The Inflaton Potential from Present Day Observations Edward W

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 background 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 background 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

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. 4 adiabatically. For the standard calculation to be This was done through the identity strictly valid H must be constant, but provided it varies sufficiently slowly (characterized by small Ag~q = e [ l _ C (5) • and I~71), it can be evaluated separately at each epoch. This injects a scale dependence into the which follows from the expressions for As and AG spectra. There is a special case corresponding in Eq. (4). Now we develop the evolution equation to power-law inflation for which • and q are pre- by taking the derivative of As(¢~): cisely constant and equal to each other. In this case there are exact expressions for the perturba- As 1 (1~) 1 tion [6,17]. Furthermore, the corrections to each As - •1/2 •_ ~ 3¢- •1/--"~ spectrum are the same and they cancel when the [ ()2] ratio is taken. In the general case • and ~ may be × C ~ 1)ee -7C treated as different constants if it is assumed that the timescale for their evolution is much longer than the tim•scale for perturbations to be im- x [l+(C+l)e - C~e ] . (6) printed on a given scale. This assumption worsens as y is removed from •, which would be charac- Now in addition to the expansion in e and its terized by the next order terms becoming large. derivatives, a truncation is necessary. The trun- It is useful to define the dimensionless quanti- cation here is to assume ~/e << ~/v/~. With this ties ¢ and v, and a dimensionless derivative de- truncation, to second order noted by a dot:

~4 dX As - ~l/2 •- ~ . (7) = ; x = (3) Now we can express .4s/As in terms of the In addition, we can use the identity y = • + spectral index 1 - n defined to be dlnA2/dlnA, ~/2v/e and adopt as the expansion variables and the evolution equation becomes ~,=o •-"/2dne/d¢ "" In terms of these variables, - the expressions for As(A), Aa(A), v, and the ¢-A relation become t~ dA - (1+ •)[2e- (1 - n)], (8) Aa - 4ra/sH[1- (C+ 1)el 4Of course if the consistency equation is only used to evolve e, it can not be used as a check of inflation as dis- 4~r"'*VEn 1 [ - 1)E + ~Ct ] As = -;--474,~H--~ i (C + cussed in [3]. 122 K l~ Kolb et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 118 125 where for the second equality we have used the the initial value Aa()`0), yielding: de/d)` expression. This evolution equation serves --1 two purposes. It removes t/V ~ from the equation for v, and it is a differential equation that can v[~()`)l = [A~2(),0) - 2 0 A' As(M) be evolved to give e as a function of )`. To solve dM vl/2 [(~()`t)] the equation it is necessary to know the spectral = (11) index as a function of )`, along with the initial o )`' As()`') condition e()`o) as a function of 1 - no, AG()`o) 3.1. A Worked Example and As(~o). Let's assume a simple power-law potential of So the system to be solved can be expressed as the form V(¢) = )`¢¢4 with )`¢ = 4 x 10 -14. This generates perturbation spectra of the form (eval- = uated at horizon crossing after inflation)

× {l + 3e(A) + C[l - n()`)l } As(),) - 4 x 10 -s [50 Tin()`/)`0)] 3/2 ; Aa(),) = 4 x 10 -s [50 + ln(),/)~0)]. (12)

_ [1 + e()`)l d)` )` On any scale, the number of statistically inde- de ¢ pendent sample measurements of the spectra that - + [2e - - n)l. (9) can be made is finite. Given that the underlying inflationary fluctuations are stochastic, one ob- In the next section we discuss the simplification tains only a limited set of realizations from the of the above expressions obtained by dropping the complete probability distribution function. Such second-order terms and working to first order. In a subset may insufficiently specify the underly- the section after that we solve an example first- ing distribution, which is the quantity predicted order problem. by an inflationary model. The cosmic variance is an important matter of principle, being a source 3. First-Order Approximation to Recon- of uncertainty which remains even if perfectly struction accurate experiments could be carried out. At any stage in the history of the Universe, it is To first order in the slow-roll expansion vari- impossible to specify accurately the properties ables the expressions simplify considerably. For (most significantly the variance, which is what example, to first order, E = A at2/A2 s, v = A~, and the spectrum specifies assuming gaussian statis- )`de/d)` = Aa/As. The evolution-consistency tics) of the probability distribution function per- equation is also quite simple. It can be written as taining to perturbations on scales close to that of the observable Universe. )` dAa(A) A2()`) Even assuming "perfect" observations, cosmic (10) AG()`) d)` AUs()`) " variance sets a lower limit on the uncertainty at any one scale. Assuming that the only errors Again, the procedure will be the same as in come from cosmic variance, the determination of the second-order case. The potential depends the spectra might look like in Fig. 2. In the real- upon Aa()`), about which we will have infor- ization generated by the random number genera- mation only on the largest scales (possibly only tor, the value of Aa()`o) is 1.87 x 10 -6, slightly on one scale), so we specify the initial value of below the ensemble mean of 2 x 10 -6. AG()`), and use the consistency-evolution equa- As a first exercise, we simply perform a first- tion to evolve Aa(A) in terms of As()`). We can order reconstruction by doing a simple trape- thus express the system to be solved in terms of zoidal integration, and making the naive assump- two equations and a single first-order differential tion that the errors are uncorrelated. If we do equation which can easily be solved in terms of that we obtain the reconstructed potential shown E.~ Kolb et al./Nuclear Physics B (Proc, Suppl,) 43 (1995) 118 125 123

' I''' * ' ' • " i''" " ' " " " I''" ' " " '

] t , I0-6

10-6 , i I|ml • i i J | Jill, I J |,,. 0 • , I ~o 10s 10s 101 ), (h-IMpc)

Figure 2. An illustration of an anticipated data set limited by cosmic variance. The data was generated with a A6¢ 4 potential with A¢ = 4 x 10 -14. The upper points are As(A), while the single lower point is AG(A). The solid lille is the mean As(A), while the mean AG(A0) is 2 x 10 -6. in Fig. 3. Also shown in Fig. 3 by the solid curve if we look at the slope of v -1, the initial value is the actual potential used to generate the syn- of As(A) drops out, and the contribution comes thetic data from which the potential was recon- from adding together a large number of different structed. As(A). Since we are combining a large number There are several things we can notice in Fig. 3. of data points, the central limit theorem tells us First of all, reconstruction works: the true poten- that the errors in the reconstructed potential will tial is within the error bars. The second obvious become small. feature is that the slope of the reconstructed data is better than one might expect given the errors. 4. Conclusions This feature can be explored by taking another approach to the uncertainty introduced in At(A0) The quantum-mechanical fluctuations im- by cosmic variance. Let's ignore that error, and pressed upon the metric during inflation depend pick three realizations of At(A0), one at the upon the inflaton potential. During inflation the "measured" value, one la above the measured Hubble expansion takes microscopic fluctuations value, and one la below the measured value. of wavelength of order 10-2Scm and stretches (Here "a" is the value determined by cosmic vari- them to super-Hubble-radius size where they are ance.) If we do that, we generate the three curves frozen. Today they appear on scales as large as shown in Fig. 4. Although we can't tell which of the observable Universe, 10+2Scm. It is possi- the curves is the true potential, we know that the ble to read the fossil record of the fluctuations true po~tential is one of a family of curves bounded by observing cosmic microwave background fluc- by the two extremes in the figure. tuations and the power spectrum of large-scale We can understand why this occurs, because structure. If the tensor perturbations are large enough to 124 E.W Kolb et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 118 125

50L~ .... I " ' " ' I .... '1 .... I .... I .... I .... I ~40~-

> 30

0 ~20

i I i i I , , , I I i i ' ' I , . . . I . . , . I . . . , I . , , . 0 0 0.5 1 1.5 2 2.5 3 3.5 I~-¢ol/10 t' GeV

Figure 3. First-order reconstruction of the example )~,~4 potential. The solid line is the actual potential, while the points and associated errors were generated from the data of Fig. 2.

50 ''''l''''l''''l .... I .... I''''I'''" 40 I 3O

~20 _------.-______.9- v:> 10

0 0 0.5 1 Iqb-~1.bSlOtSol/ 2 OeV 2.5 3 3.5

Figure 4. The reconstructed potential ignoring uncertainty in Aa(Ao) for three choices of Aa(A0) corresponding to the midpoint and -t-la. E.W Kolb et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 118 125 125 be identified, and if the scalar power spectrum is . T. Gaier, J. Schuster, J. Gunderson, T. determined, the inflaton potential may be recon- Koch, M. Seiffert, P. Meinhold and P. Lu- structed. bin, Astrophys. J. Lett. 398, L1 (1992). Hence cosmology and astrophysics may provide the first concrete piece of the potential of energy 10. S. J. Maddox, G. Efstathiou, W. J. Suther- scales of 1016GeV or so. land and J. Loveday, Mon. Not. R. astr. Not discussed here is the possibility of pertur- Soc. 242, 43p (1990). bative reconstruction, where the potential and its 11. M. J. Geller and J. P. Huchra, Science 246, first few derivatives are reconstructed about a sin- 879 (1989); M. Ramella, M. J. Geller and J. gle point [3, 18, 19]. P. Huchra, Astrophys. J. 384, 396 (1992).

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