÷ñ H lòø/÷bp 8 õeôc<JLKNMZUM=];ZUMnUKaZUMMTX];VgOQP;gwUZeV`af/UVYP;gaR8ePUKNM=UMQTX-MQZ?]\efNZUM];gNVYRUPeZUPSNVYMR*OrPSTVYgNcIZeP;T

eKNM§~W];RRUO]\eMQZeVgacRUfNZ¡]SOrMSÎFeKNM VgwUZeVYgaRVWO h/MQgaRUVx®afaOrUfa]UVYP;gqwUKaMµ§P;aN~MZLM>.CMO=];gah

x âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éeKNMZeMhaRKNV3UVYgNcPeKNMaKNPeP;g];~PSgNcUKaM~YVgaMP;¯RVYc;Kw :§Ê g¥RxdgaO?KNZeP;gaP;faRcw]fNcSM;q6UKNMsRMO]g `6M[ZeVUUMg ]SR

δT

T=

1

4δγ + vr · n−

1

2

∫ τf

τrec

hijninjdτ,

[KNMZUM¯UKaM=h/P?R]ZeMLhNMQZeV^\]\eV^SMR8[VUKXZUMsR-MOr*UPOrP;ëlTXP\^dVgac=UVYTXM=]gahnVWR] fNgNV^;MOrUPSZ

];~PSgNcXUKNM~YVgNMPRUVcSKw : JLKaVYR§VYR§bdgNP\[g1];R=UKNM ]SO?KaRëì|¥P;~IMMwfa]UVYP;g : i¡·=P;UMUKa]§VYgeKNMNVWOUfaZUM[KNMZUMZeMOQP;T`aVga]UVYP;gPdOOrfNZ?R]\îadMsheMQTX-MQZ?]\UfaZUMSq/UKNMZUM];ZUMgNPh/MgaRUVx®af6OUf6]\UVYP;g6RQqC`Nf/ UKaVYRUMZUT VWRZeMQa~Y]SOrMh1`wx] δa

a

¡]SOeP;Z§RUVYgaOrMNKNPeP;gaRIZeP;T h/V/.-MZUMgSZeMQcSVPSgaRLOQ]g Ka]_^SM§TXP;ZeMP;Z~YMReRnUVYTXMUPXZUMshNRUKNV3 : pJLKNMOQP;gwUZeVY`Nf/UVYP;gXIZeP;TZeMhNRUKNV3UVYgNc]~YP;gacUKNM=aKNPeP;ga]UKOQ];g`6M§fNgah/MZeRUPdP/hVg]

¡];VZe~xRVYTXN~M[L]_x : k¯PSgaRUVYh/MZ=]XRT]~Y~4RUMQcSTMgwLP;FeKNMa]UK¤UK6]\]NKNP;UP;g eZe];gaR^SMQZ?RMsRVYg¥UVYTM

δτ.JLKNVWRRUMQcSTMgwKa];R]¤~YMQgacUKcSV^SMQg`dx

δL = (gijdxidxj)1/2 = (a2(δij +

hij)ninj)1/2δτ ' a(1 + 1

2hijninj)δτ.JLKNM RUKNV3VgUKNMNKaPUPSgeMQTX6MZe]UfNZeM P\^;MQZUKNVWR

h/VWRe]g6OrMVYRLcSV^SMQg `dxUKNMIZ?];OrUVYP;ga];~O?Ka]gac;M P8UKNMRUMQcSTMgwn~YMQgNc;UK¤VYgeKNM-MQZeVP/hδτq

δT

T|δL = − 1

δL

∂(δL)

∂τδτ = −

[a

a+

1

2hijninj

]δτ.

JLKNMXî6ZeRUMZUTOrPSZUZeMRU-P;gahNR§ePUKNMfaRf6]~KNPSTXP;c;MgNMQPSfaRZeMhNRUKNV3UVYgNc1PLNKNP;UPSgaR];gaheKNMRUMOQP;gah,q-VYgSeMQcSZe]UMh¤P\^SMQZeKNM[KNP;~YMNKNP;UPSg1a]\eKqCc;VY^;MsRUKaMUMZUT VYg ]SO?KaRëì|¥P;~IMMQdaZUMsRURUVPSg :Ê L[¯MIP/OrfaRPSg1eKNMRUO]~W]Z NVYMOrMXP;UKaMTMQUZeVYO;q,UKaMQZeMX];ZUM[nP h/Mc;ZeMQMsR§P;IZeMQMsh/P;T

];gaheKNMTXMreZUVWO§®afaOefa]\eVPSg¤VYRLa];Ze];TXMrUMZUVYzQMshVYgRxdgaO?KNZeP;gaP;faRncS];fNc;M];R

hij =1

3hδij + (kikj −

1

3δij)h

S .

JLKdfaRqdVg ² P;faZUVYMQZRUa];OQM=UKaM 6]\UK h/MQ-MQg6h/MQgw=OrPSgSeZUVY`Nf/eVPSgePeKNMeMQTX-MQZ?]\UfaZUM§®afaOrUf/ë]UVYP;g VYR

δT

T= −1

2

∫dτ

∑

k

(1

3h− + (k · n)2hS

)eik·nτ

_ : a:9S:Ay ô>zö 3H ñ 8H l 8 øNlwô2B_ôQ÷?òñ 3H <=|¥MOQ];g¤gNP\[¦f6RMeKNVYRMr/NZeMReRVYP;gUPXeZe];gaRIP;ZeTÂUKNM ]SO?KaRël|¥P;~IMMwfa]UVYP;gVYgSePUKaM·=M[LUPSgNVY];gcS];fNc;Mi

h00 = 2Ψ]gah

hij = 2Φδij).JLKNM

cw]fNcSMeZe];gaRIPSZUT]UVYP;gaRnOQ];g`-MRKaP\[geP`6M

Ψ = 12k2

[hS + a

a hS]

Φ = − 16h

− − 12k2

aa h

S

δ = δsyn + 32k2

aa (1 + w)hS

v = vsyn + ik2k2 h

S

[KNMZUMw ≡ p/ρ = 1/3

VWRnUKNM Mwfa]UVYP;gP8Re]UM§IP;ZnUKaM§®6fNVYhVYg wfNMsReVPSg :| RUVgNcUKaMRUMZeMQ~W]\eVPSgaRQqweKNM ];O?K6Rëì|PS~IM Mwfa]UVYP;g Vg·MQ[LUPSgNVW]g¤cS];fNc;MOQ];g`-MRKaP\[geP`-M

δT

T=

1

4δγ −Ψ + vr · n +

∫ τf

τrec

(Φ + Ψ)dτ.

JLKNVWRVWRRUVYTVY~W]ZePUKNM RxdgaO?KaZUPSgNP;faRcw]fNcSMMr/NZeMReRVYP;g4q4MrNOQMQ/eKNMVYgwUZeVg6RVWOXNVYMOrMVYRTXP/h/VîaMhUPXVYgaOr~Yfah/MeKNMVgNVUVW]~,cSZe]_^dVe]UVYP;ga];~6-PeMQgwUVW]~ : JLKNM îaga];~CUMQZeT¤q/[KNVWO?KVWRnUKNMOQP;gwUZeV`af/UVYP;g¤]~YP;gNceKNMa]\eK¤P8UKNMaKNPeP;gqNVWRbdgNP\[g¤];RneKNMñ H ö ôl÷ 8 ö ô 6 ]SO?KaRëì|¥P;~IMOQP;gwUZeV`af/UVYP;g : ·=P;UMUKa]nVYgeKNM§];`aRMgaOrM§P4];gNVYRUPeZUPSNVWO=ReZUMsRURqSUKNM§´Vg6ReMQVYgMwfa]UVYP;gaRVYTXN~x

Ψ = Φ.

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê ~_ : _ : ÄdºF½QÇ\¾¡¼6ÈSÉ JLKaM¯gNMf/UZeVYgNPSR4IZUMMRUZeM];T]SRFRUPdP;g]SReKNMQxMgSeMQZFUKNMLKNPSZUVYzQPSg : JLKNMVZh/MgaRUVxOQ];g¢RUP;TXMreVTXMR`-MTXP/h/MQ~YMhj`wx¥eZUMs]\UVYgNcUKNMT];R]^dVYReOrPSfaR®6fNVYh : ­ TXP;ZeMMQ/]SOeZUMs]\eTMgwZeMwfNVYZUMsRIP;~Y~P\[VYgNceKNMIfa~~LgNMf/UZeVgaP1NKa]SRM Ra]SOrMhNMQgaRUVxSq];R[VeKeKNMNKNP;UP;g6R]\3UMZ=h/MOQP;fNa~VYgNc :

v/:=ýE4B@BF>sGd@C@,þNA,H3>HbQ@B4ANÊ eRfaRMQIfN~*UP1wfa];gwUVIx1KNP\[P;3UMgeKNMknm1o°NKaPUPSgaRReOQ]\UUMZ :ü MZUMX[nMX[V~Y~*IP/OrfaR

PSgJLKNP;TRUP;g¥RUO]\eMQZeVgacIZUPST IZUMMM~MsOeZUPSgaRQqC`NfN ¬ ];~MVcSKReOQ]UMZUVYgNcIZUPSTKdx/h/ZeP;cSMQg]UP;TRLOQ];g]~WRP`6MVYTX6PSZ?]gwLIP;Z=RUP;TXM IZUMswfNMQgaOQVMsR :vN:9S: ÆF½s¾#;»P-FÄwÆ8½Ã¯É JLKNMXIfagahN]TXMgS?]~wfa];gwUVx1UPOQP;gaRUVYhNMQZVWRUKNMNZUPS`a]`aV~YVxPReOQ]UMZUVYgNcVYg]XeVTXM

dtq6[KaVYO?K1VWR=cSV^SMQg`dx

σTnedt,[KNMQZeM

σTVWReKNMJLKNPSTRPSg1OrZePSReR

RUMOrUVYP;g]gahneVWRXUKNMh/MQgaRUVx¢PIZeMQMM~MsOUZeP;g6R : JLKaM1VgweMQc;Z?]\eMhÍNZeP;`a];`NV~YVxjP\^;MZ

RUP;TXM UVYTXMVgweMQZe^_];~,VYRbdgNP\[g ]SRLUKaM 3 ó-ölñ¡õ 85|6 ô ó-öbn"Iκ =

∫dtσTne.

iØ·=P;UM;qaUKNVWR=VWR=];~YRUPh/MgNPeMh]SR

τVYgUKaM§~YVeMQZ?]\efNZUM : pJLKNVWRVWRZeMQ~W]\eMhePUKNM TXM];gXIZUMM=UVYTXM=IPSZnReOQ]UMZUVYgNc6q

[KNVWO?K VYRtc = 1/σTne.·=MQdOrP;g6RVWh/MQZ¯UKaMNZUPS`a]`aV~YVxXeKa]\=]NKNP;UP;g Ka]SRnUZ?]_^;MQ~YMhIZUMMQ~YxIZUPST¶RPSTXM Ms]Ze~x

eVTXMeP=UKaMNZeMRUMQgw4[VUKNPSf/8ReOQ]UMZUVYgNc6q Ps(t).JLKNM¯NZUPS`a]`aV~YVx ]\eKNMaKNPeP;gUZ?]_^;M~Msh

IZeP;T]RU~YVcSKSe~xM];ZU~YVMZ*UVYTXM;qt−dt, VWRc;VY^;Mg`dx Ps(t−dt) = Ps(t)(1−σTnedt),

[KNMZUMeKNMRUMOrPSgah¥¡];OrUPSZVWR faReKNMNZeP;`6]`NVY~Vx1eKa]\UKNMNKNP;UP;geZe]_^SMQ~YMhUKaMVgweMQZe^_];~

dt[VUKNPSf/RUO]\eMQZeVgac : JLKNVYRVYTXN~VYMReKa]\ Ps(t) = e−κ(t),[KNMQZeMeKNM VgwUMc;Z?]~¯IP;ZUKNM

PS/UVWOQ];~h/MQ/eK¤VYRLM^_];~f6]\UMshIZUPSTtfNgweV~,eKNMNZeMRUMQgw :

² Vg6]~Y~xSq[¯M¤OQ];gÍO]~WOrfN~W]\eMUKaMuBsñWòñ¡ðrñ 5 ñ3öp?ø H õQölñ 3H Ig(t),

[KaVYO?KÍVYRUKaM aZUPS`a]`NVY~YVxeKa]\¯]NKNPeP;gX[L];R~W];R¯ReOQ]\UUMZUMsh]¯]eVTXM

t.JLKNVYRVYRc;VY^;Mg`dx

g(t)dt = Ps(t)σT nedt,eKa]\ VYR§UKaMNZeP;`a];`NVY~Vx UKa]§Vh/VWh1gaP ReOQ]eMQZ]\3UMZeVTXMtTfN~UVYN~YVMsh`dx UKaMNZeP;`/ë

];`NV~YVxeKa]\VhNVYh¥RUO]\UUMQZh/fNZeVYgNceKNMUVYTMXVYgwUMQZe^\]~[t, t + dt].

JLKdfaRQq,PSgNMîagahNR§UK6]\g(t) = σTnee

−κ(t).ï MQgNMZe];~~Yx;qw]ZeP;fNgaheKNM§UVYTM P~W];RnReOQ]eMQZeVgNc6qeKNM hNMQgaRUVxXP;,IZUMM

M~MsOUZeP;g6RVWRX^;MZUxwfNVWO?bw~Yxjh/MsOrZeM];RUVYgNcaq[KaV~YM Ps(t)VWRXc;ZeP\[VgacwfNVYO?bd~Yx;qn]NNZePS]SO?KNVYgNc

fNgaVx ]\~W]\eMUVYTMsR : JLKdfaRLeKNM^dVYRUVY`NV~YVxIfNg6OUVYP;gVWR-M];b;Mh¤]~Y]SRReOQ]eMQZeVgNc];gah¤VYRP;3UMQg¤]aNZUP_/VYT]\UMsh`dx]XRUVTXN~YMï]faReRUVY];g : i¡·=PSZUT];~VYzQMsh,qN];R ∫

dtg(t) = 1.p

vN: : Ä"96Åu*¾¡¼8»/½¾M6¼¯É ­ h/Mr?]VY~MshbdgNP\[~YMh/cSMP4UKNMRUO]\UUMQZeVYgNcKNVWRUP;ZexXZUMswfNVZeMR¯fNg/ëh/MZeRe];gah/VYgNc NZUMsOrVWRM~xKaP\[Ì]g6h[KNMg¥UKNMM~MsOUZeP;g6R ];gahNZUP;UPSgaROrPST`NVYgNMhUP IPSZUTKdx/h/ZeP;c;Mg : JLKaVYRPdOOrfNZeZeMh ZUPSfNc;Ka~x]§]ZeMhNRUKNV3P

z ∼ 1100,[KNMQgeKNMeMQTX6MZe]UfNZeM

P;FeKNMfNgNVY^;MZeRUM[L];R=];`6PSf/]UKNVYZ?h¤P]geV: i Ê: M : q6RUP;TXMQ[K6]\=~YMReRLUKa];g¤UKNMM~MsOeZUPSg

`NVYgah/VYgNcXMQgaMQZec;x : p­ LM]Ze~YxeVTXMsRQq/[KNMgMgNMQZec;MQUVWO=NKNP;UPSgaR[nMQZeMa~MgSeVIfa~Øq/VeRL]cSPdPdh]NNZeP_/VT]UVYP;gePX]SRURUfNTXM UKNMZeM]SOeVPSg

e+ p←→ H + γ[L];RVYgUKaMQZeTX];~MsSfaV~YV`aZUVYfNT :Ê gUKNVWROQ];RUM;qFUKaMO?KNMQTXVWOQ];~-PUMgwUVW]~WRIP;ZUKNM^_];ZUVYP;f6RRU6MsOrVYMRL`a];~Y];gaOrMSqNRPeKa]\

µH = µe + µp.k¯PST`NVYgNVYgNcUKNVWR[VUK UKNMMQ/NZUMsRURUVYP;gIPSZLUKNM

MsSfaV~YV`aZUVYfNTh/MgaRUVxPFgaP;g/ëìZUM~Y]UVY^dVYRUVWO§RU-MOrVYMRqN[nMOQ]g¤h/MZUVY^;M UKaM ]Ka]Mwfa]UVYP;gÎ

1− xe

x2e

=4√

2ζ(3)√π

η

(T

me

)3/2

eB/T .

ü MQZeMxe = ne/nb

VYRUKNMIZ?];OrUVYP;g©PIZeMQM1M~MsOeZUPSgaRQq¯UKaMKdx/h/ZUPSc;MgÍ`NVYgah/VYgNcjMgNMQZec;xB = 13.6 eV, η = nb/nγ

]gahXVWROQP;gaRUZ?]VYgNMh`dx`aVc`6]gNcgdfaOQ~MPSRUxwgweKNMRUVYRq;];gahζ(3) =

1.20206.

à âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/é´^SMQgwUfa];~~YxUKNMhNMQgaRUVxPMQgaMQZec;MreVYOXaKNPeP;gaRhNMOrZeM]SRMsR]gaheKNMc;ZeP;fagahR?]\UMP

Kdx/h/ZeP;c;Mg¡]~Y~YR¯P;f/nPMsSfaV~YV`aZUVYfNT : ­ gdxOQP;T`NVg6]\UVYP;gVgwUPUKNM c;ZeP;fNg6hRe]UM§NZUP/h/f6OrMR]NKNPeP;g[KNVYO?KMrNOrVUMsR§UKNMgNMrdc;ZeP;fagah1Re]\eMVRMMRq,~Ms];h/VYgNcePgaPgNMQO?K6]gNcSMVYgeKNMnIZe]SOUVYP;gP6IZUMMMQ~YMOrUZeP;gaR :Fü P\[¯M^;MQZsq_c;ZeP;fNg6hRe]\eMKdxdhNZUPSc;MQgOQ]gXRUVY~~N`-MNZeP/h/faOrMsh^dVW]eZe];gaRUVeVPSgaR¯UKNZeP;fNcSKeKNMKNVYc;KNMZMrNOrVUMshRe]UMsRLP*Kwx/h/ZeP;cSMQg :JFPIP;~Y~P\[¢UKNMZeMOQP;T`aVga]UVYP;gNZUP/OQMReR*Vgh/MQe]VY~lq[¯MTfaRb;MQMeZe]SO?bPCgNPPSgN~YxUKNM

gdfNT`6MZ¯P;c;ZeP;fNgahRe]UM]\eP;TRQqw`Nf/]~WRUPUKNM gdfNT`-MQZnP4]UPSTXR¯VYgM]SO?KMrNOrVUMshRe]UM : f6O?K¥]¤OQ];~YOQfN~W]\UVYP;gVYRgaP\[VgNZeVYgaOrVYN~YMT];ga]cSM]`a~M i3IP;Z]îagaVeMgwfaT`-MQZPnMrNOrVUMsh~YMQ^SMQ~WReprq`Nf/VVYROQP;TXN~YMr :©ü P\[nMQ^;MZqTfaO?K©OQ];g¢`-M~Ms]ZegNMhjIZUPSTû]¥^;MQZexj];OOrfNZ?]\eM];NNZeP_dVYT]\eVPSgVYgwUZePdhNfaOrMsh`dx y*MQMQ`a~MsRQq6[KNVYO?KTXP/h/M~YRLeKNMKdx/h/ZeP;cSMQg¤]\UPST]SR=Ka]_^dVYgNcM=.-MsOUVY^;M~x[¯PR?]\UMsR : JLKNVWR=VWR=-PSReRUV`N~YM`-MO]faRUMUKNMKNVYc;KNMZ=MrNOrVUMshR?]\eMR=ZeMQT];Vg¤VYgMsSfaV~YV`aZUVYfNT~PSgNc;MZ*UKa];gUKNMc;ZeP;fNgahXRe]UM;qd]g6hX[KNV~YM=VYgMwfNVY~VY`NZeVfaTeKNMQVYZP/OQOQfNa];gaOrxZ?]\eVPwR¯];ZUM]îadMshIfagaOeVPSgP;4UKaM§eMQTX-MQZ?]\UfaZUM : JLKaM MQNOrVUMh Re]\eMRq/[KNVY~M ZUMT]VYgNVgacVYgeKNMQZeT]~,MsSfaV~YV`aZUVYfNT¤qN];Or];RL]XOrP;g6h/fNVIP;ZnIZUMMMQ~YMOeZUPSgaRnUPXTXP\^;M VYgSePUKNMcSZUPSfNgahRe]UM :JLKNMZUM]ZeM[nPNZeVg6OrVYN~MZUPSf/UMsR=ePaZUP/h/faOQVgacUKNMc;ZeP;fagah1Re]\eMM~MsOUZeP;g6R : µVYZeMO

h/MsOQ]_xIZeP;T°eKNMR8eP UKNM 9 R8Re]UMNZeP/h/faOQMR*]xdT]gαNKNP;UP;g[KNVWO?KMQNOrVUMR];gNPeKNMQZ

]UP;T :*ü P\[nMQ^SMQZsqVCeKNMNKaPUPSgXNZUP/h/f6OrMhXVWRZeMhaRKNV3UMshXP;f/nPCUKaM]xdT]gαZUMsRPSga]gaOQM

`dx UKNMLOrPwRTXVWOMQd6]gaRUVPSg`6MQIP;ZeMV8KaV?R8];gNPeKNMQZ8c;ZeP;fagahR?]\eMn]UP;T¤qsUKaMQg]§gNMQ8cSZUPSfNgahRe]UMn[V~Y~N`-MnaZUP/h/faOQMh : ­ ~UMZUg6]\UVY^;M~xSq\V-UKaM;RR?]\eMLhNMOQ]_x/RF[VeK[¯P NKNP;UP;g6R*Ze]UKNMZeKa]g]RUVgac;~YMNKNPeP;gqagNMQVUKNMZP;8eKNMNKaPUPSgaR[VY~~F`6M]`N~YMUPMQ/OQVeM]gaPUKaMQZ=cSZUPSfNgahRe]UM :² PS~~YP\[VYgNcMQ^;MgËUKNVWRRUVTXN~YM]aNZUP_/VYT]\UVYP;gÍZUMswfNVZeMRXM^;PS~^dVYgNc¥]RUV/.hNVm.CMQZeMQgwUVW]~

MsSf6]\UVYP;g : JLKNM`6];RUVYOZUMsRfa~VYR§eKa]\UKaMQZeMZeMQT]VYgaR] RT]~Y~8IZe]SOeVPSg1PIZeMQMXM~MsOUZeP;g6ReKa]\VWRLIZUPSzQMgVYg]\=]~M^;MQ~,P;

10−3 − 10−4.Ê eR8]~WRUP[¯PSZeK gaPUVYgNceKa]\4UKNMUMQTX-MQZ?]\efNZUMPdUKNMMQ~YMOeZUPSgaRq[KNVY~YMVgaVeVY];~~YxOrP;faN~MshePeKNMNKNPeP;gUMT-MQZ?]\efNZeM;qdMQ^SMQgwUfa];~~Yx`-MQc;VYgaRnePh/ZeP;¡];RUMZL]3UMZMwfNVY~VY`NZeVfNTMQgahNR(z < 400).

|KNV~YMeKNMNKNP;UP;g eMQTX6MZe]UfNZeMh/ZeP;aRLVYgd^;MZeRUMQ~Yx[VeK¤UKNMRUO]~YM§¡]SOeP;Zsq/V?ReKNM^SMQ~YP/OrVxP;UKNMMQ~YMOrUZeP;gaR[KNVWO?Kh/ZeP;aR ];RUKaMVYgd^;MQZ?RUMRUO]~YM¡];OrUPSZ : JLKNMMQ~YMOeZUPSgeMQTX6MZe]UfNZeMVYRLaZUPS6PSZeVPSga]~-UPeKNMMQgNMZUcSx;q/PSZ¯eKNM^;M~P/OQVxRewfa]ZeMh : MM]~WRPy*MQMQ`a~MsRi 9st;t; pRUMOrUVYP;gaR u V N: Ms]cSMQZsq ];ReRM~P\^];gah OrPUi 9st;t;t p :

vN: a: ÄNR?¾#-¼F¾ML;»/½s¾#-¼É |M`-MQ~YVM^;MUKaMMQ~YMOeZUPSgaR]gah¥NZePeP;gaROQP;T`aVgNMsh1eP IP;ZeTíKdxSëh/ZeP;cSMQg]\]¤ZeMhNRUKNV3P

zrec ∼ 1100.ü P\[¯M^;MZq,eKNMQZeMVWRc;PdP/h¥MQ^dVYhNMQgaOQMXUKa]UKNVWRVWR

gNP~YP;gac;MQZ=UKNMOQ];RUMUP/hN]_x : |MZUMeKNMQZeM~W]Zec;M]TXPSfNgweR=P;gNMQf/eZe];~4Kdx/h/ZeP;c;Mgq/V§[nP;fN~Wh`-MP;`6RMZU^SMh¥VgjUKNM R-MOrUZ?]PKNVYc;KjZeMhaRKNV3wfa]SRU];ZeR`dxxdT]g

α]`aRUP;Ze/eVPSg©iIUKNM

ï fNgag/ëly*MQUMQZ?RUP;gM=.-MsO : p efah/VYMRLP;*wfa]SRU];ZeRnRKaP\[¸TXPSRLP;4eKNMfNgNVY^;MZeRUM§[L];RnZeMQVYP;gNVYzQMsh`dx

zrei > 5,MrNOQMQ/IPSZOr~YP;f6hNRPLcS]SRh/MgaRMMgNP;fac;K¥UP`-MRUMQ~RUKNVYMQ~Wh/MhiMxdT]g

αOQ~PSfahNR : pJLKNMQZeM§];ZUMeMQgwe]UVY^;M=PS`aRUMQZe^_]UVYP;gaRP,UKaM ï fNgNgNëly*MreMQZ?RPSgM>.CMOrnVYgwfa];Re]Z?R]z ' 6,

[KNVWO?KVYgah/VWOQ]UMsR[¯MT]_x`-MRMMQVYgNcUKNM?]VY~MQgah P;FZeMQVYP;gNVYz]UVYP;g :¬ MVPSgNVzs]\eVPSg1TXPSR NZeP;`6]`N~Yx¤PdOOrfNZeZeMh[KNMg1eKNMîaZeR§cSMQgNMZe]UVYP;g1P;¯Re]Z?RIP;ZeTMsh

];gahNZeP/h/faOrMsh]~W]Zec;M®6f/PVYP;gNVYzQVYgNcZ?];hNVY]UVYP;g1iEγ > 13.6

MvW : pyZeMOQVYRUMQ~YxX[KNMQgUKNVWRP/OQOQfNZeZUMshh/MQ-MQgahaRFPSgKNP\[TfaO?KRUT]~Y~dReOQ];~M¯6P\[nMQZ4UKNMZUMnVWR :FÊ gxdNVWOQ]~Nknµ mOrPwRTXPS~P;ëcSVMsRQqNZeMQVYP;gaVzs]\UVYP;g VWRgNP=MQ/6MsOUMsh UPKa];N-MQgfNgwUVY~F]XZeMhaRKNV3P;

zrei < 20.ü P\[nMQ^;MZq

VTXVcSKSK6]N-MQgTfaO?KMs]Ze~VYMQZVCUKNMVYgNVUVW]~N-MQZUUfaZU`a]UVYP;gaR[¯MZUMLgNPSg/ë ï];faReRVW]gP;Z[nMQZeMVWRP/OQfNZU^\]UfNZeM§VYg¤ga]\efNZeM :¬ MVPSgNVzs]\eVPSg VWR§VTX-P;ZUe]gwIPSZ=UKaMknm1o¦`6MsOQ]f6RMVVKa]a6MgaR=Ms]Ze~xMQgNPSfNc;KqaUKNM

NKaPUPSgaRknm1o©NKNP;UP;g6RnO]g`6M ZUMsRUO]\UUMQZeMhqw[KaVYO?K[¯PSfN~YhMQZ?];RUM[Ka]UMQ^SMQZnVgaVeVY];~C®afaOëefa]\eVPSgaRLNZeMQ^dVPSfaRU~xMQ/VYRUMsh ];gah¤OrZeM]UMgNMQ[P;gaMR : J4PM^_];~f6]\UM[KNMQUKNMZ=ZeMQVYP;gNVYz]UVYP;g

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê àUàVWRLVTX-P;ZUe];gSsq/[¯MTfaRL~YPdP;b]LUKNMPS/UVWOQ];~h/MQ/eK

κ =

∫ t0

trei

σTnecdt =0.88cσTH0ΩB

4πGmpΩ−2

[(2− 3Ω)− (2− 3Ω− Ωzrei)(1 + Ωzrei)

1

2

],

[KNVWO?KVWRLZeMQ~W]\UMshePeKNMNZeP;`a];`NV~YVxXUKa]]NKNP;UPSgVWRLZUMsRUO]\UUMQZeMh :FÊ κ > 1,

eKNMQg V=VYRwfNVUM~VYb;M~x UK6]\NKNPeP;gaR§[VY~~*RUO]\UUMQZ P.PUKNMVPSgNVYzQMh1TMsh/VYfNT : JLKaVYR§[V~Y~h/MQ-MQg6hPSgeKNM`a]Zex;PSgh/MgaRUVxSq4`Nf/VLUKNMVVWROrP;g6RVWReMQgw[VUKgdfaOr~YMQPwRxdgwUKNMsRVWROrPSgaReZe];Vgw?RiΩb ∼ 0.05),

ZeMQVYP;gNVYz]UVYP;g¥[¯PSfN~YhKa]_^SMePP/OQOrfaZ]\]¤ZeMhNRUKNV3Pzrei ∼ 60

VYgP;Z?h/MQZePK6]_^;M]RUVYc;gNVî6OQ];gwnVYTXa];Or=P;gUKNMknm1o : ­ RUVcSgNVî6OQ];gSLP;NUVWOQ]~hNMQ/eK [VY~Y~RfNaNZUMsRURRUZefaOefNZeM¯PSgReOQ]~YMR*RUT]~Y~MZFeKa]g

θ ∼ 2(1000/zrei)1/2 ]g6h[VY~~N];~YRUP§OQZUMs]\eM¯~W]Zec;MnRUO]~YM-P;~W]ZeVzs]\eVPSg :

² PSZxdNVWOQ];~4knµ§m TP/h/M~YRn[KNMZUM§ZUMVPSgNVYz]\eVPSge];b;MsRN~W];OQM§]zrei ∼ 10,

UKNM P;/eVYO]~h/M/UK©VWRP; PSZeh/MZ]IMQ[ 6MZeOQMQgw : JLKNVYRUMgahNRXePc;VY^;M]RfaNNZeMReRVYP;gÍUPeKNMRUTX];~~ReOQ];~M];gNVWRP;UZeP;NVYMR : JLKNMZUMsOrMgS§|¸m ­ y6PS~Y];ZUVYz]UVYP;gZeMRUfN~eR§VTXN~Yx κ = 0.17± 0.04,RUfNc;cSMRUVYgNc=eKa]\ZeMQVYP;gNVYz]UVYP;gTVYc;KwKa]_^SM¯P/OQOQfNZeZUMshMQ^SMQgM]Ze~YVMZq]]§ZeMhNRUKNV3*P;

zrei =20± 10.² Vg6]~Y~xSqL];gNPeKNMQZ-PeMQgwUVW]~Y~Yx¢VYT-P;ZUe];gwVYReRfaMVYRKaP\[wfNVWO?bw~Yx¢ZeMQVYP;gaVzs]\UVYP;g¢e];b;MRN~W];OQM :Ê VVYR gNP; VYgaRe];gS?]gNMP;faRqC`Nf/VYRª a]\?O?Kdx « qCUKaMQg1eKNVWROQ]g]~WRPZUMsRfN~VgUMQTë-MQZ?]\efNZUM]gaVYRUPeZUPSNVMsRn`6MVgNcOQZUMs]\eMh :

ua:a` B'"'SHbHIBF>'dB'@fg]4>>jþi4Ed@,HIBF>ua:9S: Èc6½QÇaÆ*¾#Èc;»/½½ÄwÇ\¾¡¼8Á4É ·=P\[UKa][nMfNgah/MZeRe];gah1[KNMg¥ReOQ]eMQZeVgNc VWR§~YVYb;MQ~YxePP/OQOQfNZsqN[¯MOQ]g¤eZUxUPVYgwUZeP/h/faOrMVVYgwUPonPS~ezQT]gNgMswfa]\eVPSgePîagah]XTPSZUMZeM];~VWRUVWOM^;P;~Yf/eVPSgP;¯eKNMh/VWReZUVY`Nf/eVPSgIfagaOeVPSg :ü MQZeM[nM[VY~~MQN]TXVgaMVYRUPeZUPSNVYOXReOQ]UMZUVYgNc6q];gahVYgUKNMgNMQdRUMOrUVYP;gTXP\^;MLP;gUPZeM]~YVWReVYOnJLKNP;TRUP;gXRUO]\UUMQZeVYgNc?]bdVYgNc§VYgwUP];OOrPSfNgw-PSReRVY`N~YM§-P;~W]ZeVYz]\eVPSgP;4UKaMknmo : |¥M[VY~~RUMQM UK6]\LUKNMT];Vg VTXN~YVWOQ]\eVPSgP;8]TXP;ZeMZeM];~VWReVYO ReOQ]\UUMZUVYgNcXKNVWReP;ZexVYRLePhN]TX UKNMRUT]~Y~MZRUO]~YM TXP/h/MsR :JLKNMXM>.CMOrPLRUO]\eMQZeVgacVWR UP¤~P/O]~Y~xZeMh/VWRUZeV`NfNUMXa]ZUUVWOr~YMR VYgaKa];RUMRa]SOrM :Ê g]

cSV^SMQg¤MQ~YMQTXMQgw=P;*TXPSTMgwUfNT R6];OrMSq6RUP;TXMa];ZeVYOQ~MsR=]ZeMRUO]\UUMQZeMh PSf/=PVq-[KNV~YM]eKNMRe]TXMeVTXMa]ZUUVWOr~YMR[VeK1PeKNMQZ§TPSTXMQgwe]]ZeMRUO]\eMQZeMhVYgwUPV : onPUKNZUP/OQMReRMsRKa];N-MQg[VUK¥]Z?]\eMNZeP;-P;ZUUVYP;g6]~ePeKNMNZUPS`a]`aV~YVx¤P;¯RUO]\UUMQZeVYgNc : JLKNMXonP;~UzQT];gNgMsSf6]\UVYP;g `-MOrPSTXMR

df

dτ=dκ

dτ(fin − f) ,

[KNMZUMfin = 1

4π

∫dΩP (Ω,Ω′)f(Ω′)

VWRUKaM gdfNT`6MZP a]ZUUVWOr~YMRReOQ]UMZUMshjVYgwUP¥UKNMNK6];RUMRa]SOrMP;

f,]gah

P (Ω,Ω′)ZeMQNZeMRUMQgw?R,]6PwRURUVY`N~M];gNc;fa~Y];Z4h/MQ-MQg6h/MQgwFOQZUPwRURRUMOrUVYP;g :

JLKNM~W];RLUMQZeTZeMQaZUMsRMgS?R¯UKNMReOQ]UMZUVYgNcP;f/=P;4eKNMNKa]SRMRUa];OQM P;f.Ê g UMZUTRLP8UKaMNKNPeP;g `NZeVcSKSegNMReR¯IfNgaOrUVYP;gq6[¯M îagaheKNMonP;~UzQT];gNgMswfa]\eVPSgVWR

∆ + ikµ∆ + 2hij ninj = σTnea (δγ + 4µvb −∆)

ü MQZeMeKNM IP;ZeT¶PeKNMVYgaOrPSTVYgNcZe]Sh/VW]\UVYP;g];ReRfNTXMsReKa]\=ReOQ]UMZUVYgNcVYRLVWRUPUZeP;aVYOVgUKNMZeMRIZe];TXMP*UKNMRUO]\eMQZeMQZ : ­ §M]Ze~YxUVYTXMRq σTnea

VYR~Y];ZUcSM]gahReOQ]\UUMZUVYgNcXîN/MRUKNMNKaPUPSg]gah`6]Zex;P;g^SMQ~YPdOQVeVMsRCUP`6MUKNMnRU];TM : ·=Ms]ZeMQZ,eP§h/MsOrPSfNN~YVgNc6q[nMO]gRUP;~Y^;M*UKaVYR`dxMQ/a]gahNVgNcVYg6P\[nMQZ?R*P-UKaM=TXM]gIZeMQM=ReOQ]UMZUVYgNc UVYTXM : ³§gaOQMLUKNM=TXMs]gRUO]\UUMQZeVYgNceVTXMc;ZeP\[=RUP`-M^;MQZex~W]Zec;M;qd[nMOQ]g¤faRUM UKNMOQP;~Y~VWRUVPSgN~MsRURLonP;~UzT]gNg Mwfa]UVYP;g :

àá âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éua: : ¾#P# & »NÅÆ*¾¡¼8Á4É J;faRNZeVPSZXUPZUMsOrPST`NVYga]\eVPSgqeKNMNKNP;UP;g6R]gahÍ`a]Zex;PSgaR]ZeMgNP;-MQZUIMOrU~Yx¸OrP;faN~Msh :Ê g¸¡];OrqUKNM¥NKNP;UP;g6RcSMQgNMZe];~~YxËO]geZe]_^SMQ~ RPSTM¥h/VWR?]gaOQM`-Mr[nMQMgReOQ]eMQZeVgNcwR IZUPSTM~MsOeZUPSgaR : WMQZexZeP;fNcSKN~Yx;qeKNMQVYZTXM];gIZUMMa]\eK~MgNceKVYRλc = (σTnbxe)

−1,[KNMQZeM

xeVWRUKaMIZe]SOeVPSgP;¯IZeMQMM~MsOUZeP;g6R]gah

nbVWReKNM`a];ZUxSP;g

h/MgaRUVx : JLKaVYRTXMs]gIZUMM 6]\UK¤Ka]SRn[¯PXVYTXN~VWOQ]UVYP;gaRnIPSZLUKNMknmo ]gaVYRUPeZUPSNVMsR :² VZ?Rq§UKNVWR¤]~Y~P\[=RUKNMNKNP;UPSgaRePËhNVm.CfaRUM¥PSf/¤PeKNM-PeMQgwUVW]~[nMQ~Y~YRUKaMQx¸Ka]Sh

`-MQMgÍeZe];N-MhÍVgqO]faRUVgaceKNM6MZefNZe`a]\eVPSgaRUP`-MOrPSTXMMr/6PSgNMQgweVY];~~Yx¢hN];T-Mh©]RUT]~Y~=RUO]~YMR : ­ ZeMOQP;T`NVg6]\UVYP;g4qM]SO?K¢NKaPUPSg©Ka];RXKa]ShÍ]gdfNT`6MZPVgweMQZ?];OeVPSgaRqcSV^SMQg`dx

N ∼ cτrec/λc.JLKNMaKNPeP;gh/Vm.CfaRUVPSg~MgNceK¥VWR UKdfaR]NNZeP_/VT]UMQ~Yx

λD ∼√Nλc =

√cτrecλc.

JLKNVWRn~Ms];hNR¯UP]hN]TXNVYgNcPFUKNMRUKNP;ZUUMZ¯[L]_^;M~MgNceK6P\[nMQZnc;VY^;Mg`dx

T (k) ' e−k2/k2

D .JLKNVYRLVWRbdgNP\[g ];Rh/V/.-f6RVYP;ghN];TaVgNc6q/P;Z VY~YbhN];TaVgNc :

­ TXP;ZeMMQ/]SO=ZeMRUfN~IP;~Y~P\[=RLIZeP;TMr/a];gah/VYgNcXUKaMOrPS~~YVYRUVYP;ga];~4onP;~UzQT];gNg¤Mwfa]\eVPSgeP¤RUMOQP;gahP;Z?h/MZ];gahRPS~^dVYgNcIP;ZKNP\[UKNMh/MgaRUVx1M^;PS~^SMR Vg¥eVTXM : i MQM´¡R?]\UKaVPSfi 9t;tS pP;Zy*MQM`N~YMRi 9stNYS pIP;ZTXP;ZeMh/Mr?]VY~YR : pua: a: Ã8¾#c¼8ÄdÈÈu6¿½Ã8Ä&PØ»NȽ1ÈS»/½Q½sÄSÇ\¾Ø¼FÁ¦ÈºFÇ\¿l»;Ä-É JLKNMRMsOrPSgah¥M>.CMOPLUKNMgNP;g/ëgNMc;~YVcSV`a~MNKaPUPSgTXM]gIZUMMa]\eKVYR4UK6]\eKNMZeMOQP;T`aVga]UVYP;gVWRgNP;FVYgaR?]gwe];gNMQPSfaRqs];gaheKNM~VYc;Kw[nM=RMMP;ZeVcSVg6]\UMshIZeP;T]Ze];gNc;MLPCZeMhNRUKNV3eR=i

∆z ' 80.p*JLKNVWR[VY~~-]~WRPRUTMs]Z

eKNM=RT];~~aReOQ];~MLTXVWOrZeP\[n]_^SM¯]gaVYRUPeZUPSNVMsR :FÊ -[nM=];ReRUfNTXMLUKNMLeVTXM=P;-~W];RRUO]\UUMQZeVYgNc[n]SRï];faRURUVW]gN~Yxh/VWReZUVY`Nf/eMh,qF[VeKj]R?]gaha]Z?hh/M^dVY]UVYP;g¥c;VY^;MQg¥`dx

στrec,eKNMQg¥eKNMRUTX];~~

ReOQ];~M];gNVWRP;UZeP;NVYMR[V~Y~6`-MRfNaNZUMsRURUMhX`dxT (k) ∼ e−k2σ2τ2

recc2/2.J¯xwaVYO]~Y~xSq;eKNM§[VWhdeK

VWRσ ∼ 0.03.

d:=ý§B'94A,Hbfd@,HIB4>/:9S: Ç_¾ØÁ6¾Ø¼U-¿ ! ÆPØ»/Ç\¾#L;»N½¾#-¼¯É y*P;~W]ZeVYz]\eVPSg]ZeVYRUMRIZUPSTJLKNPSTRPSgRUO]\UUMQZeVYgNcPN.P,MQ~YMOrUZeP;gaR*VYgXUKaM=NZeMRUMQgaOQMnP;]Sf6];h/ZefN-P;~YM=]gNVWRP;UZeP;dxVYgXUKNM=VYgaOQVYh/MgwZe]Sh/VW]\UVYP;g :JLKNMZe]Sh/VY]UVYP;gUKa] ];ZUVWRMsRLVYR~VYgNMs]Ze~x6PS~Y];ZUVYzQMsh,qagNP;§OrVYZ?OrfN~W]Zsqa]g6h¤VYR=xwaVYO]~Y~x v ë 9"P;UKNM~YMQ^SMQ~-P4eKNM]gNVWRUPUZeP;aVMsRQq;UKNPSfNc;KV?RLNZUMsOrVWRM§~M^;M~ChNMQ-MQgahNRLPSgeKNM UKNVWO?bdgNMReRnPeKNM~Y]SR=ReOQ]UMZUVYgNcXRfaZ¡]SOrM :JLKNM¯VgaOQVYhNMQgw8wfa]Sh/ZUfa6PS~Mn]gNVWRP;UZeP;dx=VWR4cSMQgNMZe];~~Yxh/fNMeP=eKNMcSZe]Sh/VMgw4PdUKaM¯NKaPUPSg

^SMQ~YPdOQVx : JLKNM-P\[¯MZ§RU6MsOUZefNT P¯6PS~Y];ZUVYz]UVYP;g¤®afaOrUfa]UVYP;gaR ]~WRUPZUMQ®aMOreRUKNMPSReOrVY~Y~Y]ëeVPSgaRPnUKNMaKNPeP;g/ëì`a]Zex;PSg®afNVWh :ü P\[¯M^;MZq,-M];bdRVYg¥UKNM-P;~W]ZeVYz]\eVPSgP/OQOQfNZ[KNMgeKNM=^;M~P/OQVx[n]SRT]\/VYTfNT¤qd]gah];ZUMLeKdfaRPSf/P;NKa];RUM=[VUKeKNM§µ§P;aN~MZ-M];bRUZefaOëefNZUMVg¤eKNM]gaVYRUPeZUPSwxR-MOrUZefNT : JLKNMQZeMVYRc;MgNMQZ?]~Y~xgNP-P\[¯MZP;gTXP/h/MR=UKa][nMQZeMPSf/eRUVYhNMUKNMKNP;ZeVYzQP;g]§~W];R§RUO]\UUMQZeVYgNcaqaUKNPSfNc;K~W]Zec;MReOQ];~M-P;~W]ZeVYz]\eVPSgOQ]gZUMsRfN~VZeMQVYP;gNVYz]UVYP;gPdOOrfNZeZeMhRUf/¨OrVYMQgwe~xM]Ze~Yx :y*P;~W]ZeVzs]\eVPSgO]gNZeP\^dVYhNMVgahNMQ-MQgah/MgwOrPSg/îaZeT]\UVYP;gP;UKNM ]gNVWRP;UZeP;dx1TXM]SRfaZUMQë

TXMQgw?R]gahX];~YRUPP.CMQZ][L]_xIPSZ`NZeM]bdVYgNca]Z?]TXMQUMQZhNMQc;MgNMQZ?];OQVMsR8UKa]P/OQOrfaZ[VeKXUKNM];gNVYRUPeZUPSNVYMRn]~YP;gNM :FÊ g a]ZUUVWOrfa~Y];ZqdVLVWRwfNVUMfaRUMrIfN~CIPSZLOQP;gaRUZ?]VYgNVYgNcUKNMZeMQVYP;gNVYz]UVYP;gKNVWRUP;ZexP8UKaMfNgNVY^;MQZ?RUM ];gahIP;Z=h/MQUMOrUVYgNcUKNMaZUMsRMgaOrMP8UMgaRUP;Zn®afaOrUfa]UVYP;gaR :Ê gRUP;TXM=[L]_xdRqTXM]SRfaZUVYgNcUKaM=6PS~Y];ZUVYz]UVYP;gXVWRRUVTXN~YMQZeKa]gTXMs];RUfNZUVYgNceKNM=eMQTX6MZë

]UfNZeM]gNVWRUPUZeP;aVMsRLUKNMTRM~^SMRq6]\§~Ms];R=IZUPSTeKNMcSZUPSfNgah : JLKNVWR=VWR=`-MO]faRUMTXM]SRfaZUMQëTXMQgw?R O]g1`-MXT];h/M`wx¤î6gah/VYgNceKNMXh/Vm.CMQZeMQg6OrMVg[¯P6PS~Y];ZUVYz]UVYP;gaR§]\]îN/Msh1-P;VYgSVYgjeKNMRbdx : o¯xjRU[VeO?KNVYgNcSfaVYO?bd~Yx`6MQ[¯MMQgj[nP¥6PS~Y];ZUVYzQMZeRq]gdxOrPSgSeZUVY`Nf/eVPSgjIZeP;T]UTXPSRUNKNMZUVWOM=.-MsO?R=OQ];g`-MOQ];gaOrM~MshPN.NZeMOrVWRUMQ~Yx :­ gdfNT`6MZPgNM[Mr/6MZUVYTXMQgweR]ZeMgNP\[eZUxdVYgNc¤UPRMs]Z?O?KIPSZknm1oÌ6PS~Y];ZUVYz]UVYP;g];gah©[nPjKa]_^;Mh/MreMOeMhM^dVYh/MgaOrMIP;Z V : JLKaMµ ­ dÊ VYgSeMQZUIMQZeP;TXMreMQZ ]\eKNM P;f/eK-P;~YMh/MreMOeMh¥-P;~W]ZeVzs]\UVYP;g¥h/VYZUMsOU~Yx]RUfN`/ë h/Mc;ZeMQMXRUO]~YMR :Ê g];hah/VeVPSgq|¸m ­ y°Ka];R

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê àh/MQUMsOUMsh¥UKNMUMT-MQZ?]\efNZeMrëì6PS~Y];ZUVYz]UVYP;gOrZePSReROrP;ZeZeMQ~W]\UVYP;g4q4NZeVT];ZUVY~xP;g~W]Zec;MReOQ];~MsR];gahf6RMshVUPOQP;gaRUZ?]VYgeKNMRUO]\UUMQZeVYgNcXP;/eVYO]~h/M/UK :/: : ½6ÄdÈÆ*»/Ç\»NÅjÄw½sÄSÇ\ÈSÉ 4Mr¯f6RgNP\[¸OQP;gaRUVWh/MQZeKNM NKdxdRUVWOQRP;6PS~Y];ZUVYz]UVYP;gXVYgTXP;ZeMh/MQe];V~ : k¯P;gaRUVWh/MQZZ?];h/VW]\eVPSgPIZeMwfNMgaOrx

ωeZe]_^SMQ~YVgacVYg1UKaM

zh/VYZUMsOUVYP;g :Ê eR MQ~YMOrUZeVYO

îaM~Yh OQ];g¤`6M[ZeVUUMg ]SREx = ax cos(ωt− θx)Ey = ay cos(ωt− θy).|¥M§O]ga];Ze];TMQUMZUVYzQMLeKNM§h/Mc;ZeMQMP;6PS~Y];ZUVYz]UVYP;gXVgUMQZeTRP;CIP;fNZ UPSb;MRa]Z?]TXMQUMQZ?RÎI =

⟨a2

x

⟩+

⟨a2

y

⟩

Q =⟨a2

x

⟩−

⟨a2

y

⟩

U = 〈2axay cos(θx − θy)〉V = 〈2axay sin(θx − θy)〉ü MQZeM;qwUKNM]_^;MZe];c;MsR]ZeMP\^SMQZeVTXMsR¯~YP;gacOQP;TXa]ZeMhUPUKNM 6MZUVYP/hP;UKNM [L]_^;M :

ITXM]\ë

RUfNZeMR¯UKNM ePe];~CVgwUMgaRUVxSq/[KNV~YMQ];gah

Ua]Z?]TXMQUMQZeVYzQM=eKNMT]cSgNVefah/M]gah h/VYZUMsOeVPSg

P;UKaM~VYgNMs]Z-P;~W]ZeVzs]\UVYP;g :VTXMs];RUfNZUMsR UKNM~YMQ^SMQ~P;nOQVZ?OrfN~W]Z-P;~W]ZeVYz]\eVPSgq[KNVWO?KVYR

gNP;faRf6]~Y~xMQNOrVUMh VYg UKNMknm1o OrP;gweMrd :Ê [¯MOrP;g6RVWh/MQZZUP;e]\eVgaceKNMOQPwPSZehNVga]UM`a]SRVWR`dxRUP;TXM];gNc;~YM

φ,eKNMTXM]SRfNZeMRP

~YVgaM]ZL-P;~W]ZeVzs]\eVPSgUZ?]g6RIPSZUT]SR

Q′ = Q cos 2φ+ U sin 2φU ′ = U cos 2φ−Q sin 2φ.¬ P;e]UVYgNc`dx

45e];b;MsR

Q → U]g6h

U → −Q, [KNVY~M] ZUP;e]UVYP;gP; 180ZUMQUfNZegaR eP

eKNMRU];TM^\]~YfNMR : JLKNVWRVWR§`-MO]faRUMUKNMX-P;~W]ZeVzs]\UVYP;gVYRZeMQNZeMRUMQgweMh`dxVeR]TXN~YVUfah/M];gah V?RN~W]gaMZe]UKNMZUKa];g`dx ];gP;Z?h/Vg6]Zex^;MsOUPSZ :L² PSZUKaVYR§ZUMs];RUP;gq/-P;~W]ZeVYz]\eVPSg¤VYR§]RUNVYg/ë îaM~Yh¤]g6hV=VWRLP3UMg¤ZUMNZUMsRMgwUMh`dx]

2× 2RUxwTXTXMreZUVWO§UZ?];OQMQ~YMReR¯T]UZeV,Î

P =

(Q UU −Q

).

/: a: Ã8Ä ÆQP¡»/Ç\¾#LSÄ*- ! QPI½cL;Åj»a¼F¼jÄ*!,º8»/½¾M6¼¯É J4PIP;~Y~P\[©eKNM OQZUMs]\eVPSg]gahM^;PS~f/eVPSgP;--P;~W]ZeVzs]\eVPSg®6faOefa]\eVPSgaRQqS[¯MTfaRTXPdhNVIxUKNM§o¯PS~ezQT]gagMwfa]\eVPSgaR*ePIP;~Y~P\[ÍUKNM-P;~W]ZeVzs]\eVPSgh/Mc;ZeMQMR*P;6IZeMQMsh/P;T]gahX];~YRUPVgaOQ~f6h/MLUKNM§]gNVWRP;UZeP;NVWOnga]\efNZeMnP;CJLKaP;TRPSgReOQ]UMZUVYgNc : JLKNPSTRPSg ReOQ]UMZUVYgNcKa];RneKNMh/V/.-MZUMgwUVW]~4OQZUPwRURLRUMOeVPSgcSV^SMQg `dx;q

dσ

dΩ=

3σT

8π|ε′ · ε∗|2 ,

[KNMZUMε]gah

ε′]ZeMeKNM§-P;~W]ZeVYz]\eVPSg^;MsOUPSZeRP,eKNM§VYgaOQP;TXVgac]g6hP;fNUc;PSVgacZe]Sh/VW]\UVYP;g4q

ZeMRU6MsOeV^SMQ~Yx : ´^SMQgVeKNMVYgaOQP;TXVgacZe]Sh/VW]\UVYP;gVWR§fNgN-P;~W]ZeVzMh,q-Ze]Sh/VW]\UVYP;g1ReOQ]eMQZeMh1]90

[V~Y~`-M~Ms]_^;MOrPSTa~MQUMQ~Yx-P;~W]ZeVYzQMh :JLKNVWRVWRVYgaOr~Yfah/MshXVgUKNMonPS~ezQT]gNgXMsSf6]\UVYP;g`dxUZeM]\eVgacUKNM§h/VYRUZeVY`Nf/UVYP;gXIfagaOeVPSg

]SR§]^;MOrUPSZq-[VeKMs];O?K1M~MTMgw ZeMQNZeMRUMQgweVgNc]h/V/.CMQZeMQgw eP;bSMR§a]Z?]TXMreMQZ : ² P;Z§UKNMO];RUM¯P;-ReOQ]~W]Z8®af6OUf6]\UVYP;g6RQqgaP

UTXP/h/MR];ZUMnc;MgNMQZ?]\eMh]gahUKNM=hNVYRUZeV`af/UVYP;gIfNgaOrUVYP;gaR

O]gj`-M¤ZUMNZUMsRMgwUMhj`wx¥[nP`aZUVYc;KwUgaMReRIfNg6OUVYP;g6RQq∆T

]g6h∆P .

JLKNM]gNVWRP;UZeP;dx¥PJLKNPSTRPSg¥ReOQ]UMZUVYgNc1OQP;fNN~YMRUKNMsRMMsSf6]\UVYP;g6RQqF[VeKUKNM-P;~W]ZeVYz]\eVPSg`-MQVYgNcRUP;faZeOQMh`dx¤UKaMSf6];h/ZefN-P;~YMP;UMTX6MZe]UfNZeMh/VYRUZeVY`Nf/UVYP;g4q

∆T2.JLKNM-P;~W]ZeVzs]\UVYP;g1IMQMhaR`a];O?b

VYgwUPUKNM MQ^SP;~Yf/UVYP;gPeKNM UMQTX-MQZ?]\efNZUM§®afaOrUfa]UVYP;gaRq/hN]TXNVYgNcUKNMT]\RUTX];~~YMQZReOQ];~MsROQP;TXa]ZeMhUPUKaMVYRUPeZUPSNVYOReOQ]UMZUVYgNcZUMsRfa~?R :Ê g¦UKNM¥UVYc;Kwe~x OQP;fNa~Msh ~YVTXVq UKaM¥aKNPeP;g/ëìMQ~YMOrUZeP;g®afaVYh ];OreR¤];R¤];g VWh/M];~®afNVWh

h/MsRUOQZUVY`-Mh©`dx©V?R hNMQgaRUVx©]g6h^;M~P/OrVx]~YP;gNM : JLKNM6PS~Y];ZUVYz]UVYP;g¸]g6h©UMT-MQZ?]\efNZeM

à 0 âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éwfa]Sh/ZUfa6PS~M]ZeMzQMZUP : ­ RUKNMUVYTMP~Y]SRReOQ]eMQZeVgNc];NNZePS];O?KaMRq[¯MO]gjRUP;~Y^;MIP;ZeKNM-P;~W]ZeVYz]\eVPSg`dx¥MQ/a]gahNVgNcVgÍ6P\[nMQZ?RP;=UKaM OQP;g/IPSZUT]~¯TXM]gÍRUO]\eMQZeVgac1UVYTXM;qτc.J4PîaZ?RPSZehNMQZsq6[nMîagah

∆T2,∆P ∝ ikµvτc;UK6]\§VWRq6[nMîag6hUKaMwfa];hNZUfN-P;~YMTXPë

TXMQgw];gahUKNMVYgah/faOQMh -P;~W]ZeVzs]\UVYP;g ]ZeMNZUPS6PSZeVPSga]~-ePXUKNMcSZe]Sh/VYMQgwnP;FeKNM^;M~P/OQVxeVTXMRUKNMTMs]gIZeMQMReOQ]UMZUVYgNceVTXM :*Ê ,[nMVgwUMc;Z?]\eMLUKNVWReKNZUPSfNc;KeKNM=~W];R¯RUO]\UUMQZeVYgNcRUfNZU¡];OrMSq6[nMîagah

∆T2,∆P ∝ ikµv∆τrec,[KNMQZeMeKNM[VWhdeKP;UKNM^dVYRUV`aV~YVx¤IfagaOeVPSg

Ka]SRZeMQa~Y]SOrMheKNM TMs]g RUO]\eMQZeVgacUVYTXM : JLKNVWR¯]SRURUfNTXMRUKaM§^SMQ~YPdOQVxXVWR¡];VZe~YxXOQP;gaRe];gwP\^SMQZUKNMRfNZU¡];OQM¤P~Y]SRReOQ]UMZUVYgNcwIP;ZRUT]~Y~ReOQ]~YMReKNVWRVYRgNPeKNMO];RUM¤];gahÍUKNM-P;~W]ZeVzs]\eVPSg`-MOrPSTXMRRUfNNNZeMReRMsh`dxeKNMUKaVYO?bdgNMsRURLP;4UKaM~Y]SR=ReOQ]\UUMZUVYgNcRfaZ¡]SOrM :JLKdfaR§UKNM-P;~W]ZeVzs]\UVYP;g¤eZe]SOrMsR=eKNM^SMQ~YP/OrVxî6MQ~Wh1P;g1UKNMRUfNZ¡]SOrMP;~W];RReOQ]UMZUVYgNc :

Ê eKNM ^;MQ~YP/OrVUVYMRn]ZeM=VYgw[L]Z?h/~YxOQP;gd^;MZUcSVgacVgwUPX]-P;VYgwqwUKaMQgVL[V~Y~C`6MRfaZUZeP;fNg6h/Mh`dx]Ze]Sh/VY];~F-P;~W]ZeVYz]\eVPSg1a]UMZUg :Ê VYgaReM]ShUKaM^SMQ~YP/OrVUVYMR ]ZeMh/VY^;MZUcSVgNc6q-UKNMZUM[V~Y~*`6M]X?]gNcSMQgwUVW]~F6PS~Y];ZUVYz]UVYP;g¤a]UMZUg : JLKNM-P;~W]ZeVYz]\eVPSg¤6P\[nMQZ R-MOrUZ?][V~Y~8`6Mc;ZeM]\eMRPSgReOQ]~YMRn[KNMZUM§UKaM^;MQ~YP/OrVUVYMRn[nMQZeM§`NVYc;cSMRnPSgeKNM~Y]SRReOQ]\UUMZUVYgNcXRfNZU¡];OQM : JLKdfaR¯UKNM-M];bdRn[VY~~4`6MPSf/PFaKa];RUM[VeKeKNMVgweMQgaRUVx6Ms]b/Rn[KNVWO?KIPS~~YP\[eKNMh/MQg6RVxPFUKNM®afaVYhi¡RUMQM`-MQ~YP\[ : p/: _ : 6ÇsÇ\Ä*P¡»N½¾#-¼j¿ìº8¼;½¾#-¼8È;É§Ê gc;MgNMQZ?]~lq[nMOQ];g~PdP;bIP;ZnOrP;ZeZeMQ~W]\UVYP;g6R*`6MQ[¯MMQg]gdxP;8eKNM eP;bSMR=6]Z?]TXMreMQZ?RQÎ

I,QPSZU.ü P\[nMQ^;MZq6OrMQZUe];VgOQP;ZeZUM~Y]UVYP;gaRLTfaR§`6MzQMQZePV

eKNMXfNgNVY^;MQZ?RUMZeMRU6MsO?Ra]ZeVxVYgd^\]ZeVY];gaOrM :2 g6h/MQZ] ZeMr®aMsOUVYP;g¥]`-P;f/eKNMy− ]dVWRqCUKNMePe];~/VgweMQgaRUVx]gah

Q6PS~Y];ZUVYz]UVYP;gP;-] 6PSVgwZeMQT]VYgaR*fNg6O?Ka]gNcSMh,q\`af/

U → −U. JF[¯P-P;VYgwOQP;ZeZUM~Y]UVYP;gaR§`-Mr[nMQMQgUKaVgNcwRPLP;N-PSRUVUMXa]ZeVx;qRUfaO?K¥]SR 〈TU〉 PSZ 〈QU〉 TfaR`-M§zMQZePVUKNM fNgNVY^;MZeRUMVWR¯a]ZeVxXVYgd^_];ZUVW]gw : ·PUM ]~WRPUKa] 〈TQ〉 TfaR]aNZUPw];O?KzMQZeP]RUT]~Y~RMa]Z?]\eVPSgaRn[KNMQZeM UKNM-P;~W]ZeVYz]\eVPSgOQPwPSZehNVga]UM `a];RUVWRL`6MsOrP;TXMsRLfNgah/MQîagNMh :Ê g]ShNh/VUVYP;g°eP¸UMTX6MZe]UfNZeM¢]g6h°6PS~Y];ZUVYz]UVYP;g°]fNUPë OrPSZUZeMQ~W]\eVPSgaRq PSgNM¢O]g];~YRUP

RUM];ZeO?KIP;ZOQP;ZeZUM~Y]UVYP;gaRLP;*UKNM-P;~W]ZeVYz]\eVPSg[VUK¤eKNMeMQTX-MQZ?]\UfaZUM];gNVYRUPeZUPSdx : JLKaMRUMO]g`-M-PSRUVeV^SMP;ZgNMQcw]\eV^SM]gahjPSReOrVY~~W]\eM[VeKUKaMIZeMwfNMgaOrx¥PeKNMNKaPUPSg®afNVWh,q`NfNX];ZUMPSf/P;=NKa]SRM[VeK¢MQVUKaMQZUKNMUMT-MQZ?]\efNZeMP;ZUKNM -P;~W]ZeVzs]\UVYP;gjOQP;ZeZUM~Y]UVYP;gIfNg6OUVYP;g6R : Ê gZUMs]~RUa];OQM;qFUKNMsRM[VY~Y~~YPwPSb~YVbSMgNMrZe]Sh/VY];~P;Z?]gNcSMQgwUVW]~6PS~Y];ZUVYz]UVYP;ga]eMQZegaR]ZeP;fNg6hKNP;=P;ZOrPS~Yh¤RU6P;eRLVYgeKNMUMTX6MZe]UfNZeMh/VYRUZeVY`Nf/UVYP;g :³§`aRUMQZe^\]\UVYP;g©RUZ?]\UMc;VYMRIPSZh/MreMOrUVYgNc 〈QQ〉 ]gah 〈QT 〉 h/V/.-MZRUVcSgNVî-OQ]gwe~x : JLKNMIPSZUTXMQZZeMwfNVYZeMgNP;VWRM~YMQ^SMQ~WRXUP`-M1P;PSZehNMQZXUKaMRVYzQMP UKNM1-P;~W]ZeVYz]\eVPSgRVYc;ga];~ØqLRUP

PS`aRMZU^\]UVYP;gaRTfaRIP/OrfaRP;gÍ]¤¡]VYZU~YxRT];~~LRMsOeVPSgPUKaM RUbdxePZeMh/faOQMeKNMgNPSVYRUM :JLKNM¥OrZePSReROQP;ZeZUM~Y]UVYP;gOQ];g `-M¥hNMrUMsOeMh¸[VUK ]j~W]Zec;MZgNPSVYRUM~YMQ^SMQ~lq§]SR~YP;gNcË];RUKNMeMQTX6MZe]UfNZeM®afaOefa]\eVPSgaR]ZeMTXMs];RUfNZUMshZeMQ~YVY];`N~x : JLKNMXgNPSVYRUMVYg-P;~W]ZeVYz]\eVPSgOQ];gaOrM~YRPSf/[KNMgÍRUf/¨OrVYMQgwU~Yx~Y];ZUcSM]ZeM]SRPUKNM¤Rbdx]ZeMPS`aRMZU^SMh : y8PS~Y];ZUVYz]UVYP;gT]_x[¯M~~`-M îaZeRh/MreMOeMh`dxV?ROrZePSReR¯OQP;ZeZUM~Y]UVYP;g[VUKUMQTX-MQZ?]\efNZUM§ëVYga];ZeVYOQfN~Y];ZnUKNMm ­ yRe]\eMQ~Y~VUM[VY~Y~4Ka]_^SMgNP;VWRM~M^;MQ~WRePdPKNVYc;K¤UPh/MreMO§UKNM6PS~Y];ZUVYz]UVYP;g¤]f/ePë OrPSZUZeMQ~W]\eVPSgq`NfNV¯[V~Y~6RUfNZe^;MQxUKNM=MgwUVYZUM=RUbdx]g6hUKdfaR[VY~Y~N`6M ]`N~YMLUPhNMrUMsOeKNMOQZUPwRUROQP;ZeZUM~Y]UVYP;g¡];VZe~xM];RUVY~x :/: vN: P¡Ä*½Ç\¾#»N¼-Åj»NÁ-¼FÄw½s¾#¥ÅU-8ÄwÈSÉ ­ ]gdx-P;VYgwP;gUKNMRUbwxSq,UKaM6PS~Y];ZUVYz]UVYP;gKa]SR*[nPh/MQcSZUMMRP;-IZeMQMsh/P;T¤qSV?R]TXN~YVUfah/M ]gahVYgaOr~YVYga]\eVPSg]gNcS~M :FÊ gcSMQgNMZe];~Øq\eKNM-Pë~W]ZeVzs]\eVPSgOQ];gX`6MhNMOrPSTX6PwRMshVYgwUP[nPbwVYgahNRP,TXPdhNMRÎ4MQ~YMOrUZeVYOi¡´¯pqwOrfNZe~ëØIZeMQMTXP/h/MR];gah¤T]c;gaMrUVWOiØmpq-c;Z?];h/VYMQgwUëØIZeMQMTXP/h/MR : P/OQ];~~Yx;q6´ëìTP/h/MsR];ZUMMVeKNMQZe];gNc;MgSeVY];~P;ZZ?];h/VW]~LOrPSg/îacSfNZe]UVYP;gaRP§UKNM-P;~W]ZeVYz]\eVPSgja]UMZUg4q[KNVY~Mo¯ëìTP/h/MsRXKa]_^;M¤]MQVUKNMZ]OQ~P/O?bd[VYRUM=PSZ¯OrPSfNgwUMZeOQ~P/O?bd[VYRUMKa];gah/MhNgNMReR : i ² VYc;fNZeMc_p ­ afNZUM§´ëìTP/h/M OrPSg/îac;faZe]UVYP;gO]g`-MnUZ?]gaRIP;ZeTXMhVYgwUP]afNZUMLo¯ëlTXP/h/Mn`dxZUP;e]\eVgac§Ms];O?K-P;~W]ZeVzs]\eVPSgN~W]gNMn`dx ±45iØ]gah^dVYOQM^;MQZ?Re] : p

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê à[JLKNM-P;~W]ZeVzs]\eVPSg1T]\UZeVOQ];g`-MMr/a];gah/MhVgPSZeKNP;cSP;ga];~4IfNg6OUVYP;g6R( faR~YVYb;MUKNM

eMQTX6MZe]UfNZeM;q

P =∑

l,m

(aE

lmYElm + aB

lmYBlm

),

[KNMZUMUKaMY E,B

lm

]ZeM ];g¢PSZeKNP;cSP;ga];~`a]SRVWRP RUxdTXTMQUZeVYO;qeZe]SOrMQ~YMReR2 × 2

T]\eZUVWOrMsR :JLKNMsRMO]g c;Mr=]X`NVOrPSTXN~MQ[KNMg h/Ms]~YVgac[VUK¤~W]Zec;MReOQ]~YM 6PS~Y];ZUVYz]UVYP;g :

/: ua: -ÈÅUQP#-Á-¾#;»PKN-¼FȽQÇ\»N¾Ø¼C½È¿lÇ6Å ÆPØ»/Ç\¾#LS»/½¾M6¼¯É y*MZUKa];aRUKNM¤TXPSRVYTX6PSZë?]gw4eKNVgaceP`-M¯~YM];ZUgNMshIZeP;T 6PS~Y];ZUVYz]UVYP;gVWRFUKNM¯UVYTXM¯P;a~Y]SR*RUO]\UUMQZeVYgNc : y*PS~Y];ZUVYz]UVYP;gVWR¯]OQ];faRU];~6aZUP/OrMsRUR¯]gah[¯M h/PgaPLMr/6MsOOrPSZUZeMQ~W]\eVPSgaRVYgVnPSgRUO]~YMR¯~Y];ZUcSMQZeKa]gUKNMKNPSZUVYzQPSg ]~W];R§RUO]\UUMQZeVYgNc : JLKdfaRQq66PS~Y];ZUVYz]UVYP;g¤OQ];g`6Mf6RMsh UPVYg/IMQZUKNMUVYTMP~W];RReOQ]UMZUVYgNc : y*P;~W]ZeVzs]\UVYP;gOQP;ZeZUM~Y]UVYP;gaR4P;g^;MQZex~Y];ZUcSMReOQ]~YMRq\~YVbSMUKaPSRUMnRUMQMgVYg|¸m ­ yÍqVYgah/VWOQ]UM¯UKa]8eKNMLfNgNVY^;MZeRUM¯[L];R8ZeMQVYP;gNVYzQMshM];ZU~YxMQgaP;fNcSKUP]SO?KNVM^;Mn]§RUVcSgNVî6OQ];gS*P;NUVWOQ]~h/M/UK IPSZLUKNMNKaPUPSgaRnUPXZeMReOQ]UMZ :Ê g];hNhNVeVPSgqeKNM´ë o¢h/MsOrP;TX-PSRUVeVPSgP;6eKNM-P;~W]ZeVzs]\UVYP;g[V~Y~N`-M=VYT-P;ZUe];gw : 4VgaM]Z

ReOQ];~Y];Z®af6OUf6]\UVYP;g6R OQZUMs]\eMPSgN~x´TXP/h/MR :ü P\[¯M^;MQZsq-c;Z?]_^wVe]UVYP;ga];~F[L]_^;MsRO]gOQZUMs]\eM`-PeK´]gahoTXP/h/MR : JLKdfaR]h/MreMOeVPSgPVYgSeZUVYgaRUVYOoëìTXP/h/MR[¯PSfN~Yh`-McSPwP/hM^SëVWh/MQg6OrM¤IP;ZRUVYc;gNVî6OQ];gweMQgaRUP;ZX®6faOefa]\eVPSgaR : JLKNMMr/-MOeMhË~YMQ^;M~YRXP;oTXPdhNMRXIZeP;TcSZe]_^dVe]\eVPSga]~n[L]_^;MRq¯UKNPSfNc;KqL]ZeMMr/-MOrUMhÍePj`6M1RUT]~Y~ØqLT]bdVYgNcUKaMP;`aRUMQZe^\]\UVYP;g6RwfNVUMO?Ka];~~YMQgNcSVgac :

YN:=<?>HW@,H'GCB4>4D4HW@,HIB4>·=P\[¢[¯MnKa]_^;Mn]§OQP;TXN~YMreMnRUMr*PCOrP;faN~Mshh/V/.CMQZeMQgwUVW]~dMwfa]UVYP;gaRF[KNVWO?K[nMnO]gMQ^;PS~^SM

IZeP;TM];ZU~YxeVTXMRqw[KNMQgeKNM=TXP/h/MRPVYgwUMQZeMR[nMQZeML¡]Z¯P;f/?RVWh/MeKNMKNPSZUVYzQPSgqfNgweV~6UKNMNZeMRUMQgw : JLKNMsRM§Vg6Or~Yfah/M§UKNM ´VYgaReMQVYgMswfa]\eVPSgaRQqS®afNVWhMswfa]\eVPSgaRIPSZnOQP;~WhhN];ZUbXT]\UUMQZ];gah`a]Zex;PSgaRQqSo¯PS~ezQT]gagMswfa]\eVPSgaR*IP;Z¯NKNP;UP;g6R=i3eMQTX6MZe]UfNZeM=]g6hX6PS~Y];ZUVYz]UVYP;g6p*];gahgNMf/UZeVYgNPSR :² Vg6]~Y~x¥[¯MTfaR]~WRP1OQP;gaRUVYhNMQZUKaMMQ^SP;~Yf/UVYP;g¥P[Ka]MQ^;MZhN]ZebMgNMQZec;xOQP;TX6PSgNMQgwVYR T]bdVYgNcfa " P¯UKNMT]\eMQZVYgeKNMfagNV^SMQZ?RMSq,[KNVWO?KVWR§6]ZUUVWOrfN~W]Ze~xMs];RUxVg UKaMOQ];RUMP*]OQPSRUTXP;~YP;c;VWOQ];~-OQP;gaRe];gw :­ ~Y~P;§eKNMRUMMwfa]UVYP;gaRXTfaR`6M1RUfNNa~MTMgwUMh©`dxeKNM1VYgNVUVW]~OQP;gah/VUVYP;g6RP UKNM-MQZUUfNZe`a]UVYP;gaRq*[KNVYO?KË]ZeMh/MQUMZUTXVYgNMh¢`wx]TP/h/M~LP=eKNMMs]Ze~x¥fNgNVY^;MZeRUM :ü MQZeM[nM

h/VWReOrfaReRnUKNM^\];ZUVYP;faRn-PSReRVY`NVY~VUVYMRnIPSZVgaVeVY];~4OrPSgah/VUVYP;gaR :

Ya:9S: ½s¾bO-ÄZOÈSÉ Æ*»NÈsȾ#O6Ä-É P;TXMTXPdhNMQ~WRnIP;ZLUKaMP;ZeVcSVg P*ReZUfaOrUfNZeMVg6Or~Yfah/MRUP;fNZ?OrMsR[KNVWO?KRUVY~Y~,Mr/VYRLUP/hN]_xSqw[KaPSRUM§`-MQKa]_^dVYP;fNZnTf6R=]~WRUP`-MIP;~Y~P\[nMh];~PSgNcXRVWh/M§UKNMPeKNMQZ-MQZUUfNZe`a]UVYP;gaR : faO?KRUP;fNZ?OrMsR ];ZUMXbdgNP\[g];R 8 õrölñCB\ôò 3 ø/÷Uõ?ôò=<¤³UKNMZTP/h/M~YRq*RfaO?K];RVYg/®6]UVYP;gq8[KNMQZeMP;ga~xUKNMVYgNVUVW]~OQP;gahNVeVPSgaR]ZeMO?Ka];gNc;MshMs]Ze~xVgeKNMfNgNVY^;MZeRUM;q8]ZeMbdgNP\[g ];R¯ó 8 ò?òñCB_ôò 3 ø/÷Uõ?ôrò=<JLKNM`-MRFbdgNP\[gMQ/];TXN~MsR,PN]SOUVY^;MRUP;faZeOQMR,]ZeMOrPwRTXVWOh/MQIMO4UKNMP;ZeVMsRQq_Rf6O?K]SR4OQPSRë

TXVWOReZUVYgNcSR];gahOrPwRTXVWO§UMrdefNZUMi¡M : c :L KNMQ~Y~W]Z?hVWVY~YMQgNbdVYg6p : JLKNMh/MQIMO§MQ^;PS~fNUVYP;g¤VYRgNPSg/ëì~VYgNM];Z]gahP/OQOrfaZUZeVgac OQP;gaRe];gSe~x];RFUKNM=hNMrIMOreR]eMQTX/UPfagw[VYgahUKNMTRM~^SMR];ReKNMQxMgSeMQZnUKaM KaP;ZeVzP;g : JxdNVWOQ];~~Yx;qweKNMh/MrIMsOeRLVYgwUMZe]SO¯PSgN~Yxc;Z?]_^wVe]UVYP;ga];~~YxUKNZeP;fNcSKeKNMQVYZVYg/®afNMgaOrM=PSgUKNM´VYgaRUMVgMsSf6]\UVYP;g6R : onMOQ];faRUMnP;CUKNMVZgaP;g/ëì~VYgNMs]ZMQ^SP;~Yf/UVYP;g4q\UKNMNZeMhNVYOrUVYP;gaR§IP;Zh/MQIMOreR]ZeMXOrP;g6RVWh/MQZ?]`a~x¤Ka];Zeh/MZ UPh/MreMQZeTXVgNM : ­ NZeMRUMQgwqUKNPSfNc;KqeKNMQx];N-M]ZLePX`6M];g fagN~VYb;M~xOQ]g6h/VYha]\UM IPSZLUKNMNZeVYTX];ZUxRUP;faZeOQM P;8];gNVYRUPeZUPSNVYMRo<

àj âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éYa: : »NÈsºF»POÈSɧ»;»Nº8È»PÉ k¯PSRUTXVYOLh/MQIMO?R];ZUM]~WRUP] ^SMQZexc;PdP/hMQN]TXN~YMP,]OQ]f6RU];~TXMO?K6]gNVWRTIP;ZRUZefaOefNZUM IPSZUT]\eVPSgq`-MO]faRUM UKNMxÍh/PgNP;ZUMswfNVZeMOrPSZUZeMQ~W]\eVPSgaRPSgReOQ];~MsR~Y];ZUcSMQZUKa];gUKNMKNP;ZeVYzQP;g :=Ê g/®6]UVYP;g1ga]UfNZ?]~Y~YxNZeP/h/faOQMROQP;ZeZUM~Y]UVYP;gaRLPSgRUfN6MZëKNPSZUVYzQPSgRUO]~YMRq;[KNVWO?KVYgXUKNM R?]gahN];ZehX`aVc`6]gNcOQP;gwUMQd[¯PSfN~Wh]N-M];ZUP`6M ];OQ];faRe]~ :

Ya: a: -8¾¡»*»/½¾MNÉ JLKNM¥RVYTXN~MsRbwVYgahNRPVYgNVUVW]~OrP;g6h/VeVPSgaR];ZUM[KNMQZeMUKaMgdfNT`-MQZh/MgaRUVeVMsRP];~~§UKaMOrPSgaRUVUfNMgS a]ZUUVWOr~YMR®afaOrUfa]UMVWh/MgSeVYO]~Y~x : JLKNMsRM]ZeM1bdgNP\[g]SR 8N6 ñ 8 ð 8 ölñ¡õ®af6OUf6]\UVYP;g6R§];gah1]ZeMgNMsOrMsRURe]ZeV~Yx¤OrfNZe^\]\UfaZUM®afaOefa]\eVPSgaR : ² P;Z§]Sh/VW]`a]UVWO®af6OUf6]\UVYP;g6RQq δnγ

nγ= δnc

nc= δnb

nb

]g6hIPSZ]~Y~eKNMa]ZUUVWOr~YMRU6MsOrVYMR : JLKNVWRVYTa~VYMReKa]\δc = δb = 3

4δγ ,RVYgaOrMUKNMNKNPeP;gaR§]ZeMZeMQ~W]\UVY^dVYRUVWO];gah

nγ ∝ ρ3/4γ .

Ê g]ShNh/VUVYP;gqCRVYgaOQMeKNM6MZefNZe`a]\eVPSgaRL[¯MZUMNZeP/h/faOQMhVg UKaMM];ZU~YxfNgNVY^;MZeRUM;q/VVWR];ReRfNTXMsheKa]\=PSgN~YxUKNMcSZUP\[VYgNcTXP/h/MP;4UKaM6MZefNZU`6]\UVYP;g6RLVYRLVYTX6PSZ?]gwUP/hN]_xSqNRPUKa]

δ ∝ τ2.Ê 4VWR,[nP;ZUUKh/MReOrZeVY`NVgac=]L`NVFTPSZUM];`6PSf/,UKaMR?]\eVYRUVWOQR,P;SeKNM®af6OUf6]\UVYP;g6R : JLKNZUPSfNc;KNëPSf/=[¯MK6]_^;M`6MMQg1OQP;gaRUVYhNMQZeVgNceKNMMQ^SP;~Yf/UVYP;g¤P8UKaM ² PSfNZUVYMQZLeZe];gaRIP;ZeTMsh];TXN~VUf6h/MRh/MQîagNMsh`dx

δ(x) =V

(2π)3

∫d3k eik·xδk.

JLKNM ² P;fNZeVMZTP/h/MsR]ZeMO?K6]Z?];OeMQZeVYRUMh1`wx]g];TXN~VUf6h/M]gah¥] NKa]SRM :Ê g¥UKNMX~YVgaM]ZZeMQcSVTXM UKNM¥]TXN~YVefah/MsRMQ^SP;~Y^;MVYgah/MQ-MQg6h/MQgwU~YxËPM]SO?KÍP;UKNMZqL[KNVY~M1UKNM1aKa];RUMRZeMrëT]VYgNZeMRUMQZe^;Msh :¯² P;Z ï];faReRVW]g îaM~YhNRqNeKNMNK6];RUMR];ZUMZ?]g6h/P;T : JLKNM~YVgNMs]Z=M^;PS~f/eVPSgNZeMRUMQZe^;MsR¯];gdxgNP;gNë ï];faRURUVW]gVYg/IPSZUT]\eVPSg[KaVYO?K VYR=VgUKNMaKa];RUMOrP;ZeZeMQ~W]\UVYP;g6R :² PSZXï];faReRVW]gZ?]gahNP;Tíî6MQ~WhNRQq];~~¯UKaMVYg/IP;ZeT]\UVYP;g¢VYROrP;gw?]VYgNMhVYgeKNM[nP1-P;VYgS

OQP;ZeZUM~Y]UVYP;gIfNgaOrUVYP;gq;P;Z8Vg ² PSfNZUVYMQZ*RUa];OQM¯VYgeKNML6P\[nMQZ*R-MOrUZefNT¤qP (k).

JLKNMLMQgaRUMQT`N~M]_^SMQZ?]c;MP8[¯P ² PSfNZeVMZnTXP/h/MsRLVYRcSV^SMQg `dx

〈δkδk′〉 = V

(2π)3P (k) δD(k− k′),

[KNMZUMeKNMµ§VZ?];OhNMQ~e]IfNgaOrUVYP;gVWReKNMOrPSgaRUMwfNMQg6OrMPKNPSTPSc;MgNMQVx :Ê gaZe]SOUVWOrMSqN[nMfaRUfa];~~Yx];ReRfNTXMUKa]eKNM MQgaRUMQT`N~M¤]_^;MZe];c;MsR]ZeMMsSfaV^\]~YMQgwUP]_^;MZe];c;MRP\^;MQZ~Y];ZUcSMMgNP;fNcSK^SP;~YfNTXMRi¡V : M : UKNMRa]UVW]~MZUcSP/h/VYO§eKNMQPSZUMT : pJLKNMZeM];~4RUa];OQM§[nP-P;VYgSIfagaOëeVPSg O]g `6MRUKNP\[gUP`-MUKNM ² P;fNZeVMZnUZ?]gaRIP;ZeTÂP8UKNM-P\[nMQZR-MOrUZefNT¤q

〈δ(x)δ(x′)〉 = V

(2π)3

∫d3kP (k)eik·x.

JLKNM^\];ZUVW]gaOQM;qP;ZCUKaMOrPSZUZeMQ~W]\eVPSg=IfNgaOrUVYP;g]0,VWR,c;VY^;Mg§`dx ⟨

δ2⟩

= V(2π)3 4π

∫k2dkP (k).

Ya: a:9S:] õ 85 ôñ HQ6 ô óaô H^6 ô H ö 8HQ6 ñ H B 8 ÷ñ 8H ö§òØóaô?õröl÷ 8 <=JLKNMTXPSR[VWh/MQ~YxOQP;gaRUVYhNMQZeMhOQ~Y]SRURP;,RU-MOeZe]];ZUMbdgNP\[g]SRLòQõ 85 ôñ HQ6 ôìóaô H^6 ô H ölq/]gah];ZUM=O?Ka];Ze]SOeMQZeVzMh`dxX]RUVTXN~YM=-P\[¯MZ~W]_[

P (k) ∝ kn,[KNMQZeM

nVWRLbdgNP\[g ]SRnUKNMRU-MOeZe];~,Vg6h/Mr :

­ RU6MsOrVW]~RUfN`-OQ~Y]SRURnP;FeKNMRUMVYRneKNMXòQõ 85 ôñ H B 8 ÷ñ 8H öLRU-MOeZUfaT qa[KNVYO?K VWR]~WRPbdgNP\[g]SR8UKNM=y*MMQ`N~YMRë ü ]ZeZUVWRUP;g/ë-MQ~Wh/P\^dVWO?KR-MOrUZefNT :8ü MQZeM;q;UKNMOQP;gwUZeVY`Nf/UVYP;gePUKNM=^\];ZUVW]gaOQMVWRUKNMRU];TXM1IP;Z];~~TXP/h/MR MQgweMQZeVgNc¢UKNM¥KNPSZUVYzQPSg : ³§f/eRUVYhNMeKNMKaP;ZeVzP;gq

δ ∝ τ2,RUPP (k) ∝ τ4kn.

² ZeP;TeKNM¤Mr/NZeMReRVYP;gjIP;ZUKNM¤^\]ZeVW]gaOQM;q*UKaM¤OQP;gwUZeVY`Nf/UVYP;gÍ6MZX~PScVYgwUMQZe^\]~,VWR

k3P (k) ∝ τ4k3+n [KNVYO?KRUKNP;fa~Yh `-MVgd^\]ZeVY];gw]=KNP;ZeVYzQP;g¤OrZePSReRUVgNc kτ ∼ 1.JLKNVWRLVYTa~VYMRLeKa]\]XReOQ];~MVYgw^\];ZUVW]gwR-MOrUZefNTKa]SRn = 1.Ê g/®6]UVYP;gfaRf6]~Y~xXNZeMh/VWO?R¯]gNM];ZU~YxRUO]~YMVYgd^_];ZUVW]gw¯RU-MOeZUfaT qd[VUK

n ' 0.92− 1.0.k¯fNZeZeMQgwP;`aRUMQZe^\]\eVPSga]~aOQP;gaRUZ?]VYgweRa~Y]SOrMeKNM§RU6MsOeZe];~6VYgah/MQVYgUKaMZ?]gNcSMn = 1± 0.1.

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê àsYa: _ : ÈcSº4ÇoON»/½sº4Ç\ĸÅU-FÄdÈSÉ JLKNM ];~eMQZega]\eV^SMUP];hNVY];`a]\eVYOX®6faOefa]\eVPSgaRX]ZeMbdgNP\[g]SRñWò 3 õQø/÷B 8 ölø/÷Uô®afaOefa]\eVPSgaRq8PSZ-MQZeKa]6RTXP;ZeM]SOQOrfaZe]UMQ~Yx]SRô H öl÷ 3 ó^®6faOefa]\eVPSgaR :JLKNMsRMKa]_^SM§VYgNVUVW]~4OQP;gahNVeVPSgaRL[KNMQZeM UKNMZ?]\eVPwRLPF^\];ZUVYP;faRna];ZeVYOQ~MRU6MsOrVYMRL^\]ZexIZeP;TN~W];OQMUPa~Y]SOrM :ü P\[¯M^;MQZsq/eKNMQx T]_x`-MVYgNVeVY];~~Yx¤OrP;TX-MQgaRe]\eMhRUfaO?K¤UK6]\=eKNMh/MgaRUVx®af6OUf6]\UVYP;g6R=OQ];gaOrM~ : o¯fN=UKNMTXP/h/MsRUK6]\`-MQcSVg];R=VYRUP/OrfNZe^\]\UfaZUMT]_xgNP;=ZUMT]VYg¤RUPaqa];ZeVYOQfN~W]Ze~x]3UMQZT]UMZ=h/P;TXVYga]\eVPSg :JLKNMZUM];ZUM¯];R,T]gdxh/V/.-MZUMgS8bdVgahaRP/VWRP/OQfNZU^\]UfNZeM*®afaOrUfa]UVYP;gaR8];RCeKNMQZeM¯];ZUM6]VYZeR4P

a];ZeVYOQ~MLRU6MsOrVYMR :8² P;Z*MrN]TXN~YM;q;P3UMgeKNMgdfNT`-MQZP;6NKaPUPSgaR86MZhN]ZebT]\UUMQZ6]ZUUVWOr~YMVWR];ReRUfNTXMhUP^_];ZUxSq;`NfNP;gNM§OrP;fa~Yh];~YRUPOrP;g6RVWh/MQZ^_];ZUVW]\eVPSgaRVYgXUKNMgwfaT`-MQZP;,NKNPeP;gaR-MQZ=`6]Zex;P;g4qwPSZ`a]Zex;PSgaRn6MZhN];ZUbT]UMZ=a];ZeVYOQ~M : JLKNMQZeM]ZeMTX];gdx[n]_x/RLP;8MQ/OQVeVgacVWRP/OQfNZU^\]UfNZeMTP/h/MsRQqaIZUPSTTfN~UVîaMQ~Wh¤VgN®6]\eVPSga]ZexTXP/h/MQ~WReP]\/VYP;gaRLaZUP/h/faOQMh¤VYg¤UKNMMs]Ze~xfagNV^SMQZ?RM : JLKNMsRMTXP/h/MsRT]_x]~WRUPOQPëìMr/VWR[VUK];hNVY];`a]\eVYOnTXP/h/MR];gahUKNMZUMT]_x`-MOrPSZUZeMQ~W]\eVPSgaR`-Mr[nMQMgeKNMQT :JLKNMM^;P;~Yf/eVPSgP;FeKNMTXP/h/MRLVWRTXPSR=M]SRVY~xIP;~Y~P\[nMh`dxh/Mrî6gNVgac]g]ga];~PScePXUKNM

oL]Z?h/MMQg^\];ZUVW]`N~YM IP;ZMs];O?K6]ZUUVWOr~YMRU-MOrVYMR : JLKa]LVWRqN[¯Mh/MQîagNM

ζN =1

3

(1

2h− +

δρN

ρN + pN

),

[KNVWO?KVYR*eKNM=OrfaZU^\]\efNZeM®af6OUf6]\UVYP;gXPSgRUfNZU¡];OrMsR*PCOrP;g6R?]gwh/MQgaRUVxVg] c;VY^;Mg6]ZUUVWOr~YMRU6MsOrVYMR : JLKNMsRMO]g`6MRUKNP\[g¥UP`-MOrPSgaRUMQZe^;MhP;fNeRUVYh/MeKNMKNPSZUVYzQPSgIPSZTXPSRbwVYgahNRP;¯a];ZeVYOQ~MsR`wxReZUMsRURëlMgNMQZec;xOQP;gaRUMQZe^\]\UVYP;g :Ê RUP/OrfNZe^\]\UfaZUMX-MQZUUfNZe`a]\eVPSgaR]ZeMXfaRf6]~Y~xwfa];gSeVî6Mh`wx

Sij =δni

ni− δnj

nj= 3 (ζi − ζj) ,

[KNVWO?KVWR]~WRUP1OrP;g6RMZU^SMhP;f/?RVWh/MeKNMKNPSZUVYzQPSg :ü P\[nMQ^;MZqUKNMIfN~~LoL]Z?h/MQMg^_];ZUVW]`a~MSq[KNVWO?K VYR=][nMQVYc;KwUMshRUfNT¶P;FeKNMVYgah/VY^wVWh/fa];~CUMZUTRq

ζB =∑

N

ζN (ρN + pN )/∑

N

(ρN + pN )

VWRgNP;gNMOQMReRU];ZUVY~x1OQP;gaRUMQZe^;MshPSf/eRUVWh/MXUKNMKNP;ZeVzP;g : iØ·=P;UMeKa]\IPSZ];h/VW]`6]\UVWO®afaOrUf/ë]UVYP;gaRq

ζB = ζc = ζγ . . .];gah]~Y~¯];ZUMOQP;gaRUMQZe^;MshPSf/eRUVWh/MXUKNMKNP;ZeVYzQP;g : p MQM|]gahaRQq

m1];~VYbCq xwUKVVWhNh/~YMiØ S; pIPSZLIfNZeKNMQZ=hNVYReOrfaReRUVPSg :tN:þFD4H3>Ë@þuaBQþNA&NþaG6@,A-Eg

ta:9S: Ä9¯Ä";½R ºFÆÄwÇ_Ã6Ç_¾MLN-¼¦Ä"OQPغ4½s¾#-¼É Mr faROrPSgaRVWh/MZ=UKaMa];ZeVYOQfN~Y];Z§O];RUMP;RUfN-MQZeKNP;ZeVzP;gËTXPdhNMRVYg©UKaM1T]\eMQZ h/P;TXVYga]\eMhËZUMc;VYTXM :¦¬ MOQ];~~UKa]UKaMQZeMVYR]cSZUP\[VYgNc¤TXPdhNMRPS~fNUVYP;g

δm = Akn/2τ2.JLKaM®afNVWh]g6h´Vg6ReMQVYg « RMwfa]UVYP;gaRVTXN~Yx;q

−2δm = h = hS[KNMQgeKNM^;MQ~YP/OrVUVYMR ]ZeMgNMc;~YVcSV`a~MSq-RUPeKa]\

hS = −2Akn/2τ2 + C.·Mrdq§y8PSVYReRUP;g « RMswfa]\eVPSgq ∇2Φ = 4πGδρ,VTXN~YVMsRUKNM¥c;Z?]_^dV?]\UVYP;g6]~-PeMQgwUVW]~VWR

OQP;gaRe];gSsÎΦ = − 3

2

(2τ

)2δm/k

2 = −6Akn/2−2.i¡·PeMXUKNVWRVWR UKNMRe]TXM];RPSgNMîagahNR

`dxUZ?]g6RIPSZUTXVYgNc1eP¥·MQ[LUPSgNVW]gËcS]fac;M : p³§gËRUfN-MQZeKNP;ZeVYzQP;gÍRUO]~YMRqUKNM ]SO?KaRëì|¥P;~IMMQdaZUMsRURUVPSgVYR h/PSTVYga]UMh¸`wxËeKNM

hS/2k2 UMZUT¤q RPeKNMUMQTX-MQZ?]\efNZUM¥]gNVWRUPUZeP;dxËVYRM=.-MsOUVY^;M~x

δT

T' hS/2k

2 = −2Akn/2−2 =1

3Φ.

JLKNVWRMQdaZUMsRURUVPSgVYRMs];RUVMZUPfNgah/MZeRe];gah¥VgeKNM·MQ[LUPSgNVW]gcS];fNc;M : ·§]VY^;MQ~Yx;q8P;gaMMQd-MOreR§UKaMZeMhaRKNV3UVYgNc PaKNPeP;gaR§UP `-McSV^SMQg1`dx¤eKNMh/MQ/eKP;UKaM-PeMQgwUVW]~*[nMQ~Y~ØqδT/T = Φ.

ü P\[nMQ^SMQZsq-VYg¥UKNMOQPSRUTPS~PSc;VWOQ];~8RUMrUUVYgNc¤P;gNMXTf6R]~WRUP?]b;MXVYgwUP¤];OQOQP;fNgweKNMMQd6]gaRUVPSg :² P;Z ];h/VW]`6]\UVWO®6faOefa]\eVPSgaRQqCZeMOQP;T`aVga]UVYP;gP/OQOQfNZ?R=~W]\UMZ§VYg-PeMQgwUVW]~

àx âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/é[nMQ~Y~YR`6MsOQ]f6RMUKaMNKNP;UP;g¥eMQTX6MZe]UfNZeMVWRKNVYc;KNMZUKNMZUM : JLKdfaRqeKNMRUMNKaPUPSgaRKa]_^SMKa]Sh ~MsRURLeVTXMePMr/a];gah : JLKNMh/MQ~W]_xVYg¤UVYTMVWR=cSV^SMQg `dxUKaMh/MQNUKP8UKaM6P;UMgSeVY];~Øqδt/t = Φ.

VYgaOQMeKNMRUO]~YM¡];OrUPSZ§cSZUP\[=R§];Rt

2

3

VgUKNMXT]\UUMQZh/PSTVYga]UMhZUMc;VYTXM;q-UKNM®af6OUf6]\UVYP;gVg¥RUO]~YM¡]SOUPSZ§VWR c;VY^;Mg`dx

δa/a = 2Φ/3.JLKNVWRRKNV3OQ]g6OrMQ~WR a]ZU§P;UKaM

ga];V^SM OQP;gwUZeVY`Nf/UVYP;g4q/~Ms]_^wVYgNcδT/T = Φ/3.Ê g¥UMZUTR P¯UKNMT]\UUMZ®afaOefa]\eVPSgq,eKNMUMQTX-MQZ?]\efNZUM];gNVYRUPeZUPSdx¤O]g¥`6MX[ZeVUUMg

]SRδT (n)

T=

∑

k

2

k2τ2δke

ik·nτ .

RVYgNceKNMMQd6]gaRUVPSgPN~W]gaMR[L]_^;MsRVYgRaKNMQZeVYO]~6Ka]ZeTXP;gNVWOnIfNgaOrUVYP;gaRqwUKNM§TfN~UVY6PS~MTXP;TXMgS?ROQ];g`-MRKaP\[geP`-M;q

a`m =

∫δT (n)

TY ∗

`m(n)dΩ = 8π∑

k

1

k2τ2δki

`j`(kτ)Y∗

`m(k).

² P;Z§]RUO]~YMVYgw^\];ZUVW]gw§RU6MsOeZUfNT¤qC[nMOQ];g1[ZeVeMUKaM-P\[¯MZ§RU6MsOUZefNT ];RPk ≡ |δk|2 =

Bkτ4/L3,[KNMQZeMLKa]_^;M¯¡];OrUP;ZeMhP;fN8eKNM=OrPSTXP\^wVYgNc§UVYTXM=]gah[L]_^;MQgdfNT`6MZ8hNMQ-MQgah/MgaOrM

];gaheKNML3 ¡]SOUPSZ=];ReRfaZUMsReKa]\ B VWR=h/VYTXMQgaRUVPSgN~YMReR : JLKNMknm1o¦]gNVWRP;UZeP;dxR-MOrUZefNTeKNMQg¤VWR

C` = |a`m|2 =64π2

L3B

∑

k

k−3j2` (kτ)Y`m(k)Y ∗

`m(k).

J8]bdVYgNc¥UKNM1~YVYTV]SRL ⇒ ∞, UKaMRfaTûP\^;MZTXPdhNMR`-MOQP;TXMR];gVgweMQc;Z?]~lq ∑

k⇒

(L/2π)3∫d3k.

JLKNM];gNc;fa~Y];Z§a];ZP;UKaVYRVYgwUMc;Z?]~*VYR eZUVY^dVY];~*`wx1UKNMPSZeKNP;cSP;ga];~Vx¤PeKNMRaKNMQZeVYO]~CKa];ZUTXP;gaVYORQqd[KNVWO?K¤~Ms]_^;MRq

C` =8B

π

∫dk

kj2` (kτ) =

4B

π`(`+ 1).

JLKNVWRVWRUKNMZeM]SRPSgknm1oÌR-MOrUZ?]¤]ZeMf6Rfa];~~Yx1N~YPUUMsh¥]SR`(` + 1)C`

^SMQZ?Rf6R`.m1P;ZeM

cSMQgNMZe];~~Yx;qwIP;Z=]-P\[¯MZL~Y]_[h/MQgaRUVxR-MOrUZefNTPk ∝ kn,

P;gNMO]g¤RKNP\[

C` ∝Γ

(`+ n−1

2

)

Γ(`+ n−5

2

) .ta: : ! »/Ç_À-¼ReÆ8Ã6½-¼U6Èc;¾MP#PØ»/½¾M6¼8È;É ³§gX~W]Zec;M]gac;fN~W]ZReOQ];~MsRQqcSZUMs]\eMQZ8UKa];g]h/MQcSZUMMPSZ*[¯P6qUKaM®afaOrUfa]UVYP;gaR~Ms];h/VYgNcUPeMQTX-MQZ?]\UfaZUM];gNVWRP;UZeP;NVYMR*]ZeMP;fNeRUVYh/M=UKNM=KaP;ZeVzP;g]UKNM=UVYTM§P,ZeMOrPST`NVYga]UVYP;g : ­ Rnh/MReOrZeV`-MhX];`6P\^SM;qeKNMQVYZZeMRUfN~UVYgNcRU6MsOeZUfNTVWR¡]VYZe~xIMs]\UfaZUM~MsRUR§V¯UKNMXfNg6h/MQZe~xdVYgNcVYgNVUVW]~h/MgaRUVx1RU6MsOeZUfNTíO]g`6Mh/MsRUOQZUVY`6Msh]SR ]-P\[¯MZ~W]_[ : ³§g¢RUT]~Y~MZReOQ];~MsRQq*KNP\[nMQ^SMQZsq4eKNM TP/h/MsRK6]_^;MKa]ShUVYTXMePM^;P;~Y^;M[VeKNVg¢UKNMKNPSZUVYzQPSgNZeVPSZUP~Y]SRLRUO]\eMQZeVgac : ³§gUKNMsRM RUO]~YMRq;eKNM§];gNVWRP;UZeP;NVYMR¯]ZeMh/P;TXVYga]\eMh`dxeKNMXVgwUZeVYgaRVWO]gNVWRP;UZeP;dx¤]gahUP¤] ~MsRURUMQZ MrdUMgw`dx¤eKNMµPSNN~YMQZM>.CMOr : ­ R]ZUMsRfN~qeKNMPSReOrVY~~W]\eVPSgaRnPFUKNMNKaPUPSg/ëì`a]Zex;PSg®afNVWhe]bSMP;g]b;MQxZUPS~~ :yZeVPSZLUPXZeMOrPST`NVYga]UVYP;gqdeKNMNKNP;UP;g6R]gah`a]Zex;PSgaRL]ZeM RUZeP;gac;~YxOrPSfNN~YMh ]gahUKdfaR

]SO¯]SR]RUVYgNc;~YM®afNVWh : JLKNMxeKNMQg RK6]ZeMLUKNMRU];TXM=^;M~P/OrVxvb = vγ

]g6hXUKNMVZLh/MgaRVUVYMR];ZUMXZeMQ~W]\UMsh¥`wxUKaM];h/VW]`6]\UVWOXOrPSgah/VUVYP;gUK6]\eKNMgdfNT`-MQZPNKNP;UP;g6R6MZ`a];ZUxSP;g¥VYRîN/Msh,q

δγ = 4δb/3: JLKNVWR®afaVYh]SOeR];RPSgNM[VeKNZeMReRfNZeM 1

3ργ

qa`Nf/§[KNVYO?KK6];R=h/MgaRUVxργ + ρb,

3

4δγ = δb = −1

2h− ikvγ

(ρb +4

3ργ)vγ + ρb

a

avγ = −

ikργ

3δγ

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê à~JLKNMRU6MMh P*RPSfNgah VYgeKNVWRn®afNVWhVWR

c2s =δP

δρ=

13 ργ

34 ρb + ργ

=1

3(1 +R),

[KNMZUMR ≡ 3ρb/4ργ.JLKNM^SMQ~YPdOQVxMswfa]\eVPSg O]g¤`6MZeMQ[ZeVUMgf6RVYgNcXVg eMQZeTXRLP;4eKNMRPSfNgah¤RU6MMh ];R

vγ +a

a(1− 3c2s)vγ +

3

4c2sikδγ = 0.

k¯PST`NVYgNVYgNcUKaVYR[VeKUKNMh/MQg6RVx¤MQ^;PS~fNUVYP;g¤Mwfa]UVYP;gq-[¯Mîagah¤eKa]\§UKNMh/MQgaRUVx ]SOeR]SR]IPSZeOQMh,qNha]TX6Msh Ka]ZeTXP;gNVWOPwRUOQV~Y~W]\UPSZÎ

δγ + c2sk2δγ = −2

3h.

ü MQZeMXIP;ZRUVTXN~YVWOrVx[nMKa]_^;Mh/ZeP;N-MheKNMhN];TaVgNc¤eMQZeTRQq8[KNVWO?K¥VWR];NNZeP;NZeVW]\UMVYgeKNMZ?];h/VW]\eVPSg h/PSTVYga]UMh¸ZUMc;VYTMÍi

c2s ' 1/3.p ² P;~Y~P\[VYgNc¢h/MsOrPSfNN~YVgNc6qUKaMNKaPUPSgaR

IZeMQMsReZUMs]T ]gah¸UKNM`a];ZUxSP;gaR¡]~Y~VgweP¢eKNM6P;UMQgweVY];~ [nMQ~Y~WRcSMQgNMZe]UMsh`dxUKNMhN];ZUbT]\UUMZ :

ta: :9S:k nNô| 3 óSó 5 ôQ÷4óNô 8 :\ò=< ³ ReOrVY~Y~Y]UVYP;gaR-IPSZF]Lc;VY^;Mg[n]_^SMQgdfNT`-MQZC`-MQcSVg[KNMgkτcs ' 1];gah¤UKaMQVYZRUfN`aRUMwfNMgw§MQ^SP;~Yf/UVYP;g`6MKa]_^;MsR=ZeP;fNcSKN~x]SR

δ ∼ cos(kτcs + ψ): JLKNMNKa]SRM

PN.CRUMrh/M6MgahNRPSgeKNMVgaVeVY];~OrPSgah/VUVYP;gaR];gah¥P;g¥KNP\[eKNM6MZefNZe`a]\eVPSgaR]ZeMIPSZeOQMh,q`NfN=VYRn¡];VZe~xVgahNMQ-MQgah/MgwPFUKNM[L]_^;MgwfaT`-MQZ :JLKNMknm1o°]gaVYRUPeZUPSNVMsR=NZeP;`-MeKNMUKNMh/MgaRVx ®afaOrUfa]UVYP;gaR]\]îadMsheVTXM;q

τrec.´]SO?KTXP/h/M]LUKNVWReVTXM[VY~~F`6M]]h/Vm.CMQZeMQgw=-P;VYgw=Vg¤VeRPwRUOQV~Y~Y]UVYP;gqNhNMQ-MQgah/VYgNcPSgKNP\[~YP;gNceKa]\§TP/h/MKa];R`6MMQgVYgaRVWh/MeKNMKaP;ZeVzP;g : mP/h/MR4 faR§MQgweMQZeVgNcXeKNMKaP;ZeVzP;g];ZUM`6Mc;VYgNgNVYgNc¤UKaMQVYZîaZeROrPS~~W]aRUMh/fNMXeP cSZe]_^dVe]UVYP;ga];~8]UZ?];OrUVYP;g : ~YVYc;KwU~YxRUTX];~~YMQZTXP/h/MRn[VY~~4Ka]_^;M Ka];hUVYTXMUPOrPS~~W]aRUM§UP]-P;VYgw[KNMQZeM UKNMVZNZeMReRUfNZUM§[n]SRLRf/¨OQVMgSeP¤IP;Z?OrM]¤`6PSfNgaOQM;q*]gah¥[VY~~¯`6MMQ/a]gahNVgNc¤`a]SO?b];cS]VYg : m1faO?KRT]~Y~YMQZ[L]_^;MQ~YMQgacUK6R[VY~~,K6]_^;MKa];hUVYTXMIP;ZTfN~eVa~MOQP;TXNZeMReRVYP;gaRL];gahZ?]ZeMr¡]SOeVPSgaR :JLKNM-M];bdRLVYg UKNMknmo RU6MsOUZefNTP;g¤RUO]~YMRRUTX];~~YMQZneKa]g]h/Mc;ZeMQM]ZeM]SRURUP/OrVW]\eMh

[VUK¥UKaPSRUM[L]_^;MgdfNT`-MQZ?R=eKa]\K6]_^;ME faRZeM]SO?KNMh1UKNM-M];bP;UKaMQVYZOrPSTaZUMsRURUVPSg1P;ZZ?]ZeMr¡]SOUVYP;g]\

τrec.JLKNMUMZUTµ§P;NN~YMQZ6Ms]b/RVWR]¤`NVP;nTXVWRgNPSTXMQZsq8RUVg6OrMeKNMQx¥]ZeM

OQZUMs]\UMsh]\TXVYgNVYTX]]g6hXTX]/VT]P;CUKNMh/MQg6RVx®6faOefa]\eVPSgaRQqd[KNMZUM=UKNM§^;MQ~YP/OrVUVYMR¯]ZeM]SOUf6]~Y~xzQMQZeP : JLKNMeZUfNM§µPSNN~YMQZM=.-MsOZeMRUfN~UVYgNcIZeP;TÌUKNM=^SMQ~YP/OrVUVYMRh/PdMsROrPSgwUZeV`NfNUM;q`NfN§V VWR§RUT]~Y~MZ=eKa]geKNMVYgwUZeVg6RVWONVYMOQM];gahVWR

90PSf/§P;NKa]SRM :Ê §K6];RUKNMM>.CMO

P;Fîa~Y~YVgNcfaeKNMRUa];OQM`6MQ[¯MMQg UKNMµ§P;aN~MZ=-M]b/RqNT]bdVYgNcUKNMh/VYaR§RKa];~~YP\[nMQZ¯UKa];geKNMQx[nP;fa~Yh`6MP;UKNMZU[VWRUM :·=P;UMUKa]eKNMQZeMVWR]1IfNZUUKNMZ`N~YfNZeZUVYgNcPUKaM-M];bRUZefaOrUfNZeMZeMRUfN~UVYgNcIZeP;T UKNM

NZeP MsOeVPSgPFUKNMTXP/h/MR¯IZeP;TUKaZUMM h/VYTXMQgaRUVYP;gaRh/P\[gUP[¯P : ´];O?K`TXPdhNM ZeMOQMQVY^;MsR

OQP;gwUZeV`af/UVYP;gaR=IZUPST ]Z?]gNcSMP[L]_^;MgdfNT`-MQZ?RQq-h/MQ-MQg6h/VgacPSgUKNMX]gac;~YM]\[KNVWO?KUKNMTXP/h/MVgweMQZ?RMsOeRneKNMRUNKNMQZeMPFUKNMRUbdx :

ta: a: *T½Ã8Ä©ÈQÆÄ*;½QÇ\ºFÅ-8ÄSƯÄw¼-8È-¼Æ*»/Ç\»NÅjÄw½ÄwÇ\È;É JLKNMNKdx/RUVYO]~M^;P;~Yf/eVPSgPknm1o¦]gNVWRUPUZeP;aVMsRnhNMQ-MQgahNR§RMgaRUVeV^SMQ~YxP;g]XgdfNT`6MZ=POrPwRTXP;~YP;cSVYO]~Ca];Ze];TMQUMZeR :Ê PSgNM];ReRfNTXMsR]RUVYTa~MIPSZUTIP;ZLUKaMVgNVUVW]~46MZefNZU`6]\UVYP;g6RQqdeKNMQgTXM]SRfaZUMTMgweRnP8UKNMknm1o ]gNVWRP;UZeP;NVYMRLO]g `6MfaRUMhUPOQP;gaRUZ?]VYgeKNMOrPwRTXPS~PSc;VWOQ]~-TXP/h/MQ~ :

á; âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éta: a:9S: EH l 5 ô=q 6 ñWòö 8H õeô¯÷Uô 5/8 öØñ 3H <§JLKNMNKNP;UP;gNël`a];ZUxSP;g§PSReOrVY~~W]\eVPSgaRCNZeP\^wVWh/M]LfaRUMrIfN~SNKdx/RëVWOQ];~xS];ZehaReVYO?bIP;ZUKNMM];ZU~YxfagNV^SMQZ?RM : onxbdgNP\[Vgac1[KNMQgÍ~Y]SRXRUO]\UUMQZeVYgNc1P/OQOQfNZUZeMhq[nMOQ];g¥VYg/IMZUKNMUP?]~OQfNZe^_]UfNZeMXPLUKaMfNgNVY^;MZeRUMIZUPSTíUKaM]gNcS~MQëìhNVYRe];gaOrMZUM~Y]UVYP;g :JLKNMOQP;TXP\^dVgNc¥eVTXMP ZUMsOrPST`NVYga]\eVPSgq¯]SRURUfNTXVYgNcVKa];N-MQgaRRUf/¨OrVYMQgwe~xË]\3UMZXUKNMT]\UUMZëìZe]Sh/VW]\UVYP;gMwfa];~Vx;qVWRcSV^SMQgX`dx

τrec = 2H−10

√Ω−1

m (arec/a0).Ê g] ®6]\sq;T]\UUMQZ

h/PSTXVga]UMshfNgaV^SMQZ?RMSq\UKNM=OQP;TXP\^dVgac h/VWRe]g6OrMLUPeKNM~W];RReOQ]eMQZeVgNcRUfNZU¡];OrMLVWRc;VY^;Mg`dxrrec ' τ0 = 2H−1

0

√Ω−1

m ,RUPeKNM]gNcSfN~Y];Z8ReOQ];~MLP;aUKaMLRUP;fagahKNPSZUVYzQPSg]*~W];RRUO]\UUMQZeVYgNc

VWRθ = csτrec/rdec = cs

√arec/a0 ' 1.Ê LUKNMfNgNVY^;MQZ?RUMVWRPS6Mgq4eKNMOQP;TXP\^dVgNch/VWRe]g6OrMXUP¤eKNM~W];RRUO]\UUMQZeVYgNcRfNZU¡];OQMVYR

gNP~PSgNc;MZMsSf6]~FUPeKNMXOrP;TXP\^dVYgNcXUVYTXMRUVg6OrM~Y]SRRUO]\eMQZeVgacaq-`Nf/VYR§c;VY^;Mg1`dxrrec =

2H−10 Ω−1

m .JLKaMQg UKNMZUM~M^_];gw];gNc;fa~Y];ZLReOQ];~MVWR

θ = cs√

Ωm(arec/a0): i Ê FeKNMT]\UUMQZ

h/MgaRUVx¥VYRRfN¨OQVMgwU~Yx~YP\[eKa]\UKNMTX]eMQZUëlZ?];h/VW]\eVPSg¥UZ?]gaRUVeVPSgVYROrP;g6OrfNZeZUMgw[VeKZeMOQP;T`aVga]UVYP;gqseKNMQgUKaVYR[VY~Y~NRKNV3UKaMLOQP;TXVYgNc UVYTMP;6~W];RRUO]\eMQZeVgac RUP;TXMQ[K6]\ : |MKa]_^SM§gaMQc;~YMOrUMsheKNVYR=M>.CMO=KaMQZeM : p² Vg6]~Y~xSqQIPSZfNgNVY^;MZeRUMRC[VUK]OQPSRUTPS~PSc;VWOQ];~_OQP;gaRe];gwq

rrec ' τ0 = 2H−10 Ω

−1/2m F (Ωm),[KNMZUMUKaM Kdxd-MQZUëlcSMQP;TXMQUZeVYOIfNgaOeVPSg

F (Ωm)VYRX]R~YP\[~Yx¸iI~YP;cS];ZUVUKaTVWOQ];~~YxNp^\];ZUxdVYgNc

IfNg6OUVYP;gPUKaMT]\UUMQZh/MgaRVx;qCZ?]gNcSVgacXIZUPST 9(Ωm = 1)

UP N: Y(Ωm = 0.1).

JLKdfaReKNM¯];gNc;fN~W]Z8RUO]~YMP/eKNM¯µ§P;aN~MZF-M];b/R4VWR¡];VZe~Yx§VYgah/M6Mgah/MQgw*P/eKNMT]\UUMZ8h/MgaRUVx IP;Z®6]fNgNVY^;MZeRUMRq`Nf/¯V¯O]gX`-MVgaOQZUMs];RUMhXRUP;TXMQ[Ka]*IP;Z~W]Zec;ML^\]~YfNMRP-UKNMOQPSRUTXP;~YP;c;VWOQ];~OQP;gaRe];gS :ta: a: :]\8 ölö ôr÷ 6 ô H òñ3öMN< ­ gNP;UKNMZVYT-P;ZUe];gw,¡];OeP;Z,VYgc;MgNMQZ?]\eVgNcneKNM¯knm1o¥]gNVWRP;UZeP;NVYMRVWReKNM¥NZUMsRMgwT]\UUMQZh/MgaRUVx : 4P\[¯MZUVYgNceKNMT]UMZ hNMQgaRUVxT];b;MRUKNMUVYTXMPZ?];h/VW]\eVPSg/ëìTX]eMQZMswfa]~YVx~W]\eMQZsq

zeq = 2.3× 104Ωmh2 : JLKaVYRnM>.CMOreRL`-PeKeKNMIPSZeOQVgaceMQZeTIPSZ=UKaMNKNP;UPSg/ël`6]Zex;P;g¤PwRUOQV~Y~Y]UVYP;gaR]gah eKNMOQP;gwUZeV`af/UVYP;gaR=IZUPSTUKNMVgwUMc;Z?]\eMh

]SO?KaRël|¥P;~IMM>.CMO : JLKNMIPSZeOQVgaceMQZeTVWR=RUZeP;gNcSMQZLgNM];ZZe]Sh/VW]\UVYP;gNëlT]\UUMZnMswfa]~YVxSqaRUPeKNMKNMQVYc;Kw?RLP4eKNMµ§P;NN~YMQZ-M];b/R]ZeM KaVcSKNMQZ[KNMgeKNMT]\UUMQZ§h/MQgaRUVxVYR=~P\[nMQZ :ta: a: a:¡E8 ÷ 3H6 ô H òñ3öMN< ² Vga];~~Yx;qSUKaMNKdx/RVWOQ];~N`a];ZUxSP;gXh/MQgaRUVx;q

Ωbh2,VYR¯]~WRPOQZUVUVWOQ];~NeP

eKNMMQ^SP;~Yf/UVYP;g¥P¯UKNMTXVWOrZeP\[n]_^SM]gaVYRUPeZUPSNVMsR :¬ MsOQ];~~*eKa]\eKNMNKNP;UPSg/ël`6]Zex;P;g1®afNVWh]SOeR§];R=]®afNVWh[VeK1NZeMReRfaZUM

p = ργ/3];gahh/MgaRVx

ρ = ρb + ργq,RPXeKa]\ Vg6OrZeM];RUVYgNc

eKNM`a];ZUxSP;gIZ?];OeVPSg]\nZUMsOrP;T`NVYga]\eVPSgTX];b;MsRUKaM=®afNVWhKNMs]_^dVMZ : JLKNVWR¯O]faRUMRUKNM®afNVWhePOQP;TXNZeMReR*TXP;ZeMn];gah`6PSfNgaOrM`6];O?b~YMReRQqwRKNV3UVYgNceKNMM>.CMOrUVY^;MLzQMZUP-P;VYgSP;aeKNMPSReOrVY~ë~W]\eVPSgaRRPXeKa]\

δcompression > δexpansion.JLKwf6RLUKNMiIP/hNhapµ§P;aN~MZ=-M]b/R=]SRURUP/OrVW]\eMh

[VUKeKNMOQP;TXNZeMReRVYP;gX[VY~~-`6M§KNVYc;KNMZUK6]gXUKaPSRUM];ReRP/OQVY]UMh[VUKeKNMZ?]ZeMr¡]SOeVPSg¤iIUKNMM^;MQg gdfNT`-MQZeMh-M]b/R?p :­ gaPUKaMQZ8M>.CMO*PNVYgaOQZUMs];RUVgNcLeKNM`6]Zex;P;ghNMQgaRUVxVWReKa]\FUKNMnRPSfNgahRU-MQMhVYRF~P\[nMQZeMhqRUP¤UKa]eKNM-M];b¥RUZefaOefNZeMOQ];g`6M RKNV3UMsh :ü P\[¯M^;MQZsqeKNVWRM=.-MsOVWR¡];VZe~Yx¥RUT]~Y~¯];R

c2s = R/3(1+R),[KNMQZeMLUKNMZe]UVYP VWR*xdNVWOQ];~~Yx

R = 4ργ/3ρb ∼ 3−4]ZUMsOrP;T`NVYga]\eVPSg

V6gdfaOQ~MPSRUxwgweKNMRUVYR8VWROrPSZUZeMOr :F f6O?K] KNVYc;K^\]~YfNMLTMs]gaR8eKa]\*eKNML`a]Zex;PSgh/MgaRVxgNMMhNReP`-MO?Ka];gNc;Msh]c;ZeM]LhNM]~CePM>.CMOrUKNMRU6MMhPRPSfNgah :ta: _ : & »NÅÆ*¾¡¼8ÁÄ"Ä*½ÈSÉ ­ R*h/VWRUOQfaRURUMh]`-P\^;M;q_h/V/.-f6RVYP;gM>.CMO?R*hN]TXPSf/FeKNMLRT]~Y~YMRReOQ];~MnTXPdhNMR :FÊ g];hah/VeVPSgq\eKNMP;`aRUMQZe^;MsheMQTX6MZe]UfNZeM]gNVWRUPUZeP;aVMsRF];ZUM]~WRUP§`N~YfNZeZUMsh`dxeKNM*îagNVUMeKNVWO?bwgaMReR,PwUKNM~Y]SR4ReOQ]UMZUVYgNcRUfNZ¡]SOrM : JLKNMRUMM>.CMOreR4~Ms];h§UP=];g MQ/6PSgNMQgwUVW]~RUfNNNZeMReRUVPSgPFUKNMKaVcSK

`TXPdhNMRq/IP;Z[KNVWO?K

k∆τrec ≥ 1.

ta: vN: Ã8Ĥ¾¡¼C½sÄwÁ6Ç\»/½Ä*- »sÃ8È=R QP¡¿ìÄ Ä9¯Ä";½;É JLKNMVYgwUMc;Z?]\UMsh ]SO?KaRëì|¥P;~IMOrPSgSeZUVY`Nf/ëeVPSgX];ZUVWRUMR*]~YP;gNc eKNM=NKNP;UPSg6]\UKIZeP;TÌ~Y]SRReOQ]UMZ*UPePdha]_x;q[KNMgV6];ReRMsR8UKNZeP;fNcSK

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê á?à]UVYTM^\]ZexdVgNcc;Z?]_^dV?]\UVYP;g6]~-PeMQgwUVW]~ : ´|.-MsOeV^SMQ~Yx;qCUKaMNKaPUPSgZeMOQMQVY^;MR§]RUKNV3 `-MrëO]faRUMeKNM-PeMQgwUVW]~LV¡];~~WRVgwePVWRhNVm.CMQZeMQgwIZUPSTeKNM¤6P;UMgSeVY];~¯VTfaROr~YVT`¢P;fNP; :² PSZUKaMc;ZeP\[VgNc¥TXP/h/Mh/fNZeVYgNc¥UKNM1T]UMZh/PSTVYga]UMhËM6P/O?KqnUKaMc;Z?]_^wVe]UVYP;ga];~

-PeMQgwUVW]~VWR];OrUfa];~~Yx1OrPSgaR?]gw`6MsOQ];faRMUKaMc;ZeP\[LUKP;¯-MQZUUfNZe`a]UVYP;gaR NZeMOQVYRUMQ~Yx1OrPSTë-MQgaRe]\eMR8IPSZUKNM=hNMOrZeM]SRVYgNcTXM];gMgNMQZec;xh/MQgaRUVx;qSRUP UKa]

δρ = ρδVWROrPSgaRe]gw : JLKdfaRq

eKNM Ê | M>.CMOrXOQP;gwUZeVY`Nf/UVYP;gÍVYRTXPSRXVTX-P;ZUe]gwX`6MQIP;ZeMeKNM TX]eMQZUëlZ?];h/VW]\eVPSgUZ?]g/ëRUVeVPSgÍPSZ]\3eMQZT]\UUMZh/P;TXVYga]\eVPSgËMgahNRi¡VUKaMQZeMVYR]OQPSRUTPS~PSc;VWOQ];~nOQP;gaRe];gSPSZ]wfNVYgwUMReRUMQgaOQMLîaMQ~Wh : pJLKNM=IP;ZeTMZM>.CMOr¯VYg/®afNMgaOrMsReKNM=KNMVcSKS?RPCeKNM=îaZ?RIMQ[µ§P;NN~YMQZ-M];bdRq-[KNV~YMUKNM~Y]eMQZOQ]g¥OrZeM]UM]ShNh/VUVYP;ga];~4®af6OUf6]\UVYP;g6R§PSg1^;MZUx ~W]Zec;MReOQ];~MsRi¡V : M :]L~YP\[

`.p

JLKNMX~W]Zec;MXRUO]~YMMQ/OQMReR 6P\[nMQZZeMRUfN~eVgacIZeP;T ]¤OQPSRUTPS~PSc;VWOQ];~FOrPSgaRe]gwVYRK6]Z?h1eP^SMQZeVIxh/VYZeMOe~xh/fNMLePwPOQPSRUTVWOn^\]ZeVY];gaOrM :Fü P\[nMQ^SMQZsq\VT]_x`-MTXM]SRfNZeMhVYgah/VYZUMsOU~Yx^dVW]OQP;ZeZUM~Y]UVYP;gaRP;UKaMknm1o [VUKUKaM~YP/OQ]~8T]\UUMQZh/VWRUZeV`NfNUVYP;g : JLKa] VWRq-T]UMZ[VeK]TMs]gZUMshNRKaV3 P

z ∼ 1.y*PSReRVY`N~YMNZeP;`-MRIP;Z UKaVYRRUZefaOefNZUMVg6Or~Yfah/MUKNMZe]Sh/VYP]g6h

dëìZe]_xX`a]SO?bwcSZUPSfNgahNR :ta: ua: Ä*½caÇ»N¼-½sÄd¼FÈcaÇÅU-8ÄdÈ;É WMsOeP;ZTXP/h/MsR]g6hc;Z?]_^wVe]UVYP;ga];~*Ze]Sh/VY]UVYP;gOQ];g];~YRUPOrP;gweZUVY`Nf/eM*UPLUKaMRVYc;gaVî6O]gwU~Yx=ePLUKNMLknm1o];gNVYRUPeZUPSNVYMRC^dVY]¯eKNMVYgSeMQcSZe]UMh ]SO?KaRë|¥P;~IM§M=.-MsO :8Ê gUKNM];`aRMgaOrM§PFRUP;faZeOQMRIPSZeKNMRUM TP/h/MsRQqd^SMOeP;Zn6MZefNZe`a]\eVPSgaR¯hNMOQ]_x]_[L]_x1wfNVYO?bd~Yx;q

hv ∝ t−1.JLKNMRUM]ZeMgNPMQ/6MsOUMsh¥UP1`-MVYTX6PSZ?]gwIP;ZVYg/®6]UVYP;ga];ZUx

ReOrMga]ZeVPwRQqC`NfNOrP;fa~YhNZeP\^;MVYTX6PSZ?]gwVnUKaMQZeMX];ZUMX];OrUVY^;MXRPSfNZ?OrMR§IPSZ -MQZUUfaZU`a]UVYP;gaRRUfaO?K¤];ROrPwRTXVWOh/MrIMsOeR :ï Z?]_^wVe]UVYP;ga];~C[n]_^SMRnP;`-MQxeKNM´VYgaRUMQVYg¤Mwfa]UVYP;gq

hTij + 2

a

ahT

ij + k2hTij = −16πGΘT

ij .

Ê geKNM];`aRUMQgaOQMPRUP;fNZ?OrMsRQqNUKNMXRPS~fNUVYP;gaR=UPeKNVYR ]ZeMcSV^SMQg`dxRNKaMQZeVYO]~4onMReRUMQ~IfagaOëeVPSgaRQq,V : M :

hTij ∝ j1(kτ)/kτ

VgUKaMT]UMZh/P;TXVYga]\eMhZUMc;VYTM : JLKNMsRMUMgaRUP;ZTXPdhNMR];ZUMIZeP;zMQgVYgPSf/eRUVYhNM=UKaMKNPSZUVYzQPSgqS`af/nh/MsOQ]_x]_[L]_xwfNVWO?bw~YxfN6PSgMgSeMQZeVgac : JLKwf6RQqSUKNMeMQTX6MZe]UfNZeM]gaVYRUPeZUPSNVMsRLIZeP;Tc;Z?]_^wVe]UVYP;ga];~CZe]Sh/VY]UVYP;g]ZeMNZeMh/PSTXVga]UM~x P;g1ReOQ];~MsReKa]\ [¯MZUMPSf/eRUVWh/MeKNMKNPSZUVYzQPSg1]\§hNMOrPSfNN~YVgac :Ê eKNMZeMQ~W]\eV^SMOrPSgwUZeV`NfNUVYP;gPeMQgaRUP;Z®af6OUf6]\UVYP;g6R4eP UKNM]gaVYRUPeZUPSNVMsR¯i

T/S ≡ CT2 /C

S2

pFVWRRT];~~lq_UKNMgUKNMsRMLOQ];g`-Mh/V¨OQfN~ePXhNMrUMsO=`-MO]faRUMPOrPSRUTXVYO ^\]ZeVW]gaOQM :ta:/: ÈcSº4ÇoON»/½sº4Ç\ÄX»N¼--8Äw¿ìÄ*½§ÅU-8Ä"PØÈ;É JLKdfaR4¡]Z8[nMKa]_^SM`6MMQgIPdOQfaRUVgNc NZeVT];ZUVY~xPSg]Sh/VW]`a]UVWOOrfNZe^\]\UfaZUM*®af6OUf6]\UVYP;g6RQqs[KNVWO?K]ZeMF®6faOefa]\eVPSgaRVYgUKNMP\^;MQZ?]~Y~\h/MQg6RVx;q_`Nf/[KNMZUMUKNM ZeMQ~W]\UVY^;MgwfaT`-MQZ?RP;UKNMh/V/.-MZUMgwna];ZeVYOQ~M R-MOQVMsR¯];ZUM§fNgNVIP;ZeTUKNZeP;fNcSKNP;fNRUa];OQMiIM : c : 3

4δγ = δb = δc: p|MO]g1];~YRUPOrPSgaRVWh/MZ=VYRUP/OrfNZe^\]\efNZUMSqNP;ZMQgwUZeP;dx;qa6MZefNZë

`a]UVYP;gaRqNVYg1[KNVWO?K¤UKNMh/MQgaRUVxP;*T]UMZPSg1RfN-MQZeKNPSZUVYzQPSg ReOQ];~MsRVYRfNgNVIP;ZeT¤qa`Nf/§UKNMZeMQ~W]\eV^SM h/MgaRUVeVMsRLP*a]ZUUVWOr~YMR-MOQVMsRn®afaOefa]\eM :JLKNMZeMQ~W]\eVPSg`-Mr[nMQMgeKNMXUMTX6MZe]UfNZeM]gNVWRUPUZeP;aVMsR ];gahUKNM];TXN~VUf6h/MPL~Y];ZUcSM

ReOQ];~MRUZefaOefNZUMVWRRUVcSgNVî6OQ];gSe~xh/V/.-MZUMgSIPSZVWRP/OrfaZU^\]\efNZeM¯®6faOefa]\eVPSgaR :*¬ MsOQ]~Y~/eKa]\PSg~W]Zec;MReOQ];~MsRQq_];h/VW]`6]\UVWO*-MQZUUfNZe`a]\eVPSgaRaZUMsh/VYOr

δT/T = Φ/3.ü P\[nMQ^;MZqQIP;ZFVYRUP/OrfNZe^\]\UfaZUM

TXP/h/MR=UKNMXOrPSZUZeMRU-P;gah/VYgNcXeMQTX6MZe]UfNZeM®af6OUf6]\UVYP;gVYR§KNVYc;KNMZqδT/T ∼ 2Φ.

JLKdfaRIP;Z]cSV^SMQg];TXN~VUf6h/MP;8knm1o©]gNVWRP;UZeP;NVYMRqeKNM~W]Zec;M§RUO]~YMReZUf6OUfaZUM§NZUMsh/VWOUMshVWRTfaO?K~YP\[¯MZVgVWRP/OQfNZU^\]UfNZeMUKNMP;ZeVMsRQq8]gahRUP¤~Y];ZUcSM`NVY]SR ¡];OrUP;Z?R];ZUMXc;MgNMQZ?]~Y~Yx1ZUMswfNVZeMhePMQda~Y];VgUKNMPS`aRUMQZe^_]UVYP;gaR :JLKNM¤NKa]SRM P§UKNMaKNPeP;g/ëì`a]Zex;PSgPSReOrVY~~W]\eVPSgaRVgËVWRP/OrfaZU^\]\efNZeMTXP/h/MQ~WRXVYRRUKNV3UMsh

ZeMQ~W]\eV^SMeP¤UKa]Vg]Sh/VY];`a]\eVYOXTXP/h/M~YR : JLKNVWRVWR`6MsOQ];faRMXeKNMPwRUOQV~Y~Y]UVYP;gaR];ZUMXVYgNVeVY];~~Yx

áeá âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/égNP;P.,RMQPSf/eRUVWh/MXUKNMXKNPSZUVYzQPSgq`NfN]ZeMRe];ZeMh[KNMQg¥UKaMTP/h/MsRMQgwUMZUKNMKaP;ZeVzP;gh/faMePeKNMVWRUPdOQfNZe^_]UfNZeMIPSZeOQVgacUMZUT : JLKdfaRUKNMXTXP/h/MR ]ZeM];NNZeP_/VT]\eMQ~Yx

90P;fN

P;FNKa]SRM[VUK UKaM];h/VW]`a]UVWO ZUMsRfN~ :ta: Ya: -ÈÅj¾M¢-8Äw¿ìÄ*½»N¼8¾¡Èca½Ço6Æ*¾¡ÄdÈ;É k¯PSRUTXVYOh/MQIMOreR];ZUMM=.-MsOeV^SMQ~Yx¦VWRP/OQfNZU^\]UfNZeM-MQZUUfNZe`a]UVYP;gaR[VeK¦];g¦];OrUVY^;M¥RPSfNZeOQM[KNPwRMMQ^SP;~Yf/UVYP;g¸VWR gNP;gNël~YVgaM]Z : JLKNMRPSfNZeOQMVWRXTXPSRXVYTX6PSZ?]gw faR]\3UMZXUKNM¤TXP/h/MRXMgwUMQZXeKNMKNPSZUVYzQPSgÍ]g6hjeKNM1h/MrIMsO?RX`6Mc;VYgePReZe];VcSKwUMQg¤eKNMQTRUMQ~Y^;MsR : JLKNMgNPSg/ëì~VYgNM];Z=MQ^SP;~Yf/UVYP;gP;UKNVWR NZePdOQMReRTXM];gaR=eKa]\§UKNMRUP;fNZ?OrMVWRgaPOQP;KNMZUMgwq*]gah¥eKNMh/MQIMOreR[VY~Y~nRUP;fNZ?OrMTXP/h/MsRVYg¢RPSTXMZUMc;VYP;gaR`6MQIP;ZeMP;UKNMZeR : ­ R]ZeMRUfN~qLUKaMµPSNN~YMQZ-M];bËRUZefaOrUfNZeMPUKaMRUO]~W]ZRU-MOeZUfaTO]g©`6MRUTXPwP;UKNMshPSf/Vgh/MQIMOr=TP/h/M~YR :yZeP;`a];`N~YxjUKaM1TXPSRVTX-P;ZUe]gwM=.-MsOP;§eKNM]SOUVY^;M1hNMrIMOrRUP;fNZ?OrMsRXVYRUKa]UKNMVZ

^SMOeP;Z]gah1UMgaRPSZ]gNVWRP;UZeP;NVYMR ]ZeMxdNVWOQ];~~Yx1OrPSTXa]Z?]`N~YMeP UKNMReOQ]~W]Z ]gNVWRUPUZeP;aVMsR];gahO]gNgNP; `-MgNMQcS~MsOUMsh : JLKNMsRM[VY~~cSZUMs]\U~Yx VgaOQZUMs];RUMeKNMX~Y];ZUcSMReOQ]~YM-P\[nMQZVgUKNMknm1oÌR-MOrUZefNT¤qF`af/[VY~~gNP].-MsOeKNMRUT]~Y~RUO]~YMReZUf6OUfaZUM : JLKdfaRq4eKNMµ§P;NN~YMQZ-M];bjReZUfaOrUfNZeMRqVUKaMQxjh/P¥Mr/VWRsq[V~Y~`-M RUfNNNZeMReRUMhjZeMQ~W]\eV^SMePeKNMk³ on´RUO]~YM];gNVYRUPeZUPSNVYMR :

9sN:E£ÿQþNAc¤FH3>@^4þ`a¥Ud¦$"'wE4G6@,Ed@,HIB4>9sN:Y9;: -ÈsÅj¾#kON»NÇ_¾Ø»N¼;Ä-ÉÊ FeKNM ®afaOefa]\eVPSgaR]ZeMï]faReRVW]g4qdUKNMTfa~eV-P;~YMTXP;TXMQgweR];ZUM];~~ VYgah/MQ-MQg6h/MQgw : mMs];RUfNZUMTXMQgwP;UKNMOrPSZUZeMQ~W]\eVPSgÍIfagaOeVPSg¸VYRIfNg6hN]TXMQgw?]~Y~x~YVTXVUMsh`dx UKNMn¡];Or4eKa]\*[¯MnP;`6RMZU^SMP;gN~YxP;gNMnRbdx : JLKNMn6P\[nMQZ8R-MOrUZefNT

C`VWR*h/VWRUOQZUMQUM;q

];gah¢Ms];O?K`^\]~YfNM¤VWRRU];TXN~Msh¢PSgN~Yx

2` + 1eVTXMsR : JLKNVWRZeMRUfN~eRXVYgË];gËfagNZUMsRPS~^\];`N~M

fNg6OrMQZUe];VgwxVgUKNM R-MOrUZefNTTMs];RUfNZeMQTXMQgwP;δC` '

√2C`/(2`+1)

1

2 .JLKaVYRVYR¯bdgNP\[g

]SRõ 3 òùñ¡õeB 8 ÷ñ 8H õeôc<9sN: : ¾¡Á-ÃFÄwÇu6Ç-FÄwÇÄ"Ä*½sÈ;É ·=P;gNël~YVgaM]ZM>.CMO?RO]gj]~WRUPNZUP/h/f6OrM];gNVYRUPeZUPSNVYMRVYgeKNM=TXVYOQZUP\[L]_^;Mn`a]SO?bwcSZUPSfNgah]gaheKNMRUM];ZUMfaRUfa];~~YxVYTX6PSZ?]gw¯]\nRT]~Y~YMQZReOQ];~MsRi

< 0.1h/Mc;ZeMQMsR : p ü P\[nMQ^;MZq,VLgNP;gN~YVYgNM];ZM>.CMO?R~YM];heP¤UKNMfNgaV^SMQZ?RMX`-MQVYgNcVPSgNVYzQMh]KNVcSKMgNP;fNcSKZeMhNRUKNV3eRq/UKaMMQgwUVYZUMknm1o¦R-MOrUZefNTO]g `6Mh/Z?]T]UVWOQ]~Y~Yx]~UMQZeMh :ü VYc;KaMQZPSZehNMQZM>.CMOreROQ]gaZUP/h/faOQM1]gNVWRUPUZeP;aVMsR]\~W]\eMUVYTXMRqP;gaOQMUKNM6MZefNZë

`a]UVYP;gaR `-MOQP;TXM~Y];ZUcSM : JLKNMRUMOQ];g¥OQP;TXMIZUPST UKNMknmo°NKNP;UP;g6R`6MVgNcZeMReOQ]eMQZeMh^dVW]k¯PSTX/UPSg©ReOQ]eMQZeVgNc6q¯PSgaOrM¤eKNMMQ~YMOrUZeP;gaR];ZUM¤ZeMQVYP;gNVYzQMsh,qnP;Z[KNMg©c;Z?]_^wVe]UVYP;ga];~gNPSg/ëì~VYgNM];ZUVUVYMRLO?Ka];gNc;M§UKaMMQgNMZUcSxP;Z=h/VYZUMsOeVPSgP;FeKNMNKNP;UP;g6R :9sN: :Y9;:§ naô ø H 8 ô=Bq¨8ô 5/693 Bsñ¡õrnôM©ô?õröv<§³§gaMgNP?]`N~YMTXMO?K6]gNVWRT[KNVWO?K1OrZeM]UMRUMQTë-MQZ?]\efNZUM=®afaOefa]\eVPSgaRn]gah].-MsO?ReKNMknmoËRU-MOeZUfaTÂVYR¯Vgd^SMQZ?RM k¯PSTX/UPSgRUO]\UUMQZeVYgNcP;NKNPeP;gaRP.PKaP§M~MsOeZUPSgaR=VYgOQ~faRUMZeRq-bwgaP\[g];R=UKNM fNgdxS];MQ^wë-MQ~Wh/P\^dVYO?K M>.CMOr :JLKNVWR8M>.CMOr8cSV^SMR8]=[L]_x§eP§TXMs];RUfNZUMnh/VYZUMsOU~YxUKaM¯`a];ZUxSP;gNVWOT]SRURFVgOr~YfaRUMQZ?R8]\*KNVcSKZeMhdëRUKNV3eR : OQ]\UUMZUVYgNcP.P4KaPeMQZLMQ~YMOrUZeP;gaRnOQ];faRUMRn]RUVTXN~YM§MgNMQZec;xRUKNV3L6]Z?]TXMreMQZeVzMh`dxeKNMk¯P;TX/eP;g

y− a];Ze];TXMrUMZq

y =δν

ν=

∫σTne

kTe

mec2dl

[KNMZUMσTVYRUKNMJLKNPSTRPSg¥ReOQ]UMZUVYgNcOrZePSReRRMsOeVPSgq

neVWRUKNMXIZeMQMM~MsOeZUPSg¥h/MgaRUVx

];gaheKNMVYgSeMQcSZe];~-VWR=];~PSgNceKNMNKNP;UPSga]UK UKaZUPSfNc;KUKNMOQ~f6ReMQZ :JLKNMNKaPUPSgaR=];ZUMfNaReOQ]eMQZeMh,qaO]faRUVYgNcUKNM~YP\[¦MgNMQZec;xX?]VY~P*eKNMR-MOrUZefNT¶eP`6M

RUKNV3UMhePKaVcSKNMQZLIZeMwfNMgaOrVYMRqδnν

nν= −y xex

ex − 1[4− x coth(x/2)] ,

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê á;[KNMZUM

x ≡ hν/kTγ.JLKNVWR ZUMsRfN~eR VYg]UMQTX-MQZ?]\efNZUMXh/MQî6OrV]§IZeMwfNMgaOrVYMR `-MQ~YP\[ UKNM

`N~W];O?bd`-P/h/x6Ms]b∆T/T = −2y

q]g6h]UMTX6MZe]UfNZeMXVgaOQZUMs];RUM]KNVcSKNMQZ IZeMwfNMgaOrVYMRq[VUK¢UKNMRUVcSg¢O?Ka]gac;VYgNc]

ν ∼ 220ï ü z : JxdNVWOQ];~nOQ~f6ReMQZ?RKa]_^SMeMQTX6MZe]UfNZeMRP;

PSZeh/MZ107 − 108K,

];gah xdVM~Yhk¯P;TX/eP;g/ëìxa];Ze];TMQUMZeRnP8P;Z?h/MQZy ∼ 10−4.

9sN: : :§ naô 3 ó;ó 5 ôr÷1ôM©ô?õröv< ­ gNPeKNMQZXVYT-P;ZUe];gwM>.CMOrX];ZUVWRUMRV=eKNMQZeMVWRRUVYc;gNVî6OQ];gw`Nfa~b1TXP;UVYP;g¥Vg¥UKaMVPSgNVzMhTXMh/VYfNT¤qF[KaVYO?K¥Vg6h/faOrMsR] UMT-MQZ?]\efNZeM®6faOefa]\eVPSg^dVW]eKNMµPSNN~YMQZM=.-MsOÎ

∆T

T=

∫v‖neσT dl

gN~VYb;MjUKNM ëÌM=.-MsOqUKNM͵PSNN~YMQZRUKNV3¥h/PdMsRgNP¥O?Ka]gNcSMUKNMËR-MOrUZefNT PXUKNM];gNVYRUPeZUPSNVYMRn]gah¤RUPVWRLKa];Zeh/MZnUPh/VWReVgac;fNVWRKIZeP;T¶eKNMNZeVT]Zex];gNVYRUPeZUPSNVYMR :JLKNVWRM=.-MsOVWRRUMOQP;gah¥P;Z?h/MZVg~YVgaM]ZUKNMP;Zex]g6hOQ]g`-M~W]Zec;MQZ[KaMQggNPSg/ël~YVYgNM];Z

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T ∝ λ2.1,];gahK6];R]XRVYTXV~W]ZVYgwUMgaRVx :

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TE(x) =

∫W (x− y)T (y)d2y.

JLKNVWRLOrPSgw^SP;~Yf/UVYP;gVWRnM]SRVYMRePM^;P;~Y^;M§Vg ² PSfNZUVYMQZLRUa];OQM;qd[KNMQZeMVVYRLRUVTXN~Yx]NZUP/h/f6OqV : M :

TE(k) = W (k)T (k).Ê g¢c;MgNMQZ?]~lqeKNM¤`6Ms]TûRUKa]-M¤OQ]gÍ`6MwfNVUMOrPSTXN~VWOQ]UMsh,q

VYgaOr~Yfah/VYgNcX`-MQVYgNc]gNVWRUPUZeP;aVYO ]gahKa]_^dVYgNcRUVYc;gNVî6OQ];gwRUVYc;gNVî6OQ];gwRUVWh/MQ~YP;`-MR :*Ê =VYRRUP;TXMrëeVTXMR*] faRUMrIfN~N];NNZeP_/VT]\eVPSgUP UZeM]FeKNMOrMgSeZe];~S`-M];T];R*]ï]faReRUVY];gIfNgaOrUVYP;gq;[VeKW (θ) = e−θ2/2σ2

/2πσ2.JLKNVYRFKa]SRF] RVYTXN~YM ² PSfNZeVMZFZeMQNZeMRUMQgw?]\UVYP;g4q

W (k) = e−k2σ2/2.JLKNM^\]ZeVY];gaOrM];gNc;~YMVWR=ZeMQ~W]\eMh¤ePUKaMIfN~Y~*[VYhdeK/ëìKa]~T]\/VTfNT RVYzQMPUKaM`-M];T`dxθ2FWHM = 8σ2 ln 2.

|¥MK6]_^;MeZUMs]\UMsh¤UKaMRUbdx¤]SRV¯V[¯MZUMM=.-MsOeV^SMQ~Yx¤®6]\sq-[KaVYO?KVYR]ZUMs];RUP;ga];`N~M]NNZeP_/VT]UVYP;g1]\RUTX];~~ReOQ];~MsR : JLKNMZUMSq-[nMXOQ]gTX];b;MUKNMXRUfN`aRUVUf/eVPSgkr → (`+ 1

2 )θ.² PSZcSZUPSfNgah©`a]SRMsh©Vg6ReZUfNTXMgS?RV¤VYRP3eMQg¸faRUMrIfa~§eP¢TXM]SRfaZUMeMQTX-MQZ?]\UfaZUMhNV3ëIMZUMgaOrMsRVgPSZehNMQZeP1ZUMsh/faOQMOrP;gw?]TXVg6]\UVYP;g¥IZeP;T ]\UTXPwRNKaMQZeVYOXM=.-MsOeR :¤Ê gUKNMa];Rq`-PeK1[nP]g6hUKaZUMM`-M]TíhNVm.CMQZeMQgaOQMMQ/6MZUVYTXMQgweR§[¯MZUM6MZIPSZUTXMsh :² P;ZMQ/];TXN~MSq,][nP`-M];TíMr/-MQZeVTXMgSTXM]SRfNZeMR§eKNMX^_];ZUVW]g6OrMP;eKNMUMQTX-MQZ?]\efNZUMh/V/.CMQZeMQgaOQMP;[nPa]eO?KNMsRLRMa]Z?]\eMh`wx]gUKNZeP\[¦];gNc;~YM;q

θT .Ê =VYRRUVTXN~YMUPRUKNP\[¸eKa]\

〈(T1 − T2)2〉 = 2C(0)− 2C(θT ) =

1

4π

∑

`

(2`+ 1)W`(2− 2P`(cos θT ))C`.

|KNMZUMW`VWR=eKNMRVYgNcS~M`6Ms]T [VYgah/P\[°IfNg6OUVYP;g : ­ eKNZUMM`-M];T Mr/-MQZeVTXMQgwOQ];g1`6MM^_];~f6]\UMshRUVYTVY~W]Ze~x :

9sN: uN:A«4ºF½ºFÇ_ÄÍÄ"°,ÆÄwÇ_¾ØÅjÄd¼C½ÈSÉ ­ gdfNT`-MQZP;n`6]~Y~PdP;g¥`a]SRMshMQd-MQZeVYTMgweR]ZeMXfNgah/MZë[L]_x : onPwPSTXMQZ?]gNcX]gah m1]dVYT]Ka]_^;M]~YZUMs];h/xaZUP\^dVWh/MhVTX-P;ZUe];gS=OQP;gaRUZ?]VYgweRnP;gUKNM-PSRUVeVPSg1PeKNMXµPSNN~YMQZ 6Ms]b/R : mPSZUM[V~Y~OrP;TXMIZUPSTeKNMXo¯´ ­ J];gah1JFP; ü ]\h/MrëeMOeP;Z?RQq/]SRL[¯M~~4]SRnIZUPSTÂIfNZUUKNMZ=onPwPSTXMQZ?]gNcX]gah m1]dVYT]®aVcSKweR :JLKNMTXPSR=VYT-P;ZUe];gwLfa-OQP;TXVgacc;ZeP;fNgah`a]SRMsh hNMrUMsOeP;Z?R]ZeM VYgwUMZIMZUPSTMQUMZeRLRUfaO?K

]SR=eKNM2WMZUx T];~~ ­ ZeZ?]_x¥i#W ­ p`6];RUMhVYgJFMQgNMZUVIM;qµ ­ dÊ `6];RUMh1]§eKNMRUP;fNUK1-P;~YM;q

árj âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/é];gahÍkno Ê `a];RUMhVYgjeKNMTXPSfNgwe]VYgaRP k¯KNVY~YM : JLKNMRUM];ZUM];~~nfNgahNMQZe[n]_x]gah¢]~Y~¯Ka]_^SMZeMQ-P;ZUUMshZeMRUfN~?R :JF[¯PVYTX6PSZ?]gw¯Re]\eMQ~Y~VUM§TVWReRVYP;gaR¯]ZeMN~W]gNgNMshP;Z¯VgNZUPSc;ZeMReR : · ­ ­ « R¯mVWOrZeP\[L]_^;M

­ gNVYRUPeZUPSdxyZeP;`-MiØm ­ y¯p[L];R~W]fagaO?KNMhVYg ­ NZUVY~ ;a9 ]gahjKa];RNZUP/h/f6OrMhjV?RîaZ?RZeMRUfN~eR : JLKNML´ ­ y~Y];gaO?bRU]UM~~YVeMLVYRh/faM¯eP`6ML~W]fNgaO?KaMhVg Swd: JLKNMRUM¯[VY~Y~NTXM];RUfNZeMeKNML[KNP;~YMLTVWOrZeP\[L]_^;MRUbdx[VeKXZUMsRPS~f/eVPSgP;-]aNZUP_/VYT]\UM~x a:9_v h/Mc;ZeMQMsR : y~Y];gaO?b[VY~~Ka]_^SMcSZUMs]\UMZIZeMwfNMgaOrxOQP\^;MQZ?]cSM]g6hRUKNP;fN~Wh];~YRUPaZUP\^dVWh/M¯f6RMQIfN~/TXM];RUfNZeMQTXMQgw?R4P;NUKNM-P;~W]ZeVzs]\eVPSg :

9S9;:±þU`a¥Ud²$dA,þiFEþN>4GS=aþaG6@CA-EgJLKNM¤TXVYOQZUP\[L]_^;M`a];O?bdc;ZeP;fagahKa];RX]`N~W];O?bd`6P/h/xjRU-MOeZUfaT[VeKË]UMTX6MZe]UfNZeM¤P

T0 = 2.728 ± 0.004: JLKaMk³ o¯´ ²8ʬ ­ Vg6ReZUfNTXMgS^SMQZeVî6MhUKa]UKaMknm1oÌRU-MOëeZUfNT[L];R¯OrP;g6RVWReMQgw[VUK]`N~W];O?bd`6P/h/x[VUKNVYg

50a]ZUeR6MZ¯TXVY~~YVPSgVYgUKNM§TXV~Y~VYTXMreMQZ

[L]_^;M~MgNceKaR : µ§MQ^dVW]\UVYP;g6R IZUPST `N~W];O?bd`-PdhNxOQ];g¥`-Ma];Ze];TMQUMZUVYzQMshVYgTX];gdxh/V/.-MZUMgS[L]_x/RQqeKNM¤TXPSROQP;TXTXP;gË`6MVgac¥MVeKNMQZXeKNMk¯P;TX/eP;g

y− 6]Z?]TXMreMQZP;ZXVgËeMQZeTXRXP;]O?KNMTXVYO]~=-PeMQgwUVW]~lqµ.JLKNMTP;UVY^\]\UVYP;gËIPSZUKaMRUMa]Z?]TXMreMQZeVzs]\eVPSgaRX[V~Y~`-M1MQwë

N~W]VYgNMshRUKNP;ZUU~Yx : JLKaM ²Fʬ ­ Vg6ReZUfNTXMgSa~Y]SOrMhfNN-MQZ~VYTXV?RP;g¥UKaMRUM6]Z?]TXMreMQZ?R]SR |y| < 1.5× 10−5 ]g6h µ < 9× 10−5.JLKNMIPSZUTUK6]\eKNMRU6MsOUZ?]~nh/VWReP;ZUUVYP;gaR];ZUM~YVYb;MQ~Yx1ePe];b;Mh/M6MgahPSg[KNMQgj];gahKNP\[UKaMQx]ZeMNZePdhNfaOrMsh : y8PwRURUVY`N~MRPSfNZeOQMRXP;h/VYRUPSZeVPSgaR]^;MZUxÍM]Ze~YxjUVYTXMR]ZeMVYgaOrPSgaRUMwfNMQgweVY];~8`-MrIPSZUM] ZeMhNRUKNV3P

zth ∼ 106 RUVYgaOrMUKaMfNgNVY^;MQZ?RUMZeMQT];VgaRVYg¥UKNMZëT]~8MwfNVY~YV`NZeVYfNT fNgwUVY~FeKNVWReVTXM^dVY]IZUMMrëlIZUMM]g6hZe]Sh/VW]\UVY^;MXk¯P;TX/UPSgReOQ]\UUMZUVYgNcaqC];ReKNMRUMNZeP/OrMsRURUMR¯OQ]gO?Ka]gNcSMUKaM§gdfNT`6MZ¯P;NKNPeP;gaR : ­ 3eMQZnUKNVWR¯MQ-PdO?K4qwPSgN~YxPSZehNVga];ZUxk¯PSTNUP;g¤ReOQ]\UUMZUVYgNcVYRLVYTX6PSZ?]gwqa]~Ms];RLfagSeV~F]ZeMhaRKNV3

zc ∼ 104. Vg6OrMUKaVYRaZUP;ë

OQMReROrP;g6RMZU^SMR*NKNP;UPSggdfNT`6MZq;UKNM`6];O?bdc;ZeP;fNg6hZeMQT]VYgaR*VYgbdVgNMQUVWOqS`Nf/gNP;UKaMQZeTX];~ØqMsSfaV~YV`aZUVYfNT : |KNVY~YMVgbdVgNMQUVWOMswfNV~YVY`NZUVYfNT¤qVYg MOrUMshMgNMQZec;xO]ghNVYRUPSZ8UKNMRU6MsOeZUfNT :ü P\[¯M^;MQZsqCUKaMR-MOrUZefNT Ka]SR UKNMXIPSZUTPL]o¯PwRMQël´VgaRUMVgRU6MsOUZefNT¤q46]Z?]TXMreMQZeVzMh`dx]XO?KNMTVWOQ];~,6P;UMgSeVY];~Øq

µ.JLKNMM>.CMO¤PMgNMQZec;xËVg MOeMh¦]\3UMZzch/MQ-MQg6hNRPSgV?R NZUP/h/f6OUVYP;gTXMsO?Ka]gNVWRUT :

JLKNMZUM¤]ZeM]¥^\]ZeVYMrxP 6PwRURUV`a~M¤RPSfNZ?OrMRqVYgaOr~Yfah/VYgNc¥UKaM¤hNMOQ]_xP;§T];ReRVY^;Ma]ZUUVWOr~YMRPSZRUfN-MQZ?OrPSgah/faOrUVYgNcOrPSRUTXVYOXRUZeVgacSRq4ReMQ~Y~Y];ZZe]Sh/VW]\UVYP;g4q4Rfa6MZUgNP\^\];MMQda~PwRVYP;gaRq,`N~W];O?bKNPS~M§];OQOQZUMQUVYP;gPSZMQ^\]-P;Z?]\eVPSgq;PSZ`dxh/faR : ­ gaPUKaMQZnVTX-P;ZUe]gwTXMsO?Ka]gNVWRUTÌVWRVYgd^;MZeRUMk¯PSTNUP;gReOQ]UMZUVYgNc IZeP;TKNP;eMQZIZeMQMM~MsOUZeP;g6RQq[KaVYO?KXfNaReOQ]eMQZ?RNKNP;UP;g6R]gahZeMh/VWRëeZUVY`Nf/eMRLeKNMQTIZeP;T~YP\[nMQZLIZeMwfNMQg6OrVYMRLePKNVYc;KNMZ=P;gNMsR : JLKNVWR=VWRP;`aRUMQZe^;MshePP/OQOQfNZIP;ZOQ~faRUMZeR=[KNMQZeMV§VYRbdgNP\[g]SRLUKNM fNgdxS];MQ^wëCMQ~Wh/P\^dVYO?KM>.CMOsqa`NfNV ]~WRPOQP;fN~Wh PdOOrfNZV*UKNMVYP;gaVzMh VgweMQZUëlcw]~W];OeVYO TXMhNVfNTVWR=RfN¨OQVMgwU~YxKNP; : JLKNVWR=~YM];haRLUP]k¯PSTNUP;gNëlxh/VWRUP;ZUUVYP;gP8UKNMRU-MOeZUfaT :

9 :]³6@9d@CHbQ@,HIG^eF>D¢@þU`a¥UdJLKNMZUM];ZUM]¤gdfNT`6MZP;Re]UVWReVYO]~nO?Ka]~Y~MgNc;MsRVgcSP;VYgNc¤IZeP;T P;`aRUMQZe^\]\UVYP;g6RPLUKNM

knm1o¦ePfNgahNMQZ?R?]gah/VYgNceKNMQVYZ VTXN~YVYO]\eVPSgaR=IPSZ OQPSRUTXP;~YP;c;VWOQ];~,a]Z?]TXMQUMQZ?R : ³§gNMP;3UMgcSPwMsRXUKaZUPSfNc;KT]gdxË~Y]_xSMQZ?RPhN]e]jOQP;TXNZeMReRVYP;g4Î1IZeP;TUKNMZe]_[UVYTXMRUZeM];T];gah-P;VYgwUVYgNc¥ha]\e]aqeP¥T];aR]\T]gdxjh/V/.-MZUMgwIZeMwfNMgaOrVYMRqePZUMsOrP;g6ReZUfaOrUVYgNc]RUVgac;~YMT]PUKaMfNg6h/MQZe~xdVYgNc knm1o°];gNVWRP;UZeP;NVYMRqNePVeR `NVYgNgNMh16P\[nMQZ R-MOrUZefNTí]gahUKNMgîag6]~Y~x¤eP OQP;gaRUZ?]VYgweR P;g¥OrPSRUTXP;~YP;cSVYO]~46]Z?]TXMreMQZ?R : ´]SO?KRUM¥OQP;TXNZeVYRUMR] h/V¨OQfN~Re]UVWReVYO]~,NZeP;`N~YMQT :ü MZUM Ê [VY~~4hNVYReOrfaReRRUP;TXMP8UKNMcSMQgNMZe];~CUPdP;~WRLZUMswfNVZeMhUP]ShNh/ZeMReR¯UKNMsRMNZeP;`a~MTXR

];gahT];gwxPUKaMQZNZeP;`N~YMQTRLVYgPS`aRUMQZe^_]UVYP;ga];~,OrPSRUTXP;~YP;cSx :

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê á;s9 :Y9;: ! »NÈs¾#ÍÇ\ºPØÄdÈu-¿XÆFÇ8» 8¾MP¡¾I½?ÀÉ JLKNMZefN~YMRIP;ZXT]gNVYNfN~W]\eVgacNZeP;`a];`NVY~VUVYMR]ZeMwfNVUMXRVYTXN~YM]g6h1VYgSefNVUVY^;M : JLKNMîaZ?Rsq-UKNMXRP OQ];~~YMhjò?ø/ù ÷ø 5 ô q-ZeMwfNVYZeMRUKNMXRfaT PeKNMNZeP;`a];`NV~YVUVYMR¯IP;ZLeKNM6PwRURUVY`N~MRe]UMRLP;*]^\]ZeVW]`N~YM`6MfNgaVx : JLKa]\VWRQq

∑

Xi

P (Xi|I) = 1,

[KNMZUMUKNM RfNTVWRP\^SMQZ];~~¯6PwRURUVY`N~M R?]\eMRP;X.ü MQZeM

P (Xi|I)OQP;ZeZUMsR-P;g6hNR UPUKNM

NZeP;`6]`NVY~VxeKa]\XVYg©UKaMRe]\eM

Xic;VY^;MgRPSTM`6];O?bdc;ZeP;fNg6h¢VYg/IP;ZeT]\eVPSg

I.JLKNVYR

`a]SRVWOQ];~~YxRe]_xdR¯eKa]\eKNM^\]ZeVY];`N~M Tf6R=`-MVg¤PSgNMP*V?RL6PwRURUV`a~MRe]\eMR :JLKNM RMsOrP;g6hZUfa~MVYRbdgNP\[gj];RUKNMó-÷ 3c6 øaõQö÷ø 5 ô>I¯[KNVWO?K¢Re]UMRUKaMNZeP;`6]`NVY~Vx¥P

[nPUKNVYgNcwR`-MQVYgNcUZefNMSqP (X,Y |I), VYRc;VY^;MgË`dxÍUKNMNZUPS`a]`aV~YVxÍPUKaM1îaZ?R`-MQVYgNceZUfNMSq

P (X |I), UVYTXMRUKNM§NZUPS`a]`aV~YVxP,eKNM§RUMOQP;gahXVWReZUfaMal;ñCB_ô H öCn 8 ööbnNô®´÷?ò?ö*ñWò öØ÷øaô>IP (Y |X, I). JLKa]=VYR

P (X,Y |I) = P (X |I)P (Y |X, I)= P (Y |I)P (X |Y, I),[KNMZUM eKNMRMsOrPSgahZeMQ~W]\eVPSgIPS~~YP\[=RnIZUPSTRUxwTXTXMreZUx :

|¥ZUVeMQgVYgeKNVYRF[L]_x;qUKNMnNZePdhNfaO8ZUfN~YMga]UfNZ?]~Y~Yx§~YM]ShNRVYgwUP[K6]\FVWRFbwgaP\[g];R4oL]_xSMR «JLKNMP;ZeMQT¤Î

P (X |Y, I) =P (Y |X, I)P (X |I)

P (Y |I) .

Rfa];~~YxUKaVYRXVWRX[ZUVeMQg©RPUK6]\XZeMQaZUMsRMgS?ReKNM¤a]Z?]TXMQUMQZ?RP§RUP;TXM¤TXP/h/MQ~=];gah

YZeMQNZeMRUMQgweRXRUP;TXMhN]\?]¥P;ZXP;`aRUMQZe^\]\eVPSg :

P (X |I) VWROQ];~~YMh¢UKNM óC÷ñ 3 ÷¥h/VYRUZeVY`Nf/UVYP;gIPSZeKNM6]Z?]TXMreMQZX,[KaV~YM

P (X |Y, I) VYR bdgNP\[g]SRVeRó 3 òöìôQ÷ñ 3 ÷h/VYRUZeVY`Nf/UVYP;g<JLKNMNZeP;`6]`NVY~Vx¥P=eKNMhN]\?]1c;VY^;Mg¢RUP;TXMTXP/h/M~ØqP (Y |X, I), VWRbdgNP\[gË];RV?R 5 ñ/:;ô 5 ñ/n 3c3c6 <² VYga]~Y~xXeKNMgNP;ZeT]~YVzVgac¡];OrUPSZnVYgUKNMh/MgNP;TXVYga]\eP;ZsqSeKNMNZeP;`a];`NV~YVxP;8hN]e]

P (Y |I),VWRRUP;TXMQUVYTMsRO]~Y~MshUKaMô=Bñ 6 ô H õ?ô<ÍJLKNVWRVWRIP;fNgah¢`dx¥T];ZUcSVga];~VYzQVYgNceKNM gwfaTMZe]UPSZP\^SMQZ¯UKNMµ -PSReRVY`N~YMUKNMP;ZeVMsRQq

P (Y |I) =∑

Xi

P (Y |Xi, I)P (Xi|I).|¥MP;3UMgh/Ms]~8[VeKOrPSgwUVYgwfaP;faR§^\]ZeVY];`N~MsRQq-[KNMZUMUKNMsRMNZUPS`a]`aV~YVeVMsR§];ZUM[ZUVeMQgVg/ëRUMs];h¤VYgUMZUTR=P;]NZeP;`a];`NVY~Vx hNVYRUZeV`af/UVYP;gIfagaOeVPSgqCM : c : UKaMNZeP;`6]`NVY~VxeKa]\§UKNM^\]ZeVW]`N~YMVWRVYg¤RPSTM§Ze];gNc;M OQ];g`6M[ZeVUUMg];R

P ([X,X + dX ]|I) = P (X |I)dX. Ê geKNVYRO];RUM;qdT]Zec;VYga]~YVzs]\eVPSgaRn~VYb;M UKaMP;gNM]`-P\^;MOQ];g`-M[ZeVUUMQg]SR

P (Y, I) =

∫dXP (Y |X, I)P (X |I).

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];gah RUfN` MsOUVY^;MSq§];gaheKNMQx©eMQgah¸UPÍIPdOQfaR¤NZeVT];ZUVY~x©PSgUKNM~YVbSMQ~YVKaPwP/h : oL]_xSMRUVY];gaROQP;fNgwUMZUK6]\=IZeMwfNMgSeVYReR§]ZeMfaRUVgNcaZUVYP;Z?RVYTa~VWOrVU~Yx;q-M : c : ]®6]NZeVPSZqa];gah UKaMQx¤]ZeM faR=T]bdVYgNcUKNMVZ=]SRURUfNTX/UVYP;g6RLMr/N~YVYOQV :

árx âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/éÊ g1M=.-MsOq,oL]_x;MsRVW]g6R=]gah IZeMwfNMgwUVWR?R§];ZUM]SRbdVYgNch/V/.-MZUMgw wfNMRUVYP;g6R : ­ oL]_x;MsRVW]g]SRb/Rqª KNP\[~VYb;M~x§VYRF]c;VY^;Mg 6]Z?]TXMreMQZ^\]~YfNMcSV^SMQg UKaMhN]\?]9 « qs[KaV~YM]LIZeMwfNMgSeVYRF]SRb/Rq

ª KNP\[¶NZeP;`6]`N~YMVWRUKNMhN]e]aqc;VY^;MgÍOrMZ?]VYg¢a];Ze];TXMrUMZeR¬ «Ê gËOQMQZUe];VgËOrVYZeOQfNTR?]gaOQMRqeKNMRUM]NNZePS]SO?KNMRcSV^SMeKNM1Re]TXM¤ZUMsRfN~eRqna]ZUUVWOrfN~W]Ze~YxV UKNM1ha]\e]]Ka]gahËVWR^;MZUxVYg/IP;ZeT]\eV^SM :9 : N:A)Ç_¾MaÇ\ÈSÉ VYgaOQMUKaM`NVYc;cSMRh/V/.-MZUMgaOrM`6MQ[¯MMQgeKNM];NNZePS];O?KaMRVWReKNMQVYZeZUMs]\ëTXMQgw PNZeVPSZeRqaV§VWRf6RMQIfN~4ePh/VWRUOQfaReRUKaMRUMKNMZUM : ³§gNM « R=aZUVYP;Z?R=]ZeMTXM];gw=UPZUMQ®aMOr];~~nNZeMQ^dVPSfaRVYg/IPSZUT]\eVPSgqMrNOQ~fahNVgNcUKNM¤hN]e]]Ka];gah :Ê gjNKdx/RUVYO]~LRVUfa]UVYP;gaRq[nMcSMQgNMZe];~~YxKa]_^;MX]UKNMP;ZeMreVYO]~*Mr/-MO?]\eVPSg¥PSZRPSTMXP;UKNMZ[¯Ms]b;MZ ha]\e] [KNVWO?KRKaP;fN~Whh/MQUMZUTXVYgNM UKNMRUKa]-MP*P;gNMsRLNZeVPSZeR :Ê gXUKaM]`aRUMQg6OrMLPRUfaO?KXP;`aRUMQZe^\]\UVYP;g6R8];gahUKNMP;ZeMreVYO]~NMQd-MOre]UVYP;gaRq;PSgNMVYRIP;Z?OrMheP

T]bSMRUP;TXMO?KNPSVYOQM=PNZeVYP;Zsq;c;MgNMQZ?]~Y~Yx`a]SRMshXP;geKNM§RUxdTXTMQUZexXPCUKNM NZeP;`N~YMQT : ³3UMgeKNMQZeM]ZeMRUP;TXMOrPSgaReZe];Vgw?RL[KNVYO?KUKaMNZUPS`a]`aV~YVxhNVYRUZeV`af/UVYP;g¤Tf6R=PS`6Mx :Ê g/IPSZUT]\eVPSg¥UKNMP;ZexRfac;c;MsR?R]¤[n]_x1UPO?KNPdPwRM]¤NZeVPSZ`a];RUMhP;geKNMNZeVg6OrVYN~MP

m1]/VTfNT ´gwUZeP;dx : ³§gNMXO?KNPdPSRUMR ]aZUVYP;Z [KNVWO?KT]\/VYTVYzQMsR§UKNMMQgwUZeP;dx¤PUKNMh/VWRëeZUVY`Nf/eVPSgq/[KNMQZeM UKNMMgwUZeP;dxVWRh/MrîagaMh¤];R

S = −∫dxp(x) log

(p(x)

m(x)

),

[KNMZUMm(x)

VWRLeKNMTXM];RUfNZeM :­ gdfNT`-MQZP;8-PSReRUV`N~YMh/VWReZUVY`Nf/eVPSgIfagaOeVPSgaRT]_x`6MO?KaPSRUMQgΕ P (X) ∝ constant ëeKNM®6]=NZUVYP;Z• P (X) ∝ 1/X

ëUKNMKJSM>.CZUVYMRLaZUVYP;Z• P (X) ∝ e−(X−X)2/2σ2 ëeKNMï]f6RURUVY];gNZeVYP;Z• P (N) = µN

N ! e−µ ëlUKaM y*PSVYReRPSgaZUVYP;ZnIP;Z=hNVYReOrZeMreM ^\];ZUVW]`N~YMR

JLKNM¯®6]*aZUVYP;Z8VYR*f6Rfa];~~YxfaRUMhIP;Z*~P/O]\UVYP;ga]Z?]TXMQUMQZ?Rqs[KaMQZeM¯[nMLKa]_^;M¯gNP§VYg/IPSZUT]\eVPSgePNZUMQIMQZPSgNM1^\]~YfNMP\^SMQZ]gaPUKaMQZ : JLKNMJSM>.CZeVMsRNZeVPSZVYR`a];RUVYO]~Y~xÍfNgNVIP;ZeTûVYg©UKNM~YP;cw]ZeVeKNTPeKNM¤a]Z?]TXMQUMQZX]g6h¢VWRfaRMshjIPSZRUO]~YMa];Ze];TXMrUMZeRq[KNMZUM [nMKa]_^SMgaPVYg/IP;ZeT]\eVPSg];R¯UPUKaM§P\^SMQZ¯];~~,ReOQ];~M PeKNM a]Z?]TXMreMQZ : onP;UKeKNM ®6]\]g6hUKaMaJSM>.CZUVYMRNZeVYP;Z?R]ZeM=VYTaZUPS6MZqd];RUKNMxh/V^SMQZec;M=[KaMQgVYgwUMQcSZe]UMshXP\^;MQZUKNMVZn6PwRURUVY`N~M§Ze];gNc;MsR : ­ R]ZeMRUfN~sqNP;gaMgNMQMshNRLUPh/MQîagNMfaN6MZ]g6h~YP\[nMQZL`6PSfNgahNRIP;ZeKNMRUMh/VWReZUVY`Nf/eVPSgaR[KNVWO?KZeMwfNVYZUMsRLRPSTMNKdx/RUVYO]~CVgaNf/ :JLKNMï]f6RURUVY];g];gahy*P;VWReRPSgNVY];gNZeVPSZeRn]ZeM NZUPS6MZ]gahVgwUMc;Z?]`N~YM;qd`NfNneKNMQxZUMsSfaVZeM

MQweZe]VgaNf/ : ´VeKNMQZ=]^_];ZUVW]g6OrMσ2 VYgeKNMOQ];RUMP;F]Xï]faReRVW]gh/VYRUZeVY`Nf/UVYP;g4q/P;ZL]TXM];gP/OQOQfNa]UVYP;g

µ = 〈N〉 VYgUKNMy8PSVYReRUP;g1OQ]SRM : JLKNM®6]NZeVPSZ=T]_x`-M?]bSMQg];R=UKNM~VYTXVP;4]ï]f6RURUVY];gqw[VeKUKaM^\]ZeVY];gaOrMc;P;VYgNcUPVYg/îagaVx : JLKaMï];faRURUVW]gVWR¯];~YRUPeKNM~YVTXVP]y*P;VWRURUP;gaVY];g];R

µc;MQeRL^SMQZex~Y];ZUcSM :

9 : _ : $ »NºFÈsÈs¾¡»N¼¶-F¾ØÈQ½Ç_¾#8ºF½¾M6¼8ÈSÉ=Ê g¸Refah/xdVYgNc~VYb;M~VYKNPdP/hÍIfNgaOeVPSgaRqVVWRf6RMQIfN~=ePfNg6h/MQZ?R?]gah¥eKNMTfN~UVY^_];ZUVW]\eMï]f6RURUVY];gh/VYRUZeVY`Nf/UVYP;g4q*RUVYgaOrMP;`aRUMQZe^\]\UVYP;g6]~MQZeZeP;Z?R]ZeMP;3UMQge];b;MQg1UP `6Mï]faReRUVY];g :Ê g¥]ShNh/VUVYP;gqCgNPSg/ë ï];faReRVW]ghNVYRUZeV`af/UVYP;gaR OQ];gP;3UMQg`6M];NNZeP_dVYT]\eMhgaM]ZneKNMQVYZ=-M]b/RL`dx]ï]f6RURUVY];g ];NNZeP_/VT]\eVPSg :JLKNMTXPwRcSMQgNMZe];~TfN~UVY^\]ZeVY]UMï];faRURUVW]gh/VWReZUVY`Nf/eVPSgIP;Z]

nh/VYTXMQgaRUVPSga]~¯hN]\?]

^SMOeP;ZDT]_x`-M[ZUVeMQg¤];R

P (D|X, I) = (2π)−n2 | det C|− 1

2 exp

[−1

2(D−D)T C−1(D−D)

].

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê á;~ü MQZeM

DVWReKNM=TXM];g^_];~faM=PCUKaM^;MsOeP;Z 〈D〉, [KNVY~M C VYRV?R¯OQP\^_];ZUVW]g6OrMLT]\UZeVCq Cij =

〈(Di − Di)(Dj − Dj)〉.JLKaM`NZ?];O?bSMreRhNMQgNP;UMT]Zec;VYga];~VYz]UVYP;g¥P\^;MQZUKNMNZUPS`a]`aV~YVx

IfNg6OUVYP;g4qV : M : 〈F (D)〉 ≡∫dnDF (D)P (D|X, I). Ê g¥c;MgNMQZ?]~lqCMQVUKNMZUKNMXTXM];g^\]~YfNMP;4UKaMhN]\?]PSZV?R=OrP\^\];ZUVW]gaOQMT]\eZUVT]_x`-M]IfNgaOeVPSg¤P8UKNM UKaMQP;Zex :

´^SMQgjVLUKNM hN]\?]1]ZeMï]faReRUVY];gjh/VYRUZeVY`Nf/UMsh,q8UKNMVTXN~YVYMha];Ze];TXMrUMZh/VWReZUVY`Nf/eVPSg[VY~~-gNP;¯`-MfNgN~YMReRUKNM NZeMh/VWOeVPSgaR];ZUM=~YVYgNM];ZVYgUKaM=UKaMQP;Zexa]Z?]TXMreMQZ?R :8ü P\[nMQ^SMQZsqIP;ZeKNMTXP;TXMQgw~MQeR=]SRURUfNTXM§UKa]UKNMeKNMQPSZUx6]Z?]TXMreMQZ?Rn];ZUMï]f6RURUVY];g hNVYRUZeV`af/UMsh,Î

P (X|D, I) = (2π)−m2 | det T |− 1

2 exp

[−1

2(X−X)T T−1(X−X)

].

JLKNVWRNZeP;`6]`NVY~Vxj-M];bdRIP;ZeKNM1a];Ze];TMQUMZeRX,[KNVWO?KVYRXeKNM1T]\/VYTfNTú~VYb;M~VYKNPdP/h

^\]~YfNM :JLKNM1fagaOrMZ?]VYgSx¢Vg¸]a];Ze];TXMrUMZh/MQ-MQg6hNRPSgË[KaMrUKaMQZPSgNM1OrPSgaRVWh/MZeReKNM1PeKNMQZ

a];Ze];TXMrUMZeR*]SR*îN/MhXPSZV,P;gaMO?KNPdPSRUMR8ePTX];ZUcSVg6]~YVzMnP\^SMQZ*eKNMQT :*Ê ,[¯MKNPS~YheKNM=PeKNMQZa];Ze];TXMrUMZeR¯îN/Msh,q/UKaMQg¤UKaM^_];ZUVW]g6OrMP]Xa]Z?]TXMQUMQZ

XiVWRc;VY^;Mg`dx

σ2i = 1/(T−1)ii.³§gUKNMPeKNMQZKa];gah,q,V¯P;gaMh/MsOrVWh/MR§UP T]Zec;VYga]~YVYzQMP\^SMQZUKNMXPeKNMQZa];Ze];TXMrUMZeRq6UKNMg

eKNM^\]ZeVW]gaOQMc;ZeP\[=Rh/faMePUKaM OQP\^_];ZUVW]g6OrM=`-Mr[nMQMgUKNMhN]e]-P;VYgweR :*Ê gUKaMï];faRURUVW]gO];RUM;q-UKNVWRT]Zec;VYga]~YVYz]\eVPSgVYR-MQZUIP;ZeTXMhRVYTXN~x`dxOQP;TXN~YMreVgNcUKaMXRewfa]ZeM;qCxdVYMQ~Wh/Vgacσ2

i = Tii.JLKNMRUM]ZeMXZUM~Y]UMsh`dx

(T−1)ii = 1/(Tii − Tij(Tjk)−1Tki),[KNMQZeM

j, kZUfag

P\^SMQZL]~Y~,VgahNVYOQMR`-MRUVYhNMRi.JLKNMMQZeZeP;Z?R¯O]g P;gN~YxVYgaOrZeM]SRM [KNMQg eKNMTX];ZUcSVg6]~YVzs]\UVYP;gVYR

-MQZUIP;ZeTXMh :

9 :v/: 6¼R $ »Nº8ÈsȾػN¼·-8¾¡È½QÇ\¾b*ºF½¾#-¼8È;É=Ê g¢c;MQgaMQZ?]~lq4eKNM-PSRUMZUVYP;Zh/VYRUZeVY`Nf/UVYP;g6R[VY~~~YVbSMQ~Yx1`-MgNP;g/ëï]f6RURUVY];g :XÊ g¥UKNVWROQ]SRMSqVT]_x1`6MhNV¨Orfa~`-PeK¥UP¤îagaheKNM6Ms]bPeKNM=~YVbSMQ~YVKNPdP/h]g6hXUPT];ZUcSVga];~VYzQMLePîagahVgahNV^dVWh/fa]~-h/VWReZUVY`Nf/eVPSgaR : JLKNMsRMP;-MQZ?]\UVYP;g6RfaRUfa];~~YxTfaR`-M=-MQZUIP;ZeTXMhgdfNTXMQZeVYO]~Y~xSq;MQVUKaMQZh/VYZeMOe~xP;Z`dxmP;gwUM§kn];ZU~YPTXMrUKaPdhaR :³§gNM6PSNfN~W]Z§m1P;gwUMkn];ZU~YPTXMreKNP/h,q-O]~Y~Msh UKNMm1]Zeb;P\^¤k¯Ka]VYg1TXMreKNP/h,q-Re]TXN~YMR=UKNM~YVbSMQ~YVKaPwP/hRUfNZU¡];OrM^wVW]1]Z?]g6h/P;T [n];~b : µ§MQ-MQgahNVgNcP;geKNMRUKa];6MP;UKNM~VYb;M~VYKNPdP/hRUfNZU¡];OrMSq\RfaO?KTXMrUKaPdhaR8O]gRU]_^SM¯OrPSgaRUVYh/MZe];`N~YM¯OrPSTXNf/e]UVYP;ga];~;eVTXMOrPSTXa]ZeMheP hNVZeMOrVYgwUMQcSZe]UVYP;g :Ê nO]gRUVY~Y~a`-M§faRUMrIfa~aePT]b;M§]ï]faReRUVY];g]NNZeP_/VT]UVYP;gUPeKNM~YVYb;MQ~YVYKNPdPdhRUfNZU¡];OQM

`dxMQd6]gah/VYgNcUKaM~PScP;FeKNM~VYb;M~VYKNPdP/hVYg]XJF]_xd~YP;ZRMZUVYMRq

lnL(X + δX) = lnL(X) +∑

i

∂ lnL(X)

∂XiδXi +

1

2

∑

ij

∂2 lnL(X)

∂Xi∂XjδXiδXj + · · ·

k¯f/UUVYgNcP.Ë]\UKNMXeKNVYZehUMZUT~Ms];hNR eP1]ï]faReRVW]g]aNZUP_/VYT]\UVYP;g : JLKNMXî6ZeRUMZUTVWRM>.CMOrUVY^;M~x − 1

2χ2 IP;ZeKNMTXP/h/MQ~ :¯Ê [nMMQ^\]~Yfa]UM]`-P;fN=UKNM-M];bP;FeKNM~YVbSMQ~YVKaPwP/h,q

eKNMQgUKNMn~VYgNMs]Z8UMZUTR8]ZeMzQMZUP :F² VYga]~Y~xSq −∂2 ln L(X)∂Xi∂Xj

VYR*RUP;TXMreVTXMR8OQ];~~YMhUKNMLOrfaZU^\]\efNZeMT]\eZUV IP;Z=eKNM6]Z?]TXMreMQZ?RP;ZUKNM ü MReRUVY];g :² ~Y]§h/VYZUMsOeVPSgaR=P;UKNMOrfNZe^\]\UfaZUMT]UZeVOQP;ZeZUMsR-P;gahUPa];Ze];TMQUMZLhNMQc;MgNMQZ?];OQVMsR :² Vg6]~Y~xSqS~YMrf6Rh/MQîagNM=eKNM ² VWRKNMZT]\eZUV[KaVYO?KVWRfaRUMrIfa~aIPSZT]bdVgacNZeMh/VWOeVPSgaR*IP;Z

[Ka] ]NZeP;-PSRUMh¤Mr/-MQZeVTXMgS[VY~~8UM~~*faR];`6PSf/§eKNMfNgah/MZU~YxdVgNcUKNMP;Zex : JLKNM ² VYRUKNMQZT]\eZUVVWRRVYTXN~YxUKaMMr/6MsO?]\UVYP;g PFUKNMOQfNZe^_]UfNZeMT]\UZeVCq

Fij(X) = −⟨∂2 lnL(X)

∂Xi∂Xj

⟩,

; âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/é[KNMZUM*eKNMMQ/6MsOe]UVYP;gh/MgNPeMRF]g]_^;MZe];c;VYgNcLP\^;MZhN]\?]=OrPSgaRVWRUMQgw4[VUK]c;VY^;Mg TXP/h/M~ :JLKdfaRqeKNM ² VYRUKNMQZTX]UZeVVYRh/MrîagaMhIP;Z]6]ZUUVWOrfN~W]Z*eKNMQPSZUx]gahVWRVg6h/MQ-MQgahNMQgwP,]gdx]SOUf6]~hN]e] :JLKNM ² VYRUKNMQZ¤T]\eZUVVYR faRUMrIfa~ `-MO]faRUM¥V]~Y~P\[=RfaRUPËN~W];OrM]j~YP\[nMQZ `6PSfNgah¸PSg

eKNMh/MQZeV^SMhÍa]Z?]TXMQUMQZXMZUZeP;Z?RZeMRUfN~UVYgNc¥IZUPSTû]gËMr/-MQZeVTXMgS :Ê gÍ6]ZUUVWOrfN~W]ZsqV];~~eKNMa]Z?]TXMQUMQZ?R ]ZeMîadMsh¥`Nf/P;gaM;qeKNMQgVeRMZUZeP;Z VWR`6PSfNgah/Msh¥`wx

σ2i ≥ 1/Fii.

Ê L[nMT]Zec;VYga];~VYzQM¯P\^;MQZ8eKNMPeKNMQZ*a];Ze];TMQUMZeRq

σ2i ≥ (F−1)ii.

JLKNMRUML`6PSfNgahNR];ZUMnUKaMLRe]TXM];R[Ka][¯MIP;fNgahMs]Ze~VYMQZCIPSZ,UKaM¯ï]faReRUVY];gO];RUM;qs[KNMQZeM*UKaM`6PSfNgahVWR4Re]\efNZe]UMsh :ü P\[nMQ^SMQZsqVTfaR`6MZeM];~VYzQMshUK6]\LUKNM ² VWRKNMZT]\UZeV]NNZePS]SO?KP;ga~xcSV^SMRn~YP\[¯MZ¯`-P;fNg6hNR :¬ M]~MQd-MQZeVYTMgweRFTfaR8h/Ms]~w[VeK-PSReRVY`N~YM¯RUx/ReMQT]\eVYOMQZeZUPSZeRqsRUPeKNM¯ZeMRUfN~UVYgNcMZUZeP;Z4`6PSfNgahNR[VY~~,Z?]ZeMQ~Yx];O?KaVM^;M UKNMNZeMhNVYOrUVYP;gaRLP;FeKNM ² VYRUKNMQZT]UZeV];NNZePS]SO?K :9 : uN: -ÅÆ*»/Ç\¾¡¼8Á ÅU-FÄ*P¡ÈSÉ |KNVY~MUKaM-M]bP;=UKNM ~YVbSMQ~YVKNPdP/hNZeP\^dVYhNMR]TMsO?Ka]\ëgNVWRUT¶UPO?KaPwPwRM`-Mr[nMQMg¤hNVm.CMQZeMQgw=TXP/h/MQ~WR[VUK¤UKaMRU];TM6]Z?]TXMreMQZeVzs]\UVYP;g4qwV=VWR=gNP;]Sh/Mwfa]UMIP;ZO?KNPdPwRVYgNc `6MQ[¯MMQgTXPdhNMQ~WR[KNVWO?K¥K6]_^;MXh/Vm.CMQZeMQgw^\]ZeVW]`N~YMRPSZMQ^;Mg¥hNV3ëIMZUMgSgwfaT`-MQZ?RP^\]ZeVW]`N~YMR :1² PSZUKaVYRq*P;gNMf6Rfa];~~Yx¥~PdP;b/R]\eKNMZ?]\eVP1P;6PwReMQZeVPSZNZeP;`6]`NVY~VUVYMR : fNN-PSRUMx;P;f Ka]_^SM[nPKdxd-PUKaMRUMRq ­ ];gah o§q/Ms];O?K[VeKeKNMQVYZP\[gRUMrP;4a]Z?]TXMQUMQZ?Ri

λA]gah

λBp];gahxSP;f[VWRKUPXh/VWRUOQP\^;MQZ¯[KNVWO?KVWR¯TXPwROrPSgaRVWRUMQgwn[VeK

eKNMNZeVPSZeRL]g6hPS`aRMZU^\]UVYP;gaR : k¯PSgaRUVYh/MZLUKNMZ?]\eVP

RAB =P (A|D, I)P (B|D, I) =

P (A|I)P (B|I) ×

P (D|A, I)P (D|B, I) .|KNMg

RABVWRnTfaO?K~Y];ZUcSMQZeKa]gP;gNMSqdKwxd-PeKNMRUVYR ­ VYR¡]_^;PSZUMsh,qS[KaV~YM§Kdxd-PUKaMRUVYRno©VYRNZeMrIMZUZeMh VRAB

VWR§RUT]~Y~ :§Ê UKNMZe]UVYPVWR=PPSZehNMQZ 9 qaeKNMQgP;gNMOQ]gagNP TX];b;M]Or~YM];ZNZeMrIMZUMgaOrM : JLKaMîaZ?R=eMQZeTPSg¤UKNMZUVYc;Kw=ZeMQNZeMRUMQgweRUKNMZ?]\eVPP;P;gNMsR=NZeVPSZ=`-MQ~YVMQ¡R=VYgeKNM [¯PKdxw-PeKNMRUMRq/[KNVY~M eKNMRMsOrP;g6hZeMQNZeMRUMQgweR¯eKNMZe]UVYPP;4UKaM M^dVYh/MgaOrMsRnIP;ZLM]SO?KP;6eKNM=Kdxd6P;UKNMsRMsR :*¬ MO]~Y~NeKa]\

P (D|A, I) VWRUKaM=~VYb;M~VYKNPdP/hPCUKNM§hN]\?]Nq;T];ZUcSVga];~VYzQMshP\^SMQZ¯UKNM-PSReRVY`N~YM^_];~faMRP;4UKaMa]Z?]TXMreMQZ?RλA.Ê VWRKNM~/Ifa~UPRMMXUKNVWRVYgj];OeVPSgP;g]¤^;MZUxRUVYTa~MMQ/];TXN~M : ­ RURUfNTXMUK6]\x;PSfTXM]SRfaZUMRUP;TXMhN]e]]g6h îagah

D = D ± σ. ¸ PSf[VYRUK UPOQP;TXa];ZUM [nPKdxw-PeKNMRUMRÎ ­[KNVWO?K1NZeMhNVYOreRUKa]D = 0,

]g6h1o[KNVYO?KNZUMsh/VWOeRUK6]\D = λ,

[KNMQZeMλmin > λ >

λmax.ü xw-PeKNMRUVWR ­ Ka];RLgaP6]Z?]TXMreMQZ?RQq/RUPeKNMMQ^dVWh/MQg6OrMVYRRVYTXN~x

P (D|A, I) = (2π)−1

2 σ−1e−D2/2σ2

,[KNVY~YMUKNMMQ^dVYhNMQgaOQMIP;Z§Kwxd-PeKNMRUVYRo ZeMwfNVYZUMsR=]gVYgwUMc;Z?]\UVYP;g¤P\^SMQZ-PSReRVY`N~YM^\]~YfNMRPλ :

P (D|B, I) =

∫ λmax

λmin

dλP (λ|B, I)(2π)−1

2σ−1e−(λ−D)2/2σ2

.

4Mrf6R]SRURUfNTXMUK6]\ UKNMNZUVYP;ZIP;ZλVYR®6]\ P\^;MZUKNM¥Z?]gNcSM;q=RUPUKa]

P (λ|B.I) =(λmax − λmin)−1.

² fNZeKNMQZsqV[¯M];ReRUfNTXMeKa]\XeKNMP;`aRUMQZe^\]\eVPSg¢VWRX[¯M~~[VUKNVYgËUKNM];~~YP\[¯Msh Ze];gNc;M];gah UKa]

σ λmax − λmin,UKaMQg1[nMO]g1h/PeKNMVYgwUMc;Z?]\UVYP;g¤MQ/]SOe~x

ePî6gahP (D|B, I) = P (λ = D|B, I) = (λmax − λmin)−1.JLKdfaR[nM îagah

RAB =P (A|I)P (B|I) ×

λmax − λmin

(2π)1

2 σ× e−D

2/2σ2

.

­ cS]VYgq¯UKNM¤î6ZeR¡];OrUP;ZVWRXeKNM1Z?]\UVYPP UKNMaZUVYP;Z?RQq¯[KNVY~MeKNMMr/-P;gNMgwUVW]~L¡];OrUPSZVYRM=.-MsOUVY^;M~xUKNMZe]UVYPP;*UKNM`-MR=îN§~VYb;M~VYKNPdP/hIPSZ ü xd-PeKNMRUVYR ­ UPXeKa]\=IPSZUKaM`6MsRîNTXP/h/MQ~P ü xd6P;UKNMsRVWRoíi¡V : M :λ = D.)

JLKNM1RUMOQP;gahj¡];OeP;ZVWRRPSTXMrUVYTXMRO]~Y~Msh

é,ã,æ6å ã,éæCå1çaãLèWçLèWçaâaã,å¤ä 6ç,âaãCéCê ?à³ OOQ]T « R¡]SOUPSZq[KaVYO?KjM=.-MsOeV^SMQ~Yx¥-MQg6]~YVzMR ü xd6P;UKNMsRVWRo`6MsOQ]f6RMVXKa];R]1~W]Zec;MZa];Ze];TXMrUMZ¯RUa];OQM : ï V^SMQg[¯PTP/h/M~YRn[KNVWO?Kîa¯eKNM ha]\e]Mwfa];~~YxX[¯M~~lqwUKNVWR¯¡];OrUP;Zn[VY~~eMQ~Y~6faRUPNZeMrIMZUKaM RUVTXN~YMQZnTXP/h/MQ~lqwUKNM§P;gNM§[VeK]RUT]~Y~MZa];Ze];TXMrUMZRUa];OQM :*Ê gUKaVYRO];RUM;qwVFUKNM VgaVeVY];~-6]Z?]TXMreMQZLRa]SOrMIP;Z ü xd6P;UKNMsRVWRLo©VYRLRUT]~Y~8i

λmax − λmin ∼ 20σ),eKNMQg ü xd6P;UKNMsRVWR ­ VWRNZeMrIMZUZeMhjP;gN~YxV D < 2σ.ü P\[nMQ^SMQZsq4eKNVYRXOQf/UPN.©OQP;fN~Whc;ZeP\[

RUVcSgNVî6OQ];gSe~xV8UKNMo a]Z?]TXMreMQZRUa];OQM [nMQZeM VgaOQZUMs];RUMhMr/6PSgNMQgweVY];~~Yx :þ $dþNA,þN>4G-þ

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8 Nþa>D4H9 8;:5< þ $dE' þ '9d@,HIB4>XB$ ³QNþ6G6H'%LE>FGa@,HIBF> Æ8Ã8ÄwÇ_¾M;» PLÃ8»/Ç\ÅU-¼F¾MÇ\Ä*P¡»/½s¾#-¼FÈ Î• µMQîagNVUVYP;g

Ylm(θ, φ) =

(2l+ 1

4π

(l −m)!

(l +m)!

)1/2

Pml (cos θ)eimφ

• ³§ZeKNP;cSP;ga];~Vxë∫Y`m(n)Y ∗

`′m′(n)dΩn = δ``′δmm′

• k¯P;TXN~YMreMQgNMsRURnë∑

`m

Y`m(n)Y ∗

`m(n′) = δ(φ − φ′)δ(cos θ − cos θ′)

• y~W]gNM[L]_^;M Mr/a];gaRVYP;gë

eik·r = 4π∑

`m

i`j`(kr)Y`m(k)Y ∗

`m(r) =∑

`

(2`+ 1)i`j`(kr)P`(cos θ),

[KNMQZeMj`(x)

]ZeM UKNMRUNKNMZUVWOQ];~ConMsRURUMQ~CIfNg6OUVYP;g6R := ÄwÁ-Äd¼-FÇ\ĤÇ\Ä"PØ»/½¾M6¼8È iIgaPUM

Pl(x) = P 0l (x)

prΕ Mc;MQg6h/ZUMMsSf6]\UVYP;g

(1− x2)d2

dx2Pm

l (x) − 2xd

dxPm

l (x) +

[l(l+ 1)− m2

1− x2

]Pm

l (x) = 0

eá âaã,äNå/âwæçNâ-èWæaæ6å/é-ê,å/é

• ³§ZeKNP;cSP;ga];~Vx∫ 1

−1

Pml (x)Pm

l′ (x)dx =2

2l + 1

(l +m)!

(l −m)!δll′

• ¬ P/h/ZeVcSfNMR « IP;ZeTfN~W]

Pml (x) =

(−1)m

2ll!(x2 − 1)m/2 d

l+m

dxl+m(x2 − 1)l

• µMZUVY^\]\eV^SM ZeMQ~W]\eVPSg

(1− x2)d

dxPm

l (x) = (l −m+ 1)Pml (x) − (l + 1)xPm

l (x)

• ¬ MOQfNZeZUMgaOrMZeMQ~W]\UVYP;g(2l + 1)xPm

l (x) = (l −m+ 1)Pml+1(x) + (l +m)Pm

l−1(x)

• ¬ MQ~W]\eVPSgePRaKNMQZeVYO]~CKa];ZUTXP;gaVYOR(2l+ 1)Pl(k · r) = 4π

∑

m

Ylm(k)Y ∗

lm(r)

! ÄdÈÈsÄ*PÇ\Ä*P¡»/½s¾#-¼FÈ iIf6RMQIfN~IPSZ_ë hZeMQ~W]\eVPSgaReprΕ onMReRM~,Mwfa]\eVPSg

d2

dx2Jm(x) +

1

x

d

dxJm(x) +

(1− m2

x2

)Jm(x) = 0

• ³§ZeKNP;cSP;ga];~Vx ∫kdkJm(kρ)Jm(kρ′) =

1

ρδ(ρ− ρ′)

• y~W]gNM[L]_^;M Mr/a];gaRVYP;geikρ cos φ =

∑

m

imeimφJm(kρ)

• Ê gwUMQcSZe];~CZUMNZUMsRMgwe]\eVPSg

Jm(x) =1

2πim

∫ 2π

0

dφeix cos φ−imφ

• µMZUVY^\]\eV^SM ZeMQ~W]\eVPSgd

dxJm(x) =

1

2[Jm−1(x)− Jm+1(x)]

• ¬ MOQfNZeZUMgaOrMZeMQ~W]\UVYP;g

Jm+1(x) =2m

xJm(x)− Jm−1(x)

• NKNMZUVWOQ];~onMReRM~CIfNgaOeVPSgaR

jm(x) ≡( π

2x

)1/2

Jm+ 1

2

(x)

8 aþN>D4H9d : ³,G'N4A?>@þaG6@BFA?>;±þN>sB4ABAþ6G-Bg]6BHW@,HIB4>