Cosmic (Super)String Constraints from 21 cm Radiation

Rishi Khatri*Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, Illinois 61801, USA

Benjamin D. Wandelt†

Departments of Physics and Astronomy, University of Illinois at Urbana-Champaign,1002 W. Green Street, Urbana, Illinois 61801, USA

(Received 9 September 2007; revised manuscript received 22 January 2008; published 4 March 2008)

We calculate the contribution of cosmic strings arising from a phase transition in the early Universe, orcosmic superstrings arising from brane inflation, to the cosmic 21 cm power spectrum at redshifts z � 30.Future experiments can exploit this effect to constrain the cosmic string tension G� and probe virtuallythe entire brane inflation model space allowed by current observations. Although current experiments witha collecting area of �1 km2 will not provide any useful constraints, future experiments with a collectingarea of 104–106 km2 covering the cleanest 10% of the sky can, in principle, constrain cosmic strings withtension G� * 10�10–10�12 (superstring/phase transition mass scale >1013 GeV).

DOI: 10.1103/PhysRevLett.100.091302 PACS numbers: 98.70.Vc, 11.27.+d, 98.80.Cq, 98.80.Es

Introduction.—There has been a revival of interest incosmic strings, the line like topological defects of cosmiclength, after it was found that they can arise in the super-string theories in braneworld inflation scenarios [1]. Theyare also found to be inevitable in a wide class of super-symmetric grand unified theories (GUTs) [2]. Historically,cosmic strings were found to form in GUTs during phasetransitions in the early Universe along with the othertopological defects like monopoles and domain walls [3].Unlike monopoles and domain walls, which very quicklydominate the energy density of the Universe if formed afterinflation, strings approach a scaling solution and can re-main subdominant [4]. In brane inflation only cosmicstrings and no monopoles or domain walls are produced[5].

Cosmic strings were proposed as a mechanism for gen-erating the primordial fluctuations which later grew to formthe large scale structures we see today [6]. The cosmicmicrowave background (CMB) anisotropies and the matterpower spectrum arising from the fluctuations seeded bycosmic strings are very different from those generated frominflation [7]. Inflation just prescribes the initial fluctuationsat the end of inflation generated once and for all, whichthen just evolve. Cosmic strings generate fluctuationsthroughout the history of the Universe. One effect of thisis that the fluctuations generated at different times add upout of phase and wash out the acoustic oscillations in theCMB. The discovery of the acoustic peaks by CMB experi-ments [8,9] was a major success for inflation and ruled outcosmic strings as the dominant mechanism for seeding thecosmic perturbations. A subdominant contribution fromcosmic strings to the cosmic perturbations is still not ruledout with the current constraint being G� & 10�7 for clas-sical strings [10]. Cosmic strings if discovered eitherthrough their gravitational lensing effects [11], throughthe gravitational waves produced at string cusps or decay-

ing loops [12] or through their effect on the CMB and thematter power spectrums will provide insight into the fun-damental physics at high energies which is beyond thereach of currently planned terrestrial experiments.

21 cm radiation from z � 30 is an excellent probe of thestate of the Universe at that time. This radiation can probemuch smaller scales than the CMB, in the redshift range30 � z � 200, and provides a three-dimensional view ofthe Universe before reionization [13]. It has been shown tobe an excellent probe of the fundamental physics likevariation of constants (fractional variation in the fine struc-ture constant of & 10�5 with 104 km2 collecting area),non-Gaussianity from inflation (non-Gaussianity parame-ter fnl � 0:01–1), dark matter and inflationary fundamentalphysics [14–16]. In this Letter we show that it can, inprinciple, put unprecedented tight constraints on the cos-mic string contribution to the perturbations in the matter orequivalently on the string tension G�, and possibly otherparameters, which translates into constraints on the GUTsand the superstring theory. G is the gravitational constantand � is the string mass per unit length so that G� isdimensionless.

Cosmic strings.—Cosmic strings arise in GUTs andsuperstring theories whenever there is a phase transitionin the Universe if the vacuum manifold contains unshrink-able loops, e.g., U(1). The superstring theory can produce avariety of cosmic strings, which can be fundamental (F)strings or D strings produced during annihilation of D-branes. The string tension for these strings in brane infla-tion models is 10�12 & G� & 10�6 [5,17,18]. Just like theclassical cosmic strings from GUTs, they are wiggly, canintercommute, and form loops which can decay into gravi-tational radiation or elementary particles. The main differ-ence from the classical cosmic strings of GUTs is that theirintercommuting probability can be less than unity. Alsodifferent kinds of strings can form bound states [19]. The

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superstring theory string networks have scaling solutions[20] just like classical strings [4]; i.e., the total length of thestrings inside a horizon volume is proportional to thehorizon size at any time and the string energy density isa constant fraction of the dominant energy density compo-nent of the Universe in the radiation dominated as well asmatter dominated eras, preventing them from dominating.This also makes possible to construct simpler models ofstring networks which can than be used to study theirimpact on cosmology. Also since the perturbations pro-duced by cosmic strings are independent of the inflationaryinitial conditions, the two kind of perturbations can beevolved independently and the resulting power spectrafor CMB or matter added together to get the total powerspectrum.

The long wiggly cosmic strings have a structure thatresembles a random walk on scales larger than the horizonbut are straight on small scales [21]. We use the CMBACT

code developed by Pogosian and Vachaspati [22], which isbased on CMBFAST [23]. Wiggly strings are modeled in thecode as independent pieces of small strings whose length istaken to be of the order of the correlation length of thepieces of strings derived from numerical simulations. Theintercommuting probability (P) is taken to be unity. Thewiggliness on small scales results in the effective stringtension and mass per unit length of the string observed by adistant observer to differ [24] with the equation of state~U ~T � �2, where ~U and ~T are the effective mass per unitlength and tension of the wiggly string. Thus at scalesgreater than the scale of wiggles, the matter around expe-riences a Newtonian gravitational potential in addition tothe deflection due to the conical space around the string[25]. The wiggliness can be controlled in CMBACT by awiggliness factor � defined by the equations ~U � �� and~T � �=� [22].

21 cm radiation.—After recombination, most of thebaryonic matter is in the form of neutral atomic hydrogenand helium with a very small residual ionization. Thissmall fraction of electrons couples the CMB to the matterup to z� 500 through Compton scattering and maintainsthe gas at the same temperature as the CMB. Around z�500 the Compton scattering timescale becomes larger thanthe Hubble time and the gas decouples from the CMB andcools adiabatically thereafter. The ground state of hydro-gen atom has a hyperfine splitting with an energy differ-ence of T? � 0:068 K. This corresponds to the�21 � 1420 MHz rest frequency or �21 � 21 cm restwavelength. The hydrogen gas will absorb or emit photonsof this energy depending on the population levels in thetwo states which is best parameterized by the spin tem-perature Ts defined by the equation nt=ns � gt=gse�T?=Ts .Here nt and ns are the number densities of hydrogen atomsin the excited triplet state and the ground singlet state,respectively, gt � 3 and gs � 1 are the correspondingstatistical weights. The population levels during z � 30can change through collisions or through emission and

absorption of CMB photons. Initially the collisions domi-nate over the radiative process and Ts follows the gastemperature Tg. At late times the density of gas becomestoo low for collisions to be effective and Ts approaches theCMB temperature, T� � 2:725�1� z�. Thus in the redshiftrange of about 500 � z � 30, Ts < T� and we have a netabsorption of the CMB [13,26]. At z� 30 the first stars areborn and the evolution of the Universe enters the nonlinearregime. We will focus on the redshift range 200 � z � 30in this Letter, which corresponds to the observed frequencyof �7:1 MHz � � � 47:3 MHz.

The observed intensity I� can be expressed in terms ofthe brightness temperature using the Rayleigh-Jeans for-mula, Tb � I�c2=2kB�2, with Tb given by [27]

Tb ��Ts � T���

�1� z�; � �

3c3@A10nH

16kB�221�H � �1� z�

dvdr�Ts

;

where nH is the number density of atomic hydrogen, A10 isthe Einstein A coefficient for spontaneous emission, c isthe speed of light in vacuum, kB is Boltzmann’s constantand @ � h=2� with h being Planck’s constant. H is theHubble parameter, r is the comoving distance to redshift zand v is the peculiar velocity along the line of sight. Weignore the contribution of the vector modes to peculiarvelocities since it is subdominant (< 1%) compared tothe scalar mode contribution at scales of interest (l >1000). All quantities except the physical and atomic con-stants are functions of z. There will be spatial fluctuationsin Tb and Ts caused by the fluctuations in nH and Tg, whichin the redshift range of interest are related to the linearlyevolved primordial perturbations in standard inflationarycosmology. We will see below that the cosmic strings, ifthey exist, can add a significant contribution to thesefluctuations.

Expanding the fluctuations in the brightness temperature�Tb � Tb � �Tb, where �Tb is the mean brightness tempera-ture, in spherical harmonics, we get the angular powerspectrum Cl�z� � halmalmi, where alm are the coefficientsin the spherical harmonic expansion of �Tb. Following[26] we can write,

Cil�z� �Z d3k

�2��3Pi�k; z�Sl�k; z�;

where P�k; z� is the baryon (Fourier) power spectrum,index i � ad or cs for adiabatic perturbations from infla-tion or perturbations from cosmic strings, respectively. Wehave incorporated the 21 cm physics into Sl�k; z� [26].

Results.—We calculate Pad�k; z� using CMBFAST [23]and Pcs�k; z� using CMBACT [22]. The cosmological pa-rameters are from WMAP3 assuming �CDM cosmology[9]. For the cosmic string model in CMBACT we use initialrms velocity of 0.65, wiggliness factor in the radiation eraof 1.9 and initial correlation length of 0.13 times the initialconformal time, motivated by numerical simulations[21,28]. The intercommuting probability of strings is takento be unity, which means classical strings. The effect of

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smaller intercommuting probability will translate into adenser network which will make G� smaller for thesame amplitude of string generated perturbations. Ourpredicted constraints on G� are therefore conservative.

The angular power spectra for two redshifts are plottedin Fig. 1 for G� � 10�7 along with the inflationary spec-tra. One important feature of the cosmic string powerspectrum is that it turns over at smaller scales comparedto the inflationary adiabatic power spectrum. This is due tothe fact that the strings continue to generate perturbationsactively at all times. To estimate the constraints possible onG� from these observations we calculate the Fisher matrixF for the parameter � �G�=10�7�2, assuming that allthe other parameters are known.

Fi �Xz

Xil�1

fsky�2l� 1�

2Cadl �z� � C

Nl �z��

�2

�@Cl�z�@

�2;

where Cl � Cadl � Ccsl , fsky is the sky fraction observed

and the sum is over all redshift slices and l up to amaximum l � i. We use the fact that the bandwidth ofany experiment is finite only to estimate the noise CNl andthe number of redshift slices available and ignore its damp-ing effect on Cl [29]. Bandwidth of 0.4 MHz ensures thatthe slices are seperated by * r=l for l * 1000 and are thusuncorrelated.

The noise in the frequency range 7< �< 47 MHz isdominated by the sky temperature which follows a powerlaw, �CNl �

1=2 / Tsky / ��2:5 [30]. For LOFAR [31] we use

the noise estimates of [32] at z � 10, scaled to low fre-quencies using the above mentioned power law with abandwidth �� � 0:4 MHz and integration time of 3 years.These results also apply to LWA [33] which has similarspecifications as LOFAR. Figure 2 shows the constraintson G� assuming an error of 1=2�Fl�

1=2 (the factor of 2comes in when we convert the error on to the error on1=2) for LOFAR, two futuristic experiments and the cos-mic variance limit assuming that the foregrounds can beremoved at required precision [32,34]. Most of the infor-mation onG� is at l > 104 as is clear from an inspection of

the power spectra also. For the cosmic variance limit weuse the fact that there are more independent modes at highl, �� � �2rH=�21lc [13] to calculate the number of red-shift slices. The curve labeled �100 km�2 corresponds to afuturistic telescope of size 100 km and collecting area of104 km2 that will reach out to l� 105 and G�� 10�10. A�1000 km�2 telescope will be needed to constrain G��10�12. Such a telescope might be possible not only onEarth but also in space [35] or on the far side of Moon[36]. To reachG�� 10�14 will require a collecting area of1012 km2. Reaching the cosmic variance limit of �10�15

may not be possible because the scattering of radio wavesby the ionized interstellar medium will limit the smallestangular scales ( / ��2) that can be observed [37]. Inparticular, the l > 106 modes will not be available at allredshifts.

It is clear from Fig. 2 that the information content of the21 cm signal is huge and can in principle constrain G��10�16, if we have a cosmic variance limited measurementon the full sky. This corresponds to a phase transitionenergy scale of 1011 GeV for GUT theories. If we take��2M2

s for D-brane strings, this means bounds on the super-string mass scale, Ms, down to �1011 GeV [5]. In realityonly a fraction of the sky would be available due to ourbeing confined to the galaxy and small scale modes maynot be available because of the interstellar scattering ofradio waves, but even then the 21 cm signal has impressiveconstraining power over G�. The power at small scalesdue to cosmic strings is in fact underestimated in this linearcalculation. Cosmic strings generate wakes behind them,which have a density contrast of unity, that is not taken intoaccount here and which would enhance the power due tocosmic strings at small scales through nonlinear gravita-tional evolution. This is just the information contained inthe power spectrum. Higher order correlations will provideadditional discriminating power to check for the signaturesof perturbations seeded by a network of cosmic strings.The non-Gaussianities due to cosmic strings would belarger due to the highly nonlinear nature of the perturba-

Inflationary

(Gµ =10−7)

z=100

z=40

z=40z=100

Cosmic strings

FIG. 1 (color online). Angular power spectra from inflationaryadiabatic initial conditions (COBE normalized) and cosmicstrings.

FIG. 2 (color online). Constraints from current and futureexperiments on G� for a sky fraction of 10%, bandwidth of0.4 MHz and integration time of 3 years. Also marked is theconstraint on G� achievable by a full sky cosmic variancelimited experiment for the parameters assumed in the presentcalculation.

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tions at small scales and different from those producedduring inflation [14]. This impressive constraint is due thefact that there are large number of modes available at high lso that the statistical errors become very small.

High redshift 21 cm observations thus provide a rareobservational window into the superstring theory andsupersymmetric grand unified theories through cosmicstrings.

We thank Xavier Siemens and Richard Battye for dis-cussions on the subject. We also thank Levon Pogosian andTanmay Vachaspati for making their code publicly avail-able and answering our questions related to it. We thank thereferee for comments that improved the presentation of ourresults and Nikita Sorokin for assistance in preparingFig. 2. We thank the Max Planck Institute forAstrophysics for their hospitality where most of thiswork was done. Ben Wandelt was supported by theAlexander von Humboldt Foundation.

*rkhatri2@uiuc.edu†bwandelt@uiuc.edu

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