Abstract
We compute the contribution of kinks on cosmic string loops to stochastic background of gravitational waves (SBGW). We find that kinks contribute at the same order as cusps to the SBGW. We discuss the accessibility of the total background due to kinks as well as cusps to current and planned gravitational wave detectors, as well as to the big bang nucleosynthesis (BBN), the cosmic microwave background (CMB), and pulsar timing constraints. As in the case of cusps, we find that current data from interferometric gravitational wave detectors, such as LIGO, are sensitive to areas of parameter space of cosmic string models complementary to those accessible to pulsar, BBN, and CMB bounds.
Introduction
Topological defects are remnants of spontaneously broken local or global symmetries. The simplest and the most well-known example of the former one is the Abrikosov-Nielsen-Olesen flux tube [1], which originates from spontaneously broken \(U(1)\) gauge symmetry. Most of the attention in the literature has been focused on defects originating from broken gauge symmetries, since grand unified theories have gauge symmetries which are eventually spontaneously broken down to the symmetry of the Standard Model. Cosmic strings are one dimensional topological defects predicted by a large class of unified theories [2]–[4]. Cosmic strings were first considered as the seeds of structure formation [5], [6], however, later, it was discovered that cosmic strings were incompatible with the cosmic microwave background (CMB) angular power spectrum. Cosmic strings can still contribute to structure formation, but they cannot be the dominant source. Cosmic strings are also candidates for the generation of other observable astrophysical phenomena such as high energy cosmic rays, gamma ray burst and gravitational waves [3], [7]–[9]. Furthermore, recently it has been shown that in string-theory-inspired cosmological scenarios cosmic strings may also be generated[10]. They are referred to as cosmic superstrings. This realization has revitalized interest in cosmic strings and their potential observational signatures. There are some important differences between cosmic strings and cosmic superstrings. The reconnection probability is unity for cosmic strings [3], [11]. Cosmic superstrings, on the other hand, have reconnection probability less than unity. This is a result of the probabilistic nature of their interaction and also the fact that it is less probable for strings to meet since they can live in higher dimensions[12]. The value of \(p\) ranges from \(10^{-3}\) to \(1\) in different theories [13]. Cosmic superstrings could also be unstable, decaying long before the present time. In this case, however, they may also leave behind a detectable gravitational wave signature[14].
In the early universe, a network of cosmic strings evolves toward to an attractor solution called the ``scaling regime". In the scaling regime the statistical properties of the network, such as the average distance between strings and the size of loops at formation, scale with the cosmic time. In addition, the energy density of the network remains a small constant fraction of the energy density of the universe. For cosmic superstrings in the scaling regime, the density of the network \(\rho\) is inversely proportional to the reconnection probability \(p\) , that is \(\rho\propto p^{-\beta}\) . The value of \(\beta\) is still under debate[15]–[17], and as a placeholder in our analysis we assume that \(\beta=1\) .
The gravitational interaction of strings is characterized by their tension \(\mu\) , or more conveniently by the dimensionless parameter \(G \mu\) , where \(G\) is Newton’s constant. The current CMB bound on the tension is \(G \mu<6.1\times 10^{-7}\) [18], [19]. It was first believed that gravitational radiation from cosmic strings with \(G \mu\ll 10^7\) would be too weak to observe. However it was later shown that gravitational radiation produced at cusps, which have large Lorentz boosts, could lead to a detectable signal[20]–[22]. Gravitational radiation bursts from (super)strings could be observable by current and planned gravitational wave detectors for values of \(G \mu\) as low as \(10^{-13}\) , which may provide a test for a certain class of string theories [23]. Indeed, searches for burst signals using ground-based detectors are already underway[24].
A gravitational background produced by the incoherent superposition of cusp bursts from a network of cosmic strings and superstrings was considered in [25]. In this paper we extend this computation to include kinks, long-lived sharp edges on strings that result from intercommutations, and find that kinks contribute at almost the same level as cusps. We investigate the detectability of the total background produced by cusps and kinks by a wide range of current and planned experiments. A similar calculation for the case of infinite strings has been undertaken in the recent paper [26], see also [27].
The organization of the paper is as follows: In Sect. Gravitational Radiation we consider gravitational waves generated by cusps and kinks in the weak field limit [28]. In this section we follow the conventions of [20], [21], and more details can be found in these references. In Sect. Stochastic Background we derive the expression for the stochastic background, which is a double integral over redshift and loop length. In Sect. Analytical Approximation we evaluate integral analytically with certain approximations, which results in a flat distribution for larger values of the frequency. Finally in Sec. Parameter Space Constraints we numerically evaluate the background and discuss the observability by various experiment.
Gravitational Radiation
In this section we consider gravitational waves created by cusps and kinks. For completeness we follow closely the analysis in [20], [21], and reproduce a number of their results. We begin with a derivation for the metric pertubation in terms of the Fourier transform of the stress energy tensor of the source. We then write the stress energy tensor for a relativistic string and compute its Fourier transform. Using these results we then compute the gravitational waveforms produced by cusps and kinks on cosmic strings.
Calculation of metric perturbations
Gravitational waves from a source can be calculated using the weak field approximation [28], \[\begin{equation} \tag{1} g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, \end{equation}\] where \(\eta_{\mu\nu}\) is the Minkowski metric with positive signature and \(h_{\mu\nu}\) is the metric perturbation. In the harmonic gauge, \(g^{\mu\nu}\Gamma^\lambda_{\mu\nu}=0\) , the linearized Ricci tensor is \[\begin{equation} \tag{2} R_{\mu\kappa}\simeq \frac{1}{2}\partial_\lambda\partial^\lambda h_{\mu\kappa}. \end{equation}\] Substituting into Einstein’s equations yields \[\begin{equation} \tag{3} R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R\simeq \frac{1}{2}(\partial_\lambda\partial^\lambda h_{\mu\nu}-\frac{1}{2} \eta_{\mu\nu}\partial_\lambda\partial^\lambda h)=-8\pi G {\mathcal T}_{\mu\nu}, \end{equation}\] where \(R\) is the Ricci scalar, \({\mathcal T}_{\mu\nu}\) is the energy momentum tensor of matter and \(h=\eta_{\mu\nu}h^{\mu\nu}\) . Defining \(\bar{h}_{\mu\nu}= h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\) further simplifies Eq. (3), \[\begin{equation} \tag{4} \partial_\lambda\partial^\lambda \bar{h}_{\mu\nu}=-16\pi G {\mathcal T}_{\mu\nu}, \end{equation}\] which is a wave equation with a source term. We can rewrite this equation in the frequency domain as, \[\begin{equation} \tag{5} (w^2+\nabla^2)\bar{h}_{\mu\nu}(\vec{x},w)=-16\pi G {\mathcal T}_{\mu\nu}(x,w), \end{equation}\] where \[\begin{equation} \tag{6} \bar{h}_{\mu\nu}(\vec{x},w)=\int dt\,e^{i w t}\bar{h}_{\mu\nu}(\vec{x},t). \end{equation}\] Eq. (5) can be solved by using the Green’s function for the operator \(w^2+\nabla^2\) , which is \[\begin{equation} \tag{7} {\mathcal G}(\vec{x}-\vec{x}',w)=\frac{e^{i w |\vec{x}-\vec{x}'|}}{|\vec{x}-\vec{x}'|}. \end{equation}\] Therefore the metric perturbations are given by \[\begin{eqnarray} \bar{h}_{\mu\nu}(\vec{x},w)&=& -16 \pi G \int d^3x'{\mathcal G}(\vec{x}-\vec{x}',w){\mathcal T}_{\mu\nu}(\vec{x}',w) =-16 \pi G \frac{e^{i w |\vec{x}|}}{|\vec{x}|} {\mathcal T}_{\mu\nu}(\vec{k},w), \tag{8} \end{eqnarray}\] where \(\vec{k}=w \hat{x}\) and \[\begin{eqnarray} \tag{9} {\mathcal T}_{\mu\nu}(\vec{k},w)&=&\frac{1}{T}\int_0^T dt\int d^3x'e^{i(wt- \vec{k}\cdot\vec{x}')}{\mathcal T}_{\mu\nu}(\vec{x}',t), \end{eqnarray}\] where \(T\) is the fundamental period of the source. Eq. (8) relates energy momentum tensor to gravitational waves. The next step is to calculate the energy momentum tensor of cusps and kinks on cosmic strings.
Energy Momentum Tensor of Cosmic Strings
In the thin wire approximation, the dynamics of strings is described by the Nambu-Goto action [3], [7] \[\begin{eqnarray} \tag{10} S=-\mu \int d\tau d\sigma \sqrt{-\gamma}, \end{eqnarray}\] where \(\sigma\) and \(\tau\) are world-sheet coordinates and \(\mu\) isf the string tension. \(\gamma\) is the determinant of the induced metric \[\begin{equation} \tag{11} \gamma_{a\,b}=\eta_{\mu\nu}\partial_a X^\mu\partial_b X^\nu, \end{equation}\] where \(a\) and \(b\) denote world sheet coordinates. The equation of motion following from Eq. (10) is \[\begin{eqnarray} \tag{12} &&(\partial_\tau^2-\partial_\sigma^2)X^\mu=0, \end{eqnarray}\] The solution must also satisfy Virasoro conditions \[\begin{eqnarray} \tag{13} &&\dot X\cdot\dot X+X'\cdot X'=0\;\; \rm{and} \;\;\ \dot X\cdot X'=0, \end{eqnarray}\] where \(dot\) and \(prime\) denote derivatives with respect to \(\tau\) and \(\sigma\) respectively. If we define \(\sigma_\pm= \tau\pm\sigma\) , the equation of motion becomes \[\begin{eqnarray} \tag{14} \partial_+\partial_-X^\mu&=&0, \end{eqnarray}\] which is solved by left and right moving waves, \[\begin{eqnarray} \tag{15} X^\mu&=&\frac{1}{2}\left(X^\mu_+(\sigma_+)+X^\mu_-(\sigma_-)\right). \end{eqnarray}\] Furthermore Virasoro conditions in Eq. (13) simplify to \[\begin{eqnarray} \tag{16} &&{\bf\dot X_\pm}^2=1, \end{eqnarray}\] where \(dot\) now represents the derivative with respect to the (unique) argument of the functions \(X^\mu_\pm\) . We require that \(X^\mu(\sigma,\tau)\) is periodic in \(\sigma\) with period \(l\) , which is the length of the loop. This implies that the functions \(X^\mu_\pm\) are periodic functions with the same period. The period in \(t\) is \(l/2\) since \(X^\mu(\sigma+l/2,\tau+l/2)=X^\mu(\sigma,\tau)\) . The energy momentum tensor corresponding to the Nambu-Goto action can be calculated by varying Eq. (10) with respect to the metric, which yields \[\begin{eqnarray} {\mathcal T}_{\mu\nu}(x)&=&-2 \frac{\delta S}{\delta \eta_{\mu\nu}}=\mu\int d\tau d\sigma (\dot X^\mu \dot X^\nu- X'^\nu X'^\mu)\,\delta^{(4)}(x-X) =\frac{\mu}{2}\int d\sigma_-d\sigma_+ (\dot X_+^\mu \dot X_-^\nu+ \dot X_-^\nu \dot X_-^\mu)\,\delta^{(4)}(x-X). \tag{17} \end{eqnarray}\] Inserting this expansion into Eq. (9) gives us the energy momentum tensor in momentum space \[\begin{eqnarray} \tag{18} {\mathcal T}_{\mu\nu}(k)&=&\frac{\mu}{T_l}\int d\sigma_-d\sigma_+ \dot X_+^{(\mu} \dot X_-^{\nu)}e^{-\frac{i}{2}(k\cdot X_++k\cdot X_-)}, \end{eqnarray}\] where we define \[\begin{eqnarray} \tag{19} \dot X_+^{(\mu} \dot X_-^{\nu)}&=&\frac{1}{2}(\dot X_+^{\mu} \dot X_-^{\nu}+\dot X_-^{\mu} \dot X_+^{\nu}). \end{eqnarray}\] The nice property of Eq. (18) is that two integrals can be calculated independently, \[\begin{eqnarray} \tag{20} I_\pm^\mu(k)&\equiv&\int_0^l d\sigma_\pm \dot X_\pm^{\mu} e^{-\frac{i}{2}k\cdot X_\pm}, \end{eqnarray}\] and the energy momentum tensor can be expressed in terms of \(I_\pm^\mu\) as follows; \[\begin{eqnarray} \tag{21} {\mathcal T}_{\mu\nu}(k)&=&\frac{\mu}{l}I_+^{(\mu} I_-^{\nu)}, \end{eqnarray}\] where we used \(T_l=\frac{l}{2}\) . In the following subsection we calculate \(I_\pm^\mu\) for cusps and kinks.
Cusps
Let us start with the geometrical interpretation of Eq.
(16). It tells us that \({\bf\dot X_\pm}\) trace a unit
sphere centered at the origin, which is called Kibble-Turok sphere.
Integrating \({\bf\dot X_\pm}\) and using the periodicity, we get
\[\begin{equation}
\tag{22}
\int_0^l{\bf\dot X_\pm}(\sigma_\pm)d\sigma_\pm=0,
\end{equation}\]
which implies that \({\bf\dot X_\pm}\) cannot lie completely in a
single hemisphere and therefore they intersect at some point(s). We
choose our parametrization and the coordinate system such that the
intersection occurs at the parameters \(\sigma_\pm=0\) at the origin,
that is \(X_\pm^\mu(0)=0\) . \(X_\pm(\sigma_\pm)\) and \(\dot X_\pm(\sigma_\pm)\) can be expanded around \(\sigma_\pm=0\)
\[\begin{eqnarray}
X_\pm^\mu(\sigma_\pm)&=&l_\pm^\mu \sigma_\pm+\frac{1}{2}\ddot X_\pm^\mu
\sigma_\pm^2 +\frac{1}{6}X_\pm^{(3)\mu}\sigma_\pm^3
\tag{23}
\end{eqnarray}\]
and
\[\begin{eqnarray}
\dot X_\pm^\mu(\sigma_\pm) &=&l_\pm^\mu +\ddot X_\pm^\mu \sigma_\pm
+\frac{1}{2}X_\pm^{(3)\mu}\sigma_\pm^2.
\tag{24}
\end{eqnarray}\]
where \(l_\pm^\mu=\dot X_\pm^\mu(0)\) . We can easily find the shape of
\(X_\pm^\mu\) at \(\tau=0\) (\(\sigma_\pm=\pm\sigma\) ),
\[\begin{eqnarray}
\tag{25}
X^\mu(\sigma,\tau=0)&=&\frac{1}{2}\left(X_+^\mu(\sigma)+X_-^\mu(-\sigma)\right)
=\frac{1}{4}(\ddot X_+^\mu+\ddot X_-^\mu)\sigma^2
+\frac{1}{12}(X_+^{(3)\mu}+X_-^{(3)\mu})\sigma^3.
\end{eqnarray}\]
In order to visualize the shape of the string around the origin, we
can choose the coordinate system such that \((\ddot{\vec{X}_+}+\ddot{ \vec{X}_-})\) lies on the \(x\) -axis, and define
\(x=\frac{1}{4}|\ddot{\vec{X}_+}+\ddot{ \vec{X}_-}|\sigma^2\) . Let us
also denote the direction of \(\vec{X}_+^{(3)}+\vec{X}_-^{(3)}\) by
\(\hat{y}\) , which is not necessarily orthogonal to \(\hat{x}.\) If we
define \(y=\frac{1}{12}|X_+^{(3)\mu}+X_-^{(3)\mu}|\sigma^3\) , we see
that \(y\propto x^{\frac{3}{2}}\) , which has a sharp turn at \(x=0\) , which is referred to as cusp.
We can calculate \(I_\pm^\mu\) for cusps using the expansion in Eq. (23). First of all, we note that the first
term Eq. (24) is pure gauge, it can be
removed by a coordinate transformation. Furthermore imposing
Virasoro condition in Eq. (16) gives
\[\begin{equation}
\tag{26}
l_\pm\cdot\ddot X_\pm=0,\;\;\;\rm{and}\;\;\; l_\pm\cdot X^{(3)}_\pm=-\ddot
X^2_\pm.
\end{equation}\]
When the line of sight \(k\) is in the direction of \(l\) we have \(k= w l\) , which gives
\(-i\,k\cdot X_\pm = \frac{i}{6}w\ddot X_\pm^2 \sigma_\pm^3\) .
If we plug in the expansion in Eq. (23) into Eq.
(20) we get,
\[\begin{eqnarray}
\tag{27}
I_\pm^\mu(k)&=&\ddot X_\pm^\mu\int_0^l d\sigma\, \sigma
e^{\frac{i}{12} w \ddot X_\pm^2 \sigma^3}=\frac{2 \pi i \ddot
X_\pm^\mu}{3 \Gamma(1/3)\left(\frac{1}{12} w |\ddot
X_\pm^2|\right)^{2/3}} .
\end{eqnarray}\]
Replacing \(w\) with \(2 \pi f\) gives
\[\begin{eqnarray}
I_\pm^\mu(k)&=&C_\pm^\mu f^{-\frac{2}{3}},\\
{\mathcal T}_{\mu\nu}(k)&=&\frac{\mu}{l}|f|^{-\frac{4}{3}}
C_+^{(\mu} C_-^{\nu)}
\tag{28}
\end{eqnarray}\]
where \(C_\pm^\mu= i \frac{(32 \pi/3)^{1/3}}{\Gamma(1/3)} \frac{\ddot X_\pm^\mu}{|\ddot X_\pm|^{\frac{4}{3}}}\) . Finally we need to
estimate \(|\ddot X_\pm|=|{\bf\ddot X_\pm}|\) . Since \({\bf X}_\pm\) is
periodic with period \(l\) , \({\bf \dot X}\) expanded as
\[\begin{equation}
\tag{29}
{\bf \dot X}(\sigma_\pm)=\sum_n {\bf c_n} e^{i \frac{2 \pi}{l}n
\sigma_\pm},
\end{equation}\]
where the expansion coefficients \({\bf c_n}\) are constrained by
\(|\dot {\bf X}_\pm|=1\) . If the string is not too wiggly, \({\bf c_n}\)
is nonvanishing for only small \(n\) , therefore we can estimate
\(|\ddot X_\pm|\sim \frac{2 \pi}{l}\) . Combining all the pieces
together and neglecting decimal points in the numerical coefficient,
we express the trace of the metric perturbations as
\[\begin{eqnarray}
\tag{30}
h^{(c)}(f)\equiv |\bar h^\mu_\mu|= \frac{G \mu l^{\frac{2}{3}}
}{r}|f|^{-\frac{4}{3}}.
\end{eqnarray}\]
We can express \(r\) as a function of \(z\)
\[\begin{equation}
\tag{31}
r=\frac{1}{H_0}\int_0^z\frac{dz'}{{\mathcal H}(z')}\equiv \frac{1}{H_0}\varphi_r(z),
\end{equation}\]
where \(H_0\) is the Hubble constant today and \({\mathcal H}(z)\) is
the Hubble function given by
\[\begin{equation}
\tag{32}
{\mathcal H}(z)=\left(\Omega_M(1+z)^3+\Omega_R(1+z)^4+\Omega_\Lambda\right)^{1/2}.
\end{equation}\]
The numerical values for the constants in this equation are
\(\Omega_M=0.25\) , \(\Omega_R=4.6\times 10^{-5}\) ,
\(\Omega_\Lambda=1-\Omega_R-\Omega_M\) and \(H_0= 73 \rm{km/s/Mpc}\) .\
Note that \(f\) in Eq. (30) is the frequency of the
radiation in the frame of emission. In order to convert it to the
frequency we observe today, the effect of the cosmological redshift
must be included. The frequency in the frame of emission, \(f\) , is
related to the frequency we observe now, \(f_{\rm{now}}\) , by the
relation \(f=(1+z)f_{\rm{now}}\) . After redshifting properly, Eq. (30) becomes
\[\begin{eqnarray}
\tag{33}
h^{(c)}(f,z,l)= \frac{G \mu H_0\,l^{\frac{2}{3}}
}{(1+z)^{\frac{1}{3}}\varphi_r(z)}|f|^{-\frac{4}{3}},
\end{eqnarray}\]
where we dropped the subscript now.
Kinks
Calculation of kink radiation is similar to the cusp case. The form of \(I_+^\mu\) is the same as the cusp result. \(I_-^\mu\) has a discontinuity at the cusp point and needs a different treatment. Let us describe the kink (at \(\sigma_-=0\) and \(X_\pm=0\) ) as a jump of the tangent vector from \(l_1^\mu\) to \(l_2^\mu\) . At the first order one can replace approximate \(\dot X_-^\mu\) by \(l_1^\mu\) for \(\sigma_-<0\) and \(l_2^\mu\) for \(\sigma_->0\) . At this approximation, one gets \[\begin{eqnarray} \tag{34} I_-^\mu(k)&=&\int_{-l/2}^{l/2} d\sigma_-\dot X_-^\mu e^{-\frac{i}{2} k \cdot X_-} \simeq \frac{2 i}{w}\left( \frac{l_1^\mu}{l_1\cdot \hat k}-\frac{l_2^\mu}{l_2\cdot \hat k}\right), \end{eqnarray}\] where we dropped two oscillatory terms. The exact value of Eq. (34) depends on the sharpness of the kink, \(l_1\cdot l_2\) [29], however we will assume that the average value of this quantity is of order one. Combining this result with \(I_+^\mu\) we get the frequency distribution of the radiation from a kink as \[\begin{eqnarray} \tag{35} h^{(K)}(f,z,l)= \frac{G \mu l^{\frac{1}{3}} H_0}{(1+z)^{\frac{2}{3}}\varphi_r(z)}f^{-5 /3}. \end{eqnarray}\] It is important to note that in the derivation of Eqs. (33) and (35) we assumed that the line of sight \(k^\mu\) is in the direction of the motion of the cusp or kink, \(l^\mu\) . It is easy to show that \(I_\pm\) (Eq. (20)) decay exponentially with the angle between \({\bf k}\) and \({\bf l}\) [21] . Therefore Eqs. (33) and (35) are valid for angles smaller than \[\begin{equation} \tag{36} \theta_m=\frac{1}{\left(f l (1+z)\right)^{\frac{1}{3}}}. \end{equation}\] We implement this condition with a \(\Theta\) -function in the amplitude.
Stochastic Background
The stochastic gravitational background [25] is given by
\[\begin{eqnarray}
\tag{37}
\Omega_{gw}(f)&=&\frac{4\pi^2}{3 H_0^2}f^3\int dz\int dl\,
h^2(f,z,l)\frac{d^2R(z,l)}{dz dl},
\end{eqnarray}\]
where \(h(f,z,l)\) is given in Eqs. (33)
and (35) and \(\frac{d^2R(z,l)}{dz dl}\) is
the observable burst rate per length per redshift, which will be
defined below. We take the number of cusps (kinks) to be one per
loop. If we define the density (per volume) of the loops of length
\(l\) at time \(t\) as \(n(l,t)\) , the rate of burst (per loop length per
volume) can be expressed as \(\frac{ n(l,t)}{l/2}\) , where \(l/2\)
factor is the fundamental period of the string. However, this is not
the observable burst rate since we can observe only the fraction of
bursts that is beamed toward us. Including this fraction we obtain
\[\begin{equation}
\tag{38}
\frac{dR}{dl dz}=H_0^{-3}\varphi_V(z)(1+z)^{-1}
\frac{2 n(l,t)}{l}\Delta(z,f,l),
\end{equation}\]
where \((1+z)^{-1}\) comes from converting emission rate to observed
rate, and \(H_0^{-3}\varphi_V(z)\) follows from converting
differential volume element to the corresponding function of
redshift \(z\) ,
\[\begin{equation}
\tag{39}
dV=4\pi a^3(t) r^2 dr=\frac{4\pi H_0^{-3}\varphi^2_r(z)}{(1+z)^3 {\mathcal H}(z)}dz
\equiv H_0^{-3} \varphi_V(z) dz,
\end{equation}\]
where \(a(t)\) is the cosmological scale factor. \(\Delta(z,f,l)\) is
the fraction of the bursts we can observe. Geometrically the
radiation from a cusp will be in a conic region with half opening
angle \(\theta_m\) (Eq. (36)) and outside the cone it
will decay exponentially. To simplify the calculation we assume that
the radiation amplitude vanishes outside this conic region, which
will be implemented by a \(\Theta\) -function. We can express the
corresponding solid angle in terms of the opening angle by using
the following relation
\[\begin{equation}
\tag{40}
\Omega_m=2 \pi (1-\cos\theta_m)\simeq \pi \theta_m^2.
\end{equation}\]
Thus the probability that the line of sight is within this solid
angle is
\[\begin{equation}
\tag{41}
\frac{\Omega_m}{4 \pi}\simeq \theta_m^2/4,
\end{equation}\]
which is referred to as the beaming fraction of the cusp.
We combine the cutoff for large angles and beaming effect into
\[\begin{equation}
\tag{42}
\Delta(z,f,l)\approx\frac{\theta_m^2(z,f,l)}{4}
\Theta({1-\theta_m(z,f,l)}).
\end{equation}\]
It is important to note that cusps are instantaneous events, and it is possible
to observe their radiation only if the line of sight happens to be inside
the cone of radiation. The beaming fraction, Eq.
(41), which is proportional to \(\theta_m^2\) , is the fraction of the time the line of sight is inside the cone of radiation. In contrast,
kinks radiate continuously–as kinks travel around a string loop they radiate in a fan-like pattern.
Therefore radiation cone of a kink will
sweep a strip of width \(2 \theta_m\) and an average length \(\pi\) on
the surface of the unit sphere as it travels around the cosmic string loop. That is, the probability of
observing radiation from a kink is
\[\begin{equation}
\tag{43}
\frac{\Omega^c_m}{4 \pi}\simeq \frac{2 \theta_m \pi}{4 \pi}=\frac{ \theta_m
}{2}.
\end{equation}\]
For kinks the cutoff for large angles and beaming factor that enters the rate is therefore
\[\begin{equation}
\tag{44}
\Delta^{(K)}(z,f,l)\approx\frac{\theta_m(z,f,l)}{2}
\Theta({1-\theta_m(z,f,l)}).
\end{equation}\]
Inserting this result into Eq. (37) gives the
background radiation \(\Omega_{gw}(f)\) as a double integral over
\(l\) and \(z\) , which needs to be evaluated numerically. Finally we
need to discuss the form of the loop density, \(n(l,t)\) in Eq. (38).
To do this, it is convenient to first convert the cosmic time \(t\) to a suitable
function of redshift \(z\) using the following relation
\[\begin{equation}
\tag{45}
\frac{dz}{dt}=-(1+z)H_0 \,{\mathcal H}(z),
\end{equation}\]
which can be integrated to give
\[\begin{equation}
\tag{46}
t=H_0^{-1} \int_z^\infty \frac{dz'}{(1+z'){\mathcal H}(z')}= H_0^{-1}
\varphi_t(z).
\end{equation}\]
Below we discuss the two main contending scenarios for the size
of cosmic string loops.
Small Loops
Early simulations suggested that the size of loops was dictated by
gravitational back reaction. In this case the size of the loops is
fixed by the cosmic time \(t\) , and all the loops present at a cosmic
time \(t\) , are of the same size \(\alpha\, t\) . The value of \(\alpha\)
is set by the gravitational back reaction, that is \(\alpha \propto \Gamma G \mu\) (In Sect. we parameterize
\(\alpha\) by \(\alpha =\epsilon\Gamma G \mu\) where \(\epsilon\) is a
parameter we scan over.) The constant \(\Gamma\) is the ratio of the
power radiated into gravitational waves by loops to \(G\mu^2\) .
Numerical simulation results suggest that \(\Gamma \sim 50\) .
Therefore the density is of the form
\[\begin{equation}
n(l,t) \propto(p \,\Gamma G \mu)^{-1} t^{-3} \delta(l-\alpha t),
\tag{47}
\end{equation}\]
where \(p\) is the reconnection probability. The overall coefficient
is estimated by simulations (for a review see[3]) which show
that the density in the radiation domination era is about \(10\) times
larger the one in the matter domination era. This behavior of the
density can be implemented by a function, \(c(z)\) , which converges to
\(10\) for \(z\gg z_{eq}\) and to \(1\) for \(z\ll z_{eq}\) . Therefore the
density can be written as
\[\begin{equation}
n(l,t) = c(z)(p \,\Gamma G \mu)^{-1} t^{-3} \delta(l-\alpha t),
\tag{48}
\end{equation}\]
where[20]
\[\begin{equation}
c(z)=1+\frac{9 z}{z+z_{eq}}.
\tag{49}
\end{equation}\]
Such a distribution simplifies the calculation of SBGW since the
\(l\) -integral in Eq. (37) can be evaluated trivially to
yield
\[\begin{eqnarray}
\Omega_{gw}(f)&=&\frac{4\pi^2}{3 H_0^2}\int dz\int dl\,
h^2(f,z,l)\frac{d^2R(z,l)}{dz dl}
=\frac{2 \,c \,G\mu\, \pi^2
H_0^{1/3}}{3 \,p\,\alpha^{1/3}\Gamma f^{1/3}}\int dz\frac{c(z)\varphi_V
\Theta\left(1-\left[f(1+z)\alpha \varphi_t\right]^{-1/3}\right) }
{(1+z)^{7/3}\varphi_r^2\varphi_t^{10/3}
}.
\tag{50}
\end{eqnarray}\]
For kinks, we have a similar integral ,
\[\begin{eqnarray}
\tag{51}
\Omega^K_{gw}(f)&=&\frac{4 \,c \,G\mu\, \pi^2
H_0^{1/3}}{3 \,p\,\alpha^{2/3}\Gamma f^{2/3}}\int dz\frac{c(z)\varphi_V
\Theta\left(1-\left[f(1+z)\alpha \varphi_t\right]^{-1/3}\right) }
{(1+z)^{8/3}\varphi_r^2\varphi_t^{11/3}
}.
\end{eqnarray}\]
We analytically evaluate the integrals in Eqs. (50) and (51) in Sec.
with certain approximations, and perform numerical integration in
Sec. .
Large Loops
Recent simulations [30]–[32] suggest that the size of the loops is set by the large scale dynamics of the network, and that the gravitational back-reaction scale is irrelevant. In Ref. [32] is found that the loop production functions have peaks around \(\alpha\approx0.1\) , which is the value we use below (for large loop case). For long-lived loops, the distribution can be calculated if a scaling process is assumed (see [3]). In the radiation era it is \[\begin{eqnarray} n(l,t)&=&\chi_r t^{-\frac{3}{2}} (l+\Gamma G \mu t)^{-\frac{5}{2}}, \nonumber \\ &\,& l < \alpha \,t,\,\,\, t<t_{t_{eq}} \tag{52} \end{eqnarray}\] where \(\chi_r \approx 0.4 \zeta \alpha^{1/2}\) , and \(\zeta\) is a parameter related to the correlation length of the network [22]. The numerical value of \(\zeta\) is found in numerical simulations of radiation era evolution to be about \(15\) (see Table 10.1 in [3]). The upper bound on the length arises because no loops are formed with sizes larger than \(\alpha t\) . For \(t>t_{eq}\) (the matter era) the distribution has two components, loops formed in the matter era and survivors from the radiation era. Loops formed in the matter era have lengths distributed according to, \[\begin{eqnarray} n_{1}(l,t)&=&\chi_m t^{-2} (l+\Gamma G \mu t)^{-2}, \nonumber \\ && \alpha t_{t_{eq}} - \Gamma G \mu (t-t_{t_{eq}}) < l < \alpha t,\,\,\, t>t_{t_{eq}} \tag{53} \end{eqnarray}\] with \(\chi_m \approx 0.12\zeta\) , with \(\zeta \approx 4\) (see Table 10.1 in [3]). The lower bound on the length is due to the fact that the smallest loops present in the matter era started with a length \(\alpha\, t_{eq}\) when they were formed and their lengths have since decreased due to gravitational wave emission. Additionally there are loops formed in the radiation era that survive into the matter era. Their lengths are distributed according to, \[\begin{eqnarray} n_{2}(l,t)&=&\chi_r t_{eq}^{1/2}\, t^{-2} (l+\Gamma G \mu t)^{-\frac{5}{2}}, \text{ for } l < \alpha\, t_{t_{eq}}- \Gamma G \mu (t-t_{t_{eq}}),\,\,\, \text{ and } t>t_{t_{eq}}, \tag{54} \end{eqnarray}\] where the upper bound on the length comes from the fact that the largest loops formed in the radiation era had a size \(\alpha\, t_{eq}\) but have since shrunk due to gravitational wave emission.
The cusp spectrum has been calculated in [25] and the result shows that the spectrum is flat for {} values of \(f\) . Later we will show that this is also the case
for kink spectrum. This is rather unexpected since \(\Omega(f)\) has
an explicit \(f^{-\frac{4}{3}}\) and \(f^{-\frac{1}{3}}\) dependence for
cusps and kinks, respectively. The only other \(f\) dependence comes
from the \(\Theta\) functions. In the following section we show
analytically that the \(f\) dependence coming from the \(\Theta\)
function is of the form \(f^{\frac{4}{3}}\) and \(f^{\frac{1}{3}}\) for
cusps and kinks respectively so that the spectrum is indeed flat for
large values of the frequency \(f\) .
Figure 1: Kink and Cusp spectrum for small loops: (1) \(G\mu=2 \times10^{-6}\), \(p=10^{-3}\) and \(\epsilon=10^{-4}\), (2) \(G\mu=10^{-7}\),\(p=5\times10^{-3}\) and \(\epsilon=1\)
Figure 2: Kink and Cusp spectrum for large loops: (1) \(G\mu= 10^{-7}\) and \(p=5\times 10^{-3}\) , (2) \(G\mu=10^{-9}\) and \(p=5\times 10^{-2}\) .
Here we note that for \(f\gg\frac{H_0}{G\mu}\) , the spectrum is flat for both cusps and kinks.
Analytical Approximation
In this section we evaluate the spectrum
analytically and show that the spectrum is constant for large values
of \(f\) . Our main goal is the discuss the dependence of the spectrum
on the parameters: \(G\mu\) , \(\epsilon\equiv\frac{\alpha}{\Gamma G \mu}\) and \(p\) for small loops and \(G\mu\) and \(p\) for large loops. We
limit our discussion to large values of \(f\) , for which the spectrum
gets the dominant contribution from the loops in the radiation era.
Matter era loops contribute to lower frequency part of the
spectrum.
Since we want to get an estimate of the spectrum we will neglect the
complications arising from removing rare burst.
In the radiation domination, \(z>z_{eq}=\sqrt{\Omega_R}\simeq5440\) , the Hubble function in Eq. (32), can be approximated as
\[\begin{equation}
\tag{57}
{\mathcal H}(z)\simeq\sqrt{\Omega_R}z^2=\frac{z^2}{2 \sqrt{z_{eq}}}.
\end{equation}\]
The cosmological functions that appear in the stochastic background
radiation formula can be approximated as
\[\begin{equation}
\tag{58}
\varphi_t(z)=\int_z^\infty\frac{dz'} {(1+z') {\mathcal H}(z')}\simeq\int_z^\infty\frac{dz'}
{z' {\mathcal H}(z')}\simeq\sqrt{z_{eq}}\,z^{-2}.
\end{equation}\]
\[\begin{equation}
\tag{59}
\varphi_r(z)=\int_0^z\frac{dz'} { {\mathcal H}(z')}=\int_0^{z_{eq}}\frac{dz'}
{ {\mathcal H}(z')}+\int_{z_{eq}}^z\frac{dz'} { {\mathcal H}(z')}\simeq \,3.6.
\end{equation}\]
\[\begin{equation}
\tag{60}
\varphi_V(z)=\frac{4 \pi \varphi_r^2}{(1+z)^3 {\mathcal H}(z)}\simeq 325\,
\sqrt{z_{eq}} \,z^{-5}.
\end{equation}\]
We first consider the small loop case, for which the expression for
SBGW reduces to an integral over redshift given in Eqs. (50) and (51). Inserting the
result in Eqs. (58)-(60) into Eq. (50) we get
\[\begin{eqnarray}
\tag{61}
\Omega_{gw,R}(f)&\propto&\frac{ G\mu} {p \alpha^{1/3}
f^{1/3}}\int_{z_{eq}}^{z_{max}}\frac{dz}{z^{2/3}}\Theta\left(
1-\left[\frac{f z_{eq}^{1/2} \alpha}{H_0
z}\right]^{-1/3}\right)\propto\frac{G\mu } {p},
\end{eqnarray}\]
where we dropped a term with \(1/f\) dependence since it is small in
large \(f\) limit, and the subscript \(R\) reminds us that this is the
contribution from radiation era loops. The upper limit of the
integration, \(z_{max}\) , is the redshift at the time of the creation
of the strings, which depends on the energy scale of the phase
transition. The result in Eq. (61) is
valid for \(\frac{ z_{eq}^{1/2}}{\alpha }\ll\frac{f}{H_0}<\frac{ z_{max}}{\alpha z_{eq}^{1/2}}\) , for which the upper limit of the
integral is set by the \(\Theta\) -function. If \(\frac{f}{H_0}>\frac{ z_{max}}{\alpha z_{eq}^{1/2}}\) , the integral does not depend on \(f\)
and the frequency dependence of \(\Omega_{gw,R}(f)\) is given by the
prefactor, which has \(f^{-1/3}\) behavior. For kinks we get
\[\begin{eqnarray}
\tag{62}
\Omega^K_{gw,R}(f)&\propto&\frac{ G\mu } {p \alpha^{2/3}
f^{2/3}}\int_{z_{eq}}^{z_{max}}\frac{dz}{z^{1/3}}\Theta\left(
1-\left[\frac{f z_{eq}^{1/2} \alpha}{H_0
z}\right]^{-1/3}\right)\propto\frac{G\mu } {p}.
\end{eqnarray}\]
Eqs. (61) and (62) show that for \(\frac{ z_{eq}^{1/2}}{\alpha }\ll\frac{f}{H_0}\) the spectrum is constant and it scales with
\(G\mu/p\) . The amplitude does not depend on the parameter \(\alpha\) ,
however the spectrum shifts to the right linearly in \(\alpha\) .\
This result is in perfect agreement with Fig. . For the bottom curves \(\frac{G\mu}{p}=2 \times 10^{-5}\) where
as \(\frac{G\mu}{p}=2 \times 10^{-3}\) for the top curves, which have
two orders of magnitude larger amplitude, exactly agreeing with the
figure. Furthermore, the top curves (\(\epsilon=10^{-4}\) ) are
shifted to the right compared to the bottom curves (\(\epsilon=1\) ) by
about \(4\) -orders in \(f\) as predicted by our results above.
Now we consider large loops in the radiation domination, for which the density \(n(l,t)\) is given in Eq. (52), where \(t\) is to be replaced with \(\varphi_t(z)/H_0\) .
Substituting the results in Eqs. (58)-(60) into Eq.
(37) we get
\[\begin{eqnarray}
\Omega_{gw,R}(f)&=&A(f)\int dz\int dl \frac{z (l
z)^{-\frac{1}{3}}}{(l z^2 + \beta
\delta)^{\frac{5}{2}}}\Theta(1-\frac{1}{f z
l})\Theta(\frac{\beta}{z^2}-l)
= A(f)\int_{z_{eq}}^{z^*}dz\int_{\frac{1}{f}}^{\frac{\beta}{z}} du
\frac{u^{-\frac{1}{3}}}{(u z + \beta \delta)^{\frac{5}{2}}},
\tag{63}
\end{eqnarray}\]
where we define
\[\begin{eqnarray}
\tag{64}
A(f)&=& \frac{165\, c\, \alpha^2\, \delta^2
\chi_R }{ p \,{z_{eq}}^{1/4}H_0^{\frac{3}{2}}\Gamma^2f^{\frac{1}{3}}},
\end{eqnarray}\]
with \(\delta=\frac{G \mu \Gamma}{\alpha}\) and
\(\beta=\frac{\alpha\sqrt{z_{eq}}}{H_0}\) (\(\alpha\approx0.1\) for
large loop case) and the dummy integration variable \(u=l\, z\) . The
upper limit of the \(z\) integral, \(z^*\) will be set by requiring
\(\beta/z>1/f\) , that is, \(z<f\beta\) . If \(f<z_{max}/\beta\) we have,
\[\begin{eqnarray}
\Omega(f)&=&A(f)\int_{z_{eq}}^{\beta/f}dz\int_{\frac{1}{f}}^{\frac{\beta}{z}}du\frac{u^{-\frac{1}{3}}}{(u
z + \beta \delta)^{\frac{5}{2}}}
=A(f)\int_{\frac{1}{f}}^{\frac{\beta}{z_{eq}}}du\int_{z_{eq}}^{\beta/u}dz\frac{u^{-\frac{1}{3}}}{(u
z + \beta \delta)^{\frac{5}{2}}}
=-\frac{2}{3}A(f)\int_{\frac{1}{f}}^{\frac{\beta}{z_{eq}}}\frac{du}{u^{\frac{4}{3}}}\left(\frac{1}{(\beta+\beta
\delta)^{\frac{3}{2}}}-\frac{1}{(u z_{eq}+\beta
\delta)^{\frac{3}{2}}} \right).
\tag{65}
\end{eqnarray}\]
If \(\frac{1}{f}<\frac{\delta\beta}{z_{eq}}=\frac{G\mu \Gamma}{H_0 \sqrt{z_{eq}}}\) , we can split the integration range
\([1/f,\beta/z_{eq}]\) in the second integral into
\([1/f,\delta\beta/z_{eq}]\) and \([\delta\beta/z_{eq},\beta/z_{eq}]\)
and neglect \(u z_{eq}\) and \(\beta \delta\) respectively in these two
integrals.
Combining all terms and keeping the lowest order in \(\delta\) we
get,
\[\begin{eqnarray}
\Omega_{gw,R}(f)&=&A(f)\left(\frac{2 f^{\frac{1}{3}}}{ (\delta\beta)^{\frac{3}{2}}}-
\frac{18 {z_{eq}}^{\frac{1}{3}}}{11
(\delta\beta)^{11/6}}\right)=\frac{330\, c\, \alpha^2\, \delta^{\frac{1}{2}}
\chi_R }{ p \,{z_{eq}}^{1/4}H_0^{\frac{3}{2}}\Gamma^2\beta^{\frac{3}{2}}}
\simeq 3.2 \times10^{-4} \frac{\sqrt{G \mu}}{
p}, f>\frac{3.6 \times 10^{-18}}{G\mu}Hz
\tag{66}
\end{eqnarray}\]
The calculation for the case of kink is very similar to cusp case,
following the same steps we get
\[\begin{equation}
\tag{67}
\Omega_{gw,R}^{K}(f)\simeq 3.2 \times10^{-4} \frac{\sqrt{G \mu}}{
p},\,\, f>\frac{3.6 \times 10^{-18}}{G\mu} Hz
\end{equation}\]
which is identical to the cusp result. Eqs. (66) and (67) show that the distribution
is flat for \(f>\frac{3.6 \times 10^{-18}}{G\mu} Hz\) and its
amplitude scales with \(\sqrt{G \mu}/p\) , which is in excellent
agreement with Fig. . The flat value of the
spectrum for the top curves (\(G\mu=10^{-7}\) and \(p=5\times10^{-3}\) )
is \(2.1\times 10^{-5}\) and for the bottom curve (\(G\mu=10^{-9}\) and
\(p=5\times10^{-2}\) ) is \(2.1\times 10^{-7}\) . These results are to be
compared with the analytical results \(2.0\times 10^{-5}\) and
\(2.0\times 10^{-7}\) predicted by Eqs. (66)
and (67).\
It is important to note that, in this paper we assume that the
number of kinks, \(N\) , is order of one. This assumption enters in the
estimation \(|\ddot X_\pm|\sim \frac{2 \pi}{l}\) , and if there are \(N\)
kinks on strings, it needs to be replaced by \(|\ddot X_\pm|\sim \frac{2 \pi}{l/N}\) . The replacement of \(l\) with \(l/N\) should also
be done in the opening angle of the cone of the radiation, Eq.
(36),which will result in a nontrivial dependence on
\(N\) . However we can simply convert the resultant expression to the
one we calculated in Eq. (51) ) by defining
\(\alpha=\alpha' N\) . Since we have shown that \(\alpha\) has the effect
of moving the spectrum horizontally, one effect of having \(N\) kinks
will be shifted spectrum compared to one kink spectrum. The other
effect will be an overall scaling of the spectrum by \(1/N\) .
Parameter Space Constraints
In this section we discuss certain experimental bounds on SBGW. For the case of large loops the parameters are \(G\mu\) and \(p\) , and for small loops the parameters are \(G\mu\) , \(\epsilon\) and \(p\) . It is important to note that the nontrivial dependence on \(p\) follows from excluding rare bursts as described in Eqs. (55) and (56) (if rare events were included \(\Omega(f)\) would simply scale with \(1/p\) .)
Figure 3: Top-left:Accessible regions in the \(\varepsilon-G\mu\) plane for \(p = 10^{-3}\)for small loops (loop sizes are determined by gravitational back-reaction). Top-right: Same as above for \(p=10^{-2}\). Bottom-left: Same as above for \(p=10^{-1}\). Bottom-right: Accessible regions in the \(p-G\mu\)plane for the large long-lived loop models. The accessible regions are to the right of the corresponding curves. All models are within reachof LISA and advanced LIGO, and most are within the projected pulsar bound.
Accessible regions corresponding to different
experiments and bounds are shown in Fig. 3. The shaded regions, from
darkest to lightest, are: LIGO S4 [33] limit,
LIGO S5 [34], LIGO H1H2 projected sensitivity (cross-correlating the data
from the two LIGO interferometers at Hanford, WA (H1 and H2)), and
AdvLIGO H1H2 projected sensitivity. All projections assume 1 year
of exposure and either LIGO design sensitivity or Advanced LIGO
sensitivity tuned for binary neutron star inspiral search. The
solid black curve corresponds to the BBN [35] bound, the dot-dashed curve
to the pulsar bound[36], the \(+\) s to the projected pulsar sensitivity,
the circles to the bound based on the CMB and matter spectra [37],
the \(\times\) s to the projected sensitivity of the LIGO burst [22] search,
and the \(\diamond\) -curve to the LISA projected sensitivity [38]. The BBN and CMB
bounds are integral bounds, i.e. they are upper limits for the integral of \(\Omega(f)\) over \(\ln f\) , therefore
a model is excluded if it predicts an integral larger than the limit. On the other hand,
the pulsar and LIGO bounds apply in specific frequency bands, thus
a model is excluded if it has \(\Omega(f)\) larger than the limit (or projected
sensitivity) for any \(f\) in the range of the pulsar or LIGO
experiments. The range of the redshift integral in Eq. (37) must
chosen properly for a given experiment. For BBN bound, the
integration is performed for \(z > 5.5 \times 10^9\) . Similarly, for the bound
based on the CMB and matter spectra, the integration is performed
for \(z > 1100\) . First, we note that smaller values of \(p\) are more
accessible, which follows from the fact that the loop density is
inversely proportional to \(p\) . This makes cosmic superstrings more
accessible than field theoretical strings. Second, we note that LIGO
stochastic search constrains large \(G\mu\) , small \(\epsilon\) part of
the parameter space, whereas pulsar limit constrains large \(G\mu\)
and large \(\epsilon\) part of the parameter space. Similarly, the
LIGO burst bound applies to large \(G\mu\) and intermediate \(\epsilon\)
part of the parameter space. Therefore large \(G\mu\) part of the
parameter space is covered by these three experiments. Furthermore
since they also overlap for large \(G\mu\) and intermediate
\(\epsilon\) , in the case of detection, the two LIGO searches could
potentially confirm each other. We also see that the BBN and CMB
bounds are not very sensitive to \(\epsilon\) : the corresponding
curves are rather vertical in \(\epsilon-G\mu\) plane. This result is
in perfect agreement our results (Eqs. (61) and (62)) that show
\(\Omega(f)\propto G\mu/p\) , which does not depend on \(\epsilon\) . For
the case of large loops, GW background is significantly larger than
the small loop one, see Figs. and
. Therefore more of the parameter space is
accessible to the current and proposed experiments, as depicted in
the right bottom panel of Fig. 3. The strongest constraint is the
pulsar bound, which rules out cosmic (super)string models with
\(G\mu>10^{-12}\) and \(p< 8\times 10^{-3}\) . This bound also rules out
field theoretical strings (\(p=1\) ) with \(G\mu>2\times 10^{-9}.\) One
can compare these results with the case where only cusps are
included [25]. In that case cosmic
(super)string models with \(G\mu>10^{-12}\) and \(p< 3\times 10^{-3}\)
and field theoretical strings with \(G\mu>10^{-9}\) are ruled out.
This result illustrates that kinks contribute to SBGW at the same
order as cusps.
Acknowledgment
We would like to thank Marco Peloso for useful discussions. S. Ö. is supported by the Graduate School at the University of Minnesota under the Doctoral Dissertation Fellowship, X. S. is supported in part by NSF Grant No. PHY-0758155 and the Research Growth Initiative at the University of Wisconsin-Milwaukee and V. M. is supported in part by NSF Grant No. PHY0758036.