Radiation constraints from cosmic strings

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  • EI ~S EV I ER

    UCLEAR PHYSIC~

    Nuclear Physics B (Proc. Suppl.) 43 (1995) 54-57

    PROCEEDINGS SUPPLEMENTS

    Radiation constraints from cosmic strings R.A.Battye and E.P.S.Shellard a *

    ~Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, U.K.

    We show that it is possible to evolve a network of global strings numerically including the effects of radiative backreaction, using the renormalised equations for the Kalb-Ramond action. We calculate radiative corrections to the equations of motion and deduce the effect on a network of global strings. We also discuss the implications of this work for the cosmological axion density.

    1. INTRODUCTION

    Cosmic strings can play many interesting roles in the early universe. It has been shown that a network of cosmic strings will evolve towards a self-similar scaling solution on the largest scales. However, on the smallest scales, where radiative effects are important, the evolution is less well understood. Here we report on an ongoing re- search project[l-3] to understand the true nature of these effects. Our primary motivation is to calculate accurately the amplitude and spectrum of the radiation background from both local and global strings. For the purposes of this paper we shall concentrate on Goldstone boson radiation from global strings, however analogous calcula- tions have been made for gravitational radiation from local strings[4].

    2. RADIAT IVE BACKREACTION

    The starting point for our discussion of global strings will be the Kalb-Ramond action[7]

    S = -#o d~drx /~- ~ d4xH 2

    ] B,udo "~' , (1) ~ 2 ~ ~

    which has been shown to exhibit the same dynam- ics as the U(1) Goldstone model[l] using com- parisons between numerical field theory simula-

    *We acknowledge the support of the Particle Physics and Astronomy Research Council, in particular the Cambridge Relativity rolling grant (GR/HT1550) and Computational Science Initiative grants (GR/H67652 & GR/H57585)

    0920-5632/95/$09.50 1995 Elsevier Science B.V. All rights reserved. SSDI 0920-5632(95)00450-5

    tions and analytic calculations of the radiation power. The antisymmetric tensor field Bu~ is dual to the massless Goldstone boson associated with the string, Hu~f~ is the field strength asso- ciated with B~ and da ~" = ab~aXl~bXV is the area element in terms of the worldsheet coordi- nates XU(a, T) = (~-, X(~, r)).

    Varying the action with respect to the world- sheet coordinates and the antisymmetric tensor field yields the string equations of motion and field equations. In the temporal transverse string gauge, X = t = 7- and :K. X ' = 0, the equations of motion for the worldsheet coordinates are

    where

    F" = (f0, ef + fox) = 2r r f~gu~V~, (3)

    and

    X,2 1 - X 2

    The field equation can be simplified using the analogue of the Lorentz gauge in electromag- netism, O~B ~" : 0. Using this gauge, the field equations are

    DB z = dad 64(x- (5)

    This equation can be solved easily using stan- dard retarded Green function techniques. In anal- ogy to electromagnetism, one can perform a series

  • R.A. Battye, E.P Shellard/Nuclear Physics B (Proc. SuppL) 43 (1995) 54-57 55

    of manipulations to deduce the Lienard-Wiechart potential[9],

    ja f vo< (6) 2 j \lA.Xl/b= '

    and its derivative,

    : 7 aea oe -(2:-2] , (r)

    where A , = x , - Xu(e,), A~ = 0 and T=T R

    ~-R < t. Using the sum and difference of the re- tarded and advanced time Green functions, one can calculate the self- and radiation-fields respec- tively. However, since the Lienard-Wiechart po- tential and its derivative both involve an integra- tion along the string, which in the case of long strings is infinite, we must make some approxima- tion in order to calculate the field at some point on the string. We shall employ the 'local backre- action approximation'[3]. In this approximation the integrals in (6) and (7) can be truncated at some scale A, which at present is arbitrary. One can then expand the expressions as a power se- ries in A/R (assuming A < R), where R is the average radius of curvature of the string.

    Retaining leading order terms, one can deduce expressions for the forcing terms in the equation of motion (2) as

    f0,s~f = _27rf2 log(A/5)~, (S)

    f0,rad _ 47rfa 2A [ e2l~" ~ ] (10) L I - :K2 J '

    where #(A) = #0 + 2rrf~ log(A/5) is the renor- malised string tension. The finite radiation back- reaction force experienced by the string due to frad and f0,rad is the analogue of the Abraham- Lorentz force for the point electron in classical electrodynamics.

    Performing numerical simulations of the point electron using the Abraham-Lorentz force is haz- ardous, since the triple derivative with respect to time leads to unphysical exponentially growing solutions. However, in the case of strings one can make the approximation X = X" /c 2, which re- duces the backreaction force to the analogue of a viscosity term[3]. Once one has made this approx- imation, methods similar to those used in ref.[6] can be used to evolve general string trajectories.

    For general long string perturbations parame- terized by wavelength L and relative amplitude 6 = 27rA/L, where A is the actual amplitude, the radiation power per unit length due to the back- reaction force can be seen to be

    dP /3e 2 - (13)

    dl L '

    for some constant ? ~ f2. This leads to exponen- tial decay of 6, since the energy per unit length in the perturbation is proportional to 6 2 . This was verified by numerical simulations of string tra- jectories. We also compared the decay of string trajectories to those seen in numerical field the- ory. The rate of decay of amplitude was found to agree, if the correlations in the long range fields were suppressed at the curvature radius and A ,,~ R/4. We also observed that an initially sharp kink visibly rounds under the action of this force.

    frad 47rfa~A [~ l (X"X - 3 - ; \ i - -~)X ' ] ' (11)

    where 5 ~ f-i is the string width. The self-force can be clearly seen to be a mul-

    tiple of the equations of motion[8,9], which facil- itates the well-known renormalisation,

    #(A)[ ~- l -e ( -~) ' ] = frad, #(A)~=f0'rad,(12)

    3. NETWORK EVOLUTION

    Our current understanding of the evolution of a cosmic string network is based on a marriage between analytic models and sophisticated net- work simulations[5]. However, the network sim- ulations just evolve the free equations of motion for a string in an expanding universe. In order to incorporate the radiative effects discussed in the preceding section one must modify the equations

  • 56 R.A. Battye, E, RS. Shellard/Nuclear Physics B (Proc. Suppl.) 43 (1995) 54 57

    of motion to include a radiation damping term,

    [ 2&. 1 1 ( _~) ' ] P0 2K + --~-( - X2)j( - - = f , (14) (

    where a is the scale factor However, to calculate this radiation damping

    term one must use Green functions in an expand- ing background[10]. In the radiation era, the re- tarded Green function is

    27ra(rf)a(r/) g( ( _ x,)2 )O( r / - r/) (16) Dret(X, X') -- _ \ ,X

    where x ~ = (7],x) and m '~ = (~]',x'). Applying, the 'local backreaction approximation' in this sce- nario, one finds that the forcing terms are given by,

    fraa = lflatqrad + _a g + O(1/t a) , (17) a

    f0,rad = f0, rad flat + --ago -~- O(1 / t3 ) ' (18) a

    where the flat suffix denotes the flat space back- reaction force given by (10) and (11), and (gO, g) is a correction to the force due to the expanding background.

    The forced equations of motion (15) can be used to derive equations for the evolution of the density of long strings (Poe) and loops (PL), under the influence of the expansion, Hubble damping and the radiation backreaction force. If one now inserts a term to take into account of loop pro- duction, the equations become

    boo-- 24( la + {v2))Pc CpOOL ~-'dP (19)

    35 cpoo /)L -- a ,OL + L (20)

    where

    d = do + adl + O(1/t2), (21) a

    @2) is the average string velocity, do, d1,.. are constants and c is a measure of the efficiency of loop production.

    If one now substitutes Pec = #4/t 2 and L = ~-l/2t into (19), then one can deduce a differen- tial equation for ~. This equation has an attrac- tive fixed point, which corresponds to the scaling regime. If d{ = 0 for i > 0, then one can deduce that

    c = 4-i/2(1 - @2}) _ do. (22)

    In the case where dl is non-zero, One should ob- serve transient effects in the scaling However, for large times these effects will be negligible and the attractive fixed point is exactly that for dl = 0.

    4. RADIATION CONSTRAINTS

    The evolution of a network of global strings is of particular interest, since the best motivated sce- nario for the production of global strings is at the Peccei-Quinn phase transition. In this case, the radiated Goldstone boson is the initially massless axion, which acquires mass at the QCD phase transition, and may form a substantial part of the dark matter in the universe In the light of the discussion of the previous section, we concluded that a network of global strings will evolve by the production of loops, which subsequently radi- ate into Goldstone bosons[12]. This mechanism is similar to that generally accepted to be the case for local strings. However, since the radiation into Goldstone bosons is about three orders of magni- tude stronger than that into gravitational radi- ation, the loop production size a and the string scaling density 4 are likely to be different.

    Using the scaling balance arguments of 3, one can deduce that the loop number density is given by[2]

    n(l, t) ~ 0"4z1/2~(1 + t~/(~)5/2 t3/2(l + gt)5/2 , (23)

    where ~ is the loop radiation backreaction scale. From this one can deduce that the spectral den- sity of axions emitted by the loops is

    ~( t ) ~ 450f2c~1/2 ( " \5/2 ~t2~3/2 1+~)

    x[1 - (1+-~) -a /21 . (24)

  • R.A. Bathe, E.I~S. Shellard/Nuclear Physics B (Proc. SuppL) 43 (1995) 54 57 57

    Assuming axion number conservation from the QCD phase transition to the present day one can deduce that the contribution to the density of the universe from axions produced by the loops is

    (0~3/2 [1-- ( l+7) -3 /2 ]h -2A fta, e 10.7 \~/

    ( T0 "~3( fa 1.18 x (1 + ~)st2 t 2.--.//////'.~j t. lOl~eV) ,(25)

    where To is the current temperature of CMBR, the Hubble constant is H0 = 100hkms-1Mpc -1, A ~ 10 ~=-5 takes into account the uncertainties of the QCD phase transition and 0.1

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