Gravitational waves from kinks on infinite cosmic strings
aa r X i v : . [ a s t r o - ph . C O ] M a y ICRR-Report-558IPMU 10-0017
Gravitational waves from kinks on infinite cosmic strings
Masahiro Kawasaki ( a,b ) , Koichi Miyamoto ( a ) and Kazunori Nakayama ( a ) a Institute for Cosmic Ray Research,University of Tokyo, Kashiwa, Chiba 277-8582, Japan b Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8568, Japan (Dated: October 29, 2018)
Abstract
Gravitational waves emitted by kinks on infinite strings are investigated using detailed estima-tions of the kink distribution on infinite strings. We find that gravitational waves from kinks canbe detected by future pulsar timing experiments such as SKA for an appropriate value of the thestring tension, if the typical size of string loops is much smaller than the horizon at their formation.Moreover, the gravitational wave spectrum depends on the thermal history of the Universe andhence it can be used as a probe into the early evolution of the Universe. . INTRODUCTION It is well known that cosmic strings are produced in the early Universe at the phasetransition associated with spontaneous symmetry breaking in the Grand-Unified-Theories(GUTs) [1]. Although cosmic strings formed before inflation are diluted away, some GUTspredict series of phase transitions where symmetry such as U (1) B − L ( B and L denotesbaryon and lepton number) breaks at low energy and produces strings after inflation. Cosmicstrings are also produced at the end of brane inflation in the framework of the superstringtheory [2, 3]. Once produced, they survive until now and can leave observable signatures.Thus, the cosmic strings provide us with an opportunity to probe unified theories in particlephysics which cannot be tested in terrestrial experiments.Various cosmological and astrophysical signals of cosmic strings have been intensivelystudied for decades. Especially, many authors have studied gravitational waves (GW) emit-ted from the cosmic string network, in particular, GWs from cosmic string loops. Cosmicstring loops oscillate by their tension and emit low frequency GWs corresponding to theirsize [4, 5]. Moreover, string loops generically have cusps and kinks, and these structurescause high frequency modes of GWs, i.e. GW bursts [6–9].On the other hand, GWs from infinite strings (long strings which lie across the Hubblehorizon) have attracted much less interest than those from loops. This is because thereare only a few infinite strings in one Hubble horizon, so GWs from them are much weakerthan those from loops whose number density in the Hubble volume is much larger. Loopscan emit GWs with frequency ω & ( αt ) − , where α is the parameter which represents thetypical loop size normalized by the horizon scale. Therefor, unless α is much less than 1,GWs from loops dominate in the wide range of frequency bands detected by GW detectionexperiments, e.g. pulsar timing or ground based or space-borne GW detectors.However, there are several reasons to consider GWs of infinite strings. First, recentpapers suggest that α is much less than previously thought [10, 11], and hence loops cannotemit low frequency GWs which can be detected by pulsar timing experiments. If this is For cosmic strings whose reconnection probability is small, such as cosmic superstrings, many infinitestrings can exist in a Hubble horizon. For such kinds of cosmic strings, however, the number of loopsin a horizon is accordingly large, hence the situation that the loop contribution is stronger than infinitestrings remains unchanged. Thus, we are lead to consider GWsfrom infinite strings.A structure on infinite strings which is responsible for GW emission is a kink. Kinks areproduced when infinite strings reconnect and kinks on the infinite strings can produce GWbursts. The kinks are sharp when they first appear, but are gradually smoothened as theUniverse expands. Moreover, when an infinite string self-intercommutes and produces a loop,some kinks immigrate from the infinite string to the loop, and hence the number of kinkson the infinite string decreases. Recently, Copeland and Kibble derived the distributionfunction of kinks on an infinite string taking into account these effects [13]. However, theydid not study the GWs from kinks.Therefore, in this paper, we investigate GWs from kinks using the kink distributionobtained in [13] and discuss the detectability of them by future GW experiments, especiallyby pulsar timing, such as Square Kilometer Array (SKA) and space-based detectors such asDECIGO and BBO.This paper is organized as follows. In section 2, we briefly review a part of the basisof cosmic strings. In section 3, we derive the distribution function of kinks on infinitestrings according to [13]. In section 4, we derive the formula of the energy radiated fromkinks per unit time. In section 5, we calculate the spectrum of stochastic GW backgroundoriginating from kinks, using formula derived in section 4. Section 6 is devoted to summaryand discussion. Other recent simulations imply α ∼ .
1, which is much bigger than Gµ . In this case loops can emit GWswhose wavelength is comparable to or somewhat shorter than the horizon scale [12]. Generically, infinite strings do not have any cusp since its existence depends on the boundary conditionof the string, i.e., the periodic condition. I. DYNAMICS OF COSMIC STRINGS
The dynamics of a cosmic string, whose width can be neglected, is described by theNambu-Goto action, S = − µ Z d ζ p − det( γ ab ) . (1)where ζ a ( a = 0 ,
1) are coordinates on the world sheet of the cosmic string, γ ab = g µν x µ,a x ν,b ( x µ,a = ∂x µ ∂ζ a ) is the induced metric on the world sheet, and µ is the tension of the string. Theenergy-momentum tensor is T µν ( x ) = µ Z d ζ p − det( γ ab ) γ ab x µ,a x ν,b δ ( x − X ( ζ )) , (2)where X = X ( ζ ) is embedding of the world sheet on the background metric. If the back-ground space-time is Minkowski one, we can select the coordinate system ( ζ , ζ ) = ( τ, σ )which satisfies the gauge conditions τ = t (physical time) , x ,τ · x ,σ = 0 , x ,τ + x ,σ = 0 . (3)The time scale of a GW burst is much shorter than the Hubble expansion, hence we consideran individual burst event on the Minkowskian background. The general solution of theequation of motion derived from the action (1) is x µ = 12 ( a µ ( u ) + b µ ( v )) , a ′ ( u ) = b ′ ( v ) = 1 (4)where u = σ + t, v = σ − t . We call a ( u ) ( b ( v )) the left (right)-moving mode. Then, Eq. (2)can be rewritten in terms of a ( u ) and b ( v ), T µν ( k ) = µ I µ + ( k ) I ν − ( k ) + I µ − ( k ) I ν + ( k )) , (5) I µ + ( k ) = Z dua ′ µ ( u ) e ik · a ( u ) / , I µ − ( k ) = Z dvb ′ µ ( v ) e ik · b ( v ) / , (6)where T µν ( k ) is the Fourier transform of the T µν ( x ), i.e. T µν ( k ) = R d xT µν ( x ) e ik · x . III. DISTRIBUTION FUNCTION OF KINKS
Kinks can be defined as discontinuities of a ′ or b ′ . They are produced when two infinitestrings collide and reconnect because a ′ and b ′ on the new infinite string are created by4onnecting a ′ s or b ′ s on two different strings. Let us suppose that a ′ jumps from a ′− to a ′ + at a kink. Then the “sharpness” of a kink is defined by ψ = 12 (1 − a ′ + · a ′− ) . (7)Thus the norm of the difference between a ′− and a ′ + is | ∆ a ′ | = 2 √ ψ . The production rateof kinks is given by [13] ˙ N production = ¯∆ Vγ t g ( ψ ) , (8)where N ( t, ψ ) dψ denotes the number of kinks with sharpness between ψ and ψ + dψ in thevolume V , ¯∆ and γ are constants related to string networks, whose values are [13],¯∆ r ≃ . , ¯∆ m ≃ . , γ r ≃ . , γ m ≃ . , (9)Here the subscript r ( m ) denotes the value in the radiation(matter)-dominated era. g ( ψ ) = 35256 p ψ (15 − ψ − ψ ) (10)and we set g ( ψ ) = 0 for ψ < , < ψ . The correlation length of the cosmic strings ξ is givenby ξ ≃ γt .Produced kinks are blunted by the expansion of the Universe. The blunting rate of thekink with the sharpness ψ is given by [13]˙ ψψ (cid:12)(cid:12)(cid:12)(cid:12) stretch = − ζ t − , (11)where ζ is a constant which, in the radiation(matter)-dominated era, is given by ζ r ≃ .
09 ( ζ m ≃ . NN (cid:12)(cid:12)(cid:12)(cid:12) to loop = − ηγt , (12)where η is constant which, in the radiation(matter)-dominated era, is given by η r ≃ .
18 ( η m ≃ . N obeys the followingequation, ˙ N = ¯∆ Vγ t g ( ψ ) + 2 ζt ∂∂ψ ( ψN ) − ηγt N. (13)5 efinition radiation dom. matter dom. γ - 0.3 0.55 ζ - 0.09 0.2 β - 1.1 1.2 A ζ − β − . − . B ζ − β − . − . C ( β − ζ ) / [3( β − ζ )] 0.14 − . We have to solve this equation under an appropriate initial condition. The initial con-dition to be imposed depends on how strings emerge. We consider two typical scenarios.As a first scenario, let us suppose that cosmic strings form when spontaneous symmetrybreaking (SSB) occurs in the radiation dominated Universe. After the formation, cosmicstrings interact with particles in thermal bath, which acts as friction on the string. At firstthis friction effect overcomes the Hubble expansion. (This epoch is called friction-dominatedera.) The important observation is that the friction effect smoothens strings and washes outthe small scale structure on them [14]. The temperature at which friction domination endsis given by [15] T c ∼ GµM pl , (14)where G is the Newton constant and M pl is the Planck scale. If the temperature at thestring formation is lower than this critical value, the friction effect can be neglected fromthe formation epoch to the present day, and kinks emerge at the same time as appearanceof strings.As a second scenario, let us suppose that cosmic strings are generated at the end ofinflation, like strings made by condensation of waterfall fields in supersymmetric hybridinflation [16–18], and cosmic superstrings left after annihilation of the D-brane and antiD-brane in brane inflation [2, 3]. In this case, cosmic strings form in the inflaton-oscillationdominated era, which resembles the matter-dominated era. Although strings feel frictionsfrom dilute plasma existing before the reheating completes, the effect is negligible as longas the reheating temperature after inflation T r is lower than ∼ GµM pl . In this case, kinks6egin to be formed right after the formation of strings. Otherwise, the temperature at whichkinks begin to appear is given by Eq. (14).In anyway, even if such kinks survive friction, they become extremely dense in the laterperiod, so that they do not affect the observable part of the GW spectrum, as we will seelater. On the other hand, if the reheating temperature is lower than T c , the kinks from thefirst matter era definitely continue to exist, and if the reheating temperature is extremelylow, GWs from these kinks can be observed. Therefore, we concentrate on the situationwhere the reheating temperature is low enough so that the kinks generated in the firstmatter era survive without experiencing the friction domination. We denote the time whenkinks starts to be formed by t ∗ .Then we can get the solution, but its precise form is very complicated. Here we assumethat the matter-dominated epoch follows after inflation, and the reheating completes at t = t r after which the radiation dominated era begins. If we focus on only the dominantterm and neglect O (1) numerical factor, we get dNdψ ( t, ψ ) ∼ ψ − β m / ζ m t − for ψ > (cid:0) t ∗ t (cid:1) ζ m (cid:16) tt ∗ (cid:17) β m + ζ m ψ / t − for ψ < (cid:0) t ∗ t (cid:1) ζ m (15)in the first matter era, dNdψ ( t, ψ ) ∼ ψ − β r / ζ r t − for ψ > ψ (RD)1 ( t ) (cid:16) tt r (cid:17) β r − β m ζ r /ζ m ψ − β m / ζ m t − for ψ (RD)2 ( t ) < ψ < ψ (RD)1 ( t ) ψ / (cid:16) tt r (cid:17) β r + ζ r (cid:16) t r t ∗ (cid:17) β m + ζ m t − for ψ < ψ (RD)2 ( t ) (16)in the radiation era where ψ (RD)1 ( t ) = (cid:18) t r t (cid:19) ζ r , (17) ψ (RD)2 ( t ) = (cid:18) t r t (cid:19) ζ r (cid:18) t ∗ t r (cid:19) ζ m , (18)7nd dNdψ ( t, ψ ) ∼ ψ − β m / ζ m t − for ψ > ψ (MD)1 ( t ) (cid:16) tt eq (cid:17) β m − β r ζ m /ζ r ψ − β r / ζ r t − for ψ (MD)2 ( t ) < ψ < ψ (MD)1 ( t ) (cid:16) t eq t r (cid:17) β r − β m ζ r /ζ m ψ − β m / ζ m t − for ψ (MD)3 ( t ) < ψ < ψ (MD)2 ( t ) ψ / (cid:16) tt eq (cid:17) β m + ζ m (cid:16) t eq t r (cid:17) β r + ζ r (cid:16) t r t ∗ (cid:17) β m + ζ m t − for ψ < ψ (MD)3 ( t ) (19)in the second matter era, where ψ (MD)1 ( t ) = (cid:18) t eq t (cid:19) ζ m , (20) ψ (MD)2 ( t ) = (cid:18) t eq t (cid:19) ζ m (cid:18) t r t eq (cid:19) ζ r , (21) ψ (MD)3 ( t ) = (cid:18) t eq t (cid:19) ζ m (cid:18) t r t eq (cid:19) ζ r (cid:18) t ∗ t r (cid:19) ζ m . (22)Here β is the constant related to the string network ( β r ≃ . , β m ≃ . t eq denotesthe matter-radiation equality epoch. We have converted N , distribution in the volume V ,to dN/dψ = N ( t, ψ ) / ( V /ξ ), which is the distribution per unit length. The derivation ofthe above expression of dN/dψ is described in Appendix A.Let us consider the physical meaning of this distribution function. It is not difficult toconsider how the number of kinks in the horizon changes as time goes on. When a kink isborn, its sharpness ranges from 0 to 1, but the typical value is O (0 . ψ ∼ O (0 .
1) at the very early stage. Then kinks are made bluntby the cosmic expansion. Thus newly produced kinks are sharp (large ψ ) and old ones areblunt (small ψ ). Therefore, the peak value of ψ is much smaller than 0.1 at the late stage.This peak consists of the oldest kinks. Fig. 1 roughly sketches the shape of the distributionfunction in the second matter era.It should be noted that the above distribution is derived without considering gravitationalbackreaction. Since the most abundant kinks are extremely blunt, they might be influencedby the gravitational backreaction and disappear. However, it is difficult and beyond thescope of this paper to take backreaction into account. We will comment on this issue inAppendix C. 8 IG. 1: The distribution function of kinks on infinite strings produced at the end of the inflationin the second matter era. ψ (MD)1 , ψ (MD)2 , ψ (MD)3 are given by Eqs. (20)-(22). IV. THE SPECTRUM OF GRAVITATIONAL WAVES FROM KINKS
Given the energy-momentum tensor of the source, one can calculate the energy of theGW in the direction of ˆ k with frequency ω as [19] dEd Ω (ˆ k ) = 2 G Z ∞ dωω (cid:18) T µν ∗ ( k ) T µν ( k ) − | T µµ ( k ) | (cid:19) = 2 G Λ ij.lm (ˆ k ) Z ∞ dωω T ij ∗ ( k ) T lm ( k ) (23)Λ ij,lm (ˆ k ) ≡ δ il δ jm − k j ˆ k m δ il + 12 ˆ k i ˆ k j ˆ k l ˆ k m − δ ij δ lm + 12 δ ij ˆ k l ˆ k m + 12 δ lm ˆ k i ˆ k j (24)Thus by substituting the energy-momentum tensor (5), we find the energy of the GWafter computing integral I ± in Eq. (6). It is given by dEdωd Ω ( k ) = Gµ ω (cid:18) | ~I + | | ~I − | + | ~I + · ~I − | − | ~I + · ~I ∗− | − | ~I + | | ˆ k · ~I − | − | ~I − | | ˆ k · ~I + | + | ˆ k · ~I + | | ˆ k · ~I − | − ( ~I + · ~I ∗− )(ˆ k · ~I ∗ + )(ˆ k · ~I − ) − ( ~I ∗ + · ~I − )(ˆ k · ~I + )(ˆ k · ~I ∗− )+ ( ~I + · ~I − )(ˆ k · ~I ∗ + )(ˆ k · ~I ∗− ) + ( ~I ∗ + · ~I ∗− )(ˆ k · ~I + )(ˆ k · ~I − ) (cid:19) (25)9here ~I ± ≡ ( I ± ( k ) , I ± ( k ) , I ± ( k )).The method to calculate I ± is described in [7, 20]. In the limit ω → ∞ , I ± exponentiallyreduces to 0, unless at least one of the following conditions on the integrand is met. One isthe existence of discontinuities of a ′ i ( u ) (or b ′ i ( v )). The contribution of a discontinuity of a ′ to I + ( k ) is I i + ( k ) ≃ − iω a ′ i + − ˆ k · a ′ + − a ′ i − − ˆ k · a ′− ! e iω ( u ∗ − ˆ k · a ∗ ) / (26)where u ∗ is the position of the discontinuity, a ∗ = a ( u ∗ ) and we assume a ′ jumps from a ′− to a ′ + at u = u ∗ . The region of length ∼ ω − around u = u ∗ contributes to this value. We find I i + ( k ) ∼ ψ/ω , where ψ denotes sharpness of the kink. The other condition is the existence ofstationary points of the phase of the integrand, i.e. ω ( u − ˆ k · a ( u )) / ω ( − v − ˆ k · b ( v )) / − ˆ k · a ′ ( u s ) = 0 (or − − ˆ k · b ′ ( v s ) = 0) (27)at the point u = u s ( v = v s ). The contribution of the stationary point of the phase to I + ( k )is I i + ( k ) ≃ ω / a ′′ i ( u s ) e iω ( u s − ˆ k · a ( u s )) / | ˆ k · a ′′′ ( u s ) | ! / i √ / . (28)For any value of a ′ ( b ′ ), there is one direction ˆ k that satisfies (27), i.e. ˆ k = a ′ ( − b ′ ).Therefore, every point of u can contribute to I i + ( k ) for one direction ˆ k .One can find the energy of the GW burst from ONE kink by picking up the contributionfrom the discontinuity for one of I ± (say, I + ) and from the stationary point for the other(say, I − ). The energy emitted when the kink is located at the world sheet coordinate( u, v ) = ( u ∗ , v s ) is evaluated by substituting (26) into I + and (28) into I − in (25). Then weobtain dEd Ω dω ( ω, − ω b ′ s ) = Γ (cid:0) (cid:1) Gµ ω − (cid:18) | b ′ s · b ′′′ s | (cid:19) × (cid:18) b ′ s · a ′ + ) − a ′ + · a ′− (1 + b ′ s · a ′ + )(1 + b ′ s · a ′− ) + 1(1 + b ′ s · a ′− ) (cid:19) b ′′ s , (29)where the subscript s represents the value at v = v s . The energy is radiated at everymoment toward the direction of − b ′ from the kink. This is the formula which representsenergy emitted in a short period in a small solid angle. The total energy emitted in the10hort period ∆ t is found by multiplying ∆Ω, which is the solid angle that the GW sweeps inthis short period. The extent of the radiation has the solid angle ∼ (( ω/ | b ′′ s | ) − / ) [7]. Thevariation of the direction of the GW is roughly estimated by | ∆ b ′ | ∼ | b ′′ ∆ t | . As a result,we obtain ∆Ω ∼ ( ω/ | b ′′ s | ) − / | b ′′ s | ∆ t (30)and the energy emitted per unit time is dPdω ∼ Γ (2 / Gµ ω − / (cid:18) | b ′ s · b ′′′ s | (cid:19) / × (cid:18) b ′ s · a ′ + ) − a ′ + · a ′− (1 + b ′ s · a ′ + )(1 + b ′ s · a ′− ) + 1(1 + b ′ s · a ′− ) (cid:19) | b ′′ s | / , (31)The terms in the large parenthesis in the 2nd line of Eq. (31) can be estimated by tak-ing average over the angle between the left- and right-moving mode as h b ′ s · a ′± ) i ∼h b ′ s · a ′ + )(1+ b ′ s · a ′− ) i ∼
1. The magnitudes of b ′′ s and b ′′′ s should be ξ − ( ∼ t − ) and ξ − ( ∼ t − ). After making these substitutions and neglecting O (1) numerical factors, we find dPdω (cid:12)(cid:12)(cid:12)(cid:12) one kink ∼ Gµ ψω − / t − / . (32)So far we have evaluated the GW spectrum from one kink. However, there exist manykinks on an infinite string and the final observable GW spectrum is made from sum ofcontribution from these kinks. Therefore, I + picks contributions of many kinks. Formally, I i + ( k ) = X m I i + ,m ( k ) , (33)where an integer m labels each kink. Thus Eq. (25) has cross terms of the contributions fromdifferent kinks, e.g., ~I + ,m · ~I ∗ + ,n . However, such cross terms must vanish since a GW burstfrom a kink is a local phenomenon which relates only the region around the kink. In fact,the structure of the string around a specific point arises as a result of nonlinear evolution ofthe string network, and hence the values of b ′′ s , b ′′′ s and so on, are stochastic. We show thatensemble averages of cross terms vanish as expected, i.e. h I i + ,m I j ∗ + ,n i = 0 in Appendix B,where it is also shown that the kinks that dominantly contribute to the power of GWs with Although b ′ has small and dense discontinuities which represent kinks, v s is generally not a kink point.So the rough shape of b ′ s is determined by the global appearance of the string network and hence thelength scale of its variation is roughly the curvature radius of the network. ∼ ω are ones which satisfy (cid:18) ψ dNdψ (cid:19) − ∼ ω − . (34)In other words, if the interval of kinks with sharpness ∼ ψ is similar to the period of theGW under consideration, these kinks make dominant contribution to the GW. In the firstmatter era, using Eq. (15), Eq. (34) simplifies to ψ ∼ ( ωt ) ζ m /A m (35)for ω < ( t ∗ /t ) A m t − . (Here, we set A ≡ ζ − β . A r ≃ − . , A m ≃ − . ω > ( t ∗ /t ) A m t − , (cid:16) ψ dNdψ (cid:17) − > ω − is satisfied by an arbitrary value of ψ . In Appendix B itis shown that for ω > ( t ∗ /t ) A m t − the main contribution to ω dPdω comes from the kinkscorresponding to the peak of the distribution, i.e. ψ ∼ ( t ∗ /t ) ζ m . If we denote the value ofsharpness of kinks which make dominant contribution to ω dPdω as ψ max ( ω, t ), it is given by ψ max ( ω, t ) ∼ ( ωt ) ζ m /A m for ω < ( t ∗ /t ) A m t − (cid:0) t ∗ t (cid:1) ζ m for ω > ( t ∗ /t ) A m t − (36)in the first matter era. In the radiation era, ψ max ( ω, t ) is found in a similar way as ψ max ( ω, t ) ∼ ( ωt ) ζ r /A r for ω < ω (RD)1 ( t ) (cid:0) t r t (cid:1) D/A m ( ωt ) ζ m /A m for ω (RD)1 ( t ) < ω < ω (RD)2 ( t ) (cid:0) t r t (cid:1) ζ r (cid:16) t ∗ t r (cid:17) ζ m for ω > ω (RD)2 ( t ) , . (37)where ω (RD)1 ( t ) = (cid:18) t r t (cid:19) A r t − , (38) ω (RD)2 ( t ) = (cid:18) t r t (cid:19) A r (cid:18) t ∗ t r (cid:19) A m t − . (39)In the second matter era, ψ max is estimated as ψ max ( ω, t ) ∼ ( ωt ) ζ m /A m for ω < ω (MD)1 ( t ) (cid:0) t eq t (cid:1) − D/A r ( ωt ) ζ r /A r for ω (MD)1 ( t ) < ω < ω (MD)2 ( t ) (cid:16) t r t eq (cid:17) D/A m ( ωt ) ζ m /A m for ω (MD)2 ( t ) < ω < ω (MD)3 ( t ) (cid:0) t eq t (cid:1) ζ m (cid:16) t r t eq (cid:17) ζ r (cid:16) t ∗ t r (cid:17) ζ m for ω > ω (MD)3 ( t ) , . (40) Strictly speaking, Eq. (34) has two solution for ψ , but it is sufficient to take larger one. ω (MD)1 ( t ) = (cid:18) t eq t (cid:19) A m t − , (41) ω (MD)2 ( t ) = (cid:18) t eq t (cid:19) A m (cid:18) t r t eq (cid:19) A r t − , (42) ω (MD)3 ( t ) = (cid:18) t eq t (cid:19) A m (cid:18) t r t eq (cid:19) A r (cid:18) t ∗ t r (cid:19) A m t − . (43)Here, we set D ≡ β r ζ m − β m ζ r ≃ . b ′′ s , except their sharpness, are roughly same) we can estimatethe total power of GWs with frequencies ∼ ω from all of the kinks in a horizon as, ω dPdω (cid:12)(cid:12)(cid:12)(cid:12) tot ∼ ω dPdω (cid:12)(cid:12)(cid:12)(cid:12) one kink ( ω, ψ max ( ω, t )) × ψ dNdψ (cid:12)(cid:12)(cid:12)(cid:12) ψ = ψ max ( ω,t ) × t. (44)The first factor denotes the power of GWs from one kink. The second factor denotes thenumber of kinks which satisfy ψ ∼ ψ max ( ω, t ) per unit length. The third factor is length ofan infinite string in a horizon. Then we finally obtain ω dPdω (cid:12)(cid:12)(cid:12)(cid:12) tot ∼ Gµ ( ωt ) C m for t − < ω < ( t ∗ /t ) A m t − Gµ (cid:0) t ∗ t (cid:1) B m ( ωt ) − / for ω > ( t ∗ /t ) A m t − (45)in the first matter era, ω dPdω (cid:12)(cid:12)(cid:12)(cid:12) tot ∼ Gµ ( ωt ) C r for t − < ω < ω (RD)1 ( t )10 Gµ (cid:0) t r t (cid:1) D/A m ( ωt ) C m for ω (RD)1 ( t ) < ω < ω (RD)2 ( t )10 Gµ (cid:16) t ∗ t r (cid:17) B m (cid:0) t r t (cid:1) B r ( ωt ) − / for ω > ω (RD)2 ( t ) (46)in the radiation era, and ω dPdω (cid:12)(cid:12)(cid:12)(cid:12) tot ∼ Gµ ( ωt ) C m for t − < ω < ω (MD)1 ( t )10 Gµ (cid:0) t eq t (cid:1) − D/A r ( ωt ) C r for ω (MD)1 ( t ) < ω < ω (MD)2 ( t )10 Gµ (cid:16) t eq t r (cid:17) − D/A m ( ωt ) C m for ω (MD)2 ( t ) < ω < ω (MD)3 ( t )10 Gµ (cid:16) t ∗ t r (cid:17) B m (cid:16) t r t eq (cid:17) B r (cid:0) t eq t (cid:1) B m ( ωt ) − / for ω > ω (MD)3 ( t ) (47)in the second matter era, where B ≡ ζ − β ( B r ≃ − . , B m ≃ − .
4) and C ≡ ( β − ζ ) / β − ζ ) ( C r ≃ . , C m ≃ − . t − corresponding to the horizon scale.13 . THE STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES FROMKINKS We have found Eqs. (45),(46) and (47) as the total energy radiated per unit time in ahorizon from kinks on an infinite string. Now we are in a position to calculate the densityparameter of GWs defined by Ω gw ( ω ) ≡ ωρ c dρdω ( ω ) , (48)where ρ c denotes the critical energy density of the present Universe. Noting that the energydensity of GWs decreases as a − and the frequency redshifts as a − , we getΩ gw ( ω ) ∼ ρ c Z t t ∗ dt t (cid:18) ω ′ dPdω ′ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ω ′ = ω × a /a ( t ) (cid:18) a ( t ) a (cid:19) , (49)where a ( t ) is the scale factor and a represents its present value. Using Eqs. (45),(46) and(47), we get the spectrum asΩ gw ( ω ) ∼ π ( Gµ ) ( ωt ) C m for t − < ω < ω π ( Gµ ) (cid:16) Ω r Ω m (cid:17) − D/A r ( ωt ) C r for ω < ω < ω π ( Gµ ) (cid:16) Ω r Ω m (cid:17) − D/A m (cid:16) T M pl (cid:17) D/A m (cid:16) T r M pl (cid:17) − D/A m ( ωt ) C m for ω < ω < ω π ( Gµ ) ( ωt ) − / (cid:16) Ω r Ω m (cid:17) B m / − B r (cid:16) T M pl (cid:17) B r (cid:16) T r M pl (cid:17) B m − B r (cid:16) H ∗ M pl (cid:17) − B m for ω > ω , (50)where ω = ω (MD)1 ( t ) = (cid:18) Ω r Ω m (cid:19) A m / t − , (51) ω = ω (MD)2 ( t ) = (cid:18) Ω r Ω m (cid:19) A m / − A r (cid:18) T M pl (cid:19) A r (cid:18) T r M pl (cid:19) − A r t − , (52) ω = ω (MD)3 ( t ) = (cid:18) Ω r Ω m (cid:19) A m / − A r (cid:18) T M pl (cid:19) A r (cid:18) T r M pl (cid:19) A m − A r (cid:18) H ∗ M pl (cid:19) − A m t − (53)where t and T denote the present age and temperature of the Universe, T r is the reheatingtemperature and H ∗ is the Hubble parameter at the end of inflation. This formula has Here we neglected the effect of cosmological constant. The GW spectrum will be slightly modified ifthe cosmological constant is taken into account. Detailed estimation of this effect is beyond the scopeof this paper since it needs a simulation of cosmic string network evolution in the cosmological constantdominated Universe. IG. 2: A schematic picture of gravitational wave spectrum from kinks (50). ω , ω , ω are givenby Eqs. (51)-(53). complicated exponents. Using β and ζ with values given above, Ω m / Ω r ≃ . × and T /M pl ≃ . × − , Eq. (50) simplifies toΩ gw ( ω ) ∼ ( Gµ ) ( ωt ) − . for t − < ω < ω Gµ ) ( ωt ) . for ω < ω < ω ( Gµ ) (cid:16) T r M pl (cid:17) . ( ωt ) − . for ω < ω < ω ( Gµ ) (cid:16) T r M pl (cid:17) . (cid:16) H ∗ M pl (cid:17) . ( ωt ) − / for ω > ω , . (54)where ω ∼ t − , (55) ω ∼ − (cid:18) T r T (cid:19) . t − , (56) ω ∼ (cid:18) T r M pl (cid:19) . (cid:18) H ∗ M pl (cid:19) . t − (57)We find that the integral in Eq. (49) is dominated by the contribution from the periodnear the present for the whole range of ω . In other words, almost all of the present energyof GWs from the kinks on infinite strings comes from those radiated around the present15poch. This does not mean that the kinks produced at present make dominant contributionto the energy of GWs. We see above that for given frequency ω , the kinks which dominantlycontribute to ω dPdω are determined by Eqs. (36),(37) or (40). Accordingly high frequencymodes arise from dense, blunt and old kinks, and low frequency modes arise from thin,sharp and new ones. The first line of Eq. (50) corresponds to GWs from new kinks whichwere born after the matter-radiation equality, the second line corresponds to GWs fromold kinks which were born in the radiation era, the third line corresponds to GWs fromolder kinks produced between the end of the inflation and the start of the radiation era andthe last line corresponds to GWs which came from the most abundant and oldest kinks.Figure 2 sketches the shape of the spectrum. The spectrum has three inflection points at ω , ω and ω . This is due to change of the type of kinks which mainly contribute to Ω gw .Positions of these inflection points depend on the reheating temperature T r . If T r takesthe value around its lower bound, say, 10 MeV [21], the second inflection point falls in theobservable region, as we will see. Even if T r is so low, the natural value of H ∗ makes thethird inflection far above the observable region. If the reheating ends immediately and H ∗ is as small as possible, the region between the second and third inflection is so short thatthe third inflection enters the observable region.This is a crude estimation, and in order to derive the realistic spectrum we should takeinto account subtlety described in [7, 8] where the authors claimed that GWs from kinksare burst-like and hence GW bursts with rare event rate (“isolated” GWs ) should not becounted as constituent of the stochastic GW background. We should calculate the stochasticGW background spectrum using the following formulae [7, 8]Ω gw ( f ) ∼ π f t ) h ( f ) (58) h ( f ) = Z dzz θ ( n ( f, z ) − n ( f, z ) h ( f, z ) (59) n ( f, z ) = 1 f d ˙ Nd ln z (60) d ˙ N ∼ θ m ( f, z )(1 + z ) − ψ max ( ω z , t ) ˜ N ( ψ max ( ω z , t ) , z ) t − ( z ) dV ( z ) , (61) h ( f, z ) = Gµ [ ψ max ( ω z , t )] / t ( z )[(1 + z ) f t ( z )] / zt z θ (1 − θ m ( f, z )) , (62) θ m ( f, z ) = [(1 + z ) f t ( z )] − / , (63)16 V = πt q Ω m Ω r T T r (1 + z ) − / dz (1st matter era)72 πt q Ω m Ω r (1 + z ) − dz (radiation era)54 πt ((1 + z ) / − (1 + z ) − / dz (2nd matter era) (64) t ( z ) = q Ω m Ω r T T r (1 + z ) − / t (1st matter era) q Ω m Ω r (1 + z ) − t (radiation era)(1 + z ) − / t (2nd matter era) (65)where f = ω/ π and ω z = ω (1 + z ). dV means the proper spatial volume between redshifts z and z + dz . t ( z ) represents the cosmic time at the redshift z . n ( ω, z ) represents thenumber of GW bursts at redshift ∼ z with frequencies ω superposed in a period of ∼ ω − .˜ N ( ψ, z ) dV dψ is the number of kinks with sharpness ψ ∼ ψ + dψ in the volume dV at redshift z , so ˜ N ( ψ, z ) t ( z ) ∼ dNdψ ( t ( z ) , ψ ). Isolated GW bursts are excluded from the calculation byinserting the step function in the integral in Eq. (59). h ( f, z ) is the logarithmic Fouriercomponent of the waveform of the GW burst from one kink located at redshift z and cancontribute to GW with frequency f .The results of calculation are shown in Figure 3 and Figure 4. We take Gµ ∼ − , close tothe current upper bound from CMB observation [22], and show the spectrum with frequency ω from the band of CMB experiments to that of ground-based GW detectors. These figuresinclude both our crude estimate (50) and the improved one given by Eqs. (58)-(65). We seethat the latter is much smaller than the former. This is because the spectrum is dominatedby GWs emitted recently, and recent GW bursts have a more tendency to be isolated. Thefact that the difference between the two estimates becomes larger in higher frequency bandmight disagree with intuition, since the kinks corresponding to high frequency GWs aremore abundant. However, the higher frequencies of GW are, the smaller the possibility thatthey overlap, because the period of oscillation becomes shorter and the extent of the GWbeam becomes narrower. As a result, higher frequency GWs are more likely isolated in time.In Figure 3 we assume that cosmic strings were born by SSB at GUT scale and firstkinks appeared at the temperature 10 GeV (14). In Figure 4 we assume that stringswere produced at the end of the inflation with extremely low reheating temperature ( T r =10 MeV). This assumption makes the second inflection point of the spectrum visible in theobservable frequency band. We also assume the inflation energy scale is sufficiently high so17 IG. 3: Ω gw in the case where strings emerge at the phase transition in the radiation era for Gµ = 10 − and T ∗ ∼ GeV. The upper line represents the estimate using Eq. (50), i.e. including“rare bursts”, and the lower line represents the estimate using Eqs. (58)- (63), i.e. excluding “rarebursts”. Sensitivity curves of various experiments are shown. That of DECIGO is derived from [25].That of BBO correlated is derived from [26]. Others are derived from [27]. that third inflection point on the spectrum is far from the observable region. If we couldobserve the second inflection, we can deduce the reheating temperature.Note that the spectrum depends on Gµ via the overall factor ( Gµ ) . Therefore, when wevary the value on Gµ , the spectrum only moves upward or downward, and its shape (e.g.,the position of bending) does not change. This situation is different from the case of GWsfrom cosmic string loops. In the case of cosmic string loops, since different values of Gµ givedifferent values of lifetime of loops, the resultant spectral shape is different [4, 5].Let us discuss detectability of this GW background. We have to see whether GWs fromkinks exceeds not only thresholds of various experiments but also GWs from other source.The spectrum of GWs from loops was discussed in [4–9], and the contribution of loops to18 IG. 4: Ω gw in the case where strings emerge at the end of inflation for Gµ = 10 − , T r ∼
10 MeV.The upper line represents the estimate including “rare bursts”, and the lower line represents theestimate excluding “rare bursts”.
GW background is much larger than that of kinks if they coexist in some frequency band.However, a loop cannot emit GWs with frequencies smaller than the inverse of its size. Thusthere is a cut on the low frequency side of the spectrum of GWs from loops correspondingto the inverse of the loop size ∼ ( αt ) − . If α ∼ Gµ , the spectrum of GWs from loops beginsto appear at ω & − Hz, and this covers the frequency band where both pulsar timingarrays and GW detectors have good sensitivity. However, α is one of the most unknownparameters in the cosmic string model. According to some recent simulations [12], α maybe much greater and the broader region may be covered by loops’ GW. On the other hand,some recent studies [10, 11] show the possibility that α is extremely small, say, α ∼ ( Gµ ) n with n &
1. In such a case, GWs from loops dominate only very high-frequency region andGWs from kinks may be observable at low-frequency region. For example, if α . − , theband of SKA [23] can be used for detection of GWs from kinks, and for Gµ ∼ − , Ω gw α . − , the sensitivity band ofspace-borne detectors, BBO [24] and DECIGO [25] are open for detection of kink-inducedGWs, and Ω gw exceeds the sensitivity of correlated analysis of BBO for Gµ ∼ − , andthat of ultimate-DECIGO for Gµ & − . In even more extreme case, α . − , it maybe possible to detect the inflection point in Figure 4 for Gµ & − and determine thereheating temperature as T r ∼
10 MeV.Moreover, GWs from kinks may be detected through CMB observations. As opposed toGWs from loops, kinks can emit GWs with wavelength comparable to the horizon scale.These GWs induce B-mode polarizations, which is a target of on-going and future CMBsurveys. The spectrum of GWs from kinks is quite different from inflationary GWs andhence its effect on CMB is also expected to be distinguished from that of inflationary origin.We will study this issue elsewhere.
VI. CONCLUSIONS
In this paper, we have considered gravitational waves emitted by kinks on infinite cosmicstrings. We have calculated the spectrum of the stochastic background of such gravitationalwaves and discussed their detectability by pulsar timing experiments and space-borne detec-tors. It is found that if the size of cosmic string loops is much smaller than that of Hubblehorizon, some frequency bands are open for detection of GWs originating from kinks. Itcan be detected by pulsar timing experiments for Gµ & − , and by space-borne gravita-tional wave detectors for much smaller Gµ , although the latter may be hidden by the loopcontribution unless the typical loop size is extremely small. If it is detected, it will pro-vide information on the physics of the early Universe, such as phase transition and inflationmodels. Moreover, the spectrum shape depends on the thermal history of the Universe, andhence GWs from cosmic strings can be used as a direct probe into the early evolution of theUniverse. Notice that the inflationary GWs also carry information on the thermal historyof the Universe [29–31]. Although the inflationary GWs are completely hindered by GWsfrom cosmic strings if the value of Gµ is sizable and GWs from kinks come to dominate inlow-frequency region, GWs from cosmic strings also have rich information on the physics of Note that stochastic GWs from astrophysical sources such as white-dwarf binaries make a dominantcontribution for ω . Appendix A
Here we derive the expression of dN/dψ given in section 3. First, we change the variablefrom ψ to ψ i = ψ ( t/t i ) ζ in Eq. (13), where t i is the time when we set the initial condition.Then Eq. (13) becomes t ˙ N ( ψ i , t ) + (cid:18) ¯ ηγ − ζ (cid:19) N ( ψ i , t ) = ¯∆ Vγ t g (cid:18) t i t (cid:19) ζ ψ i ! , (A1)where the dot now denotes the time derivative at constant ψ i . This equation can be easilyintegrated to obtain N ( ψ, t ) V ( t ) = ¯∆ γ t − β Z t max( t i ,ψ / ζ t ) dt ′ t ′ β g (cid:18) tt ′ (cid:19) ζ ψ ! + (cid:18) t i t (cid:19) − β N (cid:0) ( t/t i ) ζ ψ, t i (cid:1) V ( t i ) . (A2)For the distribution function during the first matter era, we set the initial condition at t i = t ∗ as N ( ψ, t ∗ ) = 0. Then Eq. (A2) becomes N ( ψ, t ) V ( t ) = ¯∆ m γ m t − β m Z t max( t ∗ ,ψ / ζm t ) dt ′ t ′ β m g (cid:18) tt ′ (cid:19) ζ m ψ ! . (A3)By substituting Eq. (10) into g in Eq. (A3), performing the integration and omitting termsexcept for dominant one, we get Eq. (15). For N/V during the radiation era, we set t i = t r and get N ( ψ, t ) V ( t ) = ¯∆ r γ r t − β r Z t max( t r ,ψ / ζr t ) dt ′ t ′ β r g (cid:18) tt ′ (cid:19) ζ r ψ ! + (cid:18) t r t (cid:19) − β r N (cid:0) ( t/t r ) ζ r ψ, t r (cid:1) V ( t r ) . (A4)We use Eq. (15) for N ( ψ, t r ). Then Eq. (A4) simplifies to Eq. (16) by picking only thedominant term. The expression during the second matter era (19) can be obtained in thesame way. Appendix B
Here, we prove kinks which dominantly contribute to GWs with frequency ω are thosewhich satisfy Eq. (34), and evaluate the integral in Eq. (6).21irst of all, we consider the situation that ω is so small that (cid:16) ψ dNdψ (cid:17) − = ω − has solu-tions. a ′ i has numerous kinks (discontinuities), from blunt ones to sharp ones, according toEqs. (15), (16) and (19). Let us consider kinks which satisfy (cid:16) ψ dNdψ (cid:17) − & ω − ( ⇔ ψ & ψ max ).From now on, we call such kinks “big” kinks. The interval between two kinks with sharpness O ( ψ ) is roughly given by (cid:16) ψ dNdψ (cid:17) − . Thus the typical interval of big kinks is about ω − .First, we divide the integration range of Eq. (6) into short intervals of length ∼ ω − aroundeach big kink as I i + ( k ) = X l I i + ,l ( k ) , (B1)where the integer l labels each big kink and I i + ,l ( k ) denotes the contribution to I i + ( k ) fromthe l -th interval. Each interval contains one big kink and numerous “small” kinks, whichsatisfy (cid:16) ψ dNdψ (cid:17) − . ω − ( ⇔ ψ . ψ max ). Let us assume that in the l -th interval a ′ i can bedecomposed as a ′ i ( u ) = ¯ a ′ il ( u ) + δa ′ il ( u ) . (B2)where ¯ a ′ i ( u ) denotes the smooth function (except one big kink) which we can get afteraveraging contributions of small kinks to a ′ i , and δa ′ il is the contribution of small kinks. δa ′ il discontinuously jumps at each small kink and the width of the jump is ∼ ψ / . Its averagevanishes ( h δa ′ il i = 0) since the jump at each kink takes random values. Then we get a i ( u ) = ¯ a il ( u ) + δa il ( u ) , (B3)after the integration of Eq. (B2). Then, I i + ,l ( k ) = Z l du ¯ a ′ il exp( iω ( u − n · ¯ a l ( u ) − n · δ a l ( u )) / Z l duδa ′ il exp( iω ( u − n · ¯ a l ( u ) − n · δ a l ( u )) / . (B4)Here the integral is performed over the l -th interval.We are interested in ensemble averages of products of two of I i + , for example, h| I i + | i . Itcontains mean squares of I i + ,l ’s and cross terms of different I i + ,l ’s. First, we evaluate meansquares of I i + ,l ’s. h| I i + ,l | i contains mean squares of the first and second terms in Eq. (B4),and the averages of the cross terms between them. The latter vanish since h δa ′ il i = 0. Inorder to estimate the mean square of the first term in Eq. (B4), we approximate it as Z l du ¯ a ′ il exp( iω ( u − n · ¯ a l ( u ) − n · δ a l ( u )) / ≃ X a ¯ a ′ il ( u l ; a ) e iω ( u l ; a − n · ¯ a ( u l ; a )) / e − iω n · δ a ( u l ; a ) / ∆ u l ; a . (B5)22ere we write the position of the a -th small kink in the l -th interval as u l ; a , and ∆ u l ; a = u l ; a +1 − u l ; a . We assume that each interval between two small kinks is so short that theintegrand can be regarded as constant. We want to evaluate the mean square of this quantity, X a,b h ¯ a ′ il ( u l ; a )¯ a ′ il ( u l ; b ) e iω ( u l ; a − n · ¯ a ( u l ; a )) / e − iω ( u l ; b − n · ¯ a ( u l ; b )) / ih e − iω n · δ a ( u l ; a ) / e iω n · δ a ( u l ; b ) / i ∆ u l ; a ∆ u l ; b (B6)Here we separate the average related to ¯ a ′ il , ¯ a il and that related to δa ′ il , δa il , assuming thatthere is no correlation between small kinks and big kinks. In order to evaluate the secondparenthesis in Eq. (B6), we decompose δa il as δa ′ il = X k F ( k ) il ( u ) , (B7)where F ( k ) il ( u ) is the contribution of kinks of sharpness ψ ∼ ψ k , so it has discontinuities atintervals ∼ (cid:16) ψ k dNdψ ( ψ k ) (cid:17) − and between two of them its absolute value ∼ ψ / k . Then,[ h| δ a l ( u l ; a ) − δ a l ( u l ; b ) | i ] / ∼ [ h| δa il ( u l ; a ) − δa il ( u l ; b ) | i ] / ∼ * X k X s F ( k ) il ( u kl,s )( u kl,s +1 − u kl,s ) ! + / ∼ "X k X s h ( F ( k ) il ( u kl,s )) i ( u kl,s +1 − u kl,s ) / ∼ "X k ψ k × (cid:18) ψ k dNdψ ( ψ k ) (cid:19) − × | u l ; a − u l ; b | / (cid:18) ψ k dNdψ ( ψ k ) (cid:19) − ! / ∼ "X k ψ k (cid:18) ψ k dNdψ ( ψ k ) (cid:19) − / | u l ; a − u l ; b | / ∼ "Z ψ max dψψ − (cid:18) dNdψ ( ψ ) (cid:19) − / | u l ; a − u l ; b | / ∼ (cid:18) dNdψ ( ψ max ) (cid:19) − / | u l ; a − u l ; b | / . (B8)To proceed from RHS of the first line to the second line, we regard F ( k ) il ( u ) as constant inthe interval between two small kinks. Here u kl,s denotes the position of s -th discontinuity of F ( k ) il ( u ). F ( k ) il ( u kl,s ) can be thought of as a probability variable whose average is 0 and whosevariance is ∼ ψ k . We can set D F ( k ) il ( u kl,s ) F ( k ′ ) il ( u k ′ l,s ′ ) E = 0 unless k = k ′ , s = s ′ , assuming23hat different kinks are not correlated. This enables the second line to be simplified to thethird line. Then we substitute ψ k into ( F ( k ) il ( u kl,s )) , and (cid:16) ψ k dNdψ ( ψ k ) (cid:17) − into u kl,s +1 − u kl,s . Thethird factor in the forth line represents the number of small kinks in the interval ( u l ; a , u l ; b ).When we proceed from the fifth line to the sixth line, we changed the sum P k to the integral R d (ln ψ ) = R dψψ − . Using ψ max dNdψ ( ψ max ) = ω and | u la − u lb | . ω − , we find( h| δ a l ( u l ; a ) − δ a l ( u l ; b ) | i ) / . ψ / ω − ≪ ω − . (B9)Therefore, ω n · ( δ a l ( u l ; a ) − δ a i ( u l ; b )) is much less than unity and h e iω n · ( δ a l ( u l ; b ) − δ a l ( u l ; a )) / i is ∼
1. Then Eq. (B6) is written as X a,b h ¯ a ′ il ( u l ; a )¯ a ′ il ( u l ; b ) e iω ( u l ; a − n · ¯ a ( u l ; a )) / e − iω ( u l ; b − n · ¯ a ( u l ; b )) / i ∆ u l ; a ∆ u l ; b = *(cid:12)(cid:12)(cid:12)(cid:12)Z l du ¯ a ′ il ( u ) e iω ( u − n · ¯ a ( u )) / (cid:12)(cid:12)(cid:12)(cid:12) + . (B10)The result is same as that derived without the contribution from small kinks. Then, theRHS of Eq. (B10) can be calculated as Eq. (26), and its magnitude is ψ / l ω ! , (B11)where ψ l denotes the sharpness of l -th big kink.In order to estimate the mean square of the second term in Eq. (B4), we approximate itas Z l duδa ′ il exp( iω ( u − n · ¯ a l ( u ) − n · δ a l ( u )) / ≃ X k X s F ( k ) il ( u kl,s ) exp( iδ kl,s )( u kl,s +1 − u kl,s ) , (B12)where δ kl,s = ω ( u kl,s − n · ¯ a l ( u kl,s ) − n · δ a l ( u kl,s )) /
2. The mean square of Eq. (B12) is * X k X s F ( k ) il ( u kl,s ) exp( iδ kl,s )( u kl,s +1 − u kl,s ) ! + = X k X s D ( F ( k ) il ( u kl,s )) E ( u kl,s +1 − u kl,s ) ∼ X k ψ k (cid:18) ψ k dNdψ ( ψ k ) (cid:19) − ω − / (cid:18) ψ k dNdψ ( ψ k ) (cid:19) − ! ∼ X k (cid:18) ψ k dNdψ ( ψ k ) (cid:19) − ω − ∼ Z ψ max dψψ − (cid:18) dNdψ ( ψ ) (cid:19) − ω − ∼ (cid:18) dNdψ ( ψ max ) (cid:19) − ω − . (B13)24emembering ψ max dNdψ ( ψ max ) = ω and ψ max < ψ l , we find (B13) < (B11). Eventually, themean square of I i + ,l is roughly estimated as (B11). In other words, in each interval aroundeach big kink, it is sufficient to consider only the isolated big kink, while neglecting smallkinks.Next, we consider the cross terms of different I i + ,l s, such as h I i + ,l I i ∗ + ,m i . This should vanish,and we can explicitly check this by straightforward calculation. To do so, we divide I i + ,l and I i + ,m as Eq. (B4) and evaluate the mean squares and the averages of the cross terms of thetwo term, using above approximations, such as Eqs. (B5) and (B12).As a result, the mean square of I i + can be evaluated by summing up Eq. (B11) for each l . Then we obtain (cid:10) | I i + | (cid:11) ∼ X l ψ l ω − ∼ Z ψ max dψ dNdψ ( ψ ) ψω − × L ∼ ψ max dNdψ ( ψ max ) × ψ max ω − × L, (B14)where L denotes the integration range of I i + . This implies that the greatest contributionto (cid:10) | I i + | (cid:11) comes from kinks which satisfy ψ ∼ ψ max . Such kinks dominantly contribute toGWs with frequency ∼ ω .So far we have discussed the case where ψ k dNdψ ( ψ k ) ∼ ω has a solution. However, if ω is solarge that ψ k dNdψ ( ψ k ) ≪ ω is satisfied for arbitrary values of sharpness, all kinks are thoughtof as “big kinks”. Therefore, the contribution from each interval of length ∼ ω around eachkink becomes (B11). That from regions far from any kinks is exponentially small when ω → ∞ . Eventually, (cid:10) | I i + | (cid:11) ∼ Z dψ dNdψ ( ψ ) ψω − × L ∼ ψ max dNdψ ( ψ max ) × ψ max ω − × L, (B15)where ψ max denotes the value of ψ at which dNdψ ( ψ ) has a peak. This implies that kinks whichsatisfy ψ ∼ ψ max dominantly contribute to (cid:10) | I i + | (cid:11) and GWs of frequency ∼ ω . Thus wehave proved the validity of Eq. (44). 25 ppendix C Here we discuss a subtlety related to validity to use the distribution function of kinks[Eqs. (15), (16) and (19)]. These formulae are derived without considering gravitationalbackreaction. The distribution may be altered if such an effect is taken into account. It maybe necessary to define the residual lifetime for blunt kinks and set lower cutoff of sharpness.It is difficult to clarify how we should take into account this effect at this moment. However,at least we can find a crude condition which must be satisfied regardless of the detail ofbackreaction; the energy of GWs emitted from strings must be less than the string energy.This condition is expressed as Z ω dω ′ dPdω ′ (cid:12)(cid:12)(cid:12)(cid:12) tot × t < µt. (C1)LHS represents the energy emitted from kinks on one infinite string in a Hubble horizonper Hubble time, and RHS denotes the energy of one infinite string in a Hubble horizon.(Note that this is only a necessary condition that the backreaction does not affect kinkdistribution.)First, let us assume that strings emerged in the radiation era. In the radiation era, thecondition (C1) is satisfied for t < (10 Gµ ) /E r t ∗ ∼ (10 Gµ ) − . t ∗ ( ⇔ T > (10 Gµ ) − / E r T ∗ ∼ (10 Gµ ) . T ∗ ) . (C2)( E ≡ ζ − β, E r ≃ − . , E m ≃ − . T ∗ < (10 Gµ ) / E r T eq ∼ (10 Gµ ) − . T eq . (C3)If we take Gµ ∼ − , this becomes T ∗ . GeV. In the matter era, the condition (C1)is written as t > (10 Gµ ) /E m (cid:18) T ∗ T eq (cid:19) − E r /E m t eq ∼ (10 Gµ ) . (cid:18) T ∗ T eq (cid:19) . t eq . (C4)The condition that the backreaction is not problematic in the matter era also leads to (C3).Eventually, if we assume that kinks had not appeared until friction domination ended orstrings emerged at low temperature at which friction can be neglected, the gravitationalbackreaction is not important.Next, let us assume that strings were born at the end of inflation. It is easy to seethat (C1) is satisfied in the first matter era, using Eq. (45). After the first matter era, the26ituation depends on whether the reheating temperature exceeds T c [Eq. (14)] or not. If T r < T c , there is no period when the friction works and kinks produced in the first matterera survive. The peak of ω dPdω | tot consists of contribution from kinks produced around theturning point from the first matter era to the radiation era. Therefore, the above discussionapplies and the condition (C1) is satisfied all the time. On the other hand, in the case of T r > T c , the friction becomes problematic in the early stage of the radiation dominatedera. If all kinks disappear in this stage and kinks restart to emerge at the end of friction-domination, the condition (C1) is never violated as discussed above. In the opposite case,where all kinks survive the friction-dominated era, (C1) is not guaranteed. In such a case,the largest contribution to R dω dPdω | tot comes from kinks produced around the end of the firstmatter era. The condition (C1) is satisfied if10 Gµ (cid:18) T r T eq (cid:19) − E r / < . (C5)For Gµ = 10 − , this leads to T r /M pl . − . Therefore the gravitational backreaction mightbe able to be neglected unless the reheating temperature is so high. Acknowledgments
K.N. would like to thank the Japan Society for the Promotion of Science for financialsupport. This work is supported by Grant-in-Aid for Scientific research from the Ministryof Education, Science, Sports, and Culture (MEXT), Japan, No.14102004 (M.K.) and No.21111006(M.K. and K.N.) and also by World Premier International Research Center Initia-tive (WPI Initiative), MEXT, Japan. [1] A. Vilenkin and E. P. S. Shellard, “Cosmic Strings and Other Topological Defects,”
CambridgeUniversity Press, Cambridge, England (1994).[2] S. Sarangi and S. H. H. Tye, Phys. Lett. B , 185 (2002) [arXiv:hep-th/0204074].[3] G. Dvali and A. Vilenkin, JCAP , 010 (2004) [arXiv:hep-th/0312007].[4] R. R. Caldwell and B. Allen, Phys. Rev. D , 3447 (1992); R. R. Caldwell, R. A. Battye andE. P. S. Shellard, Phys. Rev. D , 7146 (1996) [arXiv:astro-ph/9607130].
5] M. R. DePies and C. J. Hogan, Phys. Rev. D , 125006 (2007) [arXiv:astro-ph/0702335];arXiv:0904.1052 [astro-ph.CO].[6] T. Damour and A. Vilenkin, Phys. Rev. Lett. , 3761 (2000) [arXiv:gr-qc/0004075].[7] T. Damour and A. Vilenkin, Phys. Rev. D , 064008 (2001) [arXiv:gr-qc/0104026].[8] T. Damour and A. Vilenkin, Phys. Rev. D , 063510 (2005) [arXiv:hep-th/0410222].[9] X. Siemens, V. Mandic and J. Creighton, Phys. Rev. Lett. , 111101 (2007)[arXiv:astro-ph/0610920].[10] X. Siemens, K. D. Olum and A. Vilenkin, Phys. Rev. D , 043501 (2002)[arXiv:gr-qc/0203006].[11] J. Polchinski and J. V. Rocha, Phys. Rev. D , 083504 (2006) [arXiv:hep-ph/0606205].[12] C. Ringeval, M. Sakellariadou and F. Bouchet, JCAP , 023 (2007)[arXiv:astro-ph/0511646].[13] E. J. Copeland and T. W. B. Kibble, Phys. Rev. D , 123523 (2009) arXiv:0909.1960 [astro-ph.CO].[14] J. Garriga and M. Sakellariadou, Phys. Rev. D , 2502 (1993) [arXiv:hep-th/9303024].[15] A. Vilenkin, Phys. Rev. D , 1060 (1991).[16] G. R. Dvali, Q. Shafi and R. K. Schaefer, Phys. Rev. Lett. , 1886 (1994)[arXiv:hep-ph/9406319]; A. D. Linde and A. Riotto, Phys. Rev. D , 1841 (1997)[arXiv:hep-ph/9703209].[17] P. Binetruy and G. R. Dvali, Phys. Lett. B , 241 (1996) [arXiv:hep-ph/9606342]; E. Halyo,Phys. Lett. B , 43 (1996) [arXiv:hep-ph/9606423]; D. H. Lyth and A. Riotto, Phys. Lett.B , 28 (1997) [arXiv:hep-ph/9707273]; M. Endo, M. Kawasaki and T. Moroi, Phys. Lett.B , 73 (2003) [arXiv:hep-ph/0304126].[18] R. Jeannerot, Phys. Rev. D , 6205 (1997) [arXiv:hep-ph/9706391]; R. Jeannerot, J. Rocherand M. Sakellariadou, Phys. Rev. D , 103514 (2003) [arXiv:hep-ph/0308134].[19] S. Weinberg, “Gravitation and Cosmology,” John Wiley and Sons, 1972.[20] P. Binetruy, A. Bohe, T. Hertog and D. A. Steer, arXiv:0907.4522 [hep-th].[21] M. Kawasaki, K. Kohri and N. Sugiyama, Phys. Rev. Lett. , 4168 (1999)[arXiv:astro-ph/9811437]; Phys. Rev. D , 023506 (2000) [arXiv:astro-ph/0002127];S. Hannestad, Phys. Rev. D , 043506 (2004) [arXiv:astro-ph/0403291]; K. Ichikawa,M. Kawasaki and F. Takahashi, Phys. Rev. D , 043522 (2005) [arXiv:astro-ph/0505395].
22] M. Wyman, L. Pogosian and I. Wasserman, Phys. Rev. D , 023513 (2005) [Erratum-ibid. D , 089905 (2006)] [arXiv:astro-ph/0503364]; R. A. Battye, B. Garbrecht and A. Moss, JCAP , 007 (2006) [arXiv:astro-ph/0607339]; arXiv:1001.0769 [astro-ph.CO].[23] M. Kramer, arXiv:astro-ph/0409020.[24] J. Crowder and N. J. Cornish, Phys. Rev. D , 083005 (2005) [arXiv:gr-qc/0506015].[25] N. Seto, S. Kawamura and T. Nakamura, Phys. Rev. Lett. , 221103 (2001)[arXiv:astro-ph/0108011].[26] A. Buonanno, G. Sigl, G. G. Raffelt, H. T. Janka and E. Muller, Phys. Rev. D , 084001(2005) [arXiv:astro-ph/0412277].[27] T. L. Smith, M. Kamionkowski and A. Cooray, Phys. Rev. D , 023504 (2006)[arXiv:astro-ph/0506422].[28] A. J. Farmer and E. S. Phinney, Mon. Not. Roy. Astron. Soc. , 1197 (2003)[arXiv:astro-ph/0304393].[29] N. Seto and J. Yokoyama, J. Phys. Soc. Jap. , 3082 (2003) [arXiv:gr-qc/0305096].[30] L. A. Boyle and P. J. Steinhardt, Phys. Rev. D , 063504 (2008) [arXiv:astro-ph/0512014];L. A. Boyle and A. Buonanno, Phys. Rev. D , 043531 (2008) [arXiv:0708.2279 [astro-ph]].[31] K. Nakayama, S. Saito, Y. Suwa and J. Yokoyama, Phys. Rev. D , 124001 (2008)[arXiv:0802.2452 [hep-ph]]; JCAP , 020 (2008) [arXiv:0804.1827 [astro-ph]]; K. Nakayamaand J. Yokoyama, JCAP , 010 (2010) [arXiv:0910.0715 [astro-ph.CO]]., 010 (2010) [arXiv:0910.0715 [astro-ph.CO]].