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NATURAL INFLATION: PARTICLE PHYSICS MODELS,

arXiv:hep-ph/9207245v1 17 Jul 1992June 1992NATURAL INFLATION: PARTICLE PHYSICS MODELS,POWER LAW SPECTRA FOR LARGE-SCALE STRUCTURE,AND CONSTRAINTS FROM COBEFred C. Adams1,5, J. Richard Bond2, Katherine Freese1,5Joshua A. Frieman3,5 and Angela V. Olinto4,51Physics Department, University of MichiganAnn Arbor, MI 481092CIAR Cosmology ProgramCanadian Institute for Theoretical AstrophysicsUniversity of Toronto, Toronto, ON M5S 1A1, Canada3NASA/Fermilab Astrophysics CenterFermi National Accelerator LaboratoryP.O. Box 500, Batavia, IL 605104Department of Astronomy and AstrophysicsUniversity of ChicagoChicago, IL 606375Institute for Theoretical PhysicsUniversity of California, Santa BarbaraSanta Barbara, CAABSTRACTA pseudo-Nambu-Goldstone boson, with a potential of the form V (φ) = Λ4[1 ± cos(φ/f)],can naturally give rise to an epoch of inflation in the early universe, if f ∼MP l and Λ ∼MGUT.Such mass scales arise in particle physics models with a gauge group that becomes strongly in-teracting at the GUT scale.We explore the particle physics basis for these models, focusingon technicolor and superstring theories, and work out a specific example based on the multiplegaugino condensation scenario in string/supergravity theory.

We study the cosmological evolutionof and constraints upon these models numerically and analytically. To obtain a sufficiently highpost-inflation reheat temperature for baryosynthesis to occur we require f ∼> 0.3Mpl.

The primor-dial density fluctuation spectrum generated by quantum fluctuations in φ is a non-scale-invariantpower law, P(k) ∝kns, with ns ≃1 −(M 2P l/8πf 2), leading to more power on large length scalesthan the ns = 1 Harrison-Zeldovich spectrum. We pay special attention to the prospects of us-ing the enhanced power to explain the otherwise puzzling large-scale clustering of galaxies andclusters and their flows.

We find that the standard cold dark matter model with 0 ∼< ns ∼< 0.6could in principle explain this data. However, the microwave background anisotropies recentlydetected by COBE imply such low primordial amplitudes (that is, bias factors b8 ∼> 2) for theseCDM models that galaxy formation would occur too late to be viable and the large-scale galaxyflows would be too small; when combined with COBE, these each lead to the constraint ns ∼> 0.6,hence f > 0.3MP l, comparable to the bound from baryogenesis.

For other inflation models whichgive rise to initial fluctuation spectra that are power laws through the 3 decades in wavelength1

probed by large scale observations, such as extended inflation and inflation with exponential po-tentials, our constraint on ns is tighter, ns > 0.7. Combined with other constraints (which implyns < 0.77 −0.84), this leaves little room for most extended inflation models.

Chaotic inflationmodels with power law potentials have ns ∼> 0.95 through this band and so are not affected.2

I. INTRODUCTIONIn recent years, the inflationary universe has been in a state of theoretical limbo: it is abeautiful idea in search of a compelling model. The idea is remarkably elegant[1]: if the earlyuniverse undergoes an epoch of quasi-exponential expansion during which the Robertson-Walkerscale factor a(t) increases by a factor of at least e60, then a small causally connected region growsto a sufficiently large size to explain the observed homogeneity and isotropy of the universe, todilute any overdensity of magnetic monopoles or other unwanted relics, and to flatten the spatialhypersurfaces, Ω≡8πGρ/3H2 →1.As a bonus, quantum fluctuations during inflation cancausally generate large-scale density fluctuations, which are required for galaxy formation[2].During the inflationary epoch, the energy density of the universe is dominated by the (nearlyconstant) potential energy density V (φ) associated with a slowly rolling scalar field φ, the in-flaton[3].

To satisfy cosmic microwave background radiation (CMBR) anisotropy limits on thegeneration of density fluctuations, the potential of the inflaton must be very flat. Consequently,the field φ must be extremely weakly self-coupled, with effective quartic self-coupling constantsatisfying λφ < 10−12 −10−14 in most models [4].Density fluctuations in inflation are thus a blessing for astronomers but a curse for particlephysicists, because the theory must contain a very small dimensionless number.

Attitudes con-cerning this problem vary widely among inflation theorists: to some this represents unacceptable‘fine tuning’; to others, it is not an issue of great concern because we know there exist othersmall numbers in physics, like lepton and quark Yukawa couplings gY ∼10−5 and the ratioMweak/MP l ∼10−17. Partly as a consequence of the latter view, in recent years, it has becomecustomary to decouple the inflaton completely from particle physics models, to specify an ‘inflatonsector’ with the requisite properties, with little or no regard for its physical origin.Nevertheless, it is meaningful and important to ask whether such a small value for λφ isin principle unnatural.

Clearly, the answer depends on the particle physics model within whichφ is embedded and on one’s interpretation of naturalness. A small parameter λ is said to be“technically natural” if it is protected against large radiative corrections by a symmetry, i.e., ifsetting λ →0 increases the symmetry of the system [5].

For example, in this way, low energy su-persymmetry might protect the small ratio Mweak/MP l. However, in technically natural inflationmodels, the small coupling λφ, while stable against radiative corrections, is itself unexplained,and is generally postulated (i.e., put in by hand) solely in order to generate successful inflation.Technical naturalness is a useful concept for low energy effective Lagrangians, like the electroweaktheory and its supersymmetric extensions, but it points to a more fundamental level of theory forits origin. Since inflation takes place relatively close to the Planck scale, it would be preferable tofind the inflaton in particle physics models which are “strongly natural”, that is, which have nosmall numbers in the fundamental Lagrangian.In a strongly natural gauge theory, all small dimensionless parameters ultimately arise dy-namically, e.g., from renormalization group (or instanton) factors like exp(−1/α), where α is agauge coupling.

In particular, in an asymptotically free theory, the scale M1, at which a loga-rithmically running coupling constant becomes unity, is small, M1 ∼M2e−1/α, where M2 is thefundamental mass scale in the theory. In some models, the inflaton coupling λφ arises from aratio of mass scales, λφ ∼(M1/M2)n; for example, in the models to be discussed below, n = 4.As a result, in such models, λφ is naturally exponentially suppressed, λφ ∼e−n/α.An example of this kind, namely, a scalar field with naturally small self-coupling, is providedby the axion [6], a light pseudoscalar which arises in models introduced to solve the strong CPproblem.

In axion models, a global U(1) symmetry is spontaneously broken at some large massscale f, through the vacuum expectation value of a complex scalar field, ⟨Φ⟩= f exp(ia/f). (In this case, Φ has the familiar Mexican-hat potential, and the vacuum is a circle of radius f.)At energies below the scale f, the only relevant degree of freedom is the massless axion fielda, the angular Nambu-Goldstone mode around the bottom of the Φ potential.

However, at a3

much lower energy scale, the symmetry is explicitly broken by loop corrections. For example,the QCD axion obtains a mass from non-perturbative gluon configurations (instantons) throughthe chiral anomaly.

When QCD becomes strong at the scale ΛQCD ∼100 MeV, instanton effectsgive rise to a periodic potential of height ∼Λ4QCD for the axion. In ‘invisible’ axion models [7]with canonical Peccei-Quinn scale fP Q ∼1012 GeV, the resulting axion self-coupling is extremelysmall: λa ∼(ΛQCD/fP Q)4 ∼10−52.

This small number simply reflects the hierarchy between theQCD and Peccei-Quinn scales, which arises from the slow logarithmic running of αQCD.Pseudo-Nambu-Goldstone bosons (PNGBs) like the axion are ubiquitous in particle physicsmodels: they arise whenever an approximate global symmetry is spontaneously broken. We there-fore choose them as our candidate for the inflaton: we assume a global symmetry is spontaneouslybroken at a scale f, with soft explicit symmetry breaking at a lower scale Λ; these two scales com-pletely characterize the model and will be specified by the requirements of successful inflation,namely, a sufficent number of e-folds of inflation, sufficient reheating, and an acceptable amplitudeand spectrum of density fluctuations.

The resulting PNGB potential is generally of the formV (φ) = Λ4[1 ± cos(Nφ/f)] . (1.1)We will take the positive sign in Eq.

(1.1) (this choice has no effect on our results) and, unlessotherwise noted, assume N = 1, so the potential, of height 2Λ4, has a unique minimum at φ = πf(we assume the periodicity of φ is 2πf). In a previous paper [8] (hereafter Paper I), three of usshowed that, for f ∼MP l ∼1019 GeV and Λ ∼MGUT ∼1015 GeV, the PNGB field φ can driveinflation; in this case, the effective quartic coupling is λφ ∼(Λ/f)4 ∼10−13, as required.

In thispaper, we study this class of models and their implications in greater depth.We note that, in some cases, the potential of Eq. (1.1) is the lowest order approximation toa more complicated expression.

For inflation, the important ingredients are the height (∼Λ4)and width (∼f) of the potential, and the curvature in the vicinity of its extrema, which isdetermined by m2φ = Λ2/f. Thus, while our treatment will focus on the specific form (1.1), ourconclusions hold for more general forms of the PNGB potential which have the same overall shape(that is, same height, width, and curvature at the extrema; in addition, we assume V (φ) variesmonotonically between φ = 0 and πf, that is, we ignore higher order ripples, which might affectthe perturbation spectrum over a small range of wavelengths).In section II, we discuss the PNGB inflation scenario in the context of particle physics models.As noted above, a successful inflation scenario does not consist simply of a scalar field potentialthat does the trick; in addition, the parameters of the potential, in this case the requisite massscales f and Λ, must have a natural origin in plausible particle physics models.

PNGB potentialswith these mass scales do arise naturally in particle physics models. For example, in the hiddensector of superstring (supergravity) theories, if a non-Abelian subgroup(s) remains unbroken, therunning gauge coupling can become strong at the scale ∼1014 −1015 GeV; indeed, it is hoped thatthe resulting gaugino condensation may play a role in determining the string coupling constantand possibly in breaking supersymmetry [9].

(We note that, in such models, the only fundamentalscale is the Planck scale, f ∼MP l, and the lower scale Λ is generated dynamically.) In this case, asdiscussed in Section II, the role of the PNGB inflaton could be played by the “model-independentaxion” (the imaginary part of the dilaton) [10].In Secton III, we provide a detailed analysis of the cosmological evolution of the PNGBinflaton field.

By and large, the numerical results therein confirm the analytic treatment of paperI. In addition, we also discuss in detail constraints on the mass scales arising from the requirementof sufficient reheating, the density fluctuation amplitude, and the requirement that inflation beprobable in the sense of initial (and final) conditions.

We also discuss the issue of initial spatialgradients in the inflaton field and how they may be damped out prior to inflation.In the standard lore of inflation, the adiabatic density fluctuations generated have a nearlyscale-invariant Harrison-Zeldovich spectrum. This general statement can be violated, and an ar-bitrary perturbation spectrum ‘designed’, but at the cost of fine-tuning parameters of the inflaton4

potential (or adjusting coupling constants in models with multiple scalar fields) [11]. One canimagine corrections to the PNGB potential which would allow this behavior, but in this paper weconsider only the simplest model given by Eq.

(1.1); in this case, we have no freedom to introducefeatures into the perturbation spectrum. Nevertheless, as discussed in section IV, in this modelthe fluctuations can deviate significantly from a scale-invariant spectrum: for f <∼3Mpl/4, theperturbation amplitude at horizon-crossing grows with mass scale M as (δρ/ρ)hor ∼M m2pl/48πf2.Thus, the primordial power spectrum for density fluctuations (at fixed time) is a power law,⟨|δρ(k)/ρ|2⟩∼kns, with spectral index ns ≃1 −(M 2pl/8πf 2).

The extra power on large scales(compared to the scale-invariant ns = 1 spectrum) can have important implications for large-scalestructure, of particular interest since the scale-invariant spectrum with cold dark matter (CDM)appears to have 8too little power on large scales. Other inflation models can also give rise to non-scale-invariant power law spectra.

Therefore, in section IV, we discuss tests of non-scale-invariantpower law initial spectra with adiabatic perturbations and CDM, including the galaxy angularcorrelation function inferred from deep photometric surveys, the CMBR anisotropy detected byCOBE, large-scale peculiar velocities, and the power spectrum inferred from redshift surveys ofIRAS galaxies.II. PARTICLE PHYSICS MODELSThere are a number of ways in which massive pseudo-Nambu-Goldstone bosons with therequisite mass scales discussed above may play a role in particle physics models.

In this section,we schematically outline only a few of them. The basic idea is to build a model with a globalsymmetry spontaneously broken at a large mass scale f ∼MP l, which gives rise to one or moremassless Nambu-Goldstone bosons.

There are then several ways to introduce explicit breaking of(some or all of) the global symmetry at the scale Λ ∼MGUT, resulting in potentials for the would-be Goldstone modes. Ideally, the lower scale emerges dynamically, so that no small parametersare introduced.The most familiar example of a pseudo-Nambu-Goldstone boson in nature is the pion.

Here,the global chiral symmetry is spontaneously broken by quark condensates at the QCD scale,⟨¯qq⟩≃Λ3QCD ≃(100 MeV)3, and explicitly broken by quark masses, mu ≃md ≃10 MeV. Inthe case of the pion, these two scales are close together (they differ by a factor of about ten),so the pion gains a mass comparable to the QCD scale, m2π ∼mq⟨¯qq⟩/f 2π ∼(100 MeV)2.

Bycontrast, in invisible axion models [7], the scales of spontaneous and of explicit symmetry breakingare separated by many orders of magnitude: the spontaneous symmetry breaking scale fP Q iselevated close to the GUT scale, while the explicit breaking scale is ∼ΛQCD.The resultinghierarchy of scales yields a very light axion, m2a ∼mq⟨¯qq⟩/f 2P Q; for example, ma ≃10−5 eV forfP Q ≃1012 GeV. For the PNGB inflaton, we will be interested in models with a relatively modesthierarchy between the spontaneous and explicit global symmetry breaking scales, Λ/f ∼10−4.Such a ratio of scales is intermediate between the case of the pion (Λ/f ∼0.1) and the invisibleQCD axion (Λ/f ∼10−13).A.

PNGBs from CondensatesIn this section, we illustrate how such an intermediate mass hierarchy can arise. We consideran action that contains coupled scalar and fermion fields and exhibits a chiral U(1) symmetry.Spontaneous breaking of the chiral symmetry takes place at energy scale f (for inflation, f ∼Mpl);massless Nambu-Goldstone bosons arise at this scale.

We illustrate an additional feature that maybe attractive although not necessary to our model: if the scalar field couples non-minimally togravity, it may dynamically generate Newton’s constant at this scale (induced gravity) [12]. Next,we discuss several ways in which the symmetry can be explicitly broken at a lower energy scale∼Λ (for inflation, Λ ∼10−4Mpl).

At this scale, the Nambu-Goldstone boson acquires a mass,5

in a manner similar to the axion or schizon [13] (although at higher mass scale). We focus onaxion-like scenarios, in which a gauge group becomes strong at the scale ∼Λ.

We briefly discusshow this may arise in technicolor models and then, in somewhat more detail in Sec.IIB, insuperstring models.1) Spontaneous Symmetry BreakingTaking our cue from the axion [14], we first describe a simple model which implements themechanism described above. Consider the fundamental action for a complex scalar field Φ andfermion ψ, coupled to gravity:S =Zd4x√−ggµν∂µΦ∗∂νΦ −V (Φ∗Φ) −ξΦ∗ΦR + i ¯ψγµ∂µψ −(h ¯ψLψRΦ + h.c.)(2.1)where R is the Ricci scalar, and ψ(R,L) are respectively right- and left-handed projections ofthe fermion field, ψ(R,L) = (1 ± γ5)ψ/2.

This action is invariant under the global chiral U(1)symmetry:ψL →eiα/2ψL ,ψR →e−iα/2ψR ,Φ →eiαΦ ,(2.2)analogous to the Peccei-Quinn symmetry in axion models.We assume the global symmetry is spontaneously broken at the energy scale f in the usualway, e.g., via a potential of the formV (|Φ|) = λΦ∗Φ −f 222,(2.3)where the scalar self-coupling λ can be of order unity. The resulting scalar field vacuum expecta-tion value (vev) is ⟨Φ⟩= feiφ/f/√2.In this model, spontaneous symmetry breaking dynamically generates Newton’s constant forEinstein gravity [12].At scales below f, the non-minimal coupling of the scalar field to thecurvature induces the canonical Einstein Lagrangian, ξ⟨Φ∗Φ⟩R = (ξf 2/2)R = R/16πG, if thecoupling ξ satisfiesξ = 18πM 2P lf 2.

(2.4)Since inflation requires f ∼MP l, the above relation holds for ξ of order unity, a natural valuefor this dimensionless coupling. We note that generation of the Planck scale in this way is not anecessary ingredient of the models discussed below: since inflation takes place in the φ direction,after Φ reaches its vev, we could simply replace the non-minimal coupling term in Eq.

(2.1) withthe usual Einstein Lagrangian. On the other hand, since the mass scale f must be comparableto MP l for successful inflation, it is natural and economical to tie it directly to the gravitationalscale.

Since the gravitational sector is canonical once the temperature of the universe drops belowthe scale f, we assume ordinary Einstein gravity from now on.Below the scale f, we can neglect the superheavy radial mode of Φ (mradial = λ1/2f ∼MP l)since it is so massive that it is frozen out. The remaining light degree of freedom is the angularvariable φ, the Goldstone boson of the spontaneously broken U(1) (one can think of this as theangle around the bottom of the Mexican hat described by eqn.

(2.3)). We thus study the effectivechiral Lagrangian for φ:Leff = 12∂µφ∂µφ + i ¯ψγµ∂µψ −(m0 ¯ψLψReiφ/f + h.c.) .

(2.5)6

Here the induced fermion mass m0 ≡hf/√2; for example, for values of the Yukawa coupling10−3 ≤h ≤1, the fermion mass is in the range MGUT ≤m0 ≤Mpl. The global symmetry is nowrealized in the Goldstone mode: Leff is invariant underψL →eiα/2ψL ,ψR →e−iα/2ψR ,φ →φ + αf .

(2.6)At this stage, φ is massless because we have not yet explicitly broken the chiral symmetry.2) Explicit Symmetry BreakingSeveral options exist for explicitly breaking the global symmetry and generating a PNGBpotential at a mass scale ∼Λ several orders of magnitude below the spontaneous symmetrybreaking scale f. In a class of Z2-symmetric models studied by Hill and Ross[13], one adds a barefermion mass term m1 ¯ψLψR to Leff, which presumably arises from another sector of the theory(just as quark masses in QCD are generated in the electroweak sector). The combination of termsinvolving m0 and m1 generates a 1-loop potential for φ of the form (1.1), with Λ2 ∼m0m1; asynopsis of these ‘schizon’ models is given in refs.

[13,15].For the rest of this discussion, we focus on the simplest mechanism for explicit symmetrybreaking, by analogy with the QCD axion: dynamical chiral symmetry breaking through stronglycoupled gauge fields. Suppose the gauge symmetry of the effective theory below the scale f ∼MP lis a product group, G1 × G2, where G1 is a standard grand unified group (e.g., E6 or SU(5))which spontaneously breaks down to the standard model at some scale MGUT .

In other words,G1 describes the physics of ordinary quarks and leptons (and their heavier brethren) while G2might describe a ‘hidden sector’. At the G1 unification scale, the G1 gauge coupling is small(perturbative unification).On the other hand, let G2 be an asymptotically free non-abeliangauge theory which becomes strongly interacting at a scale κ comparable to the GUT scale.

Inaddition, we assume that ψ transforms non-trivially under G2 (ψ carries G2-‘color’). Startingwith a perturbative G2 gauge coupling at the Planck scale, α2(MP l) = g22(MP l)/4π (which is,say, comparable to α1(MP l)), the scale κ emerges from the renormalization group,κ ≃MP l exp−8π2b0g22(MP l),(2.7)where the renormalization group constant b0 determines the lowest order term in the expansion ofthe β-function of G2, β2(g) = −b0g32/(4π)2−.... For example, for G2 = SU(N) and no light matterfields with G2 charge, then b0 = 3N; if there are N matter fields (one generation) with massesm < κ in the fundamental representation of G2, then b0 = 2N.

For reasonably large groups,and therefore large b0, the gauge coupling can run sufficiently fast to generate κ ∼MGUT . Asexamples, for α2(MP l) = 1/30 and G2 = SU(5) we find κ ∼3 × 1014 GeV if there are no light(m <∼MP l) fermions transforming under G2; on the other hand, with N light fermions, the samevalue of κ arises for the larger group G2 = SU(9).Since ψ is charged under G2, we expect chiral dynamics to induce a fermion condensate,⟨¯ψψ⟩∼κ3.

(We assume the condensate can be rotated to be real; the extra phase it involvesis irrelevant for our discussion).From eqn. (2.5), the condensate explicitly breaks the globalsymmetry, giving rise to a potential for the angular PNGB field φ,V (φ) = Re[m0⟨¯ψLψR⟩eiφ/f] = m0κ3 cos(φ/f) .

(2.8)This has the form of eqn. (1.1), with Λ4 = m0κ3 = hfκ3/√2.

For inflation, we require Λ ∼MGUT.Such an energy scale can arise in at least two ways: (i) m0 ∼κ ∼MGUT; this requires theYukawa coupling h ∼10−4, or (ii) m0 ∼MP l, h = O(1), and κ is slightly below the GUT scale,κ ∼10−1MGUT. We indicated above that the running of the coupling constant for group G27

may indeed provide such a value for κ. For this second choice of parameters, we do not need tointroduce any small coupling constants in the fundamental Lagrangian near the Planck scale: thesmall ratio Λ/f emerges dynamically and is “strongly natural”.Although this model may be cosmologically appealing, we do not want to propose a newstrongly interacting gauge sector in particle physics solely to generate an inflaton potential.

Hap-pily, there is well-founded particle physics motivation for an additional gauge group which becomesstrong at the GUT scale, and this idea has a distinguished history in the particle physics literature.One possibility is that G2 is a technicolor group, and that ψ carries both G1 and G2 charge. [In this case, φ can couple through a ψ −ψ −ψ-triangle diagram to ordinary particles (e.g.,gluons and photons); this may be advantageous in that it leads to reheating of the ‘ordinary’sector of quarks and leptons.

We thank S. Dimopoulos for making this point to us.] Then, onemust introduce a source for spontaneous breaking of the standard G1 GUT group.

Here, onemay contemplate two possibilities: If the ⟨¯ψψ⟩condensate is a G1 singlet (this may happen eventhough ψ carries G1 charge), then G1 must be broken by the usual Higgs mechanism or someequivalent. Alternatively, if the condensate ⟨¯ψψ⟩is G1-non-singlet, it can spontaneously breakG1 at a scale κ ∼MGUT, by analogy with technicolor models.

If it can be implemented, the latterchoice would be most economical: a single mechanism would give rise to both GUT symmetrybreaking and inflation, and the only fundamental scalar (Φ) in the theory has Planck mass. Thisvalue of the mass for a scalar is natural, and in principle no small parameters would need to beintroduced in the theory.B.

A Superstring Model: “Supernatural” InflationA second motivation for a gauge group which becomes strongly interacting at the GUTscale comes from superstring theory[16]. In these models, the gauge symmetry of the effectivesupergravity theory below the Planck scale is again a product group, G1 ×G2, where G1 = GGUTcontains the standard model and G2 describes the hidden sector; for example, in the originalheterotic string model, G1 × G2 = E8 × E′8.In the effective field theory arising from superstrings, an important role is played by thecomplex scalar field S. The real part of this field, ReS, is the dilaton; the imaginary part ImSis the ‘model-independent axion’.

In string theory, the value of the dilaton determines the stringcoupling constant gs through the relation[17] ⟨Re(S)⟩= 1/g2s. Since gs is related by factors ofO(1) to the gauge couplings ga(Mpl) of the effective field theory at the Planck scale (a labels thegauge group Ga), the dilaton expectation value determines gauge couplings as well.

In particular,a value for the dilaton in the range ⟨ReS⟩≡ReS0 ≃1.5 −2.5 yields a phenomenologically viableG1 gauge coupling at the GUT scale, α1(MGUT ) ∼1/45. If this theory is to have predictivepower and time-independent constants of nature, one would expect that the dilaton potentialV (ReS) has a minimum in this range.

In perturbation theory, the dilaton (and axion) potentialV (S) is protected by supersymmetry: if supersymmetry is unbroken at tree level, then V (S)vanishes to all finite orders in perturbation theory, leaving the gauge couplings indeterminate[18]. However, if the hidden sector G2 is an asymptotically free non-abelian group, it will becomestrongly interacting, leading to condensation of gauginos (fermion supersymmetric partners of thegauge bosons) at a scale ⟨λλ⟩∼M 3ple−8π2/b0g22(Mpl).

Through the relation between ReS and thestring coupling constant above, this corresponds to a non-perturbative potential for the dilatonof the form V (S) ∝e−cS. As a consequence, the imaginary part of the field, the axion partnerof the dilaton, obtains a potential of the form (1.1), V (ImS) ∝cos(ImS).

Our interest in thisscenario derives from the fact that the ‘model-independent’ axion could in principle play the roleof the inflaton in natural inflation [19].Although nonperturbative effects in the hidden sector can generate a potential for the dilaton,an exponentially falling potential clearly will not by itself stabilize the dilaton in the desired rangenoted above: instead ReS runs away to infinity, yielding a free string theory. Additional physics isneeded to help pin the dilaton at the appropriate minimum; we describe this further below.

Here,8

we mention that a second important role in particle physics of hidden sector gaugino condensationis that it might break supersymmetry (SUSY) [9]. If the condensate breaks supersymmetry inthe hidden sector at the scale ⟨λλ⟩∼MGUT ∼1014 GeV, then SUSY is broken in the observablesector at the scale MSUSY ∼M 3GUT /M 2pl ∼TeV.

SUSY breaking at this scale would protectthe small Higgs mass and alleviate the heirarchy problem. Thus, the factor e−1/g2 in the scaleof gaugino condensation might lead to a large gauge hierarchy.

(It is also possible that SUSYbreaking arises from some other mechanism. )The first attempts to implement these ideas in the G1 × G2 = E8 × E′8 heterotic stringtheory relied on the hidden E′8 sector becoming strongly interacting, and generating gauginocondensation, at a scale comparable to the GUT scale [9].As noted above, in this case thegaugino condensation-generated potential for ReS decays exponentially for large values of thedilaton field.

Attempts were made [9] to stabilize the dilaton by combining gaugino condensationwith a term arising from the expectation value of the antisymmetric tensor field ⟨Hµνλ⟩. However,quantization conditions on the vacuum expectation value of this field [20] require it to be of orderunity in Planck units, implying that the resulting potential V (S) only has a minimum where ReSis small (well below the desired range above), i.e., where gs is large.

As a result, the string theorywould be strongly coupled, and the whole framework of perturbative calculations in the effectivefield theory would be unreliable [21].Recently, this problem has been reconsidered by Krasnikov [22], Casas, et al. [23], andKaplunovsky, Dixon, Louis, and Peskin (hereafter KDLP) [24], in the context of string modelswhere the hidden sector G2 is itself a product of two or more gauge groups.

They found thatthe combined effect of gaugino condensates in multiple hidden groups can generate a dilatonpotential with a weak-coupling (small perturbative gs) minimum. In some cases, supersymmetryalso appears to be broken at the requisite scale (L. Dixon, private communication; Kaplunovsky,unpublished).

Here we will briefly study the axion potential generated in these multiple gauginocondensate models and explore its suitability for inflation.The effective Lagrangian for the dilaton S can be writtenLS =M 2P l8π(S + S∗)2 ∂µS∂µS∗−V (S, S∗)(2.9)where V is the effective potential generated by gaugino condensation. In string theory, the Planckscale is derived from the fundamental string tension α′ via M 2P l = 16π/g2sα′.

At tree level, thegauge coupling of group Ga is then ga = gs/√ka, where ka is the level of the Lie algebra of Ga(a small integer). Thus, at tree level, we have ⟨ReS⟩≡ReS0 = [4πkaαGUT (MP l)]−1; assumingG1 (GGUT ), which contains the standard model, is at level one (k1 = 1), the phenomenologicallyacceptable value of the GUT gauge coupling, αGUT(MP l) ≃1/20−1/30, requires that the dilatonVEV be in the range ReS0 = 1.5 −2.5, as noted above.

As KDLP show, this large a value of thedilaton expectation value can be obtained with a hidden group structure G2 = SU(N1)×SU(N2),provided that the expression [(k1/N1)−(k2/N2)]−1 is large; e.g., for their ‘best case’, k1 = k2 = 1and N1 = 9, N2 = 10 (see below). In what follows, for simplicity, we shall follow KDLP in takingG2 to be a product of two SU(N) groups.Following KDLP and ignoring gravitational and subleading 1/N corrections (that is, con-sidering a global SUSY model with large hidden gauge groups), the effective dilaton potentialisV (S, S∗) =π2M 2P l(S + S∗)2 Xaka⟨λλ⟩a2(2.10)where subscript a = 1, 2 now refers to the hidden gauge group G2a.

When the coupling constantof group G2a = SU(Na) becomes strong, the resulting gaugino condensate is⟨λλ⟩a = NavM 3reneiθa exp−24π2kaS + 32∆ab0,a(2.11)9

Here, the renormalization mass scale (at which the effective Lagrangian is defined) is taken to beM 2ren = ρ2α′;ρ2 = e1−γ6√3π= (0.216)2 ,(2.12)where γ is Euler’s constant and α′ = 8π(S + S∗)/M 2pl is the inverse string tension; v is an N-independent constant of order unity; θa = 2πm/Na, with m integer, is an arbitrary discrete phasereflecting the Na-fold degeneracy of the vacuum states of the theory; b0,a is the renormalizationgroup constant for group G2a; and ∆a is the threshold renormalization factor of order Na. ∆a,which in general can be a function of the moduli fields T i, enters the coupling constant to one-looporder via1g2a(µ) = 12ka(S + S∗) −b0,a16π2 log M 2renµ2+ ∆a16π2 .

(2.13)Decomposing the dilaton into real and imaginary parts, and assuming no charged fermions in thehidden groups (i.e., taking b0,a = 3Na), we haveV (S) = 5 × 10−9 v2M 4P lReSk21N 21 e−16π2k1ReS+∆1N1+ k22N 22 e−16π2k2ReS+∆2N2+ 2k1k2N1N2e−8π2k1N1 + k2N2ReS−12∆1N1 + ∆2N2cos8π2 k1N1−k2N2ImS + δθ(2.14)where δθ = θ2 −θ1.To study inflation, it is preferable to work with scalar fields that have canonical kinetic termsin the Lagrangian. From eqn.

(2.9), the kinetic Lagrangian for the real and imaginary componentsof the dilaton is not of this form, sinceLkin =M 2P l32π(ReS)2 (∂µReS∂µReS + ∂µImS∂µImS)(2.15)Thus the canonically normalized real component is taken to be [19] φR = −MP l ln(ReS)/√16π.In general, the real and imaginary parts of S are interdependent, and one should follow thecoupled evolution in the two-dimensional field space. For simplicity, to focus on the imaginarycomponent, the model-independent axion, we shall assume the real component reaches its VEV,⟨ReS⟩= ReS0, well before the imaginary part does; in a chaotic scenario in which the field isinitially randomly distributed, this will always be true in some regions of space.

(Note that, nearthe Planck time, S will drop out of thermal equilibrium and its potential will be dynamicallynegligible; under these conditions, we expect no special initial value for S to be preferred.) Inthat case, we can define the canonical axion field,φa = MP lImS/√16πReS0(2.16).However, from eqns.

(1.1) and (2.14), we have φa/f = 8π2ImS[(k1/N1) −(k2/N2)]. Combiningthese two expressions, we find the equivalent global symmetry breaking scalef =MP l8π2√16πReS0 k1N1−k2N2−1(2.17).As we will see in Sections III and IV, the phenomenologically acceptable range for f is f >∼0.3MP l.From eqn.

(2.14), the potential for the real part of the dilaton is minimized atReS0 =18π2 k1N1−k2N2−1 lnk1k2+ lnN1N2+ 12δ∆(2.18)10

whereδ∆= ∆2N2−∆1N1(2.19)is the difference in threshold renormalization factors. We thus findf = MP l√16πlnk1k2+ lnN1N2+ 12δ∆−1(2.20)In order to achieve an acceptably large value for ReS0, KDLP choose, e.g., N1 = 9, N2 = 10, withk1 = k2 = 1; larger values of Na are excluded because the size of the hidden sector is constrainedby the total Virasoro central charge available.

With this choice, to obtain ReS0 > 1.5 requiresδ∆>∼2.8, which implies f/MP l =<∼0.11 ≃1/√24π. If this upper limit is saturated, a sufficientlylong epoch of slow-rollover inflation can occur (see section III), but the reheat temperature isunacceptably low and the density perturbation spectrum has too much power on large scales.

Onthe other hand, for δ∆= 1, we would have f/MP l ≃0.36, which yields a viable inflationary modelwith an interesting fluctuation spectrum. However, as eqn.

(2.18) shows, this value of δ∆wouldrequire larger groups, e.g., N1, N2 = 16, 17, to achieve ReS0 > 1.5, and this violates the centralcharge limit (however, see comment below).For these models, we can read offthe effective scale Λ, defined in eqn. (1.1), from eqns.

(2.14)and (2.18); Λ is determined up to a constant of order unity (the factor v) by the values of ka, Na,and ∆a. For the SU(9) × SU(10) example, taking δ∆= 2.8, ∆1/N1 = −∆2/N2 = −1.4, whichcorresponds to ReS0 = 1.5 and f = 0.11MP l ≃MP l/√24π, we find Λ = 6 × 10−5v1/2MP l =8× 1014v1/2 GeV, in the right vicinity for generating an acceptable density fluctuation amplitude(even though, as noted above, this value of f leads to an unacceptable fluctuation spectrum–seesection IV).

This is a pleasing feature of these models: the same physics which sets the condensatescale to be of order MGUT (∼MP lexp[−8π2/g2b0]) fixes Λ to approximately the same scale.From our perspective, the interesting result here is that a string model designed to yield aphenomenologically plausible particle physics scenario, in particular a large gauge hierarchy andpossibly supersymmetry breaking near the weak scale, implies values for the PNGB parametersf and Λ for the model-independent axion which are quite close to those needed for successfulinflation. Furthermore, as suggested in [23,24], with the inclusion of charged matter fields in oneof the hidden groups, it is possible that the value of δ∆required to fix ReS0 could be reducedfrom ∼3 to ∼1, generating a sufficiently large value of f for inflation.We end this subsection with several caveats about the treatment given here.

First, as men-tioned above, we have reduced a two-dimensional problem to a one-dimensional one by assumingthe dilaton is already pegged to its expectation value during the evolution of the axion. Althoughthis will be accurate for some region of parameter space, and for chaotic initial conditions insome regions of the universe, in general one should treat the full two-dimensional problem.

Inparticular, the possibility of inflation in the dilaton direction deserves study (the potential for thecanonical dilaton contains terms of the form exp[−ae−bφR]). Second, due to 2-loop running ofthe gauge coupling, the prefactor of the exponential in eqn.

(2.11) actually contains an additionalfactor of S [24]. This gives rise to an overall multiplicative factor of SS∗on the right side ofeqn.

(2.14), modifying the dependence of the potential on the axion field from a pure cosine.This could have interesting consequences for the cosmological evolution of the model-independentaxion. Third, in this discussion we have assumed that the dominant non-perturbative effects instring theory arise at the level of the effective supergravity Lagrangian.

It has been suggested [25]that some inherently stringy non-perturbative effects at the Planck scale are only suppressed bya factor exp(−2π/g) as opposed to the field theory factor exp(−8π2/g2). If such stringy effectscontribute to the effective dilaton potential, they could substantially modify this effective fieldtheory analysis.C.

Alternatives11

In the preceding subsections, we have outlined two particle physics models which incorporatea PNGB with the requisite parameters for inflation. Clearly there are further possibilities [13,15].For example, one can imagine doing away with fundamental scalars altogether, and having thePNGB arise as an effective field.

One choice would be a composite PNGB built from a fermioncondensate, in analogy with composite axion models [14] and the pion.A second possibility,recently discussed by Ovrut and Thomas [26], builds on the existence of instantons in the theoryof an antisymmetric tensor field Bµν (recall that such a tensor field arises, e.g., in superstringtheory.) Defining the field strengthHµνλ = ∂µBµλ + ∂λBµν + ∂νBλµ ,(2.21)the action for this theory isS =16e2Zd4xHµνλHµνλ ,(2.22)with the resulting equation of motion ∂µHµνλ = 0.As Ovrut and Thomas note, the theory(2.22) has pointlike, singular instanton solutions, analogous to Dirac monopoles in electromag-netism; evaluating their contribution to the partition function, the resulting effective action canbe expressed in terms of an effective mean scalar field φ asSeff =Zd4x14e2∂µφ∂µφ −29π2e−(α/e4)1/3 α2e41/3 f 41 ± cosφf,(2.23)where α is a number, and f is a mass scale characterizing the instanton solutions.

Clearly thisis of the form (1.1) and, for f ∼MP l, (2.23) is another potential candidate model for naturalinflation. In a variant of these models, the tensor field can be coupled to a fundamental realscalar field u with the symmetry-breaking potential V (u) = (λ/4!

)(u2 −6m2/λ)2. This also leadsto a potential of the form (1.1) for the associated scalar mean field theory; for f ∼m ∼MP land λ ∼10−4, one finds [26] Λ ∼1016 GeV, as desired for successful inflation.

In both of thesemodels, as in the string model of the previous subsection, the effective scale Λ is small comparedto f due to the exponential (instanton) suppression factor. This is the origin of the hierarchyrequired for the generation of acceptably small density fluctuations in inflation.

The advantageof these models is that this hierarchy does not need to be put in by hand.D. Other IssuesBefore leaving this survey of model-building, we note recent work drawing attention to thefact that global symmetries may be explicitly broken by quantum gravity effects [27,28] (e.g.,wormholes and black holes).If such effects are characterized by the Planck scale, they mayinduce non-renormalizable higher-dimension terms in the low-energy effective Lagrangian for Φ(see eqn.2.3), of the formVeff(Φ) = gmn|Φ|2mΦnM 2m+n−4P l.(2.24)The coefficients gmn introduced here should not be confused with the gauge and string couplingsdiscussed above.

Terms with n ̸= 0 explicitly break the global U(1) symmetry of eqn.(2.2). Takinggmn = |gmn|exp(iδmn), the induced PNGB potential is a sum of terms of the formVeff(φ) = |gmn| fMP l2m+nM 4P l cosnφf + δmn.

(2.25)Therefore, for n ̸= 0, the effective explicit symmetry breaking scale isΛeff = |gmn|1/4MP l fMP l 2m+n4,(2.26)12

where non-renormalizable terms have dimension 2m + n ≥5.Since PNGB inflation requiresf >∼0.3MP l and Λ ∼MGUT , the coefficients gmn of these terms must be relatively small; forexample, for the dimension 5 term, g <∼10−14 is required. (The upper limit on gmn is relaxed forhigher dimension terms.

)Naively, this effect appears to lead us back to the same difficulties this inflation model wasmeant to solve, namely, a small dimensionless constant of order 10−14 appearing in the Lagrangian.However, it is worth making several remarks about this problem. First, a caveat: in the discussionabove (and in refs.

(28)), it was implicitly assumed that the coefficients gmn are ‘naturally’ of orderunity. However, in the absence of a solvable quantum theory of gravity, these coefficients cannot bereliably calculated.

In model wormhole calculations, one must introduce a cutoffscale µ ≪MP l,in which case such effective operators are proportional to the tunneling factor ∼exp(−M 2P l/µ2).Thus, in the regime in which one can calculate, the coefficients gmn are highly suppressed; theassumption that they are not at all suppressed depends on an uncertain extrapolation of thecutoffscale to the Planck scale. In addition, there may be other effects which enter to suppressthese terms.

In particular, in the axion model studied in more depth in ref. [27], wormhole effectsare effectively cut offat the symmetry breaking scale f, leading to an exponential suppression∼exp(−MP l/f) in the wormhole induced axion potential.

Second, even supposing such termsare in principle unsuppressed (all gmn of order unity), there are ways in which they could beevaded. For example, for a large gauge group (as contemplated above), a global symmetry mayautomatically be present, due to the gauge symmetry and field content of the theory, preventingterms up to some relatively large value of 2m + n; in the present case, this would require thatall terms up to 2m + n ∼25 be forbidden.

Alternatively, if the field φ is an effective field whicharises below the Planck scale, as suggested above, such explicit symmetry breaking terms canbe forbidden by a local symmetry of the underlying theory, as in the superstring example ofthe previous subsection. Alternatively, as in the antisymmetric tensor model of section C above,the φ field may be unrelated to a global symmetry.

Therefore, while the arguments of [28] areprovocative, there are many examples of particle physics models which give rise to potentials ofthe form (1.1), with the requisite mass scales for inflation, which evade them.III. COSMIC EVOLUTION OF THE INFLATON FIELDWith these models as theoretical inspiration, we turn now to the cosmological dynamicsof an effective scalar field theory with a potential of the form (1.1) below the scale f.Forexample, in the model of Sec.

II.A, f is the global spontaneous symmetry breaking scale, andφ describes the phase degree of freedom around the bottom of the Mexican hat potential (2.3);in other models, however, the picture may differ. To successfully solve the cosmological puzzlesof the standard cosmology, an inflationary model must satisfy a variety of constraints, includingsufficient inflation (greater than 60 e-folds of accelerated expansion) for a reasonable range ofinitial conditions; sufficiently high reheat temperature to generate a baryon asymmetry afterinflation; and an acceptable amplitude and spectrum of density fluctuations.

In this section weexplore these constraints analytically and numerically for potentials of the form (1.1).The φ interaction cross-sections with other fields are generally of order σ ∼1/f 2, so itsinteraction rate is of order τ −1 ∼T 3/f 2. Comparing this with the expansion rate H ∼T 2/MP l,we see that the scalar inflaton field thermally decouples at a temperature T ∼f 2/MP l ∼f.We therefore assume φ is initially laid down at random between 0 and 2πf in different causallyconnected regions.

(This is the simplest but by no means only possible initial condition.) Withineach Hubble volume, (i.e., ignoring spatial gradients–see below) the evolution of the field is thendescribed by the classical equation of motion for a homogeneous field φ(t),¨φ + 3H ˙φ + Γ ˙φ + V ′(φ) = 0 ,(3.1)13

where Γ is the decay width of the inflaton, and the expansion rate H = ˙a/a is determined by theEinstein equation,H2 =8π3M 2P lV (φ) + 12˙φ2. (3.2)For completeness, it is also useful to have the second order Friedmann equation,¨aa = −8π3M 2P lh˙φ2 −V (φ)i.

(3.3)In Eqns. (3.2-3), we have assumed that the scalar field dominates the stress energy of the universe;this will hold starting near the onset of inflation.In the temperature range Λ <∼T <∼f, the potential V (φ) is dynamically irrelevant, becausethe forcing term V ′(φ) in Eq.

(3.1) is negligible compared to the Hubble-damping term. (Inaddition, for axion-like models in which V (φ) is generated by non-perturbative gauge effects,Λ →0 as T/Λ →∞due to the high-temperature suppression of instantons [14].

)Thus, inthis temperature range, aside from the smoothing of spatial gradients in φ (see below), the fielddoes not evolve.Finally, for T <∼Λ, in regions of the universe with φ initially near the topof the potential, the field starts to roll slowly down the hill toward the minimum.In thoseregions, the energy density of the universe is quickly dominated by the vacuum contribution(V (φ) ≃2Λ4 >∼ρrad ∼T 4), and the universe expands exponentially. Since the initial conditionsfor φ are random, our model is closest in spirit to the chaotic inflationary scenario [29].Insucceeding subsections, we study this evolution in more detail.A.

Standard Slow-Rollover AnalysisIn this subsection, we recapitulate the analytic treatment of PNGB inflation given in PaperI. A sufficient, but not necessary, condition for inflation is that the field be slowly rolling (SR)in its potential.

Therefore, by analyzing the conditions for, and number of e-foldings of, inflationin the SR regime, we should be at worst underestimating the true number of inflation e-folds.The field is said to be slowly rolling when its motion is overdamped, i.e.,¨φ ≪3H ˙φ, so that the¨φ term can be dropped in Eq. (3.1) (n.b., we assume Γ ≪H during this phase).

It is easy toshow that in general this SR condition is a sufficient condition for inflation. First, from the scalarequation of motion (3.1), the defining SR condition implies that ˙φ2 ≪2V (φ).

On the other hand,the universe is inflating if the Robertson-Walker scale factor a(t) is accelerating, ¨a > 0; from Eqn. (3.3), this requires ˙φ2 < V .

Thus, if the SR condition is well satisfied, we are guaranteed to bein an inflationary epoch. The converse is not necessarily true: inflation can occur even when thefield is not slowly rolling.

However, we will see in subsequent sections that, for this potential, if fis larger than about MP l/√24π, the SR epoch is roughly coincident with the inflationary epoch.Hereon, for the purposes of numerical estimates, we shall assume inflation begins at a fieldvalue 0 < φ1/f < π; since the potential is symmetric about its minimum, we could just as easilyconsider the case π < φ1/f < 2π. For the potential (1.1), the SR condition implies that twoconditions are satisfied:|V ′′(φ)| <∼9H2 , i.e.,s2 |cos(φ/f)|1 + cos(φ/f)<∼√48πfMP l(3.4a)andV ′(φ)MP lV (φ) <∼√48π , i.e.,sin(φ/f)1 + cos(φ/f)<∼√48πfMP l.(3.4b)From Eqns.

(3.4), the existence of a broad SR regime requires f ≥MP l√48π (required belowfor other reasons). The SR epoch ends when φ reaches a value φ2, at which one of the inequalities14

(3.4) is violated. In Fig.1, we show φ2/f as a function of f/MP l; as f grows, φ2/f approachesthe potential minimum at π.

For example, for f = MP l, φ2/f = 2.98, while for f = MP l/√24π,φ2/f = 1.9. For f ∼> 0.3MP l which, as we shall see below, is the mass range of greatest interest,the two inequalities (3.4a) and (3.4b) give very similar estimates for φ2.

For simplicity, we canthen use (3.4b) to obtainφ2f ≃2 arctan √48πfMP l! (f ∼> 0.3MP l) .

(3.5)Once φ grows beyond φ2, the field evolution is more appropriately described in terms of oscillationsabout the potential minimum, and reheating takes place, as described below. We note that theexpansion of the universe (the 3H ˙φ term in Eq.

3.1) acts as a strong enough source of frictionthat the field is not able to roll through the minimum at πf and back up the other side sufficientlyfar to have any further inflationary period.To solve the standard cosmological puzzles, we demand that the scale factor of the universeinflates by at least 60 e-foldings during the SR regime,Ne(φ1, φ2, f) ≡ln(a2/a1) =Z t2t1Hdt = −8πMP l2Z φ2φ1V (φ)V ′(φ)dφ= 16πf 2MP l2 lnsin(φ2/2f)sin(φ1/2f)≥60 . (3.6)Using Eqns.

(3.4,5) to determine φ2 as a function of f, the constraint (3.6) determines themaximum initial value (φmax1) of φ1 consistent with sufficient inflation, Ne(φmax1, φ2, f) = 60. Forf ∼> 0.3MP l,sin(φmax1/2f) ≃h1 + MP l248πf 2i−1/2exp−15MP l24πf 2.

(3.7)The fraction of the universe with φ1 ∈[0, φmax1] will inflate sufficiently. If we assume that φ1is randomly distributed between 0 and πf from one Hubble volume to another, the a prioriprobability of being in such a region is P = φmax1/πf.

For example, for f = 3MP l, MP l, Mpl/2,and MP l/√24π, the probability P = 0.7, 0.2, 3 × 10−3, and 3 × 10−41. The initial fraction ofthe universe that inflates sufficiently drops precipitously with decreasing f, but is large for fnear MP l. This is shown in Fig.

1, which displays log(φmax1/f) = 0.5 + logP and φ2/f. Theseconsiderations show that for values of f sufficiently near MP l, sufficient inflation takes place fora broad range of initial values of the field φ.

We note that these constraints do not determine thesecond mass scale Λ.According to some inflationists, the discussion above of the probability of sufficient inflationis overly conservative, since it did not take into account the extra relative growth of the regionsof the universe that inflate. After inflation, those initial Hubble volumes of the universe that didinflate end up occupying a much larger volume than those that did not.

Hence, below we willalso compute the a posteriori probability of inflation, that is, the fraction of the final volume ofthe universe that inflated.B. Numerical Evolution of the Scalar FieldIn this section, we expand upon the results of the preceding subsection by numerically inte-grating the equations of motion.

This yields a more accurate estimate of the time (or field value)when inflation ends and the amount of inflation that takes place, as a function of the mass scalef and the initial value of the field φ1.15

First we rewrite Eq. (3.1) in terms of more useful variables.

As a dimensionless time variable,we use the number of e-foldings of the scale factor,dn = Hdt,(3.8)and hence d/dn = H−1d/dt. We also define the dimensionless field value, field ‘velocity’, andmass ratioy ≡φ/f;v = dy/dn;γ ≡3M 2P l/8πf 2 .

(3.9)Then we can write Eqs. (3.1) and (3.2) asdvdn =hγ tan(y/2) −3vi h1 −v2/2γi−ωvh1 −v2/2γi3/2γ1/21 + cos y−1/2,(3.10)where ω = Γf/Λ2 contains the effects of dissipation.

For the purpose of numerically calculatingthe evolution of the field, we will assume as in the previous section that this dissipation term isnegligible. In this approximation, Eq.

(3.10) depends only on the shape of the potential and onγ, i.e., on the ratio f/MP l, and not explicitly on Λ.In order to solve Eq. (3.10), we must specify two initial conditions: the initial values of thefield and its time derivative.

We allow the initial field value y1 = φ1/f to range over the interval0 to π, and take the initial velocity to be v1 ≡˙φ1/Hf = 0. The assumption of zero initial velocityis the one usually made in discussions of inflationary models.

However, in the course of smoothingout gradients or due to randomness in the initial conditions, we expect the field to acquire aninitial ‘Kibble’ [30] velocity at the temperature T ∼Λ such that its kinetic energy is comparableto the potential energy ∼Λ4. Naively, this velocity effect could delay or even prevent the onset ofinflation.

This problem has been studied previously in the context of new and chaotic inflationarymodels and has been shown to be potentially problematic for new inflation [31]. Initial velocitiesin the context of the present model have been studied numerically by Knox and Olinto [32].

Theyfind that, due to the periodic nature of the potential, the effect of initial velocities is merely toshift, but not change the size of, the phase space of initial field values which lead to at least60 e-folds of inflation. That is, as ˙φ1 is increased from zero, the value of φ1 at which inflationbegins is shifted, but the fraction of initial field space which inflates is approximately invariant.Therefore, for models of the form (1.1), we lose no generality by assuming ˙φ1 = 0.

Given theseinitial conditions, we solve the equation of motion (3.10) numerically.The resulting solutiony(n, y1) provides the value of the φ-field after n e-foldings of the scale factor.As noted above, in an inflationary phase the scale factor accelerates in time, ¨a > 0. The endof the inflationary epoch thus occurs at the transition from ¨a > 0 to ¨a < 0.

We denote the fieldvalue at the end of inflation by φend. We find that the value of φend is virtually insensitive towhere the field started rolling on the potential, φ1.

In Paper I and Sect. IIIA, we used φ2, thevalue of the field at the end of the SR epoch, as an estimate of the end of inflation.

Comparing thecorrect value φend with the approximate value φ2, we find that the error is only 1% for f ≃MP l,10% for f = 0.1MP l, and rapidly gets large for smaller values of f. In particular, no slow rolloverregime exists for f ≤MP l/√48π, and yet for small enough values of φ1, significant inflation canstill occur. In practice, however, the small difference between φ2 and the exact result φend shownin Figure 1 is irrelevant, since, as we show below, values of f smaller than 0.3MP l are excludedfor other reasons.For a given initial value of the field φ1 (or y1), the solution to Eq.

(3.10) tells us the totalnumber of inflation e-foldings of the scale factor, N(φ1) (where the end of inflation is defined bythe condition ¨R = 0). Figure 2 shows the number N(φ1, f) of e-foldings as a function of the initialvalue of the field φ1 for different choices of the mass scale f. One can see that, for φ1/f < 1, thedependence is almost exactly logarithmic,N(φ1) = A −B ln(φ1/f).

(3.11)16

In the limit of small φ1/f, the analytic SR estimate of Eq. (3.6) implies this same functionaldependence and provides values for the constants A and B; in particular, Bsr = 16πf 2/M 2P l.The numerical values obtained for A and B by solving (3.10) are virtually the same as the SRestimates if f is near MP l and start to differ as f decreases.

From Fig. 2, one can read offvaluesfor ymax1= φmax1/f, the largest initial value of the field that can give rise to N(φmax1) = 60 e-foldings of inflation.

Again, the numerical results for φmax1are nearly identical to the SR estimates(shown in Fig. 1) for values of f near MP l; they differ by ∼10% for f = MP l/10, and deviatesignificantly as f approaches MP l/√24π from above.1) Analytic Solution: Small-angle approximationThe simple logarithmic behavior of the number of e-foldings N(φ1) indicates that an analyticapproximation can be found, one which differs from the SR approximation and which is moreuseful for smaller values of f/MP l. In this region of parameter space, the conditionsy ≪1andv ≪1,(3.12)always apply during the inflationary epoch, and the equation of motion (3.10) can be approximatedbydvdn = γ2 y −3v ,(3.13)where we have made the “small angle” (SA) approximation for the trigonometric functions andhave neglected higher order terms in y and v. Eqn.

(3.13) has solutions of the formy(n) = y1 eαn ,(3.14a)that is,φ = φ1eαRHdt ,(3.14b)where the constant α is given byα = 129 + 2γ1/2 −32 = 32 1 + M 2P l12πf 21/2−1!. (3.15)Thus, the total number N of e-foldings can be written asN = 1α ln(φ2/f) −1α ln(φ1/f) ,(3.16)where φ2 is the value of the field at the end of inflation.

Eqn. (3.16) provides us with an analyticsolution of the same form as Eqn.

(3.11); note that, here, the constant B of Eqn. (3.11) is givenby Bsa = 1/α, which differs in general from the value Bsr predicted by the slow rollover approxi-mation.

However, for large values of f, such that f ≫MP l/√12π, the two approximations agree,Bsa →Bsr = 16πf 2/M 2P l. Comparison with Fig. 2 shows that, unlike the SR approximation,the small angle approximation is also in excellent agreement with the numerical results for smallvalues of f.C.

Constraints: Density Fluctuations, Reheating, Sufficient InflationHaving studied the evolution of the homogeneous mode φ(t) of the scalar field and delineatedthe regions of initial field space for sufficient inflation, we now address other constraints the modelmust satisfy for successful inflation, including density fluctuations and reheating. In particular,17

these phenomena place tighter constraints on the range of allowed scales f and also limit thesecond mass scale Λ. Since, in Sec.

IIIB, we showed that the SR approximation is accurate forthe parameter range of interest, we shall rely on it throughout this discussion.1) Density Fluctuation AmplitudeQuantum fluctuations of the inflaton field as it rolls down its potential generate adiabaticdensity perturbations that may lay the groundwork for large-scale structure and leave their imprinton the microwave background anisotropy [33-36]. In this context, a convenient measure of theperturbation amplitude is given by the gauge-invariant variable ζ, first studied in ref.[36].

Wefollow ref. [11] in defining the power in ζ,P 1/2ζ(k) = 152δρρHOR= 32πH2˙φ.

(3.17)Here, (δρ/ρ)HOR denotes the perturbation amplitude (in uniform Hubble constant gauge) whena given wavelength enters the Hubble radius in the radiation- or matter-dominated era, and thelast expression is to be evaluated when the same comoving wavelength crosses outside the Hubbleradius during inflation. For scale-invariant perturbations, the amplitude at Hubble-radius-crossingis independent of perturbation wavelength.To normalize the amplitude of the perturbation spectrum, we assume that the underlyingdensity perturbations are traced by the galaxy number density fluctuations up to an overall biasfactor bg, that is, P 1/2ρ= P 1/2gal /bg.

As inferred from redshift surveys, the variance σ2gal in galaxycounts in spheres of radius 8 h−1 Mpc is about unity (where the Hubble parameter H0 = 100hkm/sec/Mpc). For a scale-invariant spectrum of primordial fluctuations with cold dark matter(CDM), this implies [11]P 1/2ζ≃10−4bg.

(3.18)As we shall see below, we will be interested in cases where the primordial spectrum may deviatesignificantly from scale-invariant, and these cases will be discussed in detail in Sec. IV; here,we will use the scale-invariant normalization to get an approximate fix on the scale Λ.

(Forvalues of f close to MP l, this approximation is very accurate. )For the scale-invariant CDMmodel, the recent COBE observation of the microwave background anisotropy [37] roughly implies7.7 × 10−5 < P 1/2ζ< 1.4 × 10−4, or 0.7 < bg < 1.3.Using the analytic estimates of Sec.

IIIA, the largest amplitude perturbations on observablescales are produced 60 e-foldings before the end of inflation, where φ = φmax1, and have amplitudeP 1/2ζ≃Λ2fM 3P l92π8π33/2 [1 + cos(φmax1/f)]3/2sin(φmax1/f). (3.19)Applying the COBE constraint above to Eq.

(3.19), we find, e.g.,Λ = 8.8 × 1015 −1.2 × 1016 GeV for f = MP l(3.20a)Λ = 1.4 × 1015 −2 × 1015 GeV for f = MP l/2 . (3.20b)Thus, to generate the fluctuations responsible for large-scale structure, Λ should be comparableto the GUT scale, and the inflaton mass mφ = Λ2/f ∼1011 −1013 GeV.We can use this to determine Λ as a function of f, shown in Fig.1.

For an analytic estimate,consider the case f <∼(3/4)MP l, for which it is a good approximation to take φmax1/πf ≪1. Asa result, in Eqn.

(3.19), we have approximatelyP 1/2ζ≈1.4Λ2fM 3P l16π33/2 fφmax1. (3.21)18

Now the last term in this expression is obtained by using Eqn. (3.6) with N(φmax1, φ2, f) = 60:φmax1f≃2 sinφ22fexp"−15M 2pl4πf 2#.

(3.22)Substituting (3.22) on the RHS of (3.21) and using Eqn. (3.18) we find the value of Λ(f) in termsof the bias parameter:Λ(f) = 1.7 × 1016b1/2gGeVMplfsinφ22f1/2exp −15M 2pl8πf 2!.

(3.23)Here, the quantity sin(φ2/2f) is determined by the slow-rollover conditions, Eqns. (3.4-3.5) andis generally of order unity.

The dominant factor in (3.23) is the exponential dependence on f 2,which is responsible for the rapid downturn as f begins to drop significantly below MP l in thecurve for Λ(f) in Fig.1. For completeness, we note that the value in Eqn.

(3.23) is strictly only anupper bound on the scale Λ, since the perturbations responsible for large-scale structure could beformed by some other (non-inflationary) mechanism.2) Density Fluctuation SpectrumUsing the approximations above, we can investigate the wavelength dependence of the per-turbation amplitude at Hubble-radius-crossing and in particular study how it deviates from thescale-invariant spectrum usually associated with inflation. Here we give a quick derivation of thespectrum, and defer a fuller discussion to Sec.

IV.Let k denote the comoving wavenumber of a fluctuation. The comoving lengthscale of thefluctuation, k−1, crosses outside the comoving Hubble radius [Ha]−1 during inflation at the timewhen the rolling scalar field has the value φk.

This occurs NI(k) ≡N(φk, φ2, f) e-folds beforethe end of inflation, where N(φk, φ2, f) is given by Eqn. (3.6) with φ1 replaced by φk.Thecorresponding comoving lengthscale (expressed in current units) isk−1 ≃(3000h−1Mpc)exp(NI(k) −60) .

(3.24)For scales of physical interest for large-scale structure, NI(k) >∼50; for f <∼(3/4)MP l, these scalessatisfy φk/f ≪1. In this limit, comparing two different field values φk1 and φk2, from Eqn.

(3.6)we haveφk2 ≃φk1exp−∆NIM 2P l16πf 2,(3.25)where ∆NI = NI(k2) −NI(k1). Thus, using Eqns.

(3.19) and (3.21), we can compare the pertur-bation amplitude at the two field values,(P 1/2ζ)k1(P 1/2ζ)k2≃φk2φk1≃exp−∆NIM 2P l16πf 2. (3.26)Now, from Eqn.

(3.24), we have the relation ∆NI = ln(k1/k2) (more precisely, ∆NI = ln(k1H2/k2H1),and we approximate H1 ≃H2); substituting this relation into (3.26), we find how the perturbationamplitude at Hubble radius crossing scales with comoving wavelength,δρρHOR,k∼(P 1/2ζ)k ∼k−M2P l/16πf2(3.27)19

By comparison, for a scale-invariant spectrum, the Hubble radius amplitude would be independentof the perturbation lengthscale k−1; the positive exponent in Eqn. (3.27) indicates that the PNGBmodels have more relative power on large scales than scale-invariant fluctuations.It is useful to transcribe this result in terms of the power spectrum of the primordial per-turbations at fixed time (rather than at Hubble-radius crossing).

Defining the Fourier transformδk of the density field, from Eqn. (3.27) the power spectrum is a power law in the wavenumber k,⟨|δk|2⟩∼kns, where the index ns is given byns = 1 −M 2P l8πf 2(f ∼< 3MP l/4) .

(3.28)For comparison, the scale-invariant Harrison-Zel’dovich-Peebles-Yu spectrum corresponds to ns =1. For values of f close to MP l, the spectrum is close to scale-invariant, as expected; however, asf decreases, the spectrum deviates significantly from scale-invariance–e.g., for f = MP l/√8π =0.2MP l, the perturbations have a white noise spectrum, ns = 0.In sec.IV, we explore theimplications of models with power law primordial spectra in depth.3) Quantum FluctuationsFor the semi-classical treatment of the scalar field used so far to be valid, the initial valueof the field should be larger than the characteristic amplitude of quantum fluctuations in φ, i.e.,φ1 ≥∆φ = H/2π.

In particular, requiring that quantum fluctuations do not reduce the numberof inflation e-folds below 60 implies that the condition φmax1> H/2π must be satisfied. Using theSR approximation and Eqn.

(3.22), we findH/2πφmax1≃13π P 1/2ζMP lf2= 10−5bgMP lf2. (3.29)Since this ratio is very small over the parameter range of interest, this constraint places nosignificant restrictions on the model.

For example, this constraint requires that φ1/f > 10−7 forf = MP l and φ1/f > 6 × 10−9 for f = MP l/2, while the corresponding values of φmax1/f are 0.63and 9.4 × 10−3. Even if φ1 is at some stage smaller than this constraint, one would expect thatquantum fluctuations might eventually bring the field into the semiclassical regime, so inflationwould begin, if it was sufficiently spatially coherent.4) Probability of Sufficient InflationArmed with the numerical and analytic results above, we now calculate the a posterioriprobability of sufficient inflation.

We consider the universe at the end of inflation, and calculatethe fraction P of the volume of the universe at that time which had inflated by at least 60e-foldings:P = 1 −R πfφmax1dφ1exp[3N(φ1)]R πfH/2π dφ1exp[3N(φ1)]. (3.30)Here, the lower limit of integration in the denominator is the limit of validity of the semiclassicaltreatment of the scalar field; the initial value of φ must exceed its quantum fluctuations, φ1 ≥∆φ = H/2π.We will use the form for N(φ1) given by Eq.

(3.11) to evaluate the integralsappearing in Eq. (3.30).

As shown previously, this approximate form for N(φ1) is only validfor φ1/f < 1. However, we will assume that it holds over the entire range of integration; in theAppendix, we show that the resulting errors are small.

Our basic result is that the a posterioriprobability for inflation is essentially unity for f larger than the critical value fc ≃0.06MP l. As20

f drops below this value, the probability given by Eq. (3.30) rapidly approaches 0.

To illustratethis result, we evaluate the integrals in (3.30); both are of the formI =Z πǫdy1e3Ae−3B ln y1=e3A3B −1n1ǫ3B−1 − 1π3B−1o,(3.31)where ǫ is a small number, ǫ = H/2πf or ymax1, and B is the f-dependent coefficient appearingin Eqn.(3.11). If 3B > 1, the integral I is dominated by the lower end of the range of integrationand only the first term in Eq.

(3.31) is significant. In this case, the probability P is given byP = 1 −H2πφmax13B−1,(3.32)where 3B −1 is positive.

The probability of sufficient inflation is close to unity as long as theratio in brackets H/2πφmax1is small; however, this is guaranteed by Eqn.(3.29). Combining Eqs.

(3.29) and (3.32) yields the probability:P ≥1 −h10−5bgMP lf2i3B−1. (3.33)This expression is valid provided that 3B −1 is positive and not extremely close to 0.As the value of 3B decreases toward unity, the probability P decreases and the approximationleading to (3.32) begins to break down.

As a reference point, consider the special case 3B = 1; thenthe integral I = e3A ln(π/ǫ) and the probability P ≈0.05. As B decreases further, the integralin Eq.

(3.31) obtains most of its contribution from the upper end of the range of integrationand hence both integrals appearing in Eq. (3.30) have nearly the same value.

As a result, theprobability P rapidly approaches 0.To summarize, we find that the probability P of sufficient inflation depends primarily onthe value of the coefficient B appearing in Eq. (3.11), which in turn determines the number ofe-foldings of the universe as a function of the initial value φ1 of the field.

For B > 1/3, theprobability P is nearly unity; for B < 1/3, the probability P quickly approaches 0. In the SRapproximation, B ≈16πf 2/M 2P l, which would imply a critical value f src= f(B = 1/3) = 1/√48π.On the other hand, the numerical calculations yield the critical value of the mass scale fc = 0.058.This discrepancy is traced to the fact that the SR approximation is invalid for such small valuesof f. In this case, the “small angle” approximation discussed in Sec.

III.B is more appropriate;using Eqn. (3.15), we can analytically determine the critical value of f for which Bsa ≡1/α = 1/3,f sac= MP l√96π.

(3.34)This is in excellent agreement with the value found numerically.5) ReheatingAt the end of the slow-rolling regime, the field φ oscillates about the minimum of the potential,and gives rise to particle and entropy production. The decay of φ into fermions and gauge bosonsreheats the universe to a temperature TRH = (45/4π3g∗)1/4√ΓMP l, where g∗is the number ofrelativistic degrees of freedom.On dimensional grounds, the decay rate is Γ ≃g2mφ3/f 2 =g2Λ6/f 5, where g is an effective coupling constant.

(For example, in the axion model [6,7],21

g ∝αEM for two-photon decay, and g2 ∝(mψ/mφ)2 for decays to light fermions ψ.) Thus, thereheat temperature isTRH =454π3g∗1/4 gΛ3f 2MP lf1/2(3.35)For example, for f = MP l, using (3.20a) for Λ, and taking g∗= 103, we find TRH ≃108g GeV,too low for conventional GUT baryogenesis, but high enough if baryogenesis takes place throughsphaleron-mediated processes at the electroweak scale.

Alternatively, the baryon asymmetry canbe produced directly during reheating through baryon-violating decays of φ or its decay products.The resulting baryon-to-entropy ratio is nB/s ≃ǫTRH/mφ ∼ǫgΛ/f ∼10−4ǫg, where ǫ is theCP-violating parameter; provided ǫg >∼10−6, the observed asymmetry can be generated in thisway.We saw above that the amplitude of density perturbations produced during inflation yieldsa bound on the scale Λ as a function of the fundamental scale f, eqn.(3.23). We can use this toexpress TRH as a function of f (which depends only weakly on g and g∗); requiring that this besufficiently high for some form of baryogenesis leads to an important lower bound on the scale f,which as we shall see below, is more restrictive than the a posteriori bound above and comparablyrestrictive with the microwave anisotropy bound on the perturbation spectrum to be discussed inSec.

IV. Since we will be interested in a lower bound on f, we consider the case f ≤(3/4)MP l sothat eqn.

(3.23) applies. Substituting (3.23) into (3.35), we find the reheat temperatureTRH = 1010 GeVb3/2gg100g∗1/4 MP lf4sin3/2φ22fexp"−45M 2pl8πf 2#.

(3.36)The important point here is that the reheat temperature drops exponentially as f drops well belowMP l. For baryogenesis to take place after inflation, at a minimum we should require TRH > 100GeV, the electroweak scale. From eqn.

(3.36), this leads to the lower boundfMP l≥0.28 . (3.37)(Here, we have set g = 1 and g∗= 100, but this limit depends only logarithmically on g andg∗.) In terms of the density perturbation spectrum given in Eqn.

(3.28), if inflation produces thedominant fluctuations on all scales, then this reheating constraint implies ns ≥0.5.One additional point concerning reheating in these models deserves mention. In the stringmodels of §II, the axion couples predominantly to the hidden sector; in such inflation models, onemight then worry that reheating would take place more efficiently in the hidden as opposed tothe ordinary sector.

(This would not be a concern in models without a hidden sector, such asthose patterned after technicolor.) In practice, this is not an insurmountable obstacle for thesemodels, because gravitational interactions lead to an effective coupling between the hidden sectorinflaton and the ordinary sector particles.

Furthermore, for f ∼MP l, the gravitationally induceddecay rate to ordinary particles, Γ ∼m3φ/M 2P l, is comparable to the axion’s decay rate to thehidden sector. Thus, we would expect the two sectors to reheat to comparable temperatures.

Itis then easy to imagine a subsequent entropy-producing ordinary particle decay which heats theordinary sector relative to the hidden sector, so that the contribution of the hidden sector to thetotal energy density at the time of big bang nucleosynthesis is negligible.6) Initial Spatial GradientsIn the previous discussion, we have focussed on the evolution of a nearly homogeneous scalarfield φ(t). However, since we expect the field initially to be laid down at random on scales largerthan the Hubble radius, spatial ‘Kibble’ [30] gradients will be present on these scales.

For inflation22

to occur, it is necessary that the stress energy tensor averaged over a Hubble volume be dominatedby the potential V (φ), not by gradient terms ((∂iφ)2). (This is of course a concern for all modelsof inflation, not just those considered here.) In paper I, we addressed this issue at some length,and argued that, when the universe has cooled to the temperature T ∼Λ at which inflation wouldotherwise begin, the energy density contributed by field gradients would be at most comparable tothat in the potential.

(During the prior radiation-dominated epoch, the gradient energy densityscales like radiation, ρgrad ∼(∂iφ)2 ∼f 2/t2 ∼T 4, where the last equality assumes f ∼MP l;thus, at T ∼Λ, we expect ρgrad ∼Λ4 ∼V (φ).) Since these gradients rapidly redshift away withthe subsequent expansion, they would typically delay only slightly the onset of inflation.Here, we point out that the canonical PNGB model has an additional automatic feature whichcan ensure that spatial gradients in the PNGB field are negligible at the onset of natural inflation.Namely, if φ is the angular component of a complex field Φ, as in the model of Eqn.

(2.3), thenthe heavier, radial component of Φ can generate an earlier period of inflation as it rolls down itspotential. If the later angular inflation leads to more than 60 e-folds of growth in the scale factor(as we have been assuming), then the only important effect of the earlier inflation epoch would beto rapidly stretch out spatial gradients in the angular φ field.

(This point was stressed to us by A.Linde, private communication.) Furthermore, as we show below, the earlier inflation period doesnot require another small coupling constant.

In particular, for the model of Eqn. (2.3), for a broadrange of initial conditions, radial inflation takes place even if the complex scalar self-coupling λ isof order unity.

In addition, only a small number of radial inflation e-folds is required to efficientlydamp spatial gradients in φ.In the usual way, we can decompose the complex field Φ into two real radial and angularcomponents η and φ,Φ = eiφ/f η√2 . (3.38)Consider the evolution of the radial mode η in the potential (2.3), V (η) = (λ/4)(η2 −f 2)2 (ingeneral the radial and angular motions are coupled; however, since the radial mode is muchheavier, its evolution can be approximately decoupled).Analyzing this motion in a manneranalagous to §III.A, and using the fact that f is comparable to MP l, we see that some amountof radial inflation is expected provided the initial value of η is sufficiently far from its minimum⟨η⟩= f. In fact, this initial period of inflation will be generic as long as gradient terms in theη energy density do not dominate over the potential V (η) near the Planck scale and the initialvalue of η is not very close to f. For example, for f = MP l, if the initial value η1 of the radialfield is greater than 2MP l, then in rolling to its minimum it will generate at least 5 e-foldings of‘chaotic’ inflation, and angular gradients would be stretched by a large factor.

Alternatively, ifη1 ≤0.3MP l, the universe would experience about the same number of e-foldings of ‘new’ inflationas the field rolls from near the local maximum of the Mexican hat at the origin. We note that,for a potential of the form (2.3), for f near MP l the SR condition holds over some range of η,independent of the value of the coupling λ (just as Eqn.

(3.6) does not depend on Λ). Therefore,radial inflation takes place even if λ is large.

The density fluctuations produced during this phaseare on unobservably large scales if the subsequent angular inflation lasts for at least 65 e-foldsof expansion, so there are no strong constraints on λ arising from density fluctuations and themicrowave anisotropy. One should, however, require√λ/ξ <∼1 to avoid fluctuations of orderunity on the Hubble radius, since these would pinch offinto black holes.IV.

POWER LAW SPECTRA AND LARGE-SCALE STRUCTURERecent observations of large scale galaxy clustering and flows suggest that there is morepower on large scales than the ns = 1 scale invariant spectrum gives for ‘standard’ cold darkmatter dominated universes (CDM models). In this section, we show the degree to which varyingthe index ns, where the primordial power spectrum |δk|2 ∼kns, while keeping all other features of23

the CDM model fixed, helps solve this large scale structure dilemma. We have shown that naturalinflation will generate such a power law perturbation spectrum over a wide range of wavenumbers,in particular over the waveband that we directly probe with observations of large scale galaxyclustering and of microwave background anisotropies.

We demonstrate this in more detail in Sec.IV.A below. In addtion, other inflation models, such as 8those with exponential potentials andmany versions of extended inflation, also predict power law spectra which can deviate from scaleinvariant.

In Sec. IV.B, we show that current data on microwave anisotropies and large-scale flows,and the requirement that structure forms sufficiently early, constrain ns to be ∼> 0.6 for CDMmodels, but values ∼< 0.6 are needed to explain the large scale clustering of galaxies.

The reasonwe put the CDM model under such scrutiny rather than other inflation-inspired models, apartfrom its having dominated the theoretical scene for the past decade, is that it is a minimal model,in the sense that it requires the least number of assumptions to specify it. For the ‘standard’ CDMmodel, one assumes a flat geometry for the Universe with Ω≈1 in non-relativistic particles andtakes h≈0.5, where h is the Hubble constant H0 in units of 100 km s−1Mpc−1.

(For values of hlarger than this, if Ω= 1 the Universe would be younger than the inferred ages of globular clusterstars.) We assume negligible baryon abundance, ΩB ≪Ω, in the following; a small value of ΩBis indicated by primordial nucleosynthesis constraints (∼< 0.07).

The rest of the non-relativisticmatter is in cold dark matter relics, Ωcdm = Ω−ΩB. Since the large scale structure dilemma hasbeen with us in one guise or another since the early 1980s, a major line of research over the pastdecade has been to invent models with scale invariant primordial spectra that have more powerthan the ns = 1 CDM model does on large scales.

These ‘nonstandard’ ns = 1 models includemodels with a non-zero cosmological constant, a larger baryon density ΩB than that inferred fromstandard nucleosynthesis, and mixtures of hot and cold dark matter, to name just a few. Oftensomewhat baroque from the particle physics prespective, such alterations would all result in morestringent constraints on ns if we allow it to vary than the ones we derive for the CDM model.

(Indeed there are models that require the effective ns to be ≫1, such as the isocurvature baryonmodel, but this is certainly not an outcome of natural inflation.)A. Inflation Models and Power Law SpectraBefore turning to the data, we first show explicitly how tiny the deviations from a powerlaw form are for natural inflation, and that Eqn.

(3.28) for ns is highly accurate. We also discussthe form that ns takes for other popular models of inflation such as power law, extended, andchaotic inflation.

Since we are dealing with spectra that can change somewhat with wavenumber,we define a ‘local’ (i.e., k-dependent) spectral index ns(k) byns(k) ≡1 + d ln Pζ(k)/d ln k ,(4.1)where the ζ-power spectrum Pζ(k) introduced in Sec. III provides a better measure of the post-inflation spectrum than does the density power spectrum.

The quantity ζ, the variation of the3-space volume on uniform Hubble parameter hypersurfaces, is gauge- and hypersurface-invariant,whereas the density is neither.1) Natural InflationFor natural inflation this local index isns(k) ≈1−MP l2(8πf 2)"1 + [1 + (MP l2/(24πf 2)]−1 exp[−MP l28πf2 NI(k)]1 −[1 + (MP l2/(24πf 2)]−1[1 + (MP l2/(16πf 2)] exp[−MP l28πf2 NI(k)]#. (4.2)Here NI(k) is the number of e-foldings between the time when the inverse wavenumber k−1 firstexceeded the comoving Hubble length (the first ‘horizon crossing’) and the end of inflation.

For24

waves on scales of observable interest, NI(k) ∼50 −60, so the fator in large brackets is alwaysvery close to unity over the entire range of values of f we are considering.The derivation of (4.2) is very similar to that given in Sec. III, so we just sketch the stepshere.

From Eqns. (3.17) and (4.1), we must evaluate ns −1 = 2d ln((3/2π)H2/| ˙φ|)/d ln Ha (sincek = Ha at horizon crossing).

If we use the slow roll approximation for ˙φ and H, we havens(k) ≈1 −MP l28πf 21 + sin2(φk/(2f))1 −sin2(φk/(2f))[1 + (MP l2/16πf 2)]. (4.3)Here φk is the value of the scalar field at which k = Ha.

As in Sec. III, we have taken the positivesign for the potential (1.1).

The ‘end of inflation’ occurs when the scalar field kinetic energy growsto the value ˙φ2 = V , i.e., the Universe passes from acceleration to deceleration (cf. Eqn.3.3).

Atthis point, the expansion rate H ≃(3/2)1/2Hsr, where the Hubble parameter during the SRepoch, Hsr =p8πV (φ)/3M 2P l. As a result, the end of inflation can also be expressed as the timewhen | ˙φ| = HMP l/√4π. Approximating ˙φ by the slow roll result, ˙φ = (−MP l2/4π)∂Hsr/∂φ,one finds that inflation ends when the field reaches the value φend ≃2farctan[√24πf/MP l].

(In Sec. III, we defined the end of SR to occur when ˙φ2 =˙φ22 = 2V , which gave the factor√48π in the argument of the arctan [Eqn.

(3.5)] rather then the√24π found here. The numericalcomputations of φend discussed in Sec.

III.B are best fit by a√34π factor in the argument, soeq. (3.5) or the value given here give about the same accuracy.

In any case, in Eqn. (4.2) this factoris multiplied by the exponential suppression factor exp[−(MP l2/8πf 2) NI(k)].) By solving theequation d ln a = (H/ ˙φ)dφ for a(φ), we can find NI(k) = ln a(φend) −ln a(φ) in terms of φ (c.f.Eq.

(3.6)):sin2(φ/(2f)) =h1 + MP l224πf 2i−1exp−MP l28πf 2 NI(k).This expression generalizes Eqn. (3.7); when it is substituted into Eq.

(4.3), Eq. (4.2) is obtained.Defining kend to be the wavenumber that equals (Ha)end at the end of inflation, and usingthe fact that NI(k) = ln(H(φk)kend/H(φend)k), the relation between NI(k) and k is given byln kkend= −NI(k) + 12 ln 1 + MP l224πf 2−1 1 −exp−MP l28πf 2 NI(k)!,Thus between the current Hubble length k−1 ∼3000 h−1Mpc and the galactic structure lengthscale, k−1 ∼0.5 h−1Mpc, the range which encompasses all of the large scale structure observations,NI(k) only changes by about 10.

Since NI(k) only enters the exponentially suppressed terms in(4.2), the index ns is quite constant at 1 −MP l2/(8πf 2) over observable scales.2) Exponential Potential InflationAlthough we view natural inflation as the best motivated model for obtaining power lawindices below unity, other possibilities for getting ns(k) significantly different from unity havebeen widely discussed in the literature. Power law inflation 8[38,39] (in which the scale factorgrows as a large power p of the time, a ∝tp, instead of quasi-exponentially) is the simplestexample of a model which predicts power law spectra.

It is realized with an exponential potentialof form V = V0exp[−p16π/p φ/MP l], and hasns = 1 −2p −1 . (4.4)25

The deceleration parameter of the universe, q = −a¨a/˙a2 is q = −(1−p−1) for power law inflation.In order to have a viable model of inflation, the universe must pass from acceleration, q < 0, todeceleration, q > 0, so that it can reheat, hence it is essential that p evolves, with inflation endingwhen p falls below unity. Thus, although power law inflation models are instructive since they areanalytically simple, the exponential part of the potential can only be valid over a limited range ofthe evolution.

Indeed, it is often convenient to characterize potentials that are not exponentialsby an index p defined by √4πp = HMP l/| ˙φ|, which reduces to the p in the exponential potentialfor that case. However, in these models structure on observable scales may be generated in aregime where p varies with k rather than being constant.

Even so, power law approximations areoften locally valid, even when rather drastic potential surfaces are adopted to ‘design’ spectra.Some examples of cases where ns changes considerably over the observable window of large scalestructure are given, for example, in [40,11].3) Extended InflationExtended inflation also leads to a power law form over a wide band in k-space [41]. In ex-tended inflation, a Brans-Dicke field, whose inverse is an effective Newton gravitational ‘constant’,is introduced as well as an inflaton.

The analysis of [41] showed that the power law index can besimply expressed in terms of the Brans-Dicke parameter ω (the coefficient of the kinetic term ofthe Brans-Dicke field),ns = 1 −82ω −1 ,p = 2ω + 34. (4.5)As far as density fluctuations are concerned, the model just mimics a power law inflation one.Indeed, the fluctuation spectrum is most easily computed in a conformally-transformed referenceframe in which the log of the Brans-Dicke field experiences an exponential potential with p asgiven in Eq.

(4.5) [41], yielding the ns relation through Eq.(4.4). We note that, in most versions ofthe theory, a value of ω ∼< 18 −25 is needed to avoid an excessive CMBR anisotropy due to largebubbles, which implies that the spectrum deviates from scale invariant, ns ∼< 0.77 −0.84.

At thesame time, it is also necessary that the effective value of ω must have evolved to a high number(> 500) by now in order to satisfy solar system tests. This can be arranged by, e.g., giving theBrans-Dicke field a mass or by other means, but at the cost of complicating the model.4) Chaotic InflationAlthough references [40,11] probed how dramatic the breaking of scale invariance could be interms of the fluctuation spectra over our observable waveband, the main conclusion was that plau-sible models of inflation were much more likely to lead to quite smooth breaking over the observ-able range.

We illustrate the level of breaking of scale invariance expected for the popular chaoticinflation models. We assume power law potentials of the form V (φ) = λeMP l4(φ/MP l)2ν/(2ν),where the power ν is usually taken to be 1 or 2.

A characteristic of such potentials is that therange of values of φ which correspond to all of the large scale structure that we observe is actuallyremarkably small. For example, for ν = 2, the region of the potential curve that corresponds toall of the structure between the scale of galaxies and the scales up to our current Hubble lengthis just 4MP l ∼< φ ∼< 4.4MP l [11].Consequently, the Hubble parameter does not evolve by alarge factor over the large scale structure region and we therefore expect near scale invariance.Although this is usually quoted in the form of a logarithmic correction to the ζ-spectrum, a powerlaw approximation is quite accurate.

Following exactly the same prescription used to evaluateEq. (4.2), we havens(k) ≈1 −ν + 1NI(k) −ν6.

(4.6)For waves the size of our current Hubble length we have the 8familiar NI(k) ∼60, hence ns ≈0.95for ν = 2 and ns ≈0.97 for ν = 1 (massive scalar field case). The relation between NI(k) and k26

is given byln kkend= −NI(k) +ν2ln1 + 3NI(k)ν,(4.7)where kend is the wavenumber that equals Ha at the end of inflation. Thus, over the range fromour Hubble radius down to the galaxy scale, ns decreases by only about 0.01.B.

Implications for Large-Scale StructureWe have discussed various inflationary models (natural, power law, extended, and chaotic),which give rise to density perturbation spectra of the form |δk|2 ∼kns, where ns ≤1. We nowturn to their implications for large scale structure.1) Galaxy and Cluster ClusteringAlthough the amplitude of the fluctuations is known once all aspects of the inflaton potentialare specified, it is more convenient to normalize the spectrum to the level of clustering we observeand use that to restrict particle physics parameter ranges, as in Sec.

III. We normalize the ampli-tude of the density perturbation spectra by setting the rms fluctuation in the mass distributionwithin spheres of radius 8 h−1Mpc, σρ,8 ≡⟨(δM/M)2⟩1/2R=8h−1Mpc, to be σ8.

The rms fluctuationin galaxy counts on this scale in the CfA survey is unity. The quantity b8 ≡σ−18is sometimescalled the ‘biasing’ factor, since roughly if b8 ≈1 we expect that galaxies would be clustered likethe mass distribution while if b8 > 1 galaxies would be more strongly clustered than the mass;this point is discussed in more detail below.

For standard CDM models with ns = 1, σ8 wasthought to lie in the range 0.4 −1.2 before the recent COBE measurement.The evolved power spectra of the linear CDM density fluctuations, dσ2ρ/d ln k = k3⟨|δk(t0)|2⟩/2π2,are shown in Figure 3(a), for spectral indices ranging from ns = −1 to 1. A transfer function T(k)relates the primordial spectrum |δk(ti)|2 ∝kns to the present spectrum, |δk(t0)|2 = T 2(k)|δk(ti)|2.For the CDM transfer function, we use the fitting formula given in Appendix G of BBKS [42],which is highly accurate in the ΩB →0 limit, but is somewhat modified for the ΩB ∼0.05 valuesmore appropriate from nucleosynthesis.

The spectra are in units of σ8. The spectra are plottedin this way to provide a measure of the contribution of a band around the given wavenumber tothe overall rms density fluctuations; the ordinate roughly gives (1 + znl(k))/σ8, where znl(k) isthe redshift at which the rms fluctuations in the band become nonlinear.

Notice that there is apeak in the CDM spectrum for ns < 1. This indicates that there is a characteristic scale, roughlythe peak, associated with the first objects that form [43].

A potential problem with these modelsthat is immediately apparent from Fig.3(a) is that the redshift of galaxy formation is lower thanthat for the scale invariant model, which, for small ns, can lead to grave difficulties in explainingwhy there are quasars at z ∼5. We discuss this point more fully below.To relate such a linear density perturbation spectrum to galaxy clustering, one must generallydo N-body calculations.

However, on large scales, the waves evolve in an essentially linear fashion,and there is an excellent approximation which relates the power spectra of galaxies and clustersof galaxies (if they arise from any function of the Gaussian process through which perturbationsarose) to that of the density field. This relation is an extension [44] of the theory which identifiesgalaxies and clusters with appropriately selected peaks of the initial density field [42, 45].

Forscales large compared with the local processes that define these objects and large enough that thewaves are evolving in the linear regime, the power spectra for galaxies and clusters are linearlyproportional to the density power spectrum, with the proportionality constants defining ‘biasingfactors’, bg for galaxies and bc for clusters (Cf. Sec.

III.C.1):dσ2gd ln k = b2gdσ2ρd ln k ,dσ2cd ln k = b2cdσ2ρd ln k . (4.8)27

In Figure 3(b), which focuses on the region of k-space in 3(a) probed by large scale structureobservations, we compare the predicted galaxy spectrum with large scale clustering data from theQDOT and UC Berkeley IRAS surveys and (less directly) the APM survey. The spectrum is inunits of bgσ8.

In the conventional BBKS peaks approach to biasing [42], we would have bg = 1/σ8,which is why σ−18 , the inverse of an amplitude measure, is often referred to as a biasing factor(e.g., in Sec. III) .

In general, bg will differ from galaxy type to galaxy type and there is noclear reason why we should suppose that bg = σ−18 ; nonetheless, it is rather remarkable that thisassumption appears to give the correct amplitude for galaxy clustering. However, we note thatthe slight differences in the power spectrum levels for the 3 surveys could be simply explainedwith slightly differing bg’s.

To compare with the data in the nonlinear regime of the spectrum,k−1 ∼< 5σ8 h−1Mpc, N-body computations are needed. However, just from the linear regime itwould appear that spectral indices in the range 0–0.6 are much preferred over the scale invariantvalue of unity.

(This point would appear even more dramatic had we forced the models to agreewith the data at the 8 h−1 Mpc normalization scale. )Probably the most reliable indication of excess large scale power is the angular correlationfunction of galaxies, wgg(θ), inferred from deep photometric surveys.

Although the angular corre-lation function suffers from having only two- rather than three-dimensional information, it gainsenormously since angular surveys currently involve a few million galaxies, while three-dimensional(redshift) surveys are still limited to samples of several thousand galaxies. Two groups have nowindependently catalogued the galaxies of the Southern Sky and have derived wgg(θ)’s in agree-ment with each other.

A Northern Sky survey is also in basic agreement. Bond and Couchman[44] showed that the angular correlation function at large angles can be evaluated using the lin-ear power spectrum for galaxies, although nonlinear effects substantially modify the estimates atsmall angles; they also showed how to evaluate the correlation function directly from the powerspectrum.

We applied these techniques to the power spectra of Figure 3(b) to compare wgg as wevary ns with the APM results in Figure 4. The dots denote the APM data for various magnitudeintervals, scaled back to the depth of the Lick survey [46].

The spread is considered to provide arough indicator of the error level. Although there is a certain amount of vertical freedom in fittingthe theory to the data, from the overall scale bgσ8, it is clear that 0 ∼< ns ∼< 0.4 is required if weare to take the spread of dots as an error estimate.

It was this graph that led to the conclusiongiven in Bond [47] that this was the allowed range. However, estimates for various corrections tothe APM catalogue such as those from plate errors and variable absorption by Galactic dust mayrevise wgg(θ) downward slightly, and the hatched region is now expected to be allowed by thedata [46].

Thus, for this paper, we consider the allowed range to be 0 ∼< ns ∼< 0.6. We note thatthis fit has been done with a CDM spectrum with h=0.5 and ΩB ≈0.

If we can contemplate has low as 0.4 or ΩB as large as 0.1, then ns ≈0.7 is feasible as well.The high degree of clustering of clusters has been a puzzle since the early 1980’s.Thecorrelation function of rich clusters was thought to be enhanced by a factor of about 11-16 overthe level of galaxy clustering, assuming both have the same power law behaviours [48]. The samplefrom which most of the estimates of clustering were derived was the Abell catalogue, which hasbeen criticized on a number of grounds.

The main problem seems to be the projection effect, inwhich clusters at different redshifts superimpose upon one another, leading one to believe that theclusters are more massive than they truly are. Recently two redshift surveys of clusters identifiedusing the Southern Sky galaxy surveys estimate correlations about half as large as the originalvalues, and have shown that they are not subject to contamination by projection effects.

Thesenew values are roughly compatible with the levels expected if one uses the power spectra suggestedby the galaxy clustering data [49]. Provided we are in the linear regime, Eq.

(4.8) shows that thecorrelations should be in the ratio (bc/bg)2. A rough estimate for this ratio can be obtained usingthe methods of [45] for a peak model of clusters, in which one can determine the combination(bc−1)σ8 just from the abundance of clusters; it is about 2.1.

Thus, (bc/bg)2 ∼(2.1+σ8)2/(bgσ8)2.Taking bg = σ−18 , the enhancement factor ranges from 6 to 10 as σ8 ranges from 0.5 to 1. Thus, ifthe new cluster correlation functions prove to be valid, they can also be explained with the same28

range of ns as the wgg data indicates.2) Constraints from Microwave Background AnisotropiesWe now determine the range of σ8 as a function of ns suggested by the COBE observations ofmicrowave background anisotropy with the Differential Microwave Radiometer experiment [37].The DMR team have given data for the rms fluctuations on the scale of 10o, σT (10o), the sum ofthe squares of the components of the quadrupole moment tensor, which we denote here by σ2T, ℓ=2,and estimates of the correlation function with the dipole and quadrupole contributions removed.Here, we express all of these values in units of ∆T/T, by dividing their results by the backgroundtemperature, 2.736K.The fwhm of the DMR beam (7o) is sufficiently large that it is quite accurate to assume forthe adiabatic fluctuations of interest here that the microwave background anisotropies arise fromcurvature fluctuations experienced by the photons as they propagate through photon decouplingto the present (Sachs-Wolfe effect). If we assume that the universe is matter dominated fromphoton decoupling to the present, the variance Cℓof the multipole coefficient aℓm in the sphericalharmonic expansion of the radiation pattern (see e.g., ref.

[50]), is given byCℓ= ⟨|aℓm|2⟩= 4π9Z ∞0d ln k dσ2Φd ln k j2ℓ(kτ0) ,(4.9a)where jℓis a spherical Bessel function and τ0 is the comoving distance to the photon decouplingregion, τ0 ≃2H−10≈6000 h−1Mpc. The comoving wavenumber k is referred to current lengthunits.

The gravitational potential spectrum is related to that for the density bydσ2Φ/d ln k = ((3/2)H20 k−2)2dσ2ρ/d ln k . (4.9b)Although we used Eq.

(4.9a) directly to evaluate the temperature power spectrum Cℓ, for powerlaw spectra on the large scales that COBE probes, there is a simple expression in terms of Gammafunctions [50] and the quadrupole power C2:Cℓ= C2Γ[ℓ+ (ns−1)2]Γ[ (9−ns)2]Γ[ℓ+ (5−ns)2]Γ[ (3+ns)2],(4.10)for ℓ≥2. In terms of Cℓ, the rms value expected in each multipole for COBE isσ2T ℓ= 2ℓ+ 14πCℓF2ℓ,(4.11)where Fℓis a filter appropriate to their beam, and is approximated by a GaussianFℓ= exp[−0.5(ℓ+ 0.5)2/(ℓdmr + 0.5)2] ,ℓdmr ≈19 ,where ℓdmr corresponds to 7◦fwhm.The strongest result to use for estimating the amplitude σ8 is provided by σT (10o), whichthe COBE team determined by evaluating the intrinsic sky dispersion after further smoothingtheir data with a 7o fwhm Gaussian filter.

To compare with this, we calculate the average valuethat our theoretical model predicts for this,σ2T (10o) =XℓF2ℓσ2T ℓ. (4.12)29

The extra filtering by F2ℓbrings the total smoothing up to a total of 10o. Since the realization ofthe Universe that we observe involves a specific set of multipole coefficients drawn from (Gaussian)distributions with variance Cℓ, there will be a theoretical dispersion in the values of σ2T (10o), whatthe COBE team refers to as cosmic variance.

For σ2T (10o), we have⟨[∆σ2T (10o)]2⟩= 2Xℓ12ℓ+ 1hF2ℓσ2T ℓi2. (4.13)An excellent fit to our calculation of Eqs.

(4.12,4.13) isσT (10o) = 0.93 × 10−5 σ8 e2.63(1−ns) [1 ± 0.1e0.42(1−ns)] . (4.14)(Since the error is for the square, σ2T (10o), there is a slight asymmetry between the upper andlower error bars for σT (10o) which we have included in Figure 5.) Eq.

(4.14) is to be comparedwith the DMR result, including their ‘1 sigma’ errors,[σT (10o)]dmr = 1.085 × 10−5 [1 ± 0.169] . (4.15)(These errors should be slightly enhanced since the detected large scale anisotropy can lead tobigger fluctuations in σT (10o) than one would get solely using single pixel errors, as the DMRteam did.

This appears to be a sufficiently small correction that it can be ignored.) The combinederror is therefore about 20% for ns = 1, rising slightly for lower values, henceσ8 = 1.17e−2.63(1−ns) [1 ± 0.2] .

(4.16)The allowed region for σ8 as a function of ns using our computed values, is shown in Figure 5. Inparticular, for ns ∼< 0.6, the DMR result requires σ8 ∼< 0.5.However, we caution that this value is for the ΩB = 0 limit.

With the value ΩB ∼0.06favoured by primordial nucleosynthesis, σT (10o) rises by about 15% and σ8 drops by this amount.The quadrupole determination by the DMR team is not nearly as restrictive, because the‘cosmic variances’ as well as the DMR error bars are quite large. Integrating Eq.

(4.9a) over allk > 10−4 h−1Mpc for C2, we obtainσT, ℓ=2 = 0.46 × 10−5 σ8 e2.94(1−ns) [1 ± 0.3] ,(4.17)to be compared with[σT, ℓ=2]dmr = 0.475 × 10−5 [1 ± 0.31] ,(4.18)henceσ8 ≈1.02e−2.94(1−ns) [1 ± 0.46] . (4.19)However, as for σT (10o), small values of σ8 are required for ns ∼< 0.6.

If we use Eq. (4.19) togetherwith Eq.

(4.16) to constrain ns, the errors on the quadrupole are such that the range is not seriouslyrestricted. (Again, we have ignored the asymmetry on the cosmic variance errors.

)One can also use the correlation function data for given ns to determine the allowed rangefor σ8. The correlation function (with quadrupole removed) and its cosmic variance are given by[50]C(θ) =Xℓ>2Pℓ(cos θ) σ2T ℓ.

(4.20a)and⟨[∆C(θ)]2⟩= 2Xℓ>212ℓ+ 1hPℓ(cos θ) σ2T ℓi2. (4.20b)30

There are also correlations from angle to angle, so a matrix is more appropriate. As well, oneshould restrict the region of correlation function estimation to that actually used by the DMRteam, which involved a cut in Galactic lattitude.

This will increase the theoretical variance. InFigure 6, we compare our theoretical correlation functions, including their errors derived fromEq.

(4.20), for the ns = 1 and ns = 0.4 cases with the DMR correlation function given in ref.[37]. We have fixed the amplitude of the theory curves by requiring that they give the DMRσT (10o) = 1.09 × 10−5.

If we vary this amplitude for fixed ns, then the theory will cease to agreewith the data. Using the error bars that the DMR team give, and calculating χ2 for the modelfits to the data assuming the errors are independent and Gaussian (which they are not), we haveconstructed an allowed range for σ8 which basically agrees with that derived from σT (10o), butwith slightly larger errors.

A more precise treatment that takes into account the correlation in thevariances of the theory C(θ) and the influence of the extra correlation over pixel noise on the dataC(θ) error bars is needed to precisely pin down the allowed range. However, we are encouragedby the general agreement between limits derived from σT (10o), C(θ) and the quadrupole.

TheDMR team derive the constraint ns = 1.1 ± 0.5 from the correlation function data. Although itcan be seen from Figure 6 that there is a slight preference for the ns = 1 case compared with thens = 0.4 case, we do not consider that the ns = 0.4 case can be ruled out by this data alone.3) Large-scale Streaming VelocitiesThere is another type of data that directly probes the amplitude of the mass density fluc-tuations as opposed to the fluctuations in galaxy or cluster number densities, namely largescale streaming velocities.

From optical surveys, Bertschinger et al. [51] estimated the three-dimensional velocity dispersions of galaxies within spheres of radius 40 h−1Mpc and 60 h−1Mpcafter the data had been smoothed with a Gaussian filter of 12 h−1Mpc,σv(40) = 388 [1 ± 0.17] km s−1;σv(60) = 327 [1 ± 0.25] km s−1 ,(4.21)which should be compared with the rms 3D velocity dispersions for power law CDM models (witherrors calculated from the variance ⟨[∆σ2v(40)]2⟩):σv(40) = 300 σ8 e1.06(1−ns) [1+.35−.57] km s−1 ;σv(60) = 238 σ8e1.19(1−ns) [1+.35−.57] km s−1 .

(4.22)The fits are good for 0 ∼< ns ∼< 1. Although we do not regard these bulk flow estimates to be onas firm a foundation as the DMR measurement of σT (10o), it is interesting to note that the rangesuggested for σ8 by the velocity data is similar,σ8 ≈1.29e−1.06(1−ns) [1+.38−.65] ,(4.23)provided ns is not very far from unity.

It can be combined with Eq. (4.16) from σT (10o) to yielda preferred value for ns of 1.07 (and σ8 = 1.4!

), and a ‘2 sigma’ lower bound of ns = 0.72. Usingthe 60 h−1Mpc σv–estimate gives a similar result.

This constraint is so restrictive because thedramatic decrease in σ8 with decreasing ns from σT (10o) more than offsets the increased velocitydue to the enhanced large scale power.4) The Epoch of Structure Formation and Other TestsGiven σ8 and the spectral index ns we can consider when structures of various types formedin the Universe.In Figure 7, we plot the range in linear rms density fluctuations σρ(M) =⟨(∆M/M)2⟩as a function of mass scale M allowed by Eq.(4.16). We actually calculate the rmsfluctuations smoothed on a ‘top hat’ filtering scale RT H which is related to the mass by M ≈1012.4(RT H/ h−1Mpc)3.

The range in RT H around Rg = 0.5 h−1Mpc corresponds to the filteringappropriate for galaxy formation (top hat mass 1011.5M⊙). The σρ(M) shown are evaluated at31

the current epoch if one extrapolates their growth by linear theory. This means that the rmsfluctuations on the scale Rg reach nonlinearity at a redshift somewhat above1 + znl(Rg) = σρ(Rg) ≈6.2σ8 e−(1−ns) ≈7.2e−3.63(1−ns) [1 ± 0.2] ,(4.24)where we have used Eq.

(4.16) for σ8. Galaxies represent a much smaller fraction of space thanthat in typical fluctuations, but there is a lag between nonlinearity and complete collapse.

Theseeffects tend to cancel each other so Eq. (4.24) gives a first reasonable, although somewhat low,estimate of the redshift of galaxy formation.A better estimate of the redshift of galaxy formation is obtained in the following way.

Wetake the observed luminosity function for galaxies [52] and assign an average mass-to-light ratio(M/L) for galaxies with luminosities above L. We then have, approximately, for the mass fractionin objects with luminosity greater than L,Ω(> L) ≈0.035exp(−L/L∗)[(M/L)/(50h)]Ω,where L∗is a fitting parameter that gives the typical luminosity for a bright galaxy. The cor-responding mass is M = 6 × 1011h−1[(M/L)/(50h)] L/L∗.

Therefore, the fraction of the massin L∗galaxies for the models we are considering is about a percent. Now consider the fractionof the mass in the Universe in collapsed objects with mass above 3 × 1011M⊙; if we choose 50hfor (M/L) and (M/L), this corresponds to the mass above L∗/4, and the expression for Ω(> L)above indicates that 2.7% of the mass should be in such objects.

We thus determine the redshiftat which the Press-Schechter mass function [53] for these models would predict that 2.7% of themass in the Universe is in collapsed objects with mass above 3 × 1011M⊙. The correspondingvalue for this redshift is just 30% higher than Eq.

(4.24) and provides a better estimate of whenpervasive galaxy formation would have occurred,(1 + zgf)P S = 8.1σ8 e−(1−ns) ≈9.5e−3.63(1−ns) [1 ± 0.2] . (4.25)The power 3.63 is so large that even if we err on the conservative side by using Eq.

(4.25) ratherthan Eq. (4.24) and take the upper limit, we obtain relatively strong limits on ns:ns ∼> 0.63 , if zgf > 2 ;ns ∼> 0.71 , if zgf > 3 .

(4.26)A more careful analysis of star formation history would be required to improve upon these limits,but they illustrate that the amplitude factors allowed by the DMR data lead to strong limits onthe spectral index to have galaxy formation occur early enough. Note that these bounds on nsare similar to those derived from the streaming velocities.A more powerful analysis of when objects of various masses form is provided by the hi-erarchical peaks method [54, 55], which identifies virialized potential wells with patches of theUniverse centred on peaks of the density field that have undergone collapse, but solves the ‘cloud-in-cloud problem’ inherent in the original BBKS peak method [42] by merging small scale peaksubstructures into the dominant peaks that contain them.

A mass function for dark matter halosat redshift z, n(M, z)dM, as well as detailed information about the spatial distribution of thehalos, can be calculated. The objects found with this method have been shown to agree wellwith groups found in N-body calculations.

Curiously, the mass function agrees reasonably wellwith that derived using the Press-Schechter approach [53], especially at the high mass end. Thisgives us some confidence in the validity of the Eq.

(4.26), ns > 0.63, constraint. However, thePress-Schechter mass function has no strong theoretical justification [56] and cannot deal withthe spatial distribution of objects.Since the total dark matter mass in galaxies is not directly measured, the mass functionn(M) is of limited diagnostic use.

On the other hand, the depth of galaxy and cluster potential32

wells can be inferred from their internal velocity dispersion v. Therefore, in Figure 8 we showthe number density of objects with velocity dispersion in excess of v, n(> v, z), for a variety ofredshifts. The ns = 1 CDM model with σ8 = 0.7 has roughly the right number of v = 200 km s−1halos at z > 3 to be a viable model of galaxy formation, and the number of clusters with 3D virialvelocity above 1500 km s−1 roughly corresponds to the number of rich Abell clusters.

Increasingσ8 for this model, as is suggested by the DMR data, might result in an excess of clusters withhigh velocity dispersions and thus high X-ray temperatures that may already be excluded by theX-ray data [55]. However, current indications from gravitational lensing observations in clusters[57] are that clusters exist with velocities in excess of v = 2000 km s−1 at z ∼> 0.2, and a z ∼0.2cluster observed with the X-ray satellite Ginga has an X-ray temperature of 13 keV [58], whichtranslates into a v ∼2500 km s−1 dispersion.

It is also possible that cluster X-ray temperaturesare below the values one would infer from the dark matter potential. Thus it may turn out thatσ8 ∼1 will be preferred over 0.7 as the data improves.

On the other hand, it is evident thatcluster velocity dispersion estimates are easily contaminated by projection effects that always giveoverestimates [59], so the lack of v ∼1500 km s−1 clusters in the σ8 = 0.5, ns = 0.6 model cannotat present be used to exclude it. Thus, although it is universally agreed that the abundance ofrich clusters as a function of velocity dispersion will be one of the most powerful measures of σ8,better data and extensive theoretical comparisons with the X-ray and optical data are requiredto test how strongly ns is constrained.

The basic conclusion of the more complete analysis ofref. [55] is that, while one may argue that low amplitude models are not excluded by the velocityor temperature data, it seems quite unlikely that the errors in the X-ray flux and luminosity data,both for nearby and distant (z ∼0.2) clusters, are so large as to allow these models to survive;explicitly, the ns = 0.6 CDM model with σ8 ≤0.5 is ruled out [55].What even more strongly rules out the ns = 0.6 model, in agreement with the analyticargument constraining ns using zgf given above, is the lack of high redshift activity, in particularthe paucity of halos with dispersion in excess of 200 km s−1 even as late as z = 2.

These are thesites of bright galaxy formation. There are some interesting differences that appear at high z evenwith the modest change in slope from ns = 1 to 0.8, with σ8 fixed: e.g., there would be an order ofmagnitude more v = 100 km s−1 ‘dwarf’ galaxies at z = 10 in the ns = 1 model than in the case ofns = 0.8.

It has been argued [60] that only those dwarf galaxies with velocities above this numberwill survive the supernova explosions that occur when galaxies assemble themselves. Having someold cores of stable objects is probably a good thing rather than a bad thing, since they could bethe birthplaces of quasars, but because of uncertainties in modelling the gas dynamical behaviourof forming galaxies and of the intergalactic medium one cannot be sufficiently definitive aboutthe high z consequences of a theory to select one model over the other at this stage.Another test which has been used to argue that σ8 ∼< 0.6 and which therefore favours ns < 1models is the velocity dispersion of pairs of galaxies over separations of order a Mpc [61].

In theearly N-body simulations of ns = 1, σ8 = 1 CDM models, the pair velocity dispersion of darkmatter halos on these scales was found to be much higher than the velocities of galaxies inferredfrom redshift surveys. However, Carlberg and Couchman [62] computed an ns = 1 CDM modelin which the relative velocity of galaxies was much less than that for the dark matter, an effecttermed ‘velocity bias’.

Coincidently, they chose σ8 = 1.17, the value suggested by DMR. Althoughhow effective this velocity biasing can have been at lowering the pair velocities is a matter of muchdebate, smaller ns will obviously help to ease the problem.Experimental upper limits on small and intermediate angle anisotropies in the microwavebackground can also be used to constrain the index ns, but require detailed computations alongthe lines of those given in ref.

[50] and we shall not undertake them here. We note however thatthe pre-COBE limits on anisotropy were already strong enough to place constraints of ns ∼> 0.6for σ8 = 1 and ns ∼> 0.3 for σ8 = 0.5 [47] at the 90% confidence level, and the constraints froman earlier DMR limit [63] also gave similar values.

(For other previous discussions of power lawCDM spectra, see [38,39,67]. )33

5) The Role of Gravitational Wave ModesStimulated by the DMR results, other groups have been independently considering inflation-inspired power law spectra [68, 69]. Davis et al.

[69] have pointed out that, although gravitationalwave modes are generally small for nearly scale invariant spectra [70], for ns ≪1 this conclusionmay not hold, amplifying upon the work of Abbott and Wise [71]. Although gravitational wavesdo not make an important contribution for natural inflation, they are significant for power lawand extended inflation models.We first sketch why they can be ignored in natural inflation.

During inflation, the samezero point quantum fluctuation phenomenon which leads to the inflaton density perturbationsalso leads to statistically independent gravitational wave perturbations. If h+ and h× are thetwo linear gravitational wave perturbations, then ϕ+,× = MP lh+,×/√16π behave just like singlemassless scalar field degrees of freedom as far as fluctuation generation is concerned.

Each ofthe fields ϕ+,× of comoving wavenumber k have power spectra P 1/2ϕ+,×(k) equal to the Hawkingtemperature H/(2π) when k = Ha, just as the inflaton fluctuations do, except that they are notamplified during subsequent evolution. With the factor given above, we therefore have for thetotal gravitational wave power, P 1/2GW ≡[Ph+ + Ph×]1/2 =√32πMP l−1H/(2π).

The ratio of thegravitational wave power spectrum to adiabatic metric perturbations, as encoded in the spectrumPζ, at horizon crossing is thereforeP 1/2GWP 1/2ζ=√2√16π| ˙φ|3MP lH,where the√2 comes from the 2 independent GW polarizations that can be generated. Using theWKB values at horizon crossing usually gives accurate estimates of final fluctuation amplitude[11].

For natural inflation, and using the slow roll approximation and the results of Sec. IV.1, wehaveP 1/2GWP 1/2ζ= 2√23√4π|∂ln Hsr/∂φ| = 2√23 MP l216πf 21/2 1 + MP l224πf 2expMP l28πf 2 NI(k)−1−1/2.

(4.27)Thus the gravity waves are exponentially suppressed relative to the adiabatic scalar fluctuationsof the inflaton over the observable large scale structure waveband. In particular, for f ≤MP l,this ratio is less than 0.04 for modes with wavelength equal to the current Hubble radius.

On theother hand, for power law inflation with an exponential potential, the ratio isP 1/2GWP 1/2ζ= 2√23√p = 2√231 +21 −ns−1/2,(4.28)which can be quite favourable to the tensor modes if ns is sufficiently small.The amplitude of gravitational wave modes decays by directional dispersions as the modesre-enter the horizon, just as waves in any relativistic collisionless matter do [43], whereas theadiabatic fluctuations maintain a constant gravitational potential. Before the gravitational wavestructure disperses however, it influences the microwave background through the Sachs-Wolfeeffect.A number of authors have calculated the magnitude of this effect [70,71].We denotethe ratio of tensor to scalar contributions to the radiation field multipole moments aLM by AL.Abbott and Wise [71] show that this ratio is not very sensitive to the multipole moment L. Daviset al.

[69] use the results of [70,71] to get the ratio for the quadrupole value; in our language, thisisA2 ≡σGWT ℓ=2σadiabT ℓ=2≃3.9P 1/2GWP 1/2ζ. (4.29)34

To estimate the correction for power law inflation, we shall assume AL is A2, which, usingEq. (4.28), is thereforeA2 ≈3.71 +21 −ns−1/2.

(4.30)The value of σT (10o) given in Eq. (4.14) should be multiplied by [1 + A22]1/2.

Thus, the rangefor σ8 as a function of ns is lowered substantially as a result of the inclusion of gravity waves,as we have shown in Figure 5; e.g., σ8 drops by a factor of 1.8 for ns = 0.6. This makes thealready strong constraints we have derived significantly stronger.

The ns-constraint we derivedby requiring that galaxies form early enough in the theory, ns > 0.63 for zgf > 2, changes tons > 0.76 for power law inflation; similarly, the bound ns > 0.71 from the requirement zgf > 3now becomes ns > 0.82. Also, the ‘2 sigma’ streaming velocity limit of ns > 0.72 increases tons > 0.89.For the chaotic inflation potentials used above, we haveP 1/2GWP 1/2ζ= 2√ν3hNI(k) + ν3i−1/2,A2 ≈2.63√νhNI(k) + ν3i−1/2,(4.31)hence gravity waves diminish σ8 by only 11% for a φ4 potential, and by 5.5% for a φ2 potential.Slightly higher values are obtained if we use a power law inflation formula with ns = 0.95 and 0.97,respectively.

Again motivated by COBE, various authors have been looking at the gravitationalwave contribution in these conventional inflation models anew [69,72].It is clear from this discussion that if one could unearth the gravity wave component ofanisotropy from the adiabatic component, it would 8not only allow a strong discrimination amongmodels, but it would also rule out natural inflation, which predicts no component whatsoever.6) DiscussionSince our ns ∼> 0.6 limit comes from a variety of arguments, we believe it is quite robust. Thus,unless the errors in the analysis of the large scale clustering observations are larger than currentlyestimated, a fluctuation spectrum with broken scale invariance that has a slowly changing spectralindex over the range from k−1 ∼10−104 Mpc cannot be the solution to the extra power dilemmathat the CDM model faces.However, the allowed values of ns ∼> 0.7 can help to ease therequirements on some of the extra power fixes proposed in the literature (e.g., [45]).Motivated by the DMR results and the many prospects for broken scale invariance in inflation,Cen et al.

[68] have very recently undertaken combined hydrodynamical and N-body calculationsof CDM models with ns = 0.7 and have independently come to a number of the conclusions wehave about such models, namely that they help but do not fully solve the large scale structuredilemma.Finally, our limit on ns can be translated into constraints on the parameters of inflationmodels that give rise to power law spectra. For example, it gives a very strong constraint onthe effective value of ω, the Brans-Dicke parameter which arises in extended inflation models.When the effect of tensor waves is included, the zgf > 2 constraint, ns ∼> 0.76, becomes ω ∼> 17,near the upper limit of the range ω ∼< 25 required for successful inflation [73] in most versionsof this theory.

Indeed, a closer examination [74] of the upper bound on ω, which arises from therequirement that large bubbles do not produce an excessive microwave anisotropy, suggests thatin fact ω < 18 is required if the dark matter is cold. (This number might even be slightly lower,since it is based on the older COBE data.) Combined with our lower bound on ω, this limit wouldleave little room for most extended inflation scenarios.

For natural inflation, from Eqn. (4.2), theconstraint ns > 0.63 translates into a lower bound for f of 0.33MP l. This is comparable to theconstraint (3.37) from reheating.35

V. CONCLUSIONSWe have studied an inflation scenario inspired by particle physics models with weakly self-coupled (pseudo-)scalars such as the axion. With the requisite mass scales, which can emergedynamically for plausible choices of gauge groups, PNGB inflation appears to be robust in thesense that it arises in the simplest class of models, with a potential of the form (1.1).

We haveshown how these models can arise in a variety of theoretical settings, and indeed that superstringmodels already in the literature come very close to providing the desired mass parameters fornatural inflation. Although the tendency of higher dimension operators on PNGBs arising fromwormhole effects, for example, would be to increase Λ, we discussed quite plausible ways in whichthe upward movement can be exponentially suppressed, so our model retains its naturalness.We numerically and analytically studied the cosmological dynamics of the inflaton field, andderived several constraints on the two-dimensional parameter space (f, Λ).

The allowed band ofparameter space includes models which have more relative fluctuation power on large lengthscalesthan the standard scale-invariant spectrum. We have studied in depth the consequences of thesepower law initial fluctuation spectra for large-scale structure and the microwave backgroundanisotropy.

We find that models with ns ∼< 0.6 are required to fit the large-scale galaxy angularcorrelation function wgg(θ) observed in the APM survey, but the recent COBE results requirea rather small amplitude for these models to be consistent, σ8 ∼< 0.5. This makes the epoch ofgalaxy formation uncomfortably recent and predicts large-scale flows of relatively small amplitude.Turning this argument around, combining the COBE results with the requirement of sufficientlyearly galaxy formation and large-scale flows of the inferred amplitude leads to the constraintns ∼> 0.6.

For natural inflation, this implies f ∼> 0.3MP l, virtually the same bound as we get fromthe reheating constraint. Although the simple expedient of reducing ns does not, by itself, solveall the large scale structure dilemmas for the CDM model, it can be combined with other waysto explain the extra large scale power [45], for example, by introducing into the CDM model aneutrino with a mass of a few eV, a nonzero cosmological constant (MP l2Λ/8πh = 0.2 with CDMfits for ns = 1), a smaller Hubble constant (h ∼0.4), a larger baryon abundance, or by simplysupposing that galaxies are distributed on large scales somewhat differently than the mass sothat the linear biasing assumption of Eq.

(4.8) is invalid. We conclude that inflation with pseudo-Nambu-Goldstone bosons offers an attractive model for generating curvature fluctuations whosegravitational instability can lead to all of the cosmological structure we observe around us, evenif the spectrum is nearly scale invariant.AcknowledgementsWe acknowledge useful conversations with and comments from S. Dimopoulos, L. Dixon,M.

K. Gaillard, C. T. Hill, S. Hsu, L. Knox, A. Liddle, A. Linde, S. Myers, S. J. Rey, and N.Vittorio and thank K. Fisher, M. Davis, and M. Strauss for providing the IRAS 1.2 Jansky powerspectrum. We would like to thank R. Davis and P. Steinhardt for drawing our attention to theimportant role gravity waves would play in power law and extended inflation models.

Four ofus acknowledge the hospitality of the Institute for Theoretical Physics during its workshop onCosmological Phase Transitions, where part of this work was completed. JRB was supportedby NSERC and a Canadian Institute for Advanced Research Fellowship.

KF was supported inpart by NSF grant NSF-PHY-92-96020, a Sloan Foundation fellowship, and a Presidential YoungInvestigator award. The research of JAF is supported in part by the DOE and by NASA grantNAGW-1340 at Fermilab.

AVO acknowledges support from the NSF at the University of Chicago.FCA is supported in part by NASA Grant No. NAGW–2802 and in part by funds from the PhysicsDepartment at the University of Michigan.36

APPENDIX: APPROXIMATION OF INTEGRALSIn this Appendix, we demonstrate the validity of the results presented in §III.C on the aposteriori probability of inflation. The main difficulty is that the simple logarithmic form (Eq.3.11) for the number of e-foldings as a function of y1 = φ1/f does not hold for large y1 (i.e., fory1 ≥1).

We should thus write the integral I (see Eq. 3.31) in the formI =Z 1ǫdy1e3Ae−3B ln y1 +Z π1dy1e3N(y1)=e3A3B −1n1ǫ3B−1 −1o+Z π1dy1e3N(y1).

(A1)In §III.C, we argued that when 3B > 1, the integral can be approximated by the first term above,I ≈e3A3B −11ǫ3B−1. (A2)We now calculate the relative error suffered in making this approximation.

We first note that thenumber of e-foldings N(y1) is a strictly decreasing function of the starting value y1. In particular,N(y1) ≤N(1) = A∀y1 ∈[1, π].

(A3)8We thus obtain a bound on the second integral in Eq. [A1],Z π1dy1e3N(y1) ≤e3A(π −1).

(A4)This contribution to the error is always positive, whereas the other contribution [namely −e3A/(3B−1)] is always negative. The total error E is therefore bounded from above byE ≤e3A hπ −1 −13B −1i.

(A5)The total error is also bounded from below by the second (negative) term alone, so we obtain therelation−1 ≤E(3B −1)e−3A ≤3B(π −1) −π,(A6)and hence the relative error E = E/I is bounded byE ≤ǫ3B−1 × max{1, 3B(π −1) −π}. (A7)This error is always sufficiently small for the cases of interest.For example, for f ≈MP l,3B ≃48π(f/MP l)2 ≈48π, then ǫ ≤ymax1≃0.6, and hence E ≤2 × 10−31.

For the other endof the mass range of interest (i.e., for f near fc = 0.06MP l), let 3B −1 = δ where δ is a smallpositive number. In this regime ymax1∼10−60 and hence E ≤10−60δ.

The error is thus completelynegligible until δ becomes smaller than 1/60 or so, that is, until f is very close to fc.837

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FIGURE CAPTIONSFigure 1: Plot of various field values and parameters vs. f/MP l. The upper curves show thatour estimate φ2/f (eq.3.5) for the value of the field at the end of the SR epoch (dotted) is veryclose to our numerical result φend/f for when inflation ends (solid); the middle (dotted) curveshows log(φmax1/f), the largest initial value of the field consistent with 60 e-folds of inflation; thelower (dashed) curve shows the density perturbation constraint (3.23) on the scale Λ [plotted aslog(Λ/MP l)], assuming the bias parameter bg = 1.Figure 2: Results of the numerical integration of the scalar and gravitational equations of motion.The number of inflation e-folds N(φ1) of the scale factor is shown as a function of the initial valueof the scalar field, φ1, for different values of the fundamental mass scale, f/MP l = 0.05, 0.07, 0.1,0.2, and 0.5.Figure 3(a): Power spectra for CDM models with variable spectral indices ns are plotted againstcomoving wavenumber k (referred to current length units). There is progressively more large scalepower as ns decreases through the values ns = 1, 0.6, 0.4, 0, -0.4, and -1 shown in the figure.

Thelines under the labels (whose vertical placements are arbitrary) indicate approximate regions in kspace that various probes of structure are sensitive to, such as: microwave background anisotropyexperiments of large angle (COBE, [37]) and of intermediate angle (e.g., SPole is a 1◦experiment[64]); clustering observations for galaxies in the APM Galaxy Survey (wgg, [46]) and the QDOTredshift survey [65] and for clusters (ξcc) [48]; and large scale streaming velocities (LSSV) [51].The hatched region denotes the region that the power spectrum must pass through to explain theAPM angular galaxy correlation function data [46]. The spectra are in units of σ8.Figure 3(b): Galaxy power spectra derived assuming linear dynamics (appropriate for k−1 ∼>5σ8 h−1Mpc and large scale linear biasing), in units of bgσ8, for CDM models with variablespectral indices ns.

This region of the spectrum is highlighted because that is where the largescale structure data exists. The hatched region is the APM region of (a), while the points denotethe power spectra estimated from the QDOT redshift survey [65] and the IRAS 1.2 Jansky survey[66].

The biasing factors for the (slightly) different types of galaxies probed by the APM, QDOTand 1.2 Jy surveys could explain the differences in these results.There are indications thatbg = 0.8σ−18is needed for the 1.2 Jy galaxies [66], while bg = σ−18describes the APM survey well,and this relative factor is enough to bring the required power spectra into line; i.e., the ns =0.2–0.6 range is also preferred by the 1.2Jy data if bg = 0.8σ−18 , while the ns = 1 curve falls belowthe data error bars.Figure 4: The models of Fig. 3(b) (with ns = 1, 0.8, 0.6, 0.4,..., -1) are compared with theangular correlation functions determined from the APM Galaxy Survey [46] scaled to the depthof the Lick catalogue, at which 1◦corresponds to a physical scale of ∼5h−1Mpc (dots).

Nononlinear corrections were applied to the theoretical power spectra, but for angular scales above∼1◦and for amplitude factors σ8 ∼< 1, the linear approximation is accurate [44]. The theoreticalcurves are in units of (bgσ8)2.

The straight line gives the angular correlation that would result ifthe behavior of the spatial correlation function observed over distances r ∼< 10h−1 Mpc, ξ ∼r−1.8,were extended to large separations. The hatched region corresponds to the allowed region oncecorrections for systematic errors are included.

The data therefore suggests 0 ∼< ns ∼< 0.6 is neededfor the CDM model if biasing is linear on large scales.Figure 5 The range (with ‘1 sigma error bars’) of the amplitude parameter σ8 for a standardCDM model in the limit that ΩB = 0 as a function of the power law slope ns, using the constraintfrom the rms fluctuations on 10o in COBE’s DMR experiment. Both the theoretical variance andthe quoted experimental error are included in the error bars, which are in total about ±20%.

Thevalues of σ8 drop by a further ∼15% when ΩB ∼0.06 is used rather than the zero used here. If41

the correlation function data of Fig.6 is used to determine the amplitudes, a similar constraintcurve arises. Also indicated is the range in ns suggested by the APM angular correlation functiondata, the ‘1 sigma error bars’ on ns derived using the correlation function by the DMR team, andthe σ8 range (dotted lines) that encompasses the values that have been advocated for the ns = 1CDM model by different workers, e.g., σ8 = 0.4, 0.55, 0.65, and 1.2 in [61,44,54,62] respectively.The dashed curves give the allowed range for σ8(ns) when gravitational wave modes are includedfor power law (and extended) inflation.

For natural inflation, the deviation from the solid curvesis infinitesimal.Figure 6: Comparison of the DMR 53A + B × 90A + B cross correlation function with thequadrupole removed [37] with the theoretical predictions, including variance, for (a) ns = 1 and(b) ns = 0.4 spectra. The σ8 amplitudes shown have been set by requiring the angular powerspectrum to reproduce the rms fluctuations on 10o.

Clearly, although the data is somewhat betterfit by the ns = 1 rather than the ns = 0.4 model, one cannot strongly distinguish between the2 models on the basis of shape alone. The experimental errors plus theoretical variance in thequadrupole amplitude are sufficiently large that one cannot use the comparison of the quadrupolewith σT (10o) to effectively constrain ns.

The strongest restriction comes from the consequencesof low σ8 for ns ∼< 0.6 for structure formation.Figure 7: The linear rms fluctuations averaged over spherical regions of radius RT H are plottedas a function of the mass M ≈1012.4(RT H/ h−1Mpc)3 M⊙, for CDM models with ns = 0.4, 0.6,0.8, and 1 (with ns increasing as one moves vertically up the figure). The error bars show the 1sigma range in spectrum normalization as a result of DMR and cosmic variance errors in σT (10o).Although Fig.3(a) shows that ns < 1 spectra have more power on large scales and less on smallscales than ns = 1 models with the same σ8, when σ8(ns) determined from COBE is used, theamplitude for ns < 1 is less on all mass scales.

The extreme problems with the ns = 0.4 modeland the marginality of the ns = 0.6 model are evident from this graph alone.Figure 8: In (a), we show the number density of collapsed objects with 3D virial velocity inexcess of v for the CDM model with spectral index ns = 0.8 and for the value of the amplitudeparameter σ8 = 0.7 (indicated by the DMR σT (10o) data for this model).The densities areshown as a function of redshift z, with z decreasing as one moves to the right in the figure. Thevelocities in the hierarchical peaks method [55] used for this computation could be larger by anamount given by the error bar labelled by ‘v range’; these error bars are explicitly put on thez = 0 curve.

The number densities shown should be compared with the abundances indicatedby the horizontal lines and velocity dispersions indicated by the downward arrows: for ‘bright’galaxies, ∼10−2( h−1Mpc)−3 with v ∼220 km s−1, for rich clusters, ∼6×10−6( h−1Mpc)−3 withv ∼1500 km s−1, and for at least one object between us and redshift 2, ∼10−9( h−1Mpc)−3 withv ∼2500 km s−1, according to the Ginga X-ray satellite team [58]. In (b), we choose the DMR1 sigma upper bound σ8 = 0.5 for ns = 0.6; even so, the number of ‘bright galaxy’ halos is toosmall by z = 2.

In (c), we plot the densities for ns = 1, using the DMR 2 sigma lower boundσ8 = 0.7 for the amplitude, to facilitate comparison with (a). The number densities of model (c)accord reasonably well with the hierarchy of objects in the Universe.

There is little to distinguishbetween the ns = 1 and ns = 0.8 models with the same σ8. To explicitly show this, we alsoplot with light solid curves the tails of the z = 0 abundances for cases (a) and (b).

The thirdlight curve, also for z = 0 (the highest curve at large v), shows the effect of increasing σ8 to 1for the ns = 1 model, closer to the number indicated by DMR. Although this may lead to toomany clusters with higher X-ray temperatures than observed [55], σ8 = 1 does help to explainthe Ginga event.42

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