Widdekind

The following is this writer's understanding, of the mechanism, by which a core-collapsing hyper-massive star generates, firstly, a central stellar-massed BH ('polar pancake phase'); and, secondly, a short-lived Accretion Disk, around that BH, whose 'consumption' powers the GRB jets, parallel to the rotation axis ('equatorial consumption phase'). Is such the conventional explanation ?

Type Ia SN are the evidence for the acceleration, of the expansion, of our space-time fabric. Essentially, SNIa are standard candles, of inferrable absolute brightness. And, when observed 'out there' in space, they become dramatically dim, as if they were very far away, even when the emitting SNIa occurred relatively recently (low-to-moderate z). Thus, even 'recent' SNIa are far too faint, and hence far too far away, for any kind of Cosmology, but one in which our spacetime fabric has been stretching 'way-too-fast' -- to wit, an accelerating expansion, for the past 5-7 Gyr. Question re: CMBR I understand, that analysis of CMBR data 'shows', that our space-time is 'flat', and 'un-curved'. Does that mean, that our space-time was flat, back at z=1000 when the CMBR was generated? Or, more stringently, does that mean, that our space-time both was flat, and has remained flat, ever since?? Could our Cosmos have begun quasi-flat, and then 'developed' a curvature, later on ?? Do the CMBR data deny, actively, that our space-time fabric's curvature could have changed, after the CMBR was generated ??

Astronomers refer to this routine, and relatively minor, re-calibration of SNIa light-curves, as 'stretch'. To wit, the bigger & brighter blasts, have higher peak brightness, and persist longer ("taller & wider"), and conversely. The height, and width, of the light curve scale together, so that, with a single 'stretch' parameter, all SNIa light-curves approximately coincide. The fact remains, that the typical SNIa absolute luminosity, of M = -19.3, corresponds to the luminosity L = R2 T4, of a WD / earth-radius object, at star core temperatures (10s MK). Such suggests, that what is actually happening, is the brief, runaway, fusion re-ignition-and-explosion, of a re-activated star core.

How many hours in the day were there, 65 Mya ? Ophiolite's points seem sensical. 'Anti-podal focusing' may not be particularly plausible, for a planet-sized system, which is by no means a 'single ceramic crystal', but rather a rough & fractured & disparate 'clump' of plastic & thermally deformable semi-molten rocks.

The Proper Distance, from 'here & now' (coordinate origin, at time t = t0), 'out & back' to some specified 'there & then' (co-moving coordinate, corresponding to some time t, or equivalently, some red-shift z), is (Carroll & Ostlie. Intro. Mod. Astrophys., p.1260): [math]d(t) = R(t) \int_t^{t_0} \frac{c d\tau}{R(\tau)}[/math] Now, can this equation not be re-written (??), as: [math]d(t) |_{t=t_0} = R(t)|_{t=t_0} \int_R^{R_0} \frac{c \; dR}{R \frac{dR}{d\tau} } = R_0 \int_R^{R_0} \frac{c \; dR}{R^2 \, H }[/math] If so, then, using [math]R/R_0 \equiv 1/(1+z)[/math], w.h.t.: [math]dR = - \frac{R_0}{(1+z)^2} dz = - \frac{R^2 \, dz}{R_0}[/math] s.t.: [math]\therefore d_0 = -\int_z^0 \frac{c \, dz'}{H(z')}[/math] Now, from the solutions of the GR equations, constrained by the Cosmological Principle, w.h.t.: Rewritten in terms of red-shift z, instead of normalized scale-factor a = R/R0 = 1/(1+z) (and ignoring Radiation & Curvature contributions), w.h.t.: [math]d_0 = -\int_z^0 \frac{c \, dz'}{H_0 \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_\Lambda} }[/math] Flat, Matter-only, Critical Cosmology Now, for a flat, matter-only (i.e., matter-critical) Cosmos, wherewithin [math]\Omega_M = 1[/math], w.h.t.: [math]d_0 = -\int_z^0 \frac{c \, dz'}{H_0 \, (1+z')^{\frac{3}{2}}} = \frac{2 c}{H_0} (1+z')^{- \frac{1}{2}} |_{z'=z}^{0} = \frac{2 c}{H_0} \left( 1 - \frac{1}{\sqrt{1+z}} \right)[/math] This seems to agree, w/ the afore-cited source, so can I conclude these equations are correct ?? Flat, Matter-and-Cosmological-Constant, Critical Cosmology Numerical integration, of the fuller formula, including matter ([math]\Omega_M \approx 0.3[/math]) and 'dark energy' ([math]\Omega_\Lambda \approx 0.7[/math]) yields the following relative distances (in units of c / H0 = 14 billion light-years), and fractional Visible Universe volumes, for a selection of survey red-shifts: Survey z d %VU # VU galaxies (billions) 2MRS 0.03 0.0297962 1:1,364,782 67 HDF 4 1.67424 1:7.692948 600 HUDF 7 2.0145 1:4.416153 620 -- inf 3.30508 1 The actual existence of 'dark energy' seems to be more consistent, with the recently released 2MRS survey, as compared to the HDF & HUDF surveys. 'dark energy' was "turned on" at z=1 ?? From the afore-derived formula, one can compute a 'mixed Cosmology', wherein 'dark energy' was "turned on", at z = 1, for unspecified reasons: [math]d_0 = -\int_z^0 \frac{c \, dz'}{H_0 \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_K (1+z')^2 \; + \; \Omega_\Lambda} }[/math] [math] = - \left( \int_1^0 + \int_z^1 \right) \frac{c \, dz'}{H_0 \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_K (1+z')^2 \; + \; \Omega_\Lambda} }[/math] [math] = - \frac{c}{H_0} \left( \int_1^0 \frac{dz'}{ \sqrt{\Omega_M (1+z')^3 \; + \; \Omega_\Lambda} } + \int_z^1 \frac{dz'}{\sqrt{\Omega_M (1+z')^3 \; + \; \Omega_K (1+z')^2}} \right)[/math] Survey z d %VU # VU galaxies (billions) 2MRS 0.03 0.0297962 1:1,031,997 51 HDF 4 1.484067 1:8.352197 650 HUDF 7 1.775787 1:4.875157 680 -- inf 3.011067 1 Thus, an unexplained 'dark energy was activated at about z=1' scenario is noticeably less compatible, with the comparison, between the 2MRS vs. HDF/HUDF sky surveys. What would such a scenario -- which would, presumably, have actually altered the curvature of this Cosmic space-time fabric -- imprint upon the 'here & now' observed CMBR ?? Can the current CMBR observations categorically exclude any kind of (large) global Cosmic curvature, at all previous epochs ??

(Thanks for the references) Is there an "easy" way, of "rolling" graphene sheets, into CNTs? I would want an "easy" way, of manufacturing CNTs (which are long, linear, "rolled up strips" of graphene).

According to Howell 2007: Now, WDs, which were "several Gyr" old, at time of SNIa, would be considerably older, colder & dimmer, lying lower, and to the right, on the HR-Diagram. Thus, could initial WD age / temperature, conceivably determine, the ensuing "luminosity trajectories", across the HR-diagram ?? Again, and conversely, younger, hotter (bluer) WDs, initially lying higher & and to the left on the HR-diagram, would presumably generate the brighter, longer-lasting SNIa's; whereas, colder & older WDs would, presumably, generate the dimmer & faster-fading SNIa's. To make an engine analogy, younger hotter WD "cores" could, conceivably, "combust completely"; whereas, older & colder "spent cores" might, possibly, only "partially combust", like an out-of-tune engine with old corroded spark plugs. The following "suggestive schematic" attempts to illustrate this suggestion / question:

Could you create a single sheet of graphene: and then 'hydrogenate' one edge (H-C), whilst 'hydroxylating' the other edge (C-OH), and then 'roll the sheet up' into a tube (center images), allowing the Hs & OHs to 'jump off', as waters, leaving the tube 'sutured together' ??

Thanks for the responses !

Amongst the EW force-carrying bosons, it qualitatively seems like there is a progressive 'layering' of physical characteristics -- first, the Z0 boson gets mass (over photons); then, the W+,- get electrical charge (over Z0s). And, somewhat seemingly similarly, electrons look like neutrinos, that have 'on-loaded' a 'burden' of charge. And more, are there any particles, that have electrical charge, w/o mass 'first' ? Or, are there any (fundamental) particles, having color-charge, w/o electrical-charge 'first' ? So, is it possible, to quantify, this qualitative argument, for why neutrino's 'must' have mass??

The first step, in the pp-chain, is the fusion of two protons, into a deuteron, a neutrino, and a positron: What would happen, if you "flipped" the positron, on the right (product side), to an electron, on the left (reactant side). If you conducted pp collisions, in an electron-rich environment, could you catalyze the fusion physics?

At low energies (1st generation), when the wave-functions, of fundamental particles (u+2/3,d-1/3,e-1), would be (hypothetically) 'hyper-spatially narrow' (confined close to the w=0 hyper-spatial 'mid-hyper-plane of spacetime'), then the quantum behavior appears to be dominated by the Color Force, so that the quarks resemble each other, in mass, much more than either resembles an electron. But, at high energies (2nd-3rd generations), when the wave-functions would 'hyper-spatially spread out' (away from w=0, in both the 'in' and 'out' directions), then the (allegedly) 'hyper-spatially excited states' of electrons (i.e., muons & taons) and down-quarks (i.e., strange & bottom) are much more similar, in terms of mass, than either are to the 'way-out-in-a-league-by-itself' up-quark excited hyper-states (i.e., charm & top). Thus, qualitatively, this hyper-spatial hypothesis alleges, that the up-quarks are, ultimately, distinct from down-quarks & electrons, b/c (supposedly) for up-quarks, their Color Force attraction to the 'matter-out-side' of spacetime is opposed by their EM Force attraction to the 'antimatter-in-side' of spacetime; whereas, for the others, all attraction forces are complimentary. And, quantitatively, up-quarks do indeed evidence unique behavior, if only at high (excited hyper-spatial) energy. And so, there is at least a conspicuous coincidence, between the qualitative claims of this hyperspatial hypothesis, and actual quantitative observations. Keeping in mind, that the Compton-wavelength is the characteristic range of any fundamental force (Bernstein. Intro. to Cosmology), then the Color Force is far shorter range, and far more confined, than the EM force. Perhaps, then, the CF is also so confined, in the 'in-out' hyperspatial dimension? For, then, in guestimating the shape, of the hyper-potential, combining contributions from the CF & EMF, perhaps the former is delta-function-like, whereas the latter, although shallower & flatter, is far broader?? Such a scenario could create conditions, wherein, at higher energy / generation, up-quarks became "bound to both sides of spacetime", conceivably accounting for the colossal energies of charm & top quarks. Thus, this hyper-spatial hypothesis predicts, seemingly testably, that, by virtue of such "straddling the hyper-span of spacetime", up-quarks (up-anti-quarks) will be, someway-somehow, "more interactive with antimatter" (matter), than down-quarks & electrons (down-anti-quarks & positrons), who "keep to their own side of the tracks".

The quadratic form, for quantum particle energy, E2 = m2 + px2 + py2 + pz2 suggests, prima facie, that the rest-mass term m2, can be interpreted, as a further, and fourth, dimensional momentum term, 'pw2', representing motion orthogonal to standard spatial dimensions. And, that is strongly suggestive, of hyper-dimensional momentum-carrying-motion, in the 'in-out' dimension, of the (supposed) hyper-spatial 'thickness', of spacetime. Given that the rest-mass is constant, its interpretation as a 'hyper-momentum' implies, that the particle is residing in a 'stationary state', existing as a 'standing wave' spanning 'across' spacetime, through the 'in-out' hyper-dimension. And, such suggests, that quantum particles, confined within our spacetime, are kept in such confinement, by a 'hyper potential', in whose local minimum, matter & energy, in our spacetime, are 'shackled'. Now, if our spacetime has a hyper-spatial 'thickness', and if our spacetime represents an attractive hyper-potential well along that 'in-out' dimension, then what shape is that 'hyper-potential' ? Mathematically, many potential wells, Taylor-expanded about their local minima, can be described, to lowest order, as quadratic 'harmonic' potentials V = 1/2 k x2. And, it is well known, from elementary QM, that the energy eigenstates, of such quadratic potential wells, create a 'ladder' of levels, all being odd multiples, of the lowest, ground-state, energy: E = E0, 3 E0, 5 E0, etc. Therefore, if an electron (say), is an energy wave, oscillating 'across' spacetime, through the hyperspatial 'in-out' dimension of spacetime 'thickness', in a quadratic attractive potential; and if its ground-state energy happens to be its observed rest-mass-energy of 511 KeV; then we would expect to see 'excited hyper-states' of electrons, with apparent rest-masses of roughly 1.5 MeV, 2.5 MeV, 3.5 MeV, etc. Such states are not, seemingly, observed. Yet, an intriguing explanation offers itself. For, by the time an energy-injecting event, has enough energy to 'hyper-excite' an electron, then that event embodies an amount of energy, equivalent to an even multiple of the electron rest-mass energy, which would preferentially produce electron-positron pairs. Thus, the to-date absence, of observations, of excited hyper-states, of electrons (or other fundamental particles), can be accounted for, by the pre-emptive preference, of physics, for pair production, over (alleged) hyper-spatial excitation. Perhaps careful, balanced-and-omni-directional stimulation, of a ground-state electron, in a deep 'Fermi sea' of electrons (e.g., in a metal), wherein all neighboring electron states are already occupied, might impede pair production (at least, on the electron side?), which would, then, allow energy blasts, of multiples of ~1 MeV, to excite electrons into 'anomalously heavy', supra-normal, rest-mass states?? Note, too, that the fact, that our spacetime's supposed hyper-potential well, can attractively bind both particles & anti-particles, whose charges are all opposite-if-equal, might mean, that the 'in' and 'out' skin-like-layers of spacetime, are equal-but-oppositely charged. If so, then the 'inside skin', and 'outside skin', of spacetime are a little like parallel-plate capacitors, carrying equal-but-opposite charges. Such a scenario would produce a potential, which was anti-symmetric, 'across' spacetime, through its 'in-out thickness': ---^v---- If so, electrons & positrons (say) would be separately attracted, to opposite 'sides' of that potential well, with the former attracted to the 'inside edge', and the latter the 'outside edge' (say). Moreover, the appearance of distinct 'generations' of particles (e.g., Muons & Taons) in the Standard Model, could be accounted for, by the higher-order terms, deviating from simple quadratic form, in the 'hyper-potential', as one moved away from the two local extrema. Again, near those extrema, Taylor expansions of the (alleged) hyper-potential are closely approximated by a quadratic, harmonic oscillator form, producing the standard energy 'ladder' of levels. But, at some point, as particles are hyper-excited to near 'ionization' energies, higher-order terms perturb the simple previous picture, producing unstable-and-distinct semi-stationary solutions, interpreted as Muons & Taons (say). Note, that, in this picture, particles & anti-particles 'ionize' in opposite hyper-spatial directions, each ionizing 'off into hyperspace', hyperspatially 'away' from the opposite 'side' of spacetime. Comparison of electrically uncharged Neutrinos, to electrically charged Electrons / Muons / Taons, seemingly suggests, that the hyper-potential depends strongly on particle charge. Similarly, subsequent comparison to quarks, seemingly suggests, that the hyper-potential also depends strongly on particle color-charge. If so, then perhaps one 'hump' of the hyper-potential, is positively electrically-charged x 'white' color-charged, whilst the other is negatively electrically-charged x 'black' color-anti-charged ? Then, by inspection, we observe, from the Standard Model, that: [math]m \approx \left( 0.5 MeV \times -q_e \right) + \left( 5 MeV \times q_c \right)[/math] Thereby, the mass excess, of bottom-quarks over top-quarks, can be explained, if: all Matter exists in the 'positive Electrical potential (+), positive Color potential (white) all Antimatter exists in the 'negative Electrical potential (-), negative Color potential (black)' Thus, for bottom-quarks, the Color and Electrical potentials contribute in concert, to its larger mass; whereas, for up-quarks, the dominant Color potential is opposed by the subordinate Electrical potential, slightly reducing its mass. Note, though, that this prescription implies, that Matter & Antimatter exist in 'opposite pockets' of the hyper-potential, preferring 'opposite sides' of the fabric of spacetime, the one preferring the 'in-side' half, the other the 'out-side' half. Note, too, that, since there is more Matter than Antimatter in this Cosmic spacetime fabric; and since the 'out-side' surface, of the same, is larger than the 'in-side' surface (slightly today, perhaps more so soon after the Big Bang, when spacetime was highly curved); then, perhaps Matter occupies the ever-so-slightly-larger 'outside pocket', and oppositely.

[rough draft] If our space-time has a 'hyper-spatial thickness', in the 'in-out' direction (a little like an ice-cream sandwich); and if massive quantum particles oscillate, transverse to spacetime, in that 'in-out' direction (accounting for their rest-mass); and if powerful explosions, within spacetime, briefly 'blister' spacetime, 'bubbling' apart the 'ice-cream-sandwich crusts'; then powerful explosions might modify the rest-masses of quantum particles, residing in sections of space-time, whose 'hyperspatial skin' was so affected. Such powerful explosions might induce Surface Waves, in the skin of spacetime, which would propagate away from the explosion site, albeit at presumably dissipating intensities. Such Surface Waves would modify the rest-masses of quantum particles, as the waves passed 'above-and-below' (through the ice cream sandwich crusts) the 'matter in the middle' actually inside spacetime (the ice cream). Such 'Hyper-surface Waves' might propagate at speeds dramatically different from c, which is only The Speed Limit, for 'matter in the middle'. And, such 'Hyper-surface Waves' might be exploitable, for sending signals across space (vibrating the ice-cream sandwich crust, instead of the ice-cream).

If moon-induced ionospheric tides, today, are 40 miles tall; then, 4 Gya, when the moon was >10x closer, so that its tides were >1000x stronger, would those tides have been 40 K miles tall? That would have enveloped the moon itself, so that earth's moon would have orbited inside of earth's ionosphere.

Super-Saturated solution of CO2, at high Temperature & Pressure (e.g., deep-sea volcanic vents), could "inject" CO2 into Reverse Citric-Acid Cycle, "driving" it around ?? The reverse Citric Acid Cycle incorporates CO2, from the surrounding aqueous medium. So, if that solution was super-saturated, with CO2 -- such as could have occurred, 4.5 Gya, in the earth's early oceans, pressed down by a then-still-super-dense-and-CO2-rich sky -- could CO2, "seeking out" of the super-saturated solution, "flee into" the reverse CAC, and "drive" that anabolic pathway through successive cycles ?

A pure "Hydro-carbon" comprises a "Carbon-chain backbone", where each interstitial carbon carries 2xH, and the terminal carbons are "capped" by H: H-,C'-,C'-...-,C'-H (,C' = H-C-H) Hydro-carbons are named "number-ane" (e.g., hexane). A pure carbo-Hydrate -- what I would want to call a "Hydroxl-carbon" -- comprises the same sort of Carbon-chain backbone, but each interstitial carbon carries H+OH (and the terminal carbons are, again, "capped" by H): H-,C"-,C"...-,C"-H (,C" = H-C-OH) Hydroxl-carbons (carbo-Hydrates) are named "number-ose" (e.g., hexose), where some of the middling carbons carry 2xH, and others H+OH ?? Is there such a thing, as a "mixed", "Hydro-carbo-Hydrate" (Hydro-Hydroxl-carbon) ??

Thanks for the link. Much of the "moon-air-tides" result from the swelling seas "lifting up" the sky above them. Is not a typical earth-atmospheric scale-height of order 10 km? And, are not such scale-heights inversely proportional, to surface-gravity, H ~ kT/mg ? And more, is not the differential gravity, induced at earth, by the moon -- as measured in earth surface-gravities -- of order (m/M) x (Re/D)3 ~ 10-7 ? Thus, the atmospheric expansion, "allowed" by the "easing off of downward gravity", when the moon is over-head (or far under-foot), would only be of order 1 mm. Evidently, gravity is weak, and the air is thin, giving gravity little to grip. Evidently, tides in the far-hotter-and-so-much-more-swollen ionosphere, are much more in magnitude, rising & falling fully 40 miles, 2x per day. If the ionosphere has a scale-height, of ~1000 km (x100), a 10-7 effect would swell the same by ~10 cm... what accounts, for an ~100 km effect, a million times more ??

If earth's moon causes "rock tides", and "ocean tides", does it also induce tides into our atmosphere ? Would such atmospheric tides affect satellites in LEO ?

If dense metallic materials, from the Chicxulub impactor, sank down deep into the interior, then the Yucatan region should have showed intense Hot-Spot-like geothermal & volcanic activity, long after the event, due to the released heat energy rising back towards the surface. That does not appear to be common consensus for what occurred.

A hyper-spatially "thick" space-time fabric, would have two "sides", like an ice-cream sandwich, an "in-side" skin, and an "out-side" skin. That duality, and an apparent Flatland-analogous-ability of space-time-embedded matter to "rotate" around that hyper-spatial "in-out" axis in either a "left-handed" or "right-handed" sense, seems qualitatively consistent, with the existence of two types of mirror-image-matter (matter vs. anti-matter).

When a photon Wave Function ("Wave Packet") encounters a partially-silvered mirror, it bifurcates, into two Wave Functions -- one "transmitted" wave, and one "reflected" wave. More generally, quantum waves always split, when encountering spatially sharp potential barriers. Now, considering the case of a fundamental point-like particle (e.g., electron), this "splitting" of Wave Functions, is qualitatively similar, to the "spreading" of Wave Functions -- which amounts to the successive splittings, in every direction, of the original delta-function-like, point-like particle, position eigenstate. What then is similar, between (1) a photon wave encountering a partially-silvered mirror, and "splitting"; and (2) an electron wave, in empty space, "spreading" ? Is this where the concept of "quantum foam" enters the picture? Is "empty space" actually filled with virtual particles, whose interactions with actual matter perpetually perturb the same, amounting to myriad "potential barrier encounters", each one of which further "splits" the ever-splitting-and-so-spreading Wave Function ?

According to this site, given the observed abundances of Deuterium, 3-Helium, & Lithium, Primeval Nucleosynthesis could have occurred, at almost any temperature, T < 500 MK, with next-to-no discriminative sensitivity beyond that. What is the cosmic abundance of neutrons (which would discriminate) ??

According to this site, "Helium can be formed by a set of reactions that cannot occur in the Sun's interior because it doesn't have any free neutrons" (rxn 3): Thus, could you enrich your cooling water, with tritium, to absorb all the excess neutrons, fusing into Helium, and converting wasted neutrons, into usable heat energy ??

For T < 104 K, the cooling curve of the ISM scales as ~T7 (Gaetz & Saltpeter 1983): P [ergs cm3 s-1] = -10-23 x nh x ne x (T/104K)7 And, since z ~ 2, the IGM has cooled from 10,000 to 4,000 K: From z ~ 6 to 2, Cluster gas continued to influence the IGM, maintaining its temperature through outflows. After z ~ 2, however, Clusters completely decoupled from the Hubble Flow, continuing to contract, compress, & heat Cluster gas. Meanwhile, the now isolated IGM, in the receding voids, began to cool radiatively. (Source: Majestic Universe [scientific American special report], pg. 8.) So, we seek to calculate the cooling, of the IGM, as a function of time, in a Critical, matter-dominated cosmos. Consider, then, a comoving volume, expanding with the stretching of spacetime, having a volume V(t) = R(t)3. Order-of-Magnitude Calculation In the comoving volume, where the matter only loses energy, to radiation, through aforestated cooling, w.h.t.: [math]E \approx \frac{\rho}{m_H} k T \times V(t) \approx \frac{M_{tot}}{m_H} k T[/math] [ergs] [math]\frac{dE}{dt} \approx -10^{-23} \left( \frac{\rho}{m_H} \right)^2 \left( \frac{T}{T_0} \right)^7 \times V(t) \approx -10^{-23} \left( \frac{M_{tot}}{m_H} \right)^2 \left( \frac{T}{T_0} \right)^7 \times V(t)^{-1}[/math] [ergs s-1] [math]\therefore \frac{dE}{dt} \approx \frac{M_{tot}}{m_H} k \frac{dT}{dt} \approx -10^{-23} \left( \frac{M_{tot}}{m_H} \right)^2 \left( \frac{T}{T_0} \right)^7 \times V(t)^{-1}[/math] [math]k \frac{dT}{dt} \approx -10^{-23} \left( \frac{M_{tot}}{m_H} \right) \left( \frac{T}{T_0} \right)^7 \times \frac{1}{R_0^3} \left( \frac{R_0}{R(t)} \right)^3 \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( \frac{T}{T_0} \right)^7 \times \left( \frac{R_0}{R(t)} \right)^3[/math] Then, since we assume matter-dominated cosmology, R(t) ~ t2/3: [math]\therefore k \frac{dT}{dt} \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( \frac{T}{T_0} \right)^7 \times \left( \frac{t_0}{t} \right)^2[/math] Defining normalized variables: [math]\therefore \frac{k T_0}{t_0} \frac{d\tau}{dx} \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \tau^7 \times x^{-2}[/math] This is easily integrated: [math]\frac{k T_0}{t_0} \int_1^{0.4} \frac{d\tau}{\tau^7} \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \int_{z=2}^1 \frac{dx}{x^2}[/math] [math]\frac{k T_0}{6 t_0} \left( 0.4^{-6} - 1 \right) \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( 1 - x^{-1}|_{z=2} \right)[/math] But, R(t) ~ (1+z)-1, which, when combined with the above, gives x ~ (1+z)-3/2: [math]\frac{k T_0}{6 t_0} \left( 0.4^{-6} - 1 \right) \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( 1 - 3^{3/2} \right) \approx 10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( 3^{3/2} - 1 \right)[/math] And, letting [math]\rho_0 \equiv \alpha \rho_{crit}[/math], w.h.t.: [math]\alpha \approx 10^{23} \frac{k T_0}{6 t_0} \frac{m_H}{\rho_{crit}} \frac{ 0.4^{-6} - 1 }{3^{3/2} - 1}[/math] Now, for a Critical cosmology, t0 = (2/3) H0-1. So, using the usual formula, for the Critical Density, w.h.t.: [math]\therefore \alpha \approx 10^{23} \frac{m_H k T_0}{4} \frac{8 \pi G}{3 H_0} \frac{ 0.4^{-6} - 1 }{3^{3/2} - 1} \approx 10^{23} \left( m_H k T_0 \right) \left( \frac{2 \pi G}{3 H_0}\right) \left( \frac{ 0.4^{-6} - 1 }{3^{3/2} - 1}\right) \approx 0.80 \, h_{72}^{-1}[/math] According to this calculation, and assuming a matter-dominated Critical cosmology, the IGM has been cooling, since z ~ 2, from roughly 2 to 9 Gyr, as if its density were comparable to Critical. Thus, such a cosmology could account, for the cooling, observed in the IGM (which will soon undergo a "second Recombination" event, perhaps w/in roughly +1 Gyr +10 Tyr). Numerical Factors For a multi-component gas, the energy density is E = (3/2) ntot k T. And, for 4,000 K < T < 10,000 K, and at ultra-low IGM gas pressures, Hydrogen is almost entirely ionized, whereas Helium is almost entirely neutral. So, ignoring the low IGM metallicity (Z ~ 0.007), w.h.t.: [math]n_{tot} = n_H + n_{He} + n_Z + n_e \approx n_H + n_{He} + n_H = 2 n_H + n_{He} \approx \rho_{tot} \times \left( 2 \frac{X}{m_H} + \frac{Y}{4 m_H} \right)[/math] Using canonical values, for the mass-fractions, X = 3/4 & Y = 1/4, w.h.t. [math]n_{tot} \approx \frac{\rho_{tot}}{m_H} \times 25/16[/math]. So, on the LHS, we have omitted numerical factors, of order unity, but specifically equal to 3/2 x 25/16 = 75/32. (Physically, hot, Hydrogen-ionized IGM, has ~2.5x more energy content, than was recognized above, due to having three degrees-of-freedom, and all of the extra Hydrogen-ionized electrons.) Likewise, on the RHS, since all the radiative cooling comes from the (near-)fully ionized Hydrogen, recombining with its own electrons, we require the product of: [math]n_H n_e \approx n_H^2 \approx \left( \frac{X \rho_{tot}}{m_H} \right)^2[/math] So, on the RHS, we have omitted numerical factors, of order unity, but specifically equal to X2 = 9/16. (Physically, hot, Hydrogen-ionized IGM, has only ~0.5x the cooling capacity, compared to what was recognized above, mainly due to all of the non-radiating mass 'locked away' in Helium.) And, so, our original, order-of-magnitude estimate, for the fraction of Critical density in gas [math]\alpha[/math], we neglected to multiply by the ratio, of "LHS / RHS" = 75/32 x 16/9 = 75/18 ~ 4. (Physically, hot, Hydrogen-ionized IGM, has more energy content, and less cooling capacity, than was recognized above -- and, so, must be much more dense, to decrease its temperature, at the same rate.) Thus, more accurate physics seemingly suggests more strongly, that a matter-dominated Critical cosmology can account, or is even required to account, for observed IGM cooling, since z ~ 2 (~7 Gyr ?). second Recombination Sticking with the same physics, the IGM will cool to 3000 K ([math]\tau_f = 0.3[/math]), at some future time, when the relative age of the universe is [math]x = 1 + \delta x[/math]: [math] 0.3^{-6} - 1 = 3^{3/2} - (1 + \delta x)[/math] [math]\delta x = 0.3^{-6} - 3^{3/2} \approx 1400[/math] Thus, given the expansion of space-time, and the IGM's already-quasi-cold condition, the IGM is could require trillions of years, to cool back down to 3000 K.