Constraining The Early-Universe Baryon Density And Expansion Rate
aa r X i v : . [ a s t r o - ph ] J un D RAFT VERSION O CTOBER
23, 2018
Preprint typeset using L A TEX style emulateapj v. 03/07/07
CONSTRAINING THE EARLY-UNIVERSE BARYON DENSITY AND EXPANSION RATE V IMAL S IMHA AND G ARY S TEIGMAN Draft version October 23, 2018
ABSTRACTWe explore the constraints on those extensions to the standard models of cosmology and particle physicswhich modify the early-Universe, radiation-dominated, expansion rate S ≡ H ′ / H (parametrized by the effectivenumber of neutrinos N ν ). The constraints on S (N ν ) and the baryon density parameter η B ≡ ( n B / n γ ) = 10 - η ,derived from Big Bang Nucleosynthesis (BBN, t ∼
20 minutes) are compared with those inferred from theCosmic Microwave Background Anisotropy spectrum (CMB, t ∼
400 kyr) and Large Scale Structure (LSS, t ∼
14 Gyr). At present, BBN provides the strongest constraint on N ν (N ν = 2 . ± . η = 6 . + . - . at 68% confidence), independent of N ν , at present they provide a relatively weak constrainton N ν which is, however, consistent with the standard value of N ν = 3. When the best fit values and theallowed ranges of these CMB/LSS-derived parameters are used to calculate the BBN-predicted primordialabundances, there is excellent agreement with the observationally inferred abundance of deuterium and goodagreement with He, confirming the consistency between the BBN and CMB/LSS results. However, the BBN-predicted abundance of Li is high, by a factor of 3 or more, if its observed value is uncorrected for possibledilution, depletion, or gravitational settling. We comment on the relation between the value of N ν and a possibleanomaly in the matter power spectrum inferred from observations of the Ly- α forest. Comparing our BBN andCMB/LSS results permits us to constrain any post-BBN entropy production as well as the production of anynon-thermalized relativistic particles. The good agreement between our BBN and CMB/LSS results for N ν and η B permits us to combine our constraints finding, at 95% confidence, 1.8 < N ν < < η < Subject headings: neutrinos — early universe — expansion rate — baryon density INTRODUCTION
The standard models of particle physics and of cosmology with dark energy, baryonic matter, radiation (including three speciesof light neutrinos), and dark matter is consistent with cosmological data from several widely-separated epochs. However, thereis room to accommodate some models of non-standard physics within the context of this well tested, Λ CDM cosmology. Onepossibility, explored here, is that of a non-standard expansion rate ( S ≡ H ′ / H , where H is the Hubble parameter) during theearly, radiation-dominated evolution of the Universe driven perhaps, but not necessarily, by a non-standard content of relativisticparticles ( ρ ′ R = ρ R ), parametrized by the equivalent number of additional neutrinos ( ρ ′ R ≡ ρ R + ∆ N ν ρ ν , where ∆ N ν ≡ N ν - e ± annihilation). In the epoch just prior to e ± annihilation, which is best probed by Big Bang Nucleosynthesis(BBN), ρ R = ρ γ + ρ e + ρ ν = 43 ρ γ /
8, so that S ≡ (cid:18) H ′ H (cid:19) ≡ ρ ′ R ρ R ≡ + ∆ N ν . (1)In the epoch after the completion of e ± annihilation, best probed by the Cosmic Microwave Background (CMB) and by LargeScale Structure (LSS), the relations between ρ R and ρ γ and between S and ∆ N ν differ from those in eq. 1, as described below insome detail in §2.We use the comparison between the predicted and observed abundances of the light elements produced during BBN, andbetween the predicted and observed CMB anisotropy spectrum, along with data from LSS observed in the present/recent Universe,to constrain new physics which leads to a non-standard, early-Universe expansion rate ( S ) or, equivalently, to place bounds on theeffective number of neutrinos (N ν ); for related earlier work see, e.g., Barger et al. (2003); Cyburt (2004). In addition, the baryondensity and, any variation in it over widely-separated epochs in the evolution of the Universe, are constrained simultaneouslywith N ν , thereby testing the standard-model expectation that the ratio (by number) of baryons to CMB photons ( η B ≡ n B / n γ )should be unchanged from e ± annihilation ( T . / t & T = 2 .
725 K ≈ × - MeV, t ≈
14 Gyr).In §2 a non-standard expansion rate ( S ) is related to a non-standard radiation density, parametrized by an effective number ofneutrinos N ν . In §3, the observationally inferred primordial light element (D, He, Li) abundances are used to constrain theradiation density (expansion rate) and the baryon density. In §4, we assume a flat, Λ CDM cosmology and use data from theCMB, along with a prior on the Hubble parameter from the HST key project, luminosity distances of type Ia supernovae, and thematter power spectrum to provide independent constraints on the radiation density and the baryon density. The results from thesetwo epochs, very widely-separated in time, are compared constraining any post-BBN entropy production. The good agreement Department of Astronomy, The Ohio State University, 140 West 18th Ave., Columbus, OH 43210 Departments of Physics and Astronomy and Center for Cosmology and Astro-Particle Physics, The Ohio State University, 191 West Woodruff Ave., Colum-bus, OH 43210 between them permits us to combine them to obtain a joint constraint on the baryon density parameter ( η B ) and the effectivenumber of neutrinos (N ν ). NON-STANDARD EARLY-UNIVERSE EXPANSION RATE OR RADIATION DENSITY
In the radiation-dominated early universe ( ρ TOT → ρ R ) the expansion rate ( H ) is related to the radiation density through theFriedman equation. H = 8 π G ρ R . (2)Any modification to the radiation density, or to the Friedman equation by a term which evolves like the radiation density (asthe inverse fourth power of the scale factor) can be parametrized by an equivalent number of additional neutrinos ∆ N ν where,prior to e ± annihilation, ∆ N ν ≡ N ν -
3. For the standard models of particle physics and cosmology, in the epoch after muonannihilation ( T .
100 MeV) and prior to e ± annihilation ( T & . e ± pairs, and three flavors of extremely relativistic, left-handed neutrinos (and their right-handed, antineutrinocounterparts). In this case, the total radiation density may be written in terms of the photon density as ρ R = 438 ρ γ = 5 . ρ γ . (3)In this same epoch, prior to e ± annihilation, a modified radiation density can be written as ρ ′ R = ρ R (cid:18) + ∆ N ν (cid:19) = ρ R (1 + . ∆ N ν ) (4)where ρ R is the standard-model radiation energy density and ρ ′ R is the modified, non-standard model radiation energy density. Ina sense, this modified energy density is simply a proxy for a non-standard expansion rate during the radiation-dominated epochrelevant for comparison with BBN. S ≡ H ′ H = (cid:18) ρ ′ R ρ R (cid:19) / = (cid:18) + ∆ N ν (cid:19) / . (5)After e ± annihilation the surviving relativistic particles are the photons (which will redshift to the presently observed CMB)and the now decoupled, relic neutrinos. In the approximation that the neutrinos are fully decoupled at e ± annihilation, thepost- e ± annihilation photons are hotter than the neutrinos by a factor of T γ / T ν = (11 / / and ρ R = " + × (cid:18) (cid:19) / ρ γ = (1 + × . ρ γ = 1 . ρ γ , (6)so that ρ ′ R = ρ γ (1 . + . ∆ N ν ) = ρ R (1 + . ∆ N ν ) , (7)where, as before, ∆ N ν ≡ N ν - e ± annihilation. As a result, the relic neutrinos share someof the energy/entropy released by e ± annihilation and they are warmer than in the fully decoupled approximation, increasingthe ratio of the post- e ± annihilation radiation density to the photon energy density. While the post- e ± annihilation phase spacedistribution of the decoupled neutrinos is no longer that of a relativistic, Fermi-Dirac gas, according to Mangano et al. (2005) theadditional contribution to the total energy density can be accounted for by replacing N ν = 3 with N ν = 3.046, so that ρ R → (1 + . × . ρ γ = 1 . ρ γ . (8)For deviations from the standard model that can be treated as equivalent to contributions from fully decoupled neutrinos, ρ ′ R = ρ R (1 + . ∆ N ν ) , (9)where ∆ N ν ≡ N ′ ν - .
046 in the post- e ± annihilation Universe relevant for comparison with the CMB and LSS. Note that inthe standard model, where N ν = 3 prior to e ± annihilation, the neutrino contribution to the post- e ± annihilation radiation energydensity is equivalent to N ′ ν = 3 . ν = 3, N ′ ν = 3 .
046 and, ∆ N ν = 0.We emphasize that although the non-standard radiation density (expansion rate) has been parametrized as if it were due toadditional species of neutrinos, this parametrization accounts for any term in the Friedman equation whose energy density variesas a - , where a is the scale factor. From this perspective, ∆ N ν could either be positive or negative; the latter does not necessarilyimply fewer than the standard-model number of neutrinos but could, for example, be a sign that the three standard-model neutrinosfailed to be fully populated in the early Universe or, could reflect modifications to the 3 + 1 dimensional Friedman equationsarising from higher-dimensional extensions of the standard model of particle physics (Randall & Sundrum 1999; Binetruy et al.2000; Cline et al. 2000). N ν AND η B FROM BBN
The stage is being set for BBN when the Universe is about a tenth of a second old and the temperature is a few MeV. At thistime the energy density of the universe is dominated by relativistic particles. When the temperature drops below ∼ e ± plasma. However, they do continue to interact with the neutrons and protonsthrough the charged-current weak interactions ( n + ν e ↔ p + e - , p + ¯ ν e ↔ n + e + , n ↔ p + e - + ¯ ν e ), maintaining the neutron-to-protonratio at its equilibrium value of n / p = exp( - ∆ m / kT ), where ∆ m is the neutron-proton mass difference. When the temperaturedrops below ∼ . ∼ Γ wk < H ). As a result, the neutron-proton ratio deviates from (exceeds) its equilibrium value,so that n / p > exp( - ∆ m / kT ), and the actual n / p ratio depends on the competition between the expansion rate ( H ) and thecharged-current weak interaction rate ( Γ wk ), as well as on the neutron decay rate, 1 /τ n , where τ n is the neutron lifetime.Although nuclear reactions such as n + p ↔ D + γ , proceed rapidly during these epochs, the large γ -ray background (the blue-shifted CMB) ensures that the deuterium (D) abundance is very small, inhibiting the formation of more complex nuclei. The morecomplex nuclei begin to form only when T . .
08 MeV, after e ± annihilation, when the Universe is about 3 minutes old. Atthis time the number density of photons with sufficient energy to photodissociate deuterium is comparable to the baryon numberdensity and various two-body nuclear reactions can begin to build more complex nuclei. Note that the neutron-to-proton ratio hasdecreased slightly since “freeze-out" (at T . . He, and He. The absence of a stable mass-5 nuclide presents a road-block to the synthesis of heavier elements in the expanding, cooling Universe, ensuring that the abundances of heavier nuclidesare severely depressed below those of the lighter nuclei. In standard BBN (SBBN) only D, He, He, and Li are produced inastrophysically interesting abundances (for a recent review see Steigman (2007)). While the BBN-predicted abundances of D, He, and Li are most sensitive to the baryon density, that of He is very sensitive to the neutron abundance when BBN beginsand, therefore, to the competition between the weak-interaction rate and the universal expansion rate. The primordial abundancesof D, He, or Li are baryometers, constraining η B , while He mass fraction (Y P ) is a chronometer, depending mainly on S or, N ν .In the standard model of particle physics and cosmology with three species of neutrinos and their respective antineutrinos, theprimordial element abundances depend on only one free parameter, the baryon density parameter, the post- e ± ratio (by number)of baryons to photons, η B = n B / n γ . This parameter may be related to Ω B , the present-Universe ratio of the baryon mass densityto the critical mass-energy density (see Steigman (2006)) η = 10 n B / n γ = 273 . Ω B h . (10)The abundance of He is very sensitive to the early expansion rate. Since a non-standard expansion rate ( S = 1) would result infewer or more neutrons at BBN and, since most neutrons are incorporated into He, the predicted He abundance differs fromthat in SBBN ( S = 1; ∆ N ν = 0). In contrast, the abundance of He is not very sensitive to the baryon density since, to first order,all the neutrons available at BBN are rapidly converted to He. For η ≈
6, N ν ≈ He mass fraction inthe range 0 . . Y P . .
27, to a very good approximation (Kneller & Steigman 2004; Steigman 2007),Y P = 0 . ± . + . η - + S - . (11)In eq. 11, the effect of incomplete neutrino decoupling on the He mass fraction is accounted for according to Mangano et al.(2005) and S is related to ∆ N ν ( ∆ N ν ≡ N ν - .
0) by eq. 5. As a result, for a fixed He abundance, a variation in η of ∼ ± . ∼
3% uncertainty in the baryon density) is equivalent to an uncertainty in ∆ N ν of ∼ ± . He, since the primordial abundances of D, He and Li are set by the competition between two body produc-tion and destruction rates, they are more sensitive to the baryon density than to the expansion rate. For example, for η ≈ y D ≡ (D / H) P , in the range, 2 . y D .
4, to a very good approximation(Kneller & Steigman 2004; Steigman 2007), y D = 2 . ± . (cid:20) η - S - (cid:21) . . (12)The effect of incomplete neutrino decoupling on this prediction is at the ∼ .
3% level (Mangano et al. 2005), about ten timessmaller than the overall error estimate above.
Observationally-Inferred Primordial Abundances
Given the monotonic post-BBN evolution of deuterium (as gas is cycled through stars, deuterium is destroyed) and the sig-nificant dependence of its predicted BBN abundance on the baryon density ( y DP ∝ η - . ), deuterium is the baryometer of choiceamong the light nuclides produced during primordial nucleosynthesis. While observations of D/H in the solar system and thelocal interstellar medium provide a lower limit to the relic deuterium abundance, it is the D/H ratio (by number) measured fromobservations of high redshift, low metallicity QSO absorption line systems which provide an estimate of its primordial abundance.The weighted mean of the six, high redshift, low metallicity D/H ratios from Kirkman et al. (2003) and O’Meara et al. (2006) is(Steigman 2007) y DP = 2 . + . - . . (13)Since the post-BBN evolution of He is more complex and model dependent than that of deuterium and, since He is onlyobserved in chemically-evolved H II regions in the Galaxy and, since the He primordial abundance is only weakly dependent onthe baryon density (10 ( He/H) ≡ y ∝ η - . ), its role as a baryometer is limited. F IG . 1.— The probability distribution of the baryon density parameter, η . The dashed line shows the probability distribution inferred from SBBN (N ν = 3)and the adopted primordial abundance of deuterium (see §3). The solid line is the probability distribution of η inferred for N ν = 3 from the combination of theWMAP-5yr data, small scale CMB data, matter power spectrum data from 2dFGRS and SDSS LRG, SNIa, and the HST Key Project (see §4). As for deuterium, the post-BBN evolution of He is monotonic, with Y P increasing along with increasing metallicity. Atlow metallicity, the He abundance should approach its primordial value. As a result, it is the observations of helium andhydrogen recombination lines from low-metallicity, extragalactic H II regions which are most useful in determining Y P . Atpresent, corrections for systematic uncertainties (and their uncertainties) dominate estimates of the observationally-inferred Heprimordial mass fraction and, especially, of its error. Following Steigman (2007), we adopt for our estimate here,Y P = 0 . ± . . (14)While the central value of Y P adopted here is low, the conservatively-estimated uncertainty is relatively large (some ten timeslarger than the uncertainty in the BBN-predicted abundance for a fixed baryon density). In this context, it should be noted thatalthough very careful studies of the systematic errors in very limited samples of H II regions provide poor estimators of Y P as aresult of their uncertain and/or model-dependent extrapolation to zero metallicity, they are of value in providing a robust upperbound to Y P . Using the results of Olive & Skillman (2004), Fukugida & Kawasaki (2006), and Peimbert et al. (2007), wefollow Steigman (2007) in adopting, Y P < . ± . . (15)Although the BBN-predicted Li relic abundance provides a potentially sensitive baryometer ((Li/H) ∝ η , for η & Li. The complication associated with this approach is that these oldest galactic stars have had themost time to dilute or deplete their lithium surface abundances, leading to the possibility that the observed abundances requirelarge, uncertain, and model-dependent corrections in order to infer the primordial abundance of Li. In the absence of correctionsfor depletion, dilution, or gravitational settling, the data of Ryan et al. (2000) and Asplund et al. (2006) suggest[Li] P ≡ + log(Li / H) = 2 . ± . . (16)In contrast, in an attempt to correct for evolution of the surface lithium abundances, Korn et al. (2006) use their observations of asmall, selected sample of stars in the globular cluster NGC6397, along with stellar evolution models which include the effect ofgravitational settling to infer [Li] P = 2 . ± . . (17)In the following analysis, the inferred primordial abundances of D and He adopted here are used to estimate η and ∆ N ν .Given the inferred best values and uncertainties in these two parameters, the corresponding BBN-predicted abundance of Li canbe derived and compared to its observationally inferred abundance.
BBN Constraints On N ν And η B Since the primordial abundance of deuterium is most sensitive to η , while that of He is most sensitive to N ν , isoabundancecontours of D/H and Y P in the { η , ∆ N ν } plane are very nearly orthogonal; see Kneller & Steigman (2004). The analytic fitsto BBN from Kneller & Steigman (2004), updated by Steigman (2007), are used in concert with the primordial abundances ofD and He adopted here to infer the best values, and to constrain the ranges of η and ∆ N ν . While these fits do have a limitedrange of applicability, they are, in fact, accurate within their quoted uncertainties for the range of parameter values and observedabundances considered here.In SBBN, with three species of neutrinos ( ∆ N ν = 0), the primordial abundances are only functions of the baryon density, η B .For SBBN, the primordial deuterium abundance adopted in §3.1, y DP = 2 . + . - . , implies, η (SBBN) = 6 . ± . h N n F IG . 2.— The 68% and 95% contours in the N ν - η plane derived from a comparison of the observationally-inferred and the BBN-predicted primordialabundances of D and He. The shaded region is excluded by the 95% upper bound to the helium abundance in eq. 15 (see, eq. 20).
This result is in excellent agreement with the independent estimate of η = 6 . + . - . from the CMB and LSS (discussed below in§4). The probability distributions of η inferred from SBBN and from the CMB and LSS are shown in Figure 1.For non-standard BBN, with ∆ N ν = 0 ( S = 1), there is a second free parameter, N ν (or, S ). In this case, in addition to y DP , the He abundance Y P is used to constrain the { η , N ν } pair. Adopting the D and He abundances from §3.1 (Y P = 0 . ± . η = 5 . ± . , N ν = 2 . ± . . (19)In Figure 2 are shown the 68% and 95% contours in the N ν - η plane which follow from a comparison of the BBN predictionswith the observationally-inferred primordial abundances of D and He. Notice that while the best fit value of N ν is less than thestandard-model value of N ν = 3, the standard-model value is consistent with the relic abundances at the ∼
68% level.These results are sensitive to the choices of the relic abundances of D and He. We note that if deuterium is ignored and therobust upper bound to the He mass fraction, Y P < .
255 at 95% confidence (eq. 15), is adopted, then eq. 11 provides an upperlimit to S (N ν ) as a function of the baryon density, S < . - . η . (20)For η in the range 5 ≤ η ≤ ν ranging from 3.6 to 3.4. N ν AND η B FROM THE CMB AND LSS
The pattern of temperature fluctuations in the cosmic microwave background contain information about the baryon density andthe radiation density and thus serve as complementary probes of η B and N ν some ∼ years after BBN.The baryon density, parametrized by η B or Ω B h affects the relative amplitudes of the peaks in the CMB temperature powerspectrum. The ratios of the amplitudes of the odd peaks to the even peaks provide a determination of η B that is largely uncorrelatedwith N ν .The radiation density, parametrized by an effective number of neutrino species N ν , affects the CMB power spectrum primarilythrough its effect on the epoch of matter-radiation equality. There are substantial differences in amplitudes between those scalesthat enter the horizon during the radiation dominated era and those that enter the horizon later, in the matter dominated era.Increasing the radiation content delays matter-radiation equality, bringing it closer to the epoch of recombination, suppressingthe growth of perturbations. As a result, the redshift of the epoch of matter-radiation equality, z eq , is a fundamental observablethat can be extracted from the CMB Power Spectrum. z eq is related to the matter and radiation densities by,1 + z eq = ρ M /ρ R . (21)Since ρ R depends on N ν , z eq is a function of both N ν and Ω M h , leading to a degeneracy between these two parameters. For a flatuniverse, preserving the fit to the CMB power spectrum when N ν increases, requires that Ω M and/or H increase. As a result ofthis degeneracy, the CMB power spectrum alone imposes only a very weak constraint on N ν (Crotty et al. 2003; Pierpaoli 2003;Barger et al. 2003; Hannestad 2003; Ichikawa et al. 2006). Inclusion of additional, independent constraints on these parametersare needed to break the degeneracy between N ν and Ω M h .Besides affecting the epoch of matter-radiation equality, relativistic neutrinos leave a distinctive signature on the CMB powerspectrum due to their free streaming at speeds exceeding the sound speed of the photon-baryon fluid. This free streaming createsneutrino anisotropic stresses generating a phase shift of the CMB acoustic oscillations in both temperature and polarization. Thisphase shift is unique and, for adiabatic initial conditions, cannot be generated by non-relativistic matter. In principle, this effectcan be used to break the degeneracy between N ν and Ω M h , leading to tighter constraints on N ν (Bashinsky & Seljak 2004). F IG . 3.— The top panel shows the CMB power spectrum for the best fit models with N ν fixed at N ν = 1 (solid black), N ν = 3 (dashed red), and N ν = 5(dot-dashed blue) illustrating its insensitivity to N ν in the absence of an independent constraint on Ω M h . The bottom panel shows the matter power spectra forthe same set of parameter values, illustrating its sensitivity to N ν . Alternatively, since the luminosity distances of type Ia supernovae (SNIa) provide a constraint on a combination of Ω M and Ω Λ complementary to that from the assumption of flatness, they are of value in restricting the allowed values of Ω M . In concert witha bound on H , this, too, helps to break the degeneracy between N ν and Ω M h .Another way to break the degeneracy between N ν and Ω M h is to use measurements of the matter power spectrum in com-bination with the CMB power spectrum. To preserve a fit to the CMB power spectrum, an increase in N ν requires that Ω M h increase in order that the redshift of matter-radiation equality remain unchanged. The turnover scale in the matter power spec-trum is set by this connection between N ν and Ω M h . Since the baryon density is constrained by the CMB power spectrum,independently of N ν , increasing the radiation density (N ν >
3) requires a higher dark matter density in order to preserve z eq (ina flat universe, Ω M + Ω Λ = 1). Between the epoch of matter-radiation equality and recombination the density contrast in the colddark matter grows unimpeded, while the baryon density contrast cannot grow. Consequently, increasing N ν and Ω M h , increasesthe amplitude of the matter power spectrum on scales smaller than the turnover scale corresponding to the size of the horizon at z eq . Data from galaxy redshift surveys can be used to infer the matter power spectrum, thereby constraining Ω M h and N ν . Thiseffect may be seen in Figure 3 which shows that for nearly indistinguishable CMB power spectra, different values of N ν yielddistinguishable matter power spectra. The upper panel of Figure 3 shows that by making suitable adjustments to the other cosmo-logical parameters, specifically the matter density and the spectral index, CMB power spectra which are nearly degenerate up tothe third peak of the power spectrum can be produced using very different values of N ν . However, as the bottom panel of Figure3 illustrates, these models produce matter power spectra with different shapes, demonstrating that the matter power spectrum canbe used to help break the degeneracy between N ν and Ω M h . Analysis And Datasets
Before presenting our results, we describe the analysis and the datasets employed.For our analysis we assume a flat, CDM cosmology with a cosmological constant, Λ , and three flavors of active neutrinos withnegligible masses. Our cosmological model is parametrized by seven parameters: p = { Ω B h , Ω M h , h , τ , n S , A S , N ν } . (22)The contents of the universe are described by the baryon density, Ω B h and the matter (baryonic plus cold dark matter) density, Ω M h . Since a flat cosmology is assumed, the dark energy density and the matter density are related by Ω Λ = 1 - Ω M . The expan-sion rate of the universe is described by the reduced Hubble parameter, h ( H = 100 h kms - Mpc - ). Instantaneous reionizationof the universe is assumed with optical depth to last scattering, τ . A S is the amplitude of scalar perturbations and n S is the scalarspectral index.The Code for Anisotropies in the Microwave Background (CAMB) (Lewis et al. 2000) is used to compute the CMB powerspectrum for a fixed set of cosmological parameters. For a given dataset, our degree of belief in a set of cosmological parameters { p } is quantified by the posterior probability distribution, P ( p | data ) ∝ L ( data | p ) Π ( p ) . (23)The likelihood L (data | p) quantifies the agreement between the data and the set of parameter values { p } . Π (p) represents the prioron cosmological parameter values before the data are considered.Markov Chain Monte Carlo methods are used to explore the multi-dimensional likelihood surface. We use the publicly availableCOSMOMC code for our analysis (Lewis & Bridle 2002). Flat priors are adopted for all parameters, along with a prior on theage of the universe, t , of t >
10 Gyr.The mode of the marginalized posterior probability distribution is used as a point estimate and the minimum credible interval as an estimate of the uncertainty. The minimum credible interval selects the region of the parameter space around the mode, b θ ,that contains the appropriate fraction of the volume ( e.g. , 68%, 95%) of the posterior probability distribution, while minimizing b θ - θ . The minimum credible interval selects the region of the parameter space with the highest probability densities .Our primary dataset is the CMB data from the Wilkinson Microwave Anisotropy Probe (WMAP) accumulated from five yearsof observations (Hinshaw et al. 2008; Nolta et al. 2008). For the WMAP data, likelihoods are computed using the code providedon the LAMBDA webpage . There are a number of ground and balloon based CMB experiments whose high angular resolutionprobe smaller scales than those probed by WMAP. These are more sensitive to the higher order acoustic oscillations beyond thethird peak in the CMB power spectrum. In particular, the 2008 results from the Arcminute Cosmology Bolometer Array Receiver(ACBAR) (Reichardt et al. 2008) impose the strongest constraints at present on the CMB power spectrum at small angular scales.In our analysis, data from the following CMB experiments are used: BOOMERANG (Piacentini et al. 2006; Montroy et al.2006), ACBAR (Reichardt et al. 2008), CBI (Readhead et al. 2004; Mason et al. 2003), VSA (Dickinson et al. 2004), MAXIMA(Hanany et al. 2000) and DASI (Halverson et al. 2001).To help break the degeneracy between N ν and Ω M h , we adopt a Gaussian prior on the Hubble parameter, H = 72 ± - Mpc - from the Hubble Space Telescope Key Project (Freedman et al. 2001).Since the luminosity distances of type Ia supernovae (SNIa) provide a constraint on a combination of Ω M and Ω Λ , they are ofvalue in restricting the allowed values of Ω M , helping to break the degeneracy between N ν and Ω M h . We use the luminositydistances for 115 type Ia supernovae measured by the Supernova Legacy Survey (SNLS) (Astier et al. 2006). For each supernova,the observed luminosity distance is compared to that predicted for a given set of cosmological parameters.We use the matter power spectrum inferred from galaxy redshift surveys such as the Sloan Digital Sky Survey (SDSS)(Tegmark et al. 2004, 2006) and the 2dFGRS (Cole et al. 2005). In using the matter power spectrum to infer cosmologicalparameters, it has become clear that the constraint on Ω M is sensitive to the choice of length scales on which the power spec-trum is measured. The power spectrum on smaller scales prefers higher values of Ω M , provided that those scales are correctlydescribed by linear perturbation growth and scale independent galaxy bias (Cole et al. 2005; Percival et al. 2007). For example,using the SDSS LRG power spectrum (Tegmark et al. 2006) in combination with the 3 year WMAP dataset (Spergel et al. 2007),Dunkley et al. (2008) find a disagreement between the matter density inferred on scales with k ≤ . h Mpc - and k ≤ . h Mpc - .Hamann et al. (2007) have shown that the constraint on N ν obtained from the matter power spectrum is affected similarly, result-ing from the correlation between Ω M and N ν . Therefore, here we only use matter power spectrum data on scales that are likely tobe safely linear. We truncate the matter power spectrum at k = 0 . h Mpc - , keeping data only on scales with k ≤ . h Mpc - .We assume that galaxy bias is constant and scale independent for these scales. N ν And The Lyman- α Forest
Measurements of the flux power spectrum of the Lyman- α forest in QSO absorption spectra can be used to reconstruct thematter power spectrum on small scales (Croft et al. 1998; McDonald et al. 2000). Observations of the SDSS Ly- α flux powerspectrum have been used to constrain the linear matter power spectrum at z ∼ ν than those we find here (§5). Forexample, Seljak et al. (2006) found N ν = 5 . . ≤ N ν ≤ . ν = 6 . . ≤ N ν ≤
11, the difference being accounted for by their different combinations of data sets. Each of these excludesthe standard model value of N ν = 3 at more than 95% confidence.As discussed earlier and shown in the lower panel of Figure 3, the principal effect of an increase in N ν is to increase theamplitude of the matter power spectrum on scales smaller than those corresponding to the horizon at matter-radiation equality, z eq . The Λ CDM fits to the Lyman- α forest data prefer higher amplitudes of density fluctuations on small scales compared to thoseexpected from measurements of the WMAP power spectrum (Viel & Haehnelt 2006), favoring higher values of N ν . This effect isseen in Figure 3 which shows that for nearly indistinguishable CMB power spectra, different values of N ν yield distinguishablematter power spectra. To preserve z eq and the fit to the CMB, the model with N ν = 1 has a lower value of Ω M h and, therefore, The commonly used GETDIST analysis package uses the mean of the marginalized posterior probability distribution as a point estimate and gives uncertaintyestimates based on the central credible interval (Hamann et al. 2007). These estimates are identical for Gaussian probability distributions but differ significantlyfor non-Gaussian distributions, particularly for asymmetric probability distributions. http://lambda.gsfc.nasa.gov W M h N n F IG . 4.— The 68% and 95% contours in the N ν - Ω M h plane inferred from the combination of the WMAP-5yr data, small scale CMB data, luminositydistances of SNIa and the HST Key Project prior on H . The dashed line shows the locus of points corresponding to the same value of z eq (= 3144), illustratingthe degeneracy between these two parameters. As the contours reveal, this degeneracy may be broken if complementary data is used to constrain Ω M h . the matter power spectrum for that model has a lower amplitude on scales smaller than the scale corresponding to the horizonat the epoch of matter-radiation equality. Conversely, the model with N ν = 5 has a higher value of Ω M h and the matter powerspectrum for that model has a higher amplitude on scales smaller than the scale corresponding to the horizon at z eq .Before reaching any conclusions about N ν based on the Lyman- α forest data, it is worth noting that assumptions about thethermal state of the IGM play an important role in reconstructing the matter power spectrum from the Lyman- α forest flux powerspectrum. Bolton et al. (2007) compare measurements of the Lyman- α forest flux probability distribution by Kim et al. (2007)to hydrodynamic simulations of the Lyman- α forest, finding evidence for an inverted temperature-density relation for the lowdensity intergalactic medium. Bolton et al. (2007) suggest that He II reionization could be a possible physical mechanism forachieving an inverted temperature-density relation. Such an inversion would result in a smaller amplitude of the matter powerspectrum for a given observed flux power spectrum, thereby alleviating the tension with the other data sets which tended to driveN ν to high values. However, in their power spectrum fits McDonald et al. (2005) marginalize over equation of state parameters,so this explanation of the tension may not be entirely satisfactory. Future studies of the Lyman- α forest may weaken or strengthenthe evidence for a discrepancy. For these reasons, we do not use data from the Lyman- α forest in our analysis. CMB And LSS Constraints On N ν And η B According to Dunkley et al. (2008) the WMAP 5 year data only impose a lower limit on N ν of N ν > . ν and Ω M h . Inclusion of data from smallscale CMB experiments do not break this degeneracy. Figure 4 illustrates this degeneracy, as well as how it may be broken bynon-CMB constraints on Ω M h . The dashed line in Fig. 4 is the locus of points with constant z eq = 3144. Figure 4 shows the jointprobability distributions of N ν and Ω M h inferred from the WMAP 5yr data and small scale CMB experiments, supplementedby independent data from measurements of SNIa luminosity distances and the HST Key Project prior on H are used to bound Ω M h . The range of N ν is now limited.Our constraints on N ν and η from various CMB and LSS datasets and combinations of them are summarized in Table 1 andin Figures 5 – 7.Using the WMAP 5 year data in combination with data from other CMB experiments (ACBAR, BOOMERANG, CBI, DASI,MAXIMA and VSA), along with the HST Key Project prior on H and luminosity distance measurements of type Ia supernovae,we find central values N ν = 2 . η = 6 .
2, along with 68% (95%) ranges of 2 . < N ν < . . < N ν < .
4) and 6 . < η < . . < η < . ν and η for this combination of CMB datasets.Adding the 2dFGRS power spectrum (Cole et al. 2005) to the CMB power spectrum from the WMAP 5 year dataset, alongwith ground based CMB experiments (see above), the HST prior on H and, luminosity distance measurements from SNIa, wefind central values N ν = 3 . η = 6 .
1, along with 68% (95%) ranges 2 . < N ν < . . < N ν < .
2) and 6 . < η < . . < η < .
4) respectively; see Table 1. Replacing the 2dFGRS power spectrum with the SDSS LRG power spectrum(Tegmark et al. 2006), we obtain very similar results: N ν = 2 .
8, 2 . < N ν < . . < N ν < .
2) and, η = 6 .
2, 6 . < η < . . < η < . ν compared to those inferred from the 2dFGRS and the SDSS (LRG) power spectra.Our best constraints on N ν and η from the CMB and LSS data are derived by combining the following datasets - WMAP5-year data, ground and balloon based CMB experiments (BOOMERANG, ACBAR, CBI, VSA, MAXIMA and DASI), theHST prior on H , the SNIa luminosity distance measurements and, the matter power spectrum on large scales inferred from the2dFGRS and SDSS (LRG) data. Constraints on the matter power spectrum from the Lyman- α forest are not included for the h N n F IG . 5.— The 68% and 95% contours in the N ν - η plane inferred from the combination of the WMAP-5yr data, small scale CMB data, SNIa luminositydistances, and the HST Key Project prior on H (see the text and Table 1). h N n F IG . 6.— The 68% and 95% contours in the N ν - η plane derived using the WMAP 5-year data, small scale CMB data, SNIa and the HST Key Project prioron H along with matter power spectrum data from 2dFGRS and SDSS LRG(see the text and Table 1). reasons discussed above in §4.2. Using these datasets, we obtain (see Table 1): η = 6 . + . + . - . - . . (24)N ν = 2 . + . + . - . - . . (25)At present the combined CMB and LSS data provide the best baryometer, determining the baryon density to better than 3%,but only a relatively weak chronometer, still allowing a large range in S (0 . ≤ S ≤ .
14 at 95% confidence). Within theiruncertainties, the CMB/LSS data, which probe the Universe at & ,
000 years, are consistent with BBN, which provides awindow on the Universe at .
20 minutes. COMPARING THE BBN AND CMB/LSS CONSTRAINTS
Using the ranges of η and N ν allowed by the CMB and LSS data, the BBN-predicted primordial abundances of He, D and Limay be inferred. Figure 8 compares these constraints to the observationally-inferred primordial abundances adopted in §3.1. TheCMB/LSS-inferred BBN abundances of D and He are in excellent agreement, within the errors, with the observationally-inferredrelic abundances. For the central values of η = 6 .
14 and N ν = 2.9, the BBN-predicted deuterium abundance of y DP = 2 .
54 is,within the errors, in agreement with its observationally-inferred primordial value of y DP = 2 .
68. For He, the BBN-predictedmass fraction is Y P = 0.247, slightly high compared to the central value of the primordial abundance adopted in §3.1, Y P =0.240, but within 1.2 σ of it and, completely consistent with the evolution-model independent upper bound presented in eq. 15,Y P < . ± . Li the BBN-predicted best fit from the CMB/LSS data is [Li] P = 2.66, considerably higher than the value ([Li] P = 2 . ± . Li abundances may, perhaps, be reconciled, as may be seen from the lower panel of Fig. 8. It remains an openquestion whether this lithium problem is best resolved by a better understanding of stellar physics or, if it is providing a hint ofnew physics beyond the standard model.0
TABLE 1N ν AND η FROM DIFFERENT DATASETS
Dataset N ν η BBN (Y P & y DP ) 2 . + . + . - . - . . + . + . - . - . WMAP(1yr)+HST 2 . + . + . - . - . . + . + . - . - . WMAP(3yr)+HST+SN 2 . + . + . - . - . . + . + . - . - . WMAP(3yr)+CMB(07)+HST+SN 2 . + . + . - . - . . + . + . - . - . . + . + . - . - . . + . + . - . - . SDSS(DR2)+WMAP(3yr)+CMB(07)+HST+SN 3 . + . + . - . - . . + . + . - . - . SDSS(LRG)+WMAP(3yr)+CMB(07)+HST+SN 2 . + . + . - . - . . + . + . - . - . WMAP(5yr)+HST+SN 3 . + . + . - . - . . + . + . - . - . WMAP(5yr)+CMB+HST+SN 2 . + . + . - . - . . + . + . - . - . . + . + . - . - . . + . + . - . - . SDSS(LRG)+WMAP(5yr)+CMB+HST+SN 2 . + . + . - . - . . + . + . - . - . SDSS(LRG)+2dFGRS+WMAP(5yr)+CMB+HST+SN 2 . + . + . - . - . . + . + . - . - . BBN+SDSS(LRG)+2dFGRS+WMAP(5yr)+CMB+HST+SN 2 . + . + . - . - . . + . + . - . - . N OTE . — Best fits and 68% and 95% confidence intervals for N ν and η from our prin-cipal datasets. WMAP refers to the CMB power spectrum data from the WMAP experi-ment and CMB to the data from ACBAR+BOOM+CBI+VSA+MAXIMA+DASI. CMB(07)uses the 2007 ACBAR dataset (Kuo et al. 2007) while CMB uses the 2008 ACBAR dataset(Reichardt et al. 2008). HST refers to the prior on H from the HST Key Project. SN standsfor the SNIa luminosity distance measurements. SDSS (LRG) and 2dF refer to the respectiveLSS matter power spectra, truncated at k = 0 . h Mpc - . n L h L F IG . 7.—On the left (red, dashed), the probability distribution of N ν inferred from the combination of the WMAP 5-year data, small scale CMB data, SNIa and the HSTKey Project prior on H and matter power spectrum data from 2dFGRS and SDSS LRG. The solid blue curve is the BBN (D plus He) distribution. On the right(same line types and colors), are the probability distributions of η using the same data sets. CONCLUSIONS
While our CMB+LSS constraints on η B and N ν are consistent with most previous analyses (Crotty et al. 2003; Pierpaoli2003; Hannestad 2003; Barger et al. 2003; Seljak et al. 2006; Ichikawa et al. 2006; Spergel et al. 2007; Mangano et al. 2007;Hamann et al. 2007; Dunkley et al. 2008; Komatsu et al. 2008), they are tighter because we have used more and/or more recentdata. However, as discussed above in §4.2, analyses that included the Lyman- α forest data generally find higher values of N ν .Until the advent of WMAP and the other ground- and balloon-based CMB experiments, BBN provided the best baryometer(mainly from deuterium) and chronometer (mainly from helium-4). As may be seen from Table 1, while the WMAP first yeardata provided a competitive baryometer, it offered a relatively poor chronometer. This improves with the WMAP 3 and 5 yeardata, especially when they are combined with the other CMB and LSS data. These now lead to a determination of the baryondensity at the ∼ -
3% level, a factor of & ν hasimproved significantly and, it is consistent with that from BBN, it remains weaker than the BBN constraint by a factor of ∼ ν > ∼
20 minutes old. Forexample, using a slightly different estimate of the primordial helium mass fraction, Barger et al. (2003) found N ν > . F IG . 8.—The solid black curves show the probability distributions for the primordial abundances of D, He and Li derived from the values of η and N ν inferred from theCMB, LSS, SNIa and the HST prior and matter power spectrum data from 2dFGRS and SDSS LRG. The blue dashed curves show the probability distributionsfor the observationally-inferred primordial abundances of D, He and Li; see §3.1. The red, dot-dashed curve in the far right panel is the Li abundance fromKorn et al. (2006). h N n h N n F IG . 9.—(Left) In blue (solid), the 68% and 95% contours in the N ν - η plane derived from a comparison of the observationally-inferred and BBN-predicted primordialabundances of D and He (see Figure 2). In red (dashed), the 68% and 95% contours derived from the combined WMAP 5-year data, small scale CMB data,SNIa, and the HST Key Project prior on H along with matter power spectrum data from 2dFGRS and SDSS LRG (see the text and Table 1). (Right) The 68%and 95% joint BBN-CMB-LSS contours in the N ν - η plane. and LSS datasets, the CMB/LSS now confirm that N ν > ν > &
400 kyrold.As may be seen from Table 1 and from Figures 7, 8, and from the left-hand panel of Figure 9, BBN and the CMB, which probephysics at widely separated epochs in the evolution of the Universe, are in excellent agreement. This permits constraints on any differences in physics between BBN and recombination and/or the present epoch. For example, since baryons are conserved, η B relates the number of thermalized black body photons in a comoving volume at different epochs (see figure 7), constraining anypost-BBN entropy production. Our results from the CMB/LSS and from BBN imply, N CMB γ N BBN γ = 0 . ± . . (26)This ratio is consistent with 1 at ∼ σ and places an interesting upper bound on any post-BBN entropy production.Alternatively, late decaying particles could produce relativistic particles (radiation), but not necessarily thermalized black bodyphotons (see, for example, Ichikawa et al. (2007)). Deviations from the standard model radiation density can be parametrized bythe ratio of the radiation density, ρ ′ R to the standard model radiation density, ρ R . In the post- e ± annihilation universe (see eq. 9), R = S = ρ ′ R ρ R = 1 + . ∆ N ν . (27)Comparing this ratio, at BBN and at recombination (see figure 7), any post-BBN production of relativistic particles can beconstrained. R CMB R BBN = 1 . + . - . (28)This ratio, too, is consistent with 1 within 1 σ , placing an upper bound on post-BBN production of relativistic particles.2Although the non-standard expansion rate has been parametrized in terms of an equivalent number of additional species ofneutrinos, we have emphasized that a non-standard expansion rate need not be related to extra (or fewer) neutrinos. For example,deviations from the standard expansion rate could occur if the value of the early-Universe gravitational constant, G N were differentin the from its present, locally-measured value (Yang et al. 1979; Boesgaard & Steigman 1985; Accetta et al. 1990; Cyburt et al.2005). For the standard radiation density with three species of light, active neutrinos, the constraint on the expansion rate can beused to constrain variations in the gravitational constant, G N . From BBN, S = G BBNN G N = 0 . ± .
07 (29)and at the epoch of the recombination, S = G CMBN G N = 0 . + . - . , (30)consistent with no variation in G at the ∼ σ level.The agreement between η B and N ν evaluated from BBN ( ∼
20 minutes) and from the CMB/LSS ( &
400 kyr) is shown in theleft hand panel of Figure 9. As Figure 9 illustrates, BBN and the CMB/LSS, which probe the Universe at widely separated epochsin its evolution, are completely consistent. As already noted, at present the CMB is a better baryometer while BBN remains abetter chronometer. Since these independent constraints from the CMB/LSS and BBN are in very good agreement, we maycombine them to obtain the joint fit in Table 1 and shown in the right hand panel of Figure 9. We note that while the best-fit valueof N ν is less than 3, this is not statistically significant since the results are consistent with the standard model of three species ofactive neutrinos at the ∼ σ level.Of course, our BBN results are sensitive to the relic abundances we have adopted. For comparison, we have repeated ouranalysis for an alternate set of primordial abundances. For deuterium, we adopted the O’Meara et al. (2006) results based on theweighted mean of the log( y D ) values, y D = 2 . + . - . and, for He we chose, somewhat arbitrarily, Y P = 0 . ± . η B is virtually unchanged from our previous result, η = 5 . ± .
3, whileN ν = 2 . ± . R CMB / R BBN = 1 . + . - . and G BBNN / G N = 0 . + . - . . When the alternate BBN constraints are combined with thosefrom the CMB and LSS, we find η = 6 . + . - . and N ν = 2 . ± . α absorber by Pettini et al. (2008)lead to a deuterium abundance very close to the mean of the previous six abundances used in this paper. As a result, the changein y D is very small ( y D = 2 .
70 rather than y D = 2 .
68 adopted in this paper). Consequently, the change in the parameters inferredfrom it (N ν and η ) is also very small.Future CMB experiments will improve the constraint on N ν by measuring the neutrino anisotropic stress more accurately.According to Bashinsky & Seljak (2004), PLANCK should determine N ν to an accuracy of σ ( N ν ) ∼ .
24 and CMBPOL, asatellite based polarization experiment, might improve it further to σ ( N ν ) ∼ .
09, independent of the BBN constraints. In thiscontext, we note that such tight constraints on N ν will be sensitive to the value of the He abundance adopted in the CMB analysis.To achieve these projected accuracies, it will no longer be sufficient to fix Y P in advance. Rather, Y P should be solved for inconcert with the other cosmological parameters. To this end, we point out that for η ≈
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