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COBE vs Cosmic Strings: An Analytical

arXiv:hep-ph/9208247v1 25 Aug 1992COBE vs Cosmic Strings: An AnalyticalModelLeandros Perivolaropoulos∗†AbstractWe construct a simple analytical model to study the effects ofcosmic strings on the microwave background radiation.Our modelis based on counting random multiple impulses inflicted on photontrajectories by the string network between the time of recombina-tion and today. We construct the temperature auto-correlation func-tion and use it to obtain the effective power spectrum index n, therms-quadrupole-normalized amplitude Qrms−P S and the rms temper-ature variation smoothed on small angular scales.

For the values ofthe scaling solution parameters obtained in Refs. [10],[3] we obtainn = 1.14 ± 0.5, Qrms−P S = (4.5 ± 1.5)Gµ and ( ∆TT )rms = 5.5Gµ.Demanding consistency of these results with the COBE data leads toGµ = (1.7 ± 0.7) × 10−6 (where µ is the string mass per unit length),in good agreement with direct normalizations of µ from observations.1IntroductionThe recent detection of anisotropy[4] in the Microwave Background Radia-tion (MBR) by the COBE (COsmic Background Explorer) collaboration hasprovided a new powerful experimental probe for testing theoretical cosmo-logical models.

The DMR (Differential Microwave Radiometer) instrumentof COBE has provided temperature sky maps leading to the rms sky varia-tionq< (∆TT )2 > (where ∆T ≡T(θ1)−T(θ2), and θ1−θ2 = 60◦is the beamseparation in the COBE experiment) and the rms quadrupole amplitude. A∗Division of Theoretical Astrophysics, Harvard-Smithsonian Center for Astrophysics60 Garden St. Cambridge, Mass.

02138, USA.†also Visiting Scientist, Department of Physics Brown University Providence, R.I.02912, U.S.A.1

fit of the data to spherical harmonic expansions has also provided the angu-lar temperature auto-correlation function C(∆θ) ≡< δTT (θ)δTT (θ′) > where<> denotes averaging over all directions in the sky, δT(θ) ≡T(θ)−< T >and ∆θ = θ −θ′. This result was then used to obtain the rms-quadrupole-normalized amplitude Qrms−P S and the index n of the power law fluctuationspectrum assumed to be of the form P(k) ∼kn.

The published results are:(∆TT )rms = (1.1 ± 0.2) × 10−5(1)Qrms−P S = (5.96 ± 0.75) × 10−6(2)n = 1.1 ± 0.5(3)These results have imposed severe constraints on several cosmological modelsfor large scale structure formation. Even the standard CDM model with bias1.5 ≤b8 ≤2.5 is inconsistent with the COBE results for H0 > 50km/(sec ·Mpc) and is barely consistent for H0 ≃50km/(sec · Mpc) [5] [15](Tensormode perturbations have recently been shown however, to make standardCDM with specific inflationary models consistent with COBE for a widercosmological parameter region[23]).It is therefore interesting to investigate the consistency of alternativemodels with respect to the COBE measurments.

The natural alternative tomodels based on adiabatic Gaussian perturbations generated during infla-tion are models where the primordial perturbations are created by topolog-ical defects like cosmic strings global monopoles or textures.Cosmological models based on topological defects have a single free pa-rameter v, the scale of symmetry breaking leading to defect formation. Con-sistency with large scale structure observations, constrains this parameterto Gv2 ≃10−6[18][19] where G is Newton’s constant.

In what follows wewill concentrate on the case of cosmic strings.Previous analytical studies of MBR anisotropies due to cosmic strings[14]were based on the old picture of the cosmic string network evolution[16] andtherefore focused on the effects of cosmic string loops. Loops however werelater shown by more detailed simulations[3][10] to be unimportant comparedto long strings.More recently, numerical simulations have been used to investigate theMBR predictions of cosmic string models[6] and comparison of these pre-dictions has been made with the COBE data[7].It was found that forGv2 ≃10−6, (v2 = µ where µ is the mass per unit length of the string)2

cosmic strings are consistent with the COBE data for a wider range of cos-mological parameters than the standard CDM model. The numerical anal-ysis that has led to this result however, is rather complicated and currentlythere is still some controversy among the different groups[3][10][9] about thedetails of the simulations involved in the analysis.

In addition, studies basedon string simulations have necessarily fixed scaling solution parameters andtherefore, the dependence of the results on these parameters can not be re-vealed. These arguments make the construction of an analytical model forthe study of the effects of topological defects on the MBR, a particularlyinteresting prospect.

It is such an analytical model that we are constructingin this paper.In particular, we use a multiple impulse[13] approximation to obtain thetemperature auto-correlation function C(∆θ) predicted by the string model.From C(∆θ) we obtain the mass per unit length µ of strings consistent withCOBE and the effective power spectrum index n predicted by the stringmodel.Our basic assumptions and approximations are the following:1. We approximate the photon path from trec to t0 by a discrete set ofN Hubble time-steps ti such that ti+1 = 2ti.

For zrec ≃1400 we haveN ≃log2[(1400)3/2] ≃16.2. At each Hubble time-step the photon beam is affected only by the longstrings within a horizon distance from the beam.

The effects of furtherstrings are cancelled due to the compensating scalar field radiation.3. The combined effects of all strings present within a horizon distance ofthe photon beam is a linear superposition of the individual effects.[17]4.

Each string that affects the photon beam induces a temperature vari-ation of the form[12][20]:δTT = 4πGµβwithβ = ˆk · (⃗vsγs × ˆs)where ˆk, ˆs and ⃗vs are the unit wave-vector, the unit vector along thestring and the string velocity vector respectively (see Fig. 1).5.

The long strings within each horizon volume have random velocitiespositions and orientations.3

6. The effects of loops are unimportant compared to the effects of longstrings[3][10].7.

Initial temperature inhomogeneities at the last scattering surface areassumed negligible compared to those induced by the string networkat later times.We will also use the result that for any function θ1(θ) that takes randomvalues as the independent variable varies we have:< f(θ1(θ)) >θ=< f(θ1) >θ1where <>α implies averaging with respect to α. We will call this for obviousreasons the ‘ergodic hypothesis’.Some of these assumptions are similar to those made in correspondingnumerical studies.

Others (like assumption 2 which is an attempt to takeinto account compensation) are improvements over those of the numericalanalyses. We comment on the possibility of further improving some of theseassumptions in section 4.2The Temperature Correlation FunctionWe begin with a description of our model.

Consider a photon beam emergingfrom the last scattering surface at t1 = trec. This beam of fixed temperaturewill initially suffer M ‘kicks’ from the M long strings within the horizon att1.

At the Hubble time t2 ∼2trec, M strings, uncorrelated with the previousones will be within the horizon giving the photon M further ‘kicks’ and theprocess will continue until the N ≃16 Hubble time-step corresponding tothe present time t0 ( t0 ≃216trec). Therefore the total temperature shiftin the direction θ (in what follows we consider fixed ϕ and omit it whendefining direction unless otherwise needed) may be written as:δTT (θ) = 4πGµNXn=1MXm=1βmn(θ)(4)where βmn(θ) corresponds to the mth string at the nth Hubble time-step.Taking ˆvmnsand ˆsmn to be unit vectors along the string velocity and stringlength directions we may write βmn(θ) = vsγs(ˆk(θ)· ˆRmn1(θ)) where ˆRmn1(θ) =ˆvmn × ˆsmn is a unit vector that varies randomly with m and n.4

It is instructive for what follows to obtain δTT (θ) averaged over all direc-tions. Defining ξ ≡4πGµvsγs we have:< δTT (θ) >= ξXn,m< ˆk(θ) · ˆRmn1(θ) >= ξ4πXm,nZd cos θdϕ cos θ1(θ, ϕ)where θ1 is the angle between ˆRmn1(θ) and ˆk(θ) (see Fig.

3) and since ˆRmn1(θ)varies randomly with m, n and with θ, ϕ on angular scales larger than θrec,we may use the ergodic hypothesis to obtain:< δTT (θ) >= ξ4πXm,nZd cos θ1dϕ1 cos θ1 = 0which is also the naively expected result.The same result could have beenobtained without using the ergodic hypothesis by simply taking the largeM × N limit and performing the sum over m,n before the θ, ϕ integration.Our goal is to investigate correlations of fluctuations :δTT (θ)δTT (θ′). Using(4) we have:δTT (θ)δTT (θ′) = (4πGµ)2NXn,n′MXm,m′βmn(θ)βm′n′(θ′)(5)Define now ∆θ ≡θ −θ′ and focus on the case when ∆θ = θp where θp is theangular size of the horizon at the Hubble time-step tp (1 ≤p ≤16).

Longstrings present at time t will inflict ‘kicks’ on two photon beams separatedby ∆θ = θp that are uncorrelated for t ≤tp but are equal for t > tp when∆θ is within the horizon scale t (see Fig. 2).

ThereforeˆRjk1 (θ) = ˆRj′k′1(θ′) iffk > p, j = j′, k = k′ˆRjk1 (θ) ̸= ˆRj′k′1(θ′)otherwisewhere ̸= here means ‘not equal and also uncorrelated’. We may thereforesplit the sum (5) in two parts consisting of correlated and uncorrelatedproducts respectively.δTT (θ)δTT (θ′) = ξ2[NXn=n′=pMXm=m′=1(ˆk(θ) · ˆRmn1(θ))(ˆk(θ′) · ˆRm′n′1(θ′))++Xn,n′Xm,m′(ˆk(θ) · ˆRmn1(θ))(ˆk(θ′) · ˆRm′n′1(θ′))] ≡ξ2[Σ1 + Σ2]5

where Σ1 refers to the terms of the sum that are correlated on a scale ∆θwhile Σ2 refers to the uncorrelated terms. Averaging Σ1 over all directionswe obtain< Σ1(∆θ) >= 14πXθmn1Zd cos θdϕ cos θmn1(θ, ϕ) cos(θmn1(θ, ϕ) + ∆θ)where θmn1is the angle between ˆk(θ) and ˆRmn1(θ) (see Fig.3).

Since by theconstruction of the model θ1 varies randomly from one correlated patch toanother we may use the ergodic hypothesis and replace the average over alldirections with an average over θ1. It is then easy to see that< Σ1(∆θ) >= cos(∆θ)3Ncor(∆θ)where Ncor(∆θ) = M(N −p(∆θ)) is the number of terms in Σ1.

For Σ2 wehave:< Σ2(∆θ) >= 14πZd cos θdϕXθmn1cos θmn1Xθm′n′2cos(θm′n′2+ ∆θ) = 0Therefore we may write< δTT (θ)δTT (θ′) >= ξ23 Ncor(∆θ) cos(∆θ)(6)Using the relations tp = 2ptrec and ∆θ = θtp = z−1/2p(for Ω◦= 1) where zpis the redshift at tp, it is straightforward to show thatp(∆θ) = 3 log2( ∆θθrec)(7)Using (6) and (7) we obtainC(∆θ) ≡< δTT (θ)δTT (θ′) >= ξ23 M(N −3 log2( ∆θθrec)) cos(∆θ)Our assumptions for the construction of C(∆θ) clearly break down for ∆θ <θrec ≃1◦(since we must have tp ≥trec) and for ∆θ > 2π9 (since we cannot have Ncor < 0). However, the region of validity of (8) may be easilyextrapolated by slightly shifting ∆θ by θrec ≃1◦to ∆θ + θrec in the log(thus extrapolating C(∆θ) to ∆θ = 0) and keeping C(∆θ) = 0 for ∆θ > 2π9as is physically expected since Ncor goes to 0 at large angular separations.We may now write the extrapolated auto-correlation function as:6

C(∆θ) = ξ23 M(N −3 log2(1 + ∆θθrec)) cos(∆θ)0 ≤∆θ ≤2π9(8)C(∆θ) = 02π9 ≤∆θ ≤πThe validity of this result for C(∆θ) has also been verified[21] by usingthe model introduced here to calculate directly the rms sky variation andshowing that the result is identical to the one obtained below from (8) and(14) (see section 3). In what follows we will use (8) to compare COBE’sdata with the string model predictions.

A typical value for M, the numberof long strings per horizon volume, is M = 10 while for zrec = 1400 we haveN ≃16.3Predictions of the ModelConsider an expansion of the temperature pattern on the celestial sphere inspherical harmonicsδTT =Xlmaml Y ml (θ, ϕ)Defining Cl ≡< |aml |2 > it may be shown using the addition theorem thatC(∆θ) ≡< δTT (θ)δTT (θ′) >= 14πXl(2l + 1)ClPl(cos(∆θ)Assuming now a power spectrum of perturbations of the form P(k) ∼kn itmay be shown[15] thatCl = C2Γ(l + n−12 )Γ(9−n2 )Γ(l + 5−n2 )Γ(3+n2 )(9)where C2 ≡4π5 (Qrms−P S)2, Qrms−P S being the rms-quadrupole-normalizedamplitude[4].Since our model predicts C(∆θ), Cl may be obtained for any l usingthe orthogonality relations for the Legendre functions. We may then find ¯nand ¯Qrms−P S that give the best fit to (9).

Using the symbol-manipulatingpackage Mathematica[22] we calculated Cl for l = 3...30. We excluded the7

quadrupole C2 from our fit (as was done with the COBE data) and con-sidered harmonics up to lmax = 30 to account for the small angle cutoffofCOBE due to the finite antena beam size. Our results are rather insensitiveto the specific cutofflmax for lmax > 15.Minimizing the sum30Xl=3(Cl −C2Γ(l + n−12 )Γ(9−n2 )Γ(l + 5−n2 )Γ(3+n2 ))2(10)with respect to C2 and n we obtain:¯n = 1.14 ± 0.24(11)¯Qrms−P S ≡r 54π¯C2 = (4.5 ± 1.5)(Gµ)6(vsγs)0.15pM10 × 10−6(12)as the parameters giving the best fit, where (Gµ)6, (vsγs)0.15 and M10 arethe corresponding quantities in units of 10−6, 0.15 and 10 respectively.

Theerror bars werep< (nl −¯n)2 >l and similarly for C2, where nl(or Cl2) isobtained by equating to 0 the lth term of the sum (10) while fixing C2(orn) to its best fit value.In Fig. 4 we show a plot of Cl vs l obtained from C(∆θ) of (8) (con-tinous thin line), superimposed with Cl obtained from (9) for the best fitvalues n = ¯n and Qrms−P S = ¯Qrms−P S (dashed thick line).The corre-sponding results coming from the COBE data are given by (3) and (2) Theagreement between (11) and (3) is not too surprising since it is well knownthat topological defects with a scaling solution naturally lead to a slightlytilted scale invariant Harrisson-Zeldovich spectrum on large angular scales.Clearly however, the confirmed prediction of (11) provides a good test show-ing that our simple analytic model has fairly robust and realistic features.Comparison of the observational result (2) with the prediction (12) leadsto an estimate of Gµ.

According to the recent simulations of Ref. [3], preferedvalues of the scaling solution parameters on the horizon scale are vsγs ≃0.15and M ≃10.

Using these values we obtainGµ = (1.3 ± 0.5) × 10−6(13)This result is in good agreement with the result obtained numerically inRef. [7].Another way to obtain an estimate for Gµ is to compare the measured(∆TT )rms ≡q< (T(θ1)−T(θ2)T)2 > with the corresponding value predicted byour model.8

The rms temperature variation may be expressed in terms of the temper-ature auto-correlation function for a two beam experiment by the relation:(∆TT )2rms = 2(C(0) −C(α))(14)where α is the beam separation angle (60◦for COBE). Since however (∆TT )2rmsfor COBE was obtained with a 3.2◦Gaussian beam smoothing we must in-troduce a similar smoothing before we compare with the COBE result.

Thus,we first expand C(θ) in spherical harmonics and then reconstruct it usingthe window function W(l) = e−l(l+1)2(17.8)2 as was done with the COBE data forthe construction of C(∆θ). This leads toC(∆θ) = 14π30Xl=3(2l + 1)ClPl(cos(∆θ))W(l)2We now use the reconstructed C(θ) in (14) to obtain(∆TT )rms =q2.4ξ2M/3 = 0.55(Gµ)6(vsγs)0.15pM10 × 10−5As we did for the rms quadrupole variation we may compare the predictedrms variation with the observational result (1) of COBE to obtain the valueof Gµ that makes our model consistent with the COBE data.

For (vsγs) =0.15 and M = 10 the resulting value for Gµ is:Gµ = (2.0 ± 0.5) × 10−6(15)consistent with our corresponding result from Qrms−P S. An estimate of Gµmay be obtained by averaging (13) and (15) and extending the error barsto cover both (13) and (15)Gµ = (1.7 ± 0.7) × 10−6(16)The interesting consequenses of this result will be discused in the next sec-tion.4DiscussionThe main result of this letter is that the COBE data constrain the sin-gle free parameter Gµ of the cosmic string model to be in the range given9

by (16). This range is in good agreement with normalizations of Gµ fromgalaxy[18][19] and large scale structure formation[8][11].

Therefore the cos-mic string model for structure formation remains a viable model, consistentwith the COBE data.Recent studies have used large scale structure observations to derivethe dependence of Gµ on the the bias factor b8, defined such that the rmsfluctuation in a sphere of radius 8h−1 Mpc is 1/b8[1][2]. In Fig.

5 we plot theallowed range of b8 for four different cosmic string models, demanding thatthe value of Gµ in [1][2] is consistent with (16). For comparison we also plotthe allowed range of b8 for the standard CDM model, with cosmologicalparameters chosen to give the most reasonable value for it [15][5].Ourestimate for the allowed range of the bias factor in the string model is slightlylower than that of Ref.

[7] but is clearly consistent with it. Since a value ofb8 in the range 1 to 3 is generally well motivated physically, Fig.

5 suggeststhat both the string model and standard CDM (with h = 0.5) are consistentwith the COBE data.Even though our model is based on fairly simple assumptions it hasprovided results that are not only self-consistent but also in good agreementwith much more complicated numerical analyses. This fact indicates thatour assumptions even though simple are fairly realistic and economical.

Inspite of that, there is certainly room for improvement.One interestingimpovement that could be implemented in a straightforward way is theconsideration of the effects of loops. Even though we do not expect loopsto affect the shape of the auto-correlation function on the angular scalesconsidered here they may introduce an overall multiplicative factor slightlylarger than one, thus reducing the predicted value of Gµ by the same factor.Another improvement that would modify our results in a similar way is theconsideration of string induced perturbations on the last scattering surfaceby strings present in the final stages of the radiation era.

Finally, it wouldbe interesting to include the effects of compensation in a more realistic way.For example, instead of introducing an abrupt cutoffof the deficit angle onthe horizon scale we could smoothly reduce it to zero as the horizon scale isaproached. The effect of this modification would increase the predicted valueof Gµ thus tending to cancel the effect of the previous two modifications.It is straightforward to apply the analysis presented here to other seedbased cosmological models including the global monopole and texture mod-els.

Work in that direction is currently in progress.10

5AcknowledgementsI wish to thank R. Brandenberger, Richhild Moesner and T. Vachaspati forinteresting discussions and for providing heplful comments after reading thepaper. This work was supported by a CfA Postdoctoral Fellowship.Figure CaptionsFigure 1 :The geometry of the vectors ⃗v, ˆs and ˆk.Figure 2 :Two photon beam paths in directions θ and θ′ from trec to t0.

The horizonin three Hubble time-steps is also shown. The effects of strings during thefirst two Hubble time-steps are uncorrelated for the particular angular beamseparation shown.Figure 3 :The geometry of the vectors ˆk(θ) and ˆRmn1(θ) in the term Σ1(∆θ).Figure 4 :The multipole coefficients Cl (continous thin line) superimposed with thebest fit of (9) (dashed thick line), obtained by using n = 1.14.Figure 5 :The allowed range of the bias factor b8 for four cosmic string based modelsobtained using our result for Gµ and the results of Ref.

[1][2]. The followingcases are shown:a) Strings + CDM, h=1 and h=0.5b) Strings + HDM, h=1 and h=0.5The allowed range for standard CDM with h = 0.5 and Ωb = 0.1 (the mostconsistent case with COBE’s data) is also shown for comparison.11

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