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COBE, INFLATION, AND LIGHT SCALARS

arXiv:hep-ph/9207243v1 16 Jul 1992YCTP-P21-92June 1992revised July 1992COBE, INFLATION, AND LIGHT SCALARSLawrence M. Krauss1Center for Theoretical Physics and Dept of AstronomySloane LaboratoryYale University, New Haven, CT 06511AbstractComparison of the COBE observed quadrupole anisotropy with that predicted in adi-abatic CDM models suggests that much of the observed signal may be due to longwavelength gravitational waves. In inflationary models this requires the generationof tensor fluctuations to be at least comparable to scalar density fluctuations.

This isfeasible, but depends sensitively on the inflaton potential. Alternatively, isocurvaturequantum fluctuations in an axion-like field could produce a quadrupole anisotropyproportional to the gravitational wave anisotropy, independent of the inflaton poten-tial.

These could also produce large scale structure with more power on larger scalesthan their adiabatic counterparts.1Bitnet: Krauss@Yalehep. Research supported in part by the NSF,DOE, and TNRLC1

The observation of primordial fluctuations in the Cosmic Microwave Backgroundby the DMR instrument aboard the COBE [1] satellite provides one of the most im-portant and fundamental pieces of data in cosmology. In principle, the spectrum offluctuations observed on large angular scales can be extrapolated to smaller scales,where these fluctuations would presumably have been responsible for the formation ofstructure in the universe by gravitational instability.

In normalizing this spectrum tothe observed COBE anisotropy however, one should be aware of other possible contri-butions to this anisotropy. In particular, it was recently argued [2] that gravitationalwaves generated during an inflationary phase in the early universe at plausible scalesmight dominate the quadrupole anisotropy observed by COBE.

These waves con-tribute to the quadrupole and higher moments of the CMB, while shorter wavelengthmodes redshift away without affecting local observables, and are thus not directly re-lated to structure formation. Thus, for example, small scale anisotropies (on scales lessthan 1o)—which might directly probe structure related density perturbations—andcorresponding large scale structure measurements may be compared to the anisotropyobserved by COBE in trying to disentangle this situation 2.There are in fact independent reasons to suppose that a significant component ofthe COBE anisotropy is related to gravitational waves.

The standard CDM models ofstructure formation, for example, generically predicts a smaller quadrupole than theobservations, by perhaps a factor of at least O(2) [4, 5]. It would be nice, therefore, ifthere were a natural way in which scalar density fluctuations and gravitational wavesproduced by inflation could lead to comparable anisotropies on large scales.32During the preparation of this manuscript, I learned from George Smoot about work of he andcollaborators [3] in which this issue and several others I raise are treated carefully and in somedetail.He kindly made available to me their preliminary preprint prior to submission.Whilesimilarly motivated, I believe this work is largely complementary to theirs.

In any case, I have triedto keep overlap to a minimum, and have referenced accordingly.3I thank Martin Rees for stressing the interest in this possibility to me , and thereby encouraging2

The ratio of scalar to tensor modes predicted to arise from inflation is model de-pendent. The former depends upon the detailed shape of the potential—in particularon its first derivative, while the latter depends merely on the energy density duringthe inflationary epoch, and therefore on V(0), assuming this remains roughly constantduring inflation.

Utilizing standard analyses it is straightforward to derive the ratio ofthe predicted quadrupole anisotropies for various models (see also [3]). Defining in aconventional way the CMBR temperature anisotropy in terms of spherical harmonicsδTT (θ, φ) =XlmalmYlm(θ, φ)(1)and projecting out a multipole to calculate the rotationally symmetric quantityDa2lE≡*Xm|alm|2+,(2)the mean quadrupole anisotropy due to gravitational waves from inflation can besimply given as [6, 2]⟨a22⟩≈7.7v.

(3)where v = V (0)/M4P. (The expectation value is based on a statistical ensemble ofuniverses which undergo inflation, in one of which we happen to make a measurement.

)The predicted distribution of a22 is of a χ2 form with 5 degrees of freedom.Alternatively, for exponential inflation one finds that scalar density perturbationsarising from the inflaton field lead to a quadrupole anisotropy [7, 8, 9, 10]:⟨a22⟩≈26.5v3v′2. (4)where v′ = [V ′/M3P]2 and V ′ = dV/dφ where φ is the inflaton field.The ratio, R, of these two quantities is then simplyme to investigate it in more detail.3

R ≈0.29v′2v2. (5)It is clear that depending upon the nature of the potential R might vary substan-tially.

Indeed there exist a plethora of current inflationary models including ‘new’inflation, chaotic inflation and extended inflation. Starobinsky in fact calculated [11]this ratio explicitly for chaotic inflation models [10] and found R ≈0.25 for a φ4potential.

This ratio has been concurrently re-examined in detail for all inflationarymodels in [3], where it is pointed out that R > 1 is possible but one is not completelyfree to vary v′ independently of v. For example, there exists a (weak) upper boundon R coming from the requirement that there be sufficient inflation to solve the flat-ness and horizon problems. Also, the spectrum of density perturbations and CMBanisotropies will vary from an n=1 Harrison-Zel’dovich spectrum when R gets large.Because existing inflationary models can allow a wide variety of scalar to tensorinduced quadrupole anisotropies, observationally distinguishing them could yield agreat deal of information about the mechanism of inflation(i.e.

see [3]). However, forthis same reason inflation does not tend to generically predict them to be comparablein magnitude.

In addition, while inflation allows the possibility that R can be O(1)or greater, all existing models suffer from the requirement that to produce adiabaticdensity perturbations which are themselves sufficiently small to be accomodated bythe observed CMB anisotropies, various parameters of the models already have tobe tuned to be very small. Whatever mechanism assures that they are small in thefirst place might make them much smaller than the maximal value allowed by CMBmeasurements.Further tuning of the potentials to obtain comparable tensor andscalar modes does not make the situation any more palatable.It should also be noted that because the contributions to the CMB quadrupole4

anisotropy from adiabatic density perturbations and gravitational waves are indepen-dent, they should be added in quadrature to determine the final predicted result:⟨a22⟩TOT = ⟨a22⟩scalar + ⟨a22⟩tensor(6)As soon as one term becomes slightly larger than the other, it can quickly come todominate the resulting anisotropy.Thus, while inflationary potentials can produce comparable anisotropies from adi-abatic density perturbations and gravitational waves, one wonders whether anothermechanism associated with inflation might also produce such a situation. I suggestone such possibility here, which has the attractive feature that it may result in den-sity perturbations which may provide a better fit to large scale structure observationsthan that resulting from adiabatic density perturbations.The suggestion is based on the calculation of the gravitational wave anisotropyitself.

Since the work of Grischuk [12], it has been clear that gravitational wave gen-eration during inflation reduces to the calculation of the generation of fluctuations fora massless scalar field in a background de Sitter expansion. Each polarization stateof gravitons behaves as a massless, minimally coupled real scalar field, with a nor-malization factor of√16πG relating the two.

As a result, therefore, long wavelengthfluctuations generated in any massless (or light) scalar field will have an amplitudetied to that for gravitational waves, up to a normalization constant.Quantum fluctuations in fields other than the inflaton will result not in adiabaticdensity perturbations, but rather in an isocurvature fluctuation spectrum [13, 14].While the net effect on the microwave background is remarkably similar (see below),the impact for structure formation differs in important ways(i.e. see [13, 15]).

Inparticular, the final density perturbation spectrum inside the horizon is flatter with5

wavenumber, with relatively more power on larger scales than the adiabatic spectrum.This is also what recent observations suggest [16].Because isocurvature fluctuations involve no real energy density perturbations (ortheir gauge invariant generalization) on scales larger than the horizon, fluctuationsin the number density of one species must be compensated by fluctuations in theremaining species, including radiation. This results in fluctuations in the temperatureof the radiation [15].

If the scalar field X comes to dominate the energy density of theuniverse then the resulting induced temperature fluctuations are identical to thoseinduced had these fluctuations been real energy density fluctuations. NamelyδTT ≈κδρXρtot≈κΩXδρXρX(7)where δρX is the energy density density in the fluctuating field, ρtot is the total energydensity and κ ≈−1/3 (X dominated) or −1/4 (radiation dominated).For a massless field, the energy density contained in the fluctuations is sufficientlysmall so that the effect on the microwave background is minimal (The energy densitycontained in the gravitational waves generated by inflation, for example, is small[2], but because of their direct effect on the metric, they can produce an observableδT/T.) A simple estimate for the δT/T induced due to horizon sized fluctuations insuch fields today from inflation at scale v is thenδTT ≈4v3(8)However, if the scalar field contributes significantly to the energy density today,isocurvature fluctuations can contribute both to structure formation, and to the CMBanisotropy.

There are two ways in which this might come about: (a) the scalar fieldwas massless during inflation, but received a non-zero mass at late times. A canonical6

axion is a prime example; or (b) the scalar field had a non-zero, but small mass atall times.This could come about for a goldstone boson field with small explicitsymmetry breaking due to high energy effects at, say, the Planck scale. We estimatethe resulting quadrupole anisotropy below.A de-Sitter expansion will produce for a massless, minimally coupled scalar fielda spectrum of fluctuations with Fourier components of wavenumber k, which havecrossed outside the horizon, given by (i.e.

[17, 7, 15, 10, 2, 18]):k3|δφk|22π2= H2π2(9)whereδφk ≡Zd3x φ(⃗x)ei⃗k·⃗x(10)These correspond to a background of φ particles with occupation number [10]: nk =H2/2k2.The energy density of the scalar field fluctuation is proportional to φ2. As a result,the fractional energy density stored in the fluctuating field is δρ(φ)/ρ(¯φ) = 2δφ/¯φ andso the rms density fluctuations of wavenumber k, in the φ field are given by* δρ(φ)ρ(¯φ)!2+k= 4k3|δφk|22π2 ¯φ2= H2π2 ¯φ2(11)This expression can then be inserted into standard formulas for the inducedquadrupole anisotropy [5, 8] to yield⟨a22⟩= (160π29)v"MP¯φ#2(12)We can now interpret this result in terms of various possible particle physicsmodels.

First, comparing eq. (12) to eq.

(3), we see that the two results are similarup to the overall multiplicative factorhMP/¯φi2. Thus, if¯φ ≈O(3) MP,(13)7

the scalar isocurvature fluctuation induced quadrupole can be comparable to thatinduced by gravitational waves.Consider first a standard cosmic axion. One finds that in this case the condition(13) cannot be easily satisfied.

This is seen by re-writing the axion field in its canonicalangular form ¯φ = ¯θfa, where fa is the PQ symmetry breaking scale. First, in orderfor the axion to exist at the inflationary scale, the PQ symmetry must break at orabove this scale.

For gravitational waves to contribute significantly to the observedquadrupole the inflation scale must be > O(1016) GeV [2]. It is well known thatfor fa > 1012 GeV, ¯θ is required to be ≪1 in order for axions not to overclose theuniverse.

Thus, for (13) to be satisfied, apparently fa > MP is required. However,even this is not sufficient, since in this case one would have to fix ¯θ < δθ, which isnot possible [19], nor would it be allowed by CMB constraints in any case.This problem for standard axions is intimately tied to the validity of the standardrelation between the axion mass and PQ symmetry breaking scale.

This relation maynot be correct. It has been argued that it could changed due to explicit global symme-try violation at the Planck scale[20, 21].

In this case (13) might be enforceable. Moregenerally, other pseudo-goldstone fields, related to spontaneous symmetry breakingnear the Planck Scale, with some (perhaps exponentially) small explicit violationmight play a role.

In either case, what is probably required is a mass which is at least3-4 orders of magnitude larger than the canonical axion for a given scale of sponta-neous symmetry breaking. One should also note that if such a field contributed onlya fraction of the closure density, this relation will be relaxed by the same fraction.Note that the masslessness of a scalar field such as the axion at the PQ breakingscale is not essential to the argument presented here.It will apply equally wellto any scalar field, be it a pseudo-goldstone boson with a mass from small non-8

perturbative effects arising at high energy— perhaps the Planck Scale— or any other“light” scalar field. As long as m2 ≪H2, the above arguments go through essentiallyunchanged.

More specifically, if t < 3H/m2 [10], where t is the time duration of theinflationary phase, all Fourier modes pushed outside the horizon will have amplitudesidentical to the massless case. For V > 1016 GeV this condition becomes, tmax ≈2.5 × 10−11(GeV/m)2sec, which is likely to be satisfied by any inflationary phase atvery early times, even for GeV scale masses.

Of course, once the age of the universebecomes comparable to m−1, the magnitude of |δφ|2 will redshift for all such modes,as the φ field begins to oscillate in its potential, but so will any non-zero mean valueof φ. Thus, the fractional value of δφ/φ or equivalently δρ/ρ will not change.

Indeedthis same phenomenon would presumably apply to an axion background field.Finally, some general comments are in order. As I have stressed here, existinginflationary models predict three sources of CMB anisotropies: adiabatic densityfluctuations coming from quantum fluctuations in the field driving inflation as itevolves in its potential, and gravitational wave tensor perturbations and isocurvaturedensity perturbations coming from quantum fluctuations in spin 2 graviton and/orspin 0 scalar fields present during inflation respectively.

The latter two are determinedby the vacuum energy density during inflation. I have concentrated primarily on therelationship between these two components here.

Nevertheless, one must recognizethat in order for this relationship to be pertinent, the CMB quadrupole anisotropydue to adiabatic density fluctuations must be subdominant.Indeed, in the mostgeneral case, the quadrupole anisotropy should be written:⟨a22⟩TOT = ⟨a22⟩adiabatic scalar + ⟨a22⟩isocurvature scalar + ⟨a22⟩tensor(14)Thus it is not entirely correct to state that the latter two terms can both be sig-nificant and comparable, independent of the inflationary potential, because in order9

to suppress the adiabatic component, as it now stands one must inevitably tune theinflationary potential, with the consequences for the shape of the power spectrumdiscussed in [3]. Nevertheless as I earlier stressed, in all existing inflationary modelsthere is an uncomfortable fine tuning required to make adiabatic perturbations ac-ceptably small.

It does not seem unreasonable to suppose that in the “correct” modelof inflation, if indeed there is such a thing, there should be a mechanism which cannaturally suppress, without fine tuning, the density fluctuations induced as inflationends and a radiation dominated expansion begins—perhaps to values well below themaximal allowed by CMB anisotropies. Such a mechanism need not, however, affectthe latter two sources of anisotropy, which are not directly sensitive to the transitionfrom de Sitter to radiation-dominated expansion.These latter statements are more speculative.However, until we have a trulycompelling inflationary model, which may inevitably be tied to our understanding ofPlanck Scale physics, we should not rule out such possibilities, and perhaps shoulduse them to get a handle on the actual physical mechanism underlying inflation.

Itis in this spirit that the arguments I present here may be useful.To conclude, inflation at present offers both the possibility of tying, through ajudiciously chosen potential, the magnitude of anisotropies due to adiabatic densityperturbations and those due to gravitational waves, and also the possibility thatisocurvature perturbations in scalar fields can arise which may also produce compa-rable quadrupole anisotropies. The latter two are tied together independent of thepotential, but depend sensitively upon the particle physics parameters of the scalarfields.

If isocurvature fluctuations are significant they could produce a potentially in-teresting power spectrum for large scale structure. It remains to see whether naturetakes advantage of any of these possibilities, or indeed of inflation itself.10

I thank Martin Rees for sparking my interest in this issue, and George Smoot andMartin White for helpful discussions.George, Martin, and Paul Steinhardt alsooffered useful comments and corrections to the original manuscript. Note: I recentlylearned of a preprint by Dolgov and Silk [22], which confirms the ratio[11] of tensorand scalar induced CMB anistropies in chaotic inflation in the context of COBE.References[1] G.F. Smoot et al, Astrophys.

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