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Physics Letters B 565 (2003) 33-41
www. elsevier. com/locate/npe
WMAP and inflation
V. Bargera c, Hye-Sung Leea, Danny Marfatiabc
a Department of Physics, University of Wisconsin, Madison, WI53706, USA b Department of Physics, Boston University, Boston, MA 02215, USA c Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
Received 20 February 2003; received in revised form 12 May 2003; accepted 15 May 2003
Editor: M. Cvetic
Abstract
We assay how inflationary models whose properties are dominated by the dynamics of a single scalar field are constrained by cosmic microwave background (CMB) data from the Wilkinson Microwave Anisotropy Probe (WMAP). We classify inflationary models in a plane defined by the horizon-flow parameters. Our approach differs from that of the WMAP Collaboration in that we analyze only WMAP data and take the spectral shapes from slow-roll inflation rather than power-law parameterizations of the spectra. The only other information we use is the measurement of h from the Hubble Space Telescope (HST) Key Project. We find that the spectral index of primordial density perturbations lies in the 1a range 0.94 < ns < 1.04 with no evidence of running. The ratio of the amplitudes of tensor and scalar perturbations is smaller than 0.61 and the inflationary scale is below 2.8 x 1016 GeV, both at the 2a C.L. No class of inflation or ekpyrotic/cyclic model is excluded. The 4 potential is excluded at 3a only if the number of e-folds is assumed to be less than 45. © 2003 Published by Elsevier B.V.
1. Introduction
That there are multiple peaks in the CMB has recently been reinforced by data from the WMAP satellite [1-3]. This establishes that the curvature fluctuations which seed structure formation were generated at superhorizon scales. The inflationary paradigm [4], which hinges on this very fact [5], is therefore vindicated. Other generic predictions of inflation, including approximate scale-invariance of the power spectra, flatness of the Universe and adiabaticity and gaussian-ity of the density perturbations are also fully consistent with WMAP data [6].
E-mail address: marfatia@buphy.bu.edu (D. Marfatia).
Given this supporting evidence for inflation, we adopt the full set of predictions of slow-roll inflation and obtain constraints on inflationary models imposed by WMAPs data. Studies of this type have been carried out in the past with less precise data [7] and with the simple parameterization of the power spectra as power laws [8]. It has been emphasized that to extract precise information from data of WMAPs quality, high-accuracy predictions of the power spectra resulting from slow-roll inflation should be used [9]. There is a wealth of cosmological information in the CMB [10], and our approach is to employ precise theoretical expectations in its extraction.
Since the WMAP Collaboration has considered what implications their data have for inflation [11], we describe at the outset the differing elements between
0370-2693/03/$ - see front matter © 2003 Published by Elsevier B.V. doi:10.1016/S0370-2693(03)00757-3
our analysis and theirs. We elaborate on these differences later. WMAP include CBI [12], ACBAR [13], 2dFGRS [14] and Lyman-a power spectrum [15] data in addition to their own data. We restrict ourselves to WMAP data with a top-hat prior on the Hubble constant h (Ho = 100h kms-1 Mpc-1), from the HST [16]. While we use the actual theoretical predictions for the primordial power spectra from single-field slow-roll inflation, WMAP parameterize the spectra with power-laws and a running spectral index. As a result, the spectral shapes we use are different from those of WMAP and we directly fit to slow-roll parameters, while WMAP fit to derivative quantities. There are different virtues of the two approaches, and a comparison of the results obtained from them serve as a check of the robustness of the conclusions reached.
2. Primordial spectra
The primordial scalar and tensor power spectra to O(ln2 k) are1 [9,17]
P = ( ao + a\ In--1- a2 In —
Ph = A( b0+b1ln^+b2ln2^-
(1) (2)
where the pivot k+ typifies scales probed by the CMB. The constants ai and bi are functions [9,18] of the horizon-flow parameters, ei of Ref. [19], that are defined by
d In|fi|
1 = -777 '
Hi_ H '
Here, N is the number of e-folds since some moment t, during inflation, when the Hubble parameter was H,.
Note that to O(e2), the bi depend only on e1 and e2, while the ai depend on e1, e2 and e3. We initially set e3 = Q, but we then later demonstrate that the fit to the WMAP data is essentially not improved by
1 X is the intrinsic curvature perturbation and hij is the trans-
verse traceless part of the metric tensor.
including nonzero e3. The reason for not simply using the e3-independent O(e) expressions is that the O(e2) expressions are more accurate far from the pivot, and for a wider range of e1 and e2 [7,9]. It is uncertain that even high-precision data from the Planck satellite [20] can constrain e3 to be small. Including e3 in our analysis would simply enlarge the allowed parameter space to include models which are not inflationary in the sense that the horizon-flow parameters are not small.
The primary advantage of the horizon-flow parameters is that accurate predictions of the shapes and normalizations of the power spectra can be made independent of parameters describing cosmic evolution. The horizon-flow parameters e1 and e2 are related to the usual slow-roll parameters [21]
1/ V '
via the first order relations [9]
fi - e,
€2 - 2(e - n).
The normalizations of the spectra are given by
Ac = ■
and the ratio of the amplitudes of the spectra at the pivot k = k+, is
AtbQ R = —- = 16ei
I + Ce2 + \ C —— +5 JeiQ
---+ 1
where C = yE + ln2 - 2 « -0.7296. Note that a0 and b0 are O(1) and |a0 - b0| is O(e2). The spectral indices and their running can be expressed in terms of
e1, e2 and e3:
ns = 1 - 2ei - f2 -
- (2C + 3)eif2 - Cf2€3,
nt = -2ei - 2e2 - 2(C + 1)eif2,
d ln k
d ln k
= 2fi €2 - €2f3,
= -2fi€2.
(1Q) (11) (12)
We set ei = 0, i > 3. R, ns and nt are interdependent and the following consistency condition on single-field slow-roll inflation applies:
R ~ -8nt.
By our choice of formalism, we implicitly assume this condition to be satisfied.
Note that the six inflationary parameters, As, At, ns, nt, as and at are determined by just three parameters, As, e1 and e2 in our analysis. In contrast, the WMAP Collaboration parameterize the power-spectra with [11]
PY = A, —
Ph = At
They eliminate nt as a free parameter by using the consistency condition Eq. (14). Thus, they have four free parameters, As, At, ns and as and set at = 0.
A convenient classification of models based on the separate regions in the R-ns plane they populate, or equivalently relationships between the slow-roll parameters, was introduced in Ref. [22]. This classification becomes particularly simple in the e2-e1 plane, as discussed in the following section and shown in Fig. 3.
3. Inflation models
In what follows we use the common jargon, tilt" for ns < 1 and "blue-tilt" for ns > 1.
(a) Canonical potentials of large-field models are the monomial potential,
V($) = Vo($/ß)p, p > 2,
and the exponential potential V($) = V0e^l,x of power-law inflation. They are typical of chaotic inflation [23] and have V" > 0. The value of the scalar field falls O(Mpi) while the relevant perturbations are generated and thereby offer a glimpse into Planck-ian physics. In terms of the horizon-flow parameters, large-field models satisfy
0 < e2 < 4e1.
They have large R and predict a red tilt.
(b) Generic small-field potentials are of the form
V($) = Vo[1 - i$/ß)pl p > 2,
and are therefore characterized by V" < 0. The scalar field rolls from an unstable equilibrium at the origin towards a non-zero vacuum expectation value. Models relying on spontaneous symmetry breaking yield such potentials [24]. For small-field models
Q > 4e1.
The tensor fraction is small and the scalar spectrum is red-tilted.
(c) Potentials for hybrid inflation [25] are of the form
(15) V($) = Vo[1 + i$/ß)pl p > 2.
Hybrid inflation models involve multiple scalar fields. One of the fields, $, is the slowly rolling inflaton which does not carry most of the energy density ($ < Another field which has a fixed value during the slow-roll of $ provides V0. When $ falls below a critical value, the other field is destabilized and promptly ends inflation. As a result the value of $ at the end of inflation is very model-dependent, which makes the number of e-folds correspondingly uncertain. Hybrid models can be treated as single-field models because the only role of the second field is to end inflation, and the slow-roll dynamics is dominated by a single field. These potentials arise in supersymmetric and supergravity models of inflation. Hybrid models have
€2 < 0.
There is no robust prediction for R as can be seen from the above e1 -independent inequality. However, a unique prediction of these models is that the spectrum is blue-tilted if |e2| > 2e1.
The line e2 = 0 (R = 8(1 — ns)) implies e = n which occurs for the exponential potential. Thus, power-law inflation marks the boundary between large-field and hybrid models.
(d) Linear potentials,
V(0) = V0($/ß), V(t) = V)[1 -
define the boundary between large and small field models and lie at
€2 = 4e1 or 3R = 8(1 - ns). (23)
Since e1 is a constant for such potentials, inflation ends only with the help of an auxiliary field or some other physics.
To avoid overstating the comprehensiveness of this classification of potentials, we list a few potentials of different forms [26], which, however, do fit into the large-field, small-field, hybrid or linear categories according to the relationships between the horizon-flow parameters. V($) = V)[1 ± ln(0/|)], V0[1 -e-<P/i^] and V0[1 ± ($/i)-p] are hybrid in the sense that an auxiliary field is needed to end inflation, but lead to e2 > 4e1 and therefore lie in the small-field region of the parameter space. Similarly, power-law inflation does not end without a hybrid mechanism, but lies in the large-field region. Finally, let us note that the simplest models of the ekpyrotic/cyclic [27] variety are small-field [28]; however, the prediction for ns in these models is controversial [29].
4. Analysis
We compute the TT and TE power spectra ST2 = l(l + 1)Ci/2n, using the Code for Anisotropies in the Microwave Background or CAMB [30] (which is a parallelized version of CMBFAST [31]) and a supporting module that calculates the inflationary predictions for the primordial scalar and tensor power spectra [32]. We assume the Universe to be flat, in accord with the predictions of inflation, and that the neutrino contribution to the matter budget is negligible. The dark energy is assumed to be a cosmological constant A. We calculate the angular power spectrum on a grid consisting of the parameters specifying the primordial spectra,2 e1, e2 and As, and those specifying cosmic evolution,3 the Hubble constant h, the reionization optical depth t , the baryon density = QBh2 and the total matter density mm = £2Mh2 (which is comprised of baryons and cold dark matter). We choose h rather than QA because it is directly constrained by the HST [16]. We
2 We call these inflationary parameters.
3 We call these cosmological parameters.
do not include priors from supernova [33], gravitational lensing [34] or large scale structure [35] data or nucleosynthesis constraints on rnb [36] although these would somewhat sharpen the cosmological parameter determinations.
We employ the following top-hat grid:
• 0.0001 < e1 < 0.048 in steps of size 0.004;
• -0.18 < e2 < 0.14 in steps of size 0.02;
• 0.64 < h < 0.80 in steps of size 0.02;
• t = 0, 0.05, 0.1, 0.125, 0.15, 0.16, 0.17, 0.18, 0.19,0.2,0.25,0.3;
• 0.018 < Mb < 0.028 in steps of size 0.001;
• 0.06 < mm < 0.22 in steps of size 0.02;
• As is a continuous parameter.
The range of values chosen for h correspond to the HST measurement, h = 0.72 ± 0.08 [16]. This serves to break the degeneracy between QM and QA without the need for supernova data [37]. We place the pivot at k+ = 0.007 Mpc-1. The primordial spectra are evaluated to O(ln2 k) (see Eqs. (1) and (2)) with €3 = 0.
The WMAP data are in the form of 899 measurements of the TT power spectrum from l = 2 to l = 900 [2] and 449 data points of the TE power spectrum [3]. We compute the likelihood of each model of our grid using Version 1.1 of the code provided by the Collaboration [38]. The code computes the full covariance matrix under the assumption that the offdiagonal terms are subdominant. This approximation breaks down for unrealistically small amplitudes. We include the off-diagonal terms only when the first peak occurs between l = 100 and l = 400 with a height above 5000 ^K2 [38]. The restriction on the peak location may at first appear irrelevant, but for very large tensor amplitudes, the maximum height of the scalar spectrum shifts to very small l. When the height of the first peak is below 5000 ^K2 (which is many standard deviations away from the data), we only use the diagonal terms of the covariance matrix to compute the likelihood. To obtain single-parameter constraints, we plot the relative likelihood e(xmn-X )/2, for each parameter after maximizing over all the others. The x-a range is obtained for likelihoods above e-x /2. For 2-dimensional constraints, the 1a, 2a and 3a regions are defined by Ax2 = 2.3, 6.17 and 11.83, respectively, for two degrees of freedom.
Fig. 1. Relative likelihood plots for several cosmological parameters and As. fy is the age of the Universe in Gyr. We do not show the plot for h because it is not constrained by the fit; see Table 1.
5. Results
Although our primary focus is to obtain constraints on the inflationary models, we display constraints on the cosmological parameters as a consistency check with the WMAP Collaboration. This is pertinent because we have used a form for the primordial spectra (Eqs. (1) and (2)) that is specific to single-field slow-roll inflation rather than the standard power-law form.
The best-fit parameters are €\ = 0.016, €2 = -0.02, Mb = 0.023, mm = 0.12, t = 0.125 and h = 0.78 with normalization As = 22 x 10-10 with x 2 = 1428.88 for 1341 degrees of freedom.4 The results from Fig. 1 are
4 To check the validity of setting €( = Q, i > 3, we enlarged the grid to include €3 = ±0.1 and found the minimum x2 values to be x2(e3 = -0.1) = 1428.80 and x2(e3 = +0.1) = 1429.05. The small changes in the x2 values confirm that it is not necessary to include €3 in the analysis.
Fig. 2. Relative likelihood plots for some inflationary parameters.
Table 1
The 1a limits on the inflationary and cosmological parameters. Quantities below the line are not directly constrained by the data but derived from those above the line
1a lower limit 1 a upper limit
0 0.028
€2 -0.08 0.04
As x 1010 19.7 26.3
T 0.07 0.21
Mb 0.22 0.026
MM 0.11 0.15
h 0.68 -
ns 0.94 1.04
nt -0.06 0
R 0 0.47
as(at) x 103 -0.32 3.5
MA 0.31 0.53
io/Gyr 13.3 14.1
Likelihood plots for the inflationary parameters are shown in Fig. 2. We do not show the result for nt because we have imposed the consistency condition, Eq. (14). The 1a confidence limits on various parameters are provided in Table 1. We see that the spectra are consistent with scale-invariance and with a small tensor contribution; the best-fit scale-invariant spectrum with no tensor contribution has X2 = 1430.61. Also, running of the spectral indices is insignificant. (In our framework are required to be consistent with inflationary predictions as dictated by Eqs. (12) and (13); they are not free parameters.)
WMAP has provided important information about R and the energy scale of inflation:
R < 0.61 (2a limit), (24)
in good agreement with those obtained by the WMAP Collaboration [6]. This indicates that the choice of the spectral shapes does not matter at the present precision of the WMAP data.
— = ^¡TtexAs < 1.48 x KT5 (2a limit), (25)
or equivalently,
V1/4 =
i- I Mpi
< 2.8 x 10
GeV (2a limit).
Since Vj1/4 > 1015 GeV is consistent with data, it is still possible to detect inflationary gravity waves by measuring the curl component of CMB polarization [39].
6. Implications for models
The allowed regions of e1 and e2 (and equivalently ns and R) are shown in Fig. 3. The different classes of inflationary models populate distinct regions of the e2-e1 plane, as discussed above. The consistency of the models with the data can be judged from the figure. Even with the high quality of the WMAP data, no class of models is excluded. As long as e1 ^ 1, e2 = 0 is consistent with data, this will remain the case. Moreover, the allowed range for as (see Table 1) is consistent with the predictions of a wide spectrum of inflationary models, and so does not help in discriminating between them.
We now place some constraints on large-field and small-field models whose predictions do not involve too much freedom.
For the monomial potentials (p > 2) of large-field models,
fl —
where N is the number of e-folds of inflation from the time that scales probed by the CMB leave the horizon until the end of inflation.5 At least about 40 e-folds are needed for the Universe to be flat and homogeneous, and typically the largest value is 70 [40]. Since e1 ^ 1, e2 = 0 is allowed, p cannot be constrained independently of N.
5 We are reverting to the conventional definition in which the number of e-folds is counted backward in time; in the definition of the horizon-flow parameters, e-folds are counted forward in time.
Fig. 3. 1a, 2a and 3a allowed regions in the €2-e\ and R-ns planes. We have plotted the predictions for the k$4 potential with the number of e-folds N = 40, 50, 60 and 70. The prediction approaches e\ = €2 = 0 (ns = 1, R = 0) as N
The 3a exclusion of the X04 potential in Ref. [11] was based on an analysis of WMAP data in combination with higher l CMB data and large scale structure data, assuming N = 50 (for which ns = 0.94 and R = 0.32). Their results are shown in the second row of their Fig. 4. Note that the point ns = 0.94, R = 0.32 lies inside the 3 a region of our Fig. 3 and is therefore not excluded by WMAP data alone. If instead N is 60, then ns = 0.95 and R = 0.27; this point lies in the 95% C.L. allowed region of the second row of Fig. 4 of Ref. [11] and within the 2a region of Fig. 3.
If e1 = 0 (e2 > 0), a lower bound (upper bound) p/4 > emin/emax (p/4 < emax/emin) ensues; p is determined if both conditions on the e's are simultaneously satisfied.
Since we expect p « 4N,
= 12.5 (2a limit),
p < 4Ne5nax = 0.15N (2a limit). (30)
For small-field models with p > 3 [26], 2 p - 1
€1 « €2 =
N p — 2
The least stringent bound on N occurs in the limit
p ^ to,
= 25 (2a limit).
For the small-field quadratic potential (p = 2),
€1 « €2 =
2n /J,2
We find the 2a bound
tjiizc
= 1.4MPl.
Similar constraints can be placed on other models, but unfortunately, they are not very enlightening.
7. Conclusions
WMAP has provided compelling evidence for the inflationary paradigm. We have adopted the explicit predictions of single-field slow-roll inflation for the shapes of the power-spectra to analyze WMAP data. The fact that our parameter determinations are consistent with those obtained with the standard power-law parameterization by the WMAP Collaboration provides further evidence for slow-roll inflation. Since exact scale-invariance and a negligible tensor contribution to the density perturbations are adequate to describe the data, it is not presently possible to exclude classes of inflationary models.
We have shown how different classes of inflationary models can be distinguished in the e2-e1 plane of the horizon-flow parameters. If and when the horizon-flow parameters e1 or/and e2 are determined to be non-zero, large numbers of inflationary models will be ruled out. For that, we await even higher precision data from WMAP and eventually from Planck.
Acknowledgements
The computations were carried out on the CONDOR system at the University of Wisconsin, Madison with parallel processing on up to 200 CPUs. We thank S. Dasu, W. Smith, D. Bradley and S. Rader for providing access and assistance with CONDOR. We thank L. Verde for communications regarding the WMAP likelihood code. This research was supported by the US DOE under Grants No. DE-FG02-95ER40896 and No. DE-FG02-91ER40676, by the NSF under Grant No. PHY99-07949 and by the Wisconsin Alumni Research Foundation. VB. and D.M. thank the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara, for hospitality.
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