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Physics Letters B 566 (2003) 8-18

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Effective number of neutrinos and baryon asymmetry from BBN and WMAP

V. Bargera f, James P. Knellerb, Hye-Sung Leea, Danny Marfatiac f, Gary Steigmand e f

a Department of Physics, University of Wisconsin, Madison, WI53706, USA b Department of Physics, North Carolina State University, Raleigh, NC 27695, USA c Department of Physics, Boston University, Boston, MA 02215, USA d Department of Physics, The Ohio State University, Columbus, OH 43210, USA e Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA f Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

Received 12 May 2003; accepted 23 May 2003

Editor: M. Cvetic

Abstract

We place constraints on the number of relativistic degrees of freedom and on the baryon asymmetry at the epoch of Big Bang Nucleosynthesis (BBN) and at recombination, using cosmic background radiation (CBR) data from the Wilkinson Microwave Anisotropy Probe (WMAP), complemented by the Hubble Space Telescope (HST) Key Project measurement of the Hubble constant, along with the latest compilation of deuterium abundances and Hii region measurements of the primordial helium abundance. The agreement between the derived values of these key cosmological and particle physics parameters at these widely separated (in time or redshift) epochs is remarkable. From the combination of CBR and BBN data, we find the 2a ranges for the effective number of neutrinos Nv and for the baryon asymmetry (baryon to photon number ratio n) to be 1.7-3.0 and 5.53-6.76 x 10-10, respectively. © 2003 Published by Elsevier B.V.

1. Introduction

The concordance model of cosmology, with dark energy, dark matter, baryons, and three flavors of light neutrinos, provides a consistent description of BBN (-20 min), the CBR (-380 Kyr), and the galaxy formation epochs of the universe (>1 Gyr). The standard model has received recent confirmation from the

E-mail address: marfatia@buphy.bu.edu (D. Marfatia).

WMAP precision measurements of the CBR temperature and polarization anisotropy spectra [1]. However, despite the impressive successes of the standard model in describing a wide range of cosmological data, the possibility remains that there could be non-standard model contributions to the total energy density in the radiation era from additional relativistic particles.

In this Letter new constraints are placed on any physics beyond the standard model that contributes to the energy density like radiation (i.e., decreases with the expansion of the universe as the fourth power of

0370-2693/03/$ - see front matter © 2003 Published by Elsevier B.V. doi:10.1016/S0370-2693(03)00800-1

the scale factor, independent of the sign of that contribution). While such new physics may or may not be due to extra relativistic degrees of freedom, it is assumed that the non-standard contribution to the energy density may be parameterized as such. Simultaneously, constraints are placed on the baryon density at widely differing epochs in the evolution of the universe. The keys to these constraints are the recently released measurements of the CBR anisotropy spectra by the WMAP Collaboration, the most recent compilation of high redshift, low metallicity deuterium abundances [2] and 4He abundances relevant to BBN.

2. Modified relativistic energy density

The cosmology of interest here begins when the universe is already a few tenths of a second old and the temperature is a few MeV At such early epochs the total energy density receives its dominant contribution from all the relativistic particles present (the evolution of the universe is said to be "radiationdominated" (RD)). In the standard cosmology, prior to e± annihilation, these relativistic particles are: photons, e± pairs and three flavors of left-handed (i.e., one helicity state) neutrinos (and their right-handed antineutrinos). Then, the energy density is

ptot = pr = Py + Pe + 3pv = — Py , (1)

where pY is the energy density in photons (which by today have redshifted to become the CBR photons at a temperature of about 2.7 K).

In "standard" BBN (SBBN) it is assumed that the neutrinos are fully decoupled prior to e± annihilation and do not share in the energy transferred from the annihilating e± pairs to the CBR photons. In this approximation, the photons in the post-e± annihilation universe are hotter than the neutrinos by a factor TY/TV = (11/4)1/3, and the relativistic energy density is

PR = Py + 3pv = 1.6813 Py.

During the RD epoch the age and the energy density are related by f pRi2 = 1 (we have chosen units in which OnG = 1), so that once the particle content (pR) is specified, the age of the universe is known as a function of the CBR temperature. In the

standard model,

Pre-e± annihilation: T = 0.738 MeV2 s, (3)

Post-e±annihilation: tTY = 1.32 MeV2 s.

The most straightforward variation of the standard cosmology is "extra" energy contributed by new, light (relativistic at BBN) particles "X". These might, but need not be sterile neutrinos. When the X are decoupled, in the sense that they do not share in the energy released by e± annihilation, it is convenient to account for the extra contribution to the standard model energy density by normalizing it to that of an "equivalent" neutrino [3],

px = ANvpv = -ANvpy. (5)

For SBBN ANV = 0, where ANV = 3 + NV. For each additional "neutrino-like" particle (i.e., any two-component fermion), if TX = TV, then ANV = 1; if X is a scalar, ANV = 4/7. However, it may well be that the X have decoupled even earlier in the evolution of the universe and have failed to profit from the heating when various other particle-antiparticle pairs annihilated (or unstable particles decayed). In this case, the contribution to ANV from each such particle will be < 1 (< 4/7). We emphasize that, in principle, we are considering any term in the energy density which scales like a-4, where a is the scale factor. In this sense, the modification to the usual Friedman equation due to higher dimensional effects, as in the RandallSundrum model [4] (see also, [5-12]), can be included as well. An important interest in this latter case is that it permits the possibility of a negative contribution to the radiation density (ANV < 0; NV < 3).

In the presence of such a modification to the relativistic energy density, the pre-e± annihilation energy density in Eq. (1) is changed to,

(prW =

1 + ■

Any extra energy density (ANV > 0) speeds up the expansion of the universe so that the right-hand side of the time-temperature relation in Eq. (3) is smaller by the square root of the factor in parentheses in Eq. (6),

Spre = (t/t ')pre = 1 +

= (1 + 0.1628ANv)l/2,

where t' is the age of the universe with the extra energy density. In the post-e± annihilation universe the extra energy density is diluted by the heating of the photons, so that,

(PR )post = 1.6813(1 + 0.1351ANv)py, and

Spost = (t/t Vt = (1 + 0.1351 ANV)1/2.

This latter expression (Eq. (9)) is also relevant for the modification to the spectrum of temperature fluctuations in the CBR (when compared with the standard NV = 3 case).

Fig. 1. The CBR degeneracy between mm and ANV is evident from the Iff and 2ff contours from the WMAP data.

3. Constraints on Nv from the CBR

The competition between gravitational potential and pressure gradients is responsible for the peaks and troughs in the CBR power spectrum. The redshift of matter-radiation equality,

Jeq = 2.4 X 10 —— S2

affects the time (redshift) duration over which this competition occurs. Here, &>M = i2Mh2 is the total matter density (comprised, for nearly massless neutrinos, of baryons and cold dark matter) and h (Ho = 100h kms-1Mpc-1) is the normalized Hubble constant. The direct correlation between &>M and ANV [13] is evident in Fig. 1 which results from our analysis described below. The primary effects of relativistic degrees of freedom (other than photons) on the CBR power spectrum result essentially from changing the redshift of matter-radiation equality. If the radiation content is increased, matter-radiation equality is delayed, and occurs closer (in time and/or redshift) to the epoch of recombination.

The redshift of matter-radiation equality is important for two reasons [14]:

• Radiation causes potential decay which blueshifts the photons because they do not have to climb out of such deep wells. Moreover, the concurrent decay in the spatial curvature doubles the blueshift effect by contracting the wavelength of the photons relative to the pure cosmological expansion.

• In the matter dominated (MD) era before recombination, the density contrast (Sp/p) of the pres-sureless cold dark matter (CDM) grows unimpeded (as t2/3) while the density contrast of the baryons is either oscillating or decaying. The longer this pre-recombination MD era lasts, the more suppressed are the amplitudes of the peaks.

Conversely, if matter-radiation equality is delayed, the gravitational potential is dominated by the photonbaryon fluid closer to recombination resulting in a more pronounced peak structure.

An increase in the relativistic content causes the universe to be younger at recombination with a correspondingly smaller sound horizon s*. Since the location of the nth peak scales roughly as nn D* /s* (where D* is the comoving angular diameter distance to recombination), the peaks shift to smaller angular scales (larger l) and with greater separation. These features are clearly visible in Fig. 2.

The heights and locations of the peaks also depend on the history of the universe after recombination. At the end of matter domination and the onset of dark energy domination, further and much slower (compared to that in the radiation epoch) potential decay occurs. The more gradual potential decay causes the induced anisotropy to be suppressed by a factor of l. The amplification of the power in the lowest l's from this late decay serves as a probe of dark energy (or another probe of the matter content in a flat universe). In principle, the degeneracy between ANV and &>M is broken by this effect and by the

Fig. 2. The power spectrum for the best-fit (Nv = 2.75) to the WMAP data is the solid line. With all other parameters and the overall normalization of the primordial spectrum fixed, the spectra for Nv = 1, Nv = 5 and Nv = 7 are the dotted, dot-dashed and dashed lines, respectively. The data points represent the binned TT power spectrum from WMAP.

accompanying change in the redshift at which the matter dominated epoch ends. However, note that the lowest multipoles also have the largest cosmic variance.

The TT and TE power spectra are computed using the Code for Anisotropies in the Microwave Background or CAMB [15] which is a parallelized version of CMBFAST [16]. The Universe is assumed to be flat, in accord with the predictions of inflation [17], and the dark energy is assumed to behave as a cosmological constant A. The restriction of a flat geometry allows us to relate the dark energy and matter densities at the present time: QA = 1 - The angular power spectrum is calculated on a grid defined by h, the baryon density &>B = QBh2 (or n10 = 1010nB/nY = 274&>B), &>M, the number of equivalent neutrinos Nv, the reion-ization optical depth t , and the spectral index ns of the primordial power spectrum. Two priors are imposed to largely break the degeneracy between &>M and ANv. For h a top-hat distribution is chosen corresponding to the HST measurement, h = 0.72 ± 0.08 [18], and we require that the universe be older than the globular clusters (which, at 2a, are older than 11Gyr[19]).For comparison, we also consider the case when the age of the universe t0 exceeds 12 Gyr.

Our top-hat grid, consisting of over 10 million points, is:

• 0.64 < h < 0.8 in steps of size 0.02.

• 0.018 < mb < 0.028 in steps of size 0.001.

• 0.11 < &>M < 0.27 in steps of size 0.01 and &>M = 0.07, 0.3.

• 1 < Nv < 3.5 in steps of size 0.25, 4 < Nv < 9 in steps of size 0.5 and Nv = 0, 0.5.

• 0 < t < 0.3 in steps of size 0.025.

• 0.90 < ns < 1.02 in steps of size 0.01 and ns = 0.80, 0.84, 0.88, 1.04, 1.08, 1.12, 1.16, 1.20.

• The normalization of the spectrum is a continuous parameter.

The first year WMAP data are in the form of 899 measurements of the TT power spectrum from l = 2 to l = 900 [20] and 449 data points of the TE power spectrum [21]. Although the effect of relativistic degrees of freedom on the TE spectrum is insignificant, it is included in our analysis for completeness. The likelihood of each model of our grid is computed using Version 1.1 of the code provided by the WMAP Collaboration [22]. The code computes the covariance matrix under the assumption that the off-diagonal terms are subdominant. This approximation breaks down for unrealistically small amplitudes. When the height of the first peak is below 5000 ^K2 (which is many standard deviations away from the data), only the diagonal terms of the covariance matrix are used to compute the likelihood.

The best-fit parameters are h = 0.68, &>B = 0.023 (n10 = 6.3), mm = 0.14, Nv = 2.75, t = 0.13, and ns = 0.97 with a x2 = 1429.13 for 1341 degrees of freedom. The allowed parameter space in the n1o-ANv plane is shown in Fig. 3. The solid (dotted) lines correspond to the 1a - and 2a -regions1 for t0 > 11 (12) Gyr. The cross identifies the best-fit point. Note that while this best fit point lies at Nv < 3, the Nv distribution is very broad. After marginalizing over n, the 2a range in Nv extends from 0.9 (ANv = -2.1) to 8.3 (ANv = 5.3). Although similar CBR analyses (see Ref. [23]) have included different, additional data to that from WMAP alone, making direct comparisons difficult, our results are in good agreement with them. The prior on t0 has a significant effect on the allowed values of ANv [24] for a simple reason. Since flatness

1 For 2-dimensional constraints, the 1a-, 2a- and 3a-regions are defined by Ax2 = 2.3, 6.17 and 11.83, respectively.

Fig. 3. The Iff and 2a contours in the n10~ANy plane from WMAP data. The solid (dotted) lines correspond to to > 11 (12) Gyr. The cross marks the best-fit at ®b = 0.023 and ANV = -0.25.

is assumed, t0 depends only on the matter content and the Hubble parameter via

1 + ^1"%

3 s/l — V The combination of the HST prior on h and the t0 prior restricts &>M and help to break the degeneracy between ANv and &>M.

The best fit WMAP-determined baryon density is n10 = 6.30 (&>B = 0.0230), in excellent agreement with Spergel et al. [25] and other similar analyses [23]. The CBR 2a range extends from n10 = 5.58 (&>B = 0.0204) to n10 = 7.26 (&>B = 0.0265).

These CBR constraints on Nv and &>B apply to epochs in the evolution of the universe >380 Kyr. An important test of the standard models of cosmology and particle physics is to compare them with corresponding constraints from the much earlier epoch probed by BBN.

4. The roles of Nv and n10 in BBN

At T ~ few MeV, the neutrinos are beginning to

decouple from the y — e± plasma and the neutron to proton ratio, crucial for the production of primor-

dial 4He, is decreasing. As the temperature drops below ~2 MeV, the two-body collisions between neutrinos and e± pairs, responsible for keeping the neutri-

nos in thermal equilibrium with the electron-positron-photon plasma become slow compared to the universal expansion rate and the neutrinos decouple, although they do continue to interact with the neutrons and protons via the charged-current weak interactions. Prior to e± annihilation, when the temperature drops below ~0.8 MeV and the universe is «1 second old, these interactions, interconverting neutrons and protons, become too slow (compared to the universal expansion rate) to maintain n-p equilibrium and the neutron-to-proton ratio begins to deviate from (exceeds) its equilibrium value ((n/p)eq = exp(—Am/T)), where Am is the neutron-proton mass difference. Beyond this point, often described as neutron-proton "freeze-out", the n/p ratio continues to decrease, albeit more slowly than would have been the case in equilibrium. Since there are several billion CBR photons for every nucleon (baryon), the abundances of any complex nuclei are entirely negligible at these early times.

We note here that if there is an asymmetry between the numbers of ve and ve ("neutrino degeneracy"), described by a chemical potential ¡xe, then the equilibrium neutron-to-proton ratio is modified to (n/p)eq = exp(—Am/T — \xe/T). In place of the neutrino chemical potential, it is convenient to introduce the dimensionless degeneracy parameter = ^e/T. A positive chemical potential (fe > 0; more ve than ve) leads to fewer neutrons and less 4He will be synthesized in BBN.

BBN begins in earnest after e± annihilation, at T « 0.08 MeV (t « 3 min), when the number density of those CBR photons with sufficient energy to pho-todissociate deuterium (those in the tail of the black body distribution) is comparable to the baryon density. By this time the n/p ratio has further decreased (the two-body reactions interconverting neutrons and protons having been somewhat augmented by ordinary beta decay; Tn = 885.7 s), limiting (mainly) the amount of helium-4 which can be synthesized. As a result, the predictions of the primordial abundance of 4He depend sensitively on the early expansion rate and on the amount—if any—of a ve- ve asymmetry.

In contrast to 4He, the BBN-predicted abundances of deuterium, helium-3 and lithium-7 (the most abundant of the nuclides synthesized during BBN) are determined by the competition between the various two-body production/destruction rates and the universal expansion rate. As a result, the D, 3He, and 7Li abun-

dances are sensitive to the post-e± annihilation expansion rate, while that of 4He depends on both the pre-and post-e± annihilation expansion rates; the former determines the "freeze-in" and the latter modulates the importance of beta decay (see, e.g., Kneller, Steigman

[26]). Also, the primordial abundances of D, 3He, and Li, while not entirely insensitive to neutrino degeneracy, are much less effected by a non-zero (e.g.,

[27]).

Of course, the BBN abundances do depend on the baryon density which fixes the nuclear reactions rates and also, through the ratio of baryons to photons, regulates the time/temperature at which BBN begins. As a result, the abundances of at least two different relic nuclei are needed to break the degeneracy between the baryon density and a possible non-standard expansion rate resulting from new physics or cosmology, and/or a neutrino asymmetry. In this Letter only the former possibility is considered; in another publication several of us (along with Langacker) have explored the consequences of neutrino degeneracy and we studied the modifications to the constraints on ANV when both of these non-standard effects are simultaneously included.

While the abundances of D, 3He, and Li are most sensitive to the baryon density (n), the 4He mass fraction (Y) provides the best probe of the expansion rate. This is illustrated in Fig. 4 where, in the ANV-n10 plane, are shown isoabundance contours for D/H and Y (the isoabundance curves for 3He/H and for

Li/H, omitted for clarity, are similar in behavior to that of D/H). The trends illustrated in Fig. 4 are easy to understand in the context of the discussion above. The higher the baryon density (n10), the faster primordial D is destroyed, so the relic abundance of D is anticorrelated with n10. But, the faster the universe expands (ANV > 0), the less time is available for D-destruction, so D/H is positively, albeit weakly, correlated with ANV .In contrast to D (and to 3He and Li), since the incorporation of all available neutrons into 4He is not limited by the very rapid nuclear reaction rates, the 4He mass fraction is relatively insensitive to the baryon density, but it is very sensitive to both the pre- and post-e± annihilation expansion rates (which control the neutron-to-proton ratio). The faster the universe expands, the more neutrons are available for 4He. The very slow increase of Y with n10 is a reflection of the fact that for higher baryon density, BBN begins earlier, when there are more neutrons. As a result of these complementary correlations, the pair of primordial abundances yD = 105 (D/H) and the 4He mass fraction Y, provide observational constraints on both the baryon density and the universal expansion rate when the universe was some 20 minutes old. Comparing these to constraints when the universe was some 380 Kyr old, from the WMAP observations of the CBR spectra, provides a test of the consistency of the standard models of cosmology and of particle physics and further constrains the allowed range of the present baryon density of the universe.

------—/

-------,/ /

4 5 6 7 8

Fig. 4. Isoabundance curves for D and 4He in the n\0-ANV plane. The nearly horizontal curves are for 4He (from top to bottom: Y = 0.25, 0.24, 0.23). The nearly vertical curves are for D (from left to right: 105(D/H) = 3.0, 2.5, 2.0. The data point with error bars corresponds to yD = 2.6 ± 0.4 and Y = 0.238 ± 0.005; see the text for discussion of these abundance values.

5. Primordial abundances

It is clear from Fig. 4 that while D (and/or 3He and/or 7Li) largely constrains the baryon density and 4He plays the same role for ANV, there is an interplay between n10 and ANV which is quite sensitive to the adopted abundances. For example, a lower primordial D/H increases the BBN-inferred value of n10, leading to a higher predicted primordial 4He mass fraction. If the primordial 4He mass fraction derived from the data is "low", then a low upper bound on ANV will be inferred. Therefore, it is crucial to make every effort to avoid biasing any conclusions by underestimating the present uncertainties in the primordial abundances derived from the observational data. For this reason deuterium is adopted as the baryometer of choice. Primar-

ily, this is because its observed abundance should have only decreased since BBN [28], but also because the deuterium observed in the high redshift, low metallic-ity QSO absorption line systems (QSOALS) should be very nearly primordial. In contrast, the post-BBN evolution of3 He and of 7Li are considerably more complicated, involving competition between production, destruction, and survival. As a result, at least so far, the current, locally observed (in the Galaxy) abundances of these nuclides have been of less value in constraining the baryon density than has deuterium. Nonetheless, inferring the primordial D abundance from the QSOALS has not been without its difficulties, with some abundance claims having been withdrawn or revised. Presently there are 5-6 QSOALS with reasonably firm deuterium detections [2,29-33]. However, there is significant dispersion among the abundances and the data fail to reveal the anticipated "deuterium plateau" at low metallicity or at high redshift [34]. Furthermore, subsequent observations of the D'Odorico et al. [33] QSOALS by Levshakov et al. [35] revealed a more complex velocity structure and led to a revised—and more uncertain—deuterium abundance. This sensitivity to the often poorly constrained velocity structure in the absorbers is also exposed by the analyses of published QSOALS data by Levshakov and collaborators [36-38], which lead to consistent, but somewhat higher deuterium abundances than those inferred from "standard" data reduction analyses. In the absence of a better motivated choice, here we adopt the five abundance determinations collected in the recent Letter of Kirkman et al. [2]. The weighted mean value of yD is 2.6.2 But, the dispersion among these five data points is very large. For this data set x 2 = 15.3 for four degrees of freedom, suggesting that one or more of these abundance determinations may be in error, perhaps affected by unidentified and unaccounted for systematic errors. For this reason, we follow the approach advocated by [31] and [2] and adopt for the uncertainty in yD the dispersion divided by the square root of the number of data points. Thus, the primordial abundance of deuterium to be used here is chosen to be: yD = 2.6 ± 0.4. For SBBN (Nv = 3,

%e = 0), at ±1a this corresponds to a baryon density

n10 = 6.1—0.5 (^B = 0.022 ± 0.002).3

A similar, less than clear situation exists for determinations of the primordial abundance of 4He. At present there are two, largely independent, estimates based on analyses of large data sets of low-metallicity, extragalactic HII regions. The "IT" [39, 40] estimate of Y(IT) = 0.244 ± 0.002, and the "OS" determination [41-43] of Y(OS) = 0.234 ± 0.003 which differ by nearly 3a. The recent analysis of high quality observations of a relatively metal-rich (hence, chemically evolved and post-primordial) HII region in the Small Magellanic Cloud (SMC) by Pe-imbert, Peimbert, and Ruiz (PPR) [44] yields an abundance Ysmc = 0.2405 ± 0.0018. When PPR extrapolated this abundance to zero metallicity, they found Y(PPR) = 0.2345 ± 0.0026, lending support to the OS value. These comparisons of different observations and analyses suggest that unaccounted systematic errors may dominate the statistical uncertainties. Indeed, Gruenwald, Steigman, and Viegas [45] argue that if unseen neutral hydrogen in the ionized helium region of the observed Hii regions is accounted for, the IT estimate of the primordial abundance should be reduced to Y(GSV) = 0.238 ± 0.003 (see also [46,47]). Here, we adopt this latter estimate for the central value but, as we did with deuterium, the uncertainty is increased in an attempt to account for likely systematic errors:

Y = 0.238 ± 0.005, leading to a 2a range, 0.228 <

Y < 0.248; this range is in accord with the estimate adopted by Olive, Steigman, and Walker (OSW) [48] in their review of SBBN. Although we will comment on the modification to any conclusions if Y(IT) is substituted for Y(OSW), Figs. 4-8 are shown for yD = 2.6 ± 0.4 and Y(OSW) = 0.238 ± 0.005.

6. Standard BBN

Before proceeding to our main goal of constraining new physics using BBN, it is worthwhile to set the scene by considering the standard model case (N v = 3, %e = 0) first. The result of this comparison is well

2 This differs from the result quoted in Kirkman et al. because they have taken the mean of log(yD) and then used it to infer yD

(yD = 10<log(yD»).

3 We have purposely avoided quoting the baryon density to

more significant figures than is justified by the accuracy of the

D-abundance determination.

105(D/H),

Fig. 5. The band is the SBBN predicted relation between the primordial abundances of D and 4He, including the errors (±1a) in those predictions from the uncertainties in the nuclear and weak interaction rates. The point with error bars is for the relic abundances of D and 4 He adopted here (see the text).

known: there is a "tension" between the primordial abundances of D and 4He inferred from the observational data and those predicted by SBBN. For example, if yD is used to fix the baryon density (yD = 2.6 ± 0.4) then, at ±1a, nSBBN = 6.1+0.7 (^ = 0.022 ± 0.002), the corresponding predicted 4He abundance is Y = 0.248 ± 0.001 (1a), which is some 2a higher than either the OSW or the IT estimates. This is illustrated in Fig. 5 which shows the SBBN-predicted relation between the relic abundances of D and 4He along with the Y(OSW) abundance estimate adopted here. The D-inferred baryon density is in excellent agreement with the baryon density determined independently (non-BBN; Nv = 3) by Spergel et al. [25] from a combination of CBR and Large Scale Structure data (2dF + Lyman a): ^1n0on"BBN = 6.14 ± 0.25 (&>b = 0.0224 ± 0.0009).4 Thus, it appears that 4He is the problem: the primordial abundance of 4He is smaller than predicted for SBBN given either the observed deuterium abundance or the non-BBN inferred

4 It should be noted that from the CBR alone Spergel et al. find nÇBR = 6 6 ± 0.3 (MB = 0.024 ± 0.001). It is this value which is most directly comparable to our CBR and joint BBN/CBR results.

baryon density [49]. At the same time we strongly emphasize that this "discrepancy" is only at the ~ 2a level and we should celebrate the excellent agreement between the baryon density determined when the universe was only 20 minutes old and when the universe was some 380 Kyr old (CBR).

7. Non-standard BBN: ANV = 0

As noted above, for SBBN (Nv = 3) the observa-tionally inferred primordial abundance of 4He is too small (by ~ 2a) for the baryon density inferred either from the D abundance or from the non-BBN analysis of Spergel et al. [25]. This suggests that the early universe expansion rate may have been too fast, leaving too many neutrons available for the synthesis of 4He. If this tension between 4He and D (or, between 4He and &>B) should persist, it could be a signal of nonstandard physics corresponding to S < 1 (ANv < 0). Indeed, in Fig. 4 it can be seen that for the adopted primordial abundances of D and 4He, there is a "perfect" fit (x2 = 0) for ANv ^ -0.7 (Nv « 2.3) and n1o ^ 5.7. Although Nv = 3 is only disfavored by ~ 2a, any increase in the early universe expansion rate (S > 1, Nv > 3) is strongly disfavored. This is illustrated in Fig. 6 which shows the 1a -, 2a - and 3a-contours in the n10-ANv plane for the adopted D and 4He (OSW) abundances. The shape of these contours reflects our much discussed complementarity between D and 4He: D provides the best constraint on the baryon density while 4He is most sensitive to the early universe expansion rate, the latter providing an excellent probe of possible new physics.

With reference to that Fig. 6 we note that even one extra, fully thermalized neutrino (ANv = 1) is strongly disfavored. This seemingly eliminates the sterile neutrino suggested by the LSND experiment [50]. For the LSND parameters, in the absence of significant neutrino asymmetry, this "sterile" neutrino would have been mixed with the active neutrinos and thermalized prior to neutrino decoupling (prior to BBN) [51]. If such a neutrino were to exist, the "new" standard model would correspond to Nv = 4. This would be a disaster since for Nv = 4 and the OSW 4He abundance the minimum x2 (which now occurs at n10 ^ 6.1) is greater than 20. The situation is even worse for the IT 4He abundance since the smaller

Fig. 6. The 1a-, 2a- and 3a-contours in the n10-ANv plane for the adopted D and 4He (OSW) abundances (solid lines). The cross marks the best fit BBN point. The 1a - and 2a-contours from WMAP (dashed lines) are shown for comparison.

Fig. 7. The 1a-, 2a- and 3a-contours (solid lines) in the n10-ANv plane for Nv ^ 3 and the adopted D and 4He (OSW) abundances. The corresponding 1a - and 2a-contours from WMAP (dashed lines) are shown for comparison.

uncertainty in Y forces a much smaller baryon density (n10 « 4.7), with xmmin > 60!

7.1. Requiring N v ^ 3

Since, as is well known from LEP [52], there are three flavors of active, left-handed neutrinos (and their right-handed antiparticles), any extra contributions to the relativistic energy density at BBN should result in Nv > 3. Of course, "new physics" in the form of non-minimally coupled fields (e.g., [26,53] and references therein) or from higher-dimensional phenomena (e.g., [4-12]) may result in an effective N v < 3. If, however, the class of non-standard physics of interest is restricted to AN v > 0, then the BBN constraints presented above (and those from the CBR) will change. With a prior of Nv > 3, the best fit BBN-determined values of the baryon-to-photon ratio and ANv (for Y(OSW)) shift from n10 « 5.7 and ANv « —0.7, to n10 « 5.9 and ANv = 0. The value of xL changes to 4.2. The corresponding confidence contours in the n10-anv plane are shown in Fig. 7.

8. Joint constraints and summary

As may be seen from Figs. 6 and 7, the agreement between the values obtained for ANv and n10 from

Fig. 8. The 1a-, 2a- and 3a-contours in the n10-ANv plane from a combination of WMAP data and the adopted D and 4He (OSW) abundances.

WMAP and from BBN separately is excellent. Guided by this, the BBN and CBR results are combined to obtain the joint fit in the n-ANv plane shown in Fig. 8. To a good first approximation, BBN (and primordial 4He) determines ANv while WMAP fixes n10 (with some help from BBN and primordial D). The corresponding figure for Nv > 3 for the joint fit has not been shown because again the ANv range is almost identical to that from BBN alone (Fig. 7).

Table 1

The 2a ranges (for 1 degree of freedom) of Nv and nio from analyses of WMAP data, deuterium and helium abundances and their combinations. The WMAP analysis involves the assumption of a flat universe, along with the strong HST prior on h and the age constraint to > 11 Gyr. For BBN the adopted primordial abundances are: yD = 105(D/H) = 2.6 ± 0.4, Y(OSW) = 0.238 ± 0.005, and Y(IT) = 0.244 ± 0.002

Nv (2a range) >?10 (2o range)

WMAP 0.9-8.3 5.58-7.26

yD + Y(OSW) 1.7-3.0 4.84-7.11

yD + Y(IT) 2.4-3.0 5.06-7.33

WMAP + yD + Y(OSW) 1.7-3.0 5.53-6.76

WMAP + yo + Y(IT) 2.4-3.0 5.58-6.71

0.01 0.02 0.03

i la /' i) 11 l i i, [i ( 1 Vio (wb)-

' 2 a 1 . i . . . . Pw, , i . , , . i , , . ,

/' ! i ... i - - f—^ ■

Table 2

The same as Table 1, except that the constraint Nv ^ 3 is imposed Nv (2a bound) n10 (2a range)

WMAP 8.3

yD + Y(OSW) 3.3

yo + Y(IT) 3.1

WMAP + yD + Y(OSW) 3.3

WMAP + yD + Y(IT) 3.1

5.64-7.30 5.04-7.18 4.89-6.56 5.66-6.80 5.54-6.60

la 1 / V • /v \ i * / \ ' 1 / \ 1 ' / \ ' 1 1 \' 1 .... 1 ■ ' ANy:

~ 2a / / 1 \ i \ i \ r \ 1 \ ' 1 1 1 1 1 / t 1 ,, , . . \ ' \ IV N.--- .!....

Fig. 9. The marginalized likelihood distributions for ANV from the joint WMAP and BBN analysis for two choices of the primordial abundance of 4He (solid: OSW, dashed: IT).

Fig. 10. The marginalized likelihood distributions for n10 from the joint WMAP and BBN analysis for two choices of the primordial abundance of 4He (solid: OSW, dashed: IT).

for either the OSW or the IT 4He abundances, each are consistent with Nv = 3 at ~ 2a. These results are in excellent agreement with those of Hannestad [23], with which they are most directly related.

Clearly, BBN constrains Nv much more stringently than WMAP, while the measurement of n10 by WMAP is at a precision superior (by a factor ~2) to that from BBN. In this sense, the CBR and BBN are quite complimentary. Indeed, while the constraint on Nv barely changes with the inclusion of the WMAP data in a joint analysis with BBN, it is sensitive to the adopted 4He abundance; see Fig. 9. On the other hand, the joint constraint on n is extremely insensitive to the choice of the 4He abundance, being dominated by the WMAP data (and the primordial D abundance); see Fig. 10. However, BBN and WMAP do provide a very important consistency check of the standard model of cosmology at widely separated epochs, using significantly different physics. The excellent agreement between Nv and n10 when the universe was 20 minutes and 380000 years old is a major triumph for the (new) standard model of cosmology.

Our results are summarized in Tables 1 and 2 and Figs. 9 and 10. BBN and the primordial D abundance combine to provide a quite accurate determination of the baryon density (0.020 < «b < 0.025 at 2a). The currently large uncertainty in the primordial abundance of 4He is responsible for the larger allowed range of nv. While the best fit value for nv is < 3

Acknowledgements

Extensive 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 to and assistance with CONDOR. This research was supported by the US DOE under Grants Nos. DE-FG02-95ER40896, DE-FG02-91ER40676, DE-FG02-02ER41216, and DE-FG02-91ER40690, by the NSF under Grant No. PHY99-07949 and by the Wisconsin Alumni Research Foundation. V.B., D.M. and G.S. thank the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara for its support and hospitality.

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