(PDF) Bayesian analysis of inflation: Parameter estimation for single field models

Bayesian analysis of inflation: Parameter estimation for single field models Michael J. Mortonson,1, ∗ Hiranya V. Peiris,2, 3, † and Richard Easther4, ‡ 1 Center for Cosmology and AstroParticle Physics, The Ohio State University, Columbus, OH 43210, U.S.A. 2 Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Cambridge CB3 0HA, U.K. 3 Department of Physics and Astronomy, University College London, London WC1E 6BT, U.K. 4 Department of Physics, Yale University, New Haven, CT 06520, U.S.A. (Dated: February 10, 2011) arXiv:1007.4205v2 [astro-ph.CO] 9 Feb 2011 Future astrophysical datasets promise to strengthen constraints on models of inflation, and ex- tracting these constraints requires methods and tools commensurate with the quality of the data. In this paper we describe ModeCode, a new, publicly available code that computes the primordial scalar and tensor power spectra for single field inflationary models. ModeCode solves the inflation- ary mode equations numerically, avoiding the slow roll approximation. It is interfaced with CAMB and CosmoMC to compute cosmic microwave background angular power spectra and perform like- lihood analysis and parameter estimation. ModeCode is easily extendable to additional models of inflation, and future updates will include Bayesian model comparison. Errors from ModeCode contribute negligibly to the error budget for analyses of data from Planck or other next generation experiments. We constrain representative single field models (φn with n = 2/3, 1, 2, and 4, natural inflation, and “hilltop” inflation) using current data, and provide forecasts for Planck. From current data, we obtain weak but nontrivial limits on the post-inflationary physics, which is a significant source of uncertainty in the predictions of inflationary models, while we find that Planck will dra- matically improve these constraints. In particular, Planck will link the inflationary dynamics with the post-inflationary growth of the horizon, and thus begin to probe the “primordial dark ages” between TeV and GUT scale energies. I. INTRODUCTION tities form the basis of an inflationary sector in the con- cordance model, if we postulate that their values can be traced back to a phase of primordial inflation. Con- The last two decades have seen a sequence of break- versely, a complete inflationary model must account for throughs in the understanding of the physical universe. these numbers. As the data improve, this set of param- The detection of cosmic microwave background (CMB) eters can easily expand to include the properties of any anisotropies by COBE in 1992 [1], evidence for dark en- primordial gravitational wave background or departures ergy in the distance-luminosity relationship for Type Ia from Gaussianity. Given the tight agreement between the supernovae in 1998 [2, 3], and the sequence of WMAP concordance model and current data, the impact of any data releases [4–7], among others, mark turning points new parameters is necessarily subleading. Consequently, in our ability to constrain — and falsify — specific cos- additional parameters needed to describe the primordial mological models. These advances begin to fulfill the perturbations can be regarded as fingerprints of specific long-standing promise that astrophysical data can di- inflationary models, in that most of these quantities will rectly probe the first moments after the Big Bang, while be vanishingly small in most inflationary models (see e.g. simultaneously constraining models of ultra-high energy Ref. [10]). physics. Experiments underway and now being devel- Physically, inflation is characterized by a period of ac- oped guarantee that this progress will continue well into celerated expansion in the early universe, and an infla- the future. In particular, the Planck satellite [8] has com- tionary model is defined by the mechanism that drives pleted a full survey of the sky, and this data should dra- this accelerated expansion. Simple models of inflation matically improve the constraints on the free parameters can usually be described by the kinetic term and poten- in the so-called concordance cosmology. tial of a single scalar degree of freedom (the inflaton), Given the quality of present-day data, the primordial along with this field’s coupling to gravity. In this paper perturbations are fully described by two numbers (e.g. we focus on models where the field is minimally coupled Ref. [9]): the amplitude As and tilt ns of the power to gravity and has a canonical kinetic term, but will relax spectrum of density (scalar) perturbations. These quan- these restrictions in future work. The simplest approach to constraining inflation is to specify the primordial perturbations in terms of the em- ∗ Electronic address:

pirical parameters † Electronic address:

‡ Electronic address:

θemp = {As , ns , αs , ...; r, nt , ...; fnl , ...}, (1) 2 where αs is the running of the scalar spectral index, r the (given the underlying inflationary prior) is generally re- tensor-to-scalar ratio, nt the tensor spectral index, and liable. fnl parametrizes non-Gaussianity. Further, θemp may be Finally, θV explicitly accounts for the theoretical un- extended indefinitely to include higher-order terms in the certainty in inflationary predictions induced by the un- expansion of the scalar and tensor power spectra and known thermalization history of the post-inflationary various deviations from Gaussianity. These additional universe [17, 18, 60–63], in combination with the scale- parameters will generally be more difficult to measure in dependence of the spectral index. This uncertainty is any given dataset than the basic empirical parameters As significant even when the running αs ≡ dns /d ln k is not and ns . In all cases, inflationary model predictions for itself observable. This apparent paradox arises because the values of the empirical parameters are to be compared when the running is included, θemp requires at least four with the measured values of these parameters. free parameters to fix the primordial spectra. Conversely, A second approach treats the determination of the in- θV may have only a single free parameter for the infla- flationary mechanism as an inverse problem, and thus tionary potential and another for the reheating physics, reconstructs the inflaton potential from the data [11–15]. and these numbers can thus be determined with more For instance, slow roll reconstruction [16–19] uses a sys- precision than those in θemp [64]. tematic expansion based on the slow roll hierarchy [20– This paper has three primary objectives. The first is to 25] together with consistency conditions on the duration introduce ModeCode, a plug-in for CAMB and Cos- of inflation, providing a minimally parametric approach moMC [39, 65]. ModeCode provides an efficient and to the inverse problem. Variations to this scheme have robust numerical evaluation of the inflationary pertur- been widely discussed in the literature [26–33]. The in- bation spectrum, and allows the free parameters in the verse problem does not have a unique solution, and typ- potential to be estimated within an MCMC computation. ically encodes basic assumptions regarding the general Secondly, we use this code to generate constraints on rep- class of inflationary models under consideration [34]. resentative single field inflationary models using current Here we adopt a third approach, in which we assume data, and give forecasts for the constraints that can be that inflation was driven by a specific model (i.e. form of expected from Planck. Finally, this analysis underlines the potential) with one or more free parameters, the importance of assumptions regarding reheating and thermalization in studies of inflation using data from the θV = {V1 , V2 , ..., Vn ; θRH }, (2) next generation of astrophysical datasets. We will see where the Vi parametrize the potential, while θRH that current data put weak but nontrivial constraints parametrizes the post-inflationary reheating phase, as on the reheating history given an explicit inflationary we discuss later. These parameters replace the empiri- model. Further, our forecasts suggest that Planck will cal inflationary sector, θemp. We estimate their values link the post-inflationary history with the inflationary alongside other cosmological parameters, typically using epoch, with significant implications for theories of parti- Markov Chain Monte Carlo (MCMC) analysis [35–41]. cle physics between Grand Unified Theory (GUT) scales Following parameter estimation for several different in- (∼ 1015 GeV) and TeV scales. flationary models, Bayesian model selection [42–47] tech- ModeCode computes both the scalar and tensor per- niques will allow us to compare the fidelity with which turbation spectra, via the algorithm described in Ref. these different models account for observations of the sky. [50]. A number of common inflationary potentials are Working directly with inflationary models, we can already included, and new models are straightforward to compute the power spectrum numerically without re- incorporate in ModeCode by supplying the potential course to the slow roll approximation. This removes a and its derivatives. Since the CMB angular power spectra lingering uncertainty from the analysis of the inflationary and likelihood are already expressed numerically, noth- parameter space, and allows us to constrain inflationary ing is lost (other than a relatively small computational models with complicated spectra not well described by overhead) by solving directly for the mode amplitudes, θemp . While the exact computation of the power spec- rather than using an analytic approximation. Attention trum is numerically straightforward and has been used has been given to ensuring that the initial conditions extensively for slow roll-violating potentials previously for the background are self-consistent and that the code [31, 32, 48–59], such precision is only now becoming nec- “fails gracefully” for unphysical parameter combinations, essary with the arrival of higher quality data from Planck. so that such points are excluded from MCMC analyses. Even when the slow roll approximation is accurate and In addition, we accurately compute the endpoint of in- the spectra simple, there are advantages to working with flation and the evolution of the comoving horizon size θV , rather than θemp. Empirical quantities such as As , during inflation, so as to precisely match scales in the ns , r, etc. are computable functions of the free param- inflationary era with scales in the present-day universe. eters of any specific inflationary model, and thus con- In a follow-up paper, we will interface ModeCode straints on θemp can, in principle, be mapped into con- with MultiNest [47], allowing us to compute the straints on θV . However, we will see that this approach Bayesian evidence for the models we analyze. In addi- is of limited use in practice, while the opposite approach tion, it is straightforward to extend ModeCode to sce- of inferring constraints on θemp from constraints on θV narios with nonminimal kinetic terms [56, 57], multi-field 3 models, or even non-Gaussianity [66]. background equations in terms of ln a. Denoting d/d ln a with a prime (′ ) and recalling that H = d ln a/dt by def- inition, we write the Einstein equation for H ′ and the II. METHODS Klein-Gordon equation for φ as follows: A. Numerical solution M2 H ′ = − Pl (φ′ )2 H , (8) 2′ In many circumstances, the slow roll approximation ′′ H 1 dV φ + + 3 φ′ + 2 = 0, (9) provides a sufficiently accurate description of the infla- H H dφ tionary power spectra and has the advantage of express- ing them as functions of the potential and its derivatives. where MPl is the reduced Planck mass. Conveniently, However, we want to avoid the slow roll approximation to our choice of independent variable gives φ′ = z, and with maintain full generality and accuracy, and instead com- the help of these background equations, the mode equa- pute the initial curvature and tensor power spectra nu- tion (4) can be written merically given a specific inflaton potential V (φ) [48–50]. ′ 2 H ′ φ′′ This approach yields exact numerical results for arbitrary H k u′′k + + 1 u′k + 2 2 − 2−4 inflationary potentials, up to the intrinsic accuracy of H a H H φ′ first order gravitational perturbation theory.1 ′ 2 #) H H′ 1 d2 V We begin by reviewing the formalism used for the nu- −2 −5 − 2 uk = 0 , (10) H H H dφ2 merical solution of the mode equations. We describe the scalar perturbations with the gauge invariant Mukhanov potential u [67, 68], which is related to the curvature where the term in square brackets is z¨/(za2 H 2 ). perturbation R: We begin the integration of the background equations when the mode of interest is still deep inside the horizon u = −zR , (3) (i.e. k ≫ 100aH). We set the initial field velocity to its slow roll value, solving only the background equations, ˙ where z ≡ φ/H, H is the Hubble parameter, and dots ensuring that the (small) initial transient in the velocity denote derivatives with respect to conformal time. The is damped away. When the mode is roughly 1/100th of Fourier components uk obey [69–71] the horizon size we start to evolve the two orthogonal so- lutions that contribute to uk , and read off the asymptotic 2 z¨ value of |uk /z| when the mode is far outside the horizon u ¨k + k − uk = 0 , (4) z and frozen, as explained in Ref. [50]. The usual mode equation for tensor perturbations, where k is the modulus of the wavevector k. The power spectrum is defined in terms of the two point correlation a ¨ 2 function v¨k + k − vk = 0 , (11) a 2π 2 2 hRk R∗k′ i = ∆ (k) (2π)3 δ (3) (k − k′ ), (5) becomes k3 R ′ 2 H′ and is related to uk and z via ′′ H ′ k vk + + 1 vk + 2 2 − 2 + vk = 0 (12) H a H H k 3 uk 2 ∆2R (k) = . (6) 2π 2 z after transforming the independent variable. The pri- mordial tensor power spectrum is In terms of the empirical parameters As , ns , and αs , ns −1+ 12 αs ln(k/kpivot )+··· 4 k 3 vk 2 ∆2t (k) = 2 a , (13) k ∆2R (k) = As , (7) π 2 MPl kpivot and the asymptotic value of |vk /a| is again taken from where kpivot denotes the pivot scale at which the power the numerical solutions. spectrum is normalized. Equation (4) depends on the background dynamics through z and its derivatives. Since the logarithm of the B. Matching and Theoretical Uncertainties scale factor is a natural time coordinate for numerical so- lutions of the inflationary mode equations, we express the Ideally, a complete model of the early universe would be predicted by a candidate theory of fundamental 1 Second and higher order mode-mode couplings lead to both non- physics. In that case, we would know the mecha- Gaussianity and loop corrections to the two-point functions. For nism by which energy is drained from the inflaton to the models discussed here, these corrections are very small. (re)thermalize the universe, as well as the equation of 4 state and expansion rate of the primordial universe. Un- the inflaton as the kpivot leaves the horizon. However, fortunately, inflationary model building is not mature φ increases in some potentials and decreases in others, enough for this to be done on a routine basis and, as a while the mapping φ → φ + φ0 produces a new poten- consequence, the unknown expansion history of the post- tial with identical inflationary dynamics, so the numerical inflationary universe introduces a theoretical uncertainty value of φ is not informative on its own. Conversely, N (φ) into the predictions of inflationary models [64]. We must has the same interpretation in all inflationary models and therefore introduce at least one phenomenological param- is a monotonic (and usually simple) function of φ for a eter θRH to account for our ignorance of post-inflationary given potential. For these reasons we take Npivot rather physics. For reasons explained below, we work with N , than φpivot as a free parameter. The coupling between the number of e-folds of inflation between moment at Npivot and the post-inflationary universe simply reflects which a specified mode leaves the horizon (k = aH) the physical reality that the observed inflationary pertur- and the end of inflation, defined by the instant at which bation spectrum is a function of the post-inflationary ex- d2 a(t)/dt2 = 0. Further, given a specified pivot scale pansion history, but our choice ensures that the primor- (kpivot ) we can then specify a corresponding Npivot . For dial perturbation spectrum is calculated solely in terms a given inflationary model, Npivot is computed from the of parameters that describe the inflationary epoch which matching equation [60, 72, 73]. generated it.3 The matching depends on the growth of the horizon As we pointed out above, Npivot depends on the de- scale following inflation. This is a function of the detailed tailed expansion history of the post-inflationary universe. composition of the primordial universe, which is not well However, there are a huge number of possible combi- known, and the matching is thus intrinsically ambigu- nations of phases in the early universe, and these can ous. The two things we know with certainty are that have strongly degenerate predictions for Npivot .4 One can the universe is not thermalized at the end of inflation, imagine that the unknown expansion history is replicated and that it must be thermalized by MeV scales, when by an effective barotropic fluid with equation of state primordial nucleosynthesis occurs.2 Given that inflation w˜ [63, 64], which is superficially equivalent to regarding can be a GUT scale phenomenon, the energy (∼ ρ1/4 ) Npivot as a free parameter. Unfortunately, as explained may change by a factor of 1018 between inflation and nu- in Ref. [64], w˜ is an ambiguous parameter. Given an ex- cleosynthesis — a much larger factor than that between plicit inflationary potential, we can always determine the nucleosynthesis and the present day. moment at which inflation ends, but we cannot compute w˜ without specifying an energy scale at which the uni- A common assumption is that the universe is effec- verse has definitely thermalized: the numerical value of tively matter-dominated (a(t) ∝ t2/3 ) as the inflaton w˜ is a function of this choice. Admittedly, Npivot has an oscillates about the minimum of the potential, ther- 4 analogous dependence on the choice of kpivot . However, malizes at some temperature TRH (with ρRH ∝ TRH ), 1/2 assuming slow roll, Npivot ∼ log (kpivot )/(1 − ǫ) so this and remains radiation-dominated (a(t) ∝ t ) until the dependence is usually transparent. Further, for a given matter-radiation transition at z ≈ 3150 [7]. Even for combination of datasets, it is possible to determine an this assumed history, the uncertainty in TRH allows a optimal choice of kpivot [17], which is typically close to wide range in N . For example, a universe which is effec- the geometric mean of the range of scales contributing to tively matter-dominated between energies of 1015 GeV the dataset(s). to 103 GeV needs 9 e-folds less inflation after a given We note that there is a correlation between the spectral scale crosses the horizon than one which thermalizes at amplitude at kpivot (As in the usual ΛCDM parameter 1015 GeV. In many simple inflationary models, the run- set), and Npivot . As an example, consider m2 φ2 infla- ning αs is a few times 10−4 , so the resulting uncertainty tion: to first order, φend and N (φ) do not depend on m2 , in the scalar spectral index ns is ∆N αs ∼ 5 × 10−3 , while lowering m2 lowers As for fixed φ. Consequently, of the same order as the statistical error expected from Vend will also decrease, ensuring that slightly less growth Planck [8, 64]. However, far more extreme possibilities occurs between the end of inflation and some fixed ref- for the post-inflationary expansion rate exist, including erence point, such as nucleosynthesis or recombination. kination (a(t) ∝ t1/6 ) [74], frustrated cosmic string net- Given the precision with which the spectral amplitude works (a(t) ∝ t) [75], or even a short burst of thermal in- is now measured, this effect is small. Moreover, a simi- flation [76], and taking these scenarios into consideration greatly magnifies the uncertainty in inflationary predic- tions for a given model. 3 This approach works for scalar perturbations, since modes at The truly fundamental variable which specifies the por- astrophysical scales today remain outside the horizon until af- tion of the inflaton potential being traversed as the pivot ter nucleosynthesis, at which point the thermal history of the mode leaves the horizon is simply φpivot — the value of universe is well constrained. However, primordial gravitational waves seen by direct detection experiments can enter the horizon during epochs for which the expansion history is not tightly con- strained, and for these we need the full transfer function [77, 78]. 2 Recent evidence points to the existence of a cosmological neu- 4 For example, a long matter-dominated phase leads to the same trino background [7], which freezes out at temperatures slightly prediction for Npivot as a suitable combination of early radiation- higher than those that apply during nucleosynthesis. domination and a short secondary period of inflation. 5 lar degeneracy arises with w,˜ since this parameter is also pect the resulting uncertainty in the estimated variances sensitive to changes in the energy density at the end of to be exceeded by the foreground-removal uncertainties inflation, if all other parameters are held fixed. [80] in the large-angle B-mode constraint, which are not Theoretical considerations put very broad constraints taken into account in these forecasts. on Npivot . Firstly, in order to ensure that modes do ac- tually reenter the horizon, we need w ˜ ≥ −1/3, so that ρ + 3p ≥ 0 and a ¨ ≤ 0. This is not incompatible with a D. Initial conditions and reheating secondary period of inflation, but does require that the average expansion is not inflationary. Secondly, for a We will now describe the implementation of the infla- barotropic fluid, ρ ≥ p is required in order to avoid a tionary initial conditions and the reheating scenarios in superluminal sound-speed, so w ˜ ≤ 1. We are free to es- ModeCode. timate inflationary parameters for a narrower range of w ˜ Initial conditions: Inflationary potentials differ in their or Npivot , but it is important to recognize that doing so sensitivity to initial conditions (see Ref. [81] and refer- amounts to imposing a theoretical prior on the properties ences within). Thus, automatically setting self-consistent of the post-inflationary universe. initial conditions is a nontrivial issue. The initial value of In what follows, we constrain the free parameters of φ′ is set according to the slow roll equations (i.e. the in- inflationary potentials for two different reheating sce- flaton is assumed to be initially on the slow roll attractor narios, general reheating (GRH) and instant reheating solution). In addition, for a given set of potential param- (IRH). In the latter case, we assume that the universe eters, the algorithm must produce a starting field value thermalizes instantaneously as inflation ends, and re- φinit which corresponds to a time well before the modes mains thermalized until matter-radiation equality.5 In of interest leave the horizon. For the particular models contrast, GRH assumes only that the universe is ther- we consider here, the code produces a first guess for φinit malized by nucleosynthesis scales and that the aver- from the field value needed to achieve N (φinit ) = 70 in age expansion is no slower than radiation-dominated, or the slow roll approximation: (IRH) Npivot ≤ Npivot , the value computed assuming instanta- Z φinit neous thermalization. This prior is analogous to stipu- 1 V N (φinit ) = 2 dφ , (14) lating that −1/3 ≤ w ˜ ≤ 1/3, which implicitly rules out a MPl φend V,φ long kination-like phase. Lower values of w˜ correspond to smaller Npivot , which for the models considered here leads where φend is the field value at the end of inflation and to an increasingly red-tilted spectrum. Current data are V,φ ≡ dV /dφ. The code then iterates on this initial guess thus more sensitive to lower values of w ˜ for the models until a self-consistent value of φinit is found. For some considered here. combinations of potential parameters, it is possible that no set of self-consistent initial conditions exists. In such cases we reject the parameter combination in the MCMC analysis by assigning it a very small likelihood. C. MCMC methodology Reheating: ModeCode evolves the inflationary back- ground solution through to the end of inflation, defined Our Markov Chain Monte Carlo methodology is by d2 a(t)/dt2 = 0 which corresponds to based on a modified version of CosmoMC [39], using 2 Metropolis-Hastings sampling for basic parameter esti- 2 H,φ mation, and nested sampling via a MultiNest [47] plug- ǫH ≡ 2MPl = 1, (15) H in for the calculation of Bayesian Evidence (to be pre- sented in a forthcoming publication). The free parame- where ǫH is the first Hubble slow roll parameter.6 This ters in the potential V (φ) (plus the one reheating param- calculation yields the number of e-folds between the ini- eter in the GRH case) are varied in the MCMC chains, tial conditions and the end of inflation. The matching along with the other cosmological parameters of the con- equation then gives the scale factor at the end of infla- cordance model, and any nuisance parameters associated tion, aend . We connect a physical “pivot” wavenumber, with the datasets. kpivot , to a particular epoch during inflation using For all MCMC analyses of current CMB data, we run ≥ 6 chains per model/data combination, requiring the kpivot ≡ apivot Hpivot = aend e−Npivot Hpivot , (16) Gelman-Rubin [79] criterion on the eigenvalues of the co- where Hpivot is the Hubble scale corresponding to kpivot , variance matrix to be R − 1 . 0.01 for convergence. For which leaves the horizon Npivot e-folds before the end of Planck simulation runs we use 4 chains per model which satisfy R − 1 ∼ 0.1. While this convergence criterion is not as rigorous as that used in our main analysis, we ex- 6 The code also includes the capability to define the end of inflation as corresponding to a particular φend , which would be useful for implementing multi-field models. However, with the exception 5 In practice, w ˜ is not exactly 1/3, if the number of degrees of of the test in Sec. II E, we do not make use of this feature in the freedom in the thermal bath is itself a function of temperature. present work. 6 inflation. In what follows, the pivot scale is set to kpivot = 0.05 Mpc−1 . For a specific inflationary potential we can (IRH) compute Npivot from the usual matching equation; the post-inflationary expansion is known by hypothesis once we assume instantaneous reheating and " # " 1/4 # (IRH) 1016 GeV Vpivot Npivot = 55.75 − log 1/4 + log 1/4 , (17) Vpivot Vend where the above expression is drawn from Eq. (20) of Ref. [64], with appropriate substitutions.7 In the IRH (IRH) case we compute simply Npivot from this expression and the Npivot is not an independent variable in the chains. (IRH) In the GRH case we set the prior 20 < Npivot < Npivot . The lower limit comes from requiring that cosmologically relevant wavenumbers are far outside the horizon when inflation ends. This is tacitly assumed by our compu- tation of the spectrum from the asymptotic mode am- plitude in any case, and values of N near this limit are excluded by the data for all the models we consider here. The upper limit is enforced by rejecting any step to a (IRH) model for which Npivot > Npivot . FIG. 1: Test of the accuracy of ModeCode for power law inflation. Upper panel: comparison of the exact analytic so- lution (solid red curves) and the ModeCode solution (black E. Accuracy and timing points) for |uk | as a function of conformal time τ (following Ref. [13], τ is negative during inflation). For comparison, the To test the accuracy of ModeCode, we compare its evolution of |z| is plotted as a dashed green line. Lower panel: output with the analytic solution for primordial pertur- percent error in the ModeCode solution for each of the points bations in the “power law inflation” model [82], where plotted in the upper panel. the scale factor evolves as a(t) ∝ tp during inflation and the potential has an exponential form r matches the exact solution with an accuracy of about 0.01% or better. This test indicates that ModeCode 2 φ V (φ) = V0 exp . (18) does not introduce significant error in the computation p MPl of CMB angular power spectra by CAMB, which has a Power law inflation is one of the very few known models root mean square accuracy of ∼ 0.3% in the configura- for which the spectrum of primordial perturbations can tion to be used for the Planck analysis (A. Lewis, private be computed exactly, making it an ideal test case for the communication). numerical solution of ModeCode. In the course of previous work using ModeCode to Since the tensor-to-scalar ratio for this model is r = compute numerical power spectra for a potential with a 16/p, we must take a large value of p to avoid violating step-like feature [59], it was extensively compared with present upper limits on r; here we choose p = 60. For the independent code of Ref. [50], yielding agreement to p > 1, power law inflation does not end via slow roll numerical precision. It also agrees to similar precision violation [Eq. (15)], so we impose an end to the inflation- with the mode evolution code used for the same potential ary expansion at φend = MPl . We additionally assume in Refs. [53, 83], once the nonstandard choice of initial 4 log(V0 /MPl ) = −8.8 and Npivot = 50, yielding power conditions in the latter work is accounted for.8 spectra that are reasonably consistent with observations. To compute the primordial power spectra at arbitrary In Fig. 1, we compare the ModeCode solution for values of k in CAMB, ModeCode uses cubic spline in- scalar modes with the exact solution from Ref. [13]. At terpolation on a grid of k values spaced evenly in ln k. all scales, the numerical evolution of scalar perturbations The extra time required to run ModeCode with CAMB depends primarily on the number of k values in this grid. For the default setting of 500 grid points over 7 This expression makes no allowance for the changing number of 10−5 < k/Mpc−1 < 5, which provides more than suffi- relativistic degrees of freedom as the Universe cools or for dark cient accuracy for smooth primordial power spectra, us- energy, and it assumes a sharp transition between radiation and matter-dominated expansion. The resulting approximation will not significantly bias our results using present data, but may need to be addressed in the future. 8 See Appendix A in Ref. [59] for details. 7 ing ModeCode with CAMB typically requires . 15% TABLE I: Priors on model parameters and maximum likeli- more time per evaluation than the default version of hood (ML) values for WMAP7 GRH constraints. All GRH CAMB. The number of grid points can be easily adjusted (IRH) models include a uniform prior of 20 < Npivot < Npivot . in the code to accurately deal with more complicated po- Dimensionful quantities are expressed in units where the re- tentials for which finer sampling in k is required. duced Planck mass MPl is set to unity. Values of n refer to specific cases of Eq. (19). F. Data Model Priors ns,ML rML −2 ln LML n = 2/3 −11 < log λ < −7.5 0.965 0.07 7475.2 CMB Data: We use the v4 version of the 7-year WMAP likelihood function with standard options [84], n=1 −11 < log λ < −7.5 0.969 0.08 7475.4 the ACBAR bandpowers from Ref. [85] between 550 ≤ n=2 −12 < log m2 < −8 0.964 0.14 7477.3 ℓ ≤ 1950, and the Pipeline 1 QUaD bandpowers be- n=4 −13.4 < log λ < −10.4 0.949 0.27 7488.7 tween 569 ≤ ℓ ≤ 2026 from Ref. [86]. For IRH mod- Natural −5 < log Λ < 0 0.962 0.08 7475.8 els, we consider constraints from the 7-year WMAP data 0.5 < log f < 2.5 (“WMAP7”) only. In the GRH case, we present both Hilltop −8 < log Λ < −2.8 0.944 4 × 10−5 7476.2 WMAP7 results and constraints that additionally include −13.3 < log λ < −12 data from QUaD and ACBAR (“WMAP7+CMB”). Planck Simulation: For selected inflation models in the GRH case, we use an unpublished simulation kindly pro- vided by George Efstathiou and Steven Gratton. In this full form of the potential, since the modification near the simulation, the model for the “observed” power spec- origin changes φend and the matching between k and φ. tra has four components: the primordial CMB power For the single term potentials, the free parameter fixes spectra, unresolved point sources, unresolved Sunyaev- the height of the potential and the amplitude of the per- Zel’dovich (SZ) clusters, and instrumental noise. The turbations (i.e. As ). However, for these models the other input CMB power spectra are computed from a random spectral parameters (ns , r, etc.) are well approximated realization centered on the “best fit” WMAP 5-year cos- by combinations of slow roll parameters, which do not de- mological parameters, including ns = 0.963. In addi- pend directly on λ. When the potential has two or more tion the simulation includes a tensor component with free parameters, the mapping between the explicit form r = 0.1, close to the margin of detectability by Planck of the potential and the cosmological observables grows [87, 88]. The likelihood function is described by an exact more complicated, since these parameters affect both the Wishart distribution, marginalizing over the SZ model height and shape of the potential. We give constraints and the point source model as nuisance parameters in the on axion-motivated “natural inflation” [92] with MCMC. Note that while the fiducial model has r = 0.1, the particular realization used in the simulation is con- 4 φ V (φ) = Λ 1 + cos , (20) sistent with a somewhat larger tensor amplitude that is f closer to r = 0.14. The best fit values for the models we consider reflect this larger tensor-to-scalar ratio. and “hilltop inflation” [93–95] with λ 4 V (φ) = Λ4 − φ , (21) G. Models and priors 4 for which r takes almost arbitrary values while ns remains In order to illustrate our methods, we derive con- close to unity. straints on a variety of models. First we consider a se- We show the model priors used in our MCMC analysis quence of “single term” potentials, in Table I. The model parameters correspond to un- known scales in high energy particle physics, so it is nat- φn ural to sample them logarithmically. For the single pa- V =λ (19) n rameter models, we will see that the data constrain these scenarios more strongly than the priors on the height with n = 2/3, 1, 2, and 4. The last two cases correspond of the potential. On the other hand, the priors chosen to the canonical quadratic and quartic chaotic inflation for the natural and hilltop inflation models are more re- models [89], and for n = 2 we replace λ with m2 in our strictive. Both natural and hilltop inflation have limits discussion. String motivated scenarios [90, 91] can yield in which they are essentially identical to a φn model, potentials with the form Eq. (19) and non-integer values and one of their two free V (φ) parameters is irrelevant of n at large φ. We assume that these potentials are mod- [64, 73]. For natural inflation, this limit is f → ∞, such ified for φ ≤ 0 to ensure that V (φ) ≥ 0, if n is not an even that integer. For convenience, we will work with the simple monomial term; to explicitly constrain the corresponding Λ4 stringy scenarios we would instead have to work with the V (φ) ≈ (φ − φ0 )2 (22) 2f 2 8 FIG. 2: Constraints on the quadratic potential (n = 2). FIG. 3: Same as Fig. 2 for the quartic potential (n = 4). Left: constraints on log m2 and Npivot for WMAP7 (large gray contours) showing 68% CL (light shading) and 95% CL (dark shading) contours. Green dashed curves show the con- tours for WMAP7+CMB data, and the small blue contours show simulated Planck constraints. Right: marginalized 1D ns and r distributions for WMAP7 (thick solid curve, black), WMAP7+CMB (dashed curve, green), and the Planck simu- lation (thin solid curve, blue). In both panels, the sharp right edges of the distributions correspond to the IRH constraints, for which the values of Npivot (left), ns , and r (right) are nearly fixed. Here and in Figs. 5 and 6, the input parameters for the Planck simulation are ns = 0.963 and r = 0.1 (see Sec. II F). with φ0 = πf , which is a quadratic potential after a field redefinition. In the hilltop case, if Λ is very large, the astrophysically relevant portion of the potential is far from the origin. In this limit, 4 φ V (φ) ≈ 4Λ 1 − , (23) φ0 √ where φ0 = 2Λ/λ1/4 is the field value at which the po- tential crosses zero. A field redefinition yields a purely linear potential with a single free parameter. Our pri- ors are chosen to avoid the regions of parameter space where the two parameters of the natural or hilltop infla- tion model are degenerate. We stress that the goal of the current paper is to in- FIG. 4: Constraints on single term potentials with n = 1 troduce ModeCode, explore the ability of both current (top) and n = 2/3 (bottom). Conventions match Fig. 2. and anticipated datasets to distinguish between differ- ent inflationary models, and constrain the number of e- folds of inflation required by specific inflationary scenar- the inferred constraints on ns and r. The limits on log λ ios. Broader issues surrounding the choice of priors and from current data are much stronger than the priors listed model selection will be addressed in a forthcoming paper. in Table I. Stipulating instant reheating (IRH) ensures that Npivot is nearly independent of the other model pa- rameters, and only weakly dependent on the value of the III. RESULTS (IRH) exponent n: Npivot ∼ 57–59. In the general reheating (GRH) case, slow roll calculations lead us to expect that A. Single term potentials Npivot , ns , and r are all effectively functions of a single free parameter, and are thus strongly correlated; these Figures 2–4 show the constraints from WMAP7 and correlations are reflected in the more accurate Mode- WMAP7+CMB on log λ (or log m2 for n = 2) and Npivot Code constraints. The IRH upper limit from the prior for V (φ) = λφn /n with n = 2/3, 1, 2 and 4, along with on Npivot creates a sharp cutoff in the distributions for 9 ns and r. Given the prior on Npivot , each of these mod- els has a red tilt (ns < 1) and a “large” tensor-to-scalar ratio, r ∼ 0.1. If Npivot is lower than its IRH value, or w˜ < 1/3, λ (or m2 ) is larger than its IRH value: the overall range in this parameter is typically a factor of ∼ 2–3. Lower values of Npivot correspond to smaller ns and larger r, as does increasing the exponent n for fixed Npivot . Specifically, the slow roll approximation gives n 3 ζ 4n Npivot + ≈ n− (1 − ns )−1 ≈ , (24) 4 2 n−1 r where ζ = 0 for n = 1 and ζ = 1 otherwise. Thus upper bounds on 1 − ns and r from data can set lower bounds on Npivot . Except for the n = 4 case, the predicted value of r is less than the current WMAP upper bound for all values of Npivot allowed by the prior. For n = 2/3 and n = 1 the constraint on Npivot appears to be driven largely by the correlation between this parameter and ns — in all cases ns < 0.94 is strongly excluded. For n = 2, the larger value of r found with smaller Npivot provides some additional constraining power. Including CMB data on smaller angular scales from QUaD and ACBAR slightly strengthens these limits, relative to the FIG. 5: Top left: WMAP7 68% and 95% CL constraints WMAP7 constraints, but does not significantly alter our on natural inflation parameters log Λ and log f for the GRH conclusions. (large contours, gray shading) and IRH (small contours, red Of the single term potentials we consider, the quartic shading) scenarios. Top right: constraints on ns and r; points λφ4 /4 potential has the largest tensor amplitude and the show random samples of models from the MCMC analysis for which Npivot is within 0.25 of the values indicated in greatest deviation from scale invariance. In agreement the plot. Bottom: Predicted natural inflation GRH con- with previous analyses of CMB data (e.g. Ref. [51]), straints from Planck (small contours, blue shading) compared this model is excluded by WMAP7 data. Specifically, with current constraints from WMAP7 (gray shading) and −2 ln LML exceeds the values found for all of the other WMAP7+CMB (dark green, dashed contours). models considered here by & 12 (see Table I). Superfi- cially, this does not appear to be significantly stronger than the result obtained with a single year of WMAP Constraints using the simulated Planck likelihood show data [51]. However, the analysis of Ref. [51] was carried that the uncertainties in parameters of these single term out at fixed Npivot = 50, whereas Npivot is a free pa- potentials will be greatly reduced by the next generation rameter in our chains. If we impose the additional prior of cosmological datasets. For the quadratic potential, Npivot < 50, the maximum likelihood of the quadratic we see from Fig. 2 that the simulated Planck constraints potential worsens to −2 ln LML = 7499.4. This is ∼ 24 strongly disfavor (at > 95% CL) models with ns . 0.96, larger than the overall best fit and excludes such mod- corresponding to Npivot . 50 and log m2 & −10.3. Re- els with much greater confidence than the first year of call that for quadratic inflation followed by a matter- WMAP data alone. dominated phase and then thermal inflation, Npivot is In recent work, Martin and Ringeval [63] quote con- (IRH) at least 10 less than the instant reheating value Npivot straints on the reheating temperature following inflation driven by a V (φ) ∼ φn potential. The tightest con- [76]. Consequently, we predict that Planck can differ- straints they present imply that the reheat temperature entiate between these two post-inflationary scenarios for is above the TeV scale. However, this specific constraint quadratic inflation. is obtained for a prior that renders the post-inflationary expansion rate a function of n. Since scenarios for which V (φ) ∼ φn at large field values can have very different B. Natural inflation shapes near the origin, the prior could only be realized by a carefully tuned potential, as V (φ) would need to Figure 5 shows our constraints on the natural inflation be well approximated by φn at energies far below the in- [Eq. (20)] parameter space and the derived empirical pa- flationary scale. Moreover, this form of V (φ) must be rameters ns and r from ModeCode. The relationship modified near the origin if V (φ) ≥ 0 for all φ, and the between the empirical parameters and the potential pa- potential does not have a discontinuous first derivative rameters for natural inflation is discussed in detail in Ref. at φ = 0 for n 6= 2 or 4. [96], along with parameter constraints derived from the 10 3-year WMAP dataset. Unlike the single term potentials, current data permit a wide range of natural inflation parameters and, as noted in Sec. II G, there is a degeneracy between f and Λ in the limit where these parameters are large. In this re- gion of parameter space, natural inflation overlaps with the quadratic model. Our priors are chosen to exclude most of this region; given the parametrization of the po- tential, if we allowed arbitrarily large values of f and Λ (and given that the quadratic potential is not currently excluded by data) almost all points drawn by the chains would be in this degenerate region. Our adopted priors on log f and log Λ still allow a region of nearly-degenerate models that contribute to the “ridge” seen in the right panels of Fig. 5; these models closely match the values of ns and r seen in the quadratic potential constraints. The marginalized constraints on ns and r depend strongly on the prior on log f due to the projection of a large number of degenerate models into this ridge. Thus, the apparent preference for this region of parameter space over models with lower values of r is largely due to this effect, and is not driven by the data. (IRH) Instant reheating requires Npivot ∼ 58, similar to the constraint for the single term potentials, but the addi- FIG. 6: Same as Fig. 5 for the hilltop inflation model with tional inflationary degree of freedom in the potential per- parameters log Λ and log λ. Note the logarithmic scale for r mits a larger range of ns and r. More generally, for fixed in the right panels; the prior here permits very small values Npivot , decreasing Λ and f reduces both ns and r. This of r. is shown by the MCMC samples with fixed Npivot plotted in the upper right panel of Fig. 5. Thus natural inflation models can have lower values of r than the quadratic correlated with ns , but not r, over most of the allowed potential in Fig. 2, without increasing ns and Npivot . region of parameter space. The lower panels of Fig. 5 show how the natural in- Constraints on the hilltop inflation model from flation constraints improve with additional CMB data. WMAP7+CMB data and from the Planck simulation are As for the single term potentials, the difference between compared with the (GRH) WMAP7 constraints in the WMAP7 and WMAP7+CMB constraints is small, but lower panels of Fig. 6. The values of ns and r allowed Planck is expected to yield a dramatic improvement. In by the hilltop model within our chosen priors are smaller particular, the uncertainty in Npivot — which is visible in than those assumed in the Planck simulation, so the con- the width of the log Λ − log f contours — is substantially tours for Planck are concentrated at the largest allowed reduced in the Planck forecast. Since the Planck simula- values of both parameters. In fact, for this particular tion we use is not inconsistent with m2 φ2 inflation, the forecast the entire region of the hilltop inflation param- limit in which natural inflation becomes indistinguishable eter space within our priors would be strongly excluded from quadratic inflation would not be excluded in this by Planck, with −2 ln LML ∼ 75 larger than its value for particular forecast. Conversely, if quadratic inflation is the quadratic and natural inflation models. disfavored by future data, we will be able to put data- Unlike the other models considered here, current data driven upper bounds on f and Λ in addition to tightening (IRH) allow Npivot to cover a substantial range (roughly 10 the existing lower bound. e-folds), as illustrated in Fig. 7. The weak constraint (IRH) on Npivot is due to a special cancellation in the slow roll expression for the scalar spectral amplitude, which C. Hilltop inflation leaves As independent of the overall height of the po- tential.9 Meanwhile, the departure from scale invariance Figure 6 shows the constraints on log Λ, log λ, ns , and (IRH) is 1 − ns ≈ 3/Npivot in the limit of small Λ. Other log r for hilltop inflation. We impose an upper limit two-parameter models can formally support inflation at of Λ < 0.0015 to remove models where the field starts low energy scales, but the spectra of such models are far from φ = 0 and near the V = 0 crossing point, as described in Sec. II G. The remaining models have a small tensor amplitude and relatively large deviations 9 In the more general class of hilltop potentials V (φ) = Λ4 −λφn /n, from scale invariance. The MCMC samples plotted at this situation only occurs for n = 4 [93–95], so this property is fixed Npivot in Fig. 6 show that the number of e-folds is not generic. 11 FIG. 7: Marginalized 1D distributions for Npivot from WMAP7 in the GRH case (thin black curves) and IRH case (thick red curves). From left to right, the models are the quadratic potential, the quartic potential, natural inflation, and hilltop inflation. that they have no significant impact on the estimated pa- rameter values and confidence regions. The upper limit on Npivot corresponding to instant reheating, however, does significantly reduce the allowed region in parame- ter space for each of these models, thus restricting the possible values of ns and r. The resulting upper limit on ns and lower limit on r can lead to tension with observations. For example, the WMAP7 constraints are ns = 0.982+0.020 −0.019 (68% CL) and r < 0.36 (95% CL) [84], treating these as empiri- cal parameters without specifying a particular potential. For the quartic λφ4 /4 potential, the instant reheating FIG. 8: The parameter space in ns and r corresponding to limit on Npivot and the nearly perfect correlation be- uniform sampling within the priors on potential parameters tween Npivot , ns , and r result in the limits ns . 0.95 listed in Table I (unshaded curves, 68% and 95% CL regions) and r & 0.27; since both of these are in tension with the for natural inflation (left) and hilltop inflation (right). The measured values, the quartic potential provides a poor WMAP7 GRH constraints from Figs. 5 and 6 are shown again fit to the data relative to the other single term potentials for comparison as shaded gray 68% and 95% CL regions. Or- which can achieve larger ns and smaller r. ange shading shows the 95% CL region for empirical WMAP7 On the other hand, as noted earlier the natural infla- constraints with flat priors on ns and r. tion and hilltop inflation priors have been chosen specif- ically to limit the extent of parameter degeneracies that would otherwise be allowed by current data. In the lim- typically far from scale-invariant and thus disfavored by iting regions that are truncated by the priors, these two the data. On the other hand, hilltop inflation can have models are degenerate with specific monomial potentials, Λ ≪ 1016 GeV, which ensures that the tensor amplitude as discussed previously. Furthermore, the mapping be- is very low, without driving the spectral index to an ob- tween θV and θemp for these models is significantly more servationally excluded value. Consequently, both r and complicated than for the single term potentials, so the (IRH) Npivot can vary greatly, as we see in Fig. 6. Models with uniform top-hat priors on θV can correspond to highly very low values of Λ do have a lower likelihood, as the non-uniform prior distributions for θemp . This mapping stronger breaking of scale invariance in these models is at of the priors is illustrated in Fig. 8, which shows the re- odds with the spectral tilt allowed by the data. For IRH gions of the (ns , r) plane obtained by uniform sampling models there is effectively only one free parameter in the within the priors on θV specified in Table I for the natural potential after fixing As , leading to a strong correlation inflation and hilltop inflation models. between ns and r which is absent in the GRH case. Due to the natural inflation degeneracy between log Λ and log f , even in the absence of any data the priors clearly favor models along the line r ≈ 4(1 − ns ) cor- D. Impact of model priors responding to the approximately quadratic regime near the minimum of the natural inflation potential. The For the single term potentials, the priors on the ampli- WMAP7 GRH constraints in Fig. 5 are qualitatively de- tude of the potential λ (or m2 ) listed in Table I are suffi- scribed by the intersection of the priors with the empiri- ciently weak compared to the constraints from the data cal constraints on ns and r from WMAP7. Note that the 12 (IRH) upper limit on ns is typically set by the Npivot ≤ Npivot prior (the lower right edge of the 95% CL region in the left panel of Fig. 8), while the lower limit on ns — and thus Npivot — follows from the data. The 95% lower limit on r does not quite reach down to the limit allowed by the prior because of marginalization over the strong pro- jection effect described in Sec. III B, which favors models along the upper diagonal ridge of the prior. With present data, the region in the (ns , r) plane allowed by the pri- ors is fully consistent with the measured values of these parameters (and therefore the best fit ns and r values in Table I are consistent with the WMAP7 empirical con- straints), but upcoming measurements may yet rule out the natural inflation model. For the hilltop inflation model, the constraints on ns and r are partially influenced by the priors on log Λ and log λ from Table I; in particular, the range of r allowed is limited by the prior on log Λ (see Fig. 8). However, the distribution of models allowed by the priors in the (ns , log r) plane is much more uniform for hilltop inflation than it is for natural inflation. As for natural inflation, the upper limit on ns for hilltop inflation models is set FIG. 9: Approximate constraints on the quadratic potential (IRH) obtained by selecting MCMC samples from standard WMAP7 by the Npivot ≤ Npivot prior, and the lower limit on ns ΛCDM+tensor chains and mapping As , ns , and r to log m2 is determined by the data. Like the quartic potential, and Npivot using the slow roll approximation. The shaded the upper limit on ns corresponding to instant reheating gray contours are the WMAP7 GRH constraints from Fig. 2, is low compared with the preferred value from WMAP7; obtained by a direct MCMC estimate of m2 and Npivot for however, in the case of hilltop inflation, this is coupled quadratic inflation. The unshaded orange contours are de- with a small value of r, which enables the hilltop inflation rived by sampling chains for As , ns , and r, from which we model to fit the WMAP7 data reasonably well since the selected only those points with |1 − ns − r/4| < 0.005 to ap- empirical constraints on ns and r are correlated. proximately match the slow roll relation between ns and r for the quadratic potential. E. Slow roll mapping respond to the same primordial power spectra, so the Many previous constraints on inflationary models have mapping θV → θemp is not always invertible. Further, been obtained by taking the empirical parameters θemp models with sharp features or other complexities do not (e.g. As , ns , and r) predicted by a given model and com- produce a power spectrum that is easily described by the paring these with constraints on empirical parameters usual empirical parametrization. (see e.g. [7, 52]). Given current data, limits on As , ns and Despite these problems, it is instructive to attempt to r typically dominate the constraints on simple inflation- derive constraints on θV from constraints on θemp for ary models; starting from constraints on these empirical the purposes of comparison with our main results in the parameters allows constraints on a number of inflation- previous sections. Here we perform these tests using the ary models to be inferred from a single set of empirical slow roll approximation, although one could implement Markov chains. the mapping using more precise methods. In Appendix A For the simplest inflationary models, this approach al- we describe the slow roll mapping for the quadratic po- lows one to quickly constrain many potentials at once. In tential. Figure 9 shows that the constraints on log m2 and general, however, the mapping θemp → θV is not one-to- Npivot obtained by mapping from empirical parameters one; thus this method requires sampling within the space agree reasonably well with the direct MCMC constraints of potential parameters and is no more efficient than di- from ModeCode. However, they are visibly noisy due rectly constraining these parameters from the data using to the smaller number of MCMC samples, and are sys- ModeCode. For any potential parametrized by θV , one tematically shifted toward lower values of log m2 , due to can compute the primordial scalar and tensor power spec- the mapping being done at lowest order in slow roll. tra (as described in Sec. II A, or otherwise) and find the While this approach works well for the quadratic po- values of θemp by fitting to these power spectra. However, tential (and can be expected to produce similar results for any given inflationary potential many values of θemp for the n = 1 and n = 2/3 cases) it is far less efficient cannot be obtained for any combination of θV . Likewise, for the quartic (n = 4) potential. Recall that this sce- as we saw for the natural and hilltop scenarios, some nario is disfavored by the data. Consequently, using the models have degenerate combinations of θV which cor- Metropolis-Hastings algorithm to draw samples from the 13 (ns , r) parameter space results in very few (if any) ac- the universe was thermalized. Consequently, there is a cepted points in the relevant region for the quartic po- huge range of energies over which the composition and tential on the (ns , r) plane, and thus the contours of λ expansion rate of the universe are effectively undeter- and Npivot are extremely noisy. Constraining the quartic mined. potential parameters using this method requires MCMC For most inflationary models, the constraints on the sampling that is specifically designed to acquire samples total number of e-folds since the pivot scale left the hori- in the (ns , r) region spanned by this model. zon, Npivot , are noticeably tighter than the hard lower Models with multiple parameters like natural inflation bound (Npivot ≥ 20) assumed in our analysis. Thus we and hilltop inflation face an additional problem in that can be confident that our constraints on Npivot are driven the mapping from θV to θemp is not invertible in regions by the data, although these currently eliminate only rel- where there is a parameter degeneracy. For example, at atively extreme post-inflationary scenarios. By contrast, fixed values of Λ4 f −2 , all natural inflation models with the Planck forecast suggests that the next generation of large f have nearly identical power spectra (see Sec. II G). CMB data will put much tighter constraints on the re- Therefore, a single point in θemp space cannot be simply heating history, given a specific inflationary model. For mapped to the corresponding values of θV in this degen- instance, for m2 φ2 inflation, Planck should discriminate erate region. Such a mapping would require an additional between a long matter-dominated phase that extends to MCMC run or some similar method of sampling in the the TeV-scale and instant reheating at perhaps the 2 σ θV parameter space. level. This correlation between the post-inflationary dynam- ics and the inflationary epoch has significant conse- IV. DISCUSSION quences for particle physics. For instance, many super- symmetric scenarios predict that the primordial universe In this paper we introduce ModeCode, a new, pub- undergoes a period of matter domination driven by heavy licly available numerical solver for the inflationary per- moduli (e.g. [78, 97]), for which Npivot differs substan- turbation equations. ModeCode is extendable, numer- tially from the instant reheating value, and post-Planck ically efficient, and integrated with CAMB and Cos- cosmology will thus be increasingly concerned with the moMC. We demonstrate the use of ModeCode by con- full evolutionary history of the universe. Moreover, infla- straining several single-field inflationary models using tionary model builders will be able to definitively test sce- current CMB data, confirming that present-day data put narios which include predictions for the post-inflationary useful constraints on the parameters of a number of in- expansion rate. teresting and well-motivated inflationary models. Using a simulated likelihood, we present forecasts for the quality of the constraints that can be expected from Planck, showing that these will greatly strengthen limits Acknowledgments on the inflationary parameter space and possibly exclude some simple inflationary models. In the Planck forecast We are extremely grateful to George Efstathiou and analysis, our aim was not to provide “Fisher”-style fore- Steven Gratton for the permission to use their unpub- casts for each model in turn, but to take the realiza- lished Planck simulation, which was used to generate tion provided by the simulation and analyze it with our the “Planck forecast” contours in Figures 2, 5, and 6. pipeline as we would the real sky. Given the level of ten- MJM is supported by CCAPP at Ohio State. HVP sors present in this simulation, which is at the margin of is supported in part by Marie Curie grant MIRG-CT- detectability with Planck, we find that small-field mod- 2007-203314 from the European Commission, and by els would be excluded with high significance as expected. STFC and the Leverhulme Trust. RE is partially sup- Conversely, in a scenario where Planck polarization mea- ported by the United States Department of Energy (DE- surements did not find evidence of tensor fluctuations, FG02-92ER-40704) and the National Science Foundation we predict that many large-field models would be either (CAREER-PHY-0747868). HVP and RE thank the As- excluded or limited to a small region of parameter space. pen Center for Physics for hospitality during the comple- In our analysis we pay close attention to the inter- tion of part of this work. RE thanks IoA Cambridge and play between the post-inflationary expansion history and DAMTP CTC for additional hospitality. Numerical com- the inflationary observables. These are connected by the putations were performed using the Darwin Supercom- matching criterion, which determines the moment dur- puter of the University of Cambridge High Performance ing the inflationary epoch at which a given comoving Computing Service (http://www.hpc.cam.ac.uk/), pro- scale leaves the horizon. Inflation is often assumed to be vided by Dell Inc. using Strategic Research Infrastructure a GUT scale phenomenon, but the expansion rate and Funding from the Higher Education Funding Council for thermal state of the post-inflationary universe is not di- England. We acknowledge the use of the Legacy Archive rectly constrained until MeV scales, at which point the for Microwave Background Data (LAMBDA). Support success of Big Bang nucleosynthesis and evidence for a for LAMBDA is provided by the NASA Office of Space cosmological neutrino background strongly suggest that Science. 14 Appendix A: Slow roll mapping—quadratic potential more closely in the limit δ → 0, or to retain more MCMC points for better sampling of the likelihood function (and Here we describe the procedure that produced the con- thus smoother contours) by choosing a larger value of straints on the quadratic potential shown in Fig. 9 by δ; for Fig. 9 we have chosen δ = 0.005. Note that the using slow roll relations to map constraints on the em- allowed values of As are not restricted by specializing to pirical parameters θemp = {ln As , ns , r} to constraints on the case of the quadratic potential. the potential parameters θV = {Npivot , log m2 }. For the We additionally exclude points for which Npivot is large potential V = m2 φ2 /2, the slow roll approximation gives enough that ρRH is greater than (3/2)Vend , thus violating energy conservation. (The factor of 3/2 arises because m2 ≈ 24π 2 As ǫ2V , (A1) the energy density at the end of inflation contains a sig- 1 nificant contribution from the kinetic energy.) That is, Npivot ≈ (ǫ−1 − 1), (A2) (IRH) we require Npivot ≤ Npivot , where 2 V 1 r ǫV ≈ (1 − ns ) ≈ , (A3) (IRH) 1/4 4 16 Npivot = ln(Hpivot /kpivot ) − 71.1 − ln(Vend /MPl ), (A4) 2 where ǫV ≡ (MPl /2)(V,φ /V )2 . with Vend ≈ m2 MPl 2 for the quadratic potential. We begin with constraints on θemp from a standard For the MCMC samples that remain after applying MCMC analysis of WMAP7 data, which treats As , ns , the cuts on ns , r, and Npivot , we multiply the likelihood and r as free, independent parameters. We then post- of each sample by r−2 to approximate flat priors on θV process the empirical parameter chains to express the instead of the original flat priors on θemp. constraints in terms of θV by first imposing new priors After including each of these priors and mapping the on θemp, and then using Eqs. (A1)–(A3) to compute the MCMC parameters from θemp to θV using Eqs. (A1)– values of θV for each MCMC sample. (A3), the resulting constraints shown in Fig. 9 match the The first prior applied to θemp enforces the slow roll more accurate constraints from ModeCode (Sec. III A) relation between ns and r for the quadratic potential, reasonably well, albeit with noisy contours due to poor r ≈ 4(1 − ns ), by selecting only those points from the sampling along the line r ≈ 4(1 − ns ). The remaining chains that satisfy |1 − ns − r/4| < δ. 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