CMB ISW-lensing bispectrum from cosmic strings

CMB ISW-lensing bispectrum from cosmic strings

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Abstract

We study the effect of weak lensing by cosmic (super-)strings on the higher-order statistics of the cosmic microwave background (CMB). A cosmic string segment is expected to cause weak lensing as well as an integrated Sachs-Wolfe (ISW) effect, the so-called Gott-Kaiser-Stebbins (GKS) effect, to the CMB temperature fluctuation, which are thus naturally cross-correlated. We point out that, in the presence of such a correlation, yet another kind of the post-recombination CMB temperature bispectra, the ISW-lensing bispectra, will arise in the form of products of the auto- and cross-power spectra. We first present an analytic method to calculate the autocorrelation of the temperature fluctuations induced by the strings, and the cross-correlation between the temperature fluctuation and the lensing potential both due to the string network. In our formulation, the evolution of the string network is assumed to be characterized by the simple analytic model, the velocity-dependent one scale model, and the intercommutation probability is properly incorporated in order to characterize the possible superstringy nature. Furthermore, the obtained power spectra are dominated by the Poisson-distributed string segments, whose correlations are assumed to satisfy the simple relations. We then estimate the signal-to-noise ratios of the string-induced ISW-lensing bispectra and discuss the detectability of such CMB signals from the cosmic string network. It is found that in the case of the smaller string tension,  , the ISW-lensing bispectrum induced by a cosmic string network can constrain the string-model parameters even more tightly than the purely GKS-induced bispectrum in the ongoing and future CMB observations on small scales.

a]Daisuke Yamauchi b]Yuuiti Sendouda c]Keitaro Takahashi

Prepared for submission to JCAP

CMB ISW-lensing bispectrum from cosmic strings

• Research Center for the Early Universe, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

• Graduate School of Science and Technology, Hirosaki University, Hirosaki, Aomori 036-8561, Japan

• Faculty of Science, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan

1 Introduction

Topological defects, appearing as solutions to the field equation in various models of particle physics, are expected to have formed during phase transitions in the early universe through spontaneous symmetry breakings [1, 2, 3] (see [4] for a review). It has been shown that cosmic strings generally appear at the end of inflation within a various variety of supersymmetric grand unified theories [5].

In the late-time universe, intercommutation of cosmic strings serves as an essential mechanism of energy dissipation which keeps the total energy of strings within the expanding Hubble volume from growing. Early studies on this subject [3, 6] employed analytic methods and suggested formation of a stable structure with constant energy density, so-called scaling string network. Afterwards, numerical simulations of dynamical formation of a string network in the expanding universe have been performed for the Nambu-Goto strings [7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and Abelian-Higgs strings [17, 18, 19, 20, 21, 22, 23], both confirming the approach to the scaling regime.

Recently, cosmic strings have attracted a renewed interest in the context of string cosmology since it was pointed out that a new type of cosmic strings, cosmic superstrings, may be formed at the end of stringy inflation [24, 25, 26, 27]. To our knowledge, the qualitative properties of cosmic superstrings in the late-time universe should be similar to those of field-theoretic strings, except for the fact that the intercommuting probability is relatively low for cosmic superstrings. It is normally unity for field-theoretic strings, while it can be significantly smaller than unity for cosmic superstrings. The authors have extended an analytic description of a network, so-called velocity-dependent one-scale (VOS) model, to include the effect of in ref. [28]. Observables associated with the global properties of a string network, e.g. the string number density, are revealed to depend sensitively on the intercommuting probability , and searches for such signals should offer a clue to distinguish cosmic superstrings from field-theoretic strings.

The Gott-Kaiser-Stebbins (GKS) effect [29, 30] is the most characteristic post-recombination effect of a cosmic string in the cosmic microwave background (CMB) sky. The GKS effect is considered as an integrated Sachs-Wolfe (ISW) effect due to a moving cosmic string, which leads to discontinuities of the CMB temperature fluctuations across the strings with a relative amplitude typically estimated by the dimensionless string tension  . The imprint of cosmic strings on the angular power spectrum of the CMB temperature anisotropies have been studied in, e.g., [31, 32], and the current upper bound on the string tension for the strings with is in the range from to  [33]. Furthermore, cosmic strings generally create non-Gaussian signals in the CMB temperature anisotropies because topological defects are themselves highly nonlinear objects. Searches for the string-induced non-Gaussian signals in the CMB may enhance the detectability of cosmic strings, and could not only be used as a tool to prove cosmic strings, but also be helpful to check foreground or systematic contributions. Non-Gaussian signals induced by the post-recombination effect of a cosmic string network have been estimated in the literature: references [15, 28, 34] discussed one-point probability distributions of the CMB temperature fluctuations; also, the CMB temperature bispectrum and trispectrum induced by the GKS effect have been estimated analytically [35, 36, 37, 38] and numerically [33].

In this paper, we will study the effect of the weak gravitational lensing by cosmic strings on the CMB temperature anisotropies. Gravitational lensing by a cosmic string have also been previously studied in the literature [39, 40, 42, 43, 44, 41]. An observationally important feature is that the lensing events lead to deviations from Gaussianity because a lensed fluctuation is a nonlinear function of fields. It is known that, in the presence of the cross correlation between the post-recombination CMB temperature fluctuations and the lensing potential, non-vanishing bispectra, which we call the ISW-lensing bispectra hereafter, will appear even if the unlensed temperature fluctuations are exactly Gaussian [45]. As a result, we expect the appearance of the ISW-lensing bispectrum induced by a cosmic string network as yet another string-induced CMB temperature bispectrum.

This paper is organized as follows. In section 2, we begin by briefly reviewing the derivation of the ISW-lensing bispectrum and apply it to the case where various gravitational sources exist. In section 3, we introduce the ISW effect due to a cosmic string, namely GKS effect, and the lensing potential due to a cosmic string. Then we explicitly calculate the string-induced bispectra based on a simple analytic model. Based on the formulae, prospects for measuring the string-induced CMB temperature bispectra are discussed. Finally section 4 is devoted to summary and conclusion.

Throughout this paper, we focus on the small patch of sky and work in the flat-sky approximation. We use the two-dimensional Fourier transformation defined as

 f(θ)=∫d2ℓ(2π)2f(ℓ)eiℓ⋅θ. (1.1)

where we use the bold letters to label two-vectors on the sky. Here and denote the two-dimensional observed position on the sky and the two dimensional Fourier modes, respectively. Inner products of two-dimensional vectors are denoted as  . We assume a flat CDM cosmological model as a background spacetime with the cosmological parameters :  . We will work in the comoving coordinates

 gμνdxμdxν=a(η)2[−dη2+δijdridrj]=a(η)2[−dη2+dχ2+χ2dΩ2], (1.2)

where is the Cartesian coordinates centered on the observer, the comoving distance, and the line element on the unit -sphere, which is approximated by on small scales. The dot is also used to denote inner products of comoving -vectors:  .

2 ISW-lensing bispectrum

In this section we discuss the lensing effect on the CMB temperature anisotropies. The lensed temperature fluctuation in a direction  ,  , is Fourier transformed according to

 ~Θ(ℓ)=∫d2θ~Θ(θ)e−iℓ⋅θ, (2.1)

where represents the Fourier coefficients. The auto-bispectrum for the lensed temperature anisotropies in the flat sky is defined as

 ⟨~Θ(ℓ1)~Θ(ℓ2)~Θ(ℓ3)⟩=(2π)2δ2D(ℓ1+ℓ2+ℓ3)B(ℓ1,ℓ2,ℓ3), (2.2)

where the angle brackets denote the ensemble average and is the Dirac delta function. We then consider that the lensed temperature fluctuations are related to the unlensed temperature fluctuations through  , where and are the unlensed temperature anisotropies and the deflection angle, respectively. Assuming that the deflection angle is a perturbed quantity,  , we can expand the lensed temperature fluctuation as

 ~Θ(θ)=Θ(θ)+d(θ)⋅∇Θ(θ)+O(d2), (2.3)

where denotes the two-dimensional covariant derivative on the sky. Hereafter we neglect the higher-order contributions of in eq. (2.3). This equation implies that the weak gravitational lensing of the CMB can produce the non-Gaussian temperature fluctuations. The deflection angle is generally characterized by the sum of two terms: the gradient of the scalar lensing potential (gradient-mode), and the rotation of the pseudo-scalar lensing potential (curl-mode) [46, 47, 48, 40, 49, 41]:

 d(θ)=∇ϕ(θ)+(∗∇)ϖ(θ), (2.4)

where is the -degree rotation operator. The Fourier coefficients of the lensed temperature anisotropies are obtained by performing the two-dimensional Fourier transformation according to eq. (2.1). With the help of eq. (2.4), we find

 (2.5)

with

 L(ℓ,ℓ1)=[ℓ1⋅(ℓ−ℓ1)]ϕ(ℓ−ℓ1)+[(∗ℓ1)⋅(ℓ−ℓ1)]ϖ(ℓ−ℓ1), (2.6)

where and denote the Fourier components of the gradient- and curl-modes of the deflection angle, respectively. Even if the unlensed temperature fluctuations is exactly Gaussian, a non-vanishing bispectrum for the lensed temperature fluctuations will appear and its value can be evaluated as [45]

 Blens(ℓ1,ℓ2,ℓ3)=ℓ12CΘϕℓ1CΘΘℓ2+(perms), (2.7)

where  , (perms) denotes the remaining five permutations of , and we have defined the auto- and cross-angular power spectra of the unlensed temperature fluctuations and lensing potential as

 ⟨X(ℓ)Y(ℓ′)⟩=(2π)2δ2D(ℓ+ℓ′)CXYℓ, (2.8)

where and take on and  . Note that due to the parity symmetry the cross correlation between the temperature fluctuations and the curl-mode of the deflection angle does not appear.

The cross correlation may be calculated for any secondary effect once its relation to the gravitational potential is given. For an illustrative example, we shall give the explicit expression for the cross correlation between the lensing potential and the ISW effect due to primordial scalar perturbations. In a standard CDM universe, the primordial scalar perturbations give a major contribution to the gradient-mode of the deflection angle. In the Born approximation, where the lensing effect is evaluated along the unperturbed light path, the lensing potential due to primordial scalar perturbations,  , can be conveniently evaluated in terms of the Bardeen gravitational potential as

 ϕP(θ)=−2∫χCMB0dχχCMB−χχCMBχΦ(η0−χ,χ,θ), (2.9)

where is the conformal distance at the last scattering surface and the conformal time at present. On the other hand, the ISW effect due to the primordial scalar perturbations contributes to the temperature anisotropies as

 ΘP(θ)=−2∫χCMB0dχ˙Φ(η0−χ,χ,θ), (2.10)

where the dot ( ) denotes the derivative with respect to the conformal time. It follows that the flat-sky cross correlation is given by [45]

 CΘPϕPℓ=2π2ℓ3∫χCMB0dχχ(−2˙F(χ))(−2F(χ)χCMB−χχCMBχ)Δ2Φ(ℓ/χ), (2.11)

where is the dimensionless primordial power spectrum of the Bardeen potential and is given by

 F(χ)=(1+z)H(z)H0∫∞zdz′(1+z′)H−3(z′)∫∞0dz′′(1+z′′)H−3(z′′), (2.12)

where is the redshift, which is related to the conformal distance through  . We should note that the unlensed temperature power spectrum has a negligibly small amplitude on small scales due to the Silk damping, while the cross-correlation between the temperature and the lensing potential could have a non-vanishing amplitude even at small scales.

We then apply the derivation of the ISW-lensing bispectrum, originally developed in the theoretical studies of the bispectrum induced by primordial density perturbations, to the case of various gravitational sources. To evaluate the various types of bispectra, we first assume that the observed sky map of the temperature anisotropies can be regarded as a superposition of those due to each source for simplicity. Let us introduce the index to denote the contribution from each kind of sources as

 ~Θ(θ)=∑α~Θα(θ)=∑αΘα(θ+∇ϕ). (2.13)

Here we have assumed that the deflection angle can be described only by the gradient-mode of the the deflection angle although the curl-mode in general contributes the deflection angle (see eq. (2.4) and, e.g., refs. [48, 40, 49, 41] for the estimation of the string-induced curl mode). Similarly, since the gradient mode of the deflection angle strongly depends on the source gravitational potential and its distribution, we assume that the total scalar lensing potential can be decomposed into each kind of contributions as

 ϕ(θ)=∑αϕα(θ). (2.14)

Using eqs. (2.13) and (2.14) and expanding the lensed temperature anisotropies up to , we can rewrite it as

 ~Θ(θ)=∑αΘα⎛⎝θ+∑β∇ϕβ⎞⎠=∑α⎛⎝Θα(θ)+∑β∇ϕβ(θ)⋅∇Θα(θ)⎞⎠. (2.15)

Hence the flat-sky bispectrum for the lensed temperature anisotropies, eq. (2.2), can be decomposed as

 B(ℓ1,ℓ2,ℓ3)=∑αBααα(ℓ1,ℓ2,ℓ3)+∑α,βBαβ(ℓ1,ℓ2,ℓ3) (2.16)

where we have introduced to denote the bispectrum for the unlensed temperature anisotropies generated by the gravitational source  , and to denote the -type ISW-lensing bispectrum, which is defined by

 Bαβ(ℓ1,ℓ2,ℓ3)=ℓ12CΘαϕαℓ1CΘβΘβℓ2+(perms), (2.17)

with  . Here we have neglected the connected part of the four-point function of the temperature fluctuations and the lensing potential for simplicity.

In this paper we focus only on the contributions from inflationary primordial fluctuations (P) and a cosmic string network (S). The resultant bispectrum for the lensed temperature anisotropies can be decomposed as

 B=BPPP+BPP+BSSS+BSS+BSP+BPS. (2.18)

The first two terms in eq. (2.18) , and  , correspond to the standard unlensed and ISW-lensing bispectra (see eq. (2.11)) due to the primordial scalar perturbations, respectively. A recent observation [50, 51] shows that there is yet no evidence for any primordial non-Gaussianity, but the ISW-lensing bispectrum expected in the standard CDM universe has been measured at more than statistical significance. On the other hand, taking into account the contributions from a string network, we have four additional components; represents the bispectrum purely due to the GKS effect of strings (see next section for the GKS effect), which has been estimated in the literature [35, 36, 37, 38, 33], whereas the new types of the string-induced, ISW-lensing, bispectra and have appeared through the CMB lensing. The remainder of the paper will be devoted to the evaluation of these new bispectra.

Before closing this section, we should discuss possible modifications to eq. (2.18) from the string-induced non-Gaussian correlations. Since each photon scattering by a cosmic string produces strongly non-Gaussian signals in the CMB, the connected part of the four-point functions such as and higher-order correlation functions would give non-vanishing contributions in eq. (2.18). However, can be actually treated as nearly Gaussian variable and these modifications should be small. This is because a photon ray is scattered by cosmic strings many times through its way from the last scattering surface to an observer. Hence would behave like a random walk and its probability distribution function may be approximated by a Gaussian distribution [28, 34]. Although we ignore those small non-Gaussian modifications hereafter, they would rather enhance the signals, and the expected signal-to-noise ratios will be increased. In this sense, the analysis we will give later would give a rather conservative estimate for the detectability of cosmic strings.

3 String-induced bispectra and their detectability

In this section, we consider the ISW effect and the gravitational lensing due to a cosmic string network as yet another source of the CMB temperature bispectrum. After briefly reviewing the properties of the post-recombination effect of the cosmic string, namely the Gott-Kaiser-Stebbins (GKS) effect and the string-induced lensing potential, in section 3.1, we give the explicit expression for the string-induced bispectra in section 3.2. The signal-to-noise ratios for the string-induced bispectra are estimated, and the detectability of the string network is discussed in section 3.3.

3.1 GKS effect and string lensing

We first consider the GKS effect as an ISW effect due to a cosmic string. The ISW formula is given by

 ΘS(θ)=−12∫χCMB0dχdxμdχdxνdχ˙hμν(η0−χ,χ,θ), (3.1)

where is the metric perturbation caused by strings and denotes the null vector along the line of sight with being a unit vector pointing the photon propagation direction in the background spacetime. In order to evaluate the metric perturbations through the linearized Einstein equations, we write down the string stress-energy tensor. To do so, we assume that a string segment can be well approximated as a Nambu-Goto string and we introduce the three-dimensional embedding function of string position as  , where is the spacelike worldsheet coordinate. The stress-energy tensor for a string segment in the transverse gauge is described as

 (3.2)

where is the string tension, the dot ( ) and the prime ( ) denote the derivatives with respect to and  , respectively.

The stress-energy tensor should be properly evaluated along the line of sight, on which  . To do so in an analytical manner, we further impose that the string segment as seen by the observer is localized at a certain redshift, namely the distance on the lightcone between the observer and the string segment can be well approximated by a constant value, [39]. This condition is solved for the conformal distance as  , so that we can parameterize the string position as seen by the observer as  . With this approximation, let us define the two-dimensional angular position of a string by

 θS(σ)≡1χS(→e1⋅→rS(σ), →e2⋅→rS(σ)), (3.3)

where the orthogonal projectors satisfy  . Similarly, the angular velocity is defined by replacing with in the right-hand side of the above expression for  .

In figure 1 , we show the representation of the quantities used in this paper. The above approximation should be valid as long as we focus on distant strings at small patch of sky; since the correlation length of a string segment (see figure 1) is known to grow in proportion to the Hubble length, the extension of a string along the line of sight is bounded by  , so it should be much smaller than the physical distance.

Under this approximation, the stress-energy tensor of a string along the line of sight is described as [39, 43, 44]

 dxμdχdxνdχTμν(η0−χ,χ,θ)≈μχ2SδD(χ−χS)∫dσδ2D(θ−θS(σ)). (3.4)

With these notations, the temperature fluctuations due to the GKS effect is evaluated by the following formula [35, 36, 37, 38, 31]:

 ∇2ΘS(θ) =8πG∫χCMB0dχχ2dxμdχdxνdχ˙Tμν(η0−χ,χ,θ) ≈8πGμ∫dσ(˙θS(σ)⋅∇)δ2D(θ−θS(σ)), (3.5)

where we have used the linearized Einstein equations in the first line of eq. (3.5). Performing the two-dimensional Fourier transformation, we obtain the Fourier coefficients of the GKS temperature fluctuations as

 ΘS(ℓ)=i8πGμ1ℓ2∫dσ(ℓ⋅˙θS(σ))e−iℓ⋅θS(σ). (3.6)

On the other hand, the lensing potential induced by a cosmic string,  , are related to the convergence field through  . The convergence field can be described in terms of the perturbed Ricci tensor and is related to the stress-energy tensor through the perturbed Einstein equations as [39]

 κS(θ)=4πG∫χCMB0dχ(χCMB−χ)χχCMBdxμdχdxνdχTμν(η0−χ,χ,θ). (3.7)

Assuming that the string segment is localized at a certain redshift, as we mentioned above, the above expression can be reduced to

 ∇2ϕS(θ)≈8πGμχCMB−χSχCMBχS∫dσδ2D(θ−θS(σ)). (3.8)

Hence we obtain the Fourier coefficients of the string lensing potential as

 ϕS(ℓ)=−8πGμχCMB−χSχCMBχS1ℓ2∫dσe−iℓ⋅θS(σ). (3.9)

For a cosmic string network, non-vanishing cross-correlation is expected to exist between the GKS temperature fluctuations and the string-induced lensing potential.

3.2 String correlations

Before going into the details, let us discuss the dependence of the string-induced bispectra on the string tension  . From the expressions of and  , eqs. (3.6) and (3.9), we deduce that the power spectra (for ) scale as  . Therefore we find the following proportionalities of the string-induced ISW-lensing:

 BSS∝(Gμ)4,BSP∝(Gμ)2,BPS∝(Gμ)2. (3.10)

On the other hand, the purely GKS-induced bispectrum obeys [35, 37, 38, 33]. Hence, in the case of the smaller string tension, the SP- and PS-type contributions could dominate the total bispectrum rather than the SSS-type. Moreover, at small scales where the unlensed primordial fluctuations are damped, the unlensed primordial spectrum has little power and only the cross-correlation is relevant. Therefore, we expect that the SP-type bispectrum, which obeys the proportionality  , is exponentially small at small scales whereas the PS-type,  , should give the most significant contributions. According to the above observations, we shall only consider and in what follows.

For our purpose, we need to evaluate and  . In order to calculate the angular power spectrum, we follow the analytic approach [32], originally developed in the studies of the Sunyaev-Zel’dovich effect [52, 53, 54]. The GKS fluctuations and the string-induced lensing potential are characterized by the distance and the parameters for the string-segment configuration including the set of the angular parameters for the string directions and the curvature. The observed sky maps of the temperature fluctuations and the lensing potential are assumed to appear as a superposition of each contribution, namely  ,  , with “” denoting the contribution from the -th string segment. The angular power spectrum then can be decomposed into two parts: the contributions from the Poisson-distributed string segments and those from the correlations between the different segments. At small scales, the angular power spectrum will be dominated by the contribution of the sum of statistically independent segments even if the segment-segment correlation is taken into account. With the help of eqs. (2.8) , (3.6) , and (3.9) , we find

 CΘSΘSℓ =1A⟨ΘtotS(ℓ)ΘtotS(−ℓ)⟩ ≈(8πGμ)2A1ℓ4∫χCMB0dχdVdχ(∏a∫dψa)fS({ψa}) ×∫dσ1dσ2(ℓ⋅˙θS(σ1))(ℓ⋅˙θS(σ2))eiℓ⋅(θS(σ1)−θS(σ2)) (3.11)

for the angular power spectrum for the GKS temperature anisotropies, and

 CΘSϕSℓ =1A⟨ΘtotS(ℓ)ϕtotS(−ℓ)⟩ ×∫dσ1dσ2(ℓ⋅˙θS(σ1))eiℓ⋅(θS(σ1)−θS(σ2)) (3.12)

for the cross correlation between the GKS anisotropies and the string-induced lensing potential, where is the area size with being the fractional sky coverage, and denote the comoving differential volume element at a distance and the comoving number density of string segments with the string configuration parameters in the range  , respectively. It is in general difficult to evaluate the average for the string configuration parameters, though we can calculate it explicitly when we focus on the exactly straight string-segments [32].

Instead, we will use the simple analytic model to estimate the correlations within the string segment developed by [31, 55, 56]. In this model, the notion of the string-segment configuration average is introduced (to be distinguished from the usual meaning of the ensemble average ), which allows evaluation of the integration over the string configuration parameters through the correspondence  , where is the comoving number density of the strings. Furthermore, the variables and are assumed to be exactly Gaussian and isotropic with mean zero, and all the equal-time correlations can be expressed in terms of the following two point functions:

 ⟨˙ri(σ1,η)˙rj(σ2,η)⟩seg=13δijVS(σ1−σ2,η), (3.13) ⟨ri′(σ1,η)rj′(σ2,η)⟩seg=13δijTS(σ1−σ2,η), (3.14) ⟨ri′(σ1,η)˙rj(σ2,η)⟩seg=13δijMS(σ1−σ2,η). (3.15)

Then the asymptotic forms of these correlators ,  ,  , and are estimated based on the velocity dependent one-scale (VOS) model [57, 58, 59]. In VOS, a string network is characterized by the correlation length and the root-mean-square velocity  . Taking into account the probabilistic nature of the intercommuting process, for , we obtain the approximate expressions and in the matter-dominated era [28], where quantifies the efficiency of the loop formation [57], and is the intercommuting probability. Since in our calculation we consider only a string segment with length  , the correlators in eqs. (3.13)-(3.15) are expected to be damped on scales larger than the correlation length of the string network, that is for  , while have the non-vanishing expectation values for  . In terms of the scaling quantities of the string network, namely and  , the asymptotic behaviors of the two-point correlators are given by (see e.g. [31])

 VS(σ,η)={v2rms (σ≲ξ/a)0 (σ≳ξ/a), (3.16) TS(σ,η)={1−v2rms (σ≲ξ/a)0 (σ≳ξ/a), (3.17) MS(σ,η)={c0aσ/ξ (σ≲ξ/a)0 (σ≳ξ/a), (3.18)

where represents the cross correlator between the string velocity and curvature. The non-vanishing cross correlation appears in the cosmological background, while one can see it vanishes in a flat spacetime [34, 35]. In the scaling regime, can be evaluated in terms of the root-mean-square velocity through the VOS scaling equations as  .

Consequently, in the model described above, eqs. (3.11) and (3.12) can be rewritten as

 CΘSΘSℓ≈ (8πGμ)2A1ℓ4∫χCMB0dχdVdχnS (3.19) CΘSϕSℓ≈ −i(8πGμ)2A1ℓ4∫χCMB0dχdVdχnSχCMB−χχCMBχ (3.20)

The comoving string number density can be estimated in terms of the correlation length as  . By virtue of the properties of the string correlators in eqs. (3.13)-(3.15), and recalling that the distant strings can be treated as thin objects, we can evaluate the string segment configuration averages as

 ⟨(ℓ⋅˙θS(σ1))(ℓ⋅˙θS(σ2))eiℓ⋅(θS(σ1)−θS(σ2))⟩seg ≈13ℓ2χ2S{VS(σ1−σ2,η0−χS)−13ℓ2Π2S(σ1−σ2,η0−χS)} ×exp[−16ℓ2ΓS(σ1−σ2,η0−χS)], (3.21) ⟨(ℓ⋅˙θS(σ1))eiℓ⋅(θS(σ1)−θS(σ2))⟩seg ≈i3ℓ2χSΠS(σ1−σ2,η0−χS)exp[−16ℓ2ΓS(σ1−σ2,η0−χS)], (3.22)

where we have introduced and defined by

 ΓS(σ1−σ2,η)=⟨(→r(σ1,η)−→r(σ2,η)χS)2⟩seg=1χ2S∫σ1σ2dσ3dσ4TS(σ3−σ4,η), (3.23) ΠS(σ1−σ2,η)=⟨(→r(σ1,η)−→r(σ2,η)χS)⋅˙→r(σ2,η)⟩seg=1χS∫σ1σ2dσ3MS(σ3,η). (3.24)

It follows that the auto- and cross-power spectra (3.19) and (3.20) can be recast as

 ℓ2CΘSΘSℓ≈(8πGμ)23A∫χCMB0dχdVdχnS1χ2 ×∫dσ12VS(σ12,η0−χ)exp[−16ℓ2ΓS(σ12,η0−χ)]∫dσ+, (3.25) ℓ3CΘSϕSℓ≈(8πGμ)2ℓ3A∫χCMB0dχdVdχnSχCMB−χχCMBχ2 ×∫dσ12ΠS(σ12,η0−χ)exp[−16ℓ2ΓS(σ12,η0−χ)]∫dσ+, (3.26)

where we have neglected the terms and we have introduced  ,  . It is useful to introduce the angular scale corresponding to the correlation length of a string segment at  :  . Since the integral corresponds to the length of a string segment and the correlators are damped at the large angle  , we can take the regions of integration as and  . We then obtain the angular power spectra as

 ℓ2CΘSΘSℓ≈(8πGμ)2v2rms6A(1−v2rms)ℓ∫χCMB0dχdVdχnS1ℓcoU0(ℓ2ℓco), (3.27) ℓ3CΘSϕSℓ≈(8πGμ)2c012A(1−v2rms)2ℓ2∫χCMB0dχdVdχnSχCMB−χχCMBU2(ℓ2ℓco) (3.28)

with  . Once the parameters and and the scaling values of the string network are given, we can calculate the angular power spectra by performing the integrations in eqs. (3.27) and (3.28). We first evaluate the auto-power spectrum (3.27) for to see the consistency with previous works. One can see that its typical amplitude at is  , and it behaves as on small scales, while it has a plateau on large scales. It is in good agreement with our previous result found with different method in [32] and the numerical result by Fraisse et al. [15]. Hence in the subsequent analysis we use the analytic model to estimate the string correlations.

In figure 2, we plot the auto-power spectrum for the GKS temperature fluctuations and the cross correlation between the GKS fluctuations and the string-induced lensing potential. To be specific, we consider the three fiducial values of the string parameters:  . These fiducial values are still consistent with the recent observation of the small-scale CMB angular power spectrum [32]. Analytic estimation implies that the auto- and cross-power spectra, eqs. (3.27) and (3.28) , roughly scale as  , for and  , for  , respectively.

We will briefly discuss the unlensed angular bispectrum induced by the GKS effect. From eqs. (2.2) and (3.6) , the Poisson term of the SSS-type angular bispectrum can be described by

 BSSS(ℓ1,ℓ2,ℓ3) =1A⟨ΘS(ℓ1)ΘS(ℓ2)ΘS(ℓ3)⟩ ≈−i(8πGμ)3A1ℓ21ℓ22ℓ23∫χCMB0dχdVdχ(∏a∫dψa)fS({ψa}) ×∫dσ1dσ2dσ3[3∏n=1(ℓn⋅˙θS(σn))]exp[−i3∑m=1(ℓm⋅θS(σm))]. (3.29)

Following the same steps as the angular power spectra [(3.25) and (3.26)], we can write down the SSS-type bispectrum in terms of the string correlators (3.13)-(3.15) as

 BSSS(ℓ1,ℓ2,ℓ3) ≈−(8πGμ)39Aℓ12ℓ31ℓ21ℓ22ℓ23∫χCMB0dχdVdχnS1χ3∫dσ123 ×∫dσ12dσ31VS(σ12,η0−χ)ΠS(σ31,η0−χ) ×exp[−16{ℓ12ΓS(σ12,η0−χ)