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PHYS3170 CosmologyBig Bang Nucleosynthesis
John Webb,University of New South Wales
PRELIMINARIES 1
Detailed big bang nucleosynthesis (BBN) calculations weremade in the mid-1960s. Stellar nucleosynthesis was alreadyunderstood but BBN research became more active afterPenzias and Wilson discovered the CMB in 1965.
The theoretical abundance calculations (over ∼10 orders ofmagnitude!) successfully predicted subsequent accurateobservations. BBN is thus justifiably regarded today as one ofthe pillars of the big bang model.
BBN theory is now very well established.
PRELIMINARIES 2
Assumptions made in SBBN (standard big bangnucleosynthesis):
1. There was a big bang.2. “Standard” physics applies, with present-day values of thefundamental constants.3. The universe had a slight matter-antimatter asymmetry.4. The universe was in thermal equilibrium.5. General relativity + isotropy and homogeneity applies.6. There was no electron or neutrino degenerecy.
BBN SUCCESSES
I Explains the observed light element abundances (well, butnot yet perfectly), over ∼10 orders of magnitude inabundance;
I Predicts < 4 ν types, probably 3 (confirmed by CERNmeasurements of the Z0 boson lifetime).
I BBN predicted, before reliable measurements, the neutronhalf-life < 10.4 mins (current-day value is 10.28 mins).
GENESIS 1
The universe is quite “simple” in the early stages,
a(t) ∝ t1/2 (1)
The photons cool as the universe expands,
T ∝ 1/a(t) (2)
i.e. the radiation temperature cooled as time proceeded,
T (t) ≈ 1010t−1/2K (3)
where t is in seconds. In terms of energy, this is,
kT (t) ≈ t−1/2MeV (4)
where k is the Boltzmann constant.
GENESIS 2
When the Universe was less than a second old, it was VERYhot, T > 1 MeV (1010)K, and dominated by radiation.
All particles were in thermodynamic equilibrium.
Matter and energy equivalence: energy can be turned intoparticles, and vice versa (if the conditions are right, and theywere in the early universe).
There are lots of reactions going on, two important ones being:pair production and annihilation.
PAIR PRODUCTION
Figure 1. Photons convert into electrons and positrons: twohigh energy photons collide, producing an electron/positronpair.
ANNIHILATION
Figure 1. Collisions between electrons and positrons turnedthem back into radiation.
MORE REACTIONS
If Ephoton < mc2, a particle with mass m can’t be created by pairproduction. At very early times, photon energies were high, soneutrons and protons were created before the lighter particles.
Neutrons are heavier than protons, and the rest-mass energydifference between a neutron and a proton is
mnc2 −mpc2 = 939.6− 938.3 = 1.3MeV (5)
When t 1sec,T 1MeV , so reactions like these couldtherefore proceed in both directions:
e− + p + 0.8MeV νe + n (6)
andνe + p + 1.8MeV e+ + n (7)
NEUTRON TO PROTON RATIOWhen neutrons bind into nuclei they are stable, but freeneutrons decay into protons, thalf = 881.5sec. The reactionsconverting neutrons to protons are equations 6 and 7.Once the universe has cooled sufficiently such that protons andneutrons are “non-relativistic” (i.e. their velocities are smallcompared to c, i.e. 1
2mv2 = kT mc2), the number density Nis given by the Maxwell-Boltzmann distribution
N ∝ m3/2 exp[−mc2
kT
](8)
Therefore the neutron to proton ratio is
Nn
Np=
(mn
mp
)3/2
exp[−
(mn −mp)c2
kT
](9)
(this is called the Saha equation).
NEUTRON FREEZE-OUT
In equation 9, (mn/mp)3/2 = 1.002, so providedkT (mn −mp)c2, the neutron to proton ratio will be ≈ 1, butas the temperature decreases, the ratio decreases.
However, equations 6 and 7 show the ratio can’t decreaseindefinitely. Once the ambient photon energy cools below0.8MeV , reaction 7 will already have ceased, reaction 6 slowsdown, and protons to neutron conversion grinds to a halt.
The neutron to proton ratio thus “freezes out” (at a very earlystage, before anything else ”interesting” happens) at
Nn
Np=
(mn
mp
)3/2
exp[−1.3MeV
0.8MeV
]≈ 0.2 (10)
MAKING HELIUM
For the next few minutes, a complex series of fusion reactionscarries on after neutron freeze-out. Some of these are:
n + p −→ D + γ (11)
D + p −→ 3He + γ (12)
D + D −→ 3He + n (13)3He + n −→ 4He + γ (14)
D + 3He −→ 4He + p (15)
Note these are nuclear reactions. Neutral atoms cannot existuntil much later. D is deuterium. He is helium. The γ’s arephotons. There are loads of other reactions too, so the above isa simplification. The reverse reactions also take place, butthose decline as the universe cools.
TIME-STEPPING THROUGH THE UNIVERSE
Figure 4. Reaction ratesdepend on particle density,temperature (obtained fromsolving the Friedmannequation) and particlecross-sections (from atomicphysics). Detailednumerical computations“time-step” through theuniverse, computingrelative abundances ateach instant, and eventualfinal yields. Figure shown isone calculation for onevalue of the particle density.
FINAL YIELDS
Figure 5. Important things to note:
1. Observations show approx 1/4 ofthe final particles produced are 4He.The other light elements have verysmall abundances.
2. Theoretical predictions span ∼10orders of magnitude.
3. 4He is relatively insensitive to ρm.
4. D is relatively sensitive to ρm.
5. 7Li is complex - multipleabundances for single ρm.
APPROXIMATING THE HELIUM ABUNDANCELet the total number density of particles be N = Np + Nn.
The parallel reactions and neutron decay result in a fraction ofneutrons ending up in 4He nuclei of Nn/Np ≈ 1/7.
All the neutrons end up in 4He (because hydrogen has none).
Each 4He nucleus contains 2n (and 2p) so the final 4Henumber density is Nn/2.
The total number density of particles in 4He nuclei is thus 4times that, i.e. 2Nn.
The mass fraction of 4He is thus
Y4 =2Nn
Np + Nn=
2Nn
Nn + 7Nn≈ 0.25
If our understanding of neutron freeze-out and the reactionprocesses is correct, this is then a firm prediction of BBN.
UNSTABLE NUCLEI MASS 5 AND 8
Fig from http://hyperphysics.phy-astr.gsu.edu/hbase/astro/mass5w.html
OBSERVATIONS - HELIUM
Two segments of thespectrum of the dwarfirregular galaxy NGC4861 containingWolf-Rayet emissionfeatures. Upper panelshows how thesefeatures interfere withthe measurement ofnebular He I 5876.
http://ned.ipac.caltech.edu/level5/March01/Dinerstein/Diner5.html
OBSERVATIONS - HELIUM
Emission lines in the spectrum of the H II galaxy UM461 taken withthe AAT. Narrower spikes are cosmic-rays landing on the detector.The spectral resolving power is about 2000.http://ned.ipac.caltech.edu/level5/Sept01/Pagel/Pagel4 4.html
OBSERVATIONS - HELIUM
Regressions of helium massfraction against oxygen andnitrogen abundance, respectively, inirregular and blue compact (or HII)galaxies with oxygen up to 1/4solar. Maximum-likelihoodregression lines are shownindicating ±1σ errors.Yp = 0.225(5) + 169(45)(O/H)Yp = 0.229(4) + 3310(940)(N/H)
Extrapolating to zero abundances(i.e. so that second terms above goto zero) gives primordial value.
http://ned.ipac.caltech.edu/level5/Sept01/Pagel/Pagel4 4.html
OBSERVATIONS - HELIUM
Helium abundance (massfractions, Y) from the sample ofextragalactic HII regions studiedby Izotov & Thuan (2010) as afunction of the correspondingoxygen abundances (O/H bynumber). The solid line is thebest fit to a linear Y versus O/Hcorrelation.
YP = 0.2565± 0.0010(stat)± 0.0050(syst)Review article by Gary Steigman:http://adsabs.harvard.edu/abs/2013APS..APR.R3003S
OBSERVATIONS - DEUTERIUM
c© J.K. Webb, 2013
BARYOGENESIS - MATTER-ANTIMATTER ASYMMETRY(AN UNSOLVED PROBLEM)
Early on, when the photon energy density was much largerthan the baryon rest mass, i.e. when kT mpc2, pairproduction and annihilation reactions like
γ + γ p + p
would be going on.
Observations tell us the photon/baryon ration at BBN was tiny,η ≈ 10−9 (see Figure 5), i.e. very few baryons survivedannihilation, i.e. the imbalance between protons andantiprotons was extremely small (but thankfully it was there!).
But do we really know that the whole universe is made of onlymatter?
NO ANTIMATTER IN THE SOLAR SYSTEM
The Earth:
No-one hasdiscovered anantiperson, yetImage credit: http://www.atlas.ch
The planets:
Curiosity landed onMars and didn’texplodeImage credit: NASA
The Sun:
Solar wind ejects ∼1014 kg/day.Severe Solar storm, Earth’satmosphere loses ∼105 kg. Ifthe Sun was antimatter, thesolar wind would create gammarays as it hit Earth.Image credit and facts:http://www.nasa.gov/mission pages/sunearth
RESIDUAL POCKETS OF PRIMORDIAL ANTIMATTER?
Potential causality problem - how did the antimatter separatefrom the matter rapidly enough to escape annihilation in the firstplace? Ignoring that minor difficulty ...
If matter/antimatter clumps existed prior to inflation, they couldbe further apart now than the size of the observable universe(but then we’d never see it anyway).
More interestingly, could they exist on galactic scales? Couldthere be antistars or even antigalaxies?
NO ANTISTARS OR ANTIGALAXIES
Other stars:
During its orbit through theGalaxy, an antistar wouldaccrete interstellar gas.Observations of discretegamma-ray sources limit thefraction of antistars in theGalaxy to be < 10−4.Steigman, Ann. Rev. Astr.&Astroph., 14,339, 1976. Image credit:http://www.futuretimeline.net
The Bullet cluster:
Formed from collision of 2 initially separate galaxies.X-ray astronomy reveals the mass of gas present.The Compton Satellite detected no gamma rays(expected if any significant amount of annihilationtook place). If one galaxy is matter, the otherantimatter, the antimatter fraction is < 3 × 10−6.Steigman 2008, http://arxiv.org/abs/0808.1122). X-ray and opticalimage credits: NASA.
RECENT MEASUREMENT: ANTIHELIUM/HELIUM < 10−7
Nature Physics, Vol 8, 2012, commenting on Abe et al, PRL, 2012
MATTER/ANTIMATTER IMBALANCE
Could the universe really be intrinsically “symmetric” (simplestpossible model) but somehow statistical fluctuations occurredto segregate the matter and antimatter, such that it’s out there,but we haven’t yet seen it?
It seems there’s no antimatter on scales less than a galaxy. Ifit’s there, it’s likely to be segregated on much larger scales.
Suppose there are anticlusters. A galaxy cluster mass isMclust ∼ 1012 − 1014MSun.Take lower range as an example. This corresponds to a mass1012 × 2× 1030 ∼ 1042 kg, so the number of particles isNclust ∼ 1042/mp ≈ 1069.To first order, a random fluctuation corresponds to√
Nclust ∼ 1034 particles, which is ridiculous. No knownmechanism can do that. Random fluctuations don’t work.
WHAT WAS IT LIKE AT BBN?
The density parameter in baryons today is
Ωb,0 =ρb,0
ρcrit ,0
where ρ is mass density. Now
ρb,0 ≈ mpnb,0 = mpηnγ,0
where mp is the proton mass, n is the particle density, and thesubscripts b and 0 indicate ”baryons” and ”today”.
Now we know η from light element abundances, and we canmeasure nγ,0 with telescopes. This turns out to give
nb,0 ≈ 0.23± 0.02 m−3
WHAT WAS IT LIKE AT BBN?
The critical density corresponds to a flat universe (k = 0) in theFriedmann equation, so that
ρ(t)crit =3H(t)2
8πG
or
ρcrit ,0 =3H2
08πG
≈ 9.2× 10−27 kg m−3
Putting this together we get
Ωb,0 =(0.23± 0.02)mp
9.2× 10−27 = 0.04± 0.01
WHAT WAS IT LIKE AT BBN?
We can scale this back to get the mass density at the BBNepoch by multiplying the ratio of the volumes, or, equivalently
ρb,BBN = Ωb,0 ρcrit
(TBBN
T0
)3
where T is the radiation temperature. This gives
ρb,BBN ≈ (0.04)(9× 10−27)
(109
2.73
)3
≈ 0.02 kg m−3
which is not all that different to the mean density of the Earth’satmosphere (∼ 0.006 kg m−3)!
