B-mode polarization induced by gravitational wavesfrom kinks on infinite cosmic strings

Masahiro Kawasaki,1,2 Koichi Miyamoto,1 and Kazunori Nakayama3

1Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan2Institute for the Physics and Mathematics of the Universe, University of Tokyo,

Kashiwa, Chiba 277-8568, Japan3KEK Theory Center, Institute of Particle and Nuclear Studies, KEK,

Tsukuba, Ibaraki 305-0801, Japan(Received 21 April 2010; revised manuscript received 26 July 2010; published 3 November 2010)

We investigate the effect of the stochastic gravitational wave (GW) background produced by kinks on

infinite cosmic strings, whose spectrum was derived in our previous work, on the B-mode power spectrum

of the cosmic microwave background (CMB) anisotropy. We find that the B-mode polarization due to

kinks is comparable to that induced by the motion of the string network and hence the contribution of

GWs from kinks is important for estimating the B-mode power spectrum originating from cosmic strings.

If the tension of cosmic strings � is large enough, i.e., G� * 10�8, B-mode polarization induced by

cosmic strings can be detected by future CMB experiments.

DOI: 10.1103/PhysRevD.82.103504 PACS numbers: 98.80.Cq

I. INTRODUCTION

Cosmic (super)strings can be produced in the earlyUniverse at the phase transition associated with spontane-ous symmetry breaking [1], the end of supersymmetrichybrid inflation [2,3], or the end of the brane inflation[4,5]. They can be a clue to particle physics beyond thestandard model and the history of the early Universe,which is difficult to obtain in terrestrial experiments.How to find signatures of cosmic strings in cosmicmicrowave background (CMB) experiments has beenextensively discussed for decades. Especially, B-modepolarization of the CMB induced by the cosmic stringnetwork was investigated in many papers [6–10].

B-mode polarization, which has not been detected yet, ispolarization of the parity-odd type. It cannot be producedby the primordial scalar perturbation from the inflationaryera, which is widely believed to be the main origin of thepresent structure of the Universe. On the other hand, thetensor perturbation can be a source of B-mode polarization.Some inflation models can produce the intense tensorperturbation enough to generate detectable B mode, whileothers cannot.

Cosmic strings can also induce B mode. Cosmic stringsmove in the Universe in a very complicated and nonlinearway, constantly generating all types of perturbations, sca-lar, vector, and tensor ones. Therefore, dynamics of thecosmic string network induces B mode and it reaches anobservable level if the tension of cosmic strings,�, is largeenough, say, G� * 10�7 [10]. Here, G denotes theNewton constant.

In this paper, we point out that there is an additionalsource of Bmode when the cosmic string network exists. Itis the stochastic gravitational wave (GW) from kinks on

infinite strings.1 In the previous paper [11], we investigatedGWs emitted from kinks on infinitely long strings, andfound that GWs with a wavelength comparable to theHubble horizon scale are generated. These longwavelength GWs can produce an observable B mode inthe CMB. As a result, we find that the contribution ofsuch a GW background to a B mode is comparable tothat due to dynamics of the cosmic string network anddetectable by future CMB experiments, such as PLANCKor CMBpol.This paper is organized as follows. In Sec. II, we present

the formalism which we adopt in order to compute the BBpower spectrum. In Sec. III, we briefly review the result ofRef. [11] for the GW spectrum from kinks which will beused in the following analysis. In Sec. IV, we show theresulting BB power spectrum and discuss its observationalimplications. Section V is devoted to summary.

II. FORMALISM FOR COMPUTATIONOF THE BB POWER SPECTRUM

In this paper, we adopt the formalism described inRef. [12]. The tensor mode of the metric perturbation Dij

is defined as gijðt;xÞ ¼ aðtÞ2ð�ij þDijðt;xÞÞ, where aðtÞ

1The effects of the kinks on infinite strings are partiallyreflected in the calculation in Ref. [9] based on the latticesimulation. However, such a simulation covers only the limitedperiod of the evolution of the string network, so the correct kinkdistribution on infinite strings cannot be taken into account bythis method. Moreover, it is impossible to completely separatethe GWs from kinks from those emitted at the phase transition.Therefore, our calculation based on the kink distribution derivedanalytically is complementary to the calculation in Ref. [9].

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is the scale factor, and it is symmetric, transverse, andtraceless: Dij ¼ Dji, @iDij ¼ 0, Dii ¼ 0. We expand Dij

in the following form:

Dijðt;xÞ �Z

d3qeiq�xDijðt;qÞ

� X�¼�2

Zd3qeiq�xeijðq̂; �ÞDðt;q; �Þ; (1)

where q̂ � q=jqj, � denotes the helicity of the GW andeijðq̂; �Þ is the polarization tensor. We can think of

Dðt;q; �Þ as a stochastic variable since it is the productof random GW emission by kinks on infinite strings. Itsroot mean square is inferred from previous paper [11] andgiven in the next section.

As described in [12], this metric perturbation relates tothe polarization of photons. We do not explain the detailhere, but write down only several important equations. TheBB power spectrum, CBB;‘, is defined by ha�B;‘maB;‘0m0 i ¼CBB;‘�‘‘0�mm0 [13], where aB;‘m is some combination of

the coefficients of the multipole expansion of the Stokesparameters, Qðn̂Þ and Uðn̂Þ, by the spin-weighted harmon-ics. aB;‘m can be written as

aB;‘m ¼ i‘T0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ð2‘þ 1Þ

8

s X�¼�2

�Z

d3qDð‘Þm;�ðSðq̂ÞÞ

�Z t0

0dtPðtÞ�ðt;q; �Þ

���

8�þ �2 @

@�

�j‘ð�Þ�2

����������¼qrðtÞ; (2)

where T0 is the present CMB temperature, t0 is the age of

the Universe, rðtÞ ¼ Rt0t

dt0aðt0Þ , Dð‘Þ is the spin-‘ unitary

representation of the rotation group, Sðq̂Þ is the rotationwhich takes the three-axis into the direction q̂ and j‘ is

the ‘th spherical Bessel function. PðtÞ ¼ !cðtÞ�expð�Rt0

t !cðt0Þdt0Þ is the so-called visibility function,

which has sharp peaks at the moment of recombinationand reionization.!cðtÞ is the rate of Thomson scattering.�is the function, which satisfies

�ðt;q; �Þ ¼ 3

2

Z t

0dt0e�

Rt

t0 dt00!cðt00Þ

���2 _Dðt0;q; �ÞK

�qZ t

t0

dt00

aðt00Þ�

þ!cðt0ÞF�qZ t

t0

dt00

aðt00Þ��ðt0;q; �Þ

�; (3)

KðxÞ ¼ j2ðxÞ=x2;FðxÞ ¼ j0ðxÞ � 2j1ðxÞ=xþ 2j2ðxÞ=x2:

(4)

The definitions of the Stokes parameters aB;‘m and � are

found in [12]. If we know the way for D to evolveprecisely, we can get � through Eq. (3) and calculate thepower spectrum by integrating Eq. (2). However, we can-not know the phase ofD, which varies randomly, since thestochastic background of GW is formed by random andcontinuous accumulation of GWs from kinks. We can findonly the expectation value of its amplitude. Nevertheless,we can estimate the BB power spectrum using the�-function-like property of PðtÞ, as described in theAppendix. The B-mode power spectrum is calculated as

CBB;‘ ’ �2T20

Zdqq2½A‘ðqÞ _~D

2ðtrec; qÞ

þ B‘ðqÞ _~D2ðtrei; qÞ�; (5)

where_~D is defined in the next section, A‘ðqÞ and B‘ðqÞ are

defined in Appendix, and trecðtreiÞ is the cosmic time atrecombination (reionization), where the visibility functionhas a peak.

III. SPECTRUM OF THE STOCHASTICGRAVITATIONALWAVE BACKGROUND

The amplitude of the tensor perturbation is found asbelow. In [11], we derived the spectrum of the stochasticGW background, using this kink distribution function [14],which describes the abundance of kinks for a given sharp-ness. In the matter-dominated (MD) era, the energy densityof GWs of frequency �! is

d�

d ln!�

8><>:10G�2ð!tÞCmt�2 for t�1 <!<!ðMDÞ

1 ðtÞ ¼�teqt

�Am

t�1

10G�2

�teqt

��2D=Arð!tÞCrt�2 for !ðMDÞ1 ðtÞ<!<!ðMDÞ

2 ðtÞ ¼�teqt

�Am�trteq

�Ar

t�1;(6)

where Am ¼ �0:8, Ar ¼ �0:92, Cm ¼ �0:17, Cr ¼ 0:14,D ¼ 0:11, teq is the cosmic time of the matter-radiationequality, and tr is the time when the reheating completes.GWs with frequency larger than !ðMDÞ

2 ðtÞ are irrelevantbecause their wavelength is too short to affect the

large-scale density perturbation probed by CMB observa-tions. The periods concerning the CMB polarization areonly those around the recombination and the reionization,hence it is sufficient to consider the matter-dominated eraonly. This GW background consists of GWs emitted

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toward random directions from random points in theUniverse at random time. Therefore, we can think of it asbeing isotropic and homogeneous.2

We want to connect this expression of the energy densityto the amplitude of GWs D. At the small scale where thecosmic expansion can be neglected, the energy density ofGWs can be written as [16]

� ¼ 1

32�Gh _Dijðt;xÞ _Dijðt;xÞi: (7)

This expression applies to GWs from infinite strings,

whose wavelength is shorter than the Hubble radius.Under the present notation, the energy density of GWswhose frequency �! can be written as

d�

d ln!¼ 1

2G!3a3

_~D2ðt; qÞ: (8)

Here, we set h _Dðt;q; �Þ _Dðt;q0; �0Þi �_~D2ðt; qÞ���0�3ðq� q0Þ and used the fact that Dðt; qÞ is

oscillating with frequency q=a ¼ !. Eventually, fromEqs. (6) and (8) we obtain

_~Dðt; qÞ �8><>:

ffiffiffiffiffiffi20

pG�q�3=2ð!tÞCm=2t�1 for t�1 <!<!ðMDÞ

1 ðtÞffiffiffiffiffiffi20

pG�q�3=2

�teqt

��D=Arð!tÞCr=2t�1 for !ðMDÞ1 ðtÞ<!<!ðMDÞ

2 ðtÞ: (9)

IV. BB POWER SPECTRUM

Now let us calculate the BB power spectrum. As for thecosmological parameters, we used the result of the 7-yearWMAP observation [17]. Besides, we have to specify theionization history, or the shape of !cðtÞ. It is given as!cðtÞ ¼ �TneðtÞ, where �T is the Thomson scatteringcross section and ne is the number density of electrons.We calculate the time evolution of neðtÞ around the recom-bination epoch by using the RECFAST code [18,19].Concerning the reionization, we make an approximationthat the reionization occurs suddenly at some redshift zre:ne jumps from 0 to some value ne0 at zre. Thereafter, nedecreases in proportion to ð1þ zÞ�3. In short, we assume

!cðtÞ ¼�0 for z > zre!c0ð1þ zÞ�3 for z < zre;

(10)

around the reionization. We set zre ¼ 10:4 according toRef. [17]. !c0 is a constant which is determined so thatRt0trei dt!cðtÞ conforms to the reionization optical depth

� ¼ 0:087 [17].We show the resulting BB power spectrum in Fig. 1

(thick red) with the spectra produced by the string networkdynamics (blue), the inflationary tensor perturbation withtensor-to-scalar ratio of 0.1 and 0.01 (black), and the lens-ing effect (green). We also show the sensitivity curves ofPLANCK and two different realizations of plannedCMBpol satellites, EPIC-LC and EPIC-2m. The spectruminduced by the string network dynamics is drawn byCMBACT [20,21] and those originating from the inflation

are obtained by the CAMB code [22]. The value ofG� is setto be 10�7, which is close to the present observational

upper bound [23–25]. In computation using CMBACT, thenetwork parameters are set as follows : the wiggliness�r ¼ 1:8, the rms string velocity vr ¼ 0:64, and the ratioof the correlation length to the cosmic time r ¼ 0:3,where subscript r means the values in the radiation era.They are derived from the results of the simulations[26–29], and extrapolated to the matter era by the proce-dure in the code of CMBACT.We can see the spectrum induced by the GWs from

kinks on infinite strings has two peaks, one of whichis located at ‘� 100 and the other at ‘� 5. The peak at‘� 100 (‘� 5) is induced by GWs which exist at therecombination (reionization). At every moment, the lowerlimit of frequency of existing GWs is roughly the Hubbleparameter at that time. Besides, the amplitude of GWsdeclines toward higher frequency. As a result, the positionof each peak is set by the Hubble parameter at the recom-bination or the reionization. Remembering that GWs of

FIG. 1 (color online). The BB power spectrum induced byvarious processes and the sensitivity curves of the future CMBexperiments. The sensitivity curves are derived from Ref. [30].

2In Ref. [11], we omitted GWs which do not overlap othersfrom the ‘‘background,’’ following the prescription given inRef. [15]. Here, however, we do not consider this subtlety andinclude all GWs in the background, because we are payingattention to only long wavelength modes, most of which overlapothers.

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frequency comparable to the Hubble parameter are, asdiscussed in [11], emitted by new kinks, one finds thatthe peak at ‘� 100ð‘� 5Þ is due to GWs emitted slightlybefore the recombination (reionization) by kinks which areborn a little before the recombination (reionization). Sincethe BB spectrum by kinks is comparable to or somewhatgreater than that by network dynamics in some regions, thetotal BB power spectrum induced by cosmic strings isdeformed by the effect of the GWs from kinks on infinitestrings. It is natural that the effect on Bmode of GWs fromkinks and that of GWs by global dynamics of strings arecomparable. The network dynamics can induce B modealso through the vector perturbation it produces. We expectthat its contribution is also comparable to that of GWs fromkinks, although strict comparison is difficult and requiresnumerical calculation. The magnitude of the spectrum isproportional to ðG�Þ2, and hence when we take a differentstring tension the shape is unchanged but the whole spec-trum moves upward or downward. If G� * ða fewÞ �10�7, this spectrum can be observed by PLANCK, and ifG� * 10�8, it can be detected by CMBpol.

The peak around ‘� 5 associated with the reionizationhas a characteristic shape. However, it might be an artifactof the approximation that we put� out of the time integralin (2), assuming that PðtÞ has a sharp peak around thereionization3 [see (A1)]. The actual peak might besmoother. In fact, B mode is induced over the finite timearound the reionization, and those which are produced atdifferent moments have their peak at different ‘. It isexpected that the total BB spectrum, which is the envelopeof such peaks, has a smoother shape. On the other hand, weexpect that Fig. 1 shows the correct position and height ofthe peak around ‘� 5.

V. CONCLUSION

In this paper, we have studied the effect of the stochasticGW background induced by kinks on the infinite cosmicstrings on the BB power spectrum of CMB polarization.Using the GW background obtained in Ref. [11], we haveestimated the resulting BB power spectrum. We found thatthis effect is comparable to that of the vector and tensormodes induced by motion of the cosmic string network andmay leave observable signatures in the spectrum. If thecosmic string tension is large enough, the BB power spec-trum by cosmic strings will be detected by future/on-goingsatellite experiments such as PLANCK and CMBpol. If itis discovered by the CMB experiments, then the directdetection of GWs from cosmic strings by pulsar timingarrays or space-laser interferometers may further confirmthe existence of the cosmic string [11].

ACKNOWLEDGMENTS

K.N. would like to thank the Japan Society for thePromotion of Science for financial support. This work issupported by Grant-in-Aid for Scientific research from theMinistry of Education, Science, Sports, and Culture(MEXT), Japan, No. 14102004 (M.K.) andNo. 21111006 (M.K. and K.N.) and also by WorldPremier International Research Center Initiative (WPIInitiative), MEXT, Japan.

APPENDIX

In this Appendix, we derive Eq. (5) using the �-function-like property of the visibility function PðtÞ. �ðt;q; �Þvaries more slowly than PðtÞ at its peaks. Therefore, thetime integral in Eq. (2) is approximated as

Z t0

0dtPðtÞ�ðt;q; �Þ‘ðqrðtÞÞ

’ �ðtrec; q; �ÞZrec

dtPðtÞ‘ðqrðtÞÞ þ�ðtrei; q; �Þ

�ZreidtPðtÞ‘ðqrðtÞÞ; (A1)

whereRrecð

RreiÞ represents integration around the recom-

bination (reionization), and ‘ðxÞ ¼ ð8xþ x2@=@xÞ�ðj‘ðxÞ=x2Þ. Then let us estimate �ðtrec;q; �Þ and�ðtrei;q; �Þ from Eq. (3). First, we consider �ðtrec;q; �Þ.The factor expð�Rtrec

t0 dt00!cðt00ÞÞ has the property that it

rapidly increases from 0 when t0 approaches trec. The

combination !cðt0Þ expð�Rtrect0 dt00!cðt00ÞÞ also has such a

property, while _D and � vary more slowly than thesefunctions. Then, we obtain

�ðtrec;q;�Þ’�3 _Dðtrec;q;�ÞZrecdt0exp

��Z trec

t0dt00!cðt00Þ

�

�K

�qZ trec

t0

dt00

aðt00Þ�þ3

2�ðtrec;q;�Þ

�Zrecdt0!cðt0Þexp

��Z trec

t0dt00!cðt00Þ

�

�F

�qZ trec

t0

dt00

aðt00Þ�: (A2)

In contrast, expð�Rtreit0 dt00!cðt00ÞÞ is almost 1 around

t0 ¼ trei, since !c does not increase enough around the

reionization epoch. However, we can put _D out of the time

integral because of the property of KðqRtreit0

dt00aðt00ÞÞ. KðxÞ

decreases proportional to x�3 for large x. Besides, for q >

aðtreiÞ=trei, for which _Dðtrei;q; �Þ has a nonzero value,

qRtreit0

dt00aðt00Þ ’ 3 qtrei

aðtreiÞ ð1� ð t0treiÞ1=3Þ grows rapidly when t0

goes away from trei. After all, KðqRtreit0

dt00aðt00ÞÞ has a sharp

peak at t0 ¼ trei. Thus we can estimate �ðtrei;q; �Þ asabove:

3This approximation is much more valid around the recombi-nation than the reionization, therefore the peak associated withthe recombination does not have a bumplike feature.

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�ðtrei;q;�Þ’�3 _Dðtrei;q;�ÞZreidt0exp

��Z trei

t0dt00!cðt00Þ

�

�K

�qZ trei

t0

dt00

aðt00Þ�þ3

2�ðtrei;q;�Þ

�Zreidt0!cðt0Þexp

��Z trei

t0dt00!cðt00Þ

�

�F

�qZ trei

t0

dt00

aðt00Þ�: (A3)

Connecting the above estimations, we finally get

CBB;‘ ’ �2T20

Zdqq2½A‘ðqÞ _~D

2ðtrec; qÞ

þ B‘ðqÞ _~D2ðtrei; qÞ�; (A4)

where

A‘ðqÞ ¼�

CðqÞ1�DðqÞ

�2�Z

recdtPðtÞ‘ðqrðtÞÞ

�2;

CðqÞ ¼ �3Zrec

dt0e�R

trec

t0 dt00!cðt00ÞK�qZ trec

t0

dt00

aðt00Þ�;

DðqÞ ¼ 3

2

Zrec

dt0!cðt0Þe�R

trec

t0 dt00!cðt00ÞF�qZ trec

t0

dt00

aðt00Þ�;

and B‘ðqÞ is the function which we can get by substitutingtrec in A‘ðqÞ for trei.

[1] A. Vilenkin and E. P. S. Shellard, Cosmic Strings andOther Topological Defects (Cambridge University Press,Cambridge, England, 1994).

[2] R. Jeannerot, Phys. Rev. D 56, 6205 (1997).[3] D. H. Lyth and A. Riotto, Phys. Lett. B 412, 28 (1997).[4] S. Sarangi and S. H.H. Tye, Phys. Lett. B 536, 185

(2002).[5] G. Dvali and A. Vilenkin, J. Cosmol. Astropart. Phys. 03

(2004) 010.[6] U. Seljak, U. L. Pen, and N. Turok, Phys. Rev. Lett. 79,

1615 (1997).[7] K. Benabed and F. Bernardeau, Phys. Rev. D 61, 123510

(2000).[8] L. Pogosian, S. H. H. Tye, I. Wasserman, and M. Wyman,

Phys. Rev. D 68, 023506 (2003); 73, 089904(E)(2006).

[9] N. Bevis, M. Hindmarsh, M. Kunz, and J. Urrestilla, Phys.Rev. D 76, 043005 (2007).

[10] L. Pogosian and M. Wyman, Phys. Rev. D 77, 083509(2008).

[11] M. Kawasaki, K. Miyamoto, and K. Nakayama, Phys. Rev.D81, 103523 (2010).

[12] S. Weinberg, Cosmology (Oxford University Press, USA,2008).

[13] U. Seljak and M. Zaldarriaga, Phys. Rev. Lett. 78, 2054(1997); M. Zaldarriaga and U. Seljak, Phys. Rev. D 55,1830 (1997).

[14] E. J. Copeland and T.W. B. Kibble, Phys. Rev. D 80,123523 (2009).

[15] T. Damour and A. Vilenkin, Phys. Rev. D 64, 064008(2001).

[16] M. Maggiore, Phys. Rep. 331, 283 (2000).[17] N. Jarosik et al., arXiv:1001.4744.[18] S. Seager, D. D. Sasselov, and D. Scott, Astrophys. J. 523,

L1 (1999); Astrophys. J. Suppl. Ser. 128, 407 (2000).[19] http://www.cfa.harvard.edu/~sasselov/rec/.[20] L. Pogosian and T. Vachaspati, Phys. Rev. D 60, 083504

(1999).[21] http://www.sfu.ca/~levon/cmbact.html.[22] A. Lewis, A. Challinor, and A. Lasenby, Astrophys. J. 538,

473 (2000).[23] M. Wyman, L. Pogosian, and I. Wasserman, Phys. Rev. D

72, 023513 (2005); 73, 089905(E) (2006).[24] U. Seljak, A. Slosar, and P. McDonald, J. Cosmol.

Astropart. Phys. 10 (2006) 014.[25] N. Bevis, M. Hindmarsh, M. Kunz, and J. Urrestilla, Phys.

Rev. Lett. 100, 021301 (2008).[26] D. P. Bennett and F. R. Bouchet, Phys. Rev. D 41, 2408

(1990).[27] B. Allen and E. P. S. Shellard, Phys. Rev. Lett. 64, 119

(1990).[28] C. J. A. Martins, J. N. Moore, and E. P. S. Shellard, Phys.

Rev. Lett. 92, 251601 (2004).[29] C. J. A. Martins and E. P. S. Shellard, Phys. Rev. D 73,

043515 (2006).[30] D. Baumann et al. (CMBPol Study Team Collaboration),

AIP Conf. Proc. 1141, 10 (2009).

B-MODE POLARIZATION INDUCED BY . . . PHYSICAL REVIEW D 82, 103504 (2010)

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