UCLEAR PHYSICS PROCEEDINGS SUPPLEMENTS FI.SEVIER Nuclear Physics B (Proc. Suppl.) 43 (1995) 118-125 The Inflaton Potential from Present Day Observations Edward W. Kolb, a* Mark Abney, b* Edmund Copeland, ct Andrew L. Liddle, c~ and James E. Lidseya*t ~NASA/Fermilab Theoretical Astrophysics Center, Fermi National Accelerator Laboratory P.O. Box 500, Batavia, Illinois 60510-0500 USA bDepartment of Physics, The Enrico Fermi Institute, The University of Chicago 5640 South Ellis Ave., Chicago, Illinois 60637 USA CSchool of Mathematical and Physical Sciences, University of Sussex Brighton BN1 9QH United Kingdom Qumatum fluctuation during the inflationary phase in the early Universe should leave their imprint on the present-day structure of the Universe. Here we discuss how to read this imprint in the cosmic microwave back- ground and the large-scale structure of the Universe to discover something about the microphysics driving inflation. 1. INTRODUCTION influence is on the large-scale microwave back- ground anisotropies, such as those probed by There are many models for cosmic inflation. COBE [1]. The scalar density perturbations are All of them involve a rapid growth of the size thought to lead to structure formation in the of the Universe. This is most easily illustrated Universe. They produce microwave background by considering a homogeneous, isotropic Universe anisotropies across a much wider range of angular with a flat Friedmann-Robertson-Walker (FRW) scales than do the tensor modes, and constraints metric, where the "size" is parameterized by a on the scalar spectrum are also available from the scale factor a(t), and "rapid growth" means a pos- clustering of galaxies and galaxy clusters, pecu- itive value of 5/a =- --(47rGN/3)(p + 3p) where p liar velocity flows and a host of other measurable is the energy density and p the pressure. quantities [2]. It is useful to identify the energy density driv* Generally, inflation predicts a nearly Gaussian ing inflation with some sort of scalar "potential" spectrum of density perturbations that is weakly energy density V that is positive, and results in scale dependent, i.e., the amplitude of the per- an effective equation of state p ~ -p _ V, which turbation depends upon the length scale. Such satisfies 5 > 0. If one identifies the potential en- a dependence typically arises because the Hub- ergy as arising from the potential of some scalar ble expansion rate during the inflationary epoch field ¢, then ¢ is known as the inflaton field. changes, albeit slowly, as the field driving the ex- Observational consequences of inflation today pansion rolls towards the minimum of the scalar are the stochastic spectra of density (scalar) per- potential. This implies that the amplitude of the turbations and gravitational wave (tensor) modes fluctuations as they cross the Hubble radius will generated during inflation. Each stretches from be weakly time-dependent. scales of order centimeters to scales well in ex- In slow-roll inflation, any scale dependence for cess of the size of the presently observable Uni- density perturbations is possible if one considers verse. Once within the Hubble radius, gravi- an arbitrary functional form for the inflaton po- tational waves redshift away and so their main tentiai, V(¢). In this sense, inflation makes no unique prediction concerning the form of the den- *Work supported in part by the Department of Energy and NASA (NAG5-2788). sity perturbation spectrum and one is left with ?Supported by the PPARC two options. Either one can aim to find a deeper *Supported by the Royal Society 0920-5632/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0920-5632(95)00462-9 E.W Kolb et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 118-125 119 L Ao ;k 1 10 10 2 lO s 6xlO s ~ (Mpc/h) I I I I I E> I0 -z 0.I I 10 0 (deg) ! cMBR I I-- LSS I Figure 1. The basic idea of inflation is that as the field evolves under the influence of the inflaton potential, quantum fluctuations in the inflaton field produce scalar density perturbations As, while fluctuations in the transverse, traceless metric components produce tensor gravitational wave perturbations, AG. Also indicated is the main observational information from the cosmic microwave background and large-scale structure surveys. 120 E. ~g Kolb et al./Nuclear Physics B (Proc. SuppL) 43 (1995) 118-125 physical principle that uniquely determines the arate places. The first is in simplifying the classi- potential, or observations that depend on V(¢) cal inflationary dynamics of expansion, with the can be employed to limit the number of possibil- lowest-order approximation ignoring the contri- ities. One such observation is the amplitude of bution of the inflaton's kinetic energy to the ex- the tensor perturbations produced by inflation. pansion rate. The second is in the calculation In the past two years we developed a formal- of the perturbation spectra; the standard expres- ism which allows one to reconstruct the inflaton sions are true only to lowest-order in slow-roll. In potential V(¢) directly from a knowledge of these the expressions in the previous section, we uti- spectra [3]. This developed an original but incom- lized the Hamilton-Jacobi approach [5] to treat plete analysis by Hodges and Blumenthal [4]. An the dynamical evolution exactly. important result that follows from our formalism A very elegant calculation of the perturbation is that knowledge of the scalar spectrum alone is spectra to next order in slow-roll has now been insufficient for a unique reconstruction. Recon- provided by Stewart and Lyth [6]. The slow-roll struction from only the scalar spectrum leaves an approximation can be specified by parameters de- arbitrary integration constant, and since the re- fined from derivatives of H(¢). There are in gen- construction is nonlinear, different choices of this eral an infinite number of these as each derivative constant lead to different functional forms for the is independent, but usually only the first few en- potential. Some minimal knowledge of the ten- ter into any expressions. We shall require the first sor spectrum, say its amplitude at a single wave- two, which are all of the same order when defined length, is sufficient to lift this degeneracy. by The most ambitious aim of reconstruction is 2 [H'(¢)12 2 H"(¢) to employ observational data to deduce the in- e(¢) = ~- [g(¢)J ; 7/(¢) - ~2 H(¢) ' (1) flaton potential over the range corresponding to microwave fluctuations and large-scale structure, where n 2 = 87rGN, ~nd prime denotes d/d¢. The although at present the observational situation is slow-roll approximation applies when these slow- some way from providing the quality of data that roll parameters are small in comparison to unity. this would require [3]. The condition for inflation, 5 > 0, is precisely In this talk I will discuss the promise of po- equivalent to e < 1. tential reconstruction assuming one knows 1) the The lowest-order expressions for the scalar amplitude of the tensor spectrum at one point (As) and tensor (AG) amplitudes assume {e, ~/} from microwave background fluctuations, pre- are negligible compared to unity. Improved ex- sumably on quadrupole scales corresponding to pressions for the scalar and tensor amplitudes for 3000h-lMpc, and 2) the scalar spectrum from finite but small {e, r/} were found by Stewart and microwave background fluctuations and the large- Lyth [6]: scale structure investigations from quadrupole scales down to scales of several Mpc. v/2t¢ 2 H 2 As "~ 8zr3/2 g' [1 - (2C+ 1)e+ C~/] /£ 2. RECONSTRUCTION EQUATIONS AG '~' 4~.3/-----~ H [1 - (C + 1)el, (2) To some extent all inflationary calculations rely where C = -2 + In2 + 3, -~ -0.73 is a numeri- on the use of the slow-roll approximation. In the cal constant, 3' ~ 0.577 being the Euler constant. form we present here, the slow-roll approximation The right hand sides of these expressions are eval- is an expansion in terms of quantities defined from uated when the scale in question crosses the Hub- derivatives of the Hubble parameter H. In gen- ble radius during inflation, 2~r/A = all. The eral there are an infinite hierarchy of these which spectra can equally well be considered to be func- can in principle all enter at the same order in an tions of wavelength or of the scalar field value. expansion. The standard results to lowest-order are given The slow-roll approximation arises in two sep- by setting the square brackets to unity. Histori- E. Eg Kolb et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 118-125 121 cally it has been common even for this result to v be written as only an approximate equality (the ambiguity arising primarily because of a vague- ness in defining the precise meaning of the density perturbation), though the precise normalization to lowest-order was established some time ago by d~ _ V~ (l+e). (4) Lyth [16] (see also the discussion in [2]). dA A The improved expressions for the spectra in In the third expression, v depends upon AG(A) Eqs. (2) are accurate in so far as e and 1/are suffi- and •. Since we anticipate that we will only have ciently slowly varying functions that they can be information about Aa(A) at the largest scales, treated adiabatically as constants while a given we have to use the "consistency" equation (also scale crosses outside the Hubble radius. Correc- called the evolution equation) to relate AG(A) to tions to this would enter at next order. This dif- the more experimentally accessible As(A) at the fers from the usual situation in which H is treated expense of introducing the additional ~/fi term.