Effects of cosmic strings on free streaming

Effects of cosmic strings on free streaming

Tomo Takahashi1 and Masahide Yamaguchi21Department of Physics, Saga University, Saga 840-8502, Japan

2Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara 229-8558, Japan(Received 17 August 2006; published 15 September 2006)

We study the effect of free streaming in a universe with cosmic strings with time-varying tension as wellas with constant tension. Although current cosmological observations suggest that fluctuation seeded bycosmic strings cannot be the primary source of cosmic density fluctuation, some contributions from themare still allowed. Since cosmic strings actively produce isocurvature fluctuation, the damping of smallscale structure via free streaming by dark matter particles with large velocity dispersion at the epoch ofradiation-matter equality is less efficient than that in models with conventional adiabatic fluctuation. Wediscuss its implications to the constraints on the properties of particles such as massive neutrinos andwarm dark matter.

DOI: 10.1103/PhysRevD.74.063512 PACS numbers: 98.80.Cq

I. INTRODUCTION

Cosmological observations such as cosmic microwavebackground (CMB), large scale structure, and so on be-come now very precise and can constrain various cosmo-logical parameters with unprecedented accuracy. In addi-tion to the determination of cosmological parameters suchas energy densities of baryons, dark matter, and dark en-ergy, Hubble parameter, the scalar spectral index, reioni-zation optical depth, and so on, cosmological observationscan also constrain unknowns in particle physics. For ex-ample, neutrino masses can be severely constrained by cos-mology [1,2]. It is well known that massive neutrinos canerase small scale inhomogeneities via free streaming sincemassive neutrinos can have large velocity dispersion at thetime of radiation-matter equality. Because of this effect, thematter power spectrum exhibits the suppression at smallscales, which can be compared to observations of largescale structure to give the constraint on neutrino masses.1

Another such example is warm dark matter (WDM)particles. WDM particles can also have large velocitydispersion at the epoch of radiation-matter equality likemassive neutrinos. WDM scenarios have been extensivelystudied in the literature in connection with particle physicsand astrophysics. As for astrophysical aspects, WDM hasbeen discussed, in particular, as a solution of the problemof small scale structure such as the missing satellite prob-lem and the cusp problem [4,5]. From the viewpoint ofparticle physics, there exist well-motivated candidates forWDM such as a light gravitino [6,7], sterile neutrinos [8],and so on. The properties of WDM such as its mass can beconstrained by cosmological observations as the samemanner as the case with massive neutrinos since WDMparticles also erase the small scale fluctuation via the freestreaming effect. It should also be mentioned that super-weakly interacting massive particles (superWIMPs) [9]

can also erase the small scale structure as WDM particles,thus this kind of model can also be constrained by cosmo-logical observations by studying the damping of matterpower spectrum. Therefore, probing the damping of smallscale structure can be an important test for particle physics.

Constraints on the above mentioned particles such asmassive neutrinos and candidates for WDM have been wellstudied in the framework where cosmic density fluctuationis seeded by conventional adiabatic primordial fluctuationwhich is motivated by inflation driven by a scalar field.However, cosmic density fluctuation can also be producedby cosmic strings. Since fluctuation seeded by cosmicstrings is an incoherent actively generated isocurvatureone, this kind of fluctuation cannot produce observedstructure of the acoustic peaks in the CMB power spectrumbut gives rise to fairly broad acoustic peaks. Thus currentobservation suggests that cosmic strings cannot be theprimary source of density fluctuation today. However,subdominant contribution from fluctuation seeded by cos-mic strings is still allowed [10]. Furthermore cosmologicalscenarios with cosmic strings have been revived for recentyears since there have been discussions that cosmic stringscan be produced in a wide class of string theory models. Inparticular, cosmic strings can be formed at the end of braneinflation [11–13].

In light of these considerations, it is interesting to studythe effect of free streaming in a scenario with cosmicstrings. Importantly, since cosmic strings produce fluctua-tion actively, the erasure of small scale inhomogeneitiesvia free streaming can be avoided to some extent, whichresults in delayed damping of small scale power [14–18].2

Hence one may consider that some contribution from fluc-tuation seeded by cosmic strings can relax the constrains onWDM or neutrino masses since the constraint on themassesmainly comes from the effect of free streaming. This is theissue which we are going to consider in this paper.

1The mass of neutrino can also affect the CMB power spec-trum through the modification of the structure of the acousticpeaks, which can give a severe constraint on them [2,3].

2For the case with baryon isocuravture models with massiveneutrinos or hot dark matter, see Ref. [19].

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In fact, the authors of Ref. [20] have discussed thepossibilities of relaxing the constraint on neutrino massesin models with cosmic strings with constant tension. Thereit was shown by a simple analytic estimate that even withthe addition of fluctuation from cosmic strings the con-straint on neutrino masses cannot be relaxed. In this paper,first of all, we reconfirm this result with more quantitativeanalysis by numerically calculating matter power spectrumand CMB anisotropy produced by cosmic strings. We alsodo similar calculations for WDM and judge whether theconstraints on WDM can be relaxed or not.

Furthermore, recently a new class of cosmic strings hasbeen considered, which has time-varying tension. When ascalar field constituting a string couples to another oscil-lating field or a string is realized in a brane configuration,cosmic strings with time-varying tension naturally appear.Cosmological evolution of such a class of cosmic stringshas been investigated in Refs. [21,22] where it was shownthat, after some relaxation time, it goes into the scalingregime like strings with constant tension [23,24] and globalmonopoles [25]. In the scaling regime, the typical length ofthe cosmic string network grows with the horizon scale.Then, in case that string tension is constant, the ratio of theenergy density of infinite strings to that of the backgrounduniverse is also constant, which generates scale invariantdensity fluctuations. On the other hand, when the stringtension is time varying, the ratio of the energy density ofinfinite strings to that of the background universe is notnecessarily constant due to the time dependence of thetension. For example, when the tension has the time de-pendence as G� / �n with � being the conformal time,fluctuations at small scales are enhanced but those at largescales are suppressed for negative values of n. Thus theabove discussion on the effects of cosmic strings on freestreaming may be modified when we consider the case withtime-varying tension.

The purpose of this paper is to study to what extentfluctuation from cosmic string with time-varying tensionas well as with constant tension can affect to avoid theerasure of small scale inhomogeneities via free streamingby massive neutrinos and WDM. We also discuss theimplications of the above mentioned phenomenon to theconstraints on the masses of these particles.

II. FREE STREAMING EFFECT IN A UNIVERSEWITH COSMIC STRINGS

Although fluctuation from cosmic strings cannot be theprimary source of cosmic density fluctuation today, somecontributions from them are still allowed. As mentioned inthe introduction, the damping of small scale fluctuation byfree streaming can be avoided to some extent when fluc-tuation is seeded by cosmic strings. Thus, even in modelswith massive neutrinos or WDM, the erasure of small scalefluctuation can be less efficient by adding some fluctuationproduced from cosmic strings, which may have many

implications to the constraints on the masses of neutrinosand WDM.

First we discuss this issue for the case with massiveneutrinos. In fact, the discussion on the alleviation of theconstraint on neutrino masses has already been made inRef. [20], where it was shown that even with the addition offluctuation from cosmic strings the constraint on neutrinomasses cannot be relaxed. This is explained as follows. Inorder that fluctuation from cosmic strings can affect thematter power spectrum at small scales where the freestreaming effect erases the inhomogeneities, the amplitudeof the fluctuation should be as large as that of the conven-tional adiabatic one. In this case, however, the fluctuationamplitude becomes too large at larger scales where thecurrent CMB measurements are relevant. Notice that thefluctuation from cosmic string becomes an isocurvatureone which gives the Sachs-Wolfe (SW) effect on theCMB temperature anisotropies as

�TT

��SW� 2�; (1)

where � is the gravitational potential which appears in ametric perturbation in the conformal Newtonian gauge. Onthe other hand, the SW effect in the conventional adiabaticcase can be written as

�TT

��SW�

1

3�: (2)

Thus for the same magnitude of �, the isocurvature fluc-tuation can give about 6 times larger CMB temperaturefluctuation on large scales than that from the adiabatic one.In other words, fixing the amplitude to give the right size tofit the CMB data, matter power spectrum for isocurvaturefluctuation should be about 36 times smaller than that foradiabatic fluctuation. Thus, it is impossible to relax theconstraint on neutrino masses using the isocurvature fluc-tuation seeded by cosmic strings without affecting theCMB constraint.

Here we demonstrate this quantitatively by calculatingthe matter and CMB power spectra in models with massiveneutrinos and cosmic strings using the CMBACT code[26]. In Fig. 1, we plot the matter power spectra P�k� forthe case with the conventional adiabatic fluctuationP�k��adiabatic� (dashed line), cosmic strings with constanttension P�k��string� (dotted line), and the total powerP�k��adiabatic� � P�k��string� (solid line) assuming neutrinomasses as

Pm� � 5 eV.3 For reference, we also plot the

data from SDSS [27]. The other cosmological parameters

3Neutrino masses can be constrained from CMB data alonethrough the modification of the structure of acoustic peaks andthe current bound is

Pm� < 2:0 eV at 95% C.L. [2]. However

this bound may not hold true in our case with not only adiabaticfluctuation but also isocurvature fluctuation generated by cosmicstrings. Thus, for an illustration, we assumed this value forneutrino masses.

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are assumed as �mh2 � 0:139, �bh

2 � 0:021, h � 0:66,ns � 1, and �reion � 0:091 where �m;b are the presentenergy densities normalized by the critical density formatter and baryon, h is the Hubble parameter, ns is thescalar spectral index, and �reion is the reionization opticaldepth. A flat universe is assumed. The energy density ofmassive neutrinos are added by reducing that of darkenergy. We adopt these values unless otherwise stated inthis paper. As seen from the figure, although the matterpower spectrum for adiabatic fluctuation is damped onsmall scales in this case, we can compensate the dampingby adding fluctuation from cosmic strings with constanttensionG� � 5:2� 10�6. However the CMB power spec-trum generated by the cosmic strings with the size of thetension has too much power and contradicts with WMAPobservations, as shown in Fig. 2.

In the discussion above, we considered cosmic stringswith constant tension. Recently, however, cosmic stringswith time-varying tension have been studied inRefs. [21,22]. In particular, in Ref. [22], the CMB andmatter power spectra have been discussed for such models.It was explicitly shown that for the cases where the stringtension decreases with time as G� / �n or / an with abeing the scale factor and n being negative power, thefluctuation on large scales is significantly suppressed andthat on small scales is enhanced. Hence we can naivelyexpect that the above argument on the alleviation of theconstraint on neutrino masses can be modified for suchcosmic strings. We studied this issue by calculating theCMB and matter power spectrum using the modified ver-sion of CMBACT code where we have introduced the timedependence of the tension as G� / �n or / an.

As an example, in Fig. 3, we plot the matter powerspectrum in models with massive neutrinos (

Pm� �

10 eV) for the cases with conventional adiabatic fluctua-tion (dashed line), cosmic strings with time-varying ten-sion G� / ��0:4 (dotted line), and the total powerspectrum from these fluctuations (solid line). For compari-son, we also plot the case with cosmic strings with constanttension (dash-dotted line). Another case for the time de-pendence of the string tension with G� / a�0:2 is alsoshown in Fig. 4. As seen from the figure, fluctuation from

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FIG. 2 (color online). The CMB TT power spectrum in modelswith massive neutrinos (

Pm� � 5 eV) for the case with cosmic

strings with constant tension G� � 5:2� 10�6 (solid line). Thedata from WMAP3 are also plotted [1].

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P(k

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FIG. 3 (color online). Matter power spectrum in models withmassive neutrinos (

Pm� � 10 eV) for the cases with conven-

tional adiabatic fluctuation (dashed line), cosmic strings withtime-varying tension G� / ��0:4 (dotted line), and the totalpower spectrum from the adiabatic fluctuation plus cosmicstrings with time-varying tension (solid line). For comparison,the case with constant tension G� � 5:1� 10�6 is also plotted(dash-dotted line).

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tional adiabatic fluctuation (dashed line), cosmic strings withconstant tension G� � 5:2� 10�6 (dotted line), and the totalmatter spectrum (solid line). For reference, we also plot the datafrom SDSS [27].

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cosmic strings with the time dependence G� / ��0:4 andG� / a�0:2 can cancel the damping caused by free stream-ing of massive neutrinos. However, even in these cases,since adiabatic fluctuation with massive neutrinos dampsthe matter power spectrum on smaller scales than aroundk� 0:02 Mpc�1, we have to add the contribution fromcosmic strings to compensate the damping around thisscale. When we have non-negligible amplitude from cos-mic strings around this scale, it generates too much powerto fit the CMB data even though the fluctuation on largescales is reduced due to the time dependence of the stringtension. To show this, in Fig. 5, we plot the CMB powerspectrum generated by cosmic string with time-varyingtension G� / ��0:4 and a�0:2 with the same normalization(string tension) as that in Fig. 3. As seen from the figure, wecannot compensate the erasure of fluctuation on suchscales without contributing to the CMB power spectrumsignificantly, which is obviously inconsistent with currentobservations, even if we introduce the cosmic strings withtime-varying tension. We note that, depending on theneutrino masses, we can compensate the damping of smallscale power via free streaming by choosing the time de-pendence of the string tension appropriately. However, ourconclusion remains unchanged as long as adiabatic fluc-tuation with massive neutrinos damps the matter powerspectrum on small scales which correspond to ‘�O�100�.

Next we discuss the effects of cosmic strings on freestreaming for the case with WDM. For WDM, since thescales of masses are significantly larger than those ofmassive neutrinos considered above, the damping scalecan be much smaller. Thus in this case, we may have a

chance to avoid the erasure of small scale inhomogeneitiesby free streaming without affecting the CMB power spec-trum on large scales by adding isocurvature fluctuationfrom cosmic strings. In Fig. 6, we plot the matter powerspectrum for the cases with conventional adiabatic fluctua-tion (dashed line), cosmic strings with constant tensionG� � 7:8� 10�7 (dot-dashed line), and the total powerspectrum (solid line). For reference, we have also plottedthe data from SDSS [27] and Lyman alpha [28] which are

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FIG. 5 (color online). The CMB TT power spectrum in modelswith massive neutrinos (

Pm� � 10 eV) for the case with

cosmic strings with time-varying tension G� / ��0:4 (solidline) and G� / a�0:2 (dashed line). The data from WMAP3are also plotted [1].

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FIG. 6 (color online). Matter power spectrum in models withWDM with the mass mWDM � 103 eV. Here we plot P�k� fromthe conventional adiabatic fluctuation (dashed line), cosmicstrings with constant tension G� � 7:8� 10�7 (dot-dashedline), and the total of them (solid line). For reference, the datafrom SDSS [27] (circles) and Lyman alpha [28] (squares) whichare rescaled to z � 0 are also plotted.

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tional adiabatic fluctuation (dashed line), cosmic strings withtime-varying tension G� / a�0:2 (dotted line), and the totalpower spectrum from the adiabatic fluctuation plus cosmicstrings with time-varying tension (solid line). For comparison,the case with constant tension G� � 5:2� 10�6 is also plotted(dash-dotted line).


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rescaled to the present time z � 0. Here we assumed thepure WDM model. The mass of WDM is taken to bemWDM � 103 eV, which corresponds to a thermally de-coupled relic with the relativistic degrees of freedom at thetime of decoupling being g��TD� � 100. For the case withadiabatic fluctuation, WDM particles erase the structure onsmaller scales compared to the case with massive neutrinosas seen from the figure. With the parameter above, the

damping scale is k� 1 Mpc�1. On the other hand, forthe case with cosmic strings, such a damping becomesmoderate. As shown in Fig. 6, the small scale dampingfor adiabatic fluctuation can be canceled by adding thatfrom cosmic strings with constant tension. Furthermore,the CMB power spectrum generated by the above cosmicstrings can satisfy WMAP constraints, as shown in Fig. 7.Thus the constraint on WDM can be relaxed in a scenariowith cosmic strings even with constant tension.

In addition, we also show matter power spectrum for thecase with cosmic strings with time-varying tension G� /��1, conventional adiabatic fluctuation, and the total spec-trum of them in Fig. 8. Here we assume the mass of WDMparticles as mWDM � 72 eV. As seen from the figure, wecan cancel the damping on small scales in the spectrum forthe adiabatic case by adding the contribution from cosmicstrings with the time-varying tension n � �1. Importantly,in this case, the contribution from the cosmic string toCMB power spectrum up to ‘�O�1000� is negligiblesince large scale fluctuation by the cosmic strings is sig-nificantly suppressed due to the time dependence of thestring tension (Fig. 9). Hence we can have possibilities ofalleviating the constraint on the masses of WDM by addingsome fluctuation from cosmic strings with time-varyingtension in this case too.

III. CONCLUSION AND DISCUSSION

We studied the erasure of small scale structure via freestreaming in models with cosmic strings with time-varyingtension as well as with constant tension. Because cosmicstrings actively produce incoherent isocurvature fluctua-tion, free streaming effect is less efficient than that inmodels with the conventional adiabatic ones because ofits nature of the fluctuation. Since the damping of small

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FIG. 7 (color online). The CMB TT power spectrum in WDMmodels with the mass mWDM � 103 eV for the case with cosmicstrings with constant tension G� � 7:8� 10�7 (solid line). Thedata from WMAP3 are also plotted [1].

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FIG. 8 (color online). Matter power spectrum in models withWDM with the mass mWDM � 72 eV. Here we plot P�k� fromthe conventional adiabatic fluctuation (dashed line), cosmicstrings with time-varying tension G� / ��1 (dot-dashed line),and the total matter spectrum from the adiabatic fluctuation pluscosmic string with time-varying tension (solid line). The casewith constant tension G� � 9:8� 10�7 is also plotted forcomparison (dash-dashed line). For reference, the data fromSDSS [27] (circles) and Lyman alpha [28] (squares) which arerescaled to z � 0 are also plotted.

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FIG. 9 (color online). The CMB TT power spectrum in WDMmodels with the mass mWDM � 72 eV for the case with cosmicstrings with time-varying tension G� / ��1 (solid line). Thedata from WMAP3 are also plotted [1].


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scale power by free streaming is an important probe toconstrain the property of particles with large velocitydispersion at the epoch of radiation-matter equality suchas massive neutrinos and WDM, the above mentionedeffects by cosmic strings can have many implications tothe constraints on such particles.

We first studied this issue for massive neutrinos. For thecase with constant string tension, it has been already dis-cussed some time ago that, even if we add some contribu-tion from isocurvature fluctuation from cosmic strings, wecannot relax the constraint on neutrino masses. In thispaper, we explicitly calculate matter power spectrum andCMB anisotropy generated by strings and reach the sameconclusion. Furthermore, we have studied this issue for thecase with cosmic strings with time-varying tension. Wehave shown that even if we introduce cosmic strings withtime-varying tension, we cannot alleviate the constraint onthe mass without conflicting with the constraint from cur-rent CMB observations. This is mainly because adiabaticfluctuation with massive neutrinos damps the matter powerspectrum on small scales corresponding to multipoles ‘�O�100� in CMB power spectrum.

We have also discussed the possibilities of relaxingconstraints on the masses of WDM by adding fluctuationproduced by cosmic strings with time-varying tension andthose with constant tension. We explicitly showed that thedamping of the matter power spectrum on small scales

from adiabatic primordial fluctuation by free streamingcaused by WDM can be canceled by adding fluctuationseeded by cosmic strings for both cases, namely, withconstant and time-varying tension by choosing appropriateparameters. Importantly we can cancel the damping with-out conflicting the CMB constraints for both cases, whichis not the case for massive neutrinos. Although much moredetailed study is needed to state quantitatively to whatextent the constraint on WDM particles can be relaxed,in this paper, we pointed out that some contributions fromcosmic strings to small scale power can affect the con-straints without conflicting with the CMB constraint. Sincethe time dependence of the string tension is highly modeldependent and can be more complicated than that adoptedhere, various possibilities can arise and have many moreimplications to the constraints given from observationsprobing the damping of small scale fluctuation via freestreaming.

ACKNOWLEDGMENTS

We would like to acknowledge the use of CMBACTcode developed and made publicly available by L.Pogosian and T. Vachaspati. M. Y. is supported in part bythe project of the Research Institute of Aoyama GakuinUniversity and by the JSPS Grant-in-Aid for ScientificResearch No. 18740157.

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Effects of cosmic strings on free streaming