DERIVATION OF THE BIG BANG HELIUM ABUNDANCEIncluding an Analysis of The Cosmic Crunch-Bang, or
Squeeze-Boom Cosmologyby James A. Green, May 2002, updated April 22-30, May 6-27, & June 4, 2006, Oct 1, 2008 Elementary Derivation of a 30% Primordial Helium Abundance Upper LimitIn this revision I supply a simple derivation yielding an outer limit of 30% by weight for the primordial abundance of He-4 after the Big Bang. The neutron-to-proton abundance ratio of particle number densities at temp T is given by statistical mechanics and kinetic theory as N _{n}/N_{p} = exp( -((m_{n} -
m_{p})c^{2})/kT ),where m _{n} = 1.6749 x
10^{-27} Kg, and m_{p} = 1.6726 x 10^{-27} Kg are
the rest masses of the neutron and the proton. Now from the equipartition
theorem, the average energy of a particle at temp T isE_avg = (3/2)kT. If a neutron and a proton of this energy collide head-on, the total energy in the rest-frame is E_collision = 2E_avg = 3kT. Setting this collision energy equal to the binding energy of the deuteron (n,p), we find the temp T at which deuterium may begin to form in the cooling Big Bang fireball, such that E_collision = 3kT = 2.224 MeV = deuteron binding energy, and kT = 2.224 MeV/3. Dividing by k, we find T = 8.3645 x 10 ^{9} degrees Kelvin. Inserting our
result for kT into the abundance ratio for neutrons-to-protons given
above, N _{n}/N_{p} = exp(-(m_{n} -
m_{p})(c^{2})/kT) = 0.1755. Thus N _{n} =
0.1755 N_{p}, and N_{p} = (1/0.1755)N_{n} = 5.6338
N_{n}.Now N _{n} + N_{p} = N_{n} +
5.6338 N_{n} = 6.6338 N_{n}, so
thatN _{n}/(N_{n} + N_{p}) = 1/6.6338 =
0.150743.That is, 15.0743 percent by number of the nucleons formed in the Big Bang will be neutrons. These will combine with an equal number of protons to form deuterium, which will all combine in pairs to form He-4. Since the masses of neutrons and protons differ by less than a tenth of 1 percent, roughly 2(15.074) = 30.148 percent of the mass will go to deuterium and rapidly combine into He-4, so that the He-4 abundance will be roughly 30 percent by weight. Here we have neglected the fusion process mass defects, less than 1% of the original mass, so that a refined treatment based on the above observations yields 30.065477 percent He-4 by weight. Note that the masses of the proton and the neutron differ by less than 0.001: m _{p} = 1.6726 x 10^{-27} kg and
m_{n} = 1.6749 x 10^{-27} kg, so (m_{n} -
m_{p})/(m_{n} + m_{p}) =
0.00068.Photodisintegration of NucleiAlso note that we have assumed that the probability of photodisintegration of deuterium nuclei by absorption of photons is small enough at 8.3645 x 10 ^{9} degrees Kelvin to be neglected. Perhaps no neutrons
captured by protons at this temperature will be lost, even if scattered by photons, but will eventually produce deuterium
again that combines into helium. However, by Wein's Law for the maximum energy of photons in black body radiation,λ _{Max Energy} = 2.8978 x 10^{-3}/T = 3.464386 x 10^{-13} meters at T= 8.36454 x 10^{9} degrees K.Then the photon energy is hc/λ = 3.68136 MeV > 2.224 MeV binding energy of deuterium. If we lower temperature to the point where photodisintegration is not possible due to hypothetical simple absorption of such higher-energy photons, then the fraction of neutrons available according to our statistical mechanical formula will be reduced. Note that T _{no_D_photodisintegration} = (2.224/3.8136)(8.36454 x 10^{9}) = 5.053284 x 10^{9} degrees K.Then N _{n}/N_{p} = exp(-(m_{n} -
m_{p})(c^{2})/kT) = 0.055023.This fraction of neutrons would combine with an equal fraction of protons to produce about 11% deuterium by weight, yielding about 11% helium by weight. Thus the effect of photodisintegration of nuclei would be to reduce the yield of deuterium from 30% by some amount depending on the cross-section of the deuteron for absorption of a photon. Since primordial helium abundance is at least 25% according to measurements, evidently most of the deuterons are not destroyed by photodisintegration. We may do better by modeling the collision of a photon and a deuteron and insisting on the conservation of energy and momentum in the process dynamics to find the maximum amount of energy that may be transferred to the deuteron from a photon in such a Compton-process event. For the simple case of a 3.68136 MeV photon completely absorbed by a stationary deuteron, I find only 0.007226 MeV transferred as kinetic energy to the deutron, all the remaining 3.674134 MeV being potentially available for deutron excitation and photodissociation. However, the lack of a suitable deuteron electric dipole moment may hinder absorption, so that the photon is scattered or re-emitted with a consequent deuteron recoil. Some other authors maintain that D is efficiently disintegrated by photons [12]. See Google links on deuterium photodisintegration [19]. Inertial Confinement of Big Bang MaterialOtherwise, there is the issue of how long the fusion-producing confinement provided by the Big Bang is actually maintained. The above model supposes that all the neutrons are used up to create He-4, but if the appropriate conditions for fusion are not maintained long enough, the resulting density of helium could be lower. This problem is accounted for in more elaborate theories of Big Bang nucleosynthesis [9,10,11]. Also, we have neglected minor reaction channels producing He ^{3} or Li-7, so that our results have the character
of an outer limit for maximum He-4 production.
Helium/Hydrogen Ratio According to Professional AstrophysicistsI note Martin Schwarzschild [6] thought that the Population II of oldest metal-poor stars found in elliptical galaxies not much enriched by supernovae featured an abundance by weight of helium at 25%. Donald D. Clayton and Aller used a 38% abundance for stellar calculations in Clayton's Principles of Stellar Evolution and Nucleosynthesis [3] based on
Aller's observations of planetary nebulae ejected from stars with
collapsing cores, and observations of the nebulae, which may usually be
stellar or supernova ejectae, in general have often resulted in estimates
of 40% or so for helium abundance there. Clayton obtains 25% for
primordial Big Bang helium in his more recent book on the elements [7],
seeming to buck up Martin Schwarzschild, whose 25% figure was also
obtained (via relatively more elaborate calculations featuring the Saha equation) by James
Peebles in his doctoral work on Physical Cosmology [8]. Bowers and
Deeming stated that the primordial helium abundance was 30% by weight in
the 2nd volume of their 2-volume set on Astrophysics [5].Rigorous Abundance Ratio Including Mass DefectsNow we'll finish the calculation to compute the accurate density ratio for He-4 including the mass defects associated with the binding energies involved in the reactions finally producing helium from protons and neutrons. Let the total density per unit volume be ρ _{H +
He} = N_{n}m_{n} + N_{p}m_{p}, so
thatρ _{H + He} = [N_{n}m_{n} +
N_{n}m_{p}] + (N_{p} -
N_{n})m_{p}, where[N _{n}m_{n} +
N_{n}m_{p}] = ρ_{D} +
(ΔE_{D}/c^{2})N_{n}, or ρ_{D} =
[N_{n}m_{n} + N_{n}m_{p}] -
(ΔE_{D}/c^{2})N_{n}. Here ΔE _{D} =
2.224 MeV, the binding energy of Deuterium when it is made via p + n ->
D.Next Deuterium nuclei combine pairwise to form He ^{4}, so
thatρ _{He} = ρ_{D} -
(ΔE_{He}/c^{2})N_{n}/2.The factor of 2 appearing as a divisor accounts for the pairwise combination of the Deuterium nuclei. ΔE _{He} = 24.76038 MeV. Finally, by
substitution, ρ _{He} = [N_{n}m_{n} +
N_{n}m_{p}] - (ΔE_{D}/c^{2})N_{n}
- (ΔE_{He}/c^{2})N_{n}/2. This we can rewrite in the form ρ _{He} = [N_{n}m_{n} +
N_{p}m_{p}] - [N_{p}m_{p} -
N_{n}m_{p}] - (ΔE_{D}/c^{2})N_{n}
- (ΔE_{He}/c^{2})N_{n}/2. Dividing by the total density ρ _{H + He} = N_{n}m_{n} +
N_{p}m_{p}, we haveρ _{He}/ρ_{H + He} =
1 - [N_{p} -
N_{n}]m_{p}/(N_{n}m_{n} +
N_{p}m_{p}) -
(ΔE_{D}/c^{2})N_{n}/(N_{n}m_{n} +
N_{p}m_{p})- (ΔE _{He}/c^{2})(N_{n}/2)/(N_{n}m_{n}
+ N_{p}m_{p}).This can be factored to bring out the known ratios m _{n}/m_{p} = 1.6749/1.6726 =
1.001375105 and N_{p}/N_{n} = 5.6338, so that
ρ _{He}/ρ_{H + He} = 1 - [N_{p} -
N_{n}]/(N_{n}(m_{n}/m_{p}) +
N_{p}) -
(ΔE_{D}/m_{p}c^{2})N_{n}/(N_{n}(m_{n}/m_{p})
+ N_{p})- (ΔE _{He}/2m_{p}c^{2})(N_{n})/(N_{n}(m_{n}/m_{p})
+ N_{p}), orρ _{He}/ρ_{H + He} = 1 -
[N_{p}/N_{n} -1][1/((m_{n}/m_{p}) +
N_{p}/N_{n})] -
(ΔE_{D}/m_{p}c^{2})[1/((m_{n}/m_{p})
+ N_{p}/N_{n})]- (ΔE _{He}/2m_{p}c^{2})[1/((m_{n}/m_{p})
+ N_{p}/N_{n})].Inserting the known numerical ratios and simplifying, we find ρ _{He}/ρ_{H + He} = 0.3029312
- (ΔE_{D}/m_{p}c^{2})/6.64755 -
(ΔE_{He}/m_{p}c^{2})/13.2951, which
yieldsρ _{He}/ρ_{H + He} = 0.3029312 - 3.466664 x
10^{-4} - 19.2976441 x 10^{-4}, so
thatρ _{He}/ρ_{H + He} = 0.30065477.Thus the outer limit on the ratio of He-4 mass to total mass is very close to 30%. The remaining 70% of the total mass is hydrogen. Trace amounts of He-3 and Li-7 may be produced via minor reaction channels not included in this analysis. Big Bang confinement time is short enough to prevent the slow formation of heavier elements via the usual mechanism involving 3 helium nuclei first analyzed by E.E. Salpeter. Further calculations show that all of the deuterium is consumed. (And now I am He.)Overall Big Crunch and Big Bang: Cosmic CyclesAfter some calculations [22], I suspect that the universe started with roughly 10 ^{11} dead galaxies falling in with a total mass of approximately 2 x 10^{22} solar masses of 1.99 x 10^{30} Kg/solar_mass that were composed largely of white dwarf material that connected somewhat like the pieces of a uranium bomb. The infalling material was primarily cooled white dwarf remnant material with a small admixture of neutron stars, and when it slammed together the radiation-dominated era began immediately. Calculations show that the initial fireball was radially unstable, such that the energy required to compress it against radiation pressure for a meter was just equal to the work done by the gravitational field in shrinking the fireball radius another meter. Thus the universe was eventually compressed to its limiting density, that of a nucleon, as it coasted in from the ignition perimeter at R_{min_WD}, initially at white dwarf density, without gaining additional energy. This would have yielded a lower spherical radius limit at nucleon density about equal to the radius of the orbit of Mars. The black body radiation field switched on at a temperature determined roughly from_{universe}^{2}/R_{min_WD} - (3/5)GM_{universe}^{2}/R_{max} = (4/3)πR_{min}^{3}aT^{4},^{8} kg/m^{3}, the universe mass fell into a sphere of radius 2000 AU, or 2.99 x 10^{14} meters, just 0.0316 light-years, before lighting up and coasting down to nucleon density radius over a period of several months at v < c. Subsequent calculations have shown that it took over 26.7244 years for the fireball to expand and cool enough for deuterium and helium to form, kicking in fusion energy, and the galaxies formed at least 3.542 x 10^{5} years after the reflection from nucleonic density [17], following the end of the radiation-dominated era at the time when radiation was decoupled from matter and the matter-dominated era began. These times are longer than many times you may read about elsewhere [11,18] that were computed from approximate, non-relativistic models for the expansion of a universe of constant density. We find easy closed-form solutions to the approximate equations like r = a(t) ^{2/3}, with v = (2/3)a(t)^{-1/3} (matter-dominated era) andr = b(t) ^{1/2}, with v = (1/2)b(t)^{-1/2} (radiation-dominated era).For both cases, it is easy to specify early times such that v >> c. These difficulties yielding short times to key early-universe events vanish when we use [16] F = dp/dt = d/dt[m _{0}v/(1 - v^{2}/c^{2})^{1/2}] = -(GM/r^{2})[m_{0}/(1 - v^{2}/c^{2})^{1/2}] - (pressure term), orF = dp/dt = d/dt[ρ _{0}v/(1 - v^{2}/c^{2})] = -(GM/r^{2})[ρ_{0}/(1 - v^{2}/c^{2})] - dP_{total}/dr, whereρ _{0} = Σ_{i} N_{i}m_{0i}, P_{total} = P_{radiation} + P_{gas}, and ρ = γ ^{2}ρ_{0} = [ρ_{0}/(1 - v^{2}/c^{2})] due to Lorenz contraction of the density volume, so that the force is equal to the time-derivative of the relativistic momentum, and gravitating mass is equal to inertial mass. Note that the equal falling of objects of different mass is preserved, so the principle of equivalence may still be applied. On the other hand, solutions are difficult to obtain in closed form then. However, one can easily show that velocities with v > c are never realized. We have gas- and radiation-ball theorems that give the radius R as a function of the temperature T in the form R(T). Then t > R(T)/c gives the time for the expansion of the fireball to radius R(T) for any given early universe event temperature. For instance, in the radiation-dominated era, taking R_{max} as approaching infinity, and neglecting the gas pressure by comparison with the radiation pressure, and setting the collapse energy equal to the blackbody radiation energy,(3/5)GM ^{2}/R = (4/3)πR^{3}(aT^{4}), so R = (T_{0}/T)R_{0}, where T _{0} = [(9/20π)GM^{2}/aR_{0}^{4}]^{1/4}, and R_{0} = 2.99 x 10^{14} meters at white dwarf density [22].Here the radiation constant a = 4σ/c = 7.565883 x 10 ^{-16} (J/m^{3})(1/K^{4}),G = 6.6742 x 10 ^{-11} N m^{2}/kg^{2}, c = 2.9979 x 10^{8} m/sec, and M = 2 x 10^{22} x 1.99 x 10^{30} kg.Then T _{0} = [(9/20π)GM^{2}/aR_{0}^{4}]^{1/4} = (9G/20πa)^{1/4}[M^{1/2}/R_{0}] = 7.07 x 10^{12} degrees Kelvin.Deuterium, then rapidly helium, forms when T = 8.3645 x 10 ^{9} degrees Kelvin, and thenR _{He} = (T_{0}/T)R_{0} = (7.07276 x 10^{12}/8.3645 x 10^{9})2.99 x 10^{14} meters = 2.52825 x 10^{17} meters.Then t _{He} > R/c = 8.4334 x 10^{8} sec = 26.7244 years.At galaxy formation time the radiation pressure may be less overwhelming, decreasing the accuracy of our estimate. Using the same scheme, however, T _{galaxy_formation} = 6.31 x 10^{5} degrees Kelvin, the He++ to He+ transition temperature. ThenR _{galaxy} = (T_{0}/T)R_{0} = (7.07276 x 10^{12}/6.31 x10^{5})2.99 x 10^{14} meters = 3.3514 x 10^{21} meters.
Then t_{galaxy} > R/c = 11.1792742 x 10^{12} seconds = 3.542 x 10^{5} years.
Note that on the matter-dominated era time scale, galaxy separation happened
13.5 million years after the Big Bang.Penzias and Wilson noticed a 2.7 degree Kelvin Big Bang fireball relic background noise in telephone company microwave antennas. We try a relic radius given by R _{relic} = (T_{0}/T_{relic})R_{0}
= (7.07 x 10^{12}/2.7)2.99 x 10^{14} meters = 7.82937 x 10^{26} meters = 82.7 x 10^{9} LY,
somewhat too big, so that the universe would be over 82 billion years old. However, it is known to be
about 13.5 billion years old from measurements of the Hubble constant H = (2/3)/t. That the dynamics
in the matter-dominated era differs from from the radiation-dominated era dynamics is probably central to
problem resolution here, since the radius R and temperature T of the fireball relic are no longer connected
by the same power law in the matter-dominated era. A certain maximum amount of energy [26](3/5)GM ^{2}/R_{WD} = (4/3)πR^{3}aT^{4}should go into the fireball from gravitational collapse to a sphere of white dwarf density, after which it will smoothly collapse to nucleonic density without gaining further energy, then expand and cool after reflection from the nucleonic hard core without gaining further energy until a little more is picked up by thermonuclear fusion in the formation of helium. Using 2.7 degrees Kelvin for T and the present radius of the universe R = 13.5 billion light-years, we solve for R _{WD} and findR _{WD} = (3/5)GM^{2}/((4/3)πR^{3} aT^{4}) = 0.943 x 10^{3} AU. This is the same order of magnitude as the 2000 AU estimate for the radius of the universe at white dwarf density that I derived above from the somewhat indeterminate white dwarf density obtained by averaging the densities of 3 known white dwarfs. Including the fusion energy contribution of 0.3(M/M _{He})ΔE_{He} = 8.1448 x 10^{66} Joules will bring it hardly any closer to 2000 AU, as calculation shows, since
(3/5)GM^{2}/R_{WD} = 4.4937 x 10^{91} Joules. Another factor may be the inclusion of neutron stars in the material mix, increasing the average density of the collapsing pre-fireball progenitor to as high asρ = M/[(4/3)πR _{WD}^{3}] = 3.3844 x 10^{9} kg/m^{3}, where R_{WD} = 0.943 x 10^{3} AU,nearly ten times as high as my initial estimate based on observed white dwarf masses. I suspect that stars over 6.7 solar masses or so finally supernova, leaving neutron star remnants, where as stars of lesser mass down to perhaps 0.5 solar masses or so are thought to leave white dwarf remnants after shedding planetary nebulae. The gaseous remnants form more stars, and more and more mass is locked up in collapsed objects until the light of the universe grows dim. After computing densities due to neutron stars plus white dwarfs, I see that the density seems to be jacked up by the addition of supermassive black rotators ("black holes") from galactic cores, the burned-out remnants of early-universe quasars that have exhausted the fuel from their accretion disks, and from smaller black hole objects distributed throughout the galaxy and other forms of dark matter [24]. As stated, further compression does not put more collapse energy into the fireball, because the opposing radiation pressure opposes further acceleration of material as the universe coasts down to nuclear density. Thus the Penzias and Wilson 2.7 degree Kelvin fireball relic temperature tends to confirm our Big Crunch model for pre-collapse of the universe prior to the Big Bang. Otherwise, I have computed that the maximum energy of observed cosmic rays may be computed from the minimum collapse radius of the universe at nucleonic density. Squeeze-Boom Cosmology: The Alpha and the Omega of TimeOur infall-before-bounce scheme insures that the basic conservation laws always hold true. I note that matter always falls in before supernovae, novae, or planetary nebula ejection occur, so it is natural to extend this to the Big Bang, equipping it with a preliminary Big Crunch and Crunch-Bang cosmic cycles supporting a beautiful Squeeze-Boom Cosmology in a universe which is everywhere locally Squeeze-Boom. The matter-dominated era nucleons in the subsequent cooled Big Bang fireball are thought to be conserved across cosmic cycles hundreds of billions of years in length. For instance, modeling the infall as an inelastic bounce based on the equation of state of the material yields an approximate cosmic cycle time of 238 billion years [16]. Meanwhile, some still write of "creation ex nihilo", which is mildly suggestive of the compression of intercourse. - James A. Green, April 25-30, 2006.Nucleosynthesis of Heavier Elements in the Big BangThe Squeeze-Boom or Crunch-Bang process has some similarities to supernova explosions. In particular, the neutronization process after the collapse from white dwarf density on the way to nuclear density must have taken place, although the subsequent relativistic inertial confinement gives neutrons time to decay before deuterium formation takes place from neutron and proton populations in thermal equilibrium nearly 27 years later. (Neutron decay time is 886 seconds, about 15 minutes.) In a supernova, an iron core collapses inside a set of shells of heavier elements distributed in layers of lighter and lighter elements as we examine the structure from the inside outwards up into the stellar envelope. In a supernova, these elements from iron down to hydrogen are exploded outward by a shock wave from the collapsing iron core and subjected to r-process neutrons [21] creating heavy elements such as uranium and detectable radioactive isotopes. In the Big Bang, however, we find a sea of photons with energies >> 8 MeV leading to photodisintegration of all nuclei until the fireball has cooled enough at T = 8.3645 x 10 ^{9} degrees Kelvin to turn on the deuterium and helium formation process in the presence of photons with energies peaked at 3.69 MeV. The triple-alpha process through which subsequent elements such as carbon are formed is slow, requiring millions of years to produce appreciable quantities of trace elements in stars, and heavier element formation is truncated by the expansion of the fireball well before 350,000 years have passed. However, now that we have discovered longer confinement times for the Big Bang fireball originating in relativistic inertial confinement, it may be useful to recompute trace element formation in order to more accurately assess Big Bang trace element abundances. I note that Lorentz time-dilation will impede decay processes inside flows of material moving at high velocity in these early universe event scenarios, just as it does in the case of cosmic rays. - James A. Green, May 8, 2006 - 9/17/08The Big Bang and the Large Hadron ColliderThe Large Hadron Collider (see particles) will produce proton beams with energies of 7 TeV/proton. How does this compare with energies of Big Bang hadrons? It turns out that the 3-quark hadrons (baryons) are seemingly indestructibly bound, with higher-energy hyperon states that make larger tracks in cloud chambers than their lower-energy state baryons do. Baryon states with higher-energy-than-strange quarks are probably also created in the Big Crunch. Since we require the conservation of energy-momentum, and since all other cosmic explosions feature collapse prior to explosion, we may assume that the universe collapsed from far out before it exploded. My cosmic cycle calculations show that the universe cycle time is on the order of 235 billion years, long enough to turn almost all stars into white dwarf debris, except a few that are neutron star supernova remnants, plus a few black holes. Thus, when the infalling dead galaxies collide in the Big Crunch, the dead matter slams together with a kinetic energy of approximately (3/2)GM ^{2}/R, where R is the radius of the universe at white dwarf density. The mass of
the universe can be gleaned from the Hubble expansion constant, and turns out to be about 2 x 10^{22}
solar masses, so that there are about_{p} = 2 x 10^{22} x 1.99 x 10^{30} kg/ 1.6726 x 10^{-27} kg/proton= 2.3795 x 10 ^{79} nuceons in the universe.^{14} meters, so that we finally findE/N = [(3/2)GM ^{2}/R]/N = 131.77 x 10^{6} TeV/nucleon.This is nearly 19 million times the energy of a 7 Tev LHC proton! At the time when the white dwarf material slams together, it seems likely that this stupendous energy produces large, swollen hyperons (see particles) that cannot be compressed to nuclear density, as enlarged hyperon tracks in cloud chambers suggest. Just what the energy partition is and how big the average hyperon is, factors that would allow us to compute the maximum compression and minimum radius of the Big Crunch fireball before its Big Bang expansion phase...these are presently unknown. The decay times of the high-energy excited-state baryons will be fast following the ricochet from maximum compression, however. Due to relativistic infall velocities, will turn out that the infalling white dwarf material reaches densities higher than that of nuclear matter. A sea of expanding excited-state baryons including particles and anti-particles bouncing from a minimum radius much larger than ^{3}]ρ_{nucleon} = M_{universe}_{min}(ρ_{nucleon}) > 3.45689 x 10^{11} meters,or 3.45689 x 10 ^{11}/(9.4605284 × 10^{15}) light-years = 0.3654 x 10^{-4} LY,less than the radius of the orbit of Jupiter at 7.78 x 10 ^{11} meters,^{11} meters < R_{min_collapse} < 2.99 x 10^{14} meters = 0.0316 LY,limits which I suppose we can improve on. Of course, energies of 19 million times 7 TeV are never likely to be obtained in terrestrial machines. However, a complete theory of the elementary particles may make it possible to compute from relativistic wave mechanics and appropriate force laws just exactly what the excited states of hadrons are like, which would make scenario calculations for these limits much better. If we inject special relativity, then in the rest frame of the Big Crunch-Bang at the moment of impact, a white dwarf remnant must have a mass M' _{WD} = M_{WD}/(1 - v^{2}/c^{2})^{1/2}, with a densityρ' _{WD} = ρ_{WD}/(1 - v^{2}/c^{2}).^{6} TeV,when its rest mass is 0.938272 GeV, we must have ^{2})^{1/2} = 140.43 x 10^{9}, so that1/(1 - (v/c) ^{2}) = 1.9723118 x 10^{22}, andv = c(1 - 0.2535096 x 10 ^{-22}).Neutron star density is about 7.6 x 10 ^{15} kg/m^{3}, andWhite Dwarf density is about 1.27 x 10 ^{7}(M/M_{sun})^{2} kg/m^{3},so that in the rest frame of the explosion at R _{WD},ρ _{WD}/(1 - (v/c)^{2}) = 2.5019 x 10^{29}(M/M_{sun})^{2} kg/m^{3},many times the density of a neutron star in its rest frame. In fact, [ρ _{WD}/(1 - (v/c)^{2})]/ρ_{NS} = 3.2894 x 10^{13}(M/M_{sun})^{2},_{WD}. Thus the
Big Bang will take place from inside R_{RD} with so much kinetic energy that its density is
many times nuclear density, up to 3.289 x 10^{13} times nuclear density.The white dwarf remnants would come down in a special relativistic universe to the R _{WD}
radius looking like flattened pancakes in the rest frame of the explosion with^{11}.The pancaked white dwarf remnants will collide and interact at the their edges when R = R _{WD} = 0.0316 LY,
then material will tumble in until it begins to richochet. From the point of view of a man piloting a white
dwarf on in at rest with respect to the white dwarf remnant, the effective velocity will be greater
than the speed of light because meter sticks in the external coordinate system will have seemed to have
shrunk according to the Lorentz contraction rule by(1 - (v/c) ^{2})^{1/2}, so that his
effective velocity of travel according to him will be abut 140 billion times the speed of light. At that
velocity, it would take 71 microseconds to cross 0.0316 light-years.If we consider the principle of equivalence, and let the event horizon be determined by the 1911 convention, then the event horizon for the mass of the universe is reached when ^{2}) = 0, so thatR _{event} = 3.126 x 10^{8} light-years.^{2}/R_{event}]/N_{nucleons} = 1.4093 GeV,Calculation then shows that v = 0.74c. Note that the Big Crunch and the Big Bang seem to blow through the associated black hole event horizon as if it didn't exist. The above calculations should probably be revised to include relativistic mass-energy from the kinetic energy of infall in the gravitating mass. Music: 2001 Trip from 2001: A Space Oddysey by Stanley Kubrick.In the movie, this is supposed to take place as our space ship approaches the orbit of Jupiter. - Jim Green, September 17, 2008. "For the benefit of Mr. Kite, there will be show tonight on trampoline..." - Visionary Beatles.GREAT BALLS OF GAS!The Outer Limits of Big Bang Abundances! GM/R in Cosmic Sign
Language:The Compression of Intercourse is On the Way. "Ram Dass" as a mythic archetype. [1] Behold the Spirit:When I first published my papers on Big Bang abundances, these images appeared at the sime time over the Midwest. I finally came up with a rough 30% helium abundance by weight, which you might compare with the Aller-Clayton 38.25% by weight value for the Sun, although the Sun is composed of supernova debris. A few minutes after I picked up the above pictures, the wind had twisted the Old Man into an image of a Lady with long hair and a pearl necklace, so that I could judge which I liked better, it seemed. R**2 -> From Ram Dass to Rammed Ass. GM/R**2 Two! The Force! The Night They Drove Old Dixie Down {2}, by Joan Baez."Til Stoneman's Calvary Came and tore up the tracks again." REFERENCES[1] Visionary Memories: See The Joyous Cosmology by Alan Watts
& Be Here Now by Ram Dass, Still
Here by Ram Dass, and Thomas Byrom's translation of The Dhammapada:
The Sayings of the Buddha with introduction by Ram Dass, for an
introduction to spiritual treats and guidance provided by Ram Dass over
the years. Ram Dass also provides audio tapes, which is perhaps why the
cloudy spirit figure is so in a wrap.... Rap! Perhaps we may characterize Squeeze-Boom Cosmology as "The Joyous Cosmology".[2] See James A. Green, Thermonuclear
Fusion in Stars, Greenwood Research, 2nd edition.[3] Clayton, Donald D., Principles
of Stellar Evolution and Nucleosynthesis, McGraw-Hill,
1965.[4] Chiu, Hong-Yee, Stellar Physics, Blaisdell Publishing
Co., 1968.[5] Richard Bowers and Terry Deeming, Astrophysics,
Vols 1 (The Stars) and 2 (Nebulae, Galaxies, and the Big Bang).[6] Martin Schwarzschild, Structure and Evolution of the Stars, Dover
Publications.[7] Donald D. Clayton, Handbook of Isotopes in the
Cosmos : Hydrogen to Gallium, 2003.[8] James Peebles, Physical
Cosmology (1971). This book was fun to read. Dr. Peebles is more recently
the author of Principles of Physical Cosmology.[9] Big Bang Nucleosynthesis, University of California at Berkeley. [10] Big Bang Nucleosynthesis, University of California at Los Angeles. [11] Big Bang Nucleosynthesis, Wikipedia. [12] Nucleosynthesis in the Early Universe by Bengt Gustafsson. [13] Electric Dipole Moments as Probes of CPT Invariance by Pavel A. Bolokhov. Includes notes on Deutron Electric Dipole Moment Suppression.[14] Steven Weinberg, The First Three Minutes, BasicBooks, 1977, 1988.[15] Google: Big Bang links [16] The Field Equations of the Electroform Model :
From General Relativity to Unified Field Theory by James A. Green. Gives the equation of motion based on a simply-derived and robust unified quantum field theory of forces. See also the 11th edition of Gravitation and the Electroform Model: From General Relativity to Unified Field Theory by James A. Green for cosmic cycle time calculations.[17] Galaxy Formation by James A. Green. Also see Red Limit 2 with notes on galaxy formation and Galaxy Formation Links. Radiation began to decouple from matter when the He++ to He+ phase transition took place, the first Big Bang fireball phase transition to bound atomic states, causing galaxies to precipitate out of the gaseous solution of hydrogen and helium.[18] Adam Frank's article The First Billion Years that appeared in Astronomy magazine, June 2006. The author evidently uses non-relativistic models that have the material universe expanding faster than light, as is also the case in Wikipedia [11]. We use non-relativistic models to explain observations in the matter-dominated era with a fair degree of confidence. However, their extrapolation to the early details of cosmic history leads to serious errors.[19] Google links on deuterium photodisintegration and Google links on deuterium photodisintegration at low energy. [20] Charts of Photodisintegration of Helium-3 as a function of photon energy. [21] R-process Nucleosynthesis in Supernovae by John J. Cowan and Friedrich-Karl Thieleman. [22] Note that our radiation-dominated era model predicts R = (T _{0}/T)R_{0} = ((9G/20πa)^{1/4}[M^{1/2}/R_{0}]/T)R_{0} = ([(9G/20πa)^{1/4}M^{1/2}]/T),
which depends only on the mass of the universe M and the event temperature T under investigation. The universe mass is determined from measurements of H = (2/3)/t = (8πGρ/3) ^{1/2}, so that ρ = 3H^{2}/8πG,based on matter-dominated era models [23]. After calculation based on the above formula for H, I find M = (4/3)πR ^{3}ρ = 4c^{3}/(27GH) = (2c^{3}/9G)t = 1.92 x 10^{22} solar masses,
which checks out when we set R = ct at the outer limits, where t = (2/3)/H is the age of the universe in seconds. [23] See The Red Limit for more on matter-dominated era models. Consider H.^{2} = 8πρG/3 and R_{max}, the expansion limitLet d/dr(dr/dt) = -GM/r ^{2}. Multiplying both sides of the equation by dr/dt,(dr/dt)(d/dr(dr/dt)) = -(GM/r ^{2})(dr/dt). Ord/dt( (dr/dt) ^{2} ) = -2GM(dr/dt)/r^{2} = 2 (d/dt)GM/r.Integrating both sides of this equation over t, and including a constant of integration K, (dr/dt) ^{2} = 2GM/r + K.If K = 0, then (dr/dt) ^{2} = 2Gρ(4/3)πr^{2}, so thatH ^{2} = [(dr/dt)/r]^{2} = 8πρG/3.To determine K, note that if we start dropping matter in from R _{max}, then from(dr/dt) ^{2} = 2GM/r + K, we find0 = 2GM/R _{max} + K, so thatK = - 2GM/R _{max}.Thus for K < 0, the universe is bound, never reaching any further out than R _{max}.In reality, (dr/dt) ^{2} = 2GM/r - 2GM/R_{max}.Then (dr/dt) ^{2} = 2G(4/3)πr^{2}ρ - 2G(4/3)πr^{3}ρ/R_{max}.Thus H ^{2} = (8πGρ/3)(1 - r/R_{max}), where R_{max} is given in meters, andH(r) = [(8πGρ/3)(1 - r/R _{max})]^{1/2}.This equation might be used to determine R _{max} if H(r) and ρ are both known, but the huge dimensions of R_{max} make this impractical, as r/R_{max} is locally small. Note H(r) is the locally observed Hubble constant. In general, we must include observational time-delay r/c, so thatH _{observed}(r, t) = H(t - r/c) = (2/3)/(t - r/c) = [(2/3)/t](1/(1 - r/ct))
= H(t)(1/(1 - r/ct)).Then c = H _{observed}(r,t)r = (2/3)r/(t - r/c) when r = (3/5)ct, the observational limit. This is as far as telescopes can see, a point now about 8.1 billion light-years away where galaxies redden out, although near this depth of view back in time Δt = r/c they all seem to be closer together and rushing away more rapidly from each other, so that we see a number of colors in the Hubble Deep Field galaxies. The observational time-delay r/c gives rise to increases of H(t,r) with increases in r, sometimes attributed to "dark energy" [27]. Thus we may expect to measure something approximately likeH _{observed}(r, t) = H(r)[1/(1 - r/ct)] =
[(8πGρ/3)(1 - r/R_{max})]^{1/2}[1/(1 - r/ct)] when r < (3/5)ct.Usually and probably at this time R _{max} >> ct > (5/3)r. Measurements of H_{observed}(r,t) at the outer limits of observation may finally in principle determine R_{max}.I note that ρ is measured by measuring H, and that the closure constant K < 0 for the universe is such that r/R _{max} is locally small. The "missing mass"
problem [8] in physical cosmology is thatρ _{visual} < ρ(H), because
it is hard to see all the mass visually. Some of it must be locked up in dark objects. Also, 1 - (3/5)^{3} = 1 - 0.216 = 0.784 of the total mass M must be invisible because of the cosmic censorship imposed by the limit of telescopic visibility at L = (3/5)ct, the Red Limit due to observational time-delay. You see less than 22% of the mass.[24] Consider a cubic light-year of the very elderly infalling universe, and let the number densities per cubic light-year be N _{WD}, N_{NS}, N_{BH},
and N_{Q}, the number densities per cubic light-year of white dwarfs, neutron stars, black holes, and galactic core quasar remnants, respectively. Then the total mass per cubic light-year will beM/LY ^{3} = N_{WD}M_{WD} + N_{NS}M_{NS} + N_{BH}M_{BH} + N_{Q}M_{Q},where M _{WD}, M_{NS}, M_{BH}, and M_{Q} are the average masses of white dwarfs, neutron stars, small-time galactic black holes, and old galactic core giant black hole quasar remnants, respectively.Now let α = 1.1812 x 10 ^{-48} (LY/m)^{3} be the number of cubic light-years per cubic meter. Then the density of the elderly infalling universe at the collision point when all fragments come together and make contact will beρ = N _{WD}M_{WD}α
[1 + (N_{NS}/N_{WD})(M_{NS}/M_{WD}) + (N _{BH}/N_{WD})(M_{BH}/M_{WD}) +
(N_{Q}/N_{WD})(M_{NQ}/M_{WD}) ] kilograms per cubic meter.The leading term outside the square brackets is the white dwarf density we assumed would be the dominant factor. However, our subsequent calculations show that the factor in square brackets must be approximately ten in order to account for the observed magnitude of the Penzias and Wilson 2.7 degree Kelvin Big Bang fireball radiation relic. Now M _{NS}/M_{WD} < 2.5, and N_{NS}/N_{WD} < 1 seem reasonable to me, so that without black holes or other dark matter, the term in square brackets is probably less than 3.5.(N _{Q}/N_{WD})(M_{Q}/M_{WD}) < 0.2 seems reasonable when we consider the relative magnitudes of central galactic black hole masses and all the rest of the white dwarf remnants in the galaxy together. Therefore, the rest of the mass, about 0.6 of it, may be due to invisible collapsed objects corresponding to black holes or other dark matter. This seems to be in rough agreement with the "missing mass" problem as described by Dr. P.J.E.Peebles (Jim Peebles) in Physical Cosmology [8] and Principles of Physical Cosmology. That is, the universe density discovered from the Hubble constant,H = (8πGρ/3) ^{1/2},is considerably larger than the density ρ _{visible} discovered by counting visible objects and accounting for their masses. Of course it is possible that other kinds of invisible object "dark matter", such as planetary bodies, brown dwarfs, and cosmic dust grains, play a more important role than I have indicated above. The relatively low densities of these tends to indicate an important role for supermassive black collapsed objects, however, the so-called "black holes". Objects with similar properties exist in unified quantum field theory, but without distortion of the spacial part of the metric of space-time. In this case relativistic perihelion precession and light curvature amplitudes are obtained otherwise than in GR, via superposition theorems.[25] See Google links on Dark Matter. [26] The infall energy of a sphere of uniform density collapsing from infinity to R is given by the integral from 0 to R of [GM(r)/r]dM. This is the integral from 0 to R of [G((4/3)πr ^{3}ρ)(4πr^{2}ρ)/r]dr= G((4/3)πρ)(4πρ)x[integral from 0 to R of r ^{4}dr]= G((4/3)πρ)(4πρ)x[R ^{5}/5]
= (3/5)GM^{2}/R.[27] James Trefil, "Where is the Universe Heading", in Astronomy magazine,
July 2006.Dr. Trefil presents cosmology with the " dark energy" included, which is
thought to increase the Hubble constant H at large distances. However, I attribute
this to observational time-delay. My solution for H in the universe of uniform density
begins with H(t) = (2/3)/t meters/sec per meter, a formula known in the 1970s to James Peebles
and described in his 1971 book Physical Cosmology [8]. Introducing observational time-delay
to account for the finite speed of light, the observed Hubble "constant" should be H(t,r) = (2/3)/(t - r/c), which increases
with the depth of observation r at time t, where t is the age of the universe in seconds. However, the increase is not due to a mysterious "dark energy", but to the finite speed
of light. As the depth of observation increases, c = H(t,r)r for r = (3/5)ct = 8.1 billion light-years in a 13.5 billion year old universe model. This is the maximum
depth of observation, the Red Limit. The term "dark energy" was introduced by cosmologist Michael Turner of the University of Chicago.Musical Themes
from Bob Dylan: Stuck inside of Mobile with
the Memphis Blues AgainMr. Tambourine Man | Knockin' on Heavens Door | Its All Over Now, Baby Blue. Jim Green's Home Page | Einstein & WeatherVisions || Yahoo | Google | Alta Vista || Music Green's Relativity | Big Bang
| Red Limit
| Red Limit 2
| Galaxy Formation
|| Orion Sky
"The only Remedy is a most serene and quiet Air, such as may perhaps be found - Sir Isaac Newton, on the tops of the highest Mountains above the grosser Clouds." Opticks. Supernova Remnant LMC N63A in the Large Magellanic Cloud.Music: Paint it Black by the Rolling Stones. Further Notes on the Large Hadron Collider and CosmologyThe high-energy Large Hadron Collider at CERN in Geneva is about to turn on a beam of 7 TeV protons 27 kilometers in circumference tomorrow morning. We are caught up in the whirl of the high energy "world". The first attempt to circulate a beam through the entire LHC is scheduled for September 10, 2008, at 7:30 GMT and the first high-energy collisions are planned to take place after the official unveiling of the LHC on October 21, 2008. It's the best we can do since the Superconducting Super Collider, with its 20 TeV beams, was cancelled in 1993. The LHC machine is being used for a search for the Higgs boson, which I doubt exists. Instead, I suppose everything to calculable from the Rishon Model of Haim Harari [Images] and Michael Shupe [Images], with suitable assumptions about the forces between Rishon preons and the Rishon masses. The Rishon model allows us to understand that high-energy quarks associated with higher flavor generations of quarks are excited states of quark ground states that are systems of Rishon preon triplets, and to derive properties of their decays from excited states. The 4 Rishons T, V, /T, and /V are the new earth, air, fire and water of modern physics. The Rishons, their composite particles, the photon, and the graviton may include all quanta. In an isolated three quark system, all excited-state quarks but one eventually decay down to the u quark, leaving us with a uud proton. The total number of baryons (3 quarks) in the finally cooled-out universe is conserved between Big Bangs in an oscillating universe, although energy may create additional temporary baryon-antibaryon pairs and matter-antimatter decay products together with quanta of radiation. Quarks have baryon number +1/3, antiquarks baryon number -1/3, so that we say that the cyclic Big Bang conserves baryon number. My work on cosmology shows that the universe is an oscillating universe, although observational time-delay can make distant galaxies atseem to recede at the speed of light. Thus, we are now living inside "A Bigger Bang", as suggested by the 2006 Rolling Stones World Tour, which stopped here at Wichita State University. Incidentally, the proof goes like this: Let t be the age of the universe in seconds. The Newtonian solution and the Einstein-de-Sitter solution for the expanding universe of uniform density in the matter-dominated era both go like ^{2/3}, so that v = (2/3)a(t)^{-1/3}, so thatthe Hubble constant H = v/r = (2/3)/t. If we include observational time delay L/c, to compute what we observe from a distance, then v = HL = (2/3)L/(t - L/c). Setting v = c, we solve to find L = (3/5)ct when distant galaxies at L seem to recede at light speed v=c. Galaxies farther away than this exist, but are out of sight in "A Bigger Bang". Until this was understood, it was not obvious that a universe apparently expanding at v = c at its outer limits could collapse from a maximum height. Now that the Large Hadron Collider is about to turn on, we'll compute what we expect to observe! Will we encounter another quark flavor generation of [(u,d),(c, s),(t,b)], say (f,q)? If not, at what energies can we expect to observe quark flavor generation (f,q) = (freedom, queen)? What kinds of new events should we expect to observe when 7 TeV proton beams collide? Time to pull out the textbooks and the pocket calculator. The original Rishon theory leads me to expect that a 4th flavor generation (f,q) exists, but it may not be observable with 7 TeV colliding beams. New excited states of hadrons (particles containing quarks) should eventually be observed including f and q, including mesons (2 quarks) and baryons (3 quarks), although their decay times will be very short. We will try to predict what these particles will be, what their masses should be, and what the observed decay times should be, including the entire sequence of decays. Perhaps we will also observe new higher-energy states of the leptons, including the electron, the positron, the neutrino, and the antineutrino. In this case we would observe new higher excited states of the Rishon systems /T/T/T, TTT, VVV, and /V/V/V corresponding to the lepton generations ^{-}, μ^{-}, τ^{-}), (e^{+}, μ^{+}, τ^{+}),(ν _{e}, ν_{μ}, ν_{τ}), (/ν_{e}, /ν_{μ}, /ν_{τ})for the electron, positron, neutrino, and antineutrino. This corresponds to the 3 quark flavor generations ^{-}, ν_{e}), (μ^{-}, ν_{μ}), (τ^{-}, ν_{τ})] = [(/T/T/T, VVV), (/T/T/T, VVV), (/T/T/T, VVV)].I note that 3-photon jets are sometimes detected from high-energy electron-positron collisions, possibly because of the 3-Rishon triplet structure of leptons like the electron, rather like 3-jet events featuring streaming meson-antimeson pairs are detected from 3-quark baryons in high-energy experiments. Subsequent generations of quarks and leptons are merely the same quarks and leptons in higher excited states of the associated Rishon system. Note that the different quark colors come from circulating the rishons like layers in a pill. For instance, u = TTV, VTT, or TVT obtains the three u-quark colors. We'll try to compute what the particle masses of the Rishon systems should be at higher energy. Now is the time! We'll compute how hot things have to get before (f,q) can be observed. "I think it will be much more exciting if we don't find the Higgs. That will show something is wrong, and we need to think again. I have a bet of $100 that we won't find the Higgs," said Stephen Hawking [Images]. However, Peter Higgs [Images] is still hoping to find the Higgs boson. We would probably be more likely to find another generation of higher-order quarks (f,q) if the LHC featured particle-antiparticle beam collisions. As for me, I've recently been pursuing the TTAGGG award in telomere reconstruction for youthful patterns of gene expresssion in
longevity. |