Cosmic string induced CMB maps
aa r X i v : . [ a s t r o - ph . C O ] F e b Cosmic String Induced CMB Maps
M. Landriau
1, 2, ∗ and E.P.S. Shellard † Max-Planck-Institut f¨ur extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Centre for Mathematical Sciences,Wilberforce Road, Cambridge CB3 0WA, United Kingdom (Dated: October 30, 2018)We compute maps of CMB temperature fluctuations seeded by cosmic strings using high resolu-tion simulations of cosmic strings in a Friedmann-Robertson-Walker universe. We create full-sky,18 ◦ and 3 ◦ CMB maps, including the relevant string contribution at each resolution from beforerecombination to today. We extract the angular power spectrum from these maps, demonstratingthe importance of recombination effects. We briefly discuss the probability density function of thepixel temperatures, their skewness, and kurtosis.
I. INTRODUCTION
Despite improving observational limits, interest in cos-mic strings has remained durable (for a review, see [1]).Strings are a generic phenomena in fundamental theoriesand they can emerge in macroscopic form in braneworldcosmologies, for example, at the end of inflation [2, 3].They are also common to cosmologically viable super-symmetry grand unified theory models [4]. Stringent con-straints on strings are important, therefore, in restrictingthe latitude available for cosmological model building.The detection of cosmic strings would be a watershed forhigh energy theory.Despite the potential significance, the investigation ofcosmic strings and their observational consequences facesmany numerical and analytic challenges, not least in cre-ating accurate realizations of string imprints in the cos-mic microwave sky. In this paper, we take this studya step further forward by presenting full-sky and small-angle CMB maps of temperature fluctuations seeded bycosmic string networks using high resolution simulationsin an Friedmann-Robertson-Walker expanding universe(with the longest dynamic range to date). This work in-cludes all the relevant recombination physics and can beused not only to determine the angular power spectrum ofstring CMB anisotropies but also the higher order corre-lators such as the bispectrum, trispectrum, and beyond.Current constraints on cosmic strings result from line-of-sight CMB power spectrum calculations sourced eitherby unequal-time correlators obtained from field theorystring simulations [5–7] or semianalytic models of Nambustrings [8, 9]. Qualitatively these two approaches pro-duce consistent spectra, that is, without the strong co-herent acoustic peaks associated with inflation. However,quantitatively there is a mismatch between the two ap-proaches in both the shape of the primary peak and itsamplitude, which differs by a factor of 2–3. This dispar- ∗ Email: [email protected] † Email: [email protected] ity arises primarily from a difference in string networkdensities, which has been discussed at some length else-where [10] (see also [11]). Nevertheless, there is generalagreement that the relative amplitude of string inducedCMB fluctuations cannot exceed more than 10% of thosearising from adiabatic inflationary perturbations [7, 8].There have also been a number of studies going beyondthe power spectrum through map making with cosmicstrings [12–15] in order to study the degree of Gaus-sianity of the resulting CMB signatures. However, thiswork has generally only included late-time gravitationaleffects, ignoring the recombination physics which makesan important contribution to the signal over a wide rangeof multipoles l ≈ II. COSMIC STRING NETWORKSIMULATIONS
Cosmic string simulations were performed with theAllen-Shellard string network code [19]. We have usedfixed comoving resolution together with an initial stringresolution of 24 points per correlation length. Simula-tions which started in the matter era had a dynamicrange of 7.5 in conformal time, i.e. η f = 7 . η i , but inorder to use only the simulation when the network hasrelaxed into a scaling regime, we ignored the first 4% oftime steps, resulting in an effective dynamic range of 6 inconformal time. Simulations that started in the radiationera had an effective dynamic range of 5, after eliminat-ing the first 4% of time steps from a simulation with adynamic range of 6. The energy-momentum tensor ofthe network was projected onto a grid of 256 points asdescribed in [20]. The background cosmology used wasΛCDM with the WMAP 5-year data best fit parame-ters [21]: Ω CDM =0.214, Ω b =0.044, Ω Λ = 0.742, and h = 0.719.We have used three string simulations which span therange from before equality to today. The epochs ofeach simulation are (1) from η = η / . η , (2) from η = η /
45 to η /
6, and (3) from η = η /
216 to η / III. EINSTEIN-BOLTZMANN EVOLUTION
We use the Landriau-Shellard code [20] to com-pute cosmological perturbations in Fourier space. Twochanges have been made to this code since the algorithmwas presented.The first modification concerns the scalar metric equa-tions employed: Instead of solving for ˙ h and ˙ h S , we nowsolve for ˙ h and ˙ h − ≡ ˙ h − ˙ h S , which obeys the followingequation:¨ h − + 2 ˙ aa ˙ h − = − πG ( p Σ + δp ) + 8 πG (Θ + 2Θ S ) (1)The other modification concerns the inverse compu-tation of the evolution equations’ fundamental matrices:Instead of LU factorization, we use singular value de-composition, for which we employ the freely availableLAPACK routines. This has proved a more numericallystable method and enables a better treatment of nearsingular matrices [22], especially around recombination. IV. MAPS
We compute maps of CMB fluctuations by followingphoton paths through the simulation boxes. The CMBtemperature fluctuations are given by the following equa-tion, obtained by integrating the linear Boltzmann equa-tion for the Stokes parameter I : δTT = Z η (cid:0) ˙ τ e − τ (cid:0) δ γ − v B · ˆ n + Π Iij ˆ n i ˆ n j (cid:1) − e − τ ˙ h ij ˆ n i ˆ n j (cid:17) dη (2)where Π Iij is the term that couples the Stokes parameter I to Q and U and is given in Fourier space byΠ Iij ( k ) = (cid:16) ˆ k i ˆ k j − δ ij (cid:17) Π S − (cid:16) (ˆ k i ˆ e j + ˆ k j ˆ e i )Π V + (ˆ k i ˆ e j + ˆ k j ˆ e i )Π V (cid:17) +(ˆ e i ˆ e j − ˆ e i ˆ e j )Π T + + (ˆ e i ˆ e j + ˆ e i ˆ e j )Π T × (3)and all the other terms have their usual meaning. A fullderivation of this formula is given in [23]. In practice, because ˙ τ e − τ and e − τ are zero before thestart of recombination, we only output grids for all per-turbations from η = 2 η rec / η ≃ η rec /
2. For simulations 1 and 2, we onlyoutput the grids for ˙ h ij , because ˙ τ e − τ is also zero afterthe end of recombination.By putting “observers” at each apex of a cube of side L/
2, where L is the simulation box size, we produce eightall-sky maps of resolution N side = 256 from simulation1. From simulation 2, we compute six 18 ◦ × ◦ mapsof resolution of N side = 2048, by putting an observeroutside each face of the simulation box. Finally, fromsimulation 3, using the same setup as for simulation 2, wecompute six 3 ◦ × ◦ maps of N side = 8192. Figure 1 showsone map produced from each simulation; it should benoted that even the maps of patches of sky are computedusing a spherical sky.Comparing our maps with those of [15], we note thatthe 18 ◦ map shown and their 7 . ◦ map have similar fea-tures, but the former does not present as sharp linelikediscontinuities as the latter. The resolution of our mapsis effectively lower than that implied by the Healpix N side parameter used. This can be seen most directly from thepower spectra (see Sec. V): Normally, one would expect ℓ max < ∼ N side , but the power in the maps falls around ℓ ≈ N side , which shows they are over pixelized or, to putit another way, the effective resolution is about a thirdof that expected from the pixelization. For example, our18 ◦ maps have an effective resolution of about 5 ′ , com-pared to 1 . ′ expected from a map of N side = 2048. Thisexplains the difference in features with the Fraisse et al.maps, which have a resolution of 0 . ′ , an order of mag-nitude higher than ours. V. POWER SPECTRUM COMPUTATION
For the all-sky maps, we decompose the temperaturefluctuations in the sky in spherical harmonics: δTT = P a ℓm Y mℓ and the angular power spectrum is estimatedfrom C ℓ = 12 ℓ + 1 X m a ℓm a ∗ ℓm . (4)For this purpose, we use the Healpix package [24]. Forthe 3 ◦ patches, we use the flat-sky approximation (seee.g. [25]) which replaces the spherical harmonic trans-form with a 2-D Fourier transform: C ℓ ≃ C k = | a k | ,where the modes are obtained from δTT = P a k e i k . x . Forthe 18 ◦ patches, we have used both methods to comparetheir relative merit. To use a spherical transform on anincomplete sky, one must multiply the extracted spec-trum with a mode decoupling matrix [26]: C ℓ = M − ℓℓ ′ ˜ C ℓ ′ ,where M ℓℓ ′ = (2 ℓ ′ + 1) X L (2 L + 1)4 π C maskL (cid:18) ℓ ℓ ′ L (cid:19) , (5) FIG. 1: Temperature fluctuations produced by cosmic strings from simulation 1 (left), simulation 2 (middle), and simulation 3(right).
However, the resulting spectra show strong oscillationsdue to the Gibbs effect that are not corrected by theprocedure outlined above. Because of this, we have useda 2D Fourier transform even though the flat-sky approx-imation starts to break down at the lower end of themultipole range probed by these maps. In Fig. 2, weshow the angular power spectrum from each of the threesimulation sets, as well as their summation (i.e. concate-nating these consecutive but nonoverlapping time contri-butions). For the full sky, all multipoles are shown, whilefor the patches of sky, the spectra are binned. The errorson the individual parts of the spectrum are estimated bythe variance between the maps. To add up the spectra insections where two sets of maps contribute, we averagethe lower l part in bins of the same size as that of thehigher l part and then add the two contributions. Also,to reduce the error, we binned the part of the spectrumin which only the all-sky maps contribute. We normalizethe string spectra using Gµ/c = 1 × − .In addition, Fig. 3 illustrates the power spectrum fromsimulation 3 which begins before equal matter radia-tion and separates out the late-time contribution (dashedline), that is, as if the string simulation and Einstein-Boltzmann evolution were to start just before decou-pling η dec . Note that maps by other groups have beengenerated by considering contributions only after decou-pling η > η dec , e.g. [15]. Here, we see starkly illus-trated the importance of the early contributions to theCMB anisotropy which arise from matter and radiationperturbations induced by strings before decoupling (pri-marily scalar modes). For l > ◦ maps is alsoqualitatively consistent with the CMB power spectrumcalculated using a simplified analytic formula [29].Comparing this result with that of [15] – the only othergroup to have recently extracted C ℓ from simulated maps– we see some significant differences attributable both tothe inclusion of recombination physics in our work andto the different approximations used. Firstly, we notethat Ref. [15] achieves much better statistics because itis easier to simulate purely gravitational effects in a flat-sky approximation at high resolution, generating a largernumber of maps which results in smaller error bars. How-ever, for l ≤ l < ∼ l ≈ l ≫ grids) means thatthere is generically small-scale power missing at highermultipoles from the summed power spectrum in Fig. 2.For example, the gravitational contribution from stringsin simulation 1 (full-sky) is absent beyond l ≈ Gµ/c = (1 . ± . × − , in agreement with ourprevious result [14]. FIG. 2: On the left, we show the angular power spectrum of CMB temperature fluctuations produced by networks of cosmicstrings The thin lines represent the spectra from simulations 1 – 3 (left to right) and the thick line is their sum. On the right,the total spectrum with its error bars is shown again as an “expectation area.”FIG. 3: Power spectrum from simulation 3 using the full sim-ulation (solid line) and the late start evolution (dashed line)and their difference (dotted line).
VI. PROBABILITY DISTRIBUTIONFUNCTION
One of the principal aspects of cosmic string inducedCMB fluctuations is their intrinsic non-Gaussianities. Inthis section, we present preliminary results from our maprealizations for the pixel temperature distribution, itsskewness and kurtosis. In future work, we shall explorethe efficiency of different techniques and tools, such as thebispectrum [31], to better characterize the non-Gaussiansignature from strings and thus infer their detectability.We compute the pixel probability distribution functionof the rectangular map pixel grids computed to estimatethe power spectrum. To compare our results, we gener-ate an ensemble of 1000 Gaussian maps with the samepower spectrum as the string map. To do so, we usethe fact that, in Fourier space, a Gaussian field will havephases that are uniformly distributed between 0 and 2 π .Hence, for a given map, after fast Fourier transform, werandomize the phases and fast Fourier transform back toreal space. In Fig. 4, we show the temperature distribu-tion for medium- and small-angle maps along with the Gaussian ensemble for comparison. These probabilitydistribution functions are remarkably Gaussian, consis-tent with early string map results which demonstratedthat, despite the distinct signature of individual strings,the central limit theorem prevailed after many stringscontributed [13]. We note, however, that the higher res-olution 3 ◦ maps do appear to have a slightly increasedlevel of non-Gaussianity.We also computed the skewness γ = N pix P i ( T i − T ) σ (6)and the kurtosis γ = N pix P i ( T i − T ) σ − σ contour value forthe ensemble average of Gaussian realizations. This in-dicates a marginal positive 1 σ skewness and kurtosis forthe 3 ◦ maps. The absence of a strong skewness in the18 ◦ maps seems to be at variance with the significantnegative skewness γ = − .
24 found in the 7 . ◦ mapsfrom late-time gravitational effects in Ref. [15] (see alsothe analytic estimates in [29, 32, 33]). Two possible ex-planations are apparent. First, as mentioned above, thepresent 3D Einstein-Boltzmann simulations have an ef-fectively lower resolution than the 2D flat-sky approxima-tion maps, so they cannot probe as far into the wings ofthe distribution. We are missing the integrated effect ofbispectrum triangles combining disparate large and smallscales. Second, the analytic modeling in Ref. [29] in-corporates a causality or correlation-length cut-off whichprevents bispectrum and trispectrum contributions fromsuperhorizon scales, effectively flattening their accumu-lated amplitude below l ≈ − FIG. 4: Temperature distributions of the medium- (left) and small- (right) angle maps shown above. The solid lines indicatethe 1 σ level of an ensemble of Gaussian maps The tentative indications of positive skewness and kur-tosis in the 3 ◦ maps, however, are consistent with phys-ical expectations. On scales above l > VII. CONCLUSION
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