V01ume 1648, num6er 4,5,6 PHY51C5 LE77ER5 12 Decem6er 1985

6 R A V 1 7 A 7 1 0 N A L R A D 1 A 7 1 0 N F R 0 M A P A R 7 1 C U L A R C L A 5 5 0 F C 0 5 M 1 C 5 7 R 1 N 6 5

C.J. 8 U R D E N t

Department 0f Natura1 Ph11050phy, Un1ver51ty 0f 61a590w, 61a590w 612 8QQ, UK

Rece1ved 6 5eptem6er 1985

7he 9rav1tat10na1 rad1at10n fr0m c105ed 100p5 0f 5uperheavy c05m1c 5tr1n9 15 5tud1ed. F0r a 6r0ad c1a55 0f 5tr1n9 traject0r1e5 1nc1ud1n9 n0n 5e1f 1nter5ect1n9 100p5 an ana1yt1c expre5510n 15 91ven f0r the p0wer rad1ated 1n 9rav1tat10na1 wave5 per un1t 5011d an91e. 7he t0ta1 p0wer rad1ated per 100p 15 eva1uated f0r 5evera1 100p traject0r1e5 1n the c1a55, the re5u1t5 6e1n9 51m11ar t0 the va1ue5 06ta1ned 1n a prev10u5 numer1ca1 ca1cu1at10n 0f 5pec1f1c traject0r1e5 6y Vacha5pat1 and V11enk1n.

1t ha5 6een pr0p05ed [1] that 5uperheavy c05m1c 5tr1n95 f0rmed dur1n9 a pha5e tran51t10n 1n the ear1y un1ver5e c0u1d pr0v1de 5eed5 f0r the c0nden5at10n 0f 9a1ax1e5 and 9a1axy c1u5ter5. 7he 10n9 11ved 5tr1n9 100p5 re4u1red f0r the 5tr1n9 5cenar10 t0 6e 5ucce55fu1 w111 decay pr1nc1pa11y 6y 9rav1tat10na1 rad1at10n, perhap5 pr0v1d1n9 a 9rav1ta. t10na1 rad1at10n 6ack9r0und 5uff1c1ent t0 create detecta61e n015e 1n pu15ar t1m1n9 [2].

Quant1tat1ve e5t1mate5 [3,4] 0f the pred1cted 9rav1tat10na1 rad1at10n 6ack9r0und per 109ar1thm1c fre4uency 1n- terva1,

n9 = ( f[pe)dp9/df , (1)

have 6een made, and 91Ve va1Ue5 f0r [29 Wh1Ch w0U1d 6eC0me exper1menta11y deteCta61e 1n m11115eC0nd pu15ar t1m- 1n9 1n the f0r5eea61e future. 1n (1), Pc 15 the cr1t1ca1 den51ty and p9 the ener9y den51ty 0f 9rav1tat10na1 wave5. 1n part1cu1ar, Vacha5pat1 and V11enk1n [4] have ca1cu1ated numer1ca11y the 9rav1tat10na1 rad1at10n fr0m 5pec1f1c 100p traject0r1e5 and u5e the1r re5u1t5 t0 e5t1mate [29. F0r the 9rav1tat10na1 rad1at10n em1tted per 100p, they fmd

dE/d t = 76/J 2 , (2)

where/~ 15 the ma55 per un1t 1en9th 0f 5tr1n9 (typ1ca11y 6 u ~ 10 -6 f0r 9rand un1f1cat10n 5tr1n95) and 7 15 a numer1- ca1 c0eff1c1ent depend1n9 0n the part1cu1ar 100p traject0ry, typ1caUy "••100 f0r the traject0r1e5 c0n51dered. A55um- 1n9 7 d0e5 n0t d1ffer 519n1f1cant1y 0ver the p0pu1at10n 0f re1evant 100p5 1n the ear1y un1ver5e, they further arr1ve at the e5t1mate

~ 9 ( f ) ~ 45~3/2~(61a/~/)1/2p.r[p e "~ 10 -7 (f>)" 10 -7 (6# ) -1 y r - 1 ) , (3)

Where e3/2/315 a numer1Ca1 parameter "~1 de5Cr161n9 the f0rmat10n 0f 100p5 and P•r 15 the pre5ent ener9y den51ty 0f therma1 rad1at10n.

1n th15 1etter, We extend the Ca1CU1at10n 0f 7 t0 5eVera1 100p C0nf19urat10n5 n0t C0Vered 1n ref. [4]. We are a61e t0 Ca1Cu1ate ana1yt1Ca11y the 1nten51ty 0f 9rav1tat10na1 rad1at10n per Un1t 5011d an91e, dP[d12, f0r a 6r0ad C1a55 0f 5tr1n9 501Ut10n5 1nC1Ud1n9 50me 0f th05e treated numer1Ca11y 1n ref. [4]. 7he C1a55 a150 1nC1Ude5 501Ut10n5 Wh1Ch are n0t 5e1f 1nter5eCt1n9, and are theref0re 0f phy51Ca1 1ntere5t. A 5tra19htf0rward numer1Ca1 1nte9rat10n then 91Ve5 the t0- ta1 p0Wer rad1ated per 5tr1n9.

We 6e91n W1th the 9enera1 501Ut10n f0r a C105ed 100p 0f 1nVar1ant 1en9th L [5]:

1 Addre55 fr0m N0vem6er, 1985: Department 0f 7he0ret1ca1 Phy51c5, R.5. Phy5. 5., Au5tra11an Nat10na1 un1ver51ty, Can6erra, Au5tra11a.

0370-2693/85/$ 03.30 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

277

V01ume 1648, num6er 4,5,6 PHY51C5 LE77ER5 12 Decem6er 1985

r(0, t) = (L14~)[a(~) + 6(~)] , ~ = (27r/L)(0 - r ) , *7 = (21r/LX0 + r),

a •2=6•2=1, a(5+2n)=a(5), 6(5+2rr)=6(5). (4)

7he m0r10n 0f the 5tr1n915 per10d1c w1th per10d L/2.7he p0wer rad1ated 1n 9rav1tat10na1 wave5 0f fre4uency c0 n = 4rm[L per un1t 5011d an91e 1n the d1rect10n k 15 [6,3,4]

dPn/d~ = (6c02/7r)[7*uv(c0n, k)7u~(c0,, k) -- { 17."(c0,. k)[ 2 ] . (5)

1n e4. (5), 7uv(c0n, k) 15 the F0ur1er tran5f0rmed ener9y m0mentum ten50r 0f the 5tr1n9, that 15

L L (1 fd,01 .,fd001,.r 7uv(c0n, k) = ~ 0 0 f • f - r • • r•

L L = ~ f dte1~nt f dae-(1L/4~r)k~[a(~)+6(~)]

L 0 0

X k [6•(r/) - a~(~)] - • [a~(~) • 6•(r1) + 6•(r1) • d(~)] • .(6)

where ]k[ = c0n~ We have avera9ed 0ver t-~0 per10d5 0f the 100p5 m0t10n 1n5tead 0f 0ne. 7h15 a110w5 u5 t0 chan9e the var1a61e5 0f 1nte9rat10n t0 ~ and 77 and, u51n9 the per10d1c1ty 0 f a and 6 1n the var1a61e5 0 and r, t0 a1ter the re910n 0f 1nte9rat10n a5 5h0wn 1n f19. 1.7h15 91ve5

27r 21r 7~"(,~n.k) = L~u f d~ e-~e -~/~0~) f dn e~"e-~/~6(,)M""(~.,). (7)

42r2 0 0

where Muv(~, r1) 15 the matr1x appear1n9 1n e4. (6). 7uv can 6e ca1cu1ated ana1yt1ca11y f0r 5u1ta61e ch01ce5 0f a(~) and 601). 1n part1cu1ar, we ch05e a and 6 t0 6e

c1rcu1ar m0t10n 1n p1ane5 at an an91e ~0 t0 0ne an0ther. 8y ch051n9 the 2er0 0f 0 appr0pr1ate1y, we can wr1te 1n 9enera1

a(~) = M -1 c05M~ e 3 + M -1 51nM~ e 1 , 607) = N -1 c05N~ e 3 + N -1 51nN~ (c05 • e 1 + 51n 4J e2 ) . (8)

t

%

Cr

5~ F19. 1.5h1ft1n9 the 5haded part5 0f the re910n 0f 1nte9rat10n a d15tance L 1n the 0 and t d1rect10n5 t0 06ta1n e4. (7).

278

V01ume 1648, num6er 4,5,6 PHY51c5 LE77ER5 12 Decem6er 1985

M and N are re1at1ve1y pr1me, 0therw15e the pa ramete r 0 traver5e5 the 100p m0re than 0nce 6e tween 0 and L and can 6e redef1ned t0 m a k e M a n d N re1at1ve1y pr1me. 7he ca5eM = N = 1 wa5 t rea ted 1n deta11 1n ref. [4]. A t t = ~L 1t c011ap5e5 t0 a d0u61e 11ne. 1t 15 5tra19htf0rward t0 check that , 1fM 0r N = 1 (6ut n0 t 60th) , the 100p w111 n0 t 5e1f 1nter5ect, 6u t 1f 60 th M and N are 9reater than 1 the 100p d0e5 1nter5ect 1t5e1f at certa1n 1n5tant5 dur1n9 1t5 m0- t10n. A11 501ut10n5 f0rm cu5p5 m0v1n9 at the 5peed 0f 119ht, 6u t 0n1y at d15crete 1n5tant5 dur1n9 the 100p5 m0t10n, the except10n5 6e1n9 the ca5e5 ~0 = 0 and ~6 = 7r. 7he5e 5pec1a1 ca5e5 [7] are r191d1y r0tat1n9 hyp0cyc101d5 w1th N + 1 cu5p5 (when ~0 = 1r,M = 1) 0r ep1cyc101d5 w 1 t h N - 1 cu5p5 (when ~0 = 0 , M = 1), the cu5p5 a1way5 m0v1n9 at the 5peed 0 f 119ht.

7 0 eva1uate 7uv(c0n, k) f0r the 5tr1n9 c0nf19urat10n5 (8) and k = (51n 0 c05 ~, 51n 0 51n ¢, c05 0), we make u5e 0 f the f0110w1n9 re5u1t5 f0r 8e55e1 funct10n5:

7 t

f e c05x c05. dx = 1n1rjn(2), 0

Jn~1(2) +Jn+1(2) = (2n/2)Jn(2), Y~x(2)--J,+1(2)-- 2J~(2), J~(--2) = ( - - 1 ) n J , ( 2 ) .

We f1nd tha t at 1ea5t 0ne 0 f the tw0 1nte9ra15 appear1n91n (7) 91ve5 2er0 un1e55 n 15 a mu1t1p1e 0fMN. 7he rema1n- 1n9 n0n-2er0 va1ue5 0 f 7 uv are

. [11(P)12(4) • [11(P)5(4) -- 12(4)R(p)] ] 7u~(c0MN m , k) = taL[ k [11(P)5(4) - 12 ( 4 ) R ( p ) ] -• [R(p) • 5(4) + 5(4) • R(p)1 ] (9)

w h e r e p = mN, 4 = m M ( m = 1 , 2 . . . . ),

11(P) = e-1Pex(-1)PJp(PA), 12(4) = ea4e2(-1)4J4(48), • • • t

R(p) = e -•pet (--1)P [--at 1 J p ( p A ) [ A + 10t2Jp(pA)] , 5(4) = e x4e2 (--1) 4 [P1J4(48)[8 + 1112J4(48)1 ,

and we have def1ned

A = C 0 5 0 [ 1 + t a n 2 0 c 0 5 2 ¢ ] 1 / 2 , 8 = C 0 5 0 [ 1 + t a n 2 0 C 0 5 2 ( $ - - ¢ ) ] 1 / 2 ,

e 1 = arCtan(tan 0 C05 ¢ ) , e 2 = arCtan[tan 0 C05(~ -- ¢)] ,

at 1 = 51n e 1 e 1 + C05 e 1 e 3 , 0t 2 = c05 e 1 e 1 -- 51n e 1 e 3 ,

P1 = 51n e 2 (e 1 c05 ~0 + e 2 51n ~0) + c05 e 2 e 3 , 112 = c05 e 2 (e 1 C05 ¢ + e 2 51n 4J) -- 51n e 2 e 3 .

5u65t1tut1n9 e4. (9) 1nt0 e4. (5) we 06ta1n

dPMNm/d~ = 167r6/a2 ( MNm)2 [• (11112 -- 1R 12)(11212 -- 1512) + ~(1R~512 -- 1R7512) + Re(1~11Rt5 -- 1~1~R75)]

= 167r69t2(MNm) 2 (~ [(1 -- A - 2 ) j 2 • jp2] [(1 - 8 - 2 ) J 4 -- j~2]

+ 2 [ a 2 " P 2 -- (°t1 X 0t2)•(p 1 X p2)[A81JpJ4JPJ4), (10)

where the ar9ument5 0 f / 1 , / 2 , R and 5 are the 5ame a5 th05e 0ccurr1n9 1n (9), and JV = Jp(pA), J4 = J4(48). After 50me w0rk1n9, We arr1ve at the re5u1t

51n~0 51n ~ 51n(~ - ~) 0t2"~2 --(Qt 1 X ~2)~(~1 X p2)/A8 =

( 1 - - 51n20 51n2#6) 1/2 [1 -- 51n20 51n2(~ -- 4~)] 1/2 •

wh1ch a110w5 u5 t0 wr1te

279

V01ume 1648, num6er 4,5,6 PHY51C5 LE77ER5 12 Decem6er 1985

dPMNm/d[2 = 87r61~2(MNm)2 [(5152JpJ4 + 4 4 ) 2 + ( 5 1 J p 4 + 524J4)2 ] ,

where

51 = 51n 0 51n ~/(1 -- 51n20 51n2c6) 1/2 , 52 = 51n 0 51n(~6 -- ¢)/[1 -- 51n20 51n2(t.k -- ~6)] 1/2

7he t0ta1 ener9y rad1ated 6y 9rav1tat10na1 wave5 15 then 91ven 6y

(11)

dE[dt = ~ Pm= ~ f ~ 1 ~ m / d a . (12) m m

we have ca1cu1ated 7, def-med 6y e4. (2), f0 rM = 1 , N = 1, 2, 3, 5 and 15 f0r var10u5 va1ue5 0f ~. 7he 8e55e1 funct10n5 appear1n9 1n (11) are ea511y eva1uated 1n the c0mputer 6y 1terat1n9 d0wnward5 fr0m h19her va1ue5 0 f p and 4 [8], and the 1nte9ra1 1n (12) 15 then d0ne numer1ca11y. 8y a 51m11ar ar9ument t0 0ne 5et 0ut 1n ref. [4] f0r the M = N = 1 ca5e 0ne can 5h0w that

Pm "~ c0n5t, m -4/3 (13)

f0r m 1ar9e, pr0v1ded # :/: 0 0r 1r, 1n wh1ch ca5e ~mPm d1ver9e5. We eva1uatedP m t0 at 1ea5t 150 term5, 0r unt11 the 6ehav10ur de5cr16ed 6y (13) 5et5 1n, and then u5ed (13) t0 e5t1mate the rema1nder 0f the 5er1e5.7he rate 0f c0n- ver9ence 510w5 c0n51dera61y f0r h19h and 10w va1ue5 0f ~0, re4u1r1n9 t00 much c0mputer t1me, 50 n0 re5u1t5 are 5h0wn f0r ~k c105e t0 0 0r 7r. 0u r re5u1t5 are 5h0wn 1n f19.2, and c0mpare we11 w1th Vacha5pat1 and V11enk1n•5 nu- mer1ca1 re5u1t5 [4] f0r the ca5e5 M = N = 1. We f1nd further that dE[dt 1ncrea5e5 a5 N 1ncrea5e5, 6ut 1n 9enera1 n0t 50 dramat1ca11y a5 t0 effect any 0rder 0f ma9n1tude ca1cu1at10n 0f the 9rav1tat10na1 6ack9r0und rad1at10n due t0 c05m1c 5tr1n95.

P055161e cau5e5 f0r c0ncem are th05e 100p traject0r1e5 w1th ff c105e t0 0 0r 1r, f0r wh1ch dE[dt 6ec0me5 c0n51der. a61y h19her. 1f the ear1y un1ver5e ha5 an a6undance 0f 5uch traject0r1e5, the ••typ1ca1 7" va1ue 1n e4. (3) 1ncrea5e5, and the e5t1mate 0f ~29 decrea5e5 6y 0ne 0rder 0f ma9n1tude f0r every tw0 0rder5 0f ma9n1tude 1ncrea5e 1n 7. 70

200 1 1 t 1 1

x N=1 + N=2 • N=3

150 •N=5 • N=15

10¢

5C

0 1 30

/ • ~ / ~ J e /

0--.....,. 0 / • j X

1 1 1 1 60 90 120 150 180"

(De9ree5)

F19.2.7he 4uant1ty 7 = (dE/dt)/(6~ 2) f0r 5eVe~a1 0f the 100p trajeCt0r1e5(8),WJthM= 1 ,N= 1,2, 3,5 and 15.

• 60

~-- 50

30

20

10

= = 1 = 0 10 20 30 40 50 60 70 80 90

~)(de9ree5)

F19. 3.6rav1tat10ua1 p0wer rad1ated 1n the d1rect10n/c per un1t 5011d an91e f0r the f0ur-cu5ped hyp0cyc101d 501ut10n M = 1, N = 3, ~ = ~r. 8ecau5e the 501ut10n ha5 ax1a1 5ymmetry a60ut e2, dP/d6 depend5 0n1y 0n the an91e 0 6etween k a n d

the p1ane c0nta1n1n9 the 5tr1n9.

280

v01ume 1648, num6er 4,5,6 PHY51C5 LE77ER5 12 Decem6er 1985

5ee where the h19h 3• va1ue5 c0me fr0m, c0n51der the 1nten51ty 0f rad1at10n per un1t 5011d an91e, dP/d~2 wh1ch 6e- c0me5 1nffm1te (6ut 1nte9ra61e, pr0v1ded ~ ~ 0 0r ,r) 1n the d1rect10n +e 3. 7he5e are the d1rect10n5 1n wh1ch the 5tr1n9•5 ve10c1ty reache5 the 5peed 0f 119ht at certa1n 1n5tant5 dur1n9 the 5tr1n9•5 m0t10n. 1t 15 the rad1at10n 1n the5e d1rect10n5 wh1ch 15 re5p0n5161e f0r the a5ympt0t1c 6ehav10ur (13) 0fP m at h19h fre4uenc1e5. A5 ~ appr0ache5 0 0r ,r, the d1rect10n5 0f 510w c0nver9ence 0f dP[d~2 6r0aden, eventua11y c0ver1n9 the wh01e c1rc1e at 1nf1n1ty 0f the p1ane c0nta1n1n9 the ~k = 0 and rr 100p traject0r1e5. F0r 1n5tance, we have p10tted 1n f19. 3 dP[d~2 f0r the f0ur- cu5ped hyp0cyc101d (M = 1, N = 3, ~k = 7r) a5 a funct10n 0f the an91e 0 6etween k and the p1ane c0nta1n1n9 the 5tr1n9.7he d15tr16ut10n 0f dP/d~2 15 ax1a11y 5ymmetr1c f0r th15 ca5e, and the 51n9u1ar1ty at 0 = 0 15 n0t 1nte9ra61e.

1n pract1ce, h0wever, the 6ack react10n 0f the 9rav1tat10na1 rad1at10n 0n the 5tr1n9 w111 prevent 1t ever reach1n9 the 5peed 0f 119ht [3], the effect 6e1n9 t0 r0und 0ff the cu5p5 and make dE/dt f•m1te f0r the5e traject0r1e5. 1t w0u1d 6e a w0rthwh11e exerc15e t0 at tempt a fu11 ca1cu1at10n 0f, 5ay, p1anar 5tr1n9 traject0r1e5,1nc1ud1n9 the 6ack react10n, t0 ru1e 0ut the p055161e effect 0f er0510n 0f ~29 6y path01091ca1 traject0r1e5 w1th h19h va1ue5 0f 3•.

1 w0u1d 11ke t0 thank Dav1d 5uthef1and f0r 6r1n91n9 ref. [4] t0 my attent10n. 7he w0rk wa5 5upp0rted 6y the 5ERC 9rant N6. 1292.1.

Reference5

[1] A. V11enk1n, Phy5. Rev. Lett. 46 (1981) 1169, 1496(E). [2] C.J. H09an and M.J. Ree5, Nature 311 (1984) 109. [3] N. 7ur0k, NueL Phy5. 8242 (1984) 520. [4] 7. Va¢ha5pat1 and A. Vf1enk1n, 6rav1tat10na1 rad1at10n fr0m c05m1c 5tr1n95, Harvard prepr1nt HU7P-84/A065. [5] 7.W.8. K1661e and N. 7ur0k, Phy5. Lett. 1168 (1982) 141. [6] 5. we1n6er9, 6rav1tat10n and c05m0109Y: pr1nc1p1e5 and app11cat10n5 0f the 9enera1 the0ry 0f re1at1v1ty (W11ey, New Y0rk,

1972). [7] C.J. 8urden and L.J. 7a551e, Au5t. J. Phy5. 37 (1984) 1. [8] M. A6ram0w1t2 and 1.A. 5te9urh Hand600k 0f mathemat1ca1 funct10n5 (D0ver, New Y0rk, 1965).

281