Cosmic structure formation in Hybrid Inflation models

Transcription

1 Cosmic structure formation in Hybrid Infation modes Richard A. Battye and Jochen Weer 2 Department of Appied Mathematics and Theoretica Physics, University of Cambridge, Siver Street, Cambridge CB3 9EW, U.K. 2 Theoretica Physics Group, Backett Laboratory, Imperia Coege, Prince Consort Road, London SW7 2BZ, U.K. A wide cass of infationary modes, known as Hybrid Infation modes, may produce topoogica defects during a phase transition at the end of the infationary epoch. We point out that, if the energy scae of these defects is cose to that of Grand Unification, then their effect on cosmic structure formation and the generation of microwave background anisotropies cannot be ignored. Therefore, it is possibe for structure to be seeded by a combination of the adiabatic perturbations produced during infation and active isocurvature perturbations produced by defects. Since the two mechanisms are uncorreated the power spectra can be computed by a weighted average of the individua contributions. We investigate the possibe observationa consequences of this with reference to genera Hybrid Infation modes and aso a specific mode based on Supergravity. These mixed perturbation scenarios have some nove observationa consequences and these are discussed quaitativey. PACS Numbers : Cq I. INTRODUCTION The precise origin of cosmic structure is one of the most important questions facing cosmoogy today. Over the past fifteen years there have been two competing paradigms: quantum fuctuations created during infation [] a period of rapid expansion of the universe just after the Panck epoch which can sove the horizon and fatness probems of the standard Hot Big Bang mode and perturbations generated by the gravitationa effects of a network of topoogica defects [2 6], which may have formed during some cosmoogica phase transition cose to the energy scae of a Grand Unification Theory (GUT). In case of infation the fuctuations are generay adiabatic, Gaussian and passive in the sense that once created they evove in a deterministic way right up to the present day. These assumptions have simpified the process of making predictions in these modes to the point where accurate ( %) cacuations of the anisotropies in the cosmic microwave background (CMB) and the density fuctuations in cod dark matter (CDM) can be made for a given set of parameters in ess than a minute on a modern workstation [7]. Making the predictions of the same eve of accuracy for defect based modes is much more difficut since the perturbations are isocurvature, non-gaussian and are created activey throughout the whoe history of the universe, from the time of defect formation to the present day. However, recent work [8 2], has estabished a basis for future work on this subject defining what can be thought of as the standard mode, athough there sti appears to be some room for understanding more subte effects [2,3]. It was suggested in refs. [9,0] that fat universe modes with a critica matter density (Ω m = ) normaized to COBE woud require unacceptaby arge biases ( 5) between cod and baryonic matter on h Mpc scaes to be consistent with the observed gaaxy distribution, but more acceptabe modes can be constructed in an open universe or one dominated by a cosmoogica constant [4 6], abeit with a bias of 2 reative to IRAS which are usuay assumed to be good tracers of the underying mass distribution. Since COBE normaized adiabatic modes based on infation have no probem producing the requisite amount of power on these scaes, this suggests if the data is shown to be accurate that such modes may at east be partiay responsibe for the formation of structure. The idea of combining these two paradigms is a simpe one since they are far from being mutuay excusive; very simpy, if the infationary reheat temperature is greater than the GUT scae then the post-infationary universe wi encounter phase transitions, which may form topoogica defects. More specuativey, one might form defects in a non-therma phase transition induced by parametric resonance [7,8] during the reheating phase after infation. But most cosmoogists woud prefer for there to be ony a singe source of fuctuations, based on some kind of minimaist principe, and woud be sceptica of any theory which has both without further motivation. There are, however, a wide cass of infationary modes, which may produce topoogica defects usuay assumed to be strings, athough it is aso possibe to produce other kinds of defects during a phase transition which marks the end of the infationary epoch. These are known as Hybrid Infation modes [9]. Hence, there is sufficient motivation to consider mixed perturbation scenarios in which structure is formed by both adiabatic density fuctuations produced during infation and active isocurvature perturbations created by defects, without breaking any principe of minimaism, and this is the subject of this paper.

2 In section II we wi discuss the individua components the fuctuations generated by infation and defects, in particuar strings. The fact that there is no universa mode of infation makes it difficut to make very specific predictions. Therefore, we wi first treat Hybrid Infation modes in generaity by reference to a simpe mode (see ref. [20] for a compendium of infationary modes both Hybrid and otherwise), before discussing a specific mode which was put forward recenty to produce infation in the context of Supergravity [2]. We wi concentrate specificay on mixing infation with strings, since they are probaby the most obvious candidate in these scenarios, but most of the genera comments that we wi make appy equay we to the case of other topoogica defects, for exampe, the goba defect modes considered in ref. [8]. The modes for strings that we wi use are based on those aready used in refs. [9,0,3,5] and we wi make two assumptions. Firsty, we wi make the simpe assumption that the strings evove in a perfect scaing regime, from their formation to the present day, and then we wi attempt to incorporate the effects radiation-matter transition by use of the veocity dependent one scae mode [22]. We shoud note that it is not our intention in choosing these particuar modes for infation and strings to make any very specific predictions or caims as to their universa vaidity. Rather, we wish to discuss quaitativey the sort on phenomena one might possiby expect in the power spectra and their reation to the current and future observationa data. We wi then discuss how the spectra can be combined in section III. This is in fact trivia since the power spectra shoud be uncorreated and hence the two can be combined by a weighted average. Ceary, the addition of this extra degree of freedom weakens any constraint that current observations pace on each of the individua modes and we wi discuss this in four different contexts. Firsty, we consider genera Hybrid Infation modes in which the reative ampitude of the adiabatic and string induced components is arbitrary, aong with the spectra index (the initia density fuctuations created during infation are normay assumed to have a simpe power aw form P (k) k n,wherenis the spectra index). Specificay, we wi comment on the constraints which come from anaysis of the spectrum of CMB observations on arge scaes detected by COBE and aso from their combination with measurements of the density fuctuations on sma scaes, which are normay quantified in terms of σ 8, the fractiona over density in spheres with radius 8h Mpc. These are considered to be the most accurate and robust measurements in cosmoogy. Comparison to just the COBE data constrains the spectra index to be in the range n < 0.2 [23], whie a simpe comparison of the ampitude of the CMB anisotropies with that of σ 8 rues out the standard CDM scenario with n = (see, for exampe, ref. [24]), since the COBE normaized vaue of σ 8 computed for this mode is approximatey twice that which is observed, σ8 OBS 0.6 in a critica density universe whereas σ8 CDM.2, favouring a ower vaue of n 0.8. But a more detaied joint anaysis of a the avaiabe CMB data and measurements of σ 8 [25,26] suggests that something cose n = gives a better fit to a the avaiabe data. Without performing a fu ikeihood anaysis, we show quaitativey that these constraints can be reaxed since the arge ange CMB can be induced by strings, aowing for higher spectra indices to fit the data usuay at the expense of the reducing the power on arge scaes. We wi aso consider modes which use other cosmoogica parameters to fit the measurements of gaaxy custering on arge scaes ( 50 h Mpc). We shoud note that there sti remains a strong upper imit on bue spectra since arge spectra indices ead to the production of unacceptabe numbers of primordia back hoes [27]. We wi then discuss the observationa aspects of the specific mode based on Supergravity which is introduced in section II B. In this case the reative normaization of the adiabatic and string induced components, and the spectra index, which in this case is aso a function of scae, are fixed by a singe parameter of the mode. First, we show how the incusion of the string component aows the more extreme vaues of this parameter, which give very bue spectra on arge scaes, to be more compatibe with the reative ampitude of the COBE measurements and those of σ 8,than if it was absent. Then we show that simpe modifications to the cosmoogica parameters can improve the fit to the shape of the observed matter power spectra on arge scaes. Most specuativey, we examine the possibiity that there may be interesting effects in the power spectrum on sma scaes. It has been suggested [28 30] that there is a feature in the power spectrum with wavenumber k 0.hMpc, where the Hubbe constant is given by H 0 = h km sec Mpc, and such a feature in the power spectrum naturay occurs in these modes, athough not necessariy on these scaes. We iustrate this possibiity by reference to a number of simpe exampes, suggesting that the forthcoming redshift surveys (the Soan Digita Sky Survey (SDSS) and 2Df) shoud aow us to test this possibiity more accuratey. This feature in the matter power spectrum eads to more power on sma scaes than in pure adiabatic modes and hence it might possibe to effect the formation of damped Lyman-α systems and other eary objects. We wi discuss this in the context of the popuar cod pus hot dark matter mode which is thought to under produce such features. The main focus of our discussion is to reconcie the ampitude of the COBE detection with the ampitude and shape of the observed gaaxy distribution, a probem which both the standard CDM and defect modes both suffer from. However, the near future wi see an exposion in measurements of the CMB anisotropies over a wide range of scaes, for exampe, from the MAP and PLANCK sateites. To this end, we finay discuss the impications for the CMB anguar power spectra and the nove features which these modes have, in particuar the Dopper peak structure and non-gaussianity. We wi focus on the need to excude or constrain these mixed perturbation scenarios. 2

3 II. THE INDIVIDUAL FLUCTUATION SPECTRA A. Genera hybrid infation modes The proto-typica mode for Hybrid Infation is one which incudes two scaar fieds φ, a rea scaar fied known as the infaton, and ψ, a compex scaar fied which is couped to the infaton. The specific potentia usuay used is [9] V (φ, ψ) = ( M 2 λ ψ 2) 2 + 4λ 2 m2 φ g2 ψ 2 φ 2, () where λ and g are dimensioness couping constants, and M and m are the mass scaes introduced; in particuar M is that associated with spontaneous symmetry breaking, which in the case of ψ being a compex scaar fied eads to the production of goba strings at the end of infation. Themassivepartofthefiedψhas an effective mass M(ψ) 2 = g 2 φ 2 M 2 and therefore for φ>φ c =M/g there is a singe minimum of the potentia in the ψ-direction at ψ =0,whereasforφ<φ c the potentia deveops minima with ψ = M/ λ. In the case where M 2 M p m λ, M 2 m 2 /g 2 (2) and M p is the Panck mass, infation takes pace for M p >φ>φ c, with the expansion being dominated by the vacuum energy V (0, 0) = M 4 /4λ, rather than the fase vacuum. Hence, if the infaton starts at around φ M p as in the Chaotic Infation scenario [3], then infation takes pace as it ros down to φ c, at which point the fied ψ fas down into the vacuum manifod creating strings. Of course the universe may continue to infate after this point, at east partiay diuting the defects, but if the phase transition takes pace sufficienty ate, which can aways be arranged by an appropriate choice of the parameters, for exampe, by ensuring that M is greater than the Hubbe parameter during infation, then one wi be eft with a network of defects which wi subsequenty evove toward a scaing regime. The adiabatic density perturbations created in this mode on a ength scae are given by [9] δρ ρ = 2 ( 6πgM 5 5λ λmpm 3 2 c ) β 2, (3) where c is the horizon size when the infaton has vaue φ c, β = m/ 3H and H 2 = 2πM4 3λMp 2, (4) is the Hubbe parameter when φ = φ c. This mode has a spectra index n =+2β 2 >, and therefore if H m then the spectrum is amost scae free, whereas if m H then the spectra index can be much arger than one. In ref. [9] two possibe scenarios were considered. Firsty, if g 2 λ 0 and m 0 2 GeV, then normaization to the observed fuctuations [(δρ/ρ) obs 5 0 5, in these units] yieds M 0 GeV, and hence the adiabatic density fuctuations wi dominate over those produced by the strings [(δρ/ρ) str GM (δρ/ρ) obs ]. In this case the creation of strings at this scae may have other interesting cosmoogica impications the production of dark matter axions by the radiative decay of axions strings [32] but they wi not have a substantia effect on structure formation. However, if g λ andm GeV then normaization requires that M 0 5 GeV around the GUT scae with a spectra index of n.. In this case the perturbations created by the strings must be taken into account since they are of comparabe size to the adiabatic ones. Since this simpe mode was first proposed, there have been many other Hybrid Infation modes discussed in the iterature with a wide variety of predictions for the energy scae and spectra index, most of which adhere to the dogma that the natura scae for infation is beow the GUT scae and the perturbations are scae free (athough see ref. [33]). However, this simpe exampe stands as a iustration that simpe modes which create topoogica defects at the end of infation exist and moreover the defects can be sufficienty massive to seed density perturbations which are of comparabe size to the adiabatic perturbations created during infation. In the next section we wi discuss a particuar mode for Hybrid infation based on Supersymmetry and Supergravity which makes specific parameter based predictions for the energy scae and spectrum of initia fuctuations. These predictions wi be used to iustrate the nove observationa features of these mixed perturbation scenarios. We shoud, however, be mindfu of the fact that no universay accepted mode of infation exists and many more modes wi be invented to try to reconcie the theoretica and observationa prejudices of the day. Athough many modes wi not have GUT scae defects as a generic consequence, it is a very natura scae for symmetry breaking transitions to occur and modes which predict 3

4 this are we worth investigating, even if their observationa consequences appear, at first gance, to be at variance with the observations. In section III B we wi iustrate the effects of simpe modes with an arbitrary normaization, cose to the GUT scae, and spectra indices which are not one, showing that, at east quaitativey, that they may be to account for some of the current observations. B. A specific infation mode based on Supergravity In the previous section we discussed the origina Hybrid Infation mode, iustrating its saient features from the cosmoogica point of view. However, such a mode does not have any particuar motivation from the point of view of fundamenta physics the utimate aim of these endeavours. Hence, for definiteness, we woud ike to have a specific mode which has much stronger origins in the ream of high energy physics theories such as Supergravity. There are a number of popuar modes which use the fat directions of superpotentias to aow sow-ro infation, notaby the F-term [35] and D-term [36] scenarios, some of which aso produce topoogica defects during the phase transition at the end of infation and coud be candidates for the ideas that we are discussing here. However, we have chosen a mode introduced by Linde and Riotto [2] to iustrate the quaitative nature of these scenarios. Athough there are some subte phiosophica probems associated with this mode [20], for specific choice of the parameters it exhibits a number of the properties which we are interested in investigating. Here, we wi derive the initia fuctuation spectrum for this mode, foowing and extending the origina work [2], and then we wi incorporate the initia spectra into CMBFAST, the standard inear Einstein-Botzmann sover, to compute the spectrum of CMB anisotropies and CDM. From now onwards we wi work in units where the Panck mass appropriate for high energy physics appications, such as String Theory, is given by (8πG) /2 =. The mode is based on the simpest choice of the superpotentia which takes the form [37] W = S ( κ φφ γ 2), (5) where φ and φ are conjugate pairs of chira superfieds, S a neutra superfied and κ< is a parameter of the theory. If we canonicay normaize the scaar fied S = σ/ 2, then the effective potentia, excuding the D-term, is given by V = κ2 σ 2 ( φ φ 2) + κ φφ γ 2 2. (6) For σ > σ c = γ 2/κ, the mass of the chira superfieds is positive and the potentia in their directions has a singe minimum at φ = φ = 0. Aong this direction the potentia is fat and so a viabe infationary mode cannot be constructed. However, if one makes a sight modification to the potentia it is possibe, for exampe, by softy breaking Supersymmetry [37]. The nove aspect of the Linde-Riotto mode is that they suggested this may be done by the incusion of radiative and Supergravity corrections. In this case, if the fied σ starts at some arge vaue as in the Chaotic Infation scenario and the origina Hybrid Infation mode, then infation takes pace unti the fied ros down to σ = σ c, at which point the fied wi quicky ro down to the absoute minimum of the potentia where σ =0 and φ = φ = γ/ κ, producing topoogica defects if the vacuum manifod has non-trivia homotopy. It was suggested in ref. [2] that such modes can naturay have the correct symmetry breaking schemes to form cosmic strings and that the energy scae of these strings is given by γ/ κ which for an appropriate choice of parameters coud be the GUT scae. In order to compute the potentia, therefore, we must incude radiative effects and Supergravity corrections. The one oop SUSY potentia for this mode is given by [38,39] [ V = κ2 (κσ 2 28π 2 2γ 2) 2 κσ 2 2γ 2 n Λ 2 + ( κσ 2 +2γ 2) 2 κσ 2 +2γ 2 ] n Λ 2 2κ 2 σ 4 n κσ2 Λ 2, (7) where Λ is the renormaization scae which is introduced in the usua way. Since the period of infation that we are interested in takes pace for σ σ c, one can expand this correction to the potentia as V = γ 4 κ2 8π 2 n σ +. (8) σ c If the fied is very arge at the beginning of infation then Supergravity corrections wi aso be important. The SUGRA correction to the potentia is given by ( σ V SUGRA = γ 4 2 exp 2 )[ ] γ σ2 2 + σ4 4 =γ 4σ4 +, (9) 4 8 4

5 and hence by combining the two one gets the fuy corrected approximate potentia ( V = γ 4 + κ2 8π 2 n σ ) + σ4, (0) σ c 8 which is stricty vaid for σ σ c, but shoud aso be usefu for smaer vaues of σ cose to σ c. Using this potentia one can compute the evoution of the scaar fied, which we have done in terms of N the number of e-fodings unti the end of infation. During the infationary epoch 3H 2 = V and 3H σ = V, from which we can deduce that σ/h = V /V and hence dσ dn = V V, () since the scae factor a e N. If we assume that the potentia does not change appreciaby during the eary stages of infation when the cosmoogicay interesting perturbations are being created, then one can sove the equation for the fied σ in terms of N, σ 2 = κ 2π 2π ( κ ) κ σ2 end +tan 2π N + 2π ( κ ), (2) κ σ2 end tan 2π N where σ end is the vaue of the fied when infation ends, that is, when N = 0. Assuming this vaue of the fied to be reativey sma, which may not aways be the case, one can further approximate (2) to give κ ( κ ) σ 2π tan 2π N. (3) From this we can see that there is a constraint on the vaue of the parameter κ from the requirement that there have been at east 60 e-fodings, so as to make the universe amost fat by the present day. The argument of the tangent function shoud never be aowed to be greater than π/2, and hence one can deduce that κ<κ c =π 2 / The number of e-fodings N can then be reated to the waveength or wavenumber of the perturbations ( =2π/k) by ( ) ( ) k N 54 n hmpc 52 + n h, (4) Mpc where we have assumed that the size of the universe is 3000h Mpc at the present day and hence we can compute parametricay the fied σ as a function of the comoving scae during infation. The spectrum of primordia density fuctuations generated by the evoution of this fied during infation can be approximated by δρ ρ 3 V 3/2 ( 3 κ 2 5π V 5π γ2 8π 2 σ + ) 2 σ3, (5) and spectra index, which is now scae dependent, is given by n 2 V ) (3σ V = 2 κ2 4π 2 σ 2. (6) Ceary, the spectrum is not totay scae invariant since the fied evoves according to (2) during infation and in genera the effective spectra index increases as one approaches the argest scaes, but we now have a simpe parametric formua which aows us to compute the initia spectrum of density fuctuations P i (k). One might aso expect such modes to create a tensor contribution to the CMB fuctuations on arge scaes, but this woud be proportiona to V /2 and hence (δt/t) T = O()γ 2, which in the regime under consideration here is much smaer than that due to scaars [40]. Assuming that these adiabatic perturbations are the ony contribution to the scaes probed by COBE, one can use the normaization [4] V 3/2 ( κ V γ 2 2 σ 60 σ=σ60 8π 2 + ) 2 σ4 60 (7) 5

6 FIG.. The effective spectra index as function of the comoving scae during infation for κ =0.08 (soid ine), κ =0. (dotted ine), κ =0.2 (short dash ine), κ =0.3 (ong dash ine) κ =0.4 (dot-short dash ine) and κ =0.5 (dot-ong dash ine). where σ 60 is the vaue of the infaton fied when the current observabe universe eaves the horizon during infation. Therefore, using (3) one can deduce that [ ( )] /2 [ ( )] 3/ γ2 30κ 30κ sin cos, (8) κ 3/2 π π and hence if κ is sma, for exampe κ =0.08, then the symmetry breaking scae for the strings is given by γ/ κ If the phase transition which takes pace at the end of infation eads to the formation of strings, then their mass per unit ength wi be Gµ γ2 8πκ (9) Whereas, if κ is cose to κ c, for exampe κ =0.5, then one can deduce that Gµ In both cases the strings are around the scae at which they wi aso contribute substantiay to the the COBE normaization and ceary they must be taken into account. In section III C, we wi discuss the impications of this for the reative contribution of adiabatic and string induced perturbations, and hence the computed spectra. For the moment we wi ignore the strings and compute the spectrum of CMB anisotropies and fuctuations in the CDM, assuming that the adiabatic fuctuations are the ony contribution. The spectrum of initia fuctuations in this mode is not a simpe power aw, and hence the spectra index is now a function of k, thatis,p i (k) k n(k). The effective spectra index is potted in the cosmoogicay interesting range of k for various vaues of κ in fig.. For κ 0.3, the spectra index is approximatey constant, but for arger vaues of κ the spectrum rises sharpy on arge scaes (sma k). This nove feature of the spectrum makes this mode particuary interesting from the point of view of this paper since it wi ead to a heaviy bue shifted spectrum on very arge scaes and such a spectrum is tighty constrained by the observations. However, if the arge ange part of the CMB spectrum is created by the strings then it may yet be possibe for this mode to be viabe. Now that we have derived the power spectrum of initia density fuctuations, it can be incorporated into CMBFAST in order to compute the power spectra of CMB anisotropies and matter fuctuations in the CDM which we woud observe today for a given vaue of κ. The spectra are presented in fig. 2 for various vaues of κ and the standard 6

7 κ σ TABLE I. The computed vaues of σ 8 for the Linde-Riotto mode of infation using the standard cosmoogica parameters, and a varying the parameter κ. Note that we have not incuded the possibe effect of strings at this stage FIG. 2. On the eft the anguar power spectrum of CMB anisotropies and on the right the power spectrum of fuctuations in the CDM for the Linde-Riotto mode without a string induced component, using the standard cosmoogica parameters and κ =0.08 (soid ine), κ =0. (dotted ine), κ =0.2 (short dash ine), κ =0.3 (ong dash ine), κ =0.4 (dot-short dash ine) and κ =0.5 (dot-ong dash ine). In both cases the current observationa data points are aso incuded to guide the eye. Notice that the CMB anisotropies for κ =0.5 are widy at odds with the observations at a scaes and that even the modes with smaer vaues of κ are ceary at odds with the ampitude and shape of the observed matter power spectrum. cosmoogica parameters, and the corresponding vaues of σ 8 are tabuated in tabe. If κ is sma (for exampe, κ 0.08) then the initia fuctuations are amost scae invariant and the resuts are essentiay just those for standard CDM, but as κ increases the spectrum deveops a tit toward smaer scaes, this being most graphicay iustrated by the extreme case of κ = 0.5 where there are amost exacty 60 e-fodings of infation. When compared to the current observations [42 44], it is cear, just by inspection, that as they stand the modes with arger vaues of κ woud be rued out by the observations of the CMB and gaaxy correations on sma-scaes. C. String modes We have chosen strings as an exampe of topoogica defects produced at the end of infation due to the we estabished property [45 48] that they evove toward a sef-simiar scaing regime, in which the arge scae properties of the network are described by a singe scae and the density remains constant reative to the horizon. It is this property which makes them a possibe source of an amost scae invariant spectrum of density perturbations across The standard cosmoogica parameters used here and throughout this paper, uness stated otherwise, are a universe comprizing of 95% CDM and 5% baryons, with three massess neutrinos, a Hubbe constant H 0 =50kmsec Mpc and the standard recombination history. 7

8 a wide range of scaes, and hence a reaistic mode for structure formation. We coud, of course, have used other topoogica defect modes, for exampe, the goba defect modes used in ref. [8] and the quaitative predictions woud be very simiar. We shoud note that in doing this we have made the assumption that the initia distribution of strings is such that the network can achieve a scaing regime before the cosmoogicay interesting scaes come inside the horizon. We have aready pointed out that if a substantia period of infation were to take pace after the phase transition, then the strings woud become diuted possiby radicay atering the standard picture of string evoution. But, so ong as infation ends quicky enough, which is usuay possibe by tuning parameters, this diution can be reversed before the cosmoogicay interesting epoch, just before the time of radiation-matter equaity. The particuar mode we wi use to describe the two-point correation functions of the strings is that which was deveoped in refs. [49,9,0], where the string network is modeed as an ensembe of straight segments each with size ξη, whereηis the conforma time, and a random veocity chosen from a Gaussian distribution which has zero mean and variance v. The scaing regime is usuay achieved by the production of oops and subsequent emission of radiation into the preferred channe, usuay assumed to be gravitationa radiation. As a first approximation this can be accounted for by removing string segments at a rate which exacty maintains scaing. The resuts of using this approximation, which we sha ca the standard scaing source, are presented in fig. 3 aong with some data points which represent the observations. The main quaitative features of this mode are the apparent absence of any kind of Dopper peak in the CMB anisotropies and a matter power spectrum which on arge scaes ( h Mpc) appears to require a bias of around 5 and the computed vaue of σ In order to mode a network of strings more reaisticay one can do two things. Firsty, one must attempt to take into account the effects of the matter radiation transition; scaing is a baance between the rate of expansion of the universe and the efficiency by which the network can ose energy into oops. During the transition era, that is, 0.η eq <η<η eq,whereη eq is the conforma time of equa matter-radiation, the expansion rate is reaxing from the radiation era, where the scae factor is proportiona to η, to the matter era, where it is proportiona to η 2. Ceary, the nature of the scaing changes during this time and it has been suggested [22] that the change in the density of strings observed in the two different eras can be modeed using the veocity dependent one-scae mode. This mode treats the two parameters, ξ and v, used in construction of the two point functions as being dependent on the conforma time, aowing one to compute the rate at which the density changes. Another aspect of string evoution which is not described by this simpe mode is the effect of sma-scae structure. In high resoution simuations [46 48], it was found that sma-scae structure buit up cose to the resoution of the simuation due to the copious production of oops on these scaes. Athough in reaity this wi be stabiized by radiation backreaction, some structure wi remain effectivey renormaizing the mass per unit ength of these string segments to be µ 2µ, whereµis the bare mass per unit ength. Formay, this can be done by using a transonic equation of state for the string [5], that is, treating the strings as having a more compicated equation of state, rather than the usua Nambu-Goto one, where the energy per unit ength and the tension are equa. In Minkowski space, the energy momentum tensor for a genera string is [ ] T µν (x) = dσδ 4 (x X(σ, t)) UẊµẊν TX µx ν, (20) where X µ (σ, t) are the spacetime coordinates of the string at time t parameterized by σ, some arbitrary coordinate aong the string, U is the energy density of the string and T is its tension. For the specia case of a Nambu-Goto string the equation of state is T = U = µ, but for the transonic case under discussion here TU = µ 2 and, therefore, if U = µ 2µthen T/U /4. By making this simpe modification to the origina mode, one can incorporate some of the effects of sma-scae structure. In particuar, these modifications can ead to an enhanced peak structure due to the effects of an enhanced Newtonian potentia [50]. A detaied investigation of these effects is the subject of work in progress. Using the modifications described above, we have computed the spectra of CMB anisotropies and fuctuations in the CDM for the standard set of cosmoogica parameters and the resuts are aso presented in fig. 3. The main quaitative features of this mode, which we wi ca the standard string mode, are a sighty tited spectrum of CMB anisotropies which rises to a singe broad peak around = with no secondary osciations, and a matter power spectrum which appears to match the observation extremey bady both in ampitude and shape 2. The computed vaue for σ reasonaby cose to the measured vaue and that for G µ , which impies that 2 Both of these deficiencies can be rectified by the incusion of a cosmoogica constant with Ω Λ (see, for exampe, ref. [5]) 8

9 FIG. 3. On the eft the anguar power spectrum of CMB anisotropies and on the right the power spectrum of fuctuations in the CDM for the standard scaing source (soid ine) and the standard string mode (dotted ine). In both cases the standard cosmoogica parameters have been used. The current observationa data points and the equivaent spectra for the standard CDM scenario (short dash ine) are aso incuded to guide the eye. Gµ once sma-scae structure is taken into account. This we within the constraint imposed by absence of timing residuas in the observations of mii-second pusars [52]. We shoud emphasize that the predictions of these two simpe assumptions are not definitive. They appear to have very simiar predictions on arge scaes, but their predictions on smaer scaes at very different, for exampe, very different vaues of σ 8. At this stage it seems sensibe to consider the impications of both modes and hopefuy future work wi enabe us to pin down the predictions of these scenarios more fuy. III. COMBINING THE SPECTRA AND ITS OBSERVATIONAL CONSEQUENCES A. Combination by a weighted average At first sight it may appear that combining the effects of adiabatic fuctuations created by infation and those created activey by topoogica defects is a highy non-trivia task and indeed if, for exampe, one were trying to create CMB sky maps by considering the evoution of each mode, it woud be. However, computations are simpified consideraby by the fact that, at this stage, we are ony trying to compute the power spectra. Since each of the two sources are uncorreated one happening during the infationary epoch and the other after one can simpe add the correcty normaized spectra together. The ony subte aspect is to normaize each of the two contributions so that the sum is normaized reative to COBE. If one assumes, for the moment, that the normaization of each of the components is arbitrary and to be computed from the observations, one can add the spectra as C tot = αc adia +( α)c str, P tot (k) =αp adia(k) +( α)p str (k), (2) where 0 α is an arbitrary constant defining the reative normaization, C adia and P adia (k) are the spectra from adiabatic perturbations individuay normaized to COBE, and C str and P str (k) are those for strings. In a specific high energy physics motivated mode, for exampe, the one discussed in section II B, the reative normaization of the two components wi be fixed by the parameters of the mode, effectivey fixing the vaue of α. As a simpe iustration of how to use this prescription for combining the spectra, we have computed the anguar power spectrum of CMB anisotropies and the power spectrum of the matter fuctuations in the CDM for the standard CDM scenario combined with both the standard scaing source (resuts presented in fig. 4) and the standard string 9

10 FIG. 4. On the eft the anguar power spectrum of CMB anisotropies and on the right the power spectrum of fuctuations in the CDM for the standard CDM scenario mixed with standard scaing source using a ratio of α =.0 (soid ine), α =0.75 (short dash ine), α =0.5 (ong dash ine), α =0.25 (short dash-dotted ine) and α =0.0 (dotted ine). mode (fig. 5), for α =.0,0.75, 0.5, 0.25 and 0.0. These figures iustrate the generic quaitative behaviour that one might expect in these mixed perturbation scenarios. The vaue of σ 8 can be computed from P (k) via the formua σ 2 8 =4π dk k k3 P (k) W (8kh Mpc) 2, (22) where the window function W (x), given by W (x) =3(sinx xcos x) /x 3. Hence, its vaue in these mixed perturbation scenarios is given by σ 2 8 = α ( σ adia 8 ) 2 ( ) +( α) σ str 2, (23) where σ8 adia and σ8 str are the COBE normaized vaues for the individua components. Therefore, we see that the vaues of σ 8 add in quadrature weighted by the factors α and α. If one of the computed vaues for the individua components is beow the observed vaue and the other is above, then it is possibe to choose α so that the mixture gives the observed vaue, σ For the combination of standard CDM and the standard scaing source, we find that the vaue of α whichdoesthisisα 0.20, whereas for the standard string mode α 0.5. Both these vaues are ow refecting the fact that on sma scaes the strings dominate, and hence on the arger scaes where the strings appear to be deficient, the appeaing aspects of the adiabatic perturbations are ost. We sha see in the next section that this is a robust feature of modes using the standard cosmoogica parameters. 8 B. Genera hybrid infation modes combined with strings In this section we wi discuss quaitativey the observationa consequences of aowing for a string induced component to the CMB anisotropies and the fuctuations in the CDM, in addition to an adiabatic component which is assumed to come from infation. Our treatment is totay genera, appying to any infationary scenario, but specificay we have in mind the GUT scae Hybrid Infation scenario discussed in section II A. We wi aow for the moment for the ratio of the two components to be arbitrary and for the spectra index to vary in the range 0.7 <n<.3. From the observationa point of view, we wi start conservativey with simpe tests which compare the ampitude of fuctuations in the CMB and the matter distribution, before discussing the shape of the observed power spectrum. Initiay, we wi concentrate on a universe with critica matter density with a matter content which comprizes ony of CDM and 0

11 FIG. 5. On the eft the anguar power spectrum of CMB anisotropies and on the right the power spectrum of fuctuations in the CDM for the standard CDM scenario mixed with standard string source using a ratio of α =.0 (soid ine), α =0.75 (short dash ine), α =0.5 (ong dash ine), α =0.25 (short dash-dotted ine) and α =0.0 (dotted ine) FIG. 6. On the eft the anguar power spectrum of CMB anisotropies and on the right the power spectrum of the fuctuations in the CDM for modes with the spectra index n varying between 0.7 and.3. n =0.7 (dotted ine), n =0.8 (short dash ine), n =0.9 (ong dash ine), n =.0 (soid ine), n =. (dot-short dash ine), n =.2 (dot-ong dash ine), n =.3(short dash-ong dash ine). At this stage no string induced component has been incuded.

12 n σ8 adia α α TABLE II. The computed vaues of σ 8 for pure adiabatic modes using the standard cosmoogica parameters, and a varying spectra index n. Incuded aso are the vaues of the ratio, α, of the adiabatic and string induced components, if a such a mode is to give the observed vaue of σ in a critica density universe. The vaue α is the ratio when the string induced component is that of the standard scaing source, that is, σ8 str 0.3, and α 2 is that for the standard string source, that is, σ8 str baryons. We wi see that even with the extra string induced component under discussion here it is difficut to fit a the data without reaxing either of these assumptions. The COBE normaized spectra of the CMB and CDM for adiabatic component are presented in fig.6 for range of vaues of n, and ceary none of the modes does particuary we with respect to a the observations. The modes with ow n 0.8 give a good fit to the observed matter power spectrum, assuming no bias, whie giving an apparenty poor fit to the observations of the anisotropy in the CMB on sma anguar scaes. For arger vaues of n.2, the situation is reversed with the fit to the matter power spectrum requiring some kind of scae dependent bias, whie at east on smaer anguar scaes the comparison with the measurements of the CMB is much better, athough we note that the arge ange spectrum is ony marginay compatibe with the spectrum of anisotropies detected by COBE. It is this rather unsatisfactory situation, which eads joint anayses of the two different types of measurements to concude that the best fit to the data is given by something cose to n =. Probaby the most stringent and robust constraint on any mode for structure formation comes from comparing the magnitude of the CMB anisotropies detected by COBE with the ampitude of the measured matter fuctuations on 8h Mpc, σ 8. Assuming that we can estimate the observed vaue of σ 8 in the underying matter distribution, without recourse to bias or anti-biasing, then it is possibe to rue out a arge cass of modes. In particuar, as we have aready discussed this test woud rue out the standard CDM scenario, uness an exotic anti-biasing mechanism was at work. We wi assume, from the point of view of this exercise, that the observed vaue of σ 8 =0.6 and we wi attempt to construct COBE normaized modes which can fit this vaue. This can be done simpy by computing α using (23) for given vaues of σ8 adia and σ8 str. The computed vaues of α are given in tabe II for both the standard scaing and string scenarios, and the resuting CDM power spectra are presented in fig.7. We see that, in both cases, the spectrum is dominated by the adiabatic component on arge scaes and by the string induced component on sma scaes, with the transition taking pace around k 0.2hMpc. Whie the ampitude of σ 8 is fitted exacty, the shape of mixed power spectra does not correspond to that which is observed. One coud argue that the observations on arge scaes are much ess certain than the ampitude of σ 8 and that such modes wi ony be rued out once more accurate data is avaiabe on scaes greater than around 20h Mpc. We shoud note that there is an interesting quaitative difference between using the standard scaing source and the standard string source on sma scaes, we sha discuss this phenomenon further in section III D. It appears that it is not possibe to fit the exact shape of the observed power spectrum by just varying the spectra index in a universe with the standard cosmoogica parameters. A better fit to the current observations may be achieved by varying the cosmoogica parameters namey the Hubbe constant, h, the number of massive neutrinos, N ν, and the contributions to the cosmoogica density from CDM, Ω c, Hot Dark Matter (HDM) such as neutrinos, Ω ν, the baryonic matter, Ω b, and the vacuum energy in the form of a cosmoogica constant, Ω Λ. A recent anaysis of pure adiabatic modes [34] varying these parameters suggests that the modes whose parameters are tabuated in tabe III aong with the computed vaues of σ 8 (incuded aso is the Standard Cod Dark Matter scenario) give the best fit to the current observations. The anisotropies in the CMB and the fuctuations in the CDM are potted in fig. 8. Note that modes B, C, D and E a fit the shape of the observed matter power spectrum very much in contrast to mode A, but that in modes B and D anti-biasing, that is, a bias of ess than one (in fact, b CHDM 0.85 and b ΛCDM 0.7), is required to reconcie the ampitude of the spectrum with that of the observations (in refs. [43,44], the bias of IRAS gaaxies was assumed to be one which means that they are good tracers of the underying mass distribution. This may not necessariy be true). 2

13 FIG. 7. Mixed perturbation scenarios which require no bias to fit the observed vaue of σ 8 with n varying between 0.7 and.3 for the adiabatic component. On the eft the defect component is that of the standard scaing source and on the right it is that of the standard string source. The curves are abeed as in fig. 6 Mode Description Ω c Ω b Ω ν Ω Λ h n N ν σ 8 A SCDM B CHDM C TCDM D ΛCDM E hcdm TABLE III. The cosmoogica parameters of the modes whose CMB anisotropies and CDM fuctuations are potted in fig. 8. We have aready discussed the incusion of a string induced component to mode A, and modes C and E appear to fit the data extremey we without any modifications, but the anti-biasing required for modes B and D to fit the data coud be perceived as a probem for such scenarios. However, the incusion of a defect induced component (from either the standard scaing source or the standard string source) with the correct ampitude, α = 0.7 for mode B and α =0.5 for mode D, has exacty the desired effect on arge scaes, as shown in fig. 9 In summary, therefore, the introduction of this extra degree of freedom aows us to fix exacty the ampitude of σ 8, reaxing any constraint on n from the simpe test of comparing the ampitude of CMB anisotropies with σ 8, but this is at the expense of having too itte power on arge scaes. In fact, if one ony uses this simpe test, there is probaby very itte constraint on the initia power spectrum on arge scaes, since the strings can account for the CMB anisotropies observed by COBE. However, if one aows for a sma component of HDM or a non-zero cosmoogica constant, then it is possibe to fit the a the observationa data without the need to postuate any kind of bias between the observations and the computed CDM power spectrum. It is aso worth noting that each of these mixed scenarios does much better on arge scaes than the string induced spectrum by itsef, midy reieving the so caed b probem [9,0]. We wi discuss the important features of the CMB power spectrum induced in these modes in section III E. C. Observationa aspects of the Linde-Riotto mode We wi now turn our attention to the specific Supergravity-inspired mode of infation discussed in section II B. There, it was shown that for κ sma the predictions were very simiar to that of a mode with scae-free spectrum, 3

14 FIG. 8. On the eft the anguar power spectrum of CMB anisotropies and on the right the power spectrum of fuctuations in the CDM for the best fit CDM type modes whose parameters are tabuated in tabe III. These modes are designed to fit the shape of the observed power spectrum, but note that some of the modes (B and D) require anti-biasing to fit the ampitude. (A,soid ine) SCDM, (B,dot-short dash ine) CHDM, (C,ong dash ine) TCDM, (D,dotted ine) ΛCDM and (E,short dash ine) hcdm. In both cases the current observationa data points are aso incuded to guide the eye FIG. 9. The power spectrum of the fuctuations in the CDM for modes B and D, with a string induced component computed using the standard scaing source (dot-short dash ine and short dash ine respectivey) and the standard string source (ong-dash ine and dotted ine respectivey). For mode B, α = % adiabatic fuctuations and 30% from strings, whie for mode D, α =0.5 equa proportions of adiabatic and string induced fuctuations. It is cear that each of these modes fits the observations very we in the inear regime, k<0.2hmpc, without the need for bias. 4

15 FIG. 0. The reative normaization of the adiabatic and string induced components, α, in the Linde and Riotto mode potted as a function of the mode parameter κ. The case of α corresponds to most of the fuctuations being adiabatic, and α 0 corresponds to most of them being induced by strings. but that for arger vaues of κ<κ c more exotic initia spectra were possibe. We aso noted that for generic symmetry breaking schemes the mode woud ead to the production of cosmic strings whose mass per unit ength woud be arge enough for them to have a substantia effect on cosmic structure formation. If one considers these modes as candidates for the mixed perturbation scenarios parameterized by the reative normaization α, then the normaization of the strings wi be given by Gµ 0 6 α (24) and the normaization of the adiabatic perturbations requires that [ ( )] /2 [ ( )] 3/2 α γ2 30κ 30κ sin cos. (25) κ 3/2 π π The vaue of Gµ can be computed in terms of γ and κ and therefore one can use (24) to eiminate γ, togiveαin terms of κ, [ α =+0.36κ sin ( 30κ π ) ( )] 30κ cos 3. (26) π This function can be approximated in both the imit of κ sma, in which case α, and κ κ c where α 0. In these two imiting cases one or the other of the two sources dominates, but in the more genera case any reative normaization of the two components is possibe. The precise function α(κ) is potted in the range 0 <κ<κ c in fig.0 and it is tabuated for various vaues of κ in tabe IV aong with the vaues of Gµ, γ and σ 8 for the mixed scenario. Notice that a the vaues of γ, which can be computed by using the energy units E = M p / 8π = GeV, ie in the sensibe range of GeV for κ =0.08 and GeV for κ =0.5, and that the corresponding vaues of Gµ are even more favourabe reative to the constraint from mii-second pusars [52]. Using this reative normaization and the standard string source mode for the defect induced component, we have computed the CMB anisotropies and fuctuations in the CDM for the same vaues of κ used in fig. 2 and the propery normaized resuts are presented in fig.. Notice that the modes with arge vaues of κ (for exampe, κ =0.5), which were widy at odds with the observations without the incusion of the string induced components, appear to have much more acceptabe spectra, with a the computed vaues of σ 8 being around. Of course the shape of the spectrum is not quite correct, a feature which is common to most sensibe critica density CDM modes athough 5

16 κ α Gµ γ σ TABLE IV. The reative contribution from adiabatic and string induced fuctuations, α as a function of κ for the Linde-Riotto mode. Aso incuded are the corresponding vaues of Gµ and γ and the vaue of σ 8 using a mixed perturbation scenario with the computed vaue of α FIG.. The same quantities and modes as in fig. 2 except that we have incuded a string induced component (the standard string source ony) with the reative normaization given in tabe IV for each vaue of κ. 6