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COBE’s Constraints on the Global Monopole and

arXiv:hep-ph/9207244v1 16 Jul 1992CfPA-TH-92-21July 9, 1992COBE’s Constraints on the Global Monopole andTexture Theories of Cosmic Structure FormationDavid P. BennettInstitute for Geophysics and Planetary Physics, LawrenceLivermore National Laboratory, Livermore, CA 94550andSun Hong RhieCenter for Particle Astrophysics, University of California, Berkeley, CA 94720ABSTRACTWe report on a calculation of large scale anisotropy in the cosmic microwavebackground radiation in the global monopole and texture models for cosmic struc-ture formation. We have evolved the six component linear gravitational field alongwith the monopole or texture scalar fields numerically in an expanding universeand performed the Sachs-Wolfe integrals directly on the calculated gravitationalfields.

On scales > 7◦, we find a Gaussian distribution with an approximatelyscale invariant fluctuation spectrum. The ∆T/T amplitude is a factor of 4-5 largerthan the prediction of the standard CDM model with the same Hubble constantand density fluctuation normalization.

The recently reported COBE-DMR resultsimply that global monopole and texture models require high bias factors or a largeHubble constant in contrast to standard CDM which requires very low H0 andbias values. For H0 = 70kmsecMpc−1, we find that normalizing to the COBE resultsimplies b8 ≃3.2 ± 1.4 (95% c.l.

).If we restrict ourselves to the range of biasfactors thought to be reasonable before the announcement of the COBE results,1.5 <∼b8 <∼2.5, then it is fair to conclude that global monopoles and textures areconsistent with the COBE results and are a better fit than Standard CDM.Submitted to The Astrophysical Journal Letters

Topological defects which formed in a phase transition in the early universeprovide an attractive mechanism for the generation of density perturbations whichcan grow to form galaxies and large scale structure.Cosmic strings (Vilenkin,1980, Zel’dovich, 1980), global monopoles (Bariola and Vilenkin, 1989, Bennettand Rhie, 1990), and global textures (Turok, 1989) have all been proposed aspossible seeds for cosmic structure formation. These theories are characterizedby a single adjustable parameter, the Grand Unified theory symmetry breakingscale, v, and the value (v ∼1016 GeV, Gv2 ∼10−6) predicted by particle physics(Amaldi et al., 1991) also gives the correct amplitude to generate galaxies andlarge scale structure.

In contrast, inflationary models generally cannot producea reasonable perturbation amplitude without a rather extreme fine tuning of thecoupling constant (λ ∼10−12).Recent studies (Cen et al., 1991, Park et al., 1991, Gooding et al., 1991 and1992, and Turok and Spergel, 1990) of texture seeded density perturbations in auniverse dominated by cold dark matter (CDM) have indicated that this theorymay solve two of the perceived problems with the standard CDM model: thelack of sufficient large scale structure and quasars at high redshift. Their resultsgenerally agree with our calculations of global monopole (GM) and texture (T)seeded structure formation (Bennett, Rhie, and Weinberg, 1992), but we also findthat these nongaussian seeds tend to generate large galaxy pair velocities andcluster velocity dispersions (see also Bartlett, Gooding and Spergel, 1992).

Thiscan be alleviated by selecting a larger “bias” factor (i.e., a lower normalization ofthe density field).Another major difference between the nongaussian GM & T models and thegaussian inflationary models can be seen in the cosmic microwave backgroundradiation (CMB) anisotropies.In inflation scenarios, ∆T/T is due to remnantquantum fluctuations crossing the horizon at last scattering, and analytic resultsfor ∆T/T have been obtained for both the scalar mode responsible for the growth ofcosmic structure (Bond and Efstathiou, 1987) and the tensor modes (gravity waves)(Abbott and Wise, 1984). For the standard exponential inflation models, only thescalar growing mode is important.

With topological defects, metric fluctuations aregenerated by the relativistic dynamics of the defects inside the horizon, and ∆T/Tis affected by the fluctuations in all six independent components of the gravitationalfield. Thus, we expect the topological seed models to predict larger ∆T/T for afixed amplitude of density fluctuations.

The recent detection of CMB anisotropy bythe COBE DMR experiment (Smoot, et al., 1992) at a level somewhat higher thanthe prediction of the Standard CDM model should be regarded as encouraging fortopological seed models. (Some non-standard inflationary models may also havesignificant tensor mode perturbations (Davis, et al., 1992), but they are certainlynot the only theory with tensor modes.

)2

A previous estimation of ∆T/T for the texture model (Turok and Spergel, 1990)was based on a simple analytic model of a single texture evolution. They found anon-Gaussian distribution of hot and cold spots at a level that seems to conflictwith the COBE data.

In this paper, we present realistic numerical calculations of∆T/T on COBE scales in the GM&T models with no simplifying assumptions.∆T/T on COBE scales reflects the variations in time delay (frequency shift)along the photon paths from last scattering until the present. This is the general-ized Sachs-Wolfe effect (Sachs and Wolfe, 1967) where not only scalar growing modebut all gravitational field components contribute to the temperature fluctuations.

Ifwe choose a coordinate system such that the metric is gµν(x) = a(η)2[ηµν +hµν(x)],where ηµν = diag(−, +, +, +) and hµν is the metric perturbation, and choose agauge h0ν = 0 (Veeraraghavan and Stebbins, 1990), then the temperature fluctua-tion is given by∆TT= −12Zdηˆxiˆxj∂hij∂η,(1)where ˆxi is the normal vector along the line of sight. Because GM&T predict theearly formation of objects such as quasars, we assume the universe was reionizedat high redshift and take this into account by introducing a visibility function,f(z) = ehΩb(1−(1+z)3/2)/21.7 ,(2)which measures the fraction of photons present at redshift z that will reach z = 0without undergoing Compton scattering.

If we assume that the electrons whichscatter each photon see ⟨∆T/T⟩= 0, then we can account for reionization byinserting f(z) inside the “Sachs-Wolfe” integral, eq. (1).

This means that our∆T/T results will now depend on h = H0/100kmsecMpc−1 and the baryon densityΩb. We have done most of our calculations for hΩb = 0.1 which is about the largestplausible value.

For hΩb = 0.04, ∆T/T is only 3% larger on COBE scales.We evolve the source fields according to the field equations of motion in theFreedman-Robertson-Walker background and calculate the metric perturbationsdue to the energy-momentum of the scalar field by solving the linearized Einsteinequations. The field equations are¨φp + 2 ˙aa˙φp −∇2φp + a2λφ2 −v2φp = 0 ,(3)where v is the vacuum expectation value of the field, and p runs from 1-3 formonopoles and 1-4 for textures.

Because the defect core size is very much smaller3

than the our grid spacing, we can use the equation for the λ →∞and improvethe dynamic range of the calculations (Bennett and Rhie, 1990),δpq −φpφqv2 ¨φq + 2 ˙aa˙φq −∇2φq= 0 . (4)The energy-momentum tensor for the scalar fields is given byΘ00 =12˙⃗φ2+ 12∇⃗φ2,Θ0i = ˙⃗φ · ∂i⃗φ ,Θij =∂i⃗φ · ∂j⃗φ + 12δij12˙⃗φ2−12∇⃗φ2,(5)and the linearized Einstein equations are (Veeraraghavan and Stebbins, 1990)¨h + ˙aa˙h + 3 ˙a2a2δc = −8π (Θ00 + Θ) ,(6)¨˜hij −∇2˜hij + 2 ˙aa˙˜hij −13∂i∂jh + 19δij∇2h+∂i∂k˜hjk + ∂j∂k˜hik −23δij∂k∂l˜hkl = 16π ˜Θij ,(7)where h and Θ refer to the traces of hij and Θij.

The ˜ is used to refer to thetrace-free terms ˜hij = hij −13δijh and ˜Θij = Θij −13δijΘ. The perturbation in colddark matter δc obeys˙δc = −12˙h ,(8)and the following (non-dynamical) constraint equations must be satisfied,12∂i∂j˜hij + 13∇2h = 8πΘ00 + 3 ˙aa2δc −˙aa˙h ,12∂j ˙˜hij −13∂i ˙h = 8πΘ0i .

(9)In theories with “external” sources such as global monopoles or textures, theseequations impose important constraints on the initial conditions. In particular,since Θ00 cannot vanish if we are to have interesting density perturbations, theinitial energy density fluctuations in the ⃗φ field must be “compensated” by fluctu-ations of the opposite sign in the other fields.

For the calculations reported in this4

paper, we have taken h(η0) = 0, ˙h(η0) = 0, ˜hij(η0) = 0, ˙˜hij(η0) = 0, ˙⃗φ(η0) = 0,and δc(η0) = −(8π/3)(a/˙a)2Θ00(η0). Once the constraint equations, (9), are satis-fied initially, they will be satisfied at subsequent times if the equations of motion,(6)-(8), are satisfied.

Extreme care must be taken when evolving these equationsnumerically, because small numerical errors can excite growing mode solutions onscales outside the horizon where the physical modes do not grow. When thesescales finally come inside the horizon, the errors can have grown large enough tocompete with the physical perturbations that are generated by the source insidethe horizon.

We have found that we can keep these errors under control even when⃗φ takes random values on the scale of 1 grid spacing by going to extremely smalltime-steps (∆t ≈0.01∆x). In order to satisfy the second equation in (9), we findthat global monopole and texture fields must be sufficiently smooth, and this limitsus to an initial horizon size of >∼8 grid spacings.Another difficulty with evolving eqs.

(6)-(8) numerically is that the mixedpartial derivatives in (7) make it difficult to come up with an explicit differencingscheme that is stable. Implicit differencing schemes would be prohibitively expen-sive in computer time because we would still have to take very small time-stepsto avoid the super-horizon scale growing mode solution discussed above since thegrowing mode is a physical mode, not a purely numerical one.Fortunately, we have found it possible to remove the offensive mixed partialinto (7).

This substitution yields¨˜hij =∇2˜hij −2 ˙aa˙˜hij −∂i∂jh + 16π∂i⃗φ · ∂j⃗φη=η0−32πηZη0dη ˙⃗φ · ∂i∂j⃗φ+δij"13∇2h + 16π3˙⃗φ2+ 4 ˙aa2δc −43˙aa˙h#,(10)for eq. (7).

The mixed partial derivatives of h and ⃗φ in eq. (10) do not give rise tonumerical instabilities because the equations of motion for h and ⃗φ ((6) and (4))do not contain any mixed partial derivatives.Our numerical simulations use a modified second order leapfrog scheme toevolve eqns.

(4), (6), (8), and (10) in time. ∆T/T is determined by integratingeq.

(1) (with the visibility function (2) inserted) along photon trajectories thatconverge to a point at the end of the simulation. We have done about 25 simulationson 643 grids and 12 simulations on 1003 grids for each of the global monopole andtexture models.

The RMS ∆T/T values for the 1003 grids are 17.80±2.00Gv2m and10.29 ± 1.43Gv2t for monopoles and textures respectively. (vm and vt refer to the5

vacuum expectation values of the monopole and texture fields.) These numberscan be compared to the RMS fluctuation measured by COBE (∆T/T)RMS =1.10 ± 0.18 × 10−5 to yield Gv2m = 6.18 ± 1.23 × 10−7 for global monopoles orGv2t = 1.07 ± 0.23 × 10−6 for textures if we assume that monopoles or textures areresponsible for the ∆T/T observed by COBE.Figs.

1 and 2 show the simulated full-sky temperature maps for the globalmonopole and texture models respectively smoothed to the same 10◦scale as theCOBE maps. The scale on these plots is given in terms of Gv2.

Figs. 4 and 5 showhistograms of the ∆T/T distributions in the monopole and texture cases.

Notethat after the smoothing the maps contain only about 400 independent pixels.Thus, the departures from a Gaussian distribution are not significant.Fig. 3 shows the angular power spectrum, ∆T 2l , of our simulated ∆T/T maps.∆TT (θ, φ) =Xl,malmYlm(θ, φ) ,(11)∆T 2l = 14πXm|alm|2 .

(12)The solid and dashed curves give the best fit to the predicted power spectrum forHarrison-Zel’dovich (n = 1) primordial adiabatic density perturbations (Bond andEfstathiou, 1987),∆T 2l = (Qrms−PS)2(2l + 1)5Γ(l + (n −1)/2)Γ((9 −n)/2)Γ(l + (5 −n)/2)Γ((3 + n)/2) . (13)The best fit of the form, (13), gives n = 1.1 ± 0.3 for global monopoles andn = 1.2±0.2 for textures when we remove the quadrupole from the fit as was donefor the COBE data.

(The error bars in these fits reflect mainly systematic errors. )The Harrison-Zel’dovich value, n = 1, gives a good fit to our simulations and tothe COBE data, so it makes sense to compare the fit amplitudes.

The COBEvalue is Qrms−PS = 6.11 ± 1.46 × 10−6, and we obtain Qrms−PS = 8.7 ± 1.6Gv2mfor monopoles and Qrms−PS = 4.7 ± 0.5Gv2t for textures. A comparison of thesevalues gives Gv2m = 7.0 ± 2.1 × 10−7 and Gv2t = 1.30 ± 0.34 × 10−6 consistent withthe values obtained above.It is worth noting that topological defect theories generically predict ∆T/Tand δρ/ρ spectra that are slightly steeper than Harrison-Zel’dovich on very largescales (>∼1000 Mpc).

The reason for this is that with topological defects, unlikeinflation, the gravitational fields are generated inside the horizon, so that scales6

near the horizon have yet to receive their full “share” of perturbations. Thus, thelargest scales are expected to have less power than the scale-free spectrum (i.e.n > 1).

(This effect is partially compensated for by the effects of reionizationwhich reduce ∆T/T on small scales.) For cosmic strings, the effect should be evenmore pronounced since the coherence scale of the strings is smaller than that ofglobal monopoles or textures (Bennett, Stebbins, and Bouchet, 1992).

In contrast,power law inflationary models (Davis, et al., 1992) which are able to fit the COBEresults with a reasonable value for b8 predict n < 1. Thus, if the four year COBEresults converge to n = 1.50 ± 0.25, it will be fair to conclude that topological de-fect models are preferred over quantum fluctuations during inflation as the sourceof the primordial density perturbations.MonopolesTextureshGv2mb8Gv2tb80.5 2.49 × 10−6/b8 4.03 ± 0.80 4.56 × 10−6/b8 4.27 ± 0.980.6 2.19 × 10−6/b8 3.54 ± 0.70 3.93 × 10−6/b8 3.68 ± 0.840.7 1.96 × 10−6/b8 3.17 ± 0.63 3.48 × 10−6/b8 3.25 ± 0.750.8 1.82 × 10−6/b8 2.94 ± 0.58 3.17 × 10−6/b8 2.97 ± 0.681.0 1.64 × 10−6/b8 2.65 ± 0.53 2.75 × 10−6/b8 2.57 ± 0.59Table 1.

The scalar field normalizations and best fit bias parameters, b8, to theCOBE-DMR RMS anisotropy at 10◦are tabulated as a function of h. 1 σ errorsare reported.In order to translate our results into limits on theories of cosmic structureformation, we must normalize Gv2 to give a reasonable spectrum of density per-turbations. Because of uncertainties in the relationship between the number den-sity of galaxies and the mass density, this normalization is conventionally given interms of a bias factor, b8, such that the RMS mass density fluctuation = 1/b8 aftersmoothing with an 8h−1Mpc radius top hat.

Table 1 gives these normalizations asdetermined in Bennett, Rhie, and Weinberg (1992).Table 1 also lists the predicted b8 values as a function of h = H0/100kmsecMpc−1as determined by a comparisons of the predicted RMS fluctuation at 10◦.Wecan see that the predicted b8 values run very high for small values of the Hubbleconstant. Choosing a large Hubble constant in the Ω= 1 universe that we haveassumed is problematic due to the implied very short age for the universe, but withh = 0.7 (the smallest value that is consistent with most of the measured valuesof h) b8 = 2.5 is within ∼1σ of the mean in the both the global monopole and7

texture models. Thus, global monopoles and textures with a bias factor in therange of 2.5 −3 seem to be in reasonably good agreement with the COBE data.In a separate study of large scale structure in the global monopole and texturemodels, (Bennett, Rhie, and Weinberg, 1992), we find that these high bias globalmonopole and texture models do reasonably well in matching the observed largescale structure.Further work is required to see if the required biasing can beobtained dynamically, however.Finally, let us compare our results to those of other, well motivated theoriesof large scale structure formation.

We find that for similar values of h and b8,global monopole and textures predict ∆T/T on COBE scales that is a factor of4-5 larger than the standard CDM prediction (Bond and Efstathiou, 1987). Witha reasonable bias value, 1.5 >∼b8 >∼2.5, this model is inconsistent with the COBEmeasurement for h > 0.5 and but perhaps barely consistent for h = 0.5.

If wedemand that 1.5 >∼b8 >∼2.5 for global monopoles and textures, we find consistencywith COBE for the entire range, 0.5 < h < 1.0 at the 2σ confidence level and0.7 < h < 1.0 at the 1σ level. The power law inflationary models discussed byDavis, et al., (1992) can be made consistent with reasonable b8 values because theyhave significant contributions to ∆T/T from tensor modes that do not contributeto δρ/ρ.

These models do seem to have some difficulty with forming galaxies earlyenough, however (Adams, et al., 1992). Other models in with a smaller amount ofCDM, such as hot + cold DM models or low Ωmodels seem to fit the COBE resultsreasonably well (Wright, et al., 1992), but they are less attractive theoretically.Cosmic Strings + HDM seem to be a good fit to the COBE results (Bennett,Stebbins, and Bouchet, 1992), but the theoretical error bars are presently ratherlarge.

Thus, none of the best motivated models are singled out by the COBEresults, but global monopoles and textures are arguably the best fit to COBEamong the Ω= 1 pure CDM models.Further work into the details of galaxyformation and ∆T/T on smaller angular scales is certainly warranted.Acknowledgements: We would like to thank A. Stebbins and D. Weinberg for help-ful discussions. This work was supported in part the U.S. Department of Energy atthe Lawrence Livermore National Laboratory under contract No.

W-7405-Eng-48and by the NSF grant No. PHY-9109414.8

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FIGURE CAPTIONS1) A ∆T/T map for the global monopole model generated by our global fieldevolution code is displayed in a full-sky equal area projection. The scale isgiven in units of Gv2m.2) A ∆T/T map for the texture model generated by our global field evolutioncode is displayed in a full-sky equal area projection.

The scale is given inunits of Gv2t .3) The average ∆T/T power spectrum is plotted for 12 1003 monopole simula-tions and 12 1003 texture simulations. The solid and dashed curves give thebest fit to the power spectrum derived for a Harrison-Zel’dovich spectrumof primordial adiabatic density perturbations.

The error bars give the RMSdeviation from the mean, so they reflect the expected deviation for a singlerealization.4) A histogram of the pixels in convolved ∆T/T map shown in Fig. 1.

Thebins on the edges of the histogram include all the points outside the limits ofthe figure. The dashed curve is the histogram for a Gaussian with the sameRMS.5) A histogram of the pixels in convolved ∆T/T map shown in Fig.

2. Thebins on the edges of the histogram include all the points outside the limits ofthe figure.

The dashed curve is the histogram for a Gaussian with the sameRMS.10

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