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Observing the Inflaton Potential

arXiv:hep-ph/9304228v1 7 Apr 1993SUSSEX-AST 93/4-1FERMILAB–PUB–93/071–AApril 1993Observing the Inflaton PotentialEdmund J. Copeland,∗Edward W. Kolb,(†,‡)Andrew R. Liddle,§ and James E. Lidsey†∗School of Mathematical and Physical Sciences, University of Sussex,Brighton BN1 9QH, U. K.†NASA/Fermilab Astrophysics Center,Fermi National Accelerator Laboratory, Batavia, Illinois60510‡Department of Astronomy and Astrophysics, The Enrico Fermi Institute,The University of Chicago, Chicago, Illinois60637§Astronomy Centre, School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH, U. K.We show how observations of the density perturbation (scalar) spec-trum and the gravitational wave (tensor) spectrum allow a reconstructionof the potential responsible for cosmological inflation. A complete func-tional reconstruction or a perturbative approximation about a single scaleare possible; the suitability of each approach depends on the data avail-able.

Consistency equations between the scalar and tensor spectra arederived, which provide a powerful signal of inflation.PACS number(s): 98.80.–k, 98.80.Cq, 12.10.Dmemail: ∗edmundjc@central.sussex.ac.uk;(†,‡)rocky@fnas01.fnal.gov;§arl@starlink.sussex.ac.uk;†jim@fnas09.fnal.gov

One of the most exciting aspects of the recent detection of large angle microwavebackground anisotropies by COBE [1] is the possibility that part of the anisotropy ob-served is due to long wavelength gravitational wave (tensor) modes instead of (scalar)density perturbations. In general the influence of scalar and tensor modes on mi-crowave background anisotropies differs as a function of angular scale, and the useof measurements on different scales may allow one to separate the anisotropies intotheir scalar and tensor components.

This has recently been considered by Crittendenet al [2].This prospect is especially exciting for models of cosmological inflation, proposedover a decade ago [3] as a possible resolution of a number of otherwise puzzlingaspects of the standard hot big bang cosmology [4]. Inflation has long been knownto predict that both scalar modes [5] and tensor modes [6, 7, 8] should exist on allastrophysically relevant scales.

Although the generic prediction from inflation has inthe past been advertised as a flat (Harrison–Zel’dovich) scalar spectrum and a tensorspectrum of negligible amplitude, the rapid improvement of observational data hasled many researchers [9] to emphasize recently the importance of taking the detailedinflationary predictions seriously. Typically the predictions from inflation are thatthe scalar spectrum possesses a scale dependence, which is weak in many models butcan be rather marked in others.

And though the amplitude of tensors may typicallybe less than that of the scalars, this does not necessarily imply that it is negligible.This is problematic from a large scale structure viewpoint, since different inflation-ary models offer a range of predictions and there is currently no clear guidance fromparticle physics as to which inflationary models may be suitable. The correct inputone should make into a large scale structure model is therefore unknown.

However,from an inflationary viewpoint this is a promising feature, as it raises the possibilitythat improved observations may allow one to distinguish between inflationary mod-els. The aim of this Letter is to investigate the use of observations precisely to thisend, by deriving equations which allow one to proceed from a knowledge of the scalarand/or tensor spectrum to a determination of the inflaton potential.

As a very usefulby-product, we derive a consistency relation between the allowed scale-dependences ofthe scalar and tensor modes. This is a powerful discriminant for inflationary modelsin general, as it does not depend on a specific choice of inflationary model.Reconstruction of the inflaton potential in this manner was first considered byHodges and Blumenthal [10] (hereafter HB).

We improve upon their results in two1

important ways. Firstly, we consider both scalar and tensor modes, whereas theyrestricted their study to the scalars alone.This is a vital improvement, because,as HB acknowledged and we rederive, the scalars alone are insufficient to uniquelydetermine the inflaton potential — such a reconstruction is possible only up to anundetermined constant, and as the reconstruction equations are nonlinear this leadsto functionally different potentials giving rise to the same spectrum.

The tensors(even just the tensor amplitude at a single scale) provide just the extra informationneeded to lift this degeneracy. Secondly, their analysis made explicit use of the so-called slow-roll approximation.

It is well known that this approximation breaks downunless both the scalar spectrum is nearly flat and the tensor amplitude is negligible.We consider the inflation dynamics in full generality. However, general expressionsfor the perturbation spectra are not known, and one must use slow-roll there.

It isshown in an accompanying paper [11] that this hybrid approach offers substantialimprovements over pure slow-roll results.The equations of motion are most conveniently written in the H(φ) formalism[12]. An isotropic scalar field φ in a spatially flat universe satisfies(H′)2 −32κ2H2=−12κ4V (φ)(1)κ2 ˙φ=−2H′,(2)provided that ˙φ does not pass through zero, where overdots are time derivatives,primes are derivatives with respect to φ, and κ = 8πG = 8π/m2P l. The usual slow-rollapproximation amounts to neglecting the first term in Eq.

(1) and its φ-derivative.The amplitudes of the scalar and tensor modes may be written using the standardexpressions asAS(φ)=√2κ28π3/2H2(φ)|H′(φ)|(3)AG(φ)=κ4π3/2 H(φ),(4)respectively. AS is equivalent to P 1/2(k)/3√2π in HB, to δH of Ref.

[13], and for aflat spectrum equal to 4πǫH of Ref. [7].

A2G is equivalent to Pg/32π of Ref. [13].

Oneimmediately notes thatAGAS=√2κ|H′|H=√2κd ln AGdφ ,(5)2

so the inflationary condition ¨a > 0 implies AG < AS. However, the relative contribu-tion of tensors to scalars for large angle microwave background anisotropies is givenroughly (for sufficiently flat spectra) by the ratio 25A2G/2A2S [11], so it is possible forthe tensor contribution to dominate the anisotropy.The spectra are quoted above as functions of φ—that is, we are given the amplitudewhen the scalar field takes a particular value.

To compare with observations we mustrelate φ to a given cosmological scale λ. This is achieved by utilizing the formulaN(φ) ≡Z tteH(t′)dt′ = −κ22Z φeφH(φ′)H′(φ′)dφ′,(6)which gives the number of e-foldings between a scalar field value φ and the end ofinflation at φ = φe.

Each length scale λ is associated with a unique value of φ whenthat scale crossed the Hubble radius during inflation, indicated by writing λ(φ). Thatvalue of φ is also associated with a value a(φ) of the scale factor.

We can make useof Eq. (6) to relate a(φ) to the value of the scale factor at the end of inflation, ae:a(φ) = ae exp[−N(φ)], which allows us to express λ(φ) asλ(φ) = exp[N(φ)]H(φ)a0ae.

(7)Differentiating Eq. (7) with respect to φ yieldsdλ(φ)dφ= ± κ√2ASAG−AGASλ.

(8)Note that the reconstruction equation derived by HB [their Eq. (2.10)] has only thefirst term on the right hand side of Eq.

(8), indicating their assumption of slow-rollbehavior (which here amounts to neglecting terms of order A2G/A2S).Substituting Eq. (8) into Eq.

(5) givesλAG(λ)dAG(λ)dλ=A2G(λ)A2S(λ) −A2G(λ). (9)This is a very important equation, because it is valid for any inflaton potential andindicates a strong connection between the forms of the scalar and tensor spectra pro-duced by inflation.

The left hand side is essentially just half of the (scale-dependent)spectral index of the tensor spectrum. Potentially, this provides a powerful discrimi-nator as to the correctness of inflation.

We shall refer to it as the consistency equation.It highlights the asymmetry in the correspondence between the scalar and tensor spec-tra. If one were given the tensor spectrum, then a simple differentiation supplies the3

unique scalar spectrum. However, if a scalar spectrum is supplied, then this first-order differential equation must be solved to find the form of AG(λ).

This leaves anundetermined constant in the tensor spectrum and, as the consistency equation isnonlinear, this implies that the scalar spectrum alone does not uniquely specify thefunctional form of the tensors. However, knowledge of the amplitude of the tensorspectrum at one scale is sufficient to determine this constant and lift the degeneracy.It is the tensor spectrum one requires to proceed with reconstruction.

Once theform of the tensor spectrum has been obtained, either directly from observation orby integrating Eq. (9), the potential, as parametrized by λ, may be derived by sub-stituting Eqs.

(3) and (4) into Eq. (1).

This givesV [φ(λ)] = 16π3A2G(λ)κ4"3 −A2G(λ)A2S(λ)#,(10)where the final term in the square brackets again improves on HB. Finally, integrationof Eq.

(8) yields the function φ = φ(λ) asφ(λ) = ±√2κZ λ dλ′λ′AS(λ′)AG(λ′)A2S(λ′) −A2G(λ′) = ±√2κZ AGdA′GAS[A′G]A′2G,(11)where we have absorbed the integration constant by taking advantage of the freedomto shift φ by a constant. The second integral follows after substitution of the consis-tency equation and is appropriate if the functional form of AS as a function of AG isknown.

The functional form of V (φ) follows by inverting Eq. (11) and substitutingthe result into Eq.

(10).The reconstruction equations are Eqs. (9), (10) and (11).

We emphasize again thateven an arbitrarily accurate determination of the scalar spectrum will not allow oneto determine the inflaton potential — at least a minimal knowledge of the tensorsis required. Ultimately, though, one might hope to overdetermine the problem byhaving observational knowledge of both spectra over a range of scales.

The consistencyequation (9) must then be satisfied, or the inflationary hypothesis has been disproved(up to the accuracy of the slow-roll approximation for the perturbation spectra).The reconstruction equations allow a functional reconstruction of the inflatonpotential. For suitably simple spectra, this can be done analytically, and in an ac-companying paper [11] we illustrate this for the well-known cases of scalar spectrawhich are exactly scale-invariant, logarithmically corrected from scale-invariance andexact power-laws.

The earliest observations with an accuracy useful for our purposes4

are likely to only provide such simple functional fits.For advanced observations,however, one might expect that the reconstruction equations would have to be solvednumerically. There are additional issues related to observational errors which we donot investigate here (but see Ref.

[11]).An alternative approach, useful for obtaining mass scales, is to concentrate ondata around a given length scale λ0, and perturbatively derive the potential aroundits corresponding scalar field value φ0 ≡φ(λ0).If we know AG(λ0) and AS(λ0)separately, then V (φ0) follows immediately from Eq. (10).

In order to make furtherprogress, one also needs information regarding the derivatives of the spectra.Ofcourse, the measurement of these derivatives requires knowledge of the spectra overat least a limited range of scales, so this process is equivalent to a Taylor expansionof the functional reconstruction [14].To obtain V ′(φ), one needs only the derivative of the scalar spectrum, or equiv-alently its spectral index. This is fortunate, as its tensor equivalent would be muchharder to observe.

With the scalar spectral index n (in general a function of scale)defined as usual by1 −n = d ln A2S(λ)d ln λ,(12)one can show thatV ′(φ0) ≡dV (φ)dφλ=λ0= ± 16π3√2κ3A3G(λ0)AS(λ0)"7 −n0 −(5 −n0)A2G(λ0)A2S(λ0)#,(13)where n(λ0) ≡n0. If one wishes, this can be simplified into the slow-roll approxima-tion (in which n0 ≈1) by ignoring the final term in the square brackets.One can continue this process.

At no stage is knowledge of the tensor spectrumderivative required, because the consistency equation can always be used to removeit. Given the second derivative of the scalars (equivalently the first derivative of thescalar spectral index), one can derive an expression for V ′′(φ0), quoted in Ref.

[11],but it is too cumbersome to reproduce here. Its slow-roll limit does not require n′0,and isV ′′sr(φ0) = 4π3κ2A2G(λ0)A2S(λ0)h4(n0 −4)2A2G(λ0) −(1 −n0)(7 −n0)A2S(λ0)i.

(14)This offers the prospect of determining whether the inflaton potential is concave orconvex when the presently observable universe crossed outside the Hubble radius5

during inflation. We note immediately that V ′′ is positive if 1 < n0 < 7.

It is theamplitude of the tensor perturbations at a particular scale which yields informationregarding the mass scale at which these processes are occurring during inflation. Thesteepness of the potential, measured by the dimensionful parameter V (φ0)/|V ′(φ0)|,is determined by the ratio AS(λ0)/AG(λ0).Let us illustrate by example.

Within a few years a combination of microwavebackground anisotropy measurements should give us some information about thescalar and tensor amplitudes at a particular length scale λ0 (corresponding to anangular scale θ0) [2]. A hypothetical, but plausible, data set that this might providewould be AS(λ0) = 1 × 10−5; AG(λ0) = 2 × 10−6; n0 = 0.9.

This would lead toV (φ0)=(2 × 1016GeV)4±V ′(φ0)=(3 × 1015GeV)3V ′′sr(φ0)=(5 × 1013GeV)2. (15)In this way cosmology might be first to get a “piece of the action” of GUT–scalephysics.In this Letter we have discussed the promising possibility of large scale structureobservations, particularly of tensor modes, providing rather specific information asregards the physics of the Grand Unified era.

We have derived equations which allowa knowledge of either the scalar spectrum, the tensor spectrum, or preferably both,to be used to reconstruct the potential of the inflaton field. We have also noted aconsistency equation, by which the scale-dependences of the spectra must be relatedif their origin lies in an inflationary era.

This potentially provides a powerful test ofinflation; the minimum knowledge required to implement it would be knowledge ofthe scalar spectrum across a range of scales plus the amplitude of the tensor spectrumat two of the wavelengths. [Technically the minimum is the tensor spectrum plus thescalar amplitude at a single scale, but observationally that would be considerablymore demanding.

]In an accompanying paper [11], all the issues herein are discussed in greater detail.As well as providing examples of functional reconstruction, we discuss in detail theopportunities available in both presently available and expected future observationsfor carrying out the program we have outlined here. While the ambitious aim offull reconstruction appears to lie some way into the future, we are optimistic as tothe short-term possibilities, as tantalizingly indicated in [2], of obtaining at least a6

perturbative reconstruction of the inflaton potential and a window on GUT scalephysics.EJC, ARL and JEL are supported by the Science and Engineering Research Coun-cil (SERC) UK. EWK and JEL are supported at Fermilab by the DOE and NASAunder Grant NAGW–2381.

ARL acknowledges the use of the Starlink computer sys-tem at the University of Sussex. We would like to thank D. Lyth, P. J. Steinhardt,and M. S. Turner for helpful discussions.References1.

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This procedure is similar in spirit to that used by M. S. Turner, “On the Pro-duction of Scalar and Tensor Perturbations in Inflationary Models,” Fermilabpreprint FNAL-PUB-93/026-A (1993), in calculating AS and AG from the po-tential.8

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