Searching for Signatures of Cosmic String Wakes in 21cm Redshift Surveys using Minkowski Functionals
SSearching for Signatures of Cosmic String Wakes in 21cm Redshift Surveys usingMinkowski Functionals
Evan McDonough and Robert H. Brandenberger
Department of Physics, McGill University, Montr´eal, QC, H3A 2T8, Canada (Dated: November 16, 2018)Minkowski Functionals are a powerful tool for analyzing large scale structure, in particular if thedistribution of matter is highly non-Gaussian, as it is in models in which cosmic strings contributeto structure formation. Here we apply Minkowski functionals to 21cm maps which arise if structureis seeded by a scaling distribution of cosmic strings embeddded in background fluctuations, and thentest for the statistical significance of the cosmic string signals using the Fisher combined probabilitytest. We find that this method allows for detection of cosmic strings with
Gµ > × − , whichwould be improvement over current limits by a factor of about 3. I. INTRODUCTION
The interest in searching for the signatures of cosmicstrings in cosmological observations has been increasing.This is due in part to the realization that cosmic stringscan arise in a variety of cosmological contexts. For ex-ample, in many inflationary models (both supergravity-based [1] and string-based [2]) the period of inflationends with the formation of a network of cosmic strings.Such strings may also be produced in conventional phasetransitions which occur after inflationary reheating [55].If they are topologically stable, the network of cosmicstrings will persist at all times and will approach a ‘scal-ing solution’ characterized by a string distribution whichis statistically independent of time if lengths are scaledto the Hubble radius (see e.g. [4–6] for reviews of cosmicstrings and their consequences for cosmology).Cosmic strings are linear topological defects whicharise in a wide range of quantum field theories as a con-sequence of a symmetry breaking phase transition. Theyare lines of trapped energy density and tension which isequal in magnitude to the energy density. They are anal-ogous to defect lines in crystals or to vortex lines in super-conductors and superfluids, except for the fact that thecosmic strings arise in relativistic field theories and henceobey relativistic dynamical equations of motion as op-posed to the strings in condensed matter systems whosedynamics is typically friction-dominated. The trappedenergy density associated with the strings leads to con-sequences in cosmology which result in clear observa-tional signatures. Since the energy density per unit stringlength increases as η as the symmetry breaking scale η increases, cosmology provides the ideal venue to searchfor strings at very high energy scales. Thus, searchingfor strings in cosmological observations provides an av-enue complementary to accelerator experiments for look-ing for signals of physics beyond the Standard Model:accelerator signals are more easily seen for low symme-try breaking scales, while signals in cosmology are moreeasily detected for larger values of η .Because of the scaling solution of the cosmic string net-work, cosmic strings lead to a scale-invariant spectrumof cosmological perturbations, as in inflationary mod- els (see e.g. [7]). However, the string-induced densityfluctuations are highly non-Gaussian, an effect which iseliminated when computing the usual power spectrumof density fluctuations. It follows that string signals aremuch easier to detect in position space than in momen-tum space.The scaling network contains two types of strings, anetwork of ‘infinite’ strings [56] and a distribution ofstring loops with radii smaller than the Hubble radius.String loops oscillate because of their relativistic ten-sion and slowly decay by emitting gravitational radiation.Their gravitational effects are similar to those of a pointmass [7]. In this paper we are interested in the charac-teristic non-Gaussian effects of long string segments andwill not further consider cosmic string loops (which pro-duce 21cm signals which are harder to differentiate fromnoise [8]).Due to the relativistic tension of the strings, an infinitestring segment will typically have a velocity in the planeperpendicular to the string which is of the order of thespeed of light [57]. Since space perpendicular to a longstraight string is conical with a ‘deficit angle’ given by[10] α = 8 πGµ , (1)where G is Newton’s constant and µ is the mass per unitlength of the string (which is proportional to η ) , a stringmoving with velocity v s through the matter gas of theearly universe at some time t i will lead to a ‘wake’ [11], athin wedge behind the string with twice the backgrounddensity whose planar dimensions are c t i × v s γ s t i . (2)Here, γ s is the relativistic γ factor associated with thevelocity v s , and c is a constant of order one which isproportional to the curvature radius of the long stringnetwork in units of the Hubble radius. The mean thick-ness h of the wake is initially h = 4 πGµv s γ s t i (3)and for t > t i it increases by gravitationally accretingmatter from above and below the wake [12]. a r X i v : . [ a s t r o - ph . C O ] J un Current observations (in particular the oscillatory fea-tures in the angular power spectrum of cosmic microwavebackground (CMB) anisotropies) set an upper bound onthe string tension of the order
Gµ < . × − [13, 14],which corresponds to cosmic strings contributing lessthan 5% to the total spectrum of inhomogeneities. Wewill discuss this in more detail in section II. In this paper,we are therefore considering a setup in which there is acontribution of cosmic strings to the power spectrum inaddition to the dominant component of Gaussian, nearlyscale-invariant fluctuations (such as can be produced ininflationary cosmology [15, 16] or in string gas cosmology[3]).Cosmic string wakes give rise to distinctive signaturesin the large-scale distribution of matter in the universe.As discussed in [17], they lead to planar structures inthe distribution of galaxies. Since wakes present betweenthe time of last scattering of the CMB and today rep-resent overdense regions of electrons, they lead to extrascattering of CMB photons and hence to distinctive rect-angular regions in the sky with extra CMB polarization[18] (with statistically equal B-mode and E-mode com-ponents). The effect relevant to the current paper is thatwakes represent overdense regions of neutral hydrogenand hence [19] lead to wedge-like regions in 21cm red-shift surveys of extra 21cm absorption or emission [58].The amplitude of the position space signal is independentof the string tension µ and can be as large as 100mK[59] . The width of the wedge, however, depends lin-early on µ . A crucial point is that wakes exist as non-linear structures at high redshifts since as soon as thestring passes by, a wake with overdensity 2 forms. Thisis to be contrasted to the situation in Gaussian modelswith a scale-invariant spectrum of primordial cosmologi-cal perturbations in which no nonlinear structures existuntil quite low redshifts. In particular, at redshifts largerthan that of reionization, the cosmic string signal shouldstand out against the effects of Gaussian fluctuations andnoise. Thus, 21cm redshift surveys appear to be an idealwindow to search for cosmic string signals.If cosmic strings exist, then the induced 21cm mapswill not only contain the characteristic wedges of extraabsorption or emission. They will also contain noise,most importantly Gaussian noise from the primordialGaussian density fluctuations which must be present inaddition to the string-induced perturbations. Good sta-tistical tools will be required in order to extract the stringsignals in a quantitative and reliable way.In this report we study the possibility that MinkowskiFunctionals [21], a tool for characterizing structure whichis orthogonal to the usual way of characterizing maps us-ing correlation functions, can be applied to identify cos-mic string signals in 21cm redshift surveys at reshiftslarger than that of reionization. The theory behindMinkowski Functionals has its roots in Integral Geom-etry and Hadwiger’s Theorem, which states that a d -dimensional set of convex bodies can be completely de-scribed by d + 1 functionals. Minkowski functionals have been applied numerous times in cosmology, e.g. to thedistribution of galaxies on large scales [22] and the analy-sis of CMB maps [23]. There have also been attempts toapply Minkowski functionals to search for signatures ofcosmic strings in the large-scale structure of the distribu-tion of galaxies [24]. Since for 21cm redshift surveys thecontributions to the ‘noise’ from the Gaussian source ofprimordial fluctuations is in the linear regime at the highredshifts we are interested in (larger than the redshift ofreionization), we expect that Minkowski functionals willbe very powerful in extracting cosmic string signals inthe case of 21cm maps.The outline of this paper is as follows: in the followingsection we outline current observational limits on cosmicstrings as well as prospects for future detection. In Sec-tion III we briefly review the toy model for the cosmicstring scaling distribution which we use and describe thetheoretical 21cm maps induced by cosmic strings. Sec-tion IV then repeats the methodology of section III butincluding a simulation of the background noise. In Sec-tion V we review the theory and calculation of Minkowskifunctionals, before presenting our results in Section VI.We conclude with a summary of our results and a discus-sion of future work. II. OBSERVATIONAL CONSTRAINTS
The strongest constraints on cosmic strings currentlycome from measurements of the angular power spec-trum of the cosmic microwave background (CMB). TheWMAP data provided no hints of cosmic strings, andhence placed an upper bound on the string tension [14].Specifically, it constrained the string contribution to theprimordial power spectrum to be less than 10%. How-ever, there are now CMB experiments with better an-gular resolution than WMAP provided and which canyield improved bounds on the cosmic string tension. TheAtacama Cosmology Telescope (ACT) [25], a six-meteroff-axis telescope with arcminute-scale resolution locatedin the Atacama desert in northern Chile, and the SouthPole Telescope (SPT) [26] , a 10 meter telescope locatedin Antarctica which can study the small-scale angularpower spectrum of the CMB, both are providing excel-lent data. A recent analysis of the angular power spec-trum of CMB anisotropies from joint SPT and WMAP7[13] made use of a Markov Chain Monte Carlo likeli-hood analysis to place a limit on the string tension of
Gµ < . × − (at 95% confidence). This analysis wasdone in the context of the zero width cosmic string toymodel which we will also use in the following. A similarlimit was found by [27], who used combined WMAP7,QUAD and ACT data to place limits on the tension ofAbelian Higgs model strings. The resulting bound on thestring tension was Gµ < . × − .The difference in the results is mostly due to the un-certainties in the distribution of strings. Due to the verylarge range of scales involved (the width of a cosmic stringis microscopic but the length is cosmological), numericalsimulations of both field theory and Nambu-Goto stringsinvolve ad hoc assumptions and/or extrapolations. Theuncertainties in the resulting string distributions appearin free parameters (e.g. the number of long string seg-ments per Hubble volume and the value of the constant c ) of toy models distributions of cosmic strings. Theseuncertainties effect the angular power spectrum of CMBanisotropies.Local position space searches for signals of individualcosmic strings are less sensitive to the uncertainties inthe string distribution than power spectra. This is theidea behind the proposal of [28–30] to search in posi-tion space for the line discontinuities in CMB anisotropymaps which long straight cosmic strings produce due tothe Kaiser-Stebbins [31] effect. It was proposed to ana-lyze position space anisotropy maps using of the Cannyedge detection algorithm. It was shown [30] that stringswith Gµ ≥ − might be detectable with this method.Application of this method to the SPT or ACT data mayyield interesting results [60]Another avenue for detection of cosmic strings is viagravitational waves, which will be produced by oscillatingcosmic strings and/or the decay of cosmic string loops.Cosmic strings in fact [33] produce a scale-invariant spec-trum of gravitational waves with an amplitude which issimilar to or larger than the amplitude of gravitationalwaves from simple inflationary models. Pulsar timing [34]or direct detection (e.g. making use of the Laser Inter-ferometer Gravitational-Wave Observatory (LIGO) [35] )provide means for searching for the cosmic string signal.However, at the moment the bounds are not competitivewith bounds obtained from the CMB [61].We are currently entering a revolution in radio as-tronomy, with both the Square Kilometer Array (SKA)[36] and the European Extremely Large Telescope (E-ELT)[37] aiming to be operational by early 2020. Ofparticular interest (in terms of potential to observe cos-mic strings) is the SKA, which as the name implieswill consist of 1 million square meters of collecting area.There are many ongoing projects to develop the scienceand technology necessary to operate the SKA. Theseare categorized into: (1) Precursor Facilities, which willphysically be at the SKA site carrying out SKA-relatedprojects, (2) Pathfinder Experiments, which will developSKA-related science and technology off-site, and (3) De-sign Studies, which will investigate technologies and de-velop prototypes. Many of these projects are world classfacilities in there own right, and we will mention some ofthe experiments which may have a chance of observingcosmic strings.One such project is LOFAR [38], a Pathfinder exper-iment, which is measuring the neutral hydrogen frac-tion of the inter-galactic medium. This will map outthe epoch of reionization between redshifts 11.5 and 6.7,and will be capable out of arc-minute resolution. For21cm measurements, the redshift range of LOFAR ex-tends to the “Dark Ages” before reionization, redshifts where the cosmic string signal will be cleanest. AnotherSKA project with potential to observe cosmic strings isthe MEERKAT experiment [39] , which is carrying outa project titled MESMER that will track the neutralhydrogen content of the early universe by using carbonmonoxide as a proxy. To make an accurate predictionas to the clumping of carbon monoxide around a cosmicstring would require semi-analytical hydrodynamics cal-culations, but to a rough approximation we should expectto see the characteristic wedge of emission/absorption. III. THE COSMIC STRING SIGNALA. Modelling the signal
In any theory which leads to the formation of stablecosmic strings in the early universe, a network of stringswill persist at all later times, and this network will ap-proach a ‘scaling solution’, meaning that the statisticalproperties of the network of cosmic strings are indepen-dent of time if all distances are scaled to the Hubble ra-dius. The existence of the scaling solution can be arguedfor using analytical arguments (see e.g. [4–6]), and it hasbeen confirmed using extensive numerical simulations [9].The scaling distribution of strings consists of a networkof long strings of mean curvature radius ζ = c t (where c is a constant of order 1), and string loops with a muchsmaller radius that result from intersection and hence‘cutting’ of the long strings. As a consequence of theirrelativistic tension, the long strings have typical trans-lational velocities v s close to the speed of light. Hence,Hubble length string segments will intersect with othersuch segments with probability close to one on a Hub-ble time scale. By this process, string loops are formedand the correlation length of the long string network(the mean curvature radius) increases in comoving coor-dianates to keep up with the Hubble radius, as confirmedin the numerical string evolution simulations [9].As is commonly done in studies of the cosmologicalconsequences of cosmic strings, we use an analytical toymodel of the infinite string network which was first in-troduced in [40]: we divide the time interval of interest(usually the time between the time t eq of equal matterand radiation and the present time t ) into Hubble timesteps. In each time step we lay down an integer num-ber N H per Hubble volume of straight string segmentsof length c t with random midpoints and random tan-gent vectors. The string segments in neighboring Hubbletime steps are taken to be independent. The long stringsare taken to have a velocity v s orthogonal to their tan-gent vectors. Each such string will produce a wake whosethickness grows in time.We will be interested in signals which are emitted at atime t e > t i from string wakes laid down at any time t i between t eq [62] and the present time . Since the planardimensions of the wake (whose initial physical dimen-sions are given in (2)) are constant in comoving coor-dinates, their physical size grows with the scale factor.The comoving thickness of the wake, whose initial phys-ical length is given by (3), grows linearly with the scalefactor because of gravitational accretion. Hence, the di-mensions of the wake at t e in physical coordinates aregiven by: z i + 1 z e + 1 (cid:2) c t i × t i v s γ s × πGµt i v s γ s z i + 1 z e + 1 (cid:3) , (4)where z e is the emission redshift z ( t e ).The dominant form of the baryonic matter in the uni-verse before reionization is neutral hydrogen (H), whichwe detect via the 21cm hyperfine line. Measuring theintensity of the redshifted 21cm radiation from the skyhas the potential of giving us three dimensional mapsof the distribution of neutral hydrogen in the universe(see e.g. [41] for an in-depth review article on 21cm cos-mology). As explained in detail in [19], since wakes areoverdense regions of neutral hydrogen, they will lead toexcess 21cm emission or absorption. We will see this ex-cess in directions for which our past light cone intersectsa wake at some time t e > t rec , t rec being the time of re-combination. This 21cm signal has the special geometrygiven by the wake geometry - a thin wedge in the three-dimensional 21cm redshift survey (see Figure 1), and it isthis special geometry which provides the ‘smoking gun’signal for cosmic strings, the signal which we are tryingto extract using Minkowski functionals in this report.Since already at the initial time t i the wake is a non-linear density fluctuation, the matter being accreted ontothe wake will collapse onto the wake, shock heat andthermalize [63]. The hydrogen (H) atoms inside a wakelaid down at redshift z i will have a temperature at time t e given by:[19] T K ( t e ) = 16 π
75 ( Gµ ) ( v s γ s ) z i + 1 z e + 1 mk B , (5)where m is the proton mass. Inserting numerical values,and expressing Gµ in units of 10 − and so written as( Gµ ) , we obtain T K ( t e ) ≈ [20 K ]( Gµ ) ( v s γ s ) z i + 1 z e + 1 . (6)Note that the numerical simulations by [43] show that thetemperature can be taken to be approximately uniforminside of the wake.The quantity of interest is the brightness temperature T b due to 21cm transitions of the Hydrogen in the wake,or more specifically the difference δT b in brightness tem-perature between photons from the wake and those fromthe surrounding space, From this point onwards, ‘bright-ness temperature’ will refer to δT b , defined by δT b ( z e ) = T b ( z e ) − T γ ( z e )1 + z e , (7) where T γ ( z e ) is the redshifted CMB temperature, and T b ( z e ) is the brightness temperature due to 21 cm emis-sion in the wake. A full derivation which can be found in[19] yields the following expression for δT b : δT b ( z e ) = [0 . K ] x c x c + 1 (cid:18) − T γ T K (cid:19) √ z e , (8)where several constants have been absorbed into the pref-actor, which can be found in the original derivation by[19]. Note that the collision coefficient x c is given by [41]: x c = nκ HH A T (cid:63) T γ , (9)where T (cid:63) , taken to be 0.068 K, is the temperature cor-responding to the hydrogen energy splitting E , A isthe spontaneous emission coefficient of the 21cm transi-tion, n is the number density of hydrogen atoms, and κ HH is the de-excitation cross section which is given in[41]. Using equation (8), the brightness temperature fora given wake with a given value of z e can be obtained,which in conjunction with the spatial dimensions of thewake, allows a 3-D map of brightness temperature to begenerated. B. Generating Temperature Maps
In the following we will outline the steps in the con-struction of 21cm redshift maps in the case of primordialperturbations produced exclusively by strings. We willconsider hypothetical sky maps covering a patch of thesky of angular scale θ × θ . First, we must compute thenumber of strings which contribute to the sky signal inthis patch. This is the number of strings whose wakesat some point in time between t rec and t intersect theobserver’s past light cone with opening angle θ .
1. Distribution of Strings
According to our model of the distribution of a scalingstring network we divide the time between t eq and thepresent time t into Hubble time steps. For each suchtime step centered at time t i we now compute the number N ( t i ) of string wakes which at some point in the futureof t i will intersect the past light cone corresponding toan observational angle θ (a box of size θ × θ on the sky).We first note that N ( t i ) is given by the ratio of thecomoving volume of the past light cone V p ( θ ) (cid:39) (cid:0) θ π (cid:1) t o (10)divided by the comoving Hubble volume at time t i V H ( t i ) = t i t , (11)multiplied by the number N H of strings per Hubble vol-ume in the scaling solution (a number which according FIG. 1: The right panel depicts a sketch of the string wake-induced wedge of extra emission/absorption in 21cm redshiftsurveys. The horizontal axis represents the two angular directions, the vertical axis redshift. The left panel is a space-timesketch showing the position of the string which gives rise to the wedge of the right panel. Here, the vertical axis is time, and thehorizontal axis corresponds to comoving spatial coordinates. Note, in particular, the positions where the string wedge intersectsthe past light cone of an observer at the current time t . The almost horizontal lines represent the string at the time when itforms the wake ( t i ), and the wedge at the times s and s when the back (and front) of the wake intersect our past light cone. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) x x x x t i t i c c ν x x δ ν t ννγ s s t to numerical simulations is in the range 1 < N H < N i (cid:39) N H (cid:0) θ π (cid:1) z ( t i ) / . (12)Now that we know the total number of string wakesproduced at redshift z i which will be seen in the simu-lation box, we need to determine the distribution of theredshift z e at which they will cross the past light cone ofthe simulation region in the sky. This can be determinedby remembering that light travels in a straight line inconformal coordinates (comoving spatial coordinates andconformal time τ ). Hence, the number of wakes which in-tersect the past light cone at conformal time τ (where theconformal time is measured backwards, meaning τ = 0 atthe present time and τ increases as we go back in time)in a conformal time interval dτ is dN i ( τ ) = (cid:0) ττ eq (cid:1) N dτ , (13)where N is determined by demanding that the integralover τ from τ i ≡ τ ( t i ) to τ ( t ) gives N i . The result canbe integrated from 0 to τ e to yield the number of stringwakes N i ( τ e ) which intersect the past light cone at a timelater or equal to τ e : N i ( τ e ) = (cid:18) τ e τ i (cid:19) N i . (14) To determine the redshift distribution of emission times z e , we need to relate the conformal time τ e to the emissionredshift z e . We start by expressing τ e in terms of t e ,d τ e = d t e a ( t e ) , (15)which evaluates to τ e = 3 t (cid:104) − ( z e + 1) − (cid:105) . (16)An expression for the number N i ( z e ) of string wakes withan emission redshift larger or equal to z e can now be ob-tained by substituting the preceding equation into equa-tion 14, and using our expression for N i from 12. Thisgives N i ( z e ) = N H (cid:0) θ π (cid:1) z ( t i ) / (cid:34) − ( z e + 1) − − ( z i + 1) − (cid:35) (17)Due to the discrete nature of this simulation, we needto approximate this relationship by splitting the redshiftspace into slices, and laying down strings in each slice.The smallest possible slice would be the number of dis-crete values of redshift in the simulation (128, as we willdiscuss shortly) divided by the range of redshift, whichwe shall take to be 50 for this example. However, as we FIG. 2: Distribution of the redshifts at which the string wakesintersect the past light cone (for a fixed time z i = 10 at whichthe strings were produced). The horizontal axis is the emis-sion redshift z e , the vertical axis is the number ∆ N i ( z e , z e − z i (taken to be 10 ) whichintersect the past light cone of the sky area we are consideringin the redshift interval between z e − z e . must have an integer numbers of wakes, a slice this smallwould result in the majority of slices having 0 wakes.Hence we instead choose a slice thickness ∆ z e = 1, whichwill contain a number of wakes given by∆ N i ( z e , z e −
1) = (cid:2) . × (cid:3) (18) (cid:20) − x + 1) − / + 2 x − − x − / (cid:21) z e z e − , where we have included all prefactors into the constant.A plot of this distribution is shown in Figure 2. As isapparent, the largest number of wakes intersect the pastlight cone at low redshifts. This can be understood sincethe low redshift range covers most of the physical volume.
2. Angular and Redshift Scales of a Wake-Induced Wedge
As discussed in Section 2, each string wake which inter-sects the past light cone of the observer’s angular patchwill lead to a thin wedge of extra 21cm absorption oremission. We are concerned with string wakes producedat time t i which intersect the past light cone of the obser-vational patch at some time t e > t rec . Each such stringwake will lead to a thin wedge in a 21cm redshift surveyof a certain angular extent and certain thickness.First, let us consider the angular extent of the wedge.A string laid down at a time t i will create a wake withphysical dimensions given by equation 4. To calculate theangular size of a wake, one must start with the physicallength of the wake at time t i which in direction tangentto the string is: x p ( t i ) = c t i , (19)where (as we recall from before) c is a constant of or-der 1. In the direction of string motion the size is the same except that the factor c gets replaced by v s γ s . Inour work we choose the rather realistic parameter valueswhere the two constants are the same. The correspond-ing comoving distance is then given by x c ( t i ) = (cid:18) t t i (cid:19) x p ( t i ) = ( z i + 1) − / c t . (20)Hence, the corresponding angle θ i is: θ i = ( z i + 1) − / c × o . (21)For the most numerous and thickest wakes, z i +1 = 1000,and if the constant c is taken to be 1 / θ i = ( z i + 1) − / × o (22)which yields approximately 1 o as the angle of the wake-induced wedge.Since the string which produces the wake is moving,the string-induced wake is slightly “tilted” in the three-dimensional redshift-angle space, i.e. the tip of the wedge(which corresponds to the string at the latest time) is ata slightly larger redshift than the mean redshift of the tailof the wedge (where the string was at the initial time).This does not change the fact that the projection of thewedge onto the angular plane has the dimension givenabove. However, when computing the projection into theredshift direction we have to be careful. The wedge is thinin direction perpendicular to the plane of the wedge, butthis perpendicular direction is at an angle relative to theredshift direction. In the following we will compute thethickness of the wedge in perpendicular direction to thewedge plane.We now calculate the thickness of the wake in the per-pendicular direction. The finite thickness in redshift di-rection originates from the fact that photons originatingfrom different parts of the wake are redshifted by slightlydifferent amounts. The starting point of the computationis the formula z + 1 = (cid:18) tt (cid:19) (23)valid in the matter-dominated period. Taking differen-tials, and setting t = t e yields∆ z = (cid:18) (cid:19) t − / t − / e ∆ t e , (24)where ∆ t e is the time delay between photons from thetop and the bottom of the wedge. We compute ∆ t e at themidpoint of the wake (point of medium thickness). Thisis given by the thickness of the wake, which, as discussedin [19] grows linearly in comoving coordinates because ofgravitational accretion onto the initial overdense region.Using linear perturbation theory, the width of the wakeat time t e is w ( t e ) = 4 πGµv s γ s t (cid:0) z ( t i ) + 1 (cid:1) / (cid:0) z ( t ) + 1 (cid:1) − , (25)which equals ∆ t e . Hence, from (24) it follows that∆ z = 8 π Gµv s γ s (cid:0) z ( t i ) + 1 (cid:1) / (cid:0) z ( t ) + 1 (cid:1) − / . (26)The analysis of gravitational accretion performed usingthe Zel’dovich approximation instead of with naive linearcosmological perturbation theory yields the same result[19] except that the coefficient is 24 π/ π/ o in both the angular directions, with z e ranging from 5 to 50, and N H = 3. In this case, theangular resolution is 10 o /
128 = . o , while the resolu-tion in the redshift direction is 45 /
128 = .
35. To scaleup the dimensions of the wake only requires multiply-ing each dimension by the inverse of the resolution (ie:128 /
45 = 2 .
84 in the z e direction).
3. Brightness Temperature: δT b The final step in the calculation is to determine thebrightness temperature at every point of the three dimen-sional angle-redshift map ( θ, φ, z ). Any point for whichthe past light ray in angular direction ( θ, φ ) does notintersect a wake at redshift z yields zero brightness tem-perature. A point which at redshift z e is intersecting awake due to a string present at time t i is assigned a non-vanishing brightness temperature given by Equation (8),with T K obtained from Equation (6). Overlapping wakesare assumed to be non-interacting. IV. BACKGROUND NOISEA. Modelling the Signal
As discussed in the introductory section, in our setupthe cosmic strings only make a contribution of less than5% to the total power spectrum of inhomogeneities. Thedominant contribution is in the form of an approximatelyscale-invariant spectrum of Gaussian perturbations suchas can be produced in a number of cosmological scenar-ios, e.g. in inflationary cosmology [16] or in string gascosmology [3]. The dominant Gaussian fluctuations arein the linear regime until late times on large length scalesand are hence not expected to contribute a lot to 21cmfluctuations at high redshifts (redshifts before reioniza-tion). There will, however, be smaller scale fluctuationswhich become non-linear, forming so-called “mini-halos”which then contribute to the spectrum of 21cm fluctua-tions. It is crucial for us to check that the cosmic string effects can be detected above the noise from the Gaus-sian fluctuations (which we call “background noise” inthe following).The contribution of “background noise” to the 21cmfluctuations of the universe has been studied in detail(see e.g. [44, 45]). As was shown, there is an effect on21cm maps which comes from the diffuse inter-galacticmedium (IGM). However, this effect is homogeneous inspace on the cosmological scales which interest us hereand will hence not be further considered in this paper.Instead, we focus on the contribution of the inhomo-geneities mentioned above which lead to the formationof mini-halos. This contribution has been studied doneboth semi-analytically, [44], and later using large scalenumerical simulations [45].To calculate the signal from minihalos we must find themass function d n/ d M , which can, when integrated overa range of masses, give the number density for minihalosin this mass range. There are two possible mass profilesfor minihalos, the Sheth-Tormen model and the Press-Schechter model. Recent numerical simulations supportthe Press-Schecter model, (see Figure 2 of [45]). Oncewe know the mass function, the brightness temperatureis then calculated using line of sight integrals given arealization of the inhomogeneities described by the massfunction.The Press-Schecter function for the number density ofminihalos is given by [46]: N ( M )d M = − (cid:16) ¯ ρM (cid:17) (cid:18) π (cid:19) / δ c σ d σ d M e − δ c σ d M (27)where ¯ ρ is the mean (baryonic and dark) matter densityof the universe, and σ ( M, t ) = (cid:104) δ M (cid:105) ( t ) is the root meansquare mass fluctuation on the mass scale M . Halos willform in regions with a δ M above a critical fluctuation δ c ≈ .
69. Our number density simplifies to: N ( M, z )d M = Ω m (cid:18) (3 + n )6 (cid:19) (cid:16) ¯ ρM (cid:17) × (cid:18) π (cid:19) / (cid:18) MM ∗ ( z ) (cid:19) (3+ n )6 (28) × exp (cid:34) − (cid:18) MM ∗ ( z ) (cid:19) (3+ n ) / (cid:35) d M where M ∗ is the mass scale for which the fractional den-sity fluctuation equals δ c , and where we assume a primor-dial power spectrum of density fluctuations P ( k ) ≈ k n (here, P ( k ) does not contain the phase space factor k and is not dimensionless. A scale-invariant spectrum ofcurvature fluctuations corresponds to n = 1).The spatially averaged brightness temperature of a setof halos is given by [44]:¯ δT b = c (1 + z ) ν H ( z (cid:48) ) (cid:90) M max M min ∆ ν eff δT b,ν A d N d M d M (29)where ∆ ν eff is the redshifted effective line width, δT b,ν is the brightness temperature of individual minihalo atthe frequency ν , and A is the geometric cross sectionof the minihalo. The integral (worked out in [47] - seeFigure 3 in that paper) yields a noise temperature whichpeaks at a value of 4mK at a redshift of 10 and decaysat higher redshifts. B. Generating Noise Temperature Maps
Given that the signal from the IGM is spatially uni-form, we focus on the contribution from minihalos.The corresponding noise temperature maps are con-structed by modeling the spatial fluctuations as a three-dimensional Gaussian random field with a power spec-trum P T ( k ) (again defined without the phase space factor k ) with a slope which will be discussed below, and witha variance given by (29). The amplitude of the resultingnoise is rescaled in redshift direction by incorporating theredshift dependence of the variance from (29).To accomplish this, we follow the procedure laid outin [30], which we generalize to 3 spatial dimensions, Wecan expand the spatial fluctuations in temperature intohyperspherical harmonics.:∆ TT ( θ, φ, ψ ) = (cid:88) l,m,n a l,m,n Y l,m,n ( θ, φ, ψ ) , (30)where the Y l,m,n are the spherical harmonics generalizedto 3+1 dimensions, the ‘ hyperspherical harmonics ’, andthe a l,m,n are the coefficients of this expansion. We cansimplify this expression to a sum of plane waves, by mak-ing the flat sky approximation (valid for angular scales < o ). This gives us:˜ T ( (cid:126)x ) = (cid:88) (cid:126)k ˜ T ( (cid:126)k ) e i(cid:126)k · (cid:126)x (31)where we have used the abbreviation ˜ T ( (cid:126)x ) ≡ ∆ TT ( (cid:126)x ).Comparing this decomposition with our previous expres-sion in terms of hyperspherical harmonics, we see thatthe ˜ T ( (cid:126)k ) correspond to the a l,m,n and hence the ˜ T ( (cid:126)k )give us the power spectrum: (cid:104) ˜ T ( (cid:126)k ) (cid:105) = (cid:104) a l,m,n (cid:105) ∼ P T ( k ) (32)where the temperature power spectrum P T is defined bythe last equality.Since the mini-halos are produced by the density fluc-tuations we use the power spectrum of density fluctua-tions to determine the slope of P T ( k ). Thus, we take P T ( k ) to be proportional to the primordial power spec-trum: P T ( k ) ∝ k n with n = 1 . (33)and the amplitude is determined by demanding that thevariance is given by (29). First, in fact, we generate a three-dimensional Gaussian random field T GRF withvariance 1.Note that in a model with a primordial scale-invariantspectrum of curvature perturbations, the dimensionlesspower spectrum of fractional density perturbations is ap-proximately scale-invariant on scales which entered theHubble radius before the time of equal matter and ra-diation. This is a consequence of the processing of theprimordial spectrum which happens on sub-Hubble scales(see e.g. any text which discusses the Newtonian theoryof cosmological perturbations, e.g. [48]). The spectrumon small scales thus takes on a slope n = −
3. We do notconsider this effect in the current simulations.Using the ergodic hypothesis, we construct a Gaussiannoise field by drawing the ˜ T ( (cid:126)k ) for fixed k = | (cid:126)k | froma Gaussian distribution with power given by P T ( k ) andamplitude chosen to give variance 1.We do our calculations on a lattice of (˜ k , ˜ k , ˜ k ) coor-dinate values ranging from 0 to N max −
1. We convert tothe corresponding k values with: k i = 2 πL (˜ k i − k max ) (34)where k max corresponds to the angular resolution of thelattice in the k i direction.We can then calculate the ˜ T ( (cid:126)k ) using:˜ T ( (cid:126)k ) = (cid:114) P T ( k )2 (cid:16) g ( (cid:126)k ) + ig ( (cid:126)k ) (cid:17) (35)where g and g are randomly generated from a Gaussiandistribution with variance 1 and mean 0, and we enforce˜ T ( (cid:126)k ) = ˜ T ( − (cid:126)k ) to ensure that T ( (cid:126)x ) is real.We can then construct a spatial map by taking theinverse Fourier transform of the preceeding expression.The result of this is a Gaussian random field T GRF witha correlation function specified by the power spectrumand variance 1. We now wish to enforce the redshiftdependence of the amplitude, which we do by identifyingredshift z with the third spatial direction. Hence the finalnoise field is specified by: T noise ( x, y, z ) = ¯ δT b ( z ) T GRF ( x, y, z ) (36)with amplitude ¯ δT b ( z ) given by (29). Note that since thechange in amplitude is uniform over the angular direc-tions, this introduces deviations from strict Gaussianityof the the three-dimensional distribution. V. INTEGRAL GEOMETRY AND MINKOWSKIFUNCTIONALS
Minkowski Functionals are a useful tool to analyze thetopology of a d -dimensional map. Originally, these func-tionals were considered in the context of describing thetopology of a body B embedded in a d -dimensional Eu-clidean space using an approximation in which the body B was approximated by the set of convex bodies of aparticular form but varying size (hence we have func-tionals and not just functions). In our application we willconsider the Minkowski functionals as functionals of theclass of bodies (termed an ‘excursion set’) which encloseregions of the map where the value of the variable whichis mapped is larger than a cutoff value and we considervarying the cutoff value. For example, applied to CMBtemperature anisotropy maps we consider bodies whichare regions of the sky where the temperature T is largerthan a cutoff temperature T c whose amplitude we vary.In the case of interest here we consider three dimensional21cm brightness temperature maps and we consider thetopology of the volumes which contain points where thetemperature exceeds a critical temperature whose mag-nitude we vary.Minkowski first developed these functionals [49] in1903 to solve problems of stochastic geometry [50]. Thisthen led to the development of Integral Geometry in themid 1900’s. At the heart of Integral Geometry is Had-wiger’s Theorem [21], which deals with the problem ofcharacterizing the topology of the body B using scalarfunctionals V . These functionals must satisfy certain re-quirements [51]:1. Motion Invariance : The functionals should beindependent of the position and orientation in spaceof the body.2.
Additivity : The functionals applied to the unionof two bodies equal the sum of their functionalsminus the functionals of their intersection: V ( A ∪ B ) = V ( A ) + V ( B ) − V ( A ∩ B ) . (37)3. Conditional Continuity : The functionals of con-vex approximations to a convex body converge tothe functionals of the body.Hadwiger’s Theorem [21] states that for any d dimen-sional convex body, there exist d + 1 functionals that sat-isfy these requirements, denoted V j , j = 0 , ...d , for the j ’th functional. Furthermore, these functionals providea complete description of topology of the body. Math-ematically, the j ’th functional of a d -dimensional body B is an integral over a ( d − j )-dimensional surface of B .For example, in three dimensions, the first functional V is simply the volume of the body, and the second ( V ) isthe surface area. In all dimensions, the last functional V j is given by the Euler characteristic χ , which in threedimensions is defined [51] as: χ = number of components - number of tunnels+ number of cavities . (38)The simplest example is a set of balls of radius r i . When r i is very small, there are no intersections of balls, and χ is very close to the number of balls. As r i increases,the balls will begin to intersect, creating tunnels. Thus TABLE I: Geometric Interpretation of Minkowski Functionalsin 1,2, and 3 dimensions [50]d 1 2 3 V length area volume V χ circumference surface area V - χ total mean curvature V - - χ χ will become negative. At a certain threshold value of r i , χ will once again be positive, as tunnels are cut off toform cavities. Finally, when r i becomes very large, theentire space is filled, and χ is equal to 0.The geometric meaning of the functionals for d ≤ ν = f ( x ) /σ which is thenumber of standard deviations σ by which the value ofthe field f ( x ) (scaled to have zero spatial average) de-viates from zero . Hence, a rescaled threshold ν = ± ν isto remove the effect of a constant factor on the function-als, and hence to allow for a comparison with structurethat may be identical in every way but the magnitude ofthe temperature. For the purposes of this simulator, wecalculated the functionals for the range − ≤ ν ≤ Minkowski3 , written by ThomasBuchert. This piece of software takes a 3-D array of floatsas input (in binary format), and outputs two estimates(with error values) of the Minkowski Functionals as wellas the functionals for a Gaussian field. The two estimatesfor the Minkowski functionals correspond to two meth-ods, one derived from differential geometry and the otherfrom integral geometry. As a full derivation of each can0be found in [53], only a brief explanation will be providedhere.In the context of differential geometry, it is possible todescribe all local curvature in terms of geometric invari-ants (Koenderink Invariants). We then express the globalMinkowski functionals V k in terms of the local Minkowskifunctionals V ( loc ) K , which are calculated in [53]. For thethree dimensional case, the functionals of a field ν ( x ) ina volume D are given by V k ( ν ) = 1 |D| (cid:90) D d xδ ( ν − ν ( x )) |∇ ν ( x ) | V ( loc ) k ( ν, x ) . (39)In the discrete case, this is equivalent to summing overthe lattice and taking a spatial average.In the context of integral geometry, the MinkowskiFunctionals are calculated using Crofton’s Formula. Thisprovides a very elegant expression for the k ’th functionalof a d dimensional object K, in a volume that consists of L lattice points of cubic lattice spacing a : V ( d ) k ( K ) = ω d ω d − k ω k a k L k (cid:88) j =0 ( − j k !( d − k + j )! d ! j ! N d − k + j ( K ) , (40)where ω j is the volume of a j -dimensional unit sphere,and N j ( K ) is the number of j -dimensional lattice cellscontained in K. For example, N ( K ) is the number oflattice points in K, and N ( K ) gives the number of cubescontained within K. VI. RESULTSA. Functionals of Cosmic Strings
We first consider the Minkowski Functionals of thewake signal alone, by taking a scaling solution of stringswith ( Gµ ) = 0 .
17, using the notation ( Gµ ) = Gµ × − . The results are shown in Figure 3. The four boxesshow the mean Minkowski functionals after 100 simula-tions, with the errors taken to be the standard deviation.In each box, the green curve is based on calculating thefunctionals using the method of Koenderink Invariants,and the red curve is based on using the Crofton Formula(the red and green curves are almost perfect replicas ofone another, and so the green is only visible in regionswhere they differ).Since the number density of wakes is peaked aroundhigh redshift, the mean brightness temperature fluctua-tion δT b will be negative, and hence the threshold value ν = 0 corresponds to a slightly negative brightness tem-perature fluctuation. The threshold value ν c correspond-ing to δT b = 0 is a small positive number. It is atthis threshold value that the four Minkowski function-als change abruptly. For ν > ν c the volume with δT b greater than the threshold value vanishes, and hence thefunctionals V , V and V vanish. For ν < ν c the hightemperature volume does not vanish, but it corresponds to the outside of the wakes as opposed to the inside.Hence, the integrated mean curvature is negative. Thekey feature to notice in these plots is the abrupt changethat occurs at the threshold value and the asymmetryabout this threshold. B. Functionals of Noise
We now turn our attention to the Minkowski Function-als of the background noise due to minihalos, shown inFigure 4, where we again calculate the average function-als over 100 simulations and indicate the standard devi-ation with error bars. The corresponding functionals fora Gaussian random field are shown as the dashed blackcurves, whereas the functionals for the noise maps areshown in red and green. The key feature of these plots isthe symmetry (or antisymmetry in the case of V and V )about 0, which matches the results for a Gaussian ran-dom field. However the noise map functionals all sharethe common characteristic of being tightly concentratedabout 0, which allows the noise map to be differentiatedfrom a pure Gaussian random field. As mentioned above,the deviation of the noise maps from being a pure Gaus-sian random field comes from the scaling of the ampli-tude with redshift which is uniform over the two angularcoordinates. C. Functionals of Cosmic Strings Embedded inNoise
Armed with our knowledge of the behaviour of theMinkowski Functionals for both pure cosmic string wakeand background noise maps, we can start with the realfun: differentiating maps of wakes and noise from mapsof pure noise. To do this, we generate a map of wakesand noise by adding (at every point in space) the tem-perature fluctuations from wakes and the fluctuationsfrom the mini-halo noise maps. We can then comparethe Minkowski functionals as we vary the string tension,allowing us to place constraints on the minimum stringtension necessary for the strings to be detected via thismethod.For each value of the string tension Gµ we performed100 simulations. For each simulation we computed the χ significance value of the difference between the string anda pure noise map. This was done in the following way:For each threshold bin i (we used N bins = 25 thresholdbins per functional, and hence 4 · N bins = 100 bins intotal) we computed the probability p i that the Minkowskifunctional values for the string simulation come from thesame distribution as obtained from the pure noise maps.We then combined the results of different bins into a χ FIG. 3: Minkowski functionals of a pure scaling solution of cosmic string wakes, with Gµ = 0 . × − . Each of the fourboxes shows the results for one of the four functionals, averaged over 100 simulations each using a lattice of 128 pixels. Eachbox contains two curves. corresponding to computing the functionals computed making use of one of the two methods, eitherKoenderink Invariants (the green curve) or the Crofton Formula (the red curve). −5 0 500.20.40.60.81 threshold ν V −5 0 5−0.500.51 threshold ν V −5 0 5−2−101 threshold ν V −5 0 5−10−50510 threshold ν V value using the Fisher combined probability test χ = − N bins (cid:88) i =1 log ( p i ) . (41)We then computed the mean value ¯ χ and the standarddeviation ∆ χ of χ . The criterion for significance of thedifference between string wake and noise maps is thatthe value of ¯ χ − ∆ χ is larger than the representativesignificance value for an analysis with 2(4 N bins ) = 200degrees of freedom .A comparison of the functionals of (wakes+noise)against pure noise is shown in Figures 5 to 8, which weredone using a lattice size of 128 , each for a different valueof Gµ . In each set of graphs, the black curves representthe results for the pure noise maps (the mean values and standard deviations for each threshold value are shown),and the orange curves (the less symmetric - in the caseof V and V - or less antisymmetric - in the case of V and V curves) represent what is obtained for maps con-taining both strings and noise. What is shown are theMinkowski functionals for a particular cosmic string wakesimulation. The mean value of χ was found to be greaterthan 10 (the maximal value our numerics could handle)for Gµ ≥ . × − , showing that for these values of Gµ string maps can indeed be distinguished from purenoise maps. For Gµ = 4 . × − the χ drops down to223 ± Gµ . Hence from this preliminary analysiswe conclude that the minimum string tension Gµ mustbe roughly 5 × − for the Minkowski functional method2 FIG. 4: Minkowski functionals of the background noise, averaged over 100 simulations each using a lattice of 128 pixels.Each box contains three curves. Two of them are for the noise map with the functionals computed making use of one of twoestimates, either Koenderink Invariants (the green curve) or the Crofton Formula (the red curve). The dashed black curvesgive the Minkowski functionals for a Gaussian map. −5 0 500.20.40.60.81 threshold ν V −5 0 500.511.5 threshold ν V −5 0 5−2−1012 threshold ν V −5 0 5−30−20−10010 threshold ν V to be able to detect the cosmic string signals which areembedded in the noise.Note the V i values for the cosmic string simulationsin Figure 5 vanish to the level of the numerical accu-racy above a certain value of the threshold, while thosefor noise alone do not. The reason this happens is thatin the presence of string wakes, a fixed threshold valuecorresponds to a different brightness temperature than itwould in the absence of strings.Given the limited accuracy of the analysis, we knowthat the Minkowski functionals at sufficiently closethreshold values are not independent. We must thus con-sider the possibility that we might have oversampled and obtained an artificially large value of χ . Put anotherway, since the temperature resolution δT of our maps isnon-zero, we must account for this in our choice for thethreshold bin resolution δν . We can relate δν and δT by δν = δTσ T . (42)The temperature resolution can be estimated to first or-der by δT = δx · ∇ T , (43)where δx is the spatial resolution and ∇ T is the gradi-ent of the background noise. Since the resolution of themaps is the worst in the redshift direction, we focus onresolution in the z -direction. The relevant quantity then3 FIG. 5: Minkowski functionals of a combined map (wakes+noise) with Gµ = 1 . × − along with the corresponding functionalsfor a pure noise map, averaged over 100 simulations each using a lattice of 128 pixels. The black curves are for the pure noisemaps, the orange ones for the strings+noise maps. χ >1000 (G µ ) =0.17 −5 0 500.20.40.60.81 threshold ν V −5 0 500.511.5 threshold ν V −5 0 5−2−1012 threshold ν V −5 0 5−40−20020 threshold ν V is the spatial resolution δl corresponding to the redshiftresolution δz , which is given by δl ( δz, z ) ∼ c ( δz ) t z − / . (44)To then calculate the gradient ∇ T , we use the fact thatthat the matter power spectrum is dominated by modesaround k ∼ /λ eq , and hence this sets the length scale ofthe problem. We can then estimate the gradient by T = ¯ Tλ eq , (45)where λ eq is λ eq ∼ ct eq z − / eq . (46)Putting this all together we find δT ∼ ¯ T (cid:18) δzz (cid:19) (cid:16) z eq z (cid:17) / ∼ . , (47)where for the second equality we take z = 50. Usingequation 42, this corresponds to δν = 0 . − σ to 4 σ ) and find N bins = 8 δν ∼ . (48)Hence our choice of 25 threshold bins is well within thelimit from oversampling.Another way to see that the bins can be treated asindependent is to consider the correlation matrix for eachfunctional. The correlation matrix is defined by Corr ij = Cov ij σ i σ j = (cid:104) ( X i − µ i )( X j − µ j ) (cid:105) σ i σ j (49)where the X i are defined by the wake+noise signal forthe i (cid:48) th bin. Hence i runs from 1 to 25, each X i is arandom variable that we have sampled 100 times (oncefor each trial of the simulation), and (cid:104)(cid:105) denotes expecta-tion value. From each X i we define a mean (cid:104) X i (cid:105) = µ i and a standard deviation σ i . From this we can calculatethe correlation in the wake signal between the bins foreach functional, as is plotted in Figure 9. For each func-tional, the correlation matrix is concentrated in a narrow4 FIG. 6: Minkowski functionals of a combined map (wakes+noise) with Gµ = 1 . × − along with the correspondingfunctionals for a pure noise map, averaged over 100 simulations each using a lattice of 128 pixels. χ >1000 (G µ ) =0.10726 −5 0 500.20.40.60.81 threshold ν V −5 0 500.511.5 threshold ν V −5 0 5−2024 threshold ν V −5 0 5−30−20−10010 threshold ν V strip along the diagonal with a width at half max of 2-3bins. Given this diagonal structure, we are safe from anyhidden correlation that would invalidate our use of theFisher combined probability test. VII. SUMMARY AND OUTLOOK
This purpose of this investigation has been to bothdemonstrate the power of Minkowski Functionals for de-tecting non-Gaussian behaviour, and to investigate thetopology of 21cm distributions in cosmological modelswith a scaling distribution of cosmic string wakes. Wegenerated 3-D maps of the 21cm brightness temperaturein pure cosmic string models, in models with only back-ground noise from mini-halos, and in maps in which bothsources of 21cm fluctuations were present, using meth-ods similar to those employed for constructing 2-D CMBtemperature by [30]. These maps were then be inves-tigated with Minkowski functionals, using the software‘
Minkowski3 ’ made available by Buchert [54]. For largevalues of the string tension, the non-Gaussian signaturesof string wakes are clearly visible in the Minkowski func-tionals. We studied for which range of values of the string tension the difference between the strings + noise versusthe pure noise maps as seen in the Minkowski functionalsare statictically significant. A large difference was foundbetween the functionals of the string-induced brightnesstemperature map and the corresponding functionals ofbackground noise, with maps of strings embedded innoise found to be differentiable from maps of pure noisewhen
Gµ > × − .The analysis presented in this report serves as a proofof concept that the Minkowski functionals can be used asa statistical tool to search for cosmic string signals. Toextend this preliminary analysis would mean perform-ing large scale numerical simulations, as new featuresmay present themselves as we study the 21cm maps withgreater resolution. Acknowledgments
We thank Thomas Buchert, for permitting the use ofhis software, and Rebecca Danos for permission to useFigure 1 which is taken from [19]. We also wish to thankan anonymous referee for many suggestions on how toimprove this work. This research has been supported in5
FIG. 7: Minkowski functionals of a combined map (wakes+noise) with Gµ = 6 . × − along with the corresponding functionalsfor a pure noise map, averaged over 100 simulations each using a lattice of 128 pixels. χ >1000 (G µ ) =0.067678 −5 0 500.20.40.60.81 threshold ν V −5 0 500.511.5 threshold ν V −5 0 5−4−2024 threshold ν V −5 0 5−40−20020 threshold ν V part by an NSERC Discovery Grant, by funds from the CRC, and by a Killam Research Fellowship. [1] R. Jeannerot, “A Supersymmetric SO(10) Model withInflation and Cosmic Strings,” Phys. Rev. D , 5426(1996) [arXiv:hep-ph/9509365];R. Jeannerot, J. Rocher and M. Sakellariadou, “Howgeneric is cosmic string formation in SUSY GUTs,” Phys.Rev. D , 103514 (2003) [arXiv:hep-ph/0308134].[2] S. Sarangi and S. H. H. Tye, “Cosmic string productiontowards the end of brane inflation,” Phys. Lett. B ,185 (2002) [arXiv:hep-th/0204074];E. J. Copeland, R. C. Myers and J. Polchinski, “CosmicF- and D-strings,” JHEP , 013 (2004) [arXiv:hep-th/0312067].[3] R. H. Brandenberger and C. Vafa, “Superstrings in theEarly Universe,” Nucl. Phys. B , 391 (1989);A. Nayeri, R. H. Brandenberger and C. Vafa, “Pro-ducing a scale-invariant spectrum of perturbations in aHagedorn phase of string cosmology,” Phys. Rev. Lett. , 021302 (2006) [arXiv:hep-th/0511140];R. H. Brandenberger, A. Nayeri, S. P. Patil and C. Vafa,“String gas cosmology and structure formation,” Int. J.Mod. Phys. A , 3621 (2007) [hep-th/0608121]; R. H. Brandenberger, “String Gas Cosmology,”arXiv:0808.0746 [hep-th].[4] A. Vilenkin and E.P.S. Shellard, Cosmic Strings andother Topological Defects (Cambridge Univ. Press, Cam-bridge, 1994).[5] M. B. Hindmarsh and T. W. B. Kibble, “Cosmicstrings,” Rept. Prog. Phys. , 477 (1995) [arXiv:hep-ph/9411342].[6] R. H. Brandenberger, “Topological defects and struc-ture formation,” Int. J. Mod. Phys. A , 2117 (1994)[arXiv:astro-ph/9310041].[7] Y. B. Zeldovich, “Cosmological fluctuations producednear a singularity,” Mon. Not. Roy. Astron. Soc. ,663 (1980);A. Vilenkin, “Cosmological Density Fluctuations Pro-duced By Vacuum Strings,” Phys. Rev. Lett. , 1169(1981) [Erratum-ibid. , 1496 (1981)];N. Turok and R. H. Brandenberger, “Cosmic Strings AndThe Formation Of Galaxies And Clusters Of Galaxies,”Phys. Rev. D , 2175 (1986);H. Sato, “Galaxy Formation by Cosmic Strings,” Prog. FIG. 8: Minkowski functionals of a combined map (wakes+noise) with Gµ = 4 . × − along with the corresponding functionalsfor a pure noise map, averaged over 100 simulations each using a lattice of 128 pixels. χ = 223 ±
101 (G µ ) =0.0427 −5 0 500.20.40.60.81 threshold ν V −5 0 500.511.5 threshold ν V −5 0 5−2024 threshold ν V −5 0 5−30−20−10010 threshold ν V FIG. 9: Correlation of wake signal between bins for each functional, using data from 100 realizations.