Widdekind

One can imagine various mathematical forms modeling the slowing expansion, or even contraction, of regions of Structure. In the logical limit, the volume of regions of Structure vanishes: [math]V_V + V_S = \frac{1}{2} a^3 + \frac{1}{2} a^3 \longrightarrow \frac{1}{2} a^3 + 0[/math] In that limit, the scale factor begins growing nearly half-again as fast: [math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{2} \alpha^3 + \frac{1}{2} \alpha^3 } = \frac{dt}{T_0} [/math] [math] \longrightarrow[/math] [math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{2} \alpha^3 + 0 } = \frac{dt}{T_0} [/math] [math] \longrightarrow[/math] [math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{2}} \sqrt{ \alpha^3 } = \frac{dt}{T_0} [/math] So, for a given increase in the scale factor, the shrinkage of structure reduces the LHS by the square-root of two; so, the RHS is reduced, i.e. less time passes during said expansion of the scale factor, i.e. expansion accelerates. That zero-size limit for Structure is (seemingly) the upper bound, to the effective acceleration of the (growth of the) scale factor, admissible, in this "Structure stall out" model. Can not the density be treated as treated above? Could not the secession, of Structure, from the Hubble flow, cause and account for, the inferred acceleration of the expansion of the universe?

i suspect i may have made a mistake somewhere above. Please ponder the first Friedmann equation: [math]\frac{1}{a} \frac{da}{dt} = H_0 \sqrt{\frac{\Omega}{\tilde{V}_V + \tilde{V}_S}}[/math] If Structure stops expanding (as rapidly as Voids), then the volume occupied by Structure is less than it "should" be. That reduces the denominator, on the RHS. To compensate, on the LHS, the denominator (dt) must decrease, i.e. less time (dt) passes, for the same (percent) change in scale factor (da/a), i.e. the scale factor starts to "accelerate" its expansion, compared to what it "should" be doing. So, i offer, that Structure "stalling" out, could cause (from these crude calculations) the scale factor to, in turn, accelerate in its growth, so explaining the reported analyses of observations regarding SNIa. seeking simplicity, redefine "now" (z=0) so that the "present" epoch is the era when Structure (supposedly) started to slow its stretching, and "stall out" from the general expansion of the universe, afterwards occurring only in Voids. For sake of simplicity, assume a flat cosmology, wherein V & S each occupy half of all volume at "present" epoch; to "present" (z>0) the volumes occupied by V & S ~ (1/2)a3 + (1/2)a3; afterwards (z<0) the volumes increase as ~ (1/2)a3 + (1/2)a2. The equations simplify considerably; and, do indeed, show that "Structure stall out" causes the scale factor growth to accelerate:

What would happen, to the expansion of the universe, if Structure stopped stretching (as fast as the Voids)? Qualitatively, assuming a flat cosmology; and assuming that, in recent epochs (z<2) Structure has been expanding more slowly than Voids; so that by present epoch (z=0) Structure occupies only a quarter of the volume of space, whereas Voids occupy three quarters: [math]\rho = \frac{M_S + M_V}{V_S + V_V} \approx \frac{M_{tot}}{V_{tot} \left( \frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3 \right) } = \frac{\rho_0}{\frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3} [/math] where [math]\alpha \equiv a(t)/a_0[/math] is the normalized scale factor. So, from the first Friedmann equation: [math]\frac{d\alpha}{\alpha} = \frac{dt}{T_0} \sqrt{ \left( \frac{\Omega_0 = 1}{\frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3} \right)}[/math] [math]\frac{d\alpha}{\alpha} \sqrt{ \frac{1}{4} \alpha^2 + \frac{3}{4} \alpha^3 } = \frac{dt}{T_0} [/math] [math]\frac{3 d\alpha}{2} \sqrt{ \frac{1 + 3 \alpha}{4} } = \frac{3 dt}{2 T_0} \equiv d\tau [/math] [math]\Delta \left( \frac{4}{3} \left( \frac{1 + 3 \alpha}{4} \right)^{\frac{3}{2}} \right) = \Delta \tau[/math] Such is the solution, for scale factor vs. lookback time, for slowly stretching Structure, from present epoch (z=0), back to the epoch when Structures occupied the same (relative) volume as Voids (z=2). At that epoch (z=2), the combined volumes of Structure & Voids was only 1/18th that at present epoch. So, the cosmic average density was 18x that at present epoch. Thus, for earlier epochs (z>2), the above solution must be matched to a (re-scaled) standard solution, for flat matter-dominated expansion: [math]\frac{d\alpha}{\alpha} = \frac{dt}{T_0} \sqrt{ \left( \frac{\Omega_2 = 18}{ \alpha^3} \right)}[/math] [math]\Delta \left( \alpha^{\frac{3}{2}} \right) = \Delta \left( \tau \sqrt{18} \right)[/math] Calculating the lookback time to (z=2), then adding the same to the second solution, and plotting, reveals the following final evolution for the (normalized) scale factor (x-axis) vs. lookback time (y-axis): The scale factor initial grows with increasing time (decreasing lookback time) as (a~t2/3), until (z=2), when Structures start to "stall", stretching slower. The slower stretching of Structure "holds everything up"; the scale factor grows much more slowly with increasing time thereafter, e.g. space only expands by a factor of 18x from (z=(2-0)) instead of 27x. i was wondering if slowly expanding Structure could mimic the reported faster expanding Voids from current SNIa observations. However, slower stretching Structure effects the opposite of what was reported -- overall, spacetime stretches slower if Structure stalls out, not faster. So, slower stretching Structure does not match those reports. i offer the above equations for discussion -- are there any reasons why one could not make the density, in the RHS of the first Friedmann equation, any arbitrary function of the scale factor?

i understand, that the detection of (partial) polarization, in CMB photons, by the WMAP, implies a high column density, of (ions &) electrons, along the LOS to earth. So, the conclusion has been reached, that Reionization "must" have occurred early (z~(30-10)) so as to force CMB photons to propagate through hundreds of millions of years more ionized plasma, as compared to Quasar spectra, which date Reionization to later epochs (z~6). Column density is the (integrated) product of number density times path length. Would a higher baryon density ([math]\Omega_B \rightarrow 1[/math]) admit a lower Reionization redshift, i.e. shorter path length, so reconciling WMAP to Quasar spectra?

core-collapse SNII spray everything in (their) sight with neutrons, thereby neutron-enriching elements, far heavier than the iron peak elements. So, could cc SNII create Deuterium, by neutron-enriching hydrogen (the most abundant element in and near them) ? According to A Question and Answer Guide to Astronomy by Pierre-Yves Bely, Carol Christian, Jean-René Roy; the oldest known stars, in Globular Clusters, are enriched in heavy elements, implying that they formed, from intergalactic gas, already enriched, by an earlier generation of primeval stars, of immense mass & luminosity, which reionized the intergalactic medium. If primeval stars processed primordial gas, then perhaps they destroyed primordial Lithium, or created additional Deuterium? How much effect could primeval stars, have had on primordial gas composition ?

The de-ionization of pan-cosmic primordial plasma impacted space sounds, i.e. "Baryon Acoustic Oscillations (BAO)". In the early universe, ionized plasma was supported by radiation pressure from photons. The extra pressure support drastically increased sound speeds (even to trans-luminal levels), and so drastically increased Jeans' wavelengths. After universal expansion diffused radiation, and cooled matter, the de-ionized space gas, oblivious to photons, lost radiation pressure support. Jeans' wavelengths plummeted, from "large" scale wavelengths (~100Mpc = BAO = CMB 1-degree anisotropies), to "small" scale wavelengths (~1kpc = globular star clusters): [math]P = P_R + P_M[/math] Assuming that matter clumps entrained clumps of photons; and assuming that matter clumps contracted & expanded adiabatically; such that local matter temperature always equaled local radiation temperature, within contracting / expanding clumps; then: [math]P_R = \frac{a T^4}{3}[/math] [math]P_M = P_0 \left( \frac{\rho}{\rho_0} \right)^{5/3}[/math] [math]\frac{T}{T_0} = \left( \frac{\rho}{\rho_0} \right)^{2/3}[/math] [math]P = P_{R,0} \left( \frac{\rho}{\rho_0} \right)^{8/3} + P_{M,0} \left( \frac{\rho}{\rho_0} \right)^{5/3}[/math] [math]C_S^2 \equiv \frac{\partial P}{\partial \rho} |_{\rho=\rho_0} = \frac{1}{\rho_0} \left( \frac{8}{3} P_{R,0} + \frac{5}{3} P_{M,0} \right)[/math] [math] = c^2 \left( \frac{8}{9} \frac{u_R}{u_M} + \frac{5 k_B T}{3 \bar{m} c^2} \right) [/math] [math] \approx c^2 \left( 10^{-4.5} + 10^{-8} \right) \times (1+z)[/math] (ionized) [math] \approx c^2 \left( 0 + 10^{-8.5} \right) \times (1+z)[/math] (neutral) De-ionization drastically reduced the sound speed, by about two orders of magnitude, so reducing the Jeans' wavelength, by a similar amount (from ~100Mpc to ~1Mpc ?). But, how coupled was matter to energy? The Thompson cross section is so small, that the mean free photon path, through space plasma, is enormous: [math]\lambda_0 = \frac{1}{n_{e,0} \sigma_{T,0}} \approx 10^{27.5} m \approx 10^{11.5} ly \approx 10^{11} pc = 100 Gpc[/math] [math]\lambda \approx 100 Gpc \times \left( 1+z \right)^{-3}[/math] assuming full ionization. The Jeans wavelength (L) is approximately equal to the sound-speed multiplied by the free-fall time [math]\left( C_S / \sqrt{G \rho} \right) \propto (1+z)^{-1}[/math]. At redshifts z~(1000-100): [math]\lambda \approx 0.1 - 100 Kpc[/math] [math]L \approx 1 - 10 Kpc[/math] So, photon mean-free paths, through the ionized space plasma, began to become comparable to the Jeans' wavelengths, by the epoch of de-ionization (z~100). And so, radiation would have ceased supporting space plasma, against gravity collapse, at long wavelengths of space sound. At redshifts z~(100-0): [math]\lambda \rightarrow \infty[/math] [math]L \rightarrow 0.1 - 10 Kpc[/math] And so, loss of radiation pressure support, drastically reduced the capacity of space gas to propagate low frequency "bass notes". Space sounds with wavelengths longer than (0.1-10)Kpc began collapsing into clumps, instead of transmitting pressure perturbations (according to these calculations). Such size scales, resemble large star clusters, and small proto-galaxies[1]. [1] Collecting & calculating correct numerical factors, L ~ 20 Kpc / (1+z). Cp. http://www.astronomy.ohio-state.edu/~dhw/A825/notes6.pdf.

On second thought, i think the equations from the cited website, represent the comoving quantities, on the comoving grid. For the comoving quantities, on the comoving grid, i suspect that the continuity equation holds, in its regular form. For, the product of comoving velocity x comoving density represents the flux of mass, per comoving area; and the comoving divergence of the same, seemingly "must" represent the amount of mass accumulated into, or dispersed from, a given comoving grid cell. As for the "force" equation (which ultimately derives from Newton's law, ma=F), i suspect the proper procedure, is to proceed from the "real" equation, and substitute in for the "real" quantities, their corresponding equivalents, in comoving quantities x appropriate powers of the scale factor. For, the force equation models real actual physical forces applied, to fluid parcels (RHS), which generate real actual physical acceleration of said parcels (LHS). So, the equation is valid in "real" actual physical quantities & units; the proper procedure "must" then be to substitute in for the real quantities [math]\rho \left( \frac{\partial}{\partial t} + v \circ \nabla \right) v = - \nabla P[/math] [math]\frac{\tilde{\rho}}{a^3} \left( \frac{\partial}{\partial t} + a \tilde{v} \circ \frac{\tilde{\nabla}}{a} \right) a \tilde{v} = - \frac{\tilde{\nabla}}{a} P[/math] [math]a^{-2} \times \tilde{\rho} \left( \frac{\partial}{\partial t} + H + \tilde{v} \circ \tilde{\nabla} \right) \tilde{v} = - \frac{\tilde{\nabla}}{a} P[/math] The above equation more closely resembles those from said cited website, e.g. adding the term involving "H", instead of subtracting. However, on the LHS, i still do not understand why some terms would involve the scale factor, whilst others would not. And, on the RHS, considerable care must be applied, to what is meant by "Pressure", and (possibly) converting the same (Newtons per square real meter) to comoving coordinates ([comoving?] Newtons per square comoving meter). Separately, in the gravity equation: [math]- \nabla \circ g = 4 \pi G \rho[/math] the Gravity constant (G) has physical units (real meters cubed / mass / time squared). So, on a comoving grid, perhaps the real physical G must be converted, to a comoving [math]\tilde{G} = G a^{-3}[/math] ?

According to the Saha equation, applied to a pure hydrogen gas: [math]n \frac{f^2}{1-f} = \mathcal{N}_0 \tau^{\frac{3}{2}} e^{-\frac{1}{\tau}} \equiv \mathcal{N}(\tau)[/math] where: [math]f \equiv \frac{n_e}{n}[/math] [math]\mathcal{N}_0 \equiv \frac{1}{ \lambda_C^3 } \left( \frac{ 2 \pi \chi }{ m_e c^2 } \right)^{\frac{3}{2}}[/math] [math]\tau \equiv \frac{k_B T}{\chi}[/math] [math]\chi \approx 14 eV[/math] and where i have attributed ions (protons) two degrees of freedom (g+ = 2), and neutrals (hydrogens = protons + electrons) four degrees of freedom (g0 = 4). Ancient Greek scholar Pythagoras gives the ionization fraction: [math]f = \frac{\mathcal{N}(\tau)}{2 n} \left( \sqrt{1 + \frac{4 n}{\mathcal{N}(\tau)}} - 1 \right)[/math] Cosmically, [math]n \propto (1+z)^3[/math] [math]T \propto (1+z)[/math] Plugging in the appropriate numbers (perhaps correctly), and plotting the resulting ionization fraction, as a function of redshift, f(z): http://www4a.wolframalpha.com/Calculate/MSP/MSP78261a441cc510d470g50000443f9hf1bad436df?MSPStoreType=image/gif&s=58&w=300&h=189&cdf=RangeControl plot (Sqrt(1 + 4.5e-24 * z^1.5 * Exp(1/(2.3e-4 * z)) ) - 1)/(2.25e-24 * z^1.5 * Exp(1/(2.3e-4 * z))) from z=1 to 200 shows that (according to these equations) space gas de-ionized over the redshift range z~(100-80). Prof. Abraham Loeb states that "residual electrons [and ions]" existed, in sufficient number density to couple space gas to the CMB, until redshifts z~(200-160). From where arises the "canon" figure of z~1000, for the redshift of Reionization ?

i understand, from some slides prepared & published by Prof. Yicheng Guo (U.MA), that during early epochs (z~5-1), the fraction of stars residing in well-formed Elliptical galaxies increased from negligible to half; and that by that epoch (z~1), emerging Ellipticals were "born" being about 1Kpc across. At earlier epochs, Irregular galaxies were the most common. i conclude from those facts, that clumpy cloud-like Irregular galaxies (Irr-1) gave rise to both Elliptical, and disky Spiral, galaxies, during early epochs (z~5-1). Those collapsing cloud-like clumps generated compact Elliptical (cE) galaxies, ~1Kpc, i.e. about as big as a Bulge in a disk galaxy. Parsimoniously, if over-simplistically, perhaps (nearly) all clumpy cloudy Irregulars collapsed into Bulge-sized compact Elliptical galaxies, with half becoming Ellipticals, and half becoming Spirals? Perhaps the difference between the former & latter is whether the clump cloud collapsed "dry" (no remaining gas) as Ellipticals, or "wet" (lots of leftover gas) as Spirals? Parsimoniously, perhaps the Hubble sequence can be summarized, with the two kinds of Irregular galaxies being the "on-ramp" and "off-ramp" to & from the sequence: Irr-1 = proto-galactic cloud clump, which collapses into... S0 = Bulge-sized compact Elliptical (cE), ~1Kpc across... dry? no relic gassy halo => small Ellipticals wet? gassy halo "settles", "disking down" into surrounding swirling disk => Spirals tidal disruption? close encounters distort & disrupt galaxies => Irr-2 merger? mergers (in busy "urban" clusters) generally "dry" the colliding galaxies, shock-heating their gassy halos, into surrounding Intra-Cluster Medium (ICM), and mingling their ellipsoids into single, larger, super-sized cluster-central large Ellipticals (cD) If so, the Irr-1 are collapsing cloud clumps, leading "onto" the Hubble Sequence; whereas Irr-2 are tidally disrupted & distorted systems, undergoing (or emerging from a recent) close encounter, and so exiting "off of" the Hubble Sequence; and larger Ellipticals generally result from full-fledged major mergers.

The equations for Fluid Dynamics, on static grids, are well known, e.g. mass continuity: [math]\frac{\partial \rho}{\partial t} + \vec{\nabla} \circ \left( \rho \vec{v} \right) = 0[/math] Those equations must (?) remain the same, on expanding grids, for comoving quantities. For example, the amount of mass within a comoving cell (derivative of comoving density) can only change, by (net) mass flow into / out of the same cell, from surrounding cells, which (net) flow is measured by the divergence. If you measured all distances with an expanding ruler, then you would never notice that your fluids were evolving on an expanding grid. The only "clue" you might get, would be that the fluid would exhibit a seemingly strange sort of dissipation, such that comoving speeds decayed away -- for a steady absolute space speed, the expansion of space [math]\left( a(t) \rightarrow \infty \right)[/math] would translate the same absolute space speed [math]\left( v \right)[/math], to fewer and fewer (larger & larger) comoving distance units [math]\left( \tilde{v} = v / a(t) \rightarrow 0 \right)[/math]. Please ponder, that the scale factor a(t) possesses important physical units, namely "actual meters of distance per comoving meters of distance" (say). Thus, the scale factor is not dimensionless. So, to translate the equations of FD, from comoving quantities (on a comoving grid) to actual quantities, translate the comoving quantities to actual quantities, within the comoving equations: [math]\frac{\partial \tilde{\rho}}{\partial t} + \tilde{\nabla} \circ \left( \tilde{\rho} \tilde{v} \right) = 0[/math] [math]\tilde{\rho} = \rho \times a(t)^3[/math] [math]\tilde{v} = v \div a(t)[/math] [math]\tilde{\nabla} = \nabla \times a(t)[/math] [math]\therefore \frac{\partial ( \rho a^3 )}{\partial t} + a \vec{\nabla} \circ \left( a^3 \frac{v}{a} \right) = 0[/math] [math]\boxed{\frac{\partial \rho}{\partial t} + 3 H \rho + \vec{\nabla} \circ \left( \rho v \right) = 0}[/math] where the last line derives from the spatial invariance of the scale factor (which depends only upon time), and then by dividing the entire previous equation, by a couple of powers of the scale factor. Please pay particular attention, to the role of the scale factor ("actual length / comoving length"), in translating from comoving to actual quantities. Similarly, for momentum, [math]\frac{\partial \tilde{v}}{\partial t} + \left( \tilde{v} \circ \vec{\nabla} \right) \tilde{v} = -\frac{\vec{\nabla} \tilde{P}}{\tilde{\rho}}[/math] For the LHS, we proceed as per previous: [math]LHS = \frac{\partial (v a^{-1})}{\partial t} + \left( (v a^{-1}) \circ (a \nabla) \right) (v a^{-1})[/math] [math]= a^{-1} \times \left( \frac{\partial v}{\partial t} - Hv + \left( v \circ \nabla \right) v \right)[/math] For the RHS, we first notice the units -- dividing out the density from denominator & numerator, the relative pressure amounts to the comoving gradient, of comoving temperature. Thermodynamically, temperature, definable as the square of a sound speed, measures the square of the characteristic (microscopic) space speed of the particles. As alluded to above, if the actual speed is steady, the comoving speed, measured with expanding comoving rulers, seems to decay away: [math]\tilde{T} \approx \frac{<\tilde{u}^2>}{m} = a(t)^{-2} \times \frac{<u^2>}{m} = a^{-2} T[/math] So, [math]RHS = -\frac{a \nabla (a^{-2} P)}{\rho}[/math] [math]= a^{-1} \times \left( -\frac{\nabla P}{\rho} \right)[/math] And so (?) [math]\boxed{\frac{\partial v}{\partial t} - Hv + \left( v \circ \nabla \right) v = -\frac{\nabla P}{\rho}}[/math] i want to question the equations stated on the following webpage: http://ned.ipac.caltech.edu/level5/March03/Bertschinger/Bert2_2.html In particular, please pay particular attention to the units embodied in the scale factor ("actual distance / comoving distance"). So, i perceive, that the correct equations cannot rescale some quantities, and not others -- to me, such equations seem "confused", "muddling" comoving lengths, in some terms, with actual lengths, in other terms. Because the units balanced in the original equations (comoving quantities, in comoving coordinates), so all length scales must all be equally translated, from comoving distances, to actual distances, by the same number of powers of the scale factor. The correct equations cannot (?) mix and match, with some terms involving, and others omitting, the scale factor. Would somebody please explain, if (and if so, then why) my boxed equations are (or are not) correct, as compared to those in the aforecited website? i feel i have made no errors in physical analysis, or mathematical derivation.

If you ponder a flat, matter-dominated (matter-only) universe; then the proper distance to the visible edge of our universe asymptotes to 2c/H0 ~ 27Gly ~ 30Gly. And then, the one-degree CMB anisotropies represent structures which were (z~1000) about (1/60 radian) x 30Gly / (1+z) ~ 0.5Mly across; and which hypothetically are expanded to a size ~0.5Gly at present epoch (z~0), ignoring gravitational collapse. That is quite close to the ~100Mpc scale of cosmic uniformity. And there are two smaller, secondary anisotropies, at scales of a third & a quarter degree. And more, our universe is plausibly (slightly) closed; a closed cosmology would dilate distant objects larger on our skies. So a combination of "matter only" and "closed curvature" could translate directly-observed one-degree anisotropies, into smaller physical size scales, at and below the directly-observed scale of cosmic uniformity. Guided by the following figure, which maps out space within 100Mly of earth, i.e. is 200Mly across; CMB anisotropies could plausibly correspond to large galaxy super-clusters, e.g. Eridanus/Fornax, Virgo; and, oppositely, Voids: Speculating, that "matter only" + "closed curvature" + "CMB anisotropies became super-clusters & Voids" seems common-sensical & parsimonious. EDIT: The following figure, mapping space out 1Gly, seemingly shows anisotropy (Voids & Super-Clusters) on scales of ~100Mpc. Perhaps the following figure can be directly correlated to CMB one-degree anisotropies?

i learned about the "Dn-sigma" & "Holmberg" relations from Galaxy Formation & Evolution by Mo, van den Bosch, and White (p.271,718):

i understand, that the anisotropies are enormous (bigger than our sun or moon on our sky), and would correspond today to volumes ~Gly across. Meanwhile, during the epoch of "reionization" (z ~ (30-6)), stars formed in globular star cluster massed clumps, on size scales over 6 decades smaller. Then, during the Quasar & star-burst era (z ~ (6-1)), the largest of those star clusters (~108Msun) formed central IMBH, and "grew" into galaxies, by further "secondary" star formation. If so, then the "seeds" of structure were over 6 decades smaller, than the CMB fluctuations -- which, again, are larger than the size scale beyond which our universe is already considered uniform, isotropic, homogenous. i understand that current cosmology computer simulations still do not span such a dynamic range. i want to understand why inflation is "well established" -- if humans cannot in principle observe back before (z ~ 1000); then how have humans confirmed inflation at (z ~ infinity) ?

Cosmic strings & inflation are speculations, yes? No observations connect one-degree CMB anisotropies, at z~1000, to anything at present epoch, z~0? CMB anisotropies are astronomically large (one degree); at present epoch, they would be about 1Gly across.

OOPS -- Lyman-alpha forest "clouds" are large (~1Mly) (omitted baryon mass fraction) According to Mo & van den Bosch' Galaxy Formation & Evolution, from direct observations, Lyman-alpha forest "clouds" are large (~1Mly). "Clouds" cluster, and dis-cluster, on larger scales (~100Mly). Inexpertly, "clouds" are fully ionized for column densities <1014cm-2; strongly ionized for column densities <1018cm-2. "Clouds" seem associated with the peripheries of galactic structures. Separately, DLAs are weakly ionized for column densities of ~1019cm-2; and increasingly neutral for column densities >1020cm-2. DLAs seem associated with emerging spiral galaxy disks. For a characteristic "cloud" of (say) N ~ 1016cm-2, fN ~ 10-4, R ~ 1Mly: [math]N(HI) \approx f_N n R[/math] [math]n \approx \frac{N(HI)}{f_N R} \approx 10^{-4}cm^{-3}[/math] At redshift (z~5), critical density was (equivalent to) about 10-3cm-3. Separately, the hot halo of the Milky Way contains most of a hundred billion solar masses, in a volume most a million light-years across; from which numbers i calculate an average space density of 10-4cm-3. If so, then "clouds" resemble galaxy hot halos. For a maximal DLA of N ~ 1023cm-2, fN ~ 1, R ~ 1Mly: [math]N(HI) \approx f_N n R[/math] [math]n \approx \frac{N(HI)}{f_N R} \approx 10^{-1}cm^{-3}[/math] Thus, the estimated densities and temperatures (~104K) of DLAs resemble the warm (and most common) component of the Milky Way's ISM. Given those recent observations, of a vast circumgalactic halo of hot x-ray emitting gas, enveloping the Milky Way, most of 1Mly across; i understand, that low neutral density "clouds" could correspond to sightlines threading through circum-galactic hot halos; and DLAs to sightlines threading through the denser proto-galaxies themselves. Over billions of years, from (z~3-0), halos became heated, and increasingly ionized. Please ponder a sightline threading through our Local (Virgo) Cluster; absorption features from heating halos ("clouds") and proto-galaxies (DLAs) would clearly cluster together within emerging Cosmic Structure, and dis-cluster from emerging Cosmic Voids: Paradoxically, "growing" galaxies, the site of ongoing star-formation, and (hence) sources of ionizing photons, were also the last bastions of neutral hydrogen in our universe.

OOPS -- Lyman-alpha forest "clouds" are large (~1Mly) (omitted baryon mass fraction) According to Mo & van den Bosch' Galaxy Formation & Evolution, from direct observations, Lyman-alpha forest "clouds" are large (~1Mly). "Clouds" cluster, and dis-cluster, on larger scales (~100Mly). Inexpertly, "clouds" are fully ionized for column densities <1014cm-2; strongly ionized for column densities <1018cm-2. "Clouds" seem associated with the peripheries of galactic structures. Separately, DLAs are weakly ionized for column densities of ~1019cm-2; and increasingly neutral for column densities >1020cm-2. DLAs seem associated with emerging spiral galaxy disks. For a characteristic "cloud" of (say) N ~ 1016cm-2, fN ~ 10-4, R ~ 1Mly: [math]N(HI) \approx f_N n R[/math] [math]n \approx \frac{N(HI)}{f_N R} \approx 10^{-4}cm^{-3}[/math] At redshift (z~5), critical density was (equivalent to) about 10-3cm-3. Separately, the hot halo of the Milky Way contains most of a hundred billion solar masses, in a volume most a million light-years across; from which numbers i calculate an average space density of 10-4cm-3. If so, then "clouds" resemble galaxy hot halos. Given those recent observations, of a vast circumgalactic halo of hot x-ray emitting gas, enveloping the Milky Way, most of 1Mly across; i understand, that low neutral density "clouds" could correspond to sightlines threading through circum-galactic hot halos; and DLAs to sightlines threading through the denser proto-galaxies themselves. Over billions of years, from (z~3-0), halos became heated, and increasingly ionized. Please ponder a sightline threading through our Local (Virgo) Cluster; absorption features from heating halos ("clouds") and proto-galaxies (DLAs) would clearly cluster together within emerging Cosmic Structure, and dis-cluster from emerging Cosmic Voids: Simplistically, Quasar spectra are emitted "blue", and "redden" as they propagate through space & time, to earth today. As their spectra propagate whilst reddening, they occasionally encountered "clouds" containing neutral hydrogen. Those clouds absorbed out (some) photons near 1215A in the local rest frame ("their, then"). But the photons absorbed had "reddened into" (redshifted into) the 1215A near-UV band, from "bluer" higher-energies in the farther-UV. Then, once imprinted into a Quasar spectrum, the absorption lines are also redshifted out to ever longer wavelengths. Successive encounters, with a series of such "clouds", so results in a series of absorption "lines", from wavelengths near that of the Quasar's own 1215A emission peak (redshifted to 1215A x (1+zQ)), towards ever shorter wavelengths (redshifted to 1215A x (1+zc), where zc < zQ). Now, at short cosmological ranges, proper distances are (approximately) proportional to the difference in redshift, i.e. Hubble's Law: dL = c dz / H(z) = c dz / H0 h(z) = c T0 dz / h(z) = D0 dz / h(z) h(z) = 2/3 (1+z)3/2 And the redshift of, and between, "clouds" is implied by the differences between their redshifted originally-1215A wavelengths: w = 1215A x (1+z) dw = 1215A x dz dz = dw/1215A (1+z) = w/1215A And so, dL = 1.5 D0 dz / (1+z)3/2 = 20Gly x dz / (1+z)3/2 The only freely-available internet images of the spectrum of QSO 1425-6039 seems to be provided by CalTech. The following figure is reproduced in Dan Moaz' Astrophysics in a Nutshell (pg.227): (EDIT: Johan Hidding has also posted a webpage, providing a picture, of the same spectrum.) From the figure, "clouds" cluster, and dis-cluster, across changes in wavelength of ~50A, near wavelengths of ~4800A, corresponding to estimated distance scales of: dz ~ 1/25 (1+z) ~ 4 dL ~ 20Gly x (1/25) / 43/2 = 0.1Gly = 100Mly So, at that distant region, at that ancient epoch (z~3), the local "space weather" was "cloudy" and "clear", on proper distance scales, of about 100Mly. Are these calculations correct? According to Dan Maoz' seemingly cogent & clear cosmology book, the "clouds" are warm (T ~ 104K) and nearly ionized (neutral fraction f ~ 10-4), whilst the "clearings" are hot (T ~ 105-6) and fully ionized. According to Hyron Spinrad's Galaxy Formation & Evolution (pg.11), the "clouds" have neutral column densities of 1014-18cm-2, and spatial densities a hundred times the cosmic average. The latter translates into about one particle per cubic centimeter (at redshift z~5). With a neutral fraction of one ten thousandth, and so neutral space densities of one ten thousandth neutral particle per cubic centimeter, the calculated column densities demand sizes of 1 to 10,000 ly, i.e. from protostellar nebulae to globular star clusters to dwarf galaxies. Are these calculations correct? EDIT: The calculated sizes, and estimated total space densities, imply masses ranging from 0.001 to 1 billion Msun, also suggesting protostellar nebulae to globular star clusters to dwarf spheroidal galaxies. Damped Lyman-alpha Absorbers (DLAs) According to Spinrad, DLAs have neutral column densities of 1020-23cm-2, and spatial densities scaling commensurately from 102-5 times the cosmic average. The latter translates into about 1-1000 particles per cubic centimeter (at redshift z~5). DLAs are a different kind of "cloud", having higher column densities, densities, and metallicities. So, assuming a neutral fraction of one, and so neutral space densities also of 1-1000 particles per cubic centimeter, the calculated column densities, and estimated space densities, all demand sizes of 100 ly; and imply masses of 1000 to 1 million Msun; all suggesting globular star clusters; and consistent with the cold component of the Interstellar Medium (ISM). However, if DLAs have lower neutral fractions (f ~ 10-4), then they would be much bigger (1Mly), and implausibly more massive (1015-18Msun). If DLAs had intermediate neutral fractions (f ~ 10-2), then they would be bigger (10Kly), and plausibly more massive (109-12Msun). DLAs are commonly associated to spiral galaxy disks (perhaps because of their higher metallicities, Z ~ 0.01-10 Zsun, similar to spiral galaxy disks, having ongoing star-formation), favoring the intermediate model, which is consistent with the warm (and most common) component of the ISM (cp. Jedamzik & Prochaska, "velocity width dispersion favors large & thick disks with small neutral gas fraction"). physical picture -- protogalaxies & halo clusters ? i understand, that Lyman-alpha clouds correspond to star clusters & dwarf galaxies; whereas DLAs correspond to large spirals. In the following figure, freely available on the internet from Atlas of the Universe, the large spiral galaxies in our Local Group (Andromeda, Milky Way, Triangulum) could have generated DLA absorption features in ancient Quasar spectra; whereas all of their satellite globular star clusters & dwarf galaxies could have generated Lyman-alpha cloud features, in such spectra: Is my understanding consistent with current cosmologist consensus? Quasar accretion disks ? From the formula for Doppler Shift, the relative broadening of the Quasar's 1215A emission implies the velocity dispersion, of the emitting neutral hydrogen: [math]\frac{\Delta f}{f_0} \approx \frac{\Delta \lambda}{\lambda_0} \approx \frac{v}{c}[/math] For QSO 1425-6039: [math]\frac{\Delta \lambda}{\lambda_0} \approx \frac{1}{25}[/math] Assuming the emitting neutral hydrogen was swirling around a central SMBH, within an emerging "growing" protogalaxy, then the gas was orbiting with speeds around ~10,000 km/s. Such speeds, assuming a mass for said SMBH, imply the size of the assumed accretion disk: [math]\left(\frac{v}{c}\right)^2 \approx \frac{G M}{c^2 R} \approx \frac{R_S}{R}[/math] [math]\approx 10^{-4}[/math] For SMBH masses of a million to a billion solar masses, implied accretion disk radii range from 100AU to 1ly, from larger than our solar system, to nearly the distance from earth to the nearest star. Is this interpretation of the Quasar's broad Lyman-alpha emission peak appropriate? If the Quasar (jets) is aimed at earth, then the accretion disk would be seen face on, and evidence no radial, Doppler-shift-inducing, motion. If so, then actual space speeds might be higher by an inverse sin or cos function. (EDIT: The only clear & cogent picture explaining Quasars easily available on the internet comes from CalTech. Quasars are observed off-jet-axis by about 30 degrees.)

The degree-scale anisotropies observed on the CMB (few parts per million) at redshift (z~1000) correspond to volumes of space, at present epoch (z~0), hundreds of Mpc across. Please ponder, that the CMB anisotropies are a degree across, at (z~1000); in their immediate foreground (z~10), cosmically almost as far away, proto-galaxies are tens of thousands of times smaller, only tenths of arc-seconds across. That's like a coin on a football field, as compared to the entire stadium; or a transiting exoplanet, as compared to its star. Whatever were the "blobs" which generated the CMB anisotropies, they have expanded today into vast volumes of space, larger than the scale of cosmic uniformity / homogeneity / isotropy. Hot or cold, all of the "blobs" formed statistically the same number of galaxies & stars, structures & voids. What were those "blobs" then? What are their descendants now? If cosmologists say that space is uniform on 100Mpc scales; and if the CMB anisotropies correspond to even larger size scales; then what is the relation between CMB anisotropies then, and (any) known structures not?

Extrapolating from non-relativistic formulae, the pressure (P) of a potentially-relativistic ideal gas, of particle density n, in N m-2, exerted as momentum perpendicular to some unit area, per unit time, is proportional to: [math]P \propto avg \left( n \times v_{\perp} \times p_{\perp} \right)[/math] [math]\longrightarrow n mc^2 avg \left( \gamma(\beta) \beta_{\perp}^2 \right)[/math] [math]\longrightarrow \frac{1}{3} n mc^2 avg \left(\gamma(\beta) \beta^2 \right)[/math] by isotropy, ultimately resulting from [math]<\beta^2> = <\beta_x^2 + \beta_y^2 + \beta_z^2> = 3 <\beta_{1D}^2>[/math]. Now, the "average" function requires integrating over possible momentum eigenstates, weighted by the appropriate Boltzmann factor of gas temperature: [math]\longrightarrow \frac{1}{3} n mc^2 \int C 4 \pi \beta^2 d\beta \left(\gamma \beta^2 \right) e^{-\gamma \frac{mc^2}{k_B T}}[/math] [math]\propto \frac{1}{3} n mc^2 \int_1^{\infty} \frac{d \gamma}{\gamma^5} \left( \gamma^2 - 1 \right)^{\frac{3}{2}} e^{-\gamma \frac{mc^2}{k_B T}}[/math] employing the relativistic relation [math]\gamma = (1-\beta^2)^{1/2}[/math], and its (cumbersome) differential. [math]\longrightarrow \frac{1}{3} n mc^2 \tau^4 \int_{\frac{1}{\tau}}^{\infty} \frac{dx}{x^5} \left( \left( x \tau \right)^2 - 1 \right)^{\frac{3}{2}} e^{-x}[/math] seeking simplicity by substituting [math]\tau \equiv \frac{k_B T}{mc^2}[/math]. The above formula has no (obvious) closed-form solution. Is the full-fledged Relativistic equation for Pressure so complicated a function of temperature?? To solve for the integration constant requires [math]1 = C \int 4 \pi \beta^2 d\beta e^{-\frac{\gamma}{\tau}} = 4 \pi C \int \frac{dx}{x^4} \left( \left( x \tau \right)^2 - 1 \right)^{\frac{1}{2}} e^{-x}[/math]. (And all of this assumes that well-defined momenta are quantum-mechanically available to the particles.)

Billions of years ago, early spiral galaxies had more gas, fewer stars, yet much higher star formation rates. Over the eons, those galaxies have evidently "disked down & spun up", whilst converting their primordial gas into stars. Now, in protoplanetary disks swirling around protostars, the disk gas orbits slower than the protoplanets, which must stream through the disk gas. For, the disk gas is affected by pressure (and viscosity). Perhaps, then, something similar happens in the galactic disks of spiral galaxies? Perhaps gas orbits central SMBH more slowly than (newly formed) stars? Perhaps "gas is slow, stars are fast" ? Cp.: http://phys.org/news/2011-06-spitzer-distant-galaxies-grazed-gas.html

total energy in Magnetic field external field [math]\vec{B}_{out} = \frac{\mu_0}{4 \pi r^3} \left( \frac{3 \vec{r} \left( \vec{m} \circ \vec{r} \right) }{r^2} - \vec{m} \right)[/math] [math]\mathcal{E}_{out} = \frac{B^2}{2 \mu_0} = \frac{\mu_0 m^2}{2 \left( 4 \pi r^3 \right)^2} \left( 6 cos(\theta)^2 + 1\right)[/math] [math]E_{out} = \int \mathcal{E}_{out} \; r^2 dr sin(\theta) d\theta d\phi[/math] [math] = \frac{\mu_0 m^2}{2 \left( 4 \pi \right)^2} \left( 2 \pi \right) \left( 6 \right) \left( \frac{1}{3 R^3} \right)[/math] [math] = \frac{\mu_0 m^2}{8 \pi R^3}[/math] internal field [math]\vec{B}_{in} = \frac{\mu_0 \vec{m}}{\left( \frac{4 \pi}{3} R^3\right) }[/math] [math]\mathcal{E}_{in} = \frac{B^2}{2 \mu_0} = \frac{\mu_0 m^2}{2 \left( \frac{4 \pi}{3} R^3\right)^2}[/math] [math]E_{in} = \mathcal{E}_{out} \; \left( \frac{4 \pi}{3} R^3 \right)[/math] [math] = \frac{3 \mu_0 m^2}{8 \pi R^3}[/math] [math] = 3 \times E_{out}[/math] total field energy [math]\boxed{E_{tot} = \frac{\mu_0 m^2}{2 \pi R^3}}[/math] [math] = B_{pole} \times m[/math] For nucleons, having nuclear magnetons [math]\mu_q \equiv \frac{\frac{e}{3} \frac{\hbar}{2}}{2 \frac{m_P}{3}} = \frac{1}{2} \mu_N[/math] [math]\mu_P \approx 6 \mu_q[/math] [math]\mu_N \approx -4 \mu_q[/math] then [math]E_{tot} = \frac{\mu_0 g^2 \mu_q^2}{2 \pi R^3}[/math] [math]\boxed{ \begin{array}{c} E_P \approx +700 KeV \\ E_N \approx +300 KeV \end{array} }[/math] total energy in Electric field Simplistically, within nucleons, the two quarks having like electric charge would repel out into thin shells near the surface of the nucleon, whereas the lone oppositely-charged quark would reside near the center. Cp. neutrons are observed to have a positively-charged core, and negatively-charged periphery. Then: [math]E \approx + \frac{q_1^2}{4 \pi \epsilon_0 R} - 2 \frac{q_1 q_2}{4 \pi \epsilon_0 R}[/math] [math]\boxed{ \begin{array}{c} E_P \approx 0 \\ E_N \approx -500 KeV \end{array} }[/math] Thus, the total electro-magnetic (EM) energy of a neutron is about 1MeV less than that of a proton; in the absence of EM interactions, protons would be even lighter, and neutrons slightly heavier, so that neutrons would be over 2MeV more massive than protons (nearly half of which is offset, by the neutron's negative EM energy). Are these Classical equations correct ?

Lots ? (FYI, the close correspondence between the NS internal pressure, and nuclear energy density, is (on second thought) not accidental -- by definition, NS are those space objects compressed to nuclear mass(-energy) density. (And Pressure is proportional to energy density.) So the fact that all the numbers work out so (astrophysically) closely is not surprising. Nevertheless, those energy densities are order-of-magnitude 1035 J m-3.) No -- in my understanding, Gluon fields are not modeled with (1/r) potentials, like EM fields. They are modeled with linear ® potentials, so modeling a force constant with distance. The Strong (gluon) force only "kicks in" when quarks stray far from each other; at zero separation, there is zero interaction (called "asymptotic freedom"). Quarks are not photons. But, when relativistic, such that their energies >> rest-mass energies, then relativistic quarks move like photons. (They still keep their color charges, and still interact strongly)

The famous Low-Surface Brightness (LSB) galaxy Malin 1 is not actually an LSB galaxy. Instead, a normal barred-spiral (SBa) galaxy about 60K ly across spins amidst an extended disk ten times larger. Now, Malin 1 has maintained its extended disk, as an isolated field galaxy, far from disturbing tidal interactions. Thus, the material comprising the extended disk has had cosmological aeons to "disk down". Whereas, our Milky Way's galactic disk was long long ago truncated by tidal interactions, with other galaxies & satellites. However, the material which would have comprised an isolated Milky Way's extended disk, would plausibly have been shock-heated, during those interactions, and so "puffed up" into a diffuse circum-galactic halo. Perhaps the Milky Way's hot halo (per OP) comprises the material which, in Malin 1, has remained cold and "disked down" ?? http://kencroswell.com/Malin1.html

Recall, throughout, [math]\Delta p \approx \frac{\hbar}{\Delta x}[/math] [math]\frac{2}{3} E_P \approx \frac{G}{c^4} \frac{E_g^2}{R}[/math] [math]\alpha \equiv \frac{3 G m^2 N^{2/3}}{\hbar c}[/math] trans-Classical regime (with gluon field weakening): Adding the fourth-order (trans-)relativistic correction terms: [math]\rho \equiv \frac{p}{mc}[/math] [math]E - mc^2 \approx mc^2 \left( \frac{1}{2} \rho^2 + \frac{3}{8} \rho^4 \right)[/math] With gluon field weakening, [math]m \approx m_0 \times \frac{R}{R_0}[/math]. Redefining [math]r \equiv \frac{\Delta x}{\lambda_0}[/math], then [math]m \approx m_0 \frac{N^{1/3} \lambda_0 r}{R_0} \equiv m_0 \frac{r}{r_0}[/math], where [math]r_0 \equiv \frac{R_0}{N^{1/3} \lambda_0}[/math] is the initial compression factor. Now, [math]\rho \equiv \frac{\Delta p}{m c} \approx \frac{h c}{m c^2} \frac{1}{\Delta x} = \frac{m_0}{m} \frac{\lambda_{C,0}}{\Delta x} \equiv \frac{r_0}{r} \frac{1}{r} = \frac{r_0}{r^2}[/math] [math]\boxed{\rho = \frac{r_0}{r^2}}[/math] [math]E_P \approx N mc^2 \left( \frac{1}{2} \rho^2 + \frac{3}{8} \rho^4 \right)[/math] [math]= N m_0c^2 \frac{r}{r_0} \left( \frac{r_0^2}{2 r^4} \right) \left( 1+ \frac{3 r_0^2}{4 r^4} \right)[/math] [math]\boxed{E_P \approx \left( N m_0c^2\right) \left( \frac{r_0}{2 r^3} \right) \left( 1+ \frac{3 r_0^2}{4 r^4} \right)}[/math] [math]E_g^2 \approx N^2 \left( \left(m c^2 \right)^2 + \left( c \Delta p \right)^2 \right) [/math] [math]= N^2 \left(m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{\lambda_{C,0}}{\Delta x} \right)^2 \right) [/math] [math]\boxed{E_g^2 \approx \left(N m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right)}[/math] Thus [math]\frac{2}{3} \left( N m_0c^2\right) \left( \frac{r_0}{2 r^3} \right) \left( 1+ \frac{3 r_0^2}{4 r^4} \right) \approx \frac{G}{c^4} \frac{1}{R} \left(N m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\left( \frac{r_0}{r^3} \right) \left( 1+ \frac{3 r_0^2}{4 r^4} \right) \approx \frac{3 G}{c^4} \frac{1}{N^{1/3} \Delta x} \frac{\lambda_{C,0}}{\lambda_{C,0}} \left(N m_0 c^2 \right) \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\left( \frac{r_0}{r^3} \right) \left( 1+ \frac{3 r_0^2}{4 r^4} \right) \approx \frac{3 G m_0^2 N^{2/3}}{\hbar c} \frac{1}{r} \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\left( \frac{r_0}{r^2} \right) \left( 1+ \frac{3 r_0^2}{4 r^4} \right) \approx \alpha \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\boxed{r_0 \left( 1+ \frac{3 r_0^2}{4 r^4} \right) \approx \alpha \left( r^2 \left( \frac{r}{r_0} \right)^2 + 1 \right)}[/math] [math]r_0^3 \left( r^4 + \frac{3 r_0^2}{4} \right) \approx \alpha \left( r^8 + r_0^2 r^4 \right)[/math] [math]r^4 \left( r_0^3 - \alpha r_0^2 \right) \approx r^8 \alpha - \frac{3 r_0^5}{4}[/math] Roots: [math]r^4 = \frac{r_0^2 \left( r_0 - \alpha \right) \pm \sqrt{r_0^4 \left( r_0 - \alpha \right)^2 + 3 \alpha r_0^5}}{2 \alpha}[/math] [math]r^4 = r_0^2 \frac{\left( r_0 - \alpha \right) \pm \sqrt{\left( r_0 - \alpha \right)^2 + 3 \alpha r_0}}{2 \alpha}[/math] [math]r^4 = r_0^2 \frac{\left( r_0 - \alpha \right) \pm \sqrt{r_0^2 + \alpha^2 + \alpha r_0}}{2 \alpha}[/math] Prima facie, stable solutions exist for any initial mass [math]\left( \propto \alpha \right)[/math], at decreasing compression ratio r. estimated Pressures inside neutron-stars [math]\frac{P}{R} \approx \frac{3 G M^2}{4 \pi R^5}[/math] [math]P \approx \frac{3 G}{4 \pi M^2} \times \left( \frac{c^2}{2 G} \right)^4 \left( \frac{R_S}{R} \right)^4[/math] [math]= \frac{c^8}{64 \pi G^3 M^2} \left( \frac{R_S}{R} \right)^4[/math] [math]\approx \frac{3 \times 10^{35} Pa}{m^2 r^4}[/math] [math]\approx \frac{3 \times 10^{30} bar}{m^2 r^4}[/math] where m measures the neutron-star mass in solar masses, and r measures the neutron-star radius in Schwarzschild radii. For a one-and-a-half solar-mass neutron-star, at three Schwarzschild radii, that would be about 1033 Pa = 1016 Tbar. By comparison, the maximum tension sustainable, by a hypothetical "gluon cable" (derived from slowly stretching a quark out of its original nucleon), would be about: F / A = (1 GeV / fm) / fm2 = 1035 Pa = 1018 Tbar Thus, as neutron-stars compress down towards their Schwarzschild radii, internal pressures approach maximum theoretical nuclear pressures.

Without speculated "gluon field weakening", the Classical-to-Relativistic equations imply maximum masses of [math]\approx \frac{1}{2} - \frac{3}{2} M_{\odot}[/math]. With those speculations, maximum (initial) masses of [math]\approx 40 M_{\odot}[/math] can survive gravitational collapse (by compressing and shedding gluon-generated mass-energy). Thus, the entire range of stellar-mass black-hole objects, up to dozens of solar masses (e.g. Cygnus X-1, SS 433), can be explained, by the "gluon field weakening" speculation. Galaxy-scale super-massive black-hole objects, may not be explicable. If so, then "star class" vs. "galaxy class" black hole objects may be qualitatively distinct. First, if gravity overcame the Strong force, then (inexpertly, simplistically) quarks would simply stop generating gluons, since gravity & pressure would keep them confined; and gluons are only generated, when quarks try to stray away from each other. Second, relatedly, the Strong force is explicitly short-ranged; gluons do not propagate more than ~1 fm. Beyond that distance, the energy in the gluon field has already become so intense, that new quark-antiquark pair production occurs. Third, you could (hypothetically) create a black-hole from photons of light, if you compressed their combined energy = mass x c2, within their combined Schwarzschild radius RS = G E / c4. Fourth, i offer that you may have heard (implicitly) of these speculated phenomena. For, current physical theory, without "gluon field weakening", only explains low-mass compact objects, barely above one solar mass. Whereas, "gluon field weakening" allows objects dozens of times more massive, to survive gravitational collapse (by shedding gluon-generated mass). Thereby, this simple parsimonious picture explains all known "star-class" stellar-massed black holes, up to dozens of solar masses, e.g. Cygnus X-1, SS 433.

Recall, throughout, [math]\Delta p \approx \frac{\hbar}{\Delta x}[/math] [math]\frac{2}{3} E_P \approx \frac{G}{c^4} \frac{E_g^2}{R}[/math] [math]\alpha \equiv \frac{3 G m^2 N^{2/3}}{\hbar c}[/math] Classical regime (no gluon field weakening): [math]E_P \approx N \left( \frac{\Delta p^2}{2 m} \right)[/math] [math]=\frac{N}{2} mc^2 \left( \frac{\lambda_C}{\Delta x} \right)^2[/math] [math]E_g^2 \approx E^2 = N^2 \left( \left(m c^2 \right)^2 + \left( c \Delta p \right)^2 \right) [/math] [math] = N^2 \left(m c^2 \right)^2 \left(1 + \left( \frac{\lambda_C}{\Delta x} \right)^2 \right) [/math] Let [math]r \equiv \frac{\Delta x}{\lambda_C}[/math], [math]R = N^{1/3} \Delta x = N^{1/3} \lambda_C r[/math]. Then [math]\therefore \frac{1}{r^2} \approx \frac{3 G N m c^2}{c^4 R} \left(1 + \left( \frac{\lambda_C}{\Delta x} \right)^2 \right)[/math] [math]= \frac{3 G N^{2/3} m}{c^2 \lambda_C r} \left(1 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]= \frac{3 G N^{2/3} m^2}{\hbar c r} \left(1 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\therefore r \approx \left( \frac{3 G m^2 N^{2/3}}{\hbar c} \right) \left( r^2 + 1 \right)[/math] [math]\boxed{r \approx \alpha \left( r^2 + 1 \right)}[/math] Roots: [math]r = \frac{1 + \sqrt{1 - 4 \alpha^2}}{2}[/math] Collapsing from [math]r \gg 1[/math], the larger radius would be reached first (and would be more likely to remain within the Classical regime). Ergo, only the larger root is considered. So, in the Classical regime (with no gluon field weakening), quantum pressure (LHS) can forestall gravity (RHS), for some compression factor [math]\frac{1}{2} < r < 1[/math]. Solutions exist, for increasing initial mass [math]N[/math], at decreasing final compaction ratio [math]r[/math], until: [math]1 - 4 \alpha^2 = 0[/math] [math]\alpha = \frac{1}{2} = \frac{3 G m^2 N^{2/3}}{\hbar c}[/math] [math]m N = m \left( \frac{1}{2} \times \frac{\hbar c}{3 G m^2} \right)^{3/2} \approx \frac{1}{2} M_{\odot}[/math] Checking assumptions, for the maximum compression ratio [math]r = \frac{1}{2}[/math], the quantum energy is twice the rest-mass energy, i.e. verging into the Relativistic regime. Relativistic corrections could apply. Classical regime (with gluon field weakening): With gluon field weakening, [math]m \approx m_0 \times \frac{R}{R_0}[/math]. Redefining [math]r \equiv \frac{\Delta x}{\lambda_0}[/math], then [math]m \approx m_0 \frac{N^{1/3} \lambda_0 r}{R_0} \equiv m_0 \frac{r}{r_0}[/math], where [math]r_0 \equiv \frac{R_0}{N^{1/3} \lambda_0}[/math] is the initial compression factor. Now, [math]E_P \approx N \left( \frac{\Delta p^2}{2 m} \right)[/math] [math]= \frac{N}{2} \left( \frac{r_0}{r} \right) m_0 c^2 \left( \frac{\lambda_0}{\Delta x} \right)^2[/math] [math]= \frac{N m_0 c^2 r_0}{2} \frac{1}{r^3}[/math] [math]E_g^2 \approx E^2 = N^2 \left( \left(m c^2 \right)^2 + \left( c \Delta p \right)^2 \right) [/math] [math]= N^2 \left(m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{\lambda_0}{\Delta x} \right)^2 \right) [/math] [math]= \left(N m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right) [/math] Thus [math]\therefore \frac{N m_0 c^2 r_0}{3} \frac{1}{r^3} \approx \frac{G}{c^4} \frac{1}{N^{1/3} \lambda_0} \frac{1}{r} \left(N m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right) [/math] [math]r_0 \approx \frac{3 G m_0^2 N^{2/3}}{\hbar c} \left( r^2 \left( \frac{r}{r_0} \right)^2 + 1 \right) [/math] [math]\boxed{r_0 \approx \alpha \left( r^2 \left( \frac{r}{r_0} \right)^2 + 1 \right)}[/math] [math]r_0 - \alpha \approx \frac{r^4}{r_0^2}[/math] Solutions exist, for increasing initial mass [math]N[/math], at decreasing final compaction ratio [math]r[/math], until [math]\alpha = r_0[/math]: [math]r_0 = \alpha = \frac{3 G m_0^2 N^{2/3}}{\hbar c}[/math] [math]m N = m \left( r_0 \times \frac{\hbar c}{3 G m_0^2} \right)^{3/2} \approx 1.4 r_0^{3/2} M_{\odot}[/math] For neutron-stars, the initial compaction ratio [math]r_0 \approx 10[/math], since nuclear density matter has a spacing of 1 nucleon per fm, whereas the Compton wavelength of 1GeV nucleons is 10% of that value. Thus, gluon field weakening permits compact objects of initial mass up to [math]\approx 40 M_{\odot}[/math]. However, at those initial masses, the final compaction ratio [math]r \rightarrow 0[/math], implying diverging quantum (Fermi/Heisenberg) energies, thus violating the assumption of Classicality. Relativistic corrections may apply. Relativistic regime (no gluon field weakening): [math]E_P \approx N \left( \Delta p c \right)[/math] [math]= N mc^2 \left( \frac{\lambda_C}{\Delta x} \right)[/math] [math]= N mc^2 \frac{1}{r}[/math] [math]E_g^2 \approx E^2 = N^2 \left( \left(m c^2 \right)^2 + \left( c \Delta p \right)^2 \right) [/math] [math]= \left(N m c^2 \right)^2 \left(1 + \left( \frac{\lambda_C}{\Delta x} \right)^2 \right)[/math] [math]= \left(N m c^2 \right)^2 \left(1 + \left( \frac{1}{r} \right)^2 \right) [/math] Thus [math]\frac{2 N mc^2}{3 r} \approx \frac{G}{c^4 R} \left(N m c^2 \right)^2 \left(1 + \left( \frac{1}{r} \right)^2 \right) [/math] [math]2 = \frac{3 G N^{2/3} m}{c^2 \lambda_C} \left(1 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]= \frac{3 G N^{2/3} m^2}{\hbar c} \left(1 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\therefore 2 r^2 \approx \left( \frac{3 G m^2 N^{2/3}}{\hbar c} \right) \left( r^2 + 1 \right)[/math] [math]\boxed{2 r^2 \approx \alpha \left( r^2 + 1 \right)}[/math] [math]\left( 2 - \alpha \right) r^2 \approx \alpha[/math] Solutions exist, for increasing initial mass [math]\alpha, N[/math], at increasing [math]r[/math], until [math]\alpha = 2[/math]. [math]\frac{3 G N^{2/3} m^2}{\hbar c} = 2[/math] [math]m N = m \left( 2 \times \frac{\hbar c}{3 G m^2} \right)^{3/2} \approx 4 M_{\odot}[/math] In the relativistic regime, Pressure can resist up to [math]2^3 \times[/math] as much mass, as compared to the Classical case. However, at those masses, the final compaction ratio diverges, implying uncompacted, i.e. non-Relativistic regimes. For a final compaction ratio of one, the limit of the Classical regime, the Relativistic equations imply [math]\alpha = 1[/math], implying an initial mass of [math]\approx 1.4 M_{\odot}[/math], nearly thrice as much as the Classical equations imply, for the same final compaction ratio. Relativistic regime (with gluon field weakening): [math]E_P \approx N \left( \Delta p c \right)[/math] [math]= N m_0c^2 \left( \frac{\lambda_0}{\Delta x} \right)[/math] [math]= N m_0c^2 \frac{1}{r}[/math] [math]E_g^2 \approx E^2 = N^2 \left( \left(m c^2 \right)^2 + \left( c \Delta p \right)^2 \right) [/math] [math]= \left(N m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{\lambda_0}{\Delta x} \right)^2 \right)[/math] [math]= \left(N m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right) [/math] Thus [math]\frac{2 N m_0c^2}{3 r} \approx \frac{G}{c^4 R} \left(N m_0 c^2 \right)^2 \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right) [/math] [math]2 \approx \frac{3 G N^{2/3} m_0}{2 c^2 \lambda_0} \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\approx \frac{3 G m_0^2 N^{2/3}}{\hbar c} \left( \left( \frac{r}{r_0} \right)^2 + \left( \frac{1}{r} \right)^2 \right)[/math] [math]\boxed{2 r^2 \approx \alpha \left( r^2 \left( \frac{r}{r_0} \right)^2 + 1 \right)}[/math] Roots: [math]r^2 = \frac{2 \pm \sqrt{ 4 - 4 \alpha^2 r_0^{-2}}}{2 \alpha r_0^{-2}}[/math] [math]r^2 = r_0^2 \left( \frac{1 \pm \sqrt{1 - \alpha^2 r_0^{-2}} }{\alpha} \right)[/math] Solutions exist, for increasing initial mass [math]N[/math], at decreasing final compaction ratio [math]r[/math], until [math]1 = \alpha^2 r_0^{-2}[/math] [math]\alpha = r_0[/math] [math]= \frac{3 G N^{2/3} m^2}{\hbar c}[/math] [math]m N = m \left( r_0 \frac{\hbar c}{3 G m^2} r_0 \right)^{3/2} \approx 1.4 r_0^{3/2} M_{\odot}[/math] For neutron-stars, beginning collapse at nuclear density (1 nucleon per fm3), [math]r_0 \approx 10[/math], since [math]\lambda_0 \approx 10^{-16}m[/math]. Thus, gluon field weakening allows pressure to overcome gravity, for larger initial masses, up to about [math]40 M_{\odot}[/math]. At that critical threshold mass, the compression ratio: [math]r^2 = \frac{r_0^2}{\alpha} = r_0 \approx 10[/math] [math]r \approx 3[/math] At that compression ratio, nucleons would be squeezed down to a third of "normal" size, and so would have been reduced to a third of their "normal" mass, i.e. only [math]\approx 15 M_{\odot}[/math] of mass would remain. Meanwhile, the nucleons' quantum (Fermi/Heisenberg) energies would only be about a third of their initial rest-mass-energies; yet, that would now equal their reduce-to-a-third-initial-rest-masses. So, the total mass-energy residing in the resulting compact object, equaling the sum of remaining mass-energy, and quantum energy, would be [math]\approx 30 M_{\odot}[/math]. Notably, the Classical & Relativistic equations both point to the same threshold initial mass of [math]\approx 40 M_{\odot}[/math].