At T = 10^19 GeV, Planck Energy.
At T = 10^16 GeV, SU(5) GUT Monopole formation ends
and the Inflationary Higgs mechanism eliminates
the relic Monopoles.
At T = 10^14 GeV, Zizzi Reheating and SU(5) Unification ends.
The phase transition at the end of inflation at about 10^15 GeV or about 10^(-34) sec sees (at 10^14 GeV) the GUT SU(5) is broken to SU(3)xU(2). In The Early Universe (paperback edition Addison-Wesley 1994) Kolb and Turner say (at p. 526): "... SU(5) GUT ... Additional Higgs bosons are required ... at the very least one complex 5-dimensional Higgs. The 5-dimensional Higgs contains the usual doublet Higgs required for SSB adn a color triplet Higgs ... which can also mediate B,L [baryon,lepton] violation. The triplet component must acquire a mass comparable to ... M = 3 x 10^14 GeV ... to guarantee the proton's longevity, while the doublet component must acquire a mass of order of a few 100 GeV to trigger electroweak SSB at the appropriate scale. ...". That indicates to me that the GUT phase transition at 10^(-34) sec does not produce electrons and protons with the customary 0.5 MeV and 1 GeV masses that we now see, but rather produces massive composite GUT monopoles with mass at 10^17 GeV, and consequently with Compton radii least about 100 Planck lengths, or 10^(-31) cm, and therefore of Compton vortex volume 10^(-93) cm, so that 10^80 of them would have volume 10^(-13) cm^3, much smaller than the 1,000 cm^3 volume of our universe at 10^(-34) sec with size 10 cm. It is only later, at the electroweak phase transition of about 100 GeV at which the electroweak U(2) is broken to U(1)xSU(2) with the SU(2) weak bosons becoming massive and the leptons and quarks getting their individual masses from the electroweak Higgs Yukawa coupling that uses the "usual doublet Higgs" described by Kolb and Turner.
At T = 100 GeV = 10^15 degK the Higgs mechanism has been effective, the SU(2) weak force symmetry breaking has occurred, and the energy level is on the order of the Truth Quark mass of 130 GeV.
Above the electroweak transition scale of about 100 GeV, at the time of about 10^(-11) sec after the big bang, when the size of the universe is about 10 x 10^(15-2) = 10^14 cm or about 10^(14-18) = 10^(-4) light years, the Higgs mechanism has not taken effect, so that there is no Higgs Yukawa coupling to give mass to leptons and quarks, so that leptons and quarks are then massless, so their compton radius is not defined and electrons are no more confined to any particular spatial volume than are individual photons in our present universe. The D4-D5-E6-E7-E8 VoDou Physics model has Compton Radius Vortex electrons, and electrons have Compton radius about 10^(-11) cm. 10^80 electrons of that size would require a volume of about 10^80 x 10^(-33) = 10^47 cm^3 which is a size of about 10^16 cm or about 10^(-2) light years which is greater than the 10^(-4) light year size of the universe at the time of the electroweak/Higgs phase transition. However, the ambient temperature of the universe is then about 100 GeV, and the electron has second and third gneration counterparts, the muon of mass about 100 MeV and the tauon of mass about 2 GeV, which are effectively excited states into which the electron will be kicked by the high 300 GeV temperature of the ambient universe. That means that any set of 10^80 electrons that found themselves in the 300 GeV universe at the time 10^(-11) sec would immediately be transfomed into muons and tauons. If all 10^80 of them were muons, then, since 100 MeV muons have a Compton radius of about 10^(-13) cm, 10^80 muons, each with volume 10(-39) cm^3, would fit into a volume of 10^(80-39) = 10^41 cm^3, or of size 10^14 cm, which is the size of the universe at the 300 GeV electroweak phase transition. It is an interesting coincidence (which I realized during a March 2001 series of e-mail conversations with Jack Sarfatti) that the electroweak phase transition occurs just when the 10^80 muon Compton Radius Vortices could fit into our universe. The size required for tauons, only an order of magnitude smaller, is still pretty close to the volume of our universe at the time of the electroweak phase transition.
Farrar and Shaposhnikov have suggested that first order phase transition processes at this stage might account for particle-antiparticle asymmetry, but Huet and Sather, and Gavela, Hernandez, Orloff, and Pene say that QCD damping effects in bubble walls would reduce the asymmetry to a negligible amount. However, Nasser and Turok point out that when such other processes as formation of longitudinal Z condensate are taken into account the observed asymmetry might indeed be produced, and Farrar and Shaposhnikov have replied to Gavela et. al. and Huet et. al., stating that the Gavela-Huet calculational scheme violates unitarity and is unreliable. A nice overview is Farrar's 1994 invited talk at Stockholm.
In hep-ph/0008142, Ayala and Pallares say "... the three well know Sakharov conditions ... (1) Existence of interactions that violate baryon number; (2) C and CP violation; (3) departure from thermal equilibrium ... are met in the standard model (SM) of electroweak interactions if the electroweak phase transition (EWPT) is of first order ... the EWPT turns out to be only too weakly first order which in turn implies that any baryon asymmetry generated at the phase transition was erased by the same mechanism that produced it, i.e. sphaleron induced transitions ... Moreover, the amount of CP violation coming from the CKM matrix alone cannot account by itself for the observed asymmetry, given that its effect shows up in the coupling of the Higgs with fermions at a high perturbative order ... giving rise to a baryon to entropy ratio at least ten orders of magnitude smaller than the observed one. Nevertheless, it has been recently pointed out that, provided enough CP violation exists, the above scenario could significantly change in the presence of large scale primordial magnetic fields ... which can be responsible for a stronger first order EWPT. The situation is similar to a type I superconductor where the presence of an external magnetic field modifies the nature of the superconducting phase transition due to the Meissner effect. ... Recall that for temperatures above the EWPT, the SU(2) x U(1)y symmetry is restored and the propagating, non-screened vector modes that represent a magnetic field correspond to the U(1)y hypercharge group. Thus, in the unbroken phase, any primordial magnetic fields belong to the hypercharge group instead of to the U(1)em group and are therefore properly called hypermagnetic fields. ... we show that the existence of such primordial hypermagnetic fields also provides a mechanism to produce a large enough amount of CP violation during the EWPT to possibly explain the observed baryon to entropy ratio in the SM. This can happen during the reflection of fermions off the true vacuum bubbles nucleated during the phase transition through an interference process equivalent to the Bohm-Aharanov effect, given that in the unbroken phase, fermions couple chirally to hypermagnetic fields with the hypercharge. The chiral nature of this coupling implies that it is possible to build a CP violating asymmetry dissociated from non-conserving baryon number processes that can then be converted to baryon number in the unbroken phase where sphaleron induced transitions are taking place with a large rate. The existence of such asymmetry provides a bias for baryon over antibaryon production. ... We estimate that for strong hypermagnetic fields By = ( 0.3 - 0.5 ) T^2 the baryon to entropy ratio can be RHO_B / S = ( 3 - 6 ) x 10^(-11) for slowly expanding bubble walls. ...".
In hep-ph/0208152 Massimo Giovanni says: "... In cosmology the possible existence of magnetic fields prior to decoupling can influence virtually all the moments in the thermodynamical history of the Universe. Big-bang nucleosynthesis (BBN), electroweak phase transition (EWPT), decoupling time are all influenced by the existence of magnetic fields at the corresponding epochs. ... The physical picture we have in mind is ... the following. Suppose that conformal invariance is broken at some stage in the evolution of the Universe, for instance thanks to the (effective) time variation of gauge couplings. Then, vacuum fluctuations will go outside the horizon and will be amplified. The amplified magnetic inhomogeneities will re-enter (crossing the horizon a second time) during different moments of the life of the Universe and, in particular, even before the BBN epoch. ... If the hypermagnetic flux lines have a trivial topology they can have an impact on the phase diagram of the electroweak phase transition ... If the topology of hypermagnetic fields is non trivial, hypermagnetic knots can be formed .... and, under specific conditions, the BAU can be generated ... A classical hypermagnetic background in the symmetric phase of the EW theory can produce interesting amounts of gravitational radiation in a frequency range between 10^(-4) Hz and the kHz. ... For the hypermagnetic background required in order to seed the BAU the amplitude of the obtained GW can be even six orders of magnitude larger than the inflationary predictions. ...
... if hypermagnetic fields are present at the EW epoch, matter-antimatter fluctuations are likely to be produced at BBN. ... the success of BBN can be used in order to bound the magnetic energy density possibly present at the time of formation of light nuclei. ...
... Before decoupling photons, baryons and electrons form a unique fluid which possesses only monopole and dipole moments, but not quadrupole. ... Large scale magnetic fields present at the decoupling epoch can have various consequences. For instance they can induce fluctuations in the CMB ... they can distort the Planckian spectrum of CMB ... they can distort the acoustic peaks of CMB anisotropies ... and they can also depolarize CMB ...".
From T about 5 GeV to T = 100 MeV = 10^12 degK the energy level goes down through the masses of the 5 lighter quarks, and down to the mass of the fundamental composite hadrons, the protons and pions. There is an SU(3) color force QCD phase transition from quark-gluon plasma to a hadronic gas.
Above the quark-hadron transition scale of about 100 MeV, where the size of the universe is about 10 x 10^(18-2) = 10^17 cm orabout 10^(17-18) = 10^(-1) light years, there are no individual protons or other hadrons, and there is only a soup of quarks and leptons.
Michael Hawkins, in his book Hunting Down the Universe (Little, Brown 1997) proposes that the QCD phase transition may be first order and so may produce density fluctuations that could create Jupiter-mass black holes. Hawkins says that David Schramm has argued that such Jupiter-mass black holes would form in catastrophic collapse around Truth Quarks, whose mass is substantially greater than the energy level of the QCD phase transition. Such Jupiter-mass black holes might be observed as gravitational microlenses that appear in every line of sight to distant quasars. They would be about the size of beach balls, would be so massive that their decay time from Hawking radiation would be about 10^57 years, and would be uniformly distributed in the universe. If all the small black holes were consolidated into the Jupiter-mass ones, and if there were enough of them to give our universe its critical mass, there would be about one every 30 light years or so. If our universe has less than critical mass, there would be fewer of them. Jedamzik has proposed that similar black holes could be formed during the later part of the QCD phase transition, when the QCD-horizon mass scale would be about one solar mass, and that such black holes might constitute the dark matter of of galactic haloes. In this case, the black holes would not be uniformly distributed throughout the universe, but would be concentrated near galaxies. However, it is my opinion that there is no requirement for the existence of galactic halo dark matter, because galactic rotation curves can be accounted for by MOND-Segal conformal gravitation. Therefore, I think that, although Jedamzik's mechanism for creation of black holes may be correct, I do not agree with his idea that they account for galactic halo dark matter (as opposed to cosmological dark matter). Another possibility for at least some dark matter is gravitational interaction from other Worlds of the Many-Worlds. From T = 1 MeV = 10^9 degK down to about T = 0.1 MeV, nucleosynthesis occurs. Neutrinos decouple before T drops below the electron mass 0.51 Mev, so that electron-positron annihilation entropy goes to photons and not to neutrinos.
According to astro-ph/0302431, by Cyburt, Fields, and Oliver: "... Big bang nucleosynthesis ...[(BBN)]... has long provided the primary determination of the cosmic baryon density OMEGA_B h^2, or equivalently the baryon-to-photon ratio n ... = n_10 / 10^10 ... With the precision of WMAP, the CMB now offers a significantly stronger constraint on n ... shown in the vertical (yellow) band ...
... the CMB ... strongly suggest[s]... that the D/H measurements are accurate, while both the 4He and 7Li abundances are systematically small. ... Primordial light element abundances as predicted by BBN and WMAP ... the dark shaded distributions ... and ... the observational abundances (... the lighter shaded distributions) ... are shown ...
According to astro-ph/0307213, by Cuoco, Iocco, Mangano, Miele, Pisanti, and Serpico: "... Theoretical estimates for nuclei abundances, along with the corresponding uncertainties, are evaluated using a new numerical code, where all nuclear rates usually considered have been updated using the most recent available data. Moreover, additional processes, neglected in previous calculations, have been included.... using the WMAP result ... at N_eff = 3.04 ... we get
we report in parenthesis the experimental value or the best estimate currently available: ...
... There are two different ... primordial abundance ... determinations of Y_p: Y_p = 0.234 +/- 0.003 ... and Y_p = 0.244 +/- 0.002 ... If the statistical error were not underestimated, this two values would be only marginally compatible. ... even the higher value of Y_p ... appears in slight disagrement (1.6 sigma effect) with standard BBN. .. Concerning 7Li experimental measurements ... the primordial origin of the Spite plateau has been recently questioned. In particular ...[there was found]... evidence for a dependence of X_7Li on metallicity. ... the discrepancy between the most recent observations ... and our theoretical value is now reduced to a less than 3 sigma effect. The average over the different observed values for X_7Li ... which are mutually compatible, gives X_7Li = 2.04 +/- 0.07. ...".
According to Two World Systems Revisited: A Comparison of Plasma Cosmology and the Big Bang, by Eric J. Lerner, author of The Big Bang Never Happened, Viking Press, New York, 1992: "... The dominant theory of cosmology, the Big Bang, is contradicted by observation, and has been for some time. The theory's predictions of light element abundance, large-scale structure, the age of the universe and the cosmic background radiation (CBR) are in clear contradiction with massive observational evidence, using almost any standard criteria for scientific validity. This situation is not new. In 1992, I reviewed these contradictions ... and concluded that theory had already been clearly falsified. Since that time, the evidence against the Big Bang has only strengthened. There is a second framework for cosmology--plasma cosmology. This approach, which assumes no origin in time for the universe and no hot, ultradense phase of universal evolution, uses the known laws of electromagnetism and the phenomena of plasma behavior to explain the main features of the universe. ...
... In contrast to the extremely bad performance of BBN [Big Bang Nucleosynthesis], the predictions of the plasma alternative have held up remarkably well. Plasma filamentation theory allows the prediction of the mass of condensed objects formed as a function of density. This leads to predictions of the formation of large numbers of intermediate mass stars during the formations of galaxies ... These stars produce and emit to the environment large amounts of 4He, but very little C, N and O. In addition cosmic rays from these stars can produce by collisions with ambient H and He the observed amounts of D and 7Li. The plasma calculations, which contained no free variables, lead to a broader range of predicted abundances than does BBN, because the plasma theory hypothesizes a process occurring in individual galaxies, so some variation is to be expected.
... the ... observations that no galaxies, indeed no stars, have been observed that are entirely free of heavier elements ...[are]... in accord with the predictions of the plasma-based stellar production of light elements. ...
The most dramatic confirmation of the predictions of the plasma-stellar model is in the discovery of large number of white dwarfs in the halo of the Milky Way. Since the theory predicts the formation of an initial population of intermediate-mass stars, it is a straightforward deduction that these stars must leave behind white dwarfs that should exist at present. Specifically the theory predicts that somewhat less than half the total mass of the galaxy should exist in the form of collapsed cores-either white dwarfs or neutron stars ... and for the intermediate stars, which are too small to become supernovae, the normal end-point would be white dwarfs. Recent observations of high proper motion stars have shown that halo white dwarfs constitute a mass of about 10^11 solar masses, comparable to about half the total estimated mass of the Galaxy ... While these observations have been sharply criticized, they have been confirmed by new observations ... Not only are the existence of these numerous white dwarfs confirmation of much earlier predictions by the plasma theory, they create new and insurmountable problems for BBN. Even if the progenitor stars were only 2-3M, a mass of He equal to about 10-15% of the mass of the remnant white dwarfs would be released into the ISM. This would account for at minimum 50% of the observed He abundance, reducing the possible contribution from the Big Bang to less than 12% of the total mass. Such a low production of 4He is impossible with BBN for a baryon/photon ratio even as low as 1 x 10^(-10). Thus the plasma model has successful predicted a new phenomenon, while the BBN model has been decisively contradicted by observation. ...
The large scale structure of the universe is inhomogeneous at all scales that have been observed ... In particular, galaxies are organized into filaments and walls that surround large voids that are apparently nearly devoid of all matter. These void typically have diameters around 140-170Mpc (taking H=70 km/sec/Mpc) and occur with some regularity ... These vast structures pose acute problems for the Big Bang theory, for there simply is not enough time to form them in the hypothesized 14 Gy since the Big Bang, given the observed velocities of galaxies in the present-day universe. Measurements of the large scale bulk streaming velocities of galaxies indicate average velocities around 200-250 km/sec ... the production of the large voids observed requires three to six times as much time as that allowed by the Big Bang theory. ... An explosive mechanism that rapidly injects energy into the medium could form voids more rapidly than gravitational attraction. ... The plasma cosmology approach can, however, easily accommodate large scale structures, and in fact firmly predicts a fractal distribution of matter with density being inversely proportional to the distance of separation of objects ... This relation flows naturally from the necessity for collapsed objects to be collisional, and from the scale invariance of the critical velocities of magnetic vortex filaments, which are crucial to gravitational collapse. This fractal scaling relationship (fractal dimension=2) has been borne out by many studies on all observable scales of the universe ... In the plasma model, where superlcusers, clusters and galaxies are formed from magnetically confined plasma vortex filaments, a break in the scaling relationship is only anticipated at scales greater than approximately 3Gpc. ...
.. The plasma alternative views the energy for the CBR as provided by the radiation released by early generations of stars in the course of producing the observed 4He. The energy is thermalized and isotropized by a thicket of dense, magnetically confined plasma filaments that pervade the intergalactic medium. While this model has not been developed to the point of making detailed predictions of the angular spectrum of the CBR anisotropy, it has accurately matched the spectrum of the CBR using the best-quality data set from COBE ... Since this theory hypotheses filaments that efficiently scatter radiation longer than about 100 microns, it predicts that radiation longer than this from distant sources will be absorbed, or to be more precise scattered, and thus will decrease more rapidly with distance than radiation shorter than 100 microns. ...
... The WMAP results contradict the Big Bang theory and support the plasma cosmology theory in another extremely important respect. Tegmark et al ... have shown that the quadruple and octopole component of the CBR are not random, but have a strong preferred orientation in the sky. The quadruple and octopole power is concentrated on a ring around the sky and are essentially zero along a preferred axis. The direction of this axis is identical with the direction toward the Virgo cluster and lies exactly along the axis of the Local Supercluster filament of which our Galaxy is a part. This observation completely contradicts the Big Bang assumption that the CBR originated far from the local Supercluster and is, on the largest scale, isotropic without a preferred direction in space. ... the plasma explanation is far simpler. If the density of the absorbing filaments follows the overall density of matter, as assumed by this theory, then the degree of absorption should be higher locally in the direction along the axis of the (roughly cylindrical) Local Supercluster and lower at right angles to this axis, where less high-density matter is encountered. This in turn means that concentrations of the filaments outside the Local Supercluster, which slightly enhances CBR power, will be more obscured in the direction along the supercluster axis and less obscured at right angle to this axis, as observed. More work will be needed to estimate the magnitude of this effect, but it is in qualitative agreement with the new observations. ...".
According to astro-ph/0008212, by Tatsuno, Berezhiani, and Mahajan. "...the interaction of large amplitude electromagnetic waves with a hot electron-positron (e-p) plasma (a principal constituent of the universe in the MeV epoch) leads to a bunching of mass, energy, and angular momentum in stable, long-lived structures. Electromagnetism in the MeV epoch, then, could provide a possible route for seeding the observed large-scale structure of the universe. ...".
Kaplinghat, Steigman, Tkachev, and Walker, in astro-ph/9805114, say: "...
The present age/expansion rate (Hubble parameter) constraint ... and the SN Ia magnitude-redshift relation require ... alpha > 0.6 ... , while production of primordial helium and deuterium force alpha to be smaller. ...". Since a single power-law expansion for all ages of our universe would either produce a universe that is too young or a universe in which the temperature does not drop below the 80 keV threshhold for nucleosynthesis prior to neutron decay, a single power law expansion is not consistent with both the age of our universe and the standard model of Big Bang Nucleosynthesis.
In particular, their paper indicates that the linearly (alpha = 1) expanding universe models of
- Roland Allen, in Instanton Cosmology, astro-ph/9803079, version 1, and
- D. L. Khokhlov, in astro-ph/9809243,
which were models that I had formely found very attractive and had described favorably on earlier versions of my web pages, are probably not accurate models of our universe.
With respect to the standard cosmological model of a Radiation Era, in which the scale of the universe expands as t^1/2, followed by a Matter Era, in which the scale of the universe expands as t^2/3, their paper is consistent with it being an accurate model of our universe, as is indicated by the red line (with two slopes, changing at the Radiation/Matter Era boundary) on my modification of Fig. 1 of their paper.
At about T = 1 eV or about 10^3 to 10^4 degK the density of matter has exceeded the density of radiation; photons decouple and the sky is transparent; matter recombines into atoms; a residual ionization freezes in. According to Peebles, at decoupling the redshift z = 1400.
As Weinberg (1988) says, "... It is striking that the transition from a radiation- to a matter-dominated universe occurred at just about the same time that the contents of the universe were becoming transparent to radiation, at about 3,000 degrees K. ... We also do not really know which transition occurred first. ...".
According to Narlikar and Padmanabhan (1986) section 8.4.2, Weinberg (1972), and Weinberg (1988): Just after recombination, the Jeans mass was 1.6 x 10^5 Msun, which is the mass of globular clusters. Just before recombination, the Jeans mass was 5 x 10^18 Msun, which is the mass of a large cluster of galaxies.
For example, if the typical galaxy has mass 10^11 Msun, and if galaxies are about a million light years apart, then 5 x 10 ^18 Msun would be 50,000,000 galaxies. Since the cube root of 50,000,000 is about 370, the cluster size would be about 370 million light years across, which is consistent with the 300 million light year size of Galactic Clusters observed when Subir Sakar did a computer analysis of data from the Anglo-Australian Automatic Plate Measuring suvey (New Scientist article by Marcus Chown, 25 April 1998, page 7).
Since electromagnetic processes may well have been interesting at the time of formation of atoms and decoupling of photons, the structure formation problem may be solved
by magnetic structures in the Radiation Era of the universe at or prior to recombination,
and
by the Layzer mechanism of structure formation in a cold universe, as applied to the component of the universe consisting of cold Planck mass black holes.
In the future, when the open Robertson-Walker universe
has expanded enough to become very dilute,
it may be enough like the original flat Minkowskian vacuum
to repeat the quantum conformal fluctuation process.
(Gunzig, Geheniau, and Prigogine (1987))
Processes of universe-creation are described
by Gott and Li in their paper Can the Universe Create Itself?
Battaner, in astro-ph/9801276, and Battaner and Florido, in astro-ph/9802009, have described a set of nested egg-carton structures using the Octonionic structure of nested Onarhedral lattices, In their model, very large-scale magnetic fields may have played a very important role in building up the present large-scale structure of the Universe, particularly at the MeV Era of evolution of our universe.
In 1997 Charles Steidel of Caltech (Science 276 (4 April 1997) 36) observed walls of galaxies hundreds of millions of light years long at redshifts between 2.8 and 3.5, only 2 billion years after the Big Bang.
In 2001, a 22 May BBC article by David Whitehouse reported that "... New observations are supporting recent computer models that suggest the early Universe was "spongy",
with galaxies forming along filaments, like droplets on a spider's web. ... recent computer simulations of the early Universe have one prediction in common: the first large-scale structures to form were long filaments connected at their ends by "nodes". The models typically look like a three-dimensional spider's web, or perhaps the neuronal structure of a brain. ... It is believed that the first galaxies would have formed inside the threads of the web. When they started emitting light, they would have been seen to mark out the otherwise invisible threads, much like beads on a string. In the course of millions and billions of years, those early galaxies would stream along these threads, towards and into the "nodes". This is where galaxy clusters would be formed later. ... observations, with the European Southern Observatory's (ESO) Very Large Telescope at Paranal, of a region around a quasar, whose light set off when the Universe was only 15% of its present age, have now identified a string of galaxies that map out a tight filament in the early Universe. ...
... One of the researchers involved, Palle Moller ... said. "At this enormous distance, we see it at a time when the Universe was only about two billion years old. This is obviously in agreement with the predictions by the computer models of a web-like structure ..." ....".
Layzer's model needs no pre-existing perturbation anisotropy
to form such structures.
In the cold universe model of Layzer,
there is cluster formation on all scales
and the clustering process continues forever.
Although Layzer bases his cold universe model on hydrogen,
it should be possible to base such a model on Planck-mass
black holes as the cold dark matter in a manner consistent
with the D4-D5-E6-E7-E8 VoDou Physics model.
Another possibility for at least some dark matter
is gravitational interaction from other Worlds of the Many-Worlds.
Particle Creation in the Inflationary Universe
should be such that any inhomogeneity can be contained
in a spherical region within which the average density of mass
is the same as the average density of the entire universe.
At all times during the expansion of the universe,
the cold dark Planck mass black holes constitute
a critical point gas, and therefore unstable against
fluctuations on all scales, particularly unstable against
density fluctuations on the scale of the entire universe
at that time (Layzer (1984)).
The result is structure formation at all scales.
Layzer's model begins with the Clausius equation
2K + (B - 1)U = 3PV ,
where K=kinetic energy, U=potential energy, P=pressure,
V=volume, and B=2 for gravity, with an inverse square force law.
In adiabatic expansion,
d(K + U)/ dt + P dV/dt = 0.
Then:
d(K + U)/dt + (1/3) ((2K + U) / V) dV/dt = 0.
If V is proportional to a^3, where a(t) is the cosmic scale factor,
then
(1/V) dV/dt = (1/a^3) da^3/dt = 3a'/a = 3H
so that
d(K + U)/dt + H(2K + U) = 0
Let U = SUM(i,j) -(1/2) G m'_i m'_j / r_ij , where
G is Newton's constant, m'_i is the excess mass in a cell of
volume dV_i, and r_ij is the distance between cell i and cell j.
Let p" be the mean density,
A = {p - p"} / p" be the relative amplitude of density fluctuations,
and L be the average scale of density variations.
Then, consider V to be a spherical region enough larger than
the size L^3 that any fluctuations inside V can be considered
to be contained entirely within V. In particular, the part of
the universe outside V can then be considered to be of
uniform density and its gravitational influence
inside V can be ignored.
Then U = SUM(i,j) -(1/2) G m'_i m'_j / r_ij =
= -2 pi G p" A L^2 p" V
because INT(theta) INT(phi) INT(r) = 4 pi INT(r) ,
SUM(i in V) m'i = SUM(i in V) {p - p"} dVi =
= SUM(i in V) p" A Vi = p" A L£ ,
{r_ij} = L , and SUM(j in V) dVj = V .
If O is the temperature and N is the number of particles,
P = (2K+U) / 3V = (N O / V) - (2 pi / 3) G A p"^2 L^2 .
Now assume that V is such that, if V is compressed by dV,
p" V remains constant, A = {p - p"} / p" remains constant,
and L =prop= V^(1/3).
Then dV/V = -dp"/p" = 3dL/L and dA = 0 , and
dP = ( (N O / V) - (4/3)(2 pi /3) G A p" L^2 )(-dV/V) =
= (P - (1/9) 2 pi G A p"^2 L^2) (-dV/V)
Layzer's model is based on the expanding universe being like a
vapor at its critical point, dP = 0, unstable against the growth
of fluctuations at all scales.
This requires a cold universe, so that |K| { |U| while 2K+U } 0.
Since
d(K + U)/dt + H(2K + U) = 0
if K+U is negative and 2K+U is positive, expansion causes K+U
to decrease further, so that the magnitude of the (negative)
potential energy U increases still further. If the magnitude
of U increases enough so that 2K+U becomes negative,
K+U increases.
Physically, the increase of the magnitude of the potential
energy U causes clusters of clumps of matter to form.
The clumps within a cluster are accelerated by the fluctuating
gravitational field due to the increase in the magnitude of
the potential energy U. The motion of the clumps then increases
the kinetic energy K and the pressure, quenching the instability.
The processes act to keep the cold universe in its critical state,
in which 2K+U = 0 and dP = 0.
The critical value for the pressure, Pcrit, at dP = 0 , is
Pcrit = (1/9) 2 pi G A p"^2 L^2.
The total energy at Pcrit is E = K+U = U/3.
Then:
K =prop= a(t) ; U =prop= a(t) ; and E =prop= a(t) .
The size of the clumps is of the scale L =prop= a^2 ,
because p"V is constant in time, p" =prop= a^(-3),
U =prop= a, A is constant, and U = -2 pi G p" A L^2 p V.
The mass M of clumps is proportional to p" L^3,
so that M =prop= a^3.
In the expanding universe, a heirarchy of larger and larger
self-gravitating clusters forms, with the self-gravitating
clusters of one stage forming the clumps in the clusters
of the next stage.
The diameter of the clusters is L =prop= M^(2/3).
Layzer estimates that at the onset of instability against
formation of self-gravitating clusters of clumps of matter,
the relative amplitude of density fluctuations
A = {p - p"} / p" is of the order 1/10 to 1/100.
The energy per unit mass e of the cluster is given by
e = (K+U) / p"V = (-(2/3)U + U) / p"V =
= U / 3 p" V = (1/3)(-2 pi) p" A L^2 G .
Since p" =prop= 1/V =prop= a^(-3) and L^2 =prop= a^4 :
e =prop= a =prop= M^(1/3).
Layzer notes that the relationship e =prop= M^(1/3) is
consistent with observation from the scale of Jupiter and
its satellites to the scale of clusters of galaxies.
The clusters of clumps of matter are of the scale of volume
V =prop= a^3, while the clumps of matter within
self-gravitating cluster are of the scale of
volume L^3 =prop= (a^2)^3 = a^6.
Therefore, at some time after the beginning of the
Friedman Robertson-Walker expansion,
L will grow large enough to equal V.
So, on very large scales, larger than clusters of galaxies,
structures are formed after the universe has expanded enough
so that the clumps are so large that they will not all fall
as spherical units into a cluster in the potential wells
of the Layzer process, but some will be stretched and
pulled among nearby clusters, thus forming luminous
filaments and sheets as well as voids.
Consider the stage of the Layzer clustering heirarchy at
which the self-gravitating clusters are the size of glaxies.
The galactic-size cluster should be a self-gravitating
spherical region that is gravitationally dominated by
cold dark Planck mass black holes.
Since the Planck mass black holes have very small
(10^(-66) cm^2) cross section, they can be considered to be
collisionless within the cluster.
The cluster of dark matter can be considered to be an
isothermal ideal gas of pressure pr, density p, and
equation of state pr =prop= p.
As it is self-gravitating, its equation of hydrostatic support is
dpr/dr = (kT/m) dp/dr = -p GM(r) / r^2 ,
where k is Boltzmann's constant, T is temperature, and
m is the Planck mass of the black hole, or
d (r^2 dlnp)/dr) /dr = -(Gm/kT) 4pi r^2 p ,
which is equivalent to a collisionless system
with distribution function
(p_1/(2 pi kT/m)^(3/2)) exp((F - (1/2)V^2) / (kT/m)) ,
which gives
p = p_1 exp(F / (kT/m))
as shown in sectiion 4.4.3(b) of Binney and Tremaine (1987).
As they show, a nonsingular solution for this isothermal
sphere is given by
d (r'^2 dlnp'/dr') /dr' = -9 r'^2 p'
where p' = p/p_0 , r' = r/r_0 , and r_0 = Sqrt(9 kT / m 4 pi G p_0)
is the core radius.
At r { 2 r_0 , p'(r') = 1/(1 + r'^2)^(3/2) , correct to about 5%.
At r } 15 r_0 , p'(r') = (2/9) r'^(-2) , and
the nonsingular isothermal sphere solution approaches
the singular isothermal sphere solution
p(r) = kT / m 2 pi G r^2 .
The circular speed Vc at r is Vc^2 = G M(r) / r ,
where M(r) is the mass inside a sphere of radius r.
From d (r^2 dlnp/dr) /dr = -(Gm/kT) 4 pi r^2 p ,
Vc^2 = -(kT/m) dlnp / dlnr .
The circular speed Vc curve for the nonsingular isothermal
sphere is similar to observed galactic rotation curves
(figure 4-8, figure 10-1, and figure 10-2 of Binney and Tremaine (1987)).
In section 10.1.6, Binney and Tremaine (1987) state that the
density distribution of a dark halo that would give the observed
flat rotation curves at large r
"is also the density distribution for the isothermal sphere at large radii ... .
However, there is no compelling theoretical argument to suggest
why the dark halo should resemble an isothermal sphere."
I disagree: Layzer clustering of cold dark Planck mass
black holes is such a compelling theoretical argument.
Binney and Tremaine (1987, section 4.4.3(b)) state
"From the astrophysical point of view, the isothermal sphere
has a very serious defect: its mass is infinite. Thus from
equations 4-127 and Figure 4-7, we have that
M = 2 s^2 r / G [s^2 = kT / m] at large r.
Clearly no real astrophysical system can be modeled over
more than a limited range of radii with a divergent mass distribution.
On the other hand, the rotations curves of spiral galaxies
(section 10.1.6 and MB section 8-3) are often remarkably flat
out to great radii, and this suggests that we try to construct models
that deviate from the isothermal sphere only far from their cores."
I disagree with the statement that the isothermal sphere mass
distribution is a defect.
Dennis Zaritsky, in astro-ph/9810069, "... collate[s] published results and demonstrate[s] that they are all consistent with a Galactic halo that is nearly isothermal with a characteristic velocity of 180 to 220 km/sec and an extent greater than or equal to 200 kpc. ... All of the data ... are entirely consistent with an isothermal sphere... . There is no evidence for a significant truncation of the mass profile at large radii. ..."
The critical density sufficient to make Omega = 1 is about 1.88 x 10^(-29) h^2 gm/cm^3 (Kolb and Turner (1990)). The galactic rotation curve halo density is on the order of at least (it could be about an order of magnitude greater) 0.01 Msun/pc^3 (Binney and Tremaine (1987), section 10.1.6), or about 0.01 x 2 x 10^33 / (3 x 10^18)^3 = = .0007 x 10^(-21) = 7 x 10^(-25) gm/cm^3. The minimum rotation curve halo density is therefore at least about 3 x 10^4 times greater than the Omega = 1 critical density. In the Layzer clustering model, the isothermal sphere density p at large r varies as p =prop= 1 / r^2. In it, p at r = 3 r_0 = 10 kpc (roughly the 8.5 kpc distance from the sun to the center of our galaxy) is 4 x 10^4 times greater than p at r = 600 r_0 = 2 Mpc (the median radius of clusters of galaxies is about 3 h^(-1) Mpc (Binney and Tremaine (1987), table 1-4)). As a lot of the mass in the universe may be in cold dark matter, Layzer's model should describe structure formation on scales large enough that gravity is the dominant force (structures at planetary scale or larger). On smaller scales, where electromagnetism or other forces are stronger, the cold dark matter (being very weakly interacting with respect to forces other than gravity) should be ignored or considered as a background, with the standard hot big bang model applying to the small scale processes. Except for gravitational interaction, the cold dark matter would be decoupled from the hot ordinary matter and radiation at all times after the end of inflation. The radiation would decouple from the ordinary matter about 200,000 years after the end of inflation. Structure in the Layzer process is always subhorizon in size, so that anisotropy of the microwave background is small-angular scale, O much less than 1 deg = decoupling Hubble scale (Kolb and Turner (1990), section 9.6.2), and not measurable by COBE. With two classes of matter (cold dark matter forming structure according to Layzer's theory and ordinary matter having a lesser role to play on gravitational scales because it is much less massive), the fact that the ordinary Jeans mass after decoupling is about the mass of a globular cluster indicates an ordinary process of globular cluster formation within the structures already formed at that time by Layzer's cold matter process. de Vega, Sanchez, and Combes suggest that the fractal structure of the InterStellar Medium of our galaxy, on scales from about 20 AU to about 300 light-years, may be due to self-gravity of isothermal clouds, rather than cascades of turbulence due to galactic rotation. Shear of galactic rotation destroys fractal structure above 300 light-years in size, and stellar radiation destroys fractal structure below 20 AU and in regions of dense stellar formation and/or radiation.
Binney, J. and Tremaine, S. (1987), Galactic Dynamics (Princeton, Princeton). Dolgov, A. (1980), Sov. Phys. JETP 52, 169. Gott, J. R. (1982, 28 January), Nature 295, 304. Gunzig, E., Geheniau, J., and Prigogine, I. (1987), Nature 330, 621. Kolb, E., and Turner, M. (1988), The Early Universe: Reprints (Addison-Wesley, Redwood City, Calif.). Kolb, E., and Turner, M. (1990), The Early Universe (Addison-Wesley, Redwood City, Calif.). Kugo, T., and Townsend, P. (1983), Nuc. Phys. B221, 357. Lawson, H., and Michelsohn, M.-L. (1989), Spin Geometry (Princeton, Princeton). Layzer, D. (1984), Constructing the Universe (Freeman, N.Y.). Linde, A. (1984), Rep. Prog. Phys. 47, 925. MacGibbon, J. (1987), Nature 329, 308. Narlikar, J., and Padmanabhan, T. (1986), Gravity, Gauge Theories and Quantum Cosmology (Reidel, Boston) Peebles, P. J. E. (1993), Principles of Physical Cosmology (Princeton, Princeton). Taubes, G. (1994, 25 March), Science 263, 1682, 1683. Weinberg, S. (1972), Gravitation and Cosmology (Wiley, New York). Weinberg, S. (1988), The First Three Minutes (Basic books). Weinberg, S. (1989), Rev. Mod. Phys. 61, 1.
......