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Contribution of Long Wavelength Gravitational Waves to the Cosmic Microwave

arXiv:hep-ph/9207239v1 14 Jul 1992YCTP-P16-92May 1992Contribution of Long Wavelength Gravitational Waves to the Cosmic MicrowaveBackground AnisotropyMartin White1Center for Theoretical Physics, Sloane LaboratoryYale University, New Haven CT 06511AbstractWe present an in depth discussion of the production of gravitational waves from an inflationary phasethat could have occurred in the early universe, giving derivations for the resulting spectrum andenergy density. We also consider the large-scale anisotropy in the cosmic microwave backgroundradiation coming from these waves.Assuming that the observed quadrupole anisotropy comesmostly from gravitational waves (consistent with the predictions of a flat spectrum of scalar densityperturbations and the measured dipole anisotropy) we describe in detail how to derive a value forthe scale of inflation of (1.5 −5) × 1016GeV, which is at a particularly interesting scale for particlephysics.

This upper limit corresponds to a 95% confidence level upper limit on the scale of inflationassuming only that the quadrupole anisotropy from gravitational waves is not cancelled by anothersource. Direct detection of gravitational waves produced by inflation near this scale will have towait for the next generation of detectors.To appear in: Physical Review D1Address after September 1, Center for Particle Astrophysics, University of California, Berkeley1

1. IntroductionIt has long been realized that a period of inflation in the early universe would lead to productionof a well defined spectrum of gravitational waves [1] (while inflation is not the only method of gen-erating a stochastic background of horizon-sized waves [2], it is certainly the most well motivated).Since the waves decouple early they are a potentially good probe of conditions in the early universe.In fact as pointed out in [3, 4, 5, 7] such waves, which are fluctuations in the metric, can resultin distortions of the cosmic microwave background radiation (CMBR) thus allowing informationon the amplitude of the waves (and the parameters of the inflationary phase that produced them)to be derived from the measured CMBR anisotropy.

This program proceeds in 3 steps. First theexpected fluctuation spectrum is derived, then the consequences of such a spectrum for CMBRanisotropies are calculated.

Finally comparing these predictions with the observations allows us toinfer the parameters of an inflationary theory that produces such waves.Recently an analysis of the implications of a CMBR anisotropy connected with gravitationalwaves from inflation was presented [8]. In this paper we fill in details of the results presented in[8] and further examine the issues addressed therein.

A review of each of the 3 steps mentionedabove will be given (many of the analytic results presented here have appeared in some formscattered in the literature, we have attempted to check, unify and reconcile previous estimates).The results can be compared to the recent COBE data [9] which gives the first positive measurementof anisotropies in the CMBR (previously this anisotropy had only been limited from above [10, 11]).On the theoretical side we give the details of derivations of some of the key results in the hopethat these will be of use. We present in one place the derivation of the scale invariant spectrumpredicted by exponential inflation (and a description of the scalar analogy), the energy densityρ or equivalently Ωg, the CMBR anisotropy produced and the predicted temperature correlationfunction.

We show how “redshift during horizon crossing” dramatically reduces the predicted Ωgfrom their asymptotic value for waves re-entering the horizon during the matter dominated epoch.Our results for the multipole moments of the CMBR temperature fluctuation agree with [5] (thoughwe use an approach due to [12] to derive the scale invariant spectrum) and are a factor of ≈2 largerthan [4]. With these results in hand we describe how the recent COBE observation is consistent withinflation and can limit the “scale near the end of inflation” assuming that the CMBR anisotropy isdue, at least in part, to inflation-produced gravitational waves (the contribution from scalar densityperturbations is constrained by the dipole anisotropy as we shall discuss).

The scale turns out tobe few×1016GeV [8] – an interesting energy for particle physics.The outline of the paper is as follows: section 2 establishes the notation and conventions. Insections 3 and 4 we discuss the scale-invariant spectrum predicted by exponential inflation and thespectral energy density.

Section 5 is devoted to the derivation of the anisotropy generated by astochastic spectrum of gravitational waves. The relation to the temperature correlation functionmeasured by double- and triple-beam experiments (e.g.

COBE) is derived in section 6 and the2

current measurement of the CMB anisotropy is compared to the theoretical predictions in section7. Section 8 contains the conclusions.2.

Cosmological historyWe will be interested in a cosmology which had an early period of inflation (where the dominantenergy density resided in the vacuum). To be specific let us suppose that the universe underwentexponential inflation, then became radiation dominated and is currently in a matter dominatedphase.

The assumption of exponential inflation is a good approximation for “successful” inflationmodels [13] based on general relativity since the requirement of a “slow roll” period in the evolutionof the inflaton field requires a very flat potential. The other possibility is that of power law inflation,where the scale factor grows like a power of time.

This changes the energy density spectrum (puttingmore power in long wavelength modes) and enhances the CMBR anisotropies for small l [5]. It seemsdifficult to have consistent power law inflation even in Brans-Dicke theory and we will not considerit further.Our metric is of the usual k = 0 Robertson-Walker formds2 = −dt2 + R2(t)dx2 + dy2 + dz2= R2(τ)−dτ 2 + d⃗x2(1)where dτ = dt/R(t) is the conformal time.

For an equation of state p = qρ the scale factor is apower of the conformal time, R(τ) ∼τ 2/(1+3q). For later convenience we list below the scale factor,which we normalize to unity today, for the history described above along with the dominant formof energy density for each epoch.

We assume that the transitions are sudden and match R(τ) and˙R(τ) at the transition points (this will be a good enough approximation for our purposes) [12]. Thismatching implies that τ is discontinuous across each transition and R(τ) has the formR(τ) =−(Hτ)−1vacuumτ ∈(−∞, −τ2)2τ1τ/τ 20radiationτ ∈(τ2, τ1/2)τ 2/τ 20matterτ ∈(τ1, τ0)(2)where H is the Hubble constant during inflation.

EquatingR τx0 dτ withR tx0 dt/R(t) we find τ0 = 3t0is the conformal time today and τ1 = (36t1t20)1/3 is the conformal time at matter-radiation equality(both τ0 and τ1 refer to τ as measured in the matter dominated era – in this respect our notationdiffers from that in [12]). The conformal time at the end of inflation is τ2 = τ0/√2Hτ1.

The Hubbleconstant H and vacuum energy density V0 driving the inflation are related byH2 = 8π3V0m2P l= 8π3 m2P lv(3)where v ≡V0/m4P l.3

It is useful to have an expression for the size of a wave, relative to the horizon size. During anormal FRW expansion the coordinate radius of the horizon grows as dr = dt/R(t) = dτ.

If theperiod of inflation is long the coordinate radius of the horizon is very nearly zero at the beginningof the radiation dominated era. Thus at time t∗the proper radius of the horizon is ≈R(τ∗)τ∗.

If weconsider a wave just entering the horizon at this time we find its comoving wavenumber, k, mustsatisfyλphys = R∗τ∗⇒k ≡2πRλphys= 2πτ∗(4)This leads us to the following two limits. A wave with kτ ≪2π is well outside the horizon while awave with kτ ≫2π is well within the horizon.3.

The scalar analogyA classical gravitational wave in the linearized theory is a ripple on the background space-timegµν = R2(τ) (ηµν + hµν)where ηµν = diag(−1, 1, 1, 1),hµν ≪1(5)In what follows we will work in transverse traceless (TT) gauge and denote the two independentpolarization states of the wave as +, ×. In the linear theory the TT metric fluctuations are gaugeinvariant (they can be related to components of the curvature tensor) [15].

We can write a planewave with comoving wavenumber |⃗k|hµν(τ, ⃗x) = hλ(τ;⃗k)ei⃗k·⃗xǫµν(⃗k; λ)(6)where ǫµν(⃗k; λ) is the polarization tensor and λ = +, ×. The equation for the amplitude hλ(τ;⃗k) isobtained by requiring the perturbed metric (5) satisfy Einstein’s equations to O(h).

One finds [15]¨hλ + 2˙RR˙hλ + k2hλ = 0(7)As noticed by Grishchuk [16] this is just the massless Klein-Gordon equation for a plane wave in thebackground space-time. Thus each polarization state of the wave behaves as a massless, minimallycoupled, real scalar field, with a normalization factor of√16πG relating the two.

We can use thefollowing heuristic argument to fix the√16πG: in the linearized theory one can write the Hilbertaction for the fluctuation using the contribution to the Ricci scalar from hµν. From the derivationof the stress-energy tensor T (h)µν [15] one sees R(h) = 14hµν;ρhµν;ρ so the corresponding Hilbert actionisS(h)H =√−g16πG12h(∇h+)2 + (∇h×)2i(8)4

which is the action for two real, massless, scalar fields φ+,× = (16πG)−1/2h+,× as expected. Thusthe study of quantum mechanical graviton production reduces to the study of the fluctuations of ascalar field in the curved background space-time, a factor√16πG relating the two cases.4.

The spectrum and ΩgQuantum fluctuations have an important consequence in a cosmology with inflation. Duringinflation short wavelength quantum fluctuations will get red-shifted out of the horizon, after whichthey freeze in.

In terms of (7) the freezing in of extrahorizontal modes is the statement that whenwavelength is much larger than the horizon the k2h term is negligible and the solution is a constantamplitude: ˙h = 0 (even before this limit is reached we note that the behaviour of waves outside thehorizon is qualitatively different from the damped oscillatory behaviour of waves within it). Whenthe mode re-enters the horizon at a much later epoch it appears as a long wavelength, classicalgravitational wave (in analogy with the case of scalar fluctuations considered by [17]).

One way ofthinking of this [18] is that the number of quanta describing a state of constant amplitude growsas a power of the scale factor, the energy of each quantum being redshifted. Thus when the modere-enters the horizon it represents a very large number of quanta.The spectrum of gravitational waves generated by quantum fluctuations during the inflationaryperiod can be derived by a sequence of Bogoliubov transformations relating creation and annihilationoperators defined in the various phases: inflationary, radiation and matter dominated [12, 19].The key idea is that for modes which have inflated outside the horizon the transitions betweenthe phases are sudden and the universe will remain in the quantum state it occupied before thetransition (treating each of the transitions as instantaneous is a good approximation for all but thehighest frequency graviton modes).

However the creation and annihilation operators that describethe particles in the state are related by a Bogoliubov transformation, so the quantum expectationvalue of any string of fields is changed (see e.g. [14] for a discussion of the calculation of Bogoliubovcoefficients).We calculate the statistical average of the ensemble of classical waves by considering the corre-sponding quantum average.

The simplest examples to consider are the 2-point functions. For thequantum theory we calculate the scalar 2-point function, following [12]∆≡k3(2π)3Zd3x ei⃗k·⃗x ⟨ψ| φ(⃗x, τ)φ(⃗0, τ) |ψ⟩where|ψ⟩= |de Sitter vac.⟩(9)with the fieldφ (⃗x, τ) =Zd3kei⃗k·⃗xa(k)φk(τ) + e−i⃗k·⃗xa†(k)φ∗k(τ)(10)where φk(τ) is a properly normalized solution of (7).

For the case of interest, a matter dominated5

universe today, φk is related to a Hankel function of order 3/2φk(τ) =e−ikτ√2k (2π)3/2 R(τ)1 −ikτ(11)In the exponentially inflating phase φk has the same functional form (with different R(τ)) and inthe radiation dominated phase the form is again as (11), with appropriate R(τ) and also with thefactor in parentheses absent.The quantum 2-point function is obtained by using the Bogoliubov coefficients relating thecreation and annihilation operators a and a† of the field in the 3 phasesarad=c1(⃗k)ainf(⃗k) + c∗2(⃗k)a†inf(−⃗k)(12)amat=c3(⃗k)ainf(⃗k) + c∗4(⃗k)a†inf(−⃗k)(13)We are interested in waves which are still well outside the horizon at the time of matter-radiationequality (kτ1 ≪2π) since these will give the largest contribution to the CMBR anisotropy today.Matching the field and its first derivative at τ2, τ1 in the limit kτ ≪2π we findc1 ≈−c2 ≈−Hτ1(kτ0)2,c3 ≈c∗4 ≈−3iH2k3τ 20(14)which is, correcting for differences in conventions, in agreement with [12]. Using (11,13,14) in (9)one obtains, for waves re-entering the horizon in the matter dominated era∆=H22(2π)3"3j1(kτ)kτ#2⇒∆GW =H2π2m2P l"3j1(kτ)kτ#2δλλ′(15)where in the last step the scalar result was multiplied by 16πG to obtain the corresponding result forgravitational waves.

The corresponding expression for waves entering during the matter dominatedera can be obtained by replacing the factor in brackets by j0(kτ) as one would expect (since j0 givesthe right τ dependence from (7) and the amplitude as kτ →0 should be the same).We match this to our classical ensemble of gravitational waves, hλ(τ;⃗k), by writinghλ(τ;⃗k) = A(k)aλ(⃗k)"3j1(kτ)kτ#with λ = +, ×(16)where the term in [· · ·] is the real solution to (7) in the matter dominated phase and aλ(⃗k) is arandom variable with statistical expectation valueDaλ(⃗k)aλ′(⃗q)E= k−3δ(3)(⃗k −⃗q)δλλ′(17)6

The 2-point function analogous to ∆in this case is simply∆′λλ′ = A2(k)"3j1(kτ)kτ#2δλλ′(18)Matching the quantum and classical 2-point functions gives us the well known prediction for the(k-independent) spectrum of gravitational waves generated by inflationA2(k) =H2π2m2P l= 83πv(19)The reason for the explicit factor of k3 introduced in the definition of ∆(9) can be seen byconsidering the expression for the spectral energy density in (classical) gravitational waves. In ournotationρ = k2phys16πGDh2+ + h2×E(20)where ⟨· · ·⟩indicates an average over many wavelengths/periods as well as an average over thestochastic variable aλ(⃗k).

kphys = k/R(τ) is the physical wavenumber. For a fixed (non-stochastic)spectrum of waves we could write the average over space as an integral over Fourier modes usingParseval’s theorem and then relate the “power spectrum” |h+,×(k)|2 to the correlation function (9).With (16) and (17) as defined the stochastic average produces the same result.

Explicitlykdρλdk = k2phys4G ∆′λλλ = +, ×(21)which goes as R−4(τ) as expected (an overbar denotes averaging). Up to the time evolution factor(3j1(kτ)/kτ)2 for matter and j0(kτ) for radiation dominated phases this agrees with [20].

Includingthis factor for waves re-entering during the matter dominated phase (i.e. considering evolution asthe wave enters the horizon) leads to a significant suppression of ρ.

Writing the energy density asa fraction of the closure density one finds, for waves just entering the horizon [8],Ωg ≡Xλ=+,×kρcdρλdkk=2π/τ=(16v/9RDv/π2MD(22)In deriving this use has been made of the fact that kphys = 1(2)πHHC for waves crossing the horizonduring the matter (radiation) dominated phase (HHC is the Hubble constant at the time the wavere-enters the horizon).Will we be able to see this energy density with terrestrial or astrophysical gravitational wavedetector? The most sensitive gravitational wave limit currently comes from the timing of millisecond7

pulsars. This is sensitive to “short” wavelengths (periods of order years) and puts an upper limiton the energy density in such waves Ωg < 9 × 10−8 (68%) (this limit improves as the fourth powerof the observing time: T 4obs) [28].

Such short wavelength modes entered the horizon during theradiation dominated era. If we write the Hubble constant today H0 = 100h km/s/Mpc then thesewaves contribute to Ωg today an amount suppressed by a factor of ρrad/ρc ≈4 × 10−5h−2 comparedto horizon crossing since they redshift with one extra power of R compared to matter.

For values ofv ∼10−9, consistent with the anisotropy of the CMBR (see later), the energy density per logarithmicfrequency interval in short wavelength modes is predicted to be Ωg ∼10−13 [8], which is ∼6 ordersof magnitude lower than the millisecond pulsar limit. Even the proposed LIGO detector [29] whichhas a sensitivity of Ωg ∼10−11 is short of the mark by 2 orders of magnitude.

Thus we expect thata positive detection of the signal will require a significant advance in technology.Since the predicted CMBR anisotropy multipoles, up to l ∼9, from gravitational waves andscale invariant scalar fluctuations are very similar [7] and direct detection of gravitational wavesis still some way offthe outlook for determining unambiguously that a significant fraction of theobserved CMBR anisotropy comes from gravitational waves seems bleak.5. Anisotropy from the stochastic backgroundIt is conventional to expand the CMBR temperature anisotropy in spherical harmonicsδTT (θ, φ) =XlmalmYlm(θ, φ)(23)We can calculate the prediction of a given spectrum of gravitational waves in terms of the alm.

Thetemperature fluctuation due to a gravitational wave hµν can be found in the linearized theory to be[23]δTT = −12Z re dΛ ∂hµν(τ, ⃗x)∂τˆxµˆxν(24)where Λ is a parameter along the unperturbed path and the lower (upper) limit of integrationrepresents the point of emission (reception) of the photon.Now we project out a multipole and calculate the rotationally symmetric quantityDa2lE≡*Xm|alm|2+=XmZdΩdΩ′ Y ∗lm(Ω)Ylm(Ω′)*δTT (Ω)δTT (Ω′)+(25)where using (6,17,24) we have for a spectrum of waves*δTT (Ω)δTT (Ω′)+=14Xλ=+,×Z d3kk3ZdΛ ˙h(τ; k)ei⃗k·⃗xǫµν(⃗k; λ)ˆxµˆxν×ZdΛ′ ˙h(τ ′; k)ei⃗k·⃗x′ǫµν(⃗k; λ) ˆx′µ ˆx′ν∗(26)8

and an overdot represents a derivative with respect to τ. We use the rotational symmetry of a2l tomake the replacementXmY ∗lm(Ω)Ylm(Ω′) =XmY ∗lm(Ωkx)Ylm(Ω′kx)(27)where Ωkx indicates that the angles are defined with respect to ˆk.

We can evaluate directlyǫµν(⃗k; λ)ˆxµˆxν =δ+λ cos(2φ) + δ×λ sin(2φ)sin2 θ(28)where θ, φ are the usual spherical angles relating ˆk and ˆx.First concentrate on the angular integrations. Expanding the exponentialei⃗k·⃗x =∞Xn=0in(2n + 1)jn(kx)Pn(cos θkx)(29)it is not hard to show (formula 7.125 of Gradshteyn and Ryzhik [24] is useful) thatIlm(k, x)≡ZdΩkx Y ∗lm(Ωkx) ei⃗k·⃗xǫµν(⃗k; λ)ˆxµˆxν=πvuut2l + 14π(l + 2)!

(l −2)! HXnin(2n + 1)jn(kx)c−2δnl−2 + c0δnl + c2δnl+2(30)whereH=(δ+λ −iδ×λ )δ+2m + (δ+λ + iδ×λ )δ−2m(31)1/c−2=(2l −1)(2l + 1)(l −3/2)(32)1/c0=−12(2l −1)(2l + 3)(l + 1/2)(33)1/c+2=(2l + 1)(2l + 3)(l + 5/2)(34)Inserting this back into the expression for a2l and using (16)Da2lE= 14XmXλ=+,×Z dkdΩkkZdΛ A(k) ddτ 3j1(kτ)kτ!Ilm(k, x)2(35)The integral over the path dΛ can be parameterized by the distance from the origin along theline of sight, r, so |⃗x(r)| = r and τ(r) = τ0 −r.

We will defer consideration of the limits of the kintegral for the moment. ThusDa2lE= 36π2(2l + 1)(l + 2)!

(l −2)!Zkdk A2(k)|Fl(k)|2(36)9

where the function Fl(k) is defined asFl(k) ≡Z τ0−τ10dr dd(kτ)j1(kτ)kτ! "jl−2(kr)(2l −1)(2l + 1) +2jl(kr)(2l −1)(2l + 3) +jl+2(kr)(2l + 1)(2l + 3)#(37)Accounting for the factor of two difference between the definitions of A2(k) this is precisely theresult of [5], and is a factor of ≈2 larger than the earlier result of [4].

The integral in equation(37) can be expressed in terms of elementary functions and the sine integral but the result is verycumbersome and is not presented here.For waves entering the horizon during the matter dominated regime the upper limit of the kintegration is 2π/τ1. The results are very insensitive to the exact choice of upper limit since theintegral is dominated by small k ≈2π/τ0, i.e.

waves that have recently entered the horizon. Forvery high frequency modes the transition between phases of the universe is no longer sudden andgraviton production is suppressed.

Taking this high frequency cutoffto be k = 2π/τ1 (i.e. restrictingattention to waves that entered the horizon during the matter dominated era) introduces negligibleerror.The lower limit of the k integral is zero, since waves of arbitrarily long wavelength cancontribute2.

From the form of the integrand however one can see that the contribution to ⟨a2l ⟩tends to zero as kn (n ≥1) as kτ →0. Thus the contribution from very long wavelength modes issuppressed as one might expect on physical grounds.One can now calculate the predicted ⟨a2l ⟩for any spectrum A(k) of gravitational waves from(36).

For the scale-invariant spectrum (19) of exponential inflation the ⟨a2l ⟩are shown in table 1.The calculation of the expectation value ⟨a2l ⟩is not the end of the story however. Before we comparepredictions with observations we must also consider the statistical properties of a2l .

The fact thatthe “weakly coupled” nature of inflation predicts the alm to be independent makes this problemtractable. Given that each of the alm are independent Gaussian random variables the probabilitydistribution for each a2l , whose mean ⟨a2l ⟩we calculated, is of the χ2 form and can be writtenP(y)dy = yl−1/2e−yΓl + 12dywherey = 2l + 12a2l⟨a2l ⟩(38)This agrees with [6].

One can calculate the confidence levels for a2l in terms of the incompletegamma function (values for the 68, 90 and 95% (lower) confidence levels can be found in table 1).We note in passing that the modal value of a2l for any universe is (2l −1)/(2l + 1) times the mean.6. The correlation functionLimits quoted on the multipole moments a2l can be directly compared to (36) to constrain theinflationary theory.

However many experiments (and importantly COBE) are of the double- or2We thank Mark Wise and Vince Moncrief for very helpful discussions on this point.10

triple-beam type and do not simply measure any one multipole moment but can constrain thecorrelation function [9, 20]C(θ21; σ) ≡*δTT (ˆx1; σ)δTT (ˆx2; σ)+21(39)where the average is over all positions ˆx1, ˆx2 on the sky with ˆx1 · ˆx2 fixed, and σ is a measure of theangular response of the detector. We shall take the angular response to be Gaussian [21]dR(θ, φ) = θdθdφ2πσ2 exp"−θ22σ2#(40)where θ, φ are the angles relative to the beam direction and σ ≪1 which justifies the use of thesmall angle expansion.

We define the “smeared” δT/T byδTT (ˆx1; σ)≡XlmalmZdΩ′ R(Ω′)Ylm(Ω)(41)=XlmalmZ ∞0dx Pl(1 −σ2x)e−xYlm(ˆx1)(42)where Ω′ is the direction Ωmeasured relative to ˆx1 and we have approximated cos θ′ ≈1 −θ′2/2for small θ′. Note that the term in parenthesis suppresses the higher l modes as we would expectphysically.

We can expand this term for small σ2Z ∞0dx Pl(1 −σ2x)e−x=∞Xn=0(−σ2)nP (n)l(1)(43)=lXn=0(−σ2)n12nn! (l + n)!

(l −n)! (44)Making the approximation (l + n)!/(l −n)!

≈(l + 1/2)2n for large l we can write the suppressionterm as: exp [−(l + 1/2)2σ2/2] (and at the same level of approximation we can replace (l + 1/2)2with l(l + 1) as used by [9]). This exponential suppression proves a good approximation for σ < 0.1which is the range for most detectors.

A simple exercise in manipulating spherical tensors allowsus to write⟨Ylm(ˆx1)Y ∗l′m′(ˆx2)⟩21 = 14π δll′ δmm′ Pl(ˆx1 · ˆx2)(45)so we can express the correlation function (39) in a simple from as (see [9, 20])C (ˆx1 · ˆx2; σ) = 14π∞Xl=0a2l Pl (ˆx1 · ˆx2) e−(l+1/2)2σ2(46)11

The predicted correlation function using the moments of table 1 is shown in figure 1 for agaussian beam width of σ = 10o. The error bars represent the upper and lower limit of the 68%confidence region arrived at using (38) for the distribution of a2l /v (there is no simple analytic formfor the probability distribution of C(θ, σ) so figure 1 was obtained from a Monte-Carlo) and thesolid line is C(θ, σ) evaluated using a2l = ⟨a2l ⟩.

The correlation function inferred from the COBEsky maps [9] is in agreement with the form shown in figure 1.7. Comparison with observationsThe recent results of COBE can be summarized as a non-zero value of the quadrupole moment a22.If we require a Harrison-Zel’dovich spectrum the quadrupole, from the Qrms−P S value, is measuredto be [8, 9]a22 = (4.7 ± 2) × 10−10(47)An early inflationary phase produces not only gravitational waves but also a scale invariantspectrum of scalar density perturbations.

To limit the contribution of these latter to a22 we canrequire the induced dipole from long wavelength scalar modes not greatly exceed the observeddipole anisotropy.For a given (e.g.flat) spectrum one can use this to place limits on all thehigher multipoles.At the 90% confidence level an upper limit of a22 ≈2 × 10−10 [7] has beenderived (ignoring the transfer function corrections). Equivalently fitting the observed clustering toa primordial spectrum yields a22 ≈(2 −10) × 10−11 [25].

Such estimates suggest a large fraction ofthe quadrupole anisotropy may be due to long wavelength gravitational waves.Let us assume that the observed quadrupole is due entirely to gravitational waves. Using ⟨a22⟩/vfrom table 1 and including the distribution for the measured a22 (gaussian) and predicted a22/v (38)one can infer a distribution for v. The result obtained from a Monte-Carlo is shown in figure 2.This corresponds to a mean v = 6.1 × 10−11, and a modal v = 4 × 10−11 with non-gaussian errors.The allowed range of v from figure 2 can be summarized as [8]2.3 × 10−11

The upper limit quotedcorresponds to a strict upper limit on the scale of inflation assuming that the quadrupole momentdue to gravitational waves is not being cancelled to any significant degree by other sources (e.g.scalar density perturbations). A large cancellation between two sources of anisotropy would beunlikely.12

No realistic model of inflation currently exists, however most are based on the idea that a phasetransition occurred in the early universe to produce the vacuum energy density to drive the inflation.If such a transition were to occur then the natural scale would seem to be the scale of unificationof the coupling constants in the context of a Grand Unified Theory (GUT) (an exception is thecase of chaotic inflation). The recent precision measurements from LEP have allowed us to sharpenour predictions for this scale.

Assuming a supersymmetric GUT with the supersymmetry breakingscale ≈1TeV, one predicts [26] a unification scale of (1 −3.6) × 1016GeV. For a theory basedon SO(10) one finds M = 1015.8±.22GeV [27].

The limits from proton decay experiments are verytheory dependent, however they give a lower bound on M. The current limits are near or largerthan 1015GeV for fashionable GUT theories.Unless there is significant fine tuning or hierarchies in the theory the vacuum energy densityassociated with a transition from a GUT with scale MG is V0 = (κMG)4 where κ is a (calculable)number usually in the range κ4 = .01 −1. This puts the GUT scale very close to the scale ofinflation inferred above – a “numerical coincidence” which is both suggestive and exciting.8.

ConclusionsAs the observations of CMBR anisotropies are refined and other gravitational wave detectorscome on line the results presented here can allow a comparison of theoretical predictions andexperimental results, which can help to constrain models of inflation and early universe physics.The recent COBE results allow the possibility that the observed quadrupole anisotropy resultsfrom gravitational waves generated during inflation with a scale of (1.5 −5.2) × 1016GeV – a veryinteresting range of energies.If the anisotropy is due to gravitational waves, confirmation willprobably not come from existing or currently planned gravitational wave detectors. Further resultsare eagerly awaited.I would like to thank L. Krauss, V. Moncrief, M. Turner and M. Wise for extremely usefulconversations.

I also wish to thank E. Gates for comments on the manuscript.13

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Figure CaptionsFigure 1: Results of a Monte-Carlo for the predicted correlation function C(θ; σ) generated by aflat spectrum of gravitational waves. The error bars represent the upper and lower limits of the68% confidence region arrived at using the predicted distribution of a2l /v.

The solid curve is thecorrelation function calculated using ⟨a2l ⟩. The beam width is taken to be σ = 10o.Figure 2: The probability distribution for the scale v ≡V0/m4P l of inflation as determined by Monte-Carlo from the COBE measurement of the quadrupole anistropy assuming that the whole anistropyis due to gravitational waves.Tables16

a2l /⟨a2l ⟩l⟨a2l ⟩/v68/90/95% CL27.74.63/.32/.2334.25.69/.40/.3143.10.73/.46/.3752.50.76/.51/.4262.12.78/.54/.4571.85.80/.57/.4881.64.81/.59/.5191.48.82/.61/.53101.35.83/.63/.55Table 1: Multipole coefficients a2l for modes l = 2 −10 predicted for a stochastic background ofgravitational waves generated by exponential inflation. The 68, 90 and 95% (lower) confidencelevels are also shown as fractions of ⟨a2l ⟩.17

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