V01ume 180, num6er 3 PHY51C5 LE77ER5 8 13 N0vem6er 1986

C 0 5 M 1 C A X 1 0 N 5 F R 0 M C 0 5 M 1 C 5 7 R 1 N 6 5 "

R.L. DAV15

5tanf0rd L1near Acce1erat0r center, 5tanf0rd un1ver51ty, 5tanf0rd, cA 94305, u5A

Rece1ved 12 May 1986; rev15ed manu5cr1pt rece1ved 22 Au9u5t 1986

7he p05516111ty 0f a new c0n5tra1nt 0n the Pecce1-Qu1nn 5ymmetry 6reak1n9 5ca1e, ar151n9 fr0m the decay 0f c05m1c ax10n 5tr1n95, 15 d15cu55ed.

1. F0rew0rd. 7hat a 5p0ntane0u51y 6r0ken, an0ma10u5, 9106a1 UpQ(1) 5ymmetry can 6e 1ntr0duced 1nt0 the 5tandard m0de1 t0 501ve the 5tr0n9 CP pr061em 15 a we11 kn0wn fact [ 1]. 7h15 m0de1 ha5 a p5eud0-901d5t0ne 6050n, the ax10n [2,3]. 7he ax10n 15 phen0men01091ca11y accepta61e 1f the 5ymmetry 6reak1n9 5ca1e 15 n0t t00 10w (fa > 4 × 10 9 6eV) ,1, f0r 5tar5 w0u1d 6urn up t00 fa5t [14,5,6] ; and 1f1t 15 n0t t00 h19h (fa < 4 X 1012 6eV), f0r the ax10n5 w0u1d d0m1nate the un1ver5e [ 7 - 1 0 ] . F0r deta115 0n the5e matter5 the 1ntere5ted reader 15 referred t0 the 11terature.

Here 1 am c0ncerned w1th the upper 60und 0n the PQ 5ymmetry 6reak1n9 5ca1e. 7h15 60und wa5 der1ved 6y 100k1n9 at the 2er0 m0mentum m0de 0f the ax10n f1e1d a5 the un1ver5e pa55e5 thr0u9h the 4 u a r k - h a • 0 n pha5e tran51t10n. A60ve the pha5e tran51t10n the ax10n 15 ma551e55 and the 2er0 m0mentum m0de c0ntr16ute5 n0th1n9 t0 the ener9y den51ty, 50 the va1ue 0f the f1e1d can 6e any c0n5tant fr0m 0 t0 fa w1th0ut affect1n9 the phy51c5. Cau5a1- 1ty ar9ument5 1mp1y that 1t 15 0(fa ). At the pha5e tran51t10n 1n5tant0n phen0mena 91ve ma55 t0 the ax10n f1e1d and 1t 6e91n5 t0 under90 5pat1a11y h0m09ene0u5 c0herent 05c111at10n5 a60ut 2er0, w1th fre4uency m a. C0n51derat10n 0f the damp1n9 needed t0 prevent the a550c1ated ener9y fr0m d0m1nat1n9 the un1ver5e 91ve5 the c05m01091ca1 60und.

1n th15 1etter 1 w0u1d 11ke t0 d15cu55 0ther 50urce5 0f ax10n5 wh1ch m19ht c0ntr16ute t0 th15 60und, t1ae n0n-2er0 m0mentum m0de5.7he f1r5t 50urce 15 a therma1 d15tr16ut10n: the ax10n5 were 1n therma1 e4u1116r1um at the PQ 5ymmetry 6reak1n9, 6ut then were dec0up1ed fr0m the 5u65e4uent therma1 h15t0ry 0f the un1ver5e. 7h0u9h de- c0up1ed, the5e ax10n5 ma1nta1ned the ••therma1•• 5pectrum a5 the un1ver5e c001ed, w1th ••temperature•• 51mp1y re- 1ated t0 the temperature 0f the 5urr0und1n9 therma1 6ath a11 the way d0wn t0 the 4ua rk -hdra0n pha5e tran51t10n. 1n the ana1y515 de5cr16ed a60ve 1t wa5 a55umed that the5e ax10n5 9et red5h1fted away and are theref0re n0t c05m0- 1091ca11y 1mp0rtant. 1n 5ect10n 2 1 w111 5h0w exp11c1t1y that th15 a55umpt10n 15 c0rrect.

8ut ma1n1y 1 am 1ntere5ted 1n an0ther, h1thert0 unrec09n12ed 50urce 0f ax10n5 wh1ch c0u1d have c05m01091ca1 1mp0rt. 7he5e ar15e fr0m the 1nev1ta61e (6arr1n9 1nf1at10nary m0de15) appearance 0f ax10n 5tr1n95 after the PQ pha5e tran51t10n. 7he 5tr1n95 105e ener9y 6y em15510n 0f ax10n5 1nt0 a character15t1c ener9y 5pectrum. 0 f n0 519n1f1cance a60ve the 4ua rk -had r0n pha5e tran51t10n, the ener9y den51ty 15 enhanced when pa551n9 thr0u9h, and th15 c0ntr1- 6ute5 t0 the upper 60und 0n the PQ 5ca1e. 7h15 mater1a1 15 treated 1n 5ect10n 3, and c0mment5 0n the re1evance 0f th15 60und are 1n the c0nc1u510n. An 1mp0rtant techn1ca1 n0te 15 re5erved f0r the append1x.

W0rk 5upp0rted 6y the Department 0f Ener9y, c0ntract DE-AC03-765F00515. ~1 7h15 60und may 6e 10wered 6y a fact0r ~0.01 1n m0de15 where the ax10n d0e5 n0t c0up1e t0 e1ectr0n5. A150, 51nce the phy51c5

0f the 1nter10r5 0f 5tar5151mperfect1y under5t00d, 0ne mu5t take th15 60und w1th 50me 5kept1c15m. Pre5ent 065ervat10na111m1t5 0n the 501ar ax10n f1ux 1mp1y that fa > 107 6eV [4].

0370-2693/86/$ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

225

V01ume 180, num6er 3 PHY51C5 LE77ER5 8 13 N0vem6er 1986

2. Ener9y den51ty 0fdec0up1ed therma1 ax10n5.7he ener9y den51ty ju5t 6e10w the PQ 5ymmetry 6reak1n9 5ca1e ha5 a therma1 5pectrum

Pa 2rr2Je k / 7 - 1

50met1me after the 5ymmetry 6reak1n9 (10n9 6ef0re the 4uark--hadr0n pha5e tran51t10n) the ax10n5 dec0up1e fr0m a11 0ther matter and rad1at10n, and the ax10n f1e1d red5h1ft5 a5 the un1ver5e expand5 and c0015. At 10wer tempera- ture5 th15 ••therma1•• d15tr16ut10n 15 ma1nta1ned, 0n1y the effect1ve temperature 0f the ax10n f1e1d may 6e d1fferent fr0m the actua1 temperature 0f the un1ver5e. 7he ener9y den51ty 15

Pa = e k/7* -- 1 •

where 7* can 6e der1ved 1n an ad1a6at1ca11y expand1n9 un1ver5e,

7* = [c)~ (7)/c~ (7dec) 1 1/3 7.

c~ (7) 15 the num6er 0f 1nteract1n9 part1c1e de9ree5 0f freed0m w1th ma55e5 1e55 than 7. At the 4uark-hadr0n pha5e tran51t10n the ax10n ma55 5tart5 t0 turn 0n. H0w d0e5 th15 effect 0a • 1n the append1x 1 have 0ut11ned a WK8 ca1cu- 1at10n f0r the pr0per 1nte9ra1 1n5ert10n; the ener9y den51ty 15

, (t/7)m2a )1/2 k3 0k 0 a - 2 - - ~ t 2 J [ k ~ 2 + ~ - -- e k / ~ * - 1•

(2.1)

where ~ 15 r0u9h1y the t1me when the C0mpt0n wave1en9th 0f the ax10n c0me5 w1th1n the h0r120n. 1 ca11 the ax10n ma55 at that t1me N = 1/7". 7he va1ue 0f~" can 6e e5t1mated fr0m 1n5tant0n phy51c5: the c0rre5p0nd1n9 temperature 15 7 ~ 800 MeV - 1 6eV [7], and ~ "~ 10 -9 - 10 -8 eV.

W1th0ut perf0rm1n9 the 1nte9ra1 we can 5ee that the pha5e tran51t10n d0e5 n0t 519n1f1cant1y a1ter Pa• 7he 1n5er- t10n 1n e4. (2.1) 15 d1fferent fr0m 1 0n1y 1n the re910n k ~ m a ~ 10 -5 eV. 8ut 51nce the 4uark-hadr0n pha5e tran51- t10n 0ccur5 at 7 ~ 100 MeV, and 51nce f0r any p1au5161e va1ue 0f [c~(7)/c~(7dec)] 1/3 we mu5t have 7~>> ma, the 0ther term 1n the 1nte9rand 15 e55ent1a11y 2er0 1n that re910n. 1n 0ther w0rd5, the effect1ve temperature 0f the ax10n 6ath 15 h19h en0u9h that the ax10n5 rema1n re1at1v15t1c thr0u9h the pha5e tran51t10n. 51nce they 6ec0me n0n-re1a- t1v15t1c at near1y the 5ame t1me a5 0ther matter, and 6ecau5e the1r ma55 15 50 5ma11, the5e therma1 ax10n5 p05e n0 threat 0f d0m1nat1n9 the un1ver5e.

3. Ener9y den51ty 0fc05m1c 5tr1n9 ax10n5. At the PQ pha5e tran51t10n a rand0m netw0rk 0f c05m1c ax10n 5tr1n95 mater1a112e5. Den5e1y kn0tted, the 5tr1n95 w111 5tart t0 05c111ate under the1r 0wn ten510n after ax10n dec0up11n9, when fr1ct10na1 effect5 cea5e t0 matter. 1n a prev10u5 art1c1e [ 11 ] 1 have dem0n5trated that 5tr1n95 fr0m 6r0ken 9106a1 5ymmetr1e5105e the1r ener9y very eff1c1ent1y 6y rad1at10n 0f 601d5t0ne 6050n5. Here 1 w111 a55ume that 5tr1n9 exc1tat10n5 w111 6e damped 0ut 50 rap1d1y that 100p f0rmat10n and decay 15 n0t an 1mp0rtant effect. 51nce 05c111at10n5 can 0n1y 0ccur when a k1nk enter5 the h0r120n, the 5tr1n9 5y5tem w111 tend t0 5tra19hten 0ut 0n 5ca1e5 1e55 than the h0r120n, ma1nta1n1n9 1t5 6r0wn1an kn0tt1ne55 0n 1ar9er 5ca1e5.7he 5y5tem rema1n5 1n a tan91e, 6ut the 5tep 1en9th 9r0w5 w1th the h0r120n, wh11e the rad1at10n 0f ax10n5 15 c0nt1nu0u5 and accumu1at1n9.70 6e91n 1 w111 der1ve the ener9y den51ty 0f the5e ma551e55 ax10n5.

1t 15 4u1te 9enera1 that a 5tat1c, 5tra19ht ax10n 5tr1n9 ha5 ma55 per un1t 1en9th

u = f [ ( 0 0 ) 2 + p2/r2 + v(0)]r dr 0¢),

226

V01ume 180, num6er 3 PHY51C5 L E 7 7 E R 5 8 13 N 0 v e m 6 e r 1986

where V 15 a 5ymmetry 6reak1n9 p0tent1a1 and the f1e1d p 6ehave5 11ke ~ r 1n51de the 5tr1n9 and 90e5 exp0nent1a11y t0 fa 0ut51de. V and the funct10na1 f0rm 0f p are m0de1 dependent, and theref0re 50 15 #, 6ut 6ecau5e 0f the m1dd1e term/~ 15 d0m1nated 6y a 10n9 ran9e 1/r p0tent1a1 wh1ch 15 unam619u0u5.7hu5 we can wr1te

/a > 27rf 2 1n(A16),

where 6 15 the th1ckne55 0f the 5tr1n9 c0re and A 15 a cut0ff pr0v1ded 6y the 1nter-5tr1n9 5pac1n9,2 Let u5 d1v1de 5pace up 1nt0 expand1n9 cu61c ce115 0f 512e t, the a9e 0f the rad1at10n d0m1nated un1ver5e. A150,

1et u5 a55ume that at any 91ven t there 15 0ne 5tra19ht 5tr1n9 0f1en9th t per ce11.7hen the 5tr1n9 ma55 1n a ce11 at any t1me 15

E(t) = 27rf2t 1n(t/6 ). (3.1)

1n 0rder t0 ma1nta1n th15 f0rm 0fE(t) , ax10n5 mu5t 6e rad1ated 6ecau5e k1nk5 appear 1n a ce11 a5 1t expand5. We can ca1cu1ate the am0unt 0f rad1at10n 1f we were t0 5upp05e that the ener9y were t0 have the c0rrect f0rm at 50me t1me t0 ,6u t afterward5 the 5tr1n95 d1d n0t 05c111ate and rad1ate at a11. At 1ater t1me5 the ener9y per ce11 w0u1d 6e

F(t) : 2.f2ax177-0(t1~177-0)31n(x177-015) -- 2.fa2t 2 1n(x/77~01~),

1n wh1ch 1 have 1nc1uded 60th the 5tretch1n9 0f the 0r191na1 ce11, t 0 + t0 tx/~0, and the fact that a ce11 0f 512e t c0n- ta1n5 ( t /X~0) 3 0f the 0r191na1 ce115.7he rate at wh1ch ener9y mu5t 6e rad1ated t0 ma1nta1n the f0rm 0f e4. (3.1) 15

= D F ( t ) 1 , = , 0 - = 27rf2a[1n(t016)-~].

7he 1ncrement t0 the ener9y den51ty at t1me t 15

Ap(t) = AEra0(t)/t3 = 27rf 2 [1n(t/6) - ~] At/t 3,

and the t0ta1 ener9y den51ty that 15 accumu1ated 6etween 50me very ear1y t1me t*, when ax10n5 dec0up1ed, and a 1ater t1me t 15

t

p(t)= 2. 4 5 (r1t)2 [1n(718 ) - 1] ~3 = (7r41t2)[1n(t16 )1n(t/t*) - 1n(t/t*)], (3.2)

where the (r/ t) 2 fact0r 1n the 1nte9rand acc0unt5 f0r the c05m01091ca1 red5h1ft 1n the rad1at10n d0m1nated ep0ch. A11 that rema1n5 15 t0 wr1te e4. (3.1) a5 a 5pectra1 d15tr16ut10n.

Rad1at10n em1tted at t1me 7- w111 0r191na11y have a wave1en9th 0f ~ - and m0mentum ~2rr/r. 70 6e carefu1, 1et u5 5upp05e that rad1at10n em1tted at 7- ha5 an effect1ve m0mentum k = f2/7-, where we expect ~2 ~ 27r. At a 1ater t1me t that m0mentum 15 red5h1fted 6y x / ~ . 1n term5 0 fk , (3.1) 6ec0me5

47rf 2 a1 t ~ 0(t) = - ~ f [1n(922/tk 2•5) -11 dkk . (3.3)

a/t

7h15 f0rmu1a 15 900d f0r a11 t1me5 up t0 when ~2/t = m(t) 15 5at15f1ed, that 15, when the ax10n ma55 15 e4ua1 t0 the ener9y 0f the 10we5t ener9y ax10n.

We w111 need t0 kn0w th15 t1me, 50 1 w111 ca1cu1ate 1t here. Reca11 that ~ 15 def1ned 50 that m -- m6~) = 1/7. 1n- 5tant0n ca1cu1at10n5 [12] 1mp1y that f0r three 4uark f1av0r5

m(t) ~ 7.3(A6/7r4fa)(mumdm 5/A3)1/2m21 1n(rr2mp1/A2t)t-2,

, 2 We can redef1ne 6 50 t h a t u = 27rfa 2 1n(A/6).

227

v01ume 180, num6er 3 PHY51c5 LE77ER5 8 13 N0vem6er 1986

where A 15 the QCD 5ca1e fact0r, mp1 15 the P1anck ma55 and mu,d, 5 are the 4uark ma55e5.51nce the 109ar1thm ev01ve5 50 510w1y we may wr1te

m(t) = t2/~ 3

a5 10n9 a5 t d0e5 n0t 5tray t00 far f r0m~. 5ett1n9 th15 e4ua1 t0 ~2/t we f1nd that the 1ate5t t1me f0r wh1ch e4. (3.3) 15 va11d 15 ~21/3~ ".

After the ax10n ma55 ha5 turned 0n the p1cture 15 ent1re1y d1fferent. Any 5tr1n9 mu5t 6e attached t0 a d0ma1n wa11.7he Wa11-60unded-6y-5tr1n9 5y5tem 6reak5 up 1nt0 100p5 5panned 6y wa115, they 05c1114te and decay rad1a- t1ve1y; 6ut 51nce the ax10n 15 n0w ma551ve the rad1at10n 15 9rav1tat10na1, at 1ea5t unt11 the wa11 area 5hr1nk5 t0 a 512e 0f a few meter5-54uared, when ax10n em15510n can a9a1n take 0ver. 7he c05m0109y 0f the 5tr1n9-wa11 5y5tem 15 very 1ntere5t1n9 [ 13], 6u t here we are c0ncerned 0n1y w1th the effect5 0f th05e ax10n5 rad14ted 6ef0re ~21/3~.

After ~21/3~ the ener9y den51ty 0f the5e ax10m ha5 the f0rm [5ee (A1)]

47rf2a (525/6 /x /~ ( k 2 + (~]t~)m2 a 1/2

0 a ( t ) - t 2 ~2 /a~ • 1c2+m 2 ) [1n(~2513/t~k26)~]~k~.dk

N0t1ce that th15 ha5 the appr0pr1ate red5h1ft1n9 pr0pert1e5: ~ t -2 f0r 1ar9e k m0de5 and ~t -3/2 f0r 10w k m0de5. L00k1n9 at th15 5pectrum we 5ee that a5 10n9 a5 ~2 15 n0t t00 1ar9e there 15 a 519n1f1cant c0ntr16ut10n fr0m very 10w va1ue5 0 f k , d0wn t0 k = ~22/3/~" = ~2 /3~ , where the effect 0f the WK8 1n5ert 15 1mp0rtant. 50, c0ntrary t0 the prev10u5 examp1e, there 5h0u1d 6e a 1ar9e enhancement when the ma55 turn5 0n.

0n1y an appr0x1mate 501ut10n can 6e f0und. F1r5t, we can 5et ~ = 0 1n the den0m1nat0r, 1ntr0duc1n9 an err0r 0f at m05t a few percent. 5ec0nd, we can undere5t1mate 04 6y truncat1n9 the upper 11m1t 0f 1nte9rat10n d0wn t0 k = m a. 7h1rd, we can put k = m a 1n the 109ar1thm and pu11 1t 0ut 0f the 1nte9ra1, 06ta1n1n9

4rrf 2 f . m 2 ~ 2 1/2 dk Pa ( t ) ~ ~ 1n(~25/3/~m28) j•a ( 1+ tma]tk ) - -

~2/317 k f -

~(47rf2[t2) 1n(~25/3/~m2~)[( 1 +t~m2/~24/3)1/2 (1 +t/t~)1/21 ~ (47rf42/t2) 1n(~25/3 /tma~)(ttrna/~ 2 ~ 2 4/3)1/2 .

At th15 p01nt 1t 15 fru1tfu1 t0 ca5t 0ur re5u1t 1nt0 a f0rm that can 6e ea511y c0mpared t0 prev10u5 ca1cu1at10n5.1n the f0110w1n9 1 w111 6e c105e1y para11e11n9 the w0rk 1n ref. [7]. 7he ener9y den51ty can 6e wr1tten

5/3 2 2/3 3 1/4 3 pa(7) ~ 4~r 1n(~2 /7ma~)~2- (1/0.3)[~(7) /~(~)] (fa/mp1)(fama/~)7, where 1 have u5ed the re1at10n t7 2 = 0.3 m p 1 / ~ . Next, u51n9 7 = 800 MeV and d1v1d1n9 6y the cr1t1ca1 den51ty 0f the un1ver5e, we 9et

P/Pcr = 4rr 1n(f25/3/~tm2a ~ )~2-2/3(1/0.3)[c~(7)3/c~(~7)] 1/4fa/1013 6eV. (3.4)

0 f c0ur5e the 60und 0n the ax10n 5ca1e c0me5 fr0m the c0nd1t10n that th15 4uant1ty 6e 1e55 than 1.

4. C0nc1u510n. ct~ ( 7 ) = 61.75, wh1ch 1nc1ude5 the he11c1ty 5tate5 0f ph0t0n5, 91u0n5, three neutr1n05, tw0 char9ed 1ept0n5, and three 4uark f1av0r5. At the pre5ent temperature c~eff(7 ) = 3.4, wh1ch acc0unt5 f0r the neu- tr1n0 dec0up11n9 6ef0re e+e - ann1h11at10n. 7ak1n9 f0r n0w f~ = 27r, fa "•" 1010 6eV, m a = [(mumd)1/2/(mu + md) ] (m~f~/fa) ~ 6 X 10 -4 eV and 8 = (1/fa) , e4. (3.4) 91ve5

P/Pcr ~ 4"5fa/1011 6eV,

wh1ch 1mp11e5

fa ~ 2 X 1010 6eV, (4.1)

228

V01ume 180, num6er 3 PHY51C5 LE77ER5 8 13 N0vem6er 1986

a fact0r 0f 200 10wer than the 60und prev10u51y 06ta1ned 6y 100k1n9 0n1y at the 2er0-m0mentum m0de 0f the ax10n f1e1d.

Ev1dent1y the 60und (4.1) depend5 cr1t1ca11y 0n the va1ue 0f £2, a5 we11 a5 0n the a55umpt10n that 0ne cu61c ce11 0f 512e t c0nta1n5 0ne 5tr1n9 0f 1en9th t. 1f we a55ume that the wave1en9th 0f the rad1at10n accurate1y f0110w5 the 5ca1e 0f the 5tr1n9 netw0rk then a 1ar9er va1ue 0f f2 re4u1re5 m0re 1en9th 0f 5tr1n9 per ce11.1n fact, 51nce $2 5ca1e5 a5 the 5tr1n9 5tep-1en9th, the 1en9th per ce11 90e5 a5 ~923. 7h15 appear5 1n the numerat0r 0f e4. (3.4) 50 the ch01ce 0f ~2 = 27r 1n e4. (4.1) tru1y w0u1d 91ve an upper 60und 0 f f a. An0ther c1a55 0f uncerta1nt1e5 901n9 1nt0 e4. (4.1) ha5 t0 d0 w1th the nature 0f the 4uark-hadr0n pha5e tran51t10n. 7he5e were c0n51dered 1n ref5. [ 4 ,6 -8 ] . 1t 5uff1ce5 t0 5ay that the effect5 c0n51dered tend t0 5tren9then the 60und, except f0r ref. [8], wh1ch u5ed the c0herence 0f the 2er0- m0mentum m0de t0 ar9ue that the 60und may 6e weakened. 51nce c05m1c 5tr1n9 ax10n5 are n0t c0herent th15 ar- 9ument 5h0u1d n0t app1y.

7h15 1etter 5h0w5 that ax10n 5tr1n95 5tren9then the 60und when there 15 n0 1nf1at10n. W1th 1nf1at10n 1t 15 p055161e t0 rem0ve the 5tr1n95 fr0m the 065erva61e un1ver5e, and a5 10n9 a5 the p05t 1nf1at10nary reheat1n9 d0e5 n0t 6r1n9 the un1ver5e a60ve 7pQ =fpQ then the re5u1t pre5ented here d0e5 n0t app1y. 51nce 6 U 7 6ary05ynthe515 re4u1re5 re- heat1n9 t0 ~10146eV 5tr1n95 are unav01da61e 1n 5uch m0de15. 1t 15 0n1y p055161e t0 9et r1d 0f 5tr1n95 1f 6ary0n num- 6er 15 9enerated 6e10w 7pQ.

1ffa were c105e t0 1t5 5aturat10n va1ue (4.1), then the ax10n w0u1d 6e an 1mp0rtant c0ntr16ut10n t0 the ener9y den51ty 0f the un1ver5e. 7h15 h01d5 the attract1ve p05516111ty that 6e51de5 501v1n9 the 5tr0n9 CP pr061em, the ax10n c0u1d exp1a1n the nature 0f the dark matter. 7ak1n9 5er10u51y the 10wer 60und 0f 4 × 10 9 6eV, th15 paper 5u99e5t5 that the ax10n mu5t 6e the dark matter 1f 1t were t0 ex15t at a11.5uch c05m01091ca1 ax10n5 are a150 heav1er, and ea51er t0 detect, than prev10u51y th0u9ht.

1n 5ummary, 1 cann0t ar9ue that the a55umpt10n5 901n9 1nt0 (4.1) are anyth1n9 6ut rea50na61e. 70 6e certa1n 0f th15 60und w0u1d re4u1re deta11ed kn0w1ed9e 0f the ev01ut10n 0f the ax10n 5tr1n9 5y5tem. 7h0u9h c0mputer 51mu- 1at10n5 have 6een d0ne 1n the ca5e 0f 9au9e 5tr1n95, t0 my kn0w1ed9e n0 0ne ha5 attempted 51m11ar ca1cu1at10n5 f0r 5tr1n95 fr0m 9106a1 5ymmetr1e5.7he pr061em 15 1nherent1y m0re d1ff1cu1t 6ecau5e the 9106a1 5tr1n95 have 10n9 ran9e 1nteract10n5. What th15 1etter 5u99e5t5 unam619u0u51y 15 that 15 that ax10n5 fr0m 5tr1n95 are 0f c05m01091ca1 1mp0r- tance. 70 90 further, and accept the 60und (4.1), w0u1d 1eave 5cant r00m f0r the ax10n t0 ex15t.

1 am 06119ed t0 D1ck 80nd f0r 1mpetu5.

Append1x. 7he WK8 appr0x1mat10nf0r n0n-2er0 m0mentum m0de5. We want t0 kn0w the enhancement 0f an ener9y den51ty 1n1t1a11y 91ven 6y

p = f f ( k ) dk, (A1)

when the ax10n ma55 turn5 0n. F0r an ax10n m0de wh1ch at 50me reference t1me t 1 , 6ef0re the ma55 turn-0n, ha5 the f0rm a = A (t) c05 kx, the

e4uat10n 0f m0t10n 15

d2A/dt 2 + [ ( t J t ) k 2 + m2(t)]A + 3H(t)dA/dt = 0.

H 15 the Hu661e c0n5tant and m 15 the t1me dependent ax10n ma55. ~and r~ are def1ned 6y the e4uat10n

7= 11m(7)= 117n. 7h15 15 r0u9h1y the t1me at wh1ch the ax10n C0mpt0n wave1en9th cr055e5 the h0r120n. 7he ax10n ma55 5tart5 at 2er0 f0r t "~ ~ and r15e5 t0 1t5 f1na1 va1ue m a f0r t ~ ~. 1f m-1 dm/dt < m and H < m then the ad1a6at1c c0nd1t10n 15 met. 51nce m(t) ~ t 2 ~e91ect1n9 a 109ar1thm1c fact0r), the t1me that mark5 the 0n5et 0f the ad1a6at1c re91me 15ju5t 27. Let u5 take t 1 = t . 7he 501ut10n 15 then

229

V01ume 180, num6er 3 PHY51C5 LE77ER5 8 13 N0vem6er 1986

t A(t) ~ A(t) c05 f [6~/r)k 2 + m2(r)] 1/2 dr ,

w1th the c0nd1t10n that the num6er 0f ax10n5 per c0m0v1n9 v01ume 15 c0n5tant. 7hu5

[(7/t)k2 + m2]1/2~(t)2 = (k 2 + ~ 2 ) 1 / 2 7 2 [R/R(t)] 3,

and the ener9y den51ty 0f th15 m0de 15

~[(7"/t)k2+m21A(t)2= k2--~-~m 2 ] ~ (~/t)2 X 1(k2 * /~2)~2"

F1na11y, 51nce (7/ t) 2 X • (k 2 + ~ 2 ).~2 15 the ener9y den51ty 0f the m0de at ~, and the 5pectrum at th15 t1me 15 e4. (A1), we can 51mp1y 1n5ert the term 1n 6race51nt0 the 1nte9ra1 t0 06ta1n

72 ( ( k 2 + (t/~)m 2 )112 p = ~ , , - - - - - f (k) dk. (A2)

t 2 • k 2 + ~ 2

Reference5

[1] R.D. Pecee1 and H.R. Qu1nn, Phy5. Rev. Lett. 38 (1978) 1440. [2] 5. We1n6er9, Phy5. Rev. Lett. 40 (1978) 223. [3] F. W11c2ek, Phy5. Rev. Lett. 40 (1978) 279. [4] 5. D1m0p0u105, 6.D. 5tarkman and 8.W. Lynn, Phy5. Lett. 8 168 (1986) 145. [5] M. Fuku91ta. 5. Watamura and M. Y05h1mura, Phy5. Rev. Lett. 48 (1982) 1522. [6] D. Dear60rn, 6.5te19man and D.N. 5chramm, Phy5. Rev. Lett. 16 (1986) 16. [7] J. Pre5k1U, M.8. W15e and F. W11c2ek, Phy5. Lett. 8 120 (1983) 127. [8] L.F. A660tt and P. 51k1v1e, Phy5. Lett. 8 120 (1983) 133. [9] M. D1ne and W. F15ch1er, Phy5. Lett. 8 120 (1983) 137.

[10] W.6. Unruh and R.M. Wa1d, Phy5. Rev. D 32 (1985) 831; 7. De6rand, M. Kephart and M. We11er, Phy5. Rev. D 33 (1986) 910; D. Kap1an, Nuc1. Phy5. 8 260 (1985) 215.

[11] R. Dav15, Phy5. Rev. D 32 (1985) 3172. [12] D.J. 6r055, R.D. P15ar5k1 and L.6. Yaffe, Rev. M0d. Phy5. 53 (1981) 43. [13] F.W. 5tecker and Q. 5haf1, Phy5. Rev. Lett. 50 (1983) 928. [14] D.5. D1cu5, E.W. K016, V.L. 7ep11t2 and R.V. Wa90ner, Phy5. Rev. D 18 (1978) 1829.

230