What do WMAP and SDSS really tell about inflation?
Julien Lesgourgues, Alexei A. Starobinsky, Wessel Valkenburg
aa r X i v : . [ a s t r o - ph ] O c t LAPTH-1210/07, arXiv:0710.1630
What do WMAP and SDSS really tell about inflation?
Julien Lesgourgues a ∗ , Alexei A. Starobinsky b † and Wessel Valkenburg a ‡ a LAPTH § , Universit´e de Savoie & CNRS, 9 chemin de Bellevue,BP110, F-74941 Annecy-le-Vieux Cedex, France and b Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow 119334, Russia (Dated: October 24, 2018)We derive new constraints on the Hubble function H ( φ ) and subsequently on the inflationarypotential V ( φ ) from WMAP 3-year data combined with the Sloan Luminous Red Galaxy sur-vey (SDSS-LRG), using a new methodology which appears to be more generic, conservative andmodel-independent than in most of the recent literature, since it depends neither on the slow-rollapproximation for computing the primordial spectra, nor on any extrapolation scheme for the po-tential beyond the observable e-fold range, nor on additional assumptions about initial conditionsfor the inflaton velocity. This last feature represents the main improvement of this work, and ismade possible by the reconstruction of H ( φ ) prior to V ( φ ). Our results only rely on the assumptionthat within the observable range, corresponding to ∼
10 e-folds, inflation is not interrupted and thefunction H ( φ ) is smooth enough for being Taylor-expanded at order one, two or three. We concludethat the variety of potentials allowed by the data is still large. However, it is clear that the firsttwo slow-roll parameters are really small while the validity of the slow-roll expansion beyond themis not established. PACS numbers: 98.80.Cq
Cosmic inflation was introduced as a simple and aes-thetically elegant scenario of the early Universe evolu-tion which is capable of explaining its main propertiesobserved at the present time [1, 2, 3, 4, 5, 6]. As a veryimportant byproduct it provides a successful mechanismfor the quantum-gravitational generation of primordialscalar (density) perturbations and gravitational waves[7, 8, 9, 10, 11, 12, 13]. The Fourier power spectrum P R ( k ) of the former ones is observed today in the cos-mic microwave background (CMB) and the large scalestructure (LSS). Vice versa, at present the CMB andthe LSS provide the only quantifiable observables whichcan confirm or falsify inflationary predictions. That iswhy matching concrete inflationary models to observa-tions has become one of the leading quests in cosmology.In the simplest class of inflationary models, inflation isdriven by a single scalar field φ (an inflaton) with somepotential V ( φ ) which is minimally coupled to the Ein-stein gravity. For these models, some new conservativebounds on V ( φ ) were presented recently in [14]. Untilthen, most post-WMAP3 studies concerning V ( φ ) reliedon the slow-roll approximation in the calculation of per-turbation power spectra and their relation to values of φ during inflation [15, 16, 17, 18, 19, 20, 21, 22, 23],or made an extrapolation of V ( φ ) from the observablewindow till the end of inflation [24, 25, 26] (a numericalintegration of exact wave equations for perturbations toobtain primordial power spectra was also permormed in ∗ [email protected] † [email protected] ‡ [email protected] § Laboratoire de Physique Th´eorique d’Annecy-le-Vieux, UMR5108
Refs. [27, 28, 29] for specific inflationary models). Theextrapolation over the full duration of inflation is moreconstraining than the data alone. Instead, Ref. [14] fo-cused only on the observable part of the potential to seeup to what extent current data really constrains inflation.For this class of models, the evolution of a spatially flatFriedmann-Lemaitre-Robertson-Walker (FLRW) uni-verse can be described by [30, 31]˙ φ = − m P π H ′ ( φ ) (1) − π m P V ( φ ) = [ H ′ ( φ )] − πm P H ( φ ) . (2)whenever ˙ φ = 0 and not specifically during inflation (so H ′ ( φ ) = 0, too). Here H ( φ ( t )) ≡ ˙ a/a , a ( t ) is the FLRWscale factor, a dot denotes the derivative with respect tothe cosmic time t , a prime with respect to an argument,and we have set Gm P = ¯ h = c = 1. If V ( φ ) is consid-ered as the defining quantity, the initial conditions forgenerating the observable window are determined by theset { ˙ φ ini , V ( φ ) } . In Ref. [14], the inflaton potential wasparametrized as a Taylor expansion up to some order, tosee up to what extent the potential can be constrainedby pure observations. However, in order to reduce thenumber of free parameters, ˙ φ ini was fixed for each modelby demanding that the inflaton follows its attractor so-lution just when the observable modes exit the horizon.In practice this means that the results of Ref. [14] as-sumed that inflation started at least a few e-folds beforethe observable modes left the horizon. These precurringe-folds led to a slightly stronger bound on the potentialsthan the data itself could actually give, although thisextra constraining power stands in no proportion to anextrapolation over the full duration of inflation.Eqs. (1, 2) however show that when one considers H ( φ )as the defining quantity, all initial conditions are alreadyuniquely set by H ( φ ). Moreover, the slow-roll conditionswhich require, in particular, that the first term in therhs of Eq. (2) is much less than the last one need not beimposed ab initio . In this Letter we derive the bounds on H ( φ ) during observable inflation using its Taylor expan-sion at various orders. We infer for this some constraintson V ( φ ) under an even more conservative approach thanin Ref. [14], since the present method requires absolutelyno extrapolation outside of the observable region (eitherforward or backward in time). Our only restriction is toassume that observable cosmological perturbations orig-inate from the quantum fluctuations of a single inflatonfield, which dynamics during observable inflation is com-patible with a smooth, featureless H ( φ ). Method.
We used the publicly available code cos-momc [32] to do a Monte Carlo Markov Chain (MCMC)simulation. We added a new module (released at )which computes numerically the primordial spectrum ofscalar and tensor perturbations for each given function H ( φ − φ ∗ ), where φ ∗ is an arbitrary pivot scale in fieldspace. This module is simpler than the one in Ref. [14],since the code never needs to find an attractor solutionof the form ˙ φ ( φ ). The comoving pivot wavenumberis fixed once and for all to be k ∗ = 0 .
01 Mpc − ,roughly in the middle of the observable range. Pri-mordial power spectra are computed in the range[ k min , k max ] = [5 × − ,
5] Mpc − needed by camb ,imposing that k ∗ leaves the Hubble radius when φ = φ ∗ .In practice, this just means that for each model the codenormalizes the scale factor to the value a ∗ = k ∗ /H ∗ when φ = φ ∗ . Note that by mapping a window of infla-tion to a window of observations today, our approachis independent of the mechanism of reheating. Theevolution of each scalar/tensor mode is given by d ξ S , T dη + (cid:20) k − z S , T d z S , T dη (cid:21) ξ S , T = 0 (3)with η = R dt/a ( t ) and z S = a ˙ φ/H for scalars, z T = a for tensors. The code integrates this equation start-ing from the initial condition ξ S , T = e − ikη / √ k when k/aH = 50, and stops when the expression for the ob-served scalar/tensor power spectrum freezes out in thelong-wavelength regime. More precisely, the spectra aregiven by k π | ξ S | z S → P R , k πm P | ξ T | z T → P h , (4)and integration stops when [ d ln P R ,h /d ln a ] < − . Iffor a given function H ( φ − φ ∗ ) the product aH can-not grow enough for fullfilling the above conditions, themodel is rejected. In addition, we impose that aH growsmonotonically, which is equivalent to saying that inflation Parameter n = 1 n = 2 n = 3Ω b h . ± .
001 0 . ± .
001 0 . ± . cdm h . ± .
004 0 . ± .
004 0 . ± . θ . ± .
003 1 . ± .
004 1 . ± . τ . ± .
03 0 . ± .
03 0 . ± . h H ∗ H ′ ∗ m P i . ± .
06 3 . ± .
06 3 . ± . “ H ′∗ H ∗ ” m P . ± .
031 0 . ± .
056 0 . ± . H ′′∗ H ∗ m P − . ± . − . ± . H ′′′∗ H ∗ H ′∗ H ∗ m P . ± . − ln L max H ( φ − φ ∗ ) at order n = 1 , , is not interrupted during the observable range. If theseconditions are satisfied, the power spectra are comparedto observations.We choose to parametrize H as a Taylor expansionwith respect to φ − φ ∗ up to a given order n vary-ing between one and three (this choice of backgroundparametrization is equivalent to that in Ref. [18], as longas no extrapolation is made). Note that for n > φ and H ′ becoming zeroat some value φ = φ in the range involved since then H ( φ ) would acquire a non-analytic part beginning fromthe term ∝ | φ − φ | / (with V ( φ ) being totally analyticat this point) [41]. As a cosmological background weused the standard ΛCDM-model with the free parame-ters shown in Table I. Results for H ( φ − φ ∗ ) . In Fig. 1 we show the prob-ability distribution of each parameter marginalized overthe other parameters. The corresponding 68% confidencelimits are displayed in Table I, as well as the minimumof the effective χ for each model. This minimum doesnot decrease significantly when n increases, which reflectsthe fact that current data are compatible with the sim-plest spectra and potentials, but derivatives up to H ′′′ can be constrained with good accuracy. Note that itwould be very difficult to give bounds directly on theset { H, H ′ , H ′′ , H ′′′ , ... } : indeed, these parameters arestrongly correlated by the data, because physical effectsin the power spectra depend on combinations of them.For example, at the pivot scale, the scalar amplitudeis mainly determined by ( H ∗ /H ′∗ ) and the tensor-to-scalar ratio r ≡ P h / P R by ( H ′∗ /H ∗ ) . The scalar tilt n S further depends on H ′′∗ /H ∗ , and the scalar runningon H ′′′∗ H ′∗ /H ∗ . The Markov Chains can converge in areasonable amount of time only if the basis of parame-ters (receiving flat priors) consists in functions of eachof the above quantities, or linear combinations of them.However, we also show in the last plot of Fig. 1 the distri- Ω b h Ω c h θ τ ln A (H ′ / H * ) m P2 −0.5 0 0.5 H ′′ / H * m P2 −2 0 2 4 6 H ′′′ H ′ / H *2 m P4 −5 H Inf / m P FIG. 1: Probability distribution for the eight independentparameters of the models considered here, normalized to acommon arbitrary value of P max . The ninth plot shows a re-lated parameter (with non-flat prior): namely, the value of theexpansion rate when the pivot scale leaves the horizon duringinflation. Our three runs n = 1 , , A is a shortcutnotation for the parameter defined in the fifth line of Table I. bution of H ∗ : this information is useful since the energyscale of inflation is given by λ = (3 H ∗ m P / π ) / , butthe displayed probability should be interpreted with caresince this parameter has a non-flat prior.The run n = 1 is not very interesting. Indeed, impos-ing H ′′ and higher derivatives to vanish leads to a one-to-one correspondence (at least in the slow-roll limit) be-tween the amplitude and the tilt of the scalar spectrum.This feature is rather artificial and unmotivated. It ex-plains anyway why the parameter H ∗ has exceptionallya lower bound in the n = 1 case [42]. Much more inter-esting is the n = 2 case for which the tensor ratio, scalaramplitude and scalar tilt are completely independent ofeach other, and the n = 3 case for which even the tiltrunning has complete freedom. The runs for n = 2 and n = 3 nicely converged and constitute the main resultof this work. Note also that the middle-right and lower-right graphs in Fig. 1 are compatible with each otherin the following sense: though H ′ ∗ may not reach zerounder our assumption, the quantity H ′ ∗ /H ∗ may be arbitrarily small if H ∗ is allowed to be arbitrarily small,too. Thus, for cases n = 2 , H ∗ is not suppressedat zero argument, H ′ ∗ /H ∗ is not suppressed there, too.The probability distribution for combinations of H ∗ , H ′∗ and H ′′∗ are robust in the sense that they do notchange significantly when one extra free parameter H ′′′∗ is included: this indicates that they are directly con-strained by the data. We tried to include an additionalparameter ( H ′′′′∗ /H ∗ )( H ′∗ /H ∗ ) m P , but then our MarkovChains did not converge even after accumulating of theorder of 10 samples. We conclude that current data donot have the sensitivity required to constrain H ( φ ) be-yond its third derivative and to establish the validity ofthe slow-roll approximation beginning from this order.On the other hand, the first two slow-roll parameters ǫ ( φ ) = H ′ m P / πH and ˜ η ( φ ) = H ′′ m P / πH are reallysmall over the observed range (tilde is used here to avoidmixing with the conformal time η ). The next parame-ter ξ ≡ λ H = H ′′′ H ′ m P / (4 π ) H is also small, ∼ . ǫ and | ˜ η | , not ǫ or ˜ η aswould follow from the standard slow-roll expansion. Thissmallness explains why our results for these parametersare similar to those obtained for the same background H ( φ ) but using the slow-roll approximation to calculatethe power spectra [21] (and to those in [24], too) althoughsome important differences exist. Results for V ( φ − φ ∗ ) . We further processed our n = 1 , , φ − φ ∗ ) ↔ − ( φ − φ ∗ ). We choose to focus on onehalf of the solutions, corresponding to ˙ φ > V ′∗ >
0. Our results are shown in Fig. 2. They appearto be compatible with those of Ref. [14], although a de-tailed comparison is difficult: first, the current methodis more conservative, and second, a given order in theTaylor-expansion of H ( φ − φ ∗ ) is not equivalent to an-other order in that of V ( φ − φ ∗ ). Our results are alsodifficult to compare with those of Ref. [26], since theseauthors choose to present their full allowed potentials ex-trapolated till the end of inflation: in principle, our Fig. 2can be seen as a zoom on the directly constrained, small φ region in their Fig. 2.Our results could give the wrong impression that allpreferred potentials are concave. This comes from thefact that in the representation of Fig. 2, many interest-ing potentials are hidden, since they almost reduce to thepoint ( V ∗ , ∆ φ ) → (0 , H ∗ and H ′∗ (and hence tiny variation of the inflaton field during theobservable e-folds) are perfectly compatible with obser-vations. It is straightforward to show that models leadingto n S < r correspond to convex potentials FIG. 2: Allowed inflationary potentials V ( φ − φ ∗ ) inferredfrom each of our n = 1 , , n = 1, the intermediate (greenish) onefor n = 2 and the outer (reddish) one for n = 3. Each po-tential is plotted between the two values φ and φ corre-sponding to Hubble exit for the limits of the observable range[ k , k ]=[2 × − , .
1] Mpc − : so we only see here the ac-tual observable part of each potential. Note that this figureshows only one half of the possible solutions: the other halfis obtained by reflection around φ = φ ∗ . (like e.g. new inflation with V = V − λφ n , or one-loophybrid inflation with V = V + λ ln φ ), while models withsame n S and larger r derive from concave potentials (likee.g. monomial inflation V = λφ α ). Current data favor n S <
1, and the upper bound on r is too loose for differ-entiating between these two situations. So, our allowedpotentials can be split in two subsets: low-energy convexpotentials and high-energy concave potentials, as illus-trated in Fig. 3, in which we rescaled all allowed poten-tials to the same variation in V and φ . More generally,this large degeneracy in potential reconstruction reflectsthe fact that an infinitely precise measurement of thescalar spectrum P R would only constrain the function P R ( k ) = 4 H m P H ′ (cid:12)(cid:12)(cid:12)(cid:12) k = aH (5)(in the slow-roll approximation). This is not sufficientfor inferring the correspondence between k and φ , andhence for a unique determination of H ( φ ) and V ( φ ). Itis necessary to measure also the tensor spectrum, equalto P h ( k ) = 16 H πm P (cid:12)(cid:12)(cid:12)(cid:12) k = aH (6)in the same approximation, in order to diminish this de-generacy (see the related discussion in Ref. [37]). In theslow-roll approximation, the knowledge of P h ( k ) leads FIG. 3: Allowed inflationary potentials V ( φ − φ ∗ ) with thesame colour/shade code as in Fig. 2, but a different choice ofaxes: each potential is now rescaled to the same variation in V an φ space. This shows that many allowed potentials areactually convex. The outer region still corresponds n = 3, theintermediate one to n = 2 and the inner (quasi-linear) one to n = 1. to the unambiguous determination of H ( φ ). However,the question how unique the determination of H ( φ ) is,even from both P R ( k ) and P h ( k ) in the generic case be-yond slow-roll, is still open because of the existence ofmany H ( φ ) leading to the same perturbation spectrawhich may not be obtained from the slow-roll expansionat all [38]. Still, since the difference of these additionalsolutions from slow-roll ones is, in some sense, exponen-tially small for small slow-roll parameters, their existencemight appear not significant from the observational pointof view. Acknowledgements.
This work follows from a very nice and fruitful stayof JL and WV at the Galileo Galilei Institute for The-oretical Physics, supported by INFN. JL and WV alsowish to thank Prof. Lev Kofman for very useful discus-sions. AS was partially supported by the Russian Foun-dation for Fundamental Research, grant 05-02-17450, andby the Research programme“Elementary particles” of theRussian Academy of Sciences. He also wishes to thankProf. Alikram Aliev and the Feza G¨ursey Institute, Is-tanbul, for hospitality during the period of completionof this paper. WV is supported by the EU 6th Frame-work Marie Curie Research and Training network “Uni-verseNet” (MRTN-CT-2006-035863). Numerical simu-lations were performed on the MUST cluster at LAPP(CNRS & Universit´e de Savoie). [1] A. A. Starobinsky, Phys. Lett.
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