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Inflaton potential reconstruction without slow-roll Ian J. Grivell and Andrew R. Liddle Astrophysics Group, The Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom (February 1, 2008) Wedescribeamethodofobtainingtheinflationarypotentialfrom observationswhichdoesnotuse the slow-roll approximation. Rather, the microwave anisotropy spectrum is obtained directly from a parametrized potential numerically, with no approximation beyond linear perturbation theory. This permits unbiased estimation of the parameters describing the potential, as well as providing the full error covariance matrix. We illustrate the typical uncertainties obtained using the Fisher information matrix technique,studyingthe λφ4 potential in detail as a concrete example. PACS numbers: 98.80.Cq, 98.70.Vc astro-ph/9906327 0 0 0 2 I. INTRODUCTION scale was considerably larger than the Hubble radius), thenitisreasonabletobelievethattheyaroseviathein- n a Thedeterminationoftheinitialpowerspectrumofper- flationarymechanism[3,4],ratherthanbeinginducedby J turbationsintheUniverseisanessentialstepinusingthe topological defects or other causal mechanism. It would 9 cosmic microwave background to constrain cosmological be further encouraging if the perturbations provedto be 1 parameters. Indeed, without an understanding of these adiabatic and gaussian, because although inflation mod- initial perturbations, the cosmic microwave background els exist which violate those conditions, they are proper- 2 v in isolation says nothing about the values of parameters ties of the simplest inflation models. 7 such as the Hubble parameter h and the density param- Attempting to derive the underlying inflation model 2 eterΩ0. The reasonisthat the effectofthe cosmologyis from observations has become known as inflaton poten- 3 on the dynamics of the perturbations, and a single time- tialreconstruction[5,6]. Studieshavefocussedonmodels 6 slice, such as the perturbations at last scattering, says whereinflationisdrivenbyasinglescalarfieldφ,moving 0 nothing about the dynamics. Usually, this problem is in a potential V(φ). Such models indeed give perturba- 9 9 circumvented by assuming a parametrization of the ini- tions which are passive, adiabatic and gaussian, though / tial conditions; the cosmological parameters then enter they come in two types, scalar (density perturbations) h p via the dynamics converting these initial conditions into and tensor (gravitational waves) which need not be per- - theconditionsatlastscattering. Forexample,acommon fectpower-laws. Eventhentheyarenotthemostgeneral o assumptionisthatthereisapower-lawspectrumofgaus- class of models leading to that set of properties, because r t sian adiabatic scalar perturbations. This is a popular modelswherethereismorethanonescalarfieldcanalso s a choice because it fits current observations, and because give rise to that outcome. However, the single-field case : itisagoodapproximationtotheperturbationsproduced appears to be the largest class of models which can be v i bythesimplestinflationmodels[1,2],butisclearlyquite dealt with as a single set, where one aims to identify the X specific since it requires four descriptive qualifiers.∗ member of the set responsible for the observations. If r Given a set of microwave anisotropy measurements, it turns out that there is no such member, then the net a one must determine both the cosmological parameters must be cast wider to include more complicated models. and the parameters describing the initial perturbations Amongstmoregeneralinflationmodels,thereisnoprob- simultaneously; it is not possible to do one and not the lem in generating the requiredpredictions of the spectra other. Given the description of the initial perturbations, to testthem oneby one againstthe data,but there isno one canthen try to determine the model which gaverise knownwayoflooking atthe observationsandconstruct- tothem. Ifthebest-fitmodelprovestohavepassiveper- inganinflationarymodelwhichwillgeneratethe desired turbations, meaning that the perturbations are observed predictions. Westressagainthatunlessavalidmodelfor to be entirely in their growing mode and hence are in- theinitialperturbationscanbefound,onecannotobtain ferred to have existed since early in the Universe’s evo- thecosmologicalparametersastherewouldbenowayto lution (in particular, to have already existed when their compute a microwave anisotropy spectrum to compare with observations. Evenunderthesingle-fieldparadigm,thesituationhas not been wholely satisfactory, the reason being that the ∗Infact,asshortlydiscussed,there’sactuallyafifthqualifier analytic results available for the spectra are only ap- asitisnormallyassumedthattheseperturbationsareentirely proximate, having been calculated using the slow-roll in thegrowing mode. approximation, within which results are known only to 1 second-order. Therefore, even if unbiased estimates of Ingoinginthereversedirection,startingwithobserva- the parameters describing the perturbation spectra are tions,thestandardprocedureistousetheobservationsto obtained from observations (such as the amplitude and estimatethecoefficientsinEq.(1),alongwiththecosmo- the spectral index n), these are not translated into un- logicalparameters. Anexampleusingcurrentdataisthe biased estimators of the inflation potential. This paper analysisbyTegmark[13]. Aslongastheparametrization describes how this shortcoming can be overcome. oftheperturbationswasadequate,that’sthejobdoneas far as the cosmological parameters are concerned. How- ever, to obtain the inflationary potential, the approxi- II. DIRECT ESTIMATION OF INFLATIONARY mate slow-roll results which give the spectra in terms of PARAMETERS the inflationary potential are inverted. This procedure is reviewed in Ref. [6], but does not yield unbiased esti- Throughout,weworkwithinthesinglefieldparadigm. mates of the inflation potential. The traditional technique for obtaining observational The numerical technology now exists to circumvent predictions from an inflationary model is the follow- this problem. The key is to immediately abandon any ing. The potential is specified as an analytical function. attempt to make the calculations analytically. Instead, The perturbations are then computed using the slow- theperturbationspectraareobtainedbynumericalsolu- roll approximation, to give the perturbation spectra in tion of the relevant mode equations, which give the per- parametrized form. For example, the density perturba- turbation amplitude at a particular wavenumber. The tionspectrumδH(k)(followingthenotationofLiddleand best formalism is that of Mukhanov [14], and the only Lyth [2]) can be expanded as a Taylor series in lnk as assumption is that linear perturbation theory is valid, which is more or less guaranteed by the fact that the lnδH2(k)=lnδH2(k∗)+(n∗−1) ln k + observed (dimensionless) perturbations are order 10−5. k∗ The necessary ingredients to proceed are (cid:12) 1 dn (cid:12)(cid:12) ln2 k +··· , (1) 1. A program which can numerically solve the mode 2 dlnk(cid:12) k∗ k∗ equations wavenumber by wavenumber. We de- where k is the comoving wavenumber and k∗ is an arbi- scribed such a code in Ref. [15]. This must be able traryscalewherethecoefficientsareevaluated. Theslow- to compute both scalar and tensor perturbations. roll approximation gives the coefficients as functions of 2. A version of cmbfast which is capable of taking the potential,butonlyapproximately,andthereisafur- arbitrarypowerspectraasinput toproducetheC ther approximationwhen the seriesis truncated at some ℓ curve. The output C is the sum of the scalar and level. Further, because the expressionrelating the scalar ℓ tensor parts. Polarization anisotropies should be fieldvalueφtothescalekcrossingthehorizonisalsoap- computed as well as temperature ones. proximate, there is a problem of a ‘drift of scales’; as we move from the expansion scale k∗, we begin to misiden- We have assembled these codes into an IDL pipeline. tify the φ value corresponding to a k value by more and The input step is to supply a parametrization of the more. potential, rather than an analytic form. In this paper Often all these errors are unimportant, especially for we use the simplest version, a Taylor expansion about observationsofthecurrentquality. Thatforcertainmod- some scalar field value φ∗, with the slight subtlety of els they give an error which will be significant for fu- pulling out the overall normalization as a prefactor for tureobservationshasbeennotedbyseveralauthors[7–9]. later convenience. This leads to the C as a function of ℓ Onecanattempttoimprovethingsbygoingtothe high- the cosmological parameters and the potential parame- est possible order in the slow-roll expansion, which is ters,i.e.Cℓ(V∗,V∗′/V∗,V∗′′/V∗,...,h,Ω0,ΩB,ΩΛ,...)where unfortunately only second-order, or by taking more and primesarederivativeswithrespecttoφ,evaluatedatφ∗. more terms in Eq. (1) [10,11], which does not require The inversion is now direct; the observed anisotropy going to higher order in slow-roll. spectrum is used to directly estimate the potential pa- One does the best one can with the scalar pertur- rameters, which can be done in an unbiased way to gen- bations, and also carries out a similar process for the eratethebestpossiblereconstruction. Ifatthisstageone tensor spectrum, which is less demanding theoretically were to find that the overall best-fit model was a poor as tensors are harder to detect observationally. These fit to the data, the first thing would be to try an im- parametrized spectra are then fed into a numerical code provedparametrizationofthe potentialand/orinclusion (e.g. cmbfast [12], possibly enhanced to allow non- of extra cosmologicalparameters, and if that still fails it power-lawspectra)tocomputethemicrowaveanisotropy would be time to suspect that the single field paradigm spectrum, the Cℓ, as a function of those and the cosmo- is not correct. logical parameters. However, optimistically assuming that the best fit is adequate,wehaveourbest-fitinflationarypotential. But 2 mation variables)whateverthe evolutionis. Asthe observations Slo w-roll approxi Parsapmecettrraized CMBFAST wkconhvoeewnrVtah(elφimr)eifltoeevrdaanrltaimsncgiateeledosfrcarwnoagsvseeoonufutφmsibvdaeelrutshekes,ahbwooreiuzntoentehddeutorinimnlyge inflation. Our input assumption (ultimately to be tested Inflaton Microwave againsttheobservations)isthatthereisapotentialV(φ), potential anisotropies and we will simply need a parametrization of it which is accurate enough over the desired range. Direct reconstruction Thereisonesubtletytothis. Thescalarwaveequation Mode equations + CMBFAST is second order,so,in addition to the value of φ, it looks as if φ˙ is an arbitraryinitial condition which needs to be FIG.1. The traditional route from model to observables and back is the two-stage process at the top. The procedure considered as an extra parameter. However it has long outlined in thispaperenables a direct routewithout approx- beenknownthatthis isnotthe case,because scalarfield imations beyond linear perturbation theory. cosmologieshave an attractorbehaviour whereby allini- tial conditions quickly converge [16,17] (indeed, during inflation convergence is at least exponentially fast with that’s not all; a further advantage of this direct method φ, but even non-inflationary expansion exhibits this be- is that it immediately gives us the covariance of the un- haviour). However, this does mean that we have to be certainties on the potential parameters. For example, it surethatthesimulationhasrunforlongenoughthatthe ′ is knownthatthe errorsonV∗ andV∗ willbe highly cor- attractor is attained before the perturbations on observ- related. Usingtheoldapproach,thesecorrelationswould able scales are generated, exactly as is believed to have have to be carried through the complicated reconstruc- happened in the real Universe. tionequations,anunpleasantenoughtaskthatinRef.[8] Asinknowninflationmodelsallobservablescalescross we instead used a Monte Carlo method to illustrate the outside the horizon over a very narrow range of φ, the uncertainties of a reconstruction from simulated data. simplest approachis a simple Taylor series expansion In this approach, the consistency equation relating sccuaslsaiorna)nids taeuntsoomraptiecratlulyrbiantcioonrpso(rsaeteedR,ebf.ei[n6g] ftoerstaeddbisy- VV(∗φ) =1+ VV∗∗′ (φ−φ∗)+ 12VV∗∗′′ (φ−φ∗)2+··· , (2) whether there is a potential offering a satisfactory abso- lute fit to the data. It could of course also be tested in where φ∗ is arbitrary and can be set to zero if desired. the traditional way by power spectrum fitting, but any- We pulled out the normalization before expanding, as way it is unlikely that observations will be good enough then the normalization of the Cℓ depends only on V∗ to say anything significant. and not the other terms. In principle one could consider The two strategies are contrasted in Fig. 1. We do a more sophisticated expansion to try and improve the not view our new approach as replacing the traditional convergenceproperties such as a Padéapproximant, but one, but rather as a next step that one would take if the that can be assessed once actual data is available. traditional fitting proves successful, in order to obtain optimal results. B. Parameter uncertainty III. UNCERTAINTY AND COVARIANCE OF Having obtained the C as a function of the potential ℓ INFLATON POTENTIAL PARAMETERS and cosmological parameters, we can assess the likely accuracy with which those parameters can be found A. Parametrizing the potential by a given experiment. This is carried out using the well-established Fisher matrix technique [18,19], which Although in principle a solution of the mode equation amounts to taking the derivative of the Cℓ with respect runs fromanearlyinitialtime until the scale is wellout- to each of the parameters. The parameter uncertain- side the horizon, when considering perturbations on a ties depend on the choice of ‘correct’ model and on the given scale k, the details of how the Universe expands number of parameters allowed to vary. For illustration, are only important for a fairly brief interval around the we vary cosmological parameters about an underlying time k = aH when the scale crosses outside the hori- model with Hubble parameter h = 0.65, density param- zon. The reason is that while the scale is well inside eter Ω0 = 0.3, cosmological constant ΩΛ = 0.7, baryon the horizon the relevant timescales are much less than densityΩB =0.05andreionizationopticaldepthτ =0.1. the expansiontimescale andexpansioncanbe neglected, The potential we choose is the λφ4 potential, and we while when scales are above the horizon the perturba- take the epoch where the present Hubble radius equaled tions are frozen in at fixed values (in the appropriate the Hubble radius during inflation as being 60e-foldings 3 before the end of inflation. Numerical solution of the underlying relative equations of motion gives this as φ∗ = 4.37mPl. In fact parameter model value uncertainty theslow-rollapproximationwillworkwellforthispoten- τ 0.1 6.1% tial, and for instance can be used to show that gravita- ΩBh2 0.021 1.2% tional waves should contribute about twenty percent of ΩCDMh2 0.11 2% the signal at large angular scales. ΩΛh2 0.30 5% Following Zaldarriaga et al. [19], we consider a version of the Planck satellite which measures both tem- 1012V∗/m4Pl 2.3 22% perature and polarization anisotropies, as described in mPlV∗′/V∗ 0.92 14% Ref. [11]. We do not attempt to include the effects of m2PlV∗′′/V∗ 0.63 2× foregrounds, as extensively studied recently by Tegmark m3PlV∗′′′/V∗ 0.29 60× et al. [20], but choose to consider only one polarized m4PlV∗′′′′/V∗ 0.066 400× Planck channel (in effect assuming that the polarized TABLE I. Uncertainties for each parameter, marginaliz- foregroundscanberemovedusingalltheotherchannels) ing over the remaining parameters. We stress that these are which, while rather approximate, yields similar results. specificallyfortheλφ4model. Thesevaluescorrespondtothe diagonalentriesofthecovariancematrix. Therearesubstan- The actual data, when available, will of course merit tialcorrelationsbetweenparameters,especiallythosedescrib- more sophisticated treatment. Our numbers are there- ingthepotential,sotheoff-diagonalentriesofthecovariance fore indicative only, and more importantly they would matrix are significant. Of the potential parameters, only the vary significantly if the assumed underlying model were magnitudeandgradientofthepotentialaredetectedwithany changed — the quality of information available from re- significance. constructiondepends stronglyonwhichmodel(ifany,of course) proves to be correct. TheresultsareshowninTableI.Thehigherderivatives still be assigned values according to their upper limit. are not detected, but it is interesting to note that even Wenotethatthispotentialismuchlessfavourableforre- with this very flat potential, the variation of the poten- constructionthanonesexploredpreviously,asitismuch ′ tialduringinflation,V ,is detectedat7-sigma. However closer to the scale-invariantlimit. ′′ V isnotdetected;thismayseemalittlesurprisinggiven The reconstructions indicate the second key advan- thatboththegravitationalwaveamplitudeandthescalar tage of the method proposed here over previous ones spectral index (which depend on different combinations (e.g. Refs. [6,8]) — the reconstructed potentials are un- ′ ′′ ofV /V andV /V)areinfactdetectable forthispoten- biased estimates of the true potential, being as likely to tial [18,19,11,20], but it turns out that the combination be too high as too low. In the upper panel, we see that ′′ ofthesegivingV /V isnotdistinguishablefromzero. In theuncertaintyintheoverallnormalizationisquitelarge our approach one never needs to make the separation of (ultimately due to a degeneracy in the effect of scalars scalars and tensors explicitly. and tensors on large angular scales). The lower panel shows the combination V′/V3/2 which is primarily sen- sitive to density perturbations alone (indeed perfectly so C. Parameter uncertainty covariance in the slow-roll approximation), and which is much bet- ′ ter determined (in particular, better than V or V sep- TheFishermatrixtechnique alsogeneratesthecovari- arately). This figure allows us to see directly the range ances of the error estimates, and these are crucial in in- of φ which is constrained by the data; to highlight this terpreting the observational constraints. In particular, wehaveindicatedthe valueswhichφtakeswhile the mi- the covariance matrix is essential in illustrating the re- crowave anisotropies are being generated. At larger φ, constructionsgraphically;ifcorrelationsareignoredthen corresponding to larger scales, the perturbations are un- the reconstruction deteriorates much more quickly with observable, and even some way within the current hori- φ than the true picture (the correlations allow for the zonscale cosmic variance contributes significantly to the fact that scale of the expansion φ∗ need not be the scale spread. Near the centre of the data the determination is at which the observations are the most powerful). That atits best, andonshortscalesthe informationagainbe- ourmethodgivesthefullcorrelationmatrixoftherecon- comes poor, partly because of the dependence on all the structed potential directly is its first key advantage over cosmological parameters and partly because Silk damp- earlier techniques. ingerasestheperturbationsasonegoesbeyondℓ∼1000. Toillustratethequalityofthereconstruction,wecarry This figure highlights once again that the information out Monte Carlo reconstructions with errors drawn ac- available from reconstructionconstrains only a tiny por- cording to the covariance matrix, and plot them against tion of the scalar field potential. Nevertheless, the infor- the true potential in Fig. 2. These reconstructions in- mation available there is of good accuracy, and can be clude up to the fourth derivative; though as seen from highly constraining in instances where theoretical mo- Table I the higher derivatives are not detected, they can tivation suggests a potential containing few unspecified 4 FIG.3. Twenty Monte Carlo reconstructions of the combination V′/V3/2,as in thelower panel of Fig. 2, butfor the model investigated by Wanget al. [7]. vational errors). Figure 3 illustrates test reconstructions of this potential, using the techniques of the present paper. The uncertainties on the cosmological parameters, ′ and on V and V , are very similar to those of the quar- ′′ tic case. However, in addition V is detected at around 3-sigma and the next two derivatives have uncertainties comparable to their values. IV. DISCUSSION FIG. 2. Twenty Monte Carlo reconstructions of the po- Itmaywellbethatoneofthesimplestmodelsofinfla- tential, compared against the true potential which is shown tioniscorrect,andtheperturbationspectraareperfectly as a dashed line. The upper panel shows the potential it- self, and the lower one the combination V′/V3/2 which is a satisfactorily approximated by a power-law [or at least combinationcomingprimarilyfromthedensityperturbations somelow-ordertruncationofEq.(1)]. If so,then cosmo- alone. Thedottedverticallinesindicatetheregion ofthepo- logicalparameterestimationcanproceedasdescribedin tentialdirectlyprobedbythemicrowavebackground,ranging previous works. However, given the intellectual and fi- from the current horizon scale to the horizon scale when the nancialinvestment in pursuing cosmologicalparameters, ℓ = 1500 mode was generated (evaluated in the underlying it is vital to be aware of possible difficulties, and to an- model). The upper panel shows that the gradient is quite alyze ways of dealing with them. We have considered well recovered but the overall amplitude much less so, while one such possibility — that the single field paradigm is the lower highlights the obvious fact that the reconstruction correct but slow-roll is not very good — and explained is accurate only where there is data available toconstrain it. that this is readily dealt with using existing numerical technology. In using this technique, as with others, it is parameters. imperative to make an overall goodness-of-fit test to en- Thequarticpotentialisaninterestingtestcasebecause sure that the class of models being considered is capable it is not far from the slow-roll limit, and this is the first of adequately explaining the data. Even if the power- time such a potential has been used to test reconstruc- law approximationprovesvalid(and certainly this is the tion methods. However the true strength of the method method which should be tried first), one will want to wouldbeunveiledifthetruemodeldoesnotsatisfyslow- use these techniques to ensure that estimates of the in- rollwell,despitethepotentialbeingsmooth. Anexample flationarypotential are unbiasedones, and to obtain the is the potential introducedby Wang et al.[7], whichwas fullest possible information about the inflaton potential used to test the traditional reconstruction technique in from observations. Ref. [8]. It was shown in that latter paper that traditional reconstruction could still work well, but led to a biasinthe estimateofthe potential(albeitwithinobser- 5 ACKNOWLEDGMENTS I.J.G. was supported by PPARC. We thank Ed Cope- land and Rocky Kolb for many useful discussions and comments. [1] A. H. Guth, Phys. Rev. D 23, 347 (1981); A. D. Linde,Particle Physics andCosmology(Harwood,Chur, Switzerland, 1990). [2] A.R. Liddleand D.H. Lyth,Phys.Rep. 231, 1 (1993). [3] A.R. 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