BIG BANG NUCLEOSYNTHESIS the Binding Energies of the First Four Light Nuclei, 2H, 3H, 3He and 4He Are 2.22 Mev, 6.92 Mev, 7.72 M

BIG BANG NUCLEOSYNTHESIS

I. EQUILIBRIUM ABUNDANCES

The binding energies of the ﬁrst four light nuclei, 2H, 3H, 3He and 4He are 2.22 MeV, 6.92 MeV, 7.72 MeV and 28.3 MeV, respectively (see Table I). As the universe cools below these temperatures, one expects these bound structures to form. The abundance of light elements, which are synthesized in the early universe, can be used to obtain important constraints on the cosmological parameters. Though the energy considerations suggest that these nuclei could be formed when the temperature of the universe is in the range (1 30) MeV, the actual synthesis takes place − only at a much lower temperature, T 0.1 MeV. The main reason for this delay is the BBN ≃ ‘high entropy’ of our universe, i.e., the high value for the photon-to-baryon ratio, η−1. Let us provisionally assume that the nuclear (and other) reactions are fast enough to maintain thermal equilibrium between various species of particles and nuclei. In thermal A equilibrium, the number density of a nuclear species NZ with atomic mass A and charge Z will be 3/2 mAT µA mA nA = gA exp − . (1) 2π T In particular, the equilibrium number densities of protons and neutrons are

3/2 mBT µp mp np 2 exp − , (2) ≃ 2π T 3/2 mBT µn mn nn 2 exp − . (3) ≃ 2π T The mass diﬀerence between the proton and the neutron Q m m = 1.293 MeV has ≡ n − p 3/2 to be retained in the exponent but can be ignored in the prefactor mA . We have set in the prefactor m m m , an average value. n ≃ p ≃ B A Since the chemical potential is conserved in the reactions producing NZ out of Z protons and (A Z) neutrons, µ for any species can be expressed in terms of µ and µ : − A p n µ = Zµ +(A Z)µ . (4) A p − n Writing µ m Zµ +(A Z)µ m exp A − A = exp p − n exp A T " T # − T

1 − = [exp(µ /T )]Z [exp(µ /T )](A Z) exp( m /T ) (5) p n − A

and substituting for exp(µp/T ) and exp(µn/T ) from (2) and (3) we get

3A/2 µA mA −A Z (A−Z) 2π Zmp +(A Z)mn mA exp − = 2 np nn exp − − T mBT " T # 3A/2 −A Z (A−Z) 2π = 2 np nn exp(BA/T ), (6) mBT where B Zm +(A Z)m m (7) A ≡ p − n − A is the binding energy of the nucleus. Therefore, the number density (1) becomes

3(A−1)/2 −A 3/2 2π Z (A−Z) nA = gA2 A np nn exp(BA/T ). (8) mBT Since particle number densities in the expanding universe decrease as a−3 (for constant

number per comoving volume), it is useful to use the total baryon density, nb = nn + np +

i(AnA)i, as a ﬁducial quantity and to consider the ‘mass fraction’ contributed by nuclear Pspecies A(Z), AnA XA . (9) ≡ nb −1 Note that i Xi = 1. Substituting for nA, np and nn in (8) by nA = nb(A XA) = −1 ηnγ(A XAP), np = ηnγ Xp and nn = ηnγXn, where

nb −8 2 η =2.68 10 (Ωbh ) (10) ≡ nγ ×

2 3 is the baryon-to-photon ratio and nγ = [2ζ(3)/π ]T0 is the number density of photons, we get 3(A−1)/2 A−1 Z A−Z XA = F (A)(T/mB) η Xp Xn exp(BA/T ) (11) where 5/2 A−1 (1−A)/2 (3A−5)/2 F (A)= gAA [ζ(3) π 2 ]. (12)

Eq. (11) shows why the high entropy of the universe, i.e. small value of η, hinders the formation of nuclei. (For purposes of comparison, in a star like our sun, n /n 10−2; γ b ∼ even in the post-collapse core of a supernova, nγ /nb is only a few. Indeed, the entropy of the univesre is enormous.) To get X 1, it is not enough that the univesre cools to the A ≃ temperature T < BA; it is necessary that is cools still further so as to oﬀset the small value ∼ 2 A−1 of the η factor. The temperature TA at which the mass fraction of a particular species A will be of order unity (X 1) is given by A ≃ BA/(A 1) TA − . (13) ≃ ln(1/η)+1.5 ln(mB/T )

2 3 4 This temperature is much smaller than BA; for H, He and He the value of TA is 0.07 MeV, 0.11 MeV and 0.28 MeV, respectively. Comparison with the binding energy of these

nuclei shows that these values are lower than BA by a factor of about 10, at least.

II. FALLING OUT OF EQUILIBRIUM

Thus, even when the thermal equilibrium is maintained, significant synthesis of nuclei can occur only at T < 0.3 MeV and not at higher temperatures. If such is the case, then we would ∼ expect significant production (XA 1) of nuclear species A at temperatures T < TA. It ≃ ∼ turns out, however, that the rate of nuclear reactions is not high enough to maintain thermal equilibrium between various species. We have to determine the temperatures up to which thermal equilibrium can be maintained and redo the calculations to find non-equilibrium mass fractions.

In particular, we used the equilibrium densities for np and nn in the above analysis. In thermal equilibrium, the interconversion between n and p is possible through weak interaction processes:

n p + e− +¯ν , ↔ e ν + n p + e−, e ↔ e+ + n p +¯ν . (14) ↔ e When the rates for these interactions are rapid enough compared to the expansion rate H, chemical equilibrium obtains, µ µ = µ µ , (15) n − p e − ν from which it follows that in chemical equilibrium

nn Xn = = exp[ Q/T +(µe µν)/T ]. (16) np Xp − − Based upon the charge neutrality of the universe, we can infer that µ /T n /n = n /n e ∼ e γ p γ ∼ η, from which it follows that µ /T 10−10. The electron-neutrino number of the universe is e ∼ 3 similarly related to µν/T ; however, since this relic background has not been detected, none of the neutrino numbers is known. We will assume that the lepton numbers, like the baryon numbers, are small, so that µ /T 1. With this assumption, ν ≪ n n = exp( Q/T ). (17) np !EQ − The ratio will be maintained as long as the n p reactions are rapid enough. But when the − reaction rate Γ falls below the expansion rate, H =1.66g1/2T 2/m 5.5(T 2/m ) at some Pl ≃ Pl temperature T , the ratio (n /n ) will get frozen at the value exp( Q/T ). The only process D n p − D which can continue to change this ratio thereafter will be the beta decay, n p + e +¯ν, of → the free neutron. The neutron decay will continue to decrease the ratio until all neutrons are used up in forming bound nuclei.

To determine TD we have to estimate Γ and use the condition Γ = H. The rate Γ can be calculated from the theory of the weak interactions. The rates are found by integrating the square of the matrix element for a given process, weighted by the available phase- space densities of particles (other than the initial nucleon), while enforcing four-momentum conservation. One obtains

∞ − 2 Γ(nνe pe ) = A dqν qν qe Ee (1 fe) fν , Ee = Eν + Q, → 0 − Z ∞ + 2 Γ(ne pν¯e) = A dqe qe qν Eν (1 fν ) fe , Eν = Ee + Q, → 0 − Z q0 − 2 2 2 Γ(n pe ν¯e) = A dqe qe qν Eν (1 fν) (1 fe) , q0 = Q me. (18) → 0 − − − Z q

Here A is an eﬀective coupling while fν and fe are the distribution functions for electrons and

neutrinos. Although the weak interaction coupling GF is known quite accurately from muon decay, the value of A or, equivalently, the neutron lifetime, cannot be directly determined from this alone because neutrons and protons also interact strongly, hence the ratio of

nucleonic axial vector (GA) and vector (GV ) couplings is altered from unity. Moreover, relating these couplings to the corresponding experimentally measured couplings of the u and d quarks is complicated by weak isospin violating effects. If we assume conservation of the weak vector current (CVC), then G = G cos θ where sin θ =0.22 = V . However, V F c c | us| the weak axial current is not conserved and GA for nucleons differs from that for the first generation quarks. These non-perturbative effects cannot be calculated reliably, hence GA

(in practice, GA/GV ) must be measured experimentally. The neutron lifetime is then given

4 by 5 2 −1 me 2 GA τn = 3 GV 1+ 2 f, (19) 2π GV ! where f = 1.715 is the integral over the ﬁnal state phase space (including Coulomb corrections) and G is usually determined from superallowed 0+ 0+ pure Fermi decays of V → suitable light nuclei. It is thus more reliable to measure the neutron lifetime directly, and then relate it to the coupling A in (18) in order to obtain the other reaction rates. The experimental value is τ = 885.7 0.8 sec. (In the literature one often ﬁnds the neutron half n ± life τ1/2(n)= τn ln 2.) Thus we obtain, for example,

τ −1(T/m )3 exp( Q/T ) T Q, m , n e − ≪ e Γ(pe νn) 7π(1+3g2 ) (20) → ≃  A G2 T 5 T Q, m .  60 F ≫ e  Clearly τnΓ is a function of temperature alone. Similar calculations can be performed for all other reactions allowing us to compute τ Γ(n p) and τ Γ(p n) as functions of T . n → n → At high temperatures, T Q 1.3 MeV, both rates vary as T 5; in the range T (0.1 1) ≫ ≃ ≃ − MeV, the rate Γ(n p) T 4.42 while Γ(p n) decreases faster; for T < 0.1 MeV, → ∝ → Γ(n p) τ −1 is essentially dominated by the neutron decay while Γ(p n) drops → ≃ n → exponentially. These rates have to be compared to the Hubble factor, H(T ) 5.5(T 2/m ) ≃ Pl ≃ 2 −1 0.179(T/me) s : 3 Γ/H (T/0.8 MeV ) for T > me. (21) ∼ ∼ Thus thermal equilibrium between neutrons and protons exists only for T > TD 0.8 MeV. ∼ ≃ At TD, when the assumptions of thermal equilibrium becomes invalid, the n/p ratio will be X Q 1 n = exp , (22) Xp −TD ≃ 6 giving X (1/7), X (6/7). Since T calculated in (13) is lower than T for all the n ≃ p ≃ A D light nuclei, none of the light nuclei will exist in signiﬁcant quantities at this temperature T . For example, 2H contributes only a mass fraction X 10−12 at T . D 2 ≃ D To summarize: the time t 1 s, when T 1 MeV is a very interesing time in the ≃ ≃ history of the universe. Shortly before this epoch, the three neutrino species decouple from the plasma. A little later, the e± pairs annihilate, transferring their entropy to the 1/3 photons alone and thereby rising Tγ relative to Tν by a factor of (11/4) . At about this time the weak interactions that interconvert neutrons and protons freeze out, with

5 (n /n ) = exp( Q/T ) (1/6). After freeze-out, the neutron-to-proton ratio does not n p D − D ≃ remain truly constant, but slowly decreases due to occasional weak interactions (eventually dominated by free neutron decays). The light elements are still in statistical equilibrium with very small abundances (see Table I).

III. SYNTHESIS OF THE LIGHT ELEMENTS

As the temperature falls further to T = T 0.28 MeV, a signiﬁcant amount of 4He could He ≃ have been produced if nuclear rates were high enough. These reactions [D(D,n) 3He(D,p) 4He; D(D,p) 3H(D,n) 4He; D(D,γ) 4He] are all based on D, 3He and 3H and do not occur −12 rapidly enough because the mass fractions are still too small at T 0.3 MeV: X2 10 , ≃ H ∼ −19 −19 X3 10 , and X3 5 10 . He ∼ H ∼ × These abundances become nearly unity at T < 0.1 MeV; therefore, only at T < 0.1 ∼ ∼ MeV can these reactions be fast enough to produce an equilibrium abundance of 4He. (The delay from 0.3 to 0.1 MeV is known as ‘the deuterium bottleneck’.) When the temperature becomes T < 0.1 MeV, the abundance of 2H and 3H builds up and these elements further ∼ react to form 4He. A good fraction of 2H and 3H is converted to 4He. The resultant abundance of of 4He can be easily calculated by assuming that almost all neutrons end up 4 4 in He. Since each He nucleus has two neutrons, (nn/2) helium nuclei can be formed (per 4 unit volume) if the number density of neutrons is nn. Thus the mass fraction of He will be

4n4 4(nn/2) 2(nn/np)NS Y X4He = = = . (23) ≡ nb np + nn 1+(nn/np)NS

The ratio (nn/np) at the time of ‘freeze-out’ (tD) was (1/6); from tD till the time of nucleosynthesis (tNS) a certain fraction of neutrons would have decayed, lowering this ratio. Since the freeze-out occured at T 0.8 MeV, t 1 s and 4He synthesis occured at T 0.1 D ≃ D ≃ NS ≃ MeV, t 3 min, the decay factor will be exp( t /τ ) 0.8. Therefore, we obtain NS ≃ − NS n ≃

(n /n ) 0.8 (1/6) (1/7) = Y 0.25. (24) n p NS ≃ × ≃ ⇒ ≃

As the reactions converting 2H and 3H to 4He proceed, the number density of 2H and 3H is depleted and the reaction rates, which are proportional to Γ X (ηn ) σv , becomes ∝ A γ h i small. These reactions soon freeze-out leaving a residual fraction of 2H and 3H (a fraction of about 10−5 10−4). Since Γ η it is clear that the fraction of (2H, 3H) left unreacted − ∝

6 will decrease with η. In contrast, 4He synthesis, which is not limited by any reaction rate, is fairly independent of η and depends only on the (n/p) ratio at T 0.1 MeV. ≃ The production of still heavier elements, even those like 16C and 16O which have higher binding energies than 4He, is highly suppressed in the early universe. Two factors are responsible for this suppression: (i) Direct reactions between two He nuclei or between H and He will lead to nuclei with atomic masses 8 or 5. Since there are no tightly bound isotopes with masses 8 or 5, these reactions do not lead to any further synthesis. (The three body interaction, 4He + 4He + 4He 12C is suppressed by the low number density of 4He nuclei; it is this ‘triple-α’ reaction → which helps further synthesis in stellar interiors.) (ii) For nuclear reactions to proceed, the participating nuclei must overcome their Coulomb repulsion. The probability to tunnel through the Coulomb barrier is governed by the factor F = exp 2A¯1/3(Z Z )2/3(T/1 MeV )−1/3 (25) − 1 2 h i where A¯ = A1A2/(A1 + A2). For heavier nuclei (with larger Z), this factor suppresses the reaction rate. Small amounts (about 10−10 10−9 by mass) of 7Li are produced by 4He(3H,n)7Li or − by 4He(3He,γ)7Be followed by the decay of 7Be to 7Li. The ﬁrst process dominates if η < ∼ 3 10−10 and the second process for η > 3 10−10. In the second case, a small amount × ∼ × (10−11) of 7Be is left as a residue. A good approximation to the BBN predictions is the following:

4 η Yp( He) = 0.245+0.01 ln −10 , 5 10 × −1.6 D −5±0.06 η = 3.6 10 −10 , H p × 5 10 3 × −0.63 He −5±0.06 η = 1.2 10 − , (26) H ! × 5 10 10 p × 7 −2.38 2.38 Li −11±0.2 η η = 1.2 10 − + 21.7 − . H ! × " 5 10 10 5 10 10 # p × ×

7 IV. FROM OBSERVATIONS TO PRIMORDIAL ABUNDANCES

A. Introduction

To test the SBBN it is necessary to confront predictions of BBN with the primordial abundances of light nuclides which are not ‘observed’, but are inferred from observations. The path from observational data to primordial abundances is long and twisted and often fraught with peril. In addition to the usual statistical and insidious systematic uncertainties, it is necessary to forge the conncetion between the derived abundances and their primordial values. It is fortunate that each of the key elements is observed in different astrophysical sites using very different astronomical techniques and that the corrections for chemical evolution differ and, even more important, can be minimized. For example, deuterium is mainly observed in cool, neutral gas (HI regions) via resonant UV absorption from the ground state (Lyman series), while radio telescopes allow helium-3 to be studied via the analog of the 21 cm line for 3He+ in regions of hot, ionized gas (HII regions). Helium-4 is probed via emission from its optical recombination lines in HII regions. In cotrast, lithium is observed in the absorption spectra of hot, low-mass halo stars. With such different sites, with the mix of absorption/emission, and with the variety of telescopes involved, the possibility of correlated errors biasing the comparison with the prediction of BBN is unlikely. This favorable situation extends to the obligatory evolutionary corrections. For example, although until recently observations of D were limited to the solar system and the Galaxy, mandating uncertain corrections to infer the pregalactic abundance, the Keck and Hubble Space telescopes have opened the window to deuterium in high redshift, low metallicity, nearly primordial regions (Lyman-α clouds). Observations of 4He in low-metallicity ( 1/50 ∼ of solar) extragalactic HII regions permit the evolutionary correction to be reduced to the level of the statistical uncertainties. The abundances of lithium inferred from observations of the very metal-poor halo stars ( 1/1000 of solar and even lower) require almost no ∼ correction for chemical evolution. On the other hand, the status of 3He is in contrast to that of the other light elements. Although all prestellar D is converted to 3He during pre-main sequence evolution, 3He is burned to 4He and beyond in the hotter interiors of most stars, while it survives the cooler exteriors. For lower mass stars, a greater fraction of the prestellar 3He is expected to survive

8 and, indeed, incomplete burning leads to buildup of 3He in the interior which may, or may not, survive to be returned to the interstellar medium. In fact, some planetary nebulae have been observed to be highly enriched in 3He, with abundances 3He/H 10−3. Although ∼ such high abundances are expected in the remnants of low mass stars, if all stars in the low mass range produced comparable abundances, we would expect solar and present ISM abundances of 3He to greatly exceed their observed values. It is therefore necessary that at least some low mass stars are net destroyers of 3He. For example, there could be extra mixing below the convection zone in these stars when they are on the red giant branch. Given such complicated histories of survival, destruction and production, the use of current Galactic and solar system data to infer the primordial abundance of 3He should be taken with caution.

B. Observations

1. D

Observations of deuterium in the solar system and in the interstellar medium (ISM) of the Galaxy provide interesting lower bounds to its primordial abundance. Measuring the primeval deuterium - rather than ﬁnding a lower bound - became possible only in the last few years. The idea is that deuterium abundance in a high redshift hydrogen cloud can be measured. Distant hydrogen clouds are observed in absorption against even more distant quasars. Many absorption features are seen - the Lyman series of hydrogen and the lines of various ionization states of carbon, oxygen, silicon, magnesium and other elements. Because of the large hydrogen abundabce, Ly-α is very prominent. In the rest frame, Ly-α occurs at

1216 A,˚ so that for a cloud at redshift z, Ly-α is seen at 1216(1+ zcloud)A.˚ The isotopic shift for deuterium is 0.33(1+ z)A,˚ or expressed as a Doppler velocity, 82 km s−1. The idea is − − to detect the deuterium Ly-α feature in the wing of the hydrogen feature. For z > 3, Ly-α ∼ is shifted into the visible part of the spectrum and thus can be observed from Earth; “Ly-α clouds” are ubiquitous with hundreds being seen along the line of sight to a quasar of this redshift, and judged by their metal abundance, anywhere from 10−2 of that seen in the solar system to undetectably small levels, these clouds represent nearly pristine samples of cosmic material. There are technical challenges: Because the expected deuterium abundance is

9 −5 −4 17 −2 small, D/H 10 10 , clouds of very high column density, nH > 10 cm , are needed; ∼ − ∼ because hydrogen clouds are ubiquitous, the probability of another, low coloumn-density cloud sitting in just the right place to mimic deuterium - an interloper - is not negligible; many clouds have broad absorption features due to large internal velocities or complex velocity structure; and to ensure suﬃcient signal-to-noise bright QSOs and large-aperture telescopes are a must. At present (2007) we have six high redshift, low metallicity, QSO absorption line systems with deuterium detections leading to reasonably reliable abundance determinations. There is excessive dispersion among these determinations (central values vary between 1.6 and 8 in units of 10−5), suggesting that systematic errors, whose magnitudes are hard to estimate, may have contaminated the determinations of at least some of the D and/or H column densities. This dispersion serves to mask the anticipated primordial deuterium plateau. The weighted average of these 6 best detections in high redshift Ly-α gives

+0.27 −5 (D/H) = (2.68− ) 10 . (27) p 0.25 ×

2. D + 3He

Chemical evolution issues have been woven into the study of BBN from the start. In order to extrapolate contemporary abundances to primordial abundances the use of stellar and Galactic chemical evolution models is unavoidable. The difficulties are well illustrated by 3He: generally the idea that the sum of D+3He is constant or slowly increasing seems to be true, but the details, e.g., predicted increase during the last few Gyr, are inconsistent with a measurement of 3He in the local ISM. Beginning with deuterium, the assumed primeval abundance of Eq. (27) is a factor of two larger than the present ISM abundance, D/H=(1.5 0.2) 10−5, determined by the Hubble ± × Space Telescope observations. This implies (i) little nuclear processing over the history of the Galaxy; and/or (ii) significant infall of primordial material into the disk of the Galaxy. The metal composition of the Galaxy, which indicates significant processing through stars, together with the suggestion that even more metals may have been made and ejected into the IGM, means that option (i) is less likely than option (ii). Even more intriguing is the fact that the inferred abundance of deuterium in the pre-solar nebula, D/H=(2.6 0.4) 10−5, ± × indicates less processing in the first 10 Gyr of Galactic history than in the past 5 Gyr.

10 Moving on to 3He, the primeval value corresponding to our assumed abundance is 3He/H ≃ 10−5. The pre-solar value, measured in meteorites and more recently in the outer layer of Jupiter, is 3He/H=(1.2 0.2) 10−5, comparable to the primeval value. The value in the ± × present ISM, 3He/H=(2.2 0.8) 10−5, is about twice as large as the primeval value. On ± × the other hand, the primeval sum of deuterium and 3He,

(D + 3He)/H=(3.7 1.0) 10−5, (28) ± × is essentially equal to that determined in the pre-solar nebula and for the present ISM. This indicates little net 3He production beyond the burning of deuterium to 3He, in conﬂict with conventional models for the evolution of 3He which predict a signiﬁcant increase in D +3 He due to 3He production by low-mass stars.

3. 7Li

For the primordial 7Li abundance, one has to correct the data from measurements in the atmospheres of old halo stars by including empirical corrections for cosmic ray production, stellar depletion: 7 +0.35 −10 ( Li/H) = (1.2− ) 10 . (29) 0.20 ×

4. 4He

Helium-4 plays a diﬀerent role and presents diﬀerent challenges. First, the primeval yield of 4He is relatively insensitive to the baryon density. Second, the chemical evolution of 4He is straightforward; the abundance of 4He increases due to stellar production: As gas cycles through generations of stars, hydrogen is burned to helium-4 (and beyond), increasing the 4 4 He abundance above its primordial value. As a result, the present He mass fraction, Y0,

has received a signiﬁcant contribution from post-BBN, stellar nucleosynthesis, and Y0 >YP . However, since the “metals” such as oxygen are produced by short-lived, massive stars, and 4He is synthesized (to a greater or lesser extent) by stars of all masses, at very low metallicity the increase in Y shoould lag that in, e.g., O/H, so that as O/H 0, Y Y . → → P 4 Therefore, although He is observed in the Sun and in Galactic HII regions, the crucial data for inferring the primordial abundance of 4He comes from measurements of He/H in regions

11 of hot, ionized gas (HII regions) found in metal-poor, dwarf emission-line galaxies. One takes a sample and extrapolates to zero metallicity. The present (2007) inventory of such regions studied for their helium content exceeds 80. Thus, statistics is no longer a problem, but systemetic corrections and/or errors become the limiting factor. Indeed, many eﬀects have to be considered to achieve the desired accuracy: corrections for doubly ionized 4He and neutral 4He have to be made; absorption by dust and by stars have to be accounted for; collisional excitation must be accounted for; potential systematic errors in the input atomic physics; and extrapolation to zero metallicity must be made in the absence of a well motivated model. A recent analysis gives Y =0.238 0.005. (30) p ±

V. THE BARYON-TO-PHOTON RATIO

BBN and light element abundances can be used to ﬁx the baryon-to-photon ratio at the end of BBN (t 200 sec). Using the above values one obtains ∼

η = (5.6 0.4) 10−10. (31) ± ×

This value is driven almost entirely by deuterium. The data may hint that there has been more depletion of 7Li in old halo stars than the conventional models suggest, by a factor of about three. On the other hand, the data support the simple hypothesis of only pre-MS 3He production. 2 To relate η to the present baryon density, Ωbh , one needs to know the present photon temperature, T =2.725 0.001 K, (32) 0 ± and the average mass per baryon number,

m¯ =1.6700 10−24 g. (33) ×

With the assumption that the expansion has been adiabatic since BBN, we obtain

Ω h2 = (3.650 0.004) 107η. (34) b ± ×

12 Then the BBN determination of η, Eq. (31), translates to

Ω h2 =0.0204 0.0015 (95% CL). (35) b ±

For h =0.70 0.07, this implies a baryon fraction ±

Ω =0.041 0.009, (36) b ±

with the error dominated by that in H0. Measurements of the cosmic microwave background (CMB) anisotropy have recently also determined the baryon density. The physics underlying the method is very diﬀerent - gravity- 2 driven acoustic oscillations in the Universe at 500,000 yrs - but the result is similar, Ωbh = 0.0223 0.0008. ± The BBN measurement of η, now conﬁrmed by CMB, requires nonbaryonic dark matter with Ω h2 0.12. dm ∼

VI. BEYOND SBBN

A. Neutrino Degeneracy

We now consider the possible role of neutrino degeneracy. A chemical potential in electron neutrinos can alter neutron-proton equilibrium, as well as increase the expansion rate, the latter effect being less important. The possible excess of electron neutrinos over antineutrinos is parametrized by a dimen- sionless chemical potential, ξ µ /T, (37) νe ≡ νe which remains constant for freely expanding neutrinos. Anticipating that ξνe will be constrained to be sufficiently small, we neglect the slight increase in expansion rate due to the increased energy density of the neutrinos and consider only the effect on neutron-proton interconversions. (We do not consider a chemical potential for other neutrino types, which would only add to the energy density without affecting the weak reactions.) Consequently, only the abundance of 4He is significantly affected and the allowed range of η is still determined by the adopted primordial abundances of the other elements.

13 The resultant increase in the rate of nν pe− alters the detailed balance equation to e → Q Γp→n =Γn→p exp ξνe , (38) "−T (t) − #

and, hence, lowers the equilibrium neutron abundance to

−1 Xeq(t, ξ )= 1+ e(y+ξνe ) , y Q/T (t). (39) n νe ≡ h i This alters the asymptotic neutron abundance by the same factor,

− X (ξ ,y )= e ξνe X (y ). (40) n νe →∞ n →∞

It is now easy to see that the synthesized helium mass fraction depends on ξνe as follows:

− − Y (4He) 2X (t ) 2X (y )e tNS/τn e ξνe p ≃ n NS ≃ n →∞ − = Y ξνe =0(4He)e ξνe Y ξνe =0(4He)(1 ξ ) p ≃ p − νe 0.245 0.25ξ . (41) ≈ − νe

Imposing 0.228 Y 0.248 requires ≤ p ≤

0.012 < ξνe < 0.068. (42) − ∼ ∼ (For orientation, a value of ξ µ /T √2 is equivalent to adding an additional neutrino ν ≡ ν ≈ ﬂavor (with ξν = 0) which we consider in the next subsection.)

B. Constraints on New, Light Particles

The number of relativistic degrees of freedom at the epoch of nucleosynthesis can be constrained. The relevant eﬀects are related to the dependence of the expansion rate, H, and time, t on the number of relativistic degrees of freedom:

2 8πρr 1/2 T H = 2 =1.66 g , (43) s3mPl mPl and, using H2(t)=1/(4t2) in the RD epoch,

−2 −1/2 mPl T −1/2 t 0.3g 2 1 sec g . (44) ≃ T ∼ 1 MeV

14 The counting of the eﬀective number of degrees of freedom goes through the following summation over relativistic species:

T 4 7 T 4 g = g i + g i . (45) i T 8 i T bosonsX fermionsX In the Standard Model, the relativistic degrees of freedom at T 1 MeV include the photons ∼ (gγ = 2), electrons and positrons (ge = 4) and three neutrino generations (gν = 2), all with the same temperature. Thus

7 g (T 1 MeV )=2+ (4+3 2) = 10.75. (46) SBBN ∼ 8 ×

At T < me, the relativistic degrees of freedom include the photons and the three neutrino ∼ 1/3 generations with lower temperature (Tν/T = (4/11) ):

7 4 4/3 gSBBN(T 0.1 MeV )=2+ 3 2 3.36. (47) ∼ 8 × × × 11 ≃ The freeze-out of the weak interactions that keep the (n/p) ratio is determined by the ratio between the weak interaction rate and the expansion rate. If g at that time is diﬀerent from the standard one, eq. (21) is modiﬁed:

1/2 3 10.75 Γ/H (T/0.8 MeV ) for T > me. (48) ∼ gh ! ∼

Thus the freeze-out temperature TD becomes larger with g:

T g 1/6 D = h . (49) 0.8 MeV 10.75 Consequently, the (n/p) ratio [see eq. (22)] becomes larger with g:

n Q Q 0.8 MeV/TD = exp = exp p !D −TD −0.8 MeV 1/6 [(10.75/gh) ] n SBBN = . (50)   p !D   Nucleosynthesis starts when the deuterium bottleneck is overcome, that is T 0.086 NS ≈ MeV. If g is diﬀerent at that time from the standard one, this temperature will be reached at earlier time: t 3.36 1/2 NS = . (51) 180 sec gl !

15 Then the depletion in the (n/p) ratio due to neutron decay will be weaker:

tNS/180 sec − 180 sec e tNS/τn = exp − τn (3.36/g )1/2 − SBBN [ l ] = e tNS/τn . (52) Combining (50) and (52), we learn that eq. (24) is modiﬁed as follows:

1/2 1/6 (n /n ) 0.8(3.36/gl) 0.15(10.75/gh) . (53) n p NS ≃ ×

C. Additional Neutrino Generations

Let us consider the eﬀect of additional neutrino generations, ∆N = N 3: ν ν −

gh = 10.75+1.75∆Nν,

gl = 3.36+0.46∆Nν. (54)

A single additional neutrino generation leads to an increase of about 6% in the ratio (n/p)NS, leading to a similar increase in the helium abundance. More accurate calculations give

4 Yp( He) = 0.245+0.013∆Nν. (55)

4 Imposing, for example, Yp( He) < 0.25, we obtain

∆N N 3 < 0.4. (56) ν ≡ ν −

In particular, nucleosynthesis excludes the possibility of additional massless (or, more pre- cisely, signiﬁcantly lighter than 1 MeV) active neutrino generations. At present, the measurement of the decay width of the Z0 boson into neutrinos makes

the existence of three, and only three, light (that is, mν < mZ /2) active neutrinos an ∼ experimental fact. When expressed in units of the SM prediction for a single neutrino generation, one gets

N = 2.994 0.012 (Standard Model ﬁts to LEP data), ν ± N = 3.00 0.06 (Direct measurement of invisible Z width). (57) ν ±

16 D. Superweakly Interacting Particles

While the BBN bound on the number of active neutrino generations is, at present, weaker than the bound from laboratory measurements, the bound on the number of relativistic degrees of freedom at the time of BBN is more generic than that and applies in an interesting way to many other extensions of the Standard Model. The interaction rate keeping a superweakly interacting particle in thermal equilibrium will typically fall behind the Hubble expansion rate at much larger decoupling temperature than the value of a few MeV for active neutrinos. If the comoving entropy increases afterwards due to annihilation of massive particles, the abundance of the new particle will be diluted relative to that of neutrinos or, equivalently, their temperature will be lowered relatively since neutrinos have the same temperature as that of photons down to T m . Hence ∼ e the energy density during nucleosynthesis of new massless (or, more generally, m MeV ) ≪ particles i is equivalent to an eﬀective number ∆Nν of additional doublet neutrinos: 4 gi Ti ∆Nν = fB,F , (58) 2 Tν Xi where fB =8/7 (bosons) and fF = 1 (fermions), and

T g (T ) 1/3 i = I , (59) Tγ "gI (TD)#

and gI is found by summing over particles that are in thermal equilibrium with the photons.

Thus the number of such particles allowed by a given bound on ∆Nν depends on how small Ti/Tν is, i.e. on the ratio of number of interacting relativistic degrees of freedom at

TD (when i decouples) to its value at a few MeV (when neutrinos decouple). Using table qh III we see that TD > mµ implies Ti/Tν < 0.910 while TD > T bounds Ti/Tν < 0.594. The EW smallest possible value of Ti/Tν in the SM is 0.465, for TD > T . For example, consider a new singlet fermion F (‘sterile neutrino’). Its decouping temperature can be approximately calculated by equating the interaction rate to the Hubble rate. For deﬁniteness, consider annihilation to leptons, and parametrize the cross section as

2 2 σv ¯↔ ¯ G T . (60) h iℓℓ F F ≡ X Them equating the annihilation rate Γeq = n σv to the expansion rate H gives ann F h i (T ) G 2/3 g[(T ) ] 1/6 D F = F D F . (61) (TD)ν GX g[(TD)ν] !

17 Taking into account that G =1.166 10−5 GeV −2, (T ) 2 MeV and g[(T ) ]=10.75, F × D ν ∼ D ν we ﬁnd that

−7 −2 (TD)F > mµ GX < 3 10 GeV , ∼ ⇔ ∼ × −8 −2 (TD)F > 0.3 GeV GX < 10 GeV . (62) ∼ ⇔ ∼ The energy density of the new particle during nucleosynthesis is then equivalent to, respectively, ∆Nν =0.69 and ∆Nν =0.12. Thus, if we impose, for example, ∆Nν < 0.4, we must require that the interaction strength of the new particle, GX , is weaker than a few times 10−8 GeV −2. Taking G g2/m2 , we ﬁnd that the mass scale of new intermediate vector X ∼ X bosons should be mX > 10 T eV . ∼

E. Sterile Neutrinos

We now consider extra fermions that have no new gauge interactions. Their only interactions with Standard Model particles are through their Yukawa couplings to the active neutrinos. The consequence of that is that the neutrino mass eigenstates are mixtures of active and sterile interaction eignestates. Consider for simplicity a single active neutrino νa and a single sterile neutrino νs. Then, the two mass eigenstates, ν1 and ν2, are related to these states through a unitary mixing matrix:

ν1 cos θ sin θ νa = . (63) ν ! sin θ cos θ ! ν ! 2 − s We also deﬁne the mass-squared diﬀerence,

∆m2 = m2 m2. (64) 2 − 1

Because interaction eigenstates are not eigenstates of free propagation, an initial νa state could oscillate into a νs state. The transition probability is a function of the mixing angle sin θ, of the mass-squared diﬀerence ∆m2, of the energy E and distance traveled L: 2 2 2 ∆m L P (νa νs) = sin 2θ sin . (65) → 4E ! In dense plasma of the early universe, the interactions of the active neutrino with the plasma change this transition probability. The production of sterile neutrinos by oscillations is given by Γ sin2 θ Γ f , (66) s ∝ m 0 eq 18 TABLE I: The binding energies of light nuclei. AZ B (MeV) T (MeV) g X (T Q) X (T = T ) A A A A ≫ A D 2H 2.22 0.07 3 6 10−12 10−12 × 3H 6.92 2 3He 7.72 0.11 2 2 10−23 10−23 × 4He 28.3 0.28 1 2 10−34 10−28 × 12C 92.2 1 2 10−126 10−108 ×

TABLE II: Light element abundances - SBBN predictions and ‘observed’ values (ˆη η/5 10−10). ≡ × Predictions Observations

(D/H) 3.6 10−5±0.06 ηˆ−1.6 (2.7 0.3) 10−5 × ± × (3He/H) 1.2 10−5±0.06 ηˆ−0.63 (3.7 1.0) 10−5 (D/H) × ± × − 7 −11±0.2 −2.38 2.38 +0.35 −10 ( Li/H) 1.2 10 ηˆ + 21.7η ˆ (1.2− ) 10 × 0.20 × Y(4He) 0.245 + 0.01 lnη ˆ 0.238 0.010 ±

where sin 2θ tan2θm = 2 , (67) cos2θ + (2EVeff /∆m ) where Veff is the effective potential due to forward scattering in matter, and

180ζ(3)g Γ = a G2 T 4p. (68) 0 7π4 F

Sterile neutrinos would never be abundant in the universe if Γs/H < 1. But what we have learnt above is that can impose a stronger condition: the energy density should be smaller than the energy density of, say, one light neutrino species at BBN (T 1 MeV). ∼ This leads to a bound of the form

∆m2 sin2 2θ < 2.3 105 eV 2. (69) ∼ ×

19 TABLE III: Thermodynamic history of the RD era.

T Threshold (GeV) Particle Content gI (T ) Nγ (T0)/Nγ (T )