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Type Ia supernovae progenitor problem and the variation of fundamental constants.

1 Introduction

On account of the supposed uniformity of their absolute magnitude, the Type Ia supernovae (SNe Ia) play an important role of "standard candles" in cosmology. A tight correlation between the peak light output and the light-curve width (width-luminosity relation) results from the way SNe Ia originate from white dwarfs (WDs)--the final remnants for low and medium mass stars. According to the current understanding, the carbon-oxygen (CO) thermonuclear fusion triggering the supernova explosion takes place in compact binary systems in either of two principal progenitor channels. A single-degenerate (SD) model (Whelan & Iben [80]) predicts that CO WD accretes matter from the companion, usually the red giant or the main sequence star. Just before approaching the Chandrasekhar mass-limit [M.sub.Ch] [approximately equal to] 1.44M[dot encircle] for which electron degeneracy pressure becomes insufficient to prevent the gravitational collapse, the WD's core reaches the ignition temperature for the runaway carbon and oxygen fusion into heavier elements. In a preceding time lasting usually ~ [10.sup.6] yr WD processes the transferred matter falling onto its surface through the accretion disc. In this phase, called "nuclear-burning white dwarf" (NBWD) the hydrogen-helium fusion releases energy in a form of copious X-radiation, observed as "super-soft X-ray source" (Di Stefano [17]).

Instead, the double-degenerate (DD) model (Webbink [79], Iben & Tutukov [37]) predicts that two WDs of the combined mass [greater than or equal to] [M.sub.Ch] form a compact binary system and subsequently spiral towards each other in a common envelope. Eventually, they collide and merge and, after exceeding the Chandrasekhar limit, explode as SN Ia. Unlike in accrete scenario the merging WDs are not expected to be the source of X-radiation until a short time preceding the supernova explosion. The X-ray signatures of SD and DD channels differ significantly, which makes them easy to distinguish. The DD model admits a broader range of progenitor mass and SNe Ia luminosity; thus is thought to be responsible for the nonstandard SNe Ia explosions.

These two basic models (hereinafter collectively referred to as "SNe Ia binary paradigm") do not however provide a fair explanation to the diversity in the observed characteristics of SNe Ia and the paucity of their potential progenitors. The relevant SNe Ia progenitor problem amounts to the following two items. First is the problem of SNe Ia rate: the total number of potential progenitors seems to be inadequate to the number of observed SNe Ia events. Second is the problem of SNe Ia properties: the observed light-curves and remnants spectra do not match satisfactorily the detailed predictions of SD and DD models.

Our goal here is to provide a solution to the progenitor problem based on assumption of the varying Chandrasekhar mass, a consequence of varying constant G[c.sup.-2]. It's not been a century yet since one realized our universe has a turbulent history behind and some kind of final fate ahead. Compared with the prior model of eternal and basically invariable universe, this forms quite different ground for thinking about physical fundamental constants. One cannot ascribe logical necessity to any of fundamental constants (class C "universal" constants, according to Uzan's nomenclature(Uzan [74]) as e.g. in the case of mathematical constant n or the Euler's number e. Likewise, one cannot obtain them by pure deduction in a way similar to that Eddington tried (ineffectively) to do with the fine structure constant alpha. For the time being, they work as "free parameters". Hence, still valid is Dirac's opinion: "It is usually assumed that the laws of nature have always been the same as they are now. There is no justification for this. The laws may be changing, and in particular quantities which are considered to be constants of nature may be changing with cosmological time" (Dirac [16]). Let us complement this opinion with another one: "Ignoring the possibility of varying constants could lead to a distorted view of our universe and if such a variation is established corrections would have to be applied" (Uzan [74]).

2 The SNe Ia progenitor problem: a brief overview

The question of identity of Type Ia supernovae progenitors is widely considered as the "major unsolved problem in astrophysics" (Maoz & Mannucci [47]). The main problem is the discrepancy between the observed SNe Ia rate and the number of potential progenitors. Taking into account the estimated rate of SNe Ia (~ ([10.sup.-3]-[10.sup.-2])[yr.sup.-1] events in a typical spiral or elliptical galaxy) and the mean/median delay time for the SNe Ia progenitors (~ (0.5-1) Gyr for DD channel and ~ (2-3) Gyr for SD channel), X-ray sources should manifest in thousands in any such galaxy including the Milky Way. Meanwhile, the X-ray flux from the sample of six neighboring spiral galaxies obtained from Chandra X-ray Observatory is a factor of 30-50 times fainter than expected (Gilfanov & Bogdan [29]). In some of SNe Ia previously thought to originate in SD channel no remnants of red giant has been observed (Schaeffer & Pagnotta [70], Li et al. [42], Nugent et al. [56]). Generally, in most cases red giants have been excluded as possible ex-companions in binaries. The discrepancy between the observed amount of X-ray sources and the assessed numbers of SNe Ia led to conclusion that accrete scenario is not a primary route to supernovae, giving priority to the merger scenario. Gonzalez Hernandez et al. [30] estimate that fewer than 20% of SNe Ia is produced in SD channel. Gilfanov & Bogdan [29] opt for even more stringent limit [less than or equal to] 5% of total population. Di Stefano [17] indicates the lack of 90%-99% of the required number of X-ray sources. She argues (Di Stefano [18]) that companion stars forming the double degenerates do not age at the same rate and thus do not become WDs at the same time; for that reason the common envelope phase should be preceded by a symbiotic pre-double-degenerate phase with the hydrogen-helium fusion similar to NBWD. Thus, merger channel should also produce X-ray flux comparable to the accrete channel prior to the common envelope phase, which puts into doubt DD model as an effective explanation.

A vital problem is the paucity of the observed white dwarfs mergers. According to Gilfanov [28] "... too few double-white-dwarf systems appeared to exist". One expects the ESO Supernovae Type Ia Progenitor Survey (SPY) (Napiwotzki et al. [54, 55]) and the ongoing Sloan White dwArf Radial velocity data Mining Survey (SWARMS) (Badenes et al. [3], Mullally et al. [53]) to provide evidences for the merger channel (DD) as the main route to SNe Ia. Badenes & Maoz [4] using Doppler techniques isolated 15 WD binaries from a sample of ~ 4,000 WDs brought by Sloan Digital Sky Survey (SDSS). They compared the rate of WD binaries with the rate of SNe Ia in the Milky Way-like Sbc galaxies and found a "remarkable agreement" between them. However, a majority of these WD binaries appeared to be sub-Chandrasekhar, although usually with total mass relatively close to [M.sub.Ch] (1.1-1.2M[dot encircle]).

Some of researches (Hachisu et al. [34], Van Kerkwijk et al. [40], Zhu et al. [82], Maoz & Mannucci [47]) claim that the requirement as to the total mass of merging CO WDs (i.e 1.4M[dot encircle]) is too restrictive. This would match observations of super-Chandra WD progenitor stars with the combined mass reaching 2.4-2.8M[dot encircle]. According to the respective models, the observed number of SNe Ia can be explained provided the wider range of combined mass: smaller than [M.sub.Ch] (sub-[M.sub.Ch] merger channel) or bigger than [M.sub.Ch] (super-[M.sub.Ch] merger channel), dependently on detailed conditions such as rotation, magnetic fields, metallicity and the host galaxy population. This would account for better agreement with observations, both as to the rate of SNe Ia and to the differences in their properties. The controversial point of these models is that they require special fine-tuning to be effective. Maoz & Mannucci [48] attribute some of discrepancies as caused by "deadly sins", i.e. incorrect or inadequate methods in measuring and analyzing the SNe Ia rates. They admit however the "detailed models still falls short of the observed number (of SNe Ia) by at least factor of a few".

Di Stefano [18] suggests that, possibly, only a small fraction of accreting WDs can be detected and identified as X-ray sources. This may occur by two reasons: either the winds from a companion giant reprocess the supersoft X-ray radiation into the radiation of longer wavelengths, or the duty cycle of nuclear burning is to low to be detected. However, neither of these solutions has been properly recognized and confirmed as yet. Another proposal (Di Stefano et al. [19]) links the mass of progenitor with the angular momentum gained from the donor star together with matter. The angular momentum prevents the super-[M.sub.Ch] WD from collapse, which widens the potential range of SNe Ia progenitors. The relevant "spin-up/spin-down" models predict the existence of numerous WD "ticking bombs" waiting to explode until their rotation slows down to a proper level.

There is a broad agreement (e.g. Totani et al. [73], Maoz et al. [48], Mennekens et al. [50], Hachisu et al. [33]) as to the key role of "delay time distribution" (DTD)--the number of SNe Ia events in unit time as a function of time elapsed since starburst, in predicting the SNe Ia rates. It seems that DTD (indicated as [t.sup.-1] power law) favors the DD scenario. Hachisu et al. [33] found a good agreement of DTD with SD model either, provided the donor stars are both red giants and the main-sequence stars. Undoubtedly, DTD introduces an indispensible methodological order to the SNe Ia progenitor problem. In general however, regarding DTD did not bring a decisive breakthrough so far in the question of identity of SNe Ia progenitors.

It has gradually become evident that SNe Ia are not "standard candles" in the originally attributed sense. Their intrinsic luminosity is neither considered nor demanded to be exactly uniform, which gives priority to the more "capacious" merger channel. Instead of standard candles, SNe Ia are currently interpreted as "standarizable candles", which means that utilizing them as the correct distance indicators requires due calibration. This in turn demands better recognizing of their origin and nature. The study by Linden, Virey & Tilquin [43] revealed a likely positive correlation between the SNe Ia absolute brightness and distance, which may put in question the actually determined cosmological parameters. The observed relationship between the intrinsic color and ejecta velocity may help in reducing systematic biases in the estimates of distance (Foley et al. [25]). Instead, Sullivan et al. [72] point to the relationship between the luminosity of SNe Ia and metallicity of their hosts, while metallicity is supposed to depend on redshift. Gallagher et al. [26] comparing the spectra of a sample of 29 early elliptical galaxies of the age exceeding 5 Gyr with the general sample from SDSS including younger galaxies, find a strong correlation between the absolute magnitude of SNe Ia and the age of host galaxies while, most likely, "... the observed trend with metallicity is merely an artifact brought about the evolutionary entanglement of age and metallicity". These findings may help in recognizing the properties of SNe Ia, which is particularly important for the question of dark energy and the relevant accelerating expansion of the universe (Riess et al. [66], Perlmutter et al. [60]). The supposed correlation between the absolute magnitude and distance suggests the presence of a time dependent factor in the effective SNe Ia progenitor model.

3 Varying Chandrasekhar limit as the postulated main route to SNe Ia

The mass-limit formula for white dwarfs based on the equation of state for ideal Fermi gas (Chandrasekhar [11]) reads

[M.sub.Ch] = 4[pi] [([K.sub.2]/[pi]G).sup.3/2] [[omega].sup.0.sub.3], (1)

where [[omega].sup.0.sub.3] is the numerical constant equal to 2.018, derived from the explicit solution of the Lane-Emden equation for the polytropic index n = 3. The constant K in the general case connects pressure and density: P = K[[rho].sup.(n+1)/n] while in the case including white dwarfs (i.e. for n = 3) becomes specified as P = [K.sub.2][[rho].sup.4/3]. Since [K.sub.2] is defined as

[K.sub.2] = 1/8 [(3/[pi]).sup.1/3] hc/[([[mu].sub.e][m.sub.H]).sup.4/3] (2)

([[mu].sub.e]-mean molecular mass per electron, [m.sub.H]-mass of hydrogen atom), so substituting gives

[M.sub.Ch] = 4[pi] [[1/8 [(3/[pi]).sup.1/3] hC/[([[mu].sub.e][m.sub.H]).sup.4/3] [pi]G].sup.3/2] [[[omega].sup.0.sub.3]. (3)

Collecting the pure numbers, and considering that h = h/2[pi], one gets

[M.sub.Ch] [approximately equal to] 1.11065 [varies] [10.sup.54] [[mu].sup.-2.sub.e] [m.sup.3.sub.P], (4)

where [m.sub.P] = [(hc/G).sup.1/2] is the Planck mass. Since CO WDs are mainly composed of carbon-12 and oxygen-16, and because in both cases atomic number equal to half the atomic weight so one has [[mu].sub.e] = 2, leading to [M.sub.Ch] [approximately equal to] 1.44M[dot encircle]. It is important that Chandrasekhar mass it is proportional to the cube of Planck mass:

[M.sub.Ch] ~ [m.sup.3.sub.P]. (5)

Assuming h = con st it relates to the speed of light and to the Newton's gravitational constant as

[M.sub.Ch] ~ [(c/G).sup.3/2]. (6)

(We use tilde for linear dependence in the cases when the variability of a reference quantity [here: c and G] is hypothetical. Instead, the symbol of proportionality [exact or approximate] [varies] is used when variation of a reference quantity is obvious or certain, e.g. cosmic time t or radius of universe [R.sub.u]).

From this relationship it follows that any cosmological model postulating varying G or/and c (except the case they change accordingly) implies the postulate of varying [M.sub.Ch]. This fact has not been properly explored so far. What we propose here is the "varying Chandrasekhar mass-limit" model (VCM) in which [M.sub.Ch] decreases in cosmic time. VCM postulates that the currently known value of Chandrasekhar limit refers solely to the present epoch while in general:

[M.sub.Ch](past) > 1.44M[dot encircle] > [M.sub.Ch](future). (7)

This determines a scenario for the single WD progenitors of SNe Ia, which can be outlined as follows. Once an individual WD is formed, it keeps its mass approximately constant during the cooling process while the Chandrasekhar limit gradually decreases in time. Eventually, it equates or approaches a given WD's mass triggering the SN Ia explosion. From a logical point of view, an effect of SN Ia caused by decreasing [M.sub.Ch] reminds bringing water to a boil by reducing the atmospheric pressure without supplying heat. Hence, single WDs are, along with binary WDs, the potential progenitors of SNe Ia.

4 Varying constants and the black-hole cosmology

Varying Chandrasekhar limit, as a hypothesis based on assumption of varying constants c or/and G is closely related to the black-hole cosmology. A constitutive observation of the respective models is the coincidence between the radius of observable universe and the Schwarzschild radius, supposed to be valid over the whole course of the universe's history. According to a hypothesis advanced by Pathria [58] and Good [31], the universe is the interior of a black hole existing, among many others, within a larger structure called multiverse.

The recent multiverse model by Poplawski [64,65] uses the Einstein-Cartan-Sciama-Kibble theory removing from General Relativity the constraint of symmetry in the affine connection, and regarding the antisymmetric variable torsion tensor in the Friedmann equations. The relevant cosmological scenario takes an advantage of the fact that most stars have a non-zero angular momentum. When a massive rotating star collapses to a (Kerr) black hole, the torsion of extremely dense matter inside the horizon prevents from the point singularity (replaced by the ring singularity). As a result, the black hole becomes a wormhole to another universe thought to originate in "big bounce". As far as our own universe is the interior of a black hole existing in another universe, any black hole in our universe is thought to contain (produce) a separate universe. The new universe is interpreted as a "white hole"--a time reversal black hole whose expansion, e.g. such as observed in our universe is driven by the torsion, identified with dark energy. This model predicts the presence of traces of primordial torsion in a form of slight anisotropies in both cosmic and nanoscopic scales. Some reported evidences of the preferred handedness of spiral galaxies (dipole asymmetry of the value 0.0408 [+ or -] 0.011 based on SDSS data sample containing 15,158 spiral galaxies with the redshift < 0.085) seem to support the idea of cosmic parity violation (Longo [44]). However, the area covered by this sample is still too small to derive unambiguous conclusions. According to Neta Bahcall "The directional spin of spiral galaxies may be impacted by other local gravitational effects".

Besides, even if the filaments forming the cosmic web are uniformly distributed, anisotropy connected with rotation will break the homogeneity in a deeper sense. In the isotropic cosmic space, the "center" is a purely relative concept connected with the notion of observable universe. But it is no longer relative in the anisotropic space with the fixed axis of rotation. The spinning universe implies, besides anisotropy, the presence of preferred points. We may think about analogies between directional spin of spiral galaxies and the Coriolis effects on the Earth, e.g. manifesting itself in different spin of hurricanes in north and south hemispheres. Anyway, the question of spinning universe is, in the end, a matter of (further) observations.

The model here proposed (VCM) bases on formal resemblance of our universe with a black hole (and thus we shall use the Schwarzschild equation for radius) yet does not settle whether the universe is a black hole in the literal sense. It seems instead that crucial property of the universe conceived as the interior of a black hole is that its total energy amounts to zero. In this regard, the black-hole cosmologies are close to the "zero-energy universe" theories.

The legitimacy for interpreting the universe in terms of a black hole depends on its parameters, in particular size, density and mass. Recent estimations concerning the radius of observable universe point to the value [greater than or equal to] 14 Gpc (4.3 x [10.sup.26] m) or 28 Gpc in diameter. Cornish et al. [12] analyzing the WMAP data in search of the matched back-to-back circles predicted by various nontrivial topologies, settled the low bound of diameter of the last scattering surface of fundamental domain for 24 Gpc. Bielewicz & Banday [6], using similar methods extended this value to 27.9 Gpc. This admittedly does not prejudge the question of size, yet, provided the multi-connected space of universe, constraints the topology scale from below. An additional (though partly linked) difficulty comes out from the potential difference between the notions of entire and observable universe. In principle, entire universe may significantly surpass the observable universe (as inflationary theory predicts), but it can be as well slightly smaller due to nontrivial topology. The respective ratio may also change in time. Presumably, the black hole parameters describe the entire universe, and not just the universe currently observed. However, this distinction becomes important only insofar as "entire", by virtue of convention, denotes the biggest physically connected object defined according to the horizon problem of the early universe. Assuming the approximately linear rate of expansion after the end of inflationary epoch (or from the beginning), the parameters of the so defined "entirety" should not significantly differ from the "observable" parameters. Bearing in mind the obvious uncertainties, we shall use in calculations the value [10.sup.27] m for the universe's radius.

The critical density for a flat universe derived from Friedmann equation for the Hubble constant obtained from Planck telescope: [H.sub.0] = 67.15 [kms.sup.-1] [Mpc.sup.-1] is [[rho].sub.c] = 3[H.sup.2]/8[pi]G [approximately equal to] 0.85 [varies] [10.sup.-26] kg[m.sup.-3]. The resultant total mass for [R.sub.u] = [10.sup.27] m amounts to [M.sub.u] [approximately equal to] 1.44 [varies] [10.sup.54] kg (we shall use [10.sup.54] kg in calculations). Considering the approximated values of gravitational constant: G [approximately equal to] 6.7 [varies] [10.sup.-11] [m.sup.3][kg.sup.-1] [s.sup.-2] and the speed of light: c [approximately equal to] 3 [varies] [10.sup.8] [ms.sup.-1] ([c.sup.2] [approximately equal to] [10.sup.17] [m.sup.2] [s.sup.-2]) one obtains the numerical relationship connecting radius and mass:

[10.sup.27] = [10.sup.-10] [10.sup.54] [10.sup.-17] (m), (8)

which means that equation for the Schwarzschild radius:

[r.sub.S] = 2G[mc.sup.-2] (9)

apparently applies to the universe as

[R.sub.u] [approximately equal to] G[M.sub.u] [c.sup.-2]. (10)

We postulate that universe constantly fulfills the "black-hole condition" (BHC), which means that it is always fulfilled:

[R.sub.u] [equivalent to] [r.sub.S]. (11)

Together with assumption [M.sub.u] = con st, and the general assumption of isotropy of cosmic space, BHC implies

G[c.sup.-2] [varies] [R.sub.u]. (12)

5 Models with varying constants

In the intensive discussion on the variability of fundamental constants, variation of c is probably the leading topic. A majority of the "variable speed of light" (VSL) models conceived as a challenge to inflation restricts the variation of c to the early superluminary universe (Moffat [51], Albrecht & Magueijo [2], Magueijo & Smolin [45]). These models do not match BHC since, after restoring the local Lorentz invariance, the light is thought to travel at the presently measured speed. Likewise, assuming the change of c refers totally to the time preceding the structure formation, they would not imply the variability of [M.sub.Ch].

In some VSL models, the change in c value has been considered as a continuous process spread over the whole lifespan of the universe. Dicke's theory of gravity (Dicke [13]), developing the earlier considerations by Einstein [22,23] explains the cosmological redshift as a result of c decreasing with time, which somehow corresponds with the steady state theory. However, this model does not predict the change of c to be a measurable effect since it assumes the units of length and time to change accordingly.

In turn, variability of G has been proposed in some scalar-tensor models modifying the Einstein's General Relativity, in particular the Brans-Dicke theory [9] inspired by Mach's principle, with the time and space dependent scalar field [phi] modifying the Newton's constant. A similar as to the general structure and conclusions model by Hoyle & Narlikar [36] originates from considerations concerning the action on distance. Petit [61, 62] advanced a model with joint variation of G, c and h decoding the Hubble's law in a static universe. One of the first models postulating varying G, and likely the most influential one, is the Dirac's "large number hypothesis" (Dirac [15]). From the supposed coincidence between two ratios: radius of universe (expressed as ct) vs. radius of electron, and electrostatic force vs. gravitational force between proton and electron (both of them yielding [approximately equal to] [10.sup.40]), Dirac derived a conclusion that G changes as the inverse of cosmic time: G [varies] [t.sup.-1], while the mass of universe increases as [M.sub.u] [varies] [t.sup.2]. Provided the approximately linear relationship between time and radius ([R.sub.u] [varies] t), LNH satisfies BHC. However, LNH also implies [M.sub.Ch] [varies] [t.sup.3/2], which compared with the standard assumption of constant G makes the SNe Ia progenitor problem even more puzzling. A model proposed by the present author (Rybicki [69]) has postulated G [varies] [R.sub.u], [M.sub.u] = con st, yet then with no reference to BHC and the SNe Ia progenitor problem.

A question underlying the varying constants models is whether the postulated changes in dimensional constants are physically meaningful. A long-lasting controversy over this subject has not been concluded so far. Some physicists (e.g. Barrow [5], Duff [20]) claim that only the (potential) change in dimensionless constants matters, e.g. the coupling constants of fundamental forces such as fine structure constant a, gravitational coupling constant [[alpha].sub.G], or the masses of elementary particles related to Planck mass contributing to standard model. Instead, dimensional constants such as h, c, G, e, or k may change in value dependently on the (arbitral) choice of units, thus being merely the "human constructs" or "conversion factors". Others (Okun [57], Veneziano [76]) consider as indispensable in shaping the fundamental theories respectively three (G, c and h) and two (c and string length [[lambda].sub.s]) dimensional constants.

From the "dimensionless" point of view as applied to BHC, no matter whichever of dimensional constants is thought to vary; only what counts is the change of [[alpha].sub.G] = G[m.sup.2.sub.e]/hc. Since we discriminate here between the change of G and c treated as different solutions of BHC, so this question demands a clarifying comment. Let's start with two remarks: 1) There is no doubt that G[c.sup.-2] [varies] [R.sub.u] implies the variability of [[alpha].sub.G]; 2) The fact that dimensional constant changes its numerical value together with the change of unit is trivial, and as such contributes nothing to discussion.

Let the increase of [[alpha].sub.G] be observed, correlated with the increase of [R.sub.u]. Assuming [m.sub.e] = con st, h = const, we conclude that it is either G [varies] [R.sub.u] or c [varies] [R.sup.-1.sub,u] which, according to the "dimensionless" paradigm, we treat as fully equivalent (i.e. physically indistinguishable) interpretations of [[alpha].sub.G] [varies] [R.sub.u]. However, from G[c.sup.-2] [varies] [R.sub.u] it follows: G [varies] [R.sub.u] h [[alpha].sub.G] [varies] [R.sub.u], and c [varies] [R.sup.-1/2.sub.u] h [[alpha].sub.G] [varies] [R.sup.1/2.sub.u], which obviously differs from [[alpha].sub.G] [varies] [R.sub.u]. Thus, G and c cannot be considered as "conversion factors" within BHC.

As we show in next sections, the Planck units of length and time react differently depending on whether G or c is postulated to vary. Besides, each of respective solutions affects entropy in a different way. We thus agree with the anonymous referee cited in Duff's paper: "It is true that if the fundamental "constants" h, c, G, k ... are truly constant, then they do indeed only act as conversion factors and can e.g. be set equal to unity. However, when they are postulated (or discovered experimentally to vary) in time, then we have to take into account that varying one or the other of these constants can have significant consequences for physics" (Duff [20]).

6 Basics of the VCM hypothesis

Expressed in the here proposed nomenclature, our main idea consists in postulating VCM as being the consequence of BHC. Any model satisfying BHC makes the Planck units variable, and thus determines new parameters of the Planck era.

Identifying the mass in the equation for Schwarzschild radius with Planck mass: m [equivalent to] [m.sub.P] gives

[r.sub.S] = G[m.sub.P][c.sup.-2] = G[(hc[G.sup.-1]).sup.1/2] [c.sup.-2] = [(hG[c.sup.-3]).sup.1/2] = [l.sub.P]. (13)

Accordingly, the black hole becomes the Planck particle. Implementing the Planck mass to the reduced Compton wavelength [lambda]/2[pi] = h[m.sup.-1] [c.sup.-1] makes the Planck particle the only one black hole whose Schwarzschild radius equals the Compton wavelength

[lambda]/2[pi] = h[(G[h.sub.-1] [c.sup.-1]).sup.1/2] [c.sup.-1] [equivalent to] [(hG[c.sup.-3]).sup.1/2] = [l.sub.P]. (14)

Rewriting the Schwarzschild equation for the Planck particle:

[l.sub.P] [approximately equal to] G[m.sub.p][c.sup.-2] gives the identity

[(hG[c.sup.-3]).sup.1/2] [equivalent to] G[(hc[G.sup.-1]).sup.1/2] [c.sup.-2], (15)

which means that Planck particle's property of being a black hole is insensible to the change of G or/and c.

From G[c.sup.-2] [varies] [R.sub.u] it follows [m.sub.P] [varies] [R.sup.-1/2.sub.u]; hence for [R.sub.u] [right arrow] 0 the Planck mass tends to infinity. However, to avoid singularities (and also taking into account that Planck mass should have "realistic" reference), we assume that in the newly defined Planck era (denoted [P.sub.0]) the Planck mass coincidences with the mass of universe:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Thus, the initial value of Schwarzschild radius becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

This can be also obtained by expressing the Newton's constant in the equation [R.sub.u] = [GM.sub.u][c.sup.-2] in terms of Planck units, namely: G = [l.sub.P][m.sup.-1.sub.P] [c.sup.2]. Then

[R.sub.u] = [l.sub.P][M.sub.u][m.sup.-1.sub.P] (18)

and so

[R.sub.u][l.sup.-1.sub.P] = [M.sub.u][m.sup.-1.sub.P] (19)

meaning that identity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] becomes a consequence of the conjecture [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We have thus arrived at conclusion that the universe at its initial stage (here called "primordial Planck era"--PPE) had the form of a quantum mechanical black hole identified with a single one "primordial Planck particle" (PPP), described by equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

Accordingly, the notion of PPP becomes coherent with the concept of the universe emerging from "nothing" due to the Heisenberg uncertainty.

From [M.sub.u] [approximately equal to] [10.sup.54] kg, provided [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

a factor hereinafter denoted by [delta].

Because [M.sub.Ch] ~ [m.sup.3.sub.P] so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Obviously, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as related to the early universe, is a formal entity only. To be a physically meaningful concept, Chandrasekhar limit demands a proper physical "enviroment" (atoms, elements, stars). It belongs then to the epoch of structure formation starting from Population III stars. Provided the universe expanded in a roughly uniform rate, BHC can be expressed as the approximate function of cosmic time: G[c.sup.-2] [varies] t. From the whole range of possible BHC scenarios, the two deserve special attention, namely: 1) G [varies] [R.sub.u] i.e. G [varies] t, c = const, and 2) c [vaties] [R.sup.-1/2.sub.u], G = con st, both analyzed in the next sections.

7 Assumption c [varies] [R.sup.1/2.sub.u], G = const: collision with the second law of thermodynamics

The initial value of speed of light derived from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] becomes [c.sub.0] = [10.sup.132] [ms.sup.-1], yielding [c.sub.0]/c [approximately equal to] [10.sup.124] = [[delta].sup.2]. The respective Planck length is (hereinafter, SI units always when omitted)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

a value equal to the Schwarzschild radius

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

and to the Compton wavelength

[[lambda].sub.0] = h[M.sup.-1.sub.u] [c.sup.-1.sub.0] [approximately equal to] [10.sup.-220]. (25)

The initial Planck time would amount to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

From [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] it follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

As derived from c [varies] [t.sup.-1/2], with the age of universe [approximately equal to] 13.8 x [10.sup.9] yr the current rate of decrease in the speed of light becomes

h/c [approximately equal to] -2.7 x [10.sup.11] [yr.sup.-1]. (28)

Let us compare this prediction with the results obtained from observations of gas clouds spectra intersecting the distant quasars, the Oklo natural uranium fission reactor, and atomic clocks. In agreement with the VSL paradigm, the supposed change of a is usually interpreted as the change of c. For the approximate emission time connected with the observational data samples concerning quasars: [t.sub.EM] [approximately equal to] 0.25 [t.sub.0]/0.85 [t.sub.0] covering [approximately equal to] 8.3 Gyr (here [t.sub.0] stands for the present moment), the reported values suggesting the change are: [DELTA]c(t)/c = (-0.57 [+ or -] 0.10) x [10.sup.-5] (Webb et al. [77]), and [DELTA]c(t)/c = (-1.-9 [+ or -] 0.17) x [10.sup.-5] (Webb et al. [78]). At the same time, other groups (e.g. Chand et al. [10]) reported no detectable change in a value over the last 1o-12 billion years. In the case of Oklo, for the respective operating time [t.sub.prev]/[t.sub.0] [approximately equal to] 0.87, Petrov et al. [63] obtained [??]/[alpha] = (-4 + 3) x [10.sup.-17] [yr.sup.-1], in fact signifying no detectable change. In turn, Lamoreaux & Torgerson [41] reported a decrease in alpha at the level -4.5 x [10.sup.-8] over the last 2 billion years, which consequently should be interpreted as the increase of the speed of light. Observations based on atomic clocks give a direct insight to the possible current rate of change. Peik et al. [59], using cesium atomic clock set the limit of annual change of the present variation of alpha for [??]/[alpha] = (-1.2 [+ or -] 4.4) x [10.sup.-15] [yr.sup.-1]. In turn, Rosenband et al. [67], based on the frequency ratio of [Al.sup.+] and [Hg.sup.+] in a single ion atomic clocks obtained a bound: [??]/[alpha] = (-1.6 [+ or -] 2.3) x [10.sup.-17] [yr.sup.-1].

Except the data provided by Webb et al. suggesting decrease of c at the level [10.sup.-15] [yr.sup.-1], and the opposite one (as to general conclusion) provided by Lamoreaux & Torgerson, all other results seem to point the zero change. This suggests the failure of assumption c [varies] [R.sup.-1/2]. Besides, the question of entropy provides us with an additional argument against declining c. As is known, entropy is proportional to the horizon surface area, which normally (i.e. by assumption G = con st, c = con st) implies linear dependence on the squared mass. Let us apply the Bekenstein-Hawking formula for the entropy of black hole:

[S.sub.BH] = Ak[c.sup.3] [(4Gh).sup.-1] (29)

or, written in terms of Planck length,

[S.sub.BH] = Ak(4[l.sup.2.sub.P]).sup.-1], (30)

where A is the surface area for event horizon, and k the Boltzmann constant. For the spherically symmetrical black hole, the surface area is A = [m.sup.2]8[pi][G.sup.2] [c.sup.-2] so entropy becomes [S.sub.BH] = [m.sup.2]2[pi][Gkch.sup.-1]. Thus, despite increasing surface area A = [m.sup.2]8[pi][G.sup.2] [c.sup.-2], at the assumption m = M = const, G = con st and c [varies] [R.sup.-1/2.sub.u], the entropy decreases according to [S.sub.BH] = [m.sup.2]2[pi][Gkch.sup.-1] being dependent on the decreasing speed of light: [S.sub.BH] ~ c. One obtains therefore

[S.sub.BH(present)]/[S.sub.BH(primordial)] = [[delta].sup.2], (31)

which violates the second law of thermodynamics applied to the universe as a whole. This does not exclude VSL models in general; in particular, does not exclude VSL applied to the very early universe. However, BHC is not agreeable with VSL conceived as a continuous process. Therefore, in the further considerations, we shall specify BHC as a model defined by the assumption G [varies] [R.sub.u], c = con st. We shall also treat this model as a right basis for the VCM hypothesis and the respective quantitative predictions.

8 Assumption G [varies] [R.sub.u], c = con st: parameters of the universe at Planck era

Provided [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the initial value of Newton's constant derived from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], yielding G/[G.sub.0] = [[delta].sup.2]. The initial Planck length becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

equal to the Schwarzschild radius:

[r.sub.s] = [G.sub.0][M.sub.u][c.sup.-2] [approximately equal to] [10.sup.-97] (33)

and to the (constant) value of Compton wavelength for the universe:

[[lambda].sub.0] = h[M.sup.-1.sub.u][c.sup.-1] [approximately equal to] [10.sup.-97]. (34)

All three quantities apply to the initial size of universe [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

The initial Planck time is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The invariability of Planck constant is a consequence of the fact that, although individually Planck energy and Planck time change in time, their product remains constant:

[E.sub.P(variable)] X [t.sub.P(variable)] = [h.sub.(constant)]. (38)

In general, initial values of the base Planck units relate to their present equivalents as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

The horizon problem in PPE is solved so to speak by definition, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (40)

which means that the whole primordial universe fits in a light cone.

The density in the primordial Planck era is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

equal to initial Planck density:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (42)

Let us compare this with the critical density derived from the Friedmann equation: [[rho].sub.c] = 3[H.sup.2][(8[pi]G).sup.-1], as calculated for PPE. The current value of Hubble constant ([approximately equal to] 70 [kms.sup.-1]/Mpc) expressed in SI units amounts to

[H.sub.(now)] [approximately equal to] 2.27 X [10.sup.-18] [s.sup.-1] (43)

yielding the respective value of the Hubble constant in PPE:

[H.sub.(PPE)] = [H.sub.(now)] X [[delta].sup.2] [approximately equal to] [10.sup.106] [s.sup.-1]. (44)

Approximating 8[pi][G.sub.0] [approximately equal to] [10.sup.-133], one obtains the PPE critical density:

[[rhp].sub.c(PPE)] [approximately equal to] [10.sup.212] [10.sup.133] [approximately equal to] [10.sup.345]. (45)

Hence, it is likely that also in PPE

[[rho].sub.(PPE)] = [[rho].sub.c] (46)

which solves the flatness problem.

In contrast to the previously considered assumption c [varies] [R.sup.-1/2.sub.u], G = con st, the thermodynamic arrow of time becomes well defined. Considering G/[G.sub.0] = [[delta].sup.2], from [S.sub.BH] = [m.sup.2]2[pi][Gkch.sup.-1] it follows

[S.sub.BH(present)]/[S.sub.BH(primodial)] = [[delta].sup.2]. (47)

In the cosmological scenario based on assumption G [varies] [R.sub.u], c = con st, the expansion is linear, or roughly linear, including the early epoch. This means that G [varies] [R.sub.u] is tantamount to G [varies] t; in particular, [G.sub.0]([10.sup.-134]) coincides with [t.sub.P0] ([10.sup.-105]). At some additional assumptions, this scenario could be modified so as to regard nonlinear expansion during early epochs. However, considering that basic motives for invoking inflation (horizon problem and flatness problem) are absent in BHC scenario, inflation appears to be basically redundant.

9 Assumption G [varies] [R.sub.u], c = con st: question of consistence with observational tests of G variability

Provided the approximately uniform rate of Hubble flow, the derived from G [varies] [R.sub.u] current rate of increase of G becomes a simple inverse of the age of universe. In fact, the Hubble time does not significantly differ from estimations of the age of universe derived from Friedman equation equipped with definite values of k and [LAMBDA]. Whereas these estimations range from [approximately equal to] 13.798 Gyr (Lambda-CDM concordance model based on data from Planck satellite and WMAP) to [approximately equal to] 13.82Gyr (Planck mission), the Hubble time ranges between [approximately equal to] 13.7 Gyr and [approximately equal to] 14.26 Gyr according to the current extreme estimates of the Hubble constant: [approximately equal to] 72 and and [approximately equal to] 67 kms-1Mpc-1 respectively. Thus, on the average, the Hubble time only slightly exceeds the supposed age of universe. Interpreting G [varies] [R.sub.u] as G [varies] t and estimating the age of the universe for [approximately equal to] 13.8 x [10.sup.9] yr gives the current rate of change:

G/G [approximately equal to] 7.25 x [10.sup.-11] [yr.sup.-1]. (48)

Let us compare this prediction with the constraints put upon G variation, derived from different sources (paleontology and geophysics, celestial mechanics, stellar physics, cosmology). A handful of representative results covering the whole range are:

--paleontological data connected with Earth temperature: [absolute value of [??]/G] < 2.0 x [10.sup.-11] [yr.sup.-1] (Eichendorf & Reinhardt [21]);

--increase of Earth radius: [??]/G = (-0.5 [+ or -] 2) x [10.sup.-11] [yr.sup.-1] (Blake [8]);

--stability of the radii of Earth, Moon and Mars: -[??]/G [less than or equal to] 8 x [10.sup.-12] [yr.sup.-1] (McElhiny et al. [49]);

--stability of the orbit of Mars (Mariner 9 and Mars orbiter data): [??]/G = (-2 [+ or -] 10) x [10.sup.-12] [yr.sup.-1] (Shapiro [71]);

--systematic deviations from the Keplerian orbital periods of Moon: [??]/G [greater than or equal to] (3.2 [+ or -] 1.1) x [10.sup.-11] [yr.sup.-1] (Van Flandern [75]);

--lunar laser ranging (LLR): [absolute value of [??]/G] < 6 x [10.sup.-12] [yr.sup.-1] (Dickey et al. [14]); LLR: [??]/G [less than or equal to] (4 + 9) x [10.sup.-13] [yr.sup.-1] (Williams et al. [81]);

--spin-down of pulsar JP1953: -[??]/G < 5.8 [+ or -] 1 x [10.sup.-11] yr-1 (Mansfield [46]);

--pulsar timing PSR B1913+16: [??]/G < (4 [+ or -] 5) x [10.sup.-12] yr-1 (Kaspi et al. [38]);

--luminosity function of white dwarfs (cooling age): -[??]/G [less than or equal to] [3.sup.+1.sub.-3] x [10.sup.-11] [yr.sup.-1] (Garcia-Berro et al. [27]);

--pulsating white dwarf data G117-B15A: [absolute value of [??]/G] [less than or equal to] 4.10x [10.sup.-10] [yr.sup.-1] (Biesiada & Malec [7]);

--SNe Ia luminosity vs. redshift: [??]/G = (-3, +7.3) x [10.sup.-11] [yr.sup.-1] (Mould & Uddin [52]);

--helioseismology: [absolute value of [??]/G] [less than or equal to] 1.6 [varies] [10.sup.-12] [yr.sup.-1] (Guenther et al. [32]);

--big bang nucleosynthesis (BBN): [absolute value of [??]/G] [less than or equal to] 9. [varies] [10.sup.-13] [yr.sup.-1] (Accetta et al. [1]); BBN: [absolute value of [??]/G] [less than or equal to] 1.7 x [10.sup.-13] [yr.sup.- 1] (Rothman & Matzner [68]).

One can easily notice that BHC prediction hardly matches the minority of the above bounds. However, a closer insight into methodology reveals various circumstances hidden behind the digits. We shall discuss them now, one by one.

9.1 Accuracy of the constraints on G variation and accuracy in measurements of the value of G

Unlike in the case of other fundamental constants, the increasing precision of measurements of G value is accompanied by increasing discrepancy of the obtained results. This led the CODATA to widen the uncertainty range from 0.013% to 0.15%. We ask whether this uncertainty may impinge on the G variability tests. This question does not seem groundless taking into account the ratio between typical bound put on the annual rate of change of G (~ [10.sup.-11]) and the uncertainty range of G value (1.5 x [10.sup.-3]), roughly ten-billionth! To better realize the scale, imagine we test the Wegener's continental drift theory (btw unaccepted for a long time) by settling a constraint on the annual rate of relative motion between two continents, say, America and Europe. Assume we determine two points (measuring devices) placed on each of these continents, and estimate the distance between them for 5 thousand kilometers. However, due to hypothetic imperfection of measuring techniques, this distance is only known with the relative uncertainty 0.15%, which translates into 7.5km. Assume next that, undeterred by this immense inaccuracy, we derive the constraint for the drift rate for [10.sup.-11] [yr.sup.-1], i.e. 0.05 mm/year, while the drift rate estimated by the theory amounts to 7.25 x [10.sup.-11] [yr.sup.-1], i.e. 0.36 mm/year (in fact, Wegener estimated the speed of drift for 2.5 m/year, while the currently observed rate amounts to about 2.5 cm/year).

Obviously, measuring a given value and measuring a change in this value are, basically, two different things; yet the mentioned discrepancy is too significant to be ignored. This in particular happens when a constraint depends on assumptions that are themselves encumbered by sizeable uncertainty (see subsection 9.3). In the above fictional example, before drawing ultimate conclusions as to the correctness of Wegener's idea, one should certainly aim at eliminating the distance uncertainty or try to find its hidden sources. Otherwise, any ultimate conclusions as to the change of distance could not be considered reliable. There is no reason to assume the question of variability of Newton's constant should subject to different rules.

9.2 Differences in notation and the question of autonomy of particular constraints

There is no unique notation for the constraints on G variation; for different reasons, particular constraints (or their groups) are expressed in different mathematical forms. Revealing their meaning provides us with a better insight into the question of autonomy. The (here called) canonical form, [??]/G [less than or equal to] (a [+ or -] b) x [10.sup.-cyr-1 (a positive/negative, b, c positive) reads: "An annual rate of increase/decrease of G, not greater than a x [10.sup.-c] has been observed, with the uncertainty range equal to [+ or -] b x [10.sup.-c]". If a = 0, it means that no change has been observed, although b still describes the range of uncertainty of that finding. Expression -[??]/G, instead of [??]/G, means that given constraint concerns solely (is design to detect) the decrease rate of G. This takes place when a theory predicting the decrease of G (e.g. Dirac's LNH) is tested, and thus respective assumptions are the base of derivation. In turn, the form [absolute value of [??]/G] reads: "The possibility of G variation (including increase and decrease in equal degree) fits in the range ..." However, [absolute value of [??]/G] is sometimes used as equivalent to -[??]/G, in particular when aimed at testing Dirac's hypothesis (e.g. Eichendorf & Reinhardt [21]). This form implies a to be indistinguishable from b, i.e. treats expressions "(rate) not greater than" and "with the uncertainty range" as tantamount to each other. Another way to identify the range of possible change with the range of uncertainty is the form [??]/G = (-[b.sub.1], + [b.sub.2]) x [10.sup.-c] [yr.sup.-1], [b.sub.1] + [b.sub.2]. Although apparently similar to [absolute value of G/G], this form indicates the observed tendency (i.e. increase or decrease) and thus seems to be basically equivalent to the canonical form; e.g. the term (-2, +4) could be expressed as (1 [+ or -] 3). An alternative use of the relation symbols [greater than or equal to] and = in each of the above forms can be interpreted (dependently on the context) as a gradable expression of conviction as to the observed tendency. In particular, symbols < and [less than or equal to], when used in the canonical form, play the role of additional proviso (apart of b term) due to general uncertainty; for example, if [absolute value of a] is greater than b then using = unambiguously points to the observed change of G. Instead, using < or [less than or equal to] weakens this statement, suggesting the change to be only probable.

Let us assume that, generally, all observations meet the criteria of scientific rigor. Apart of proper methodology and precision, this would also mean the unbiased standpoint as to the principal question, i.e. whether the Newton's constant is a true constant. Provided that, the postulate of autonomy says that each constraint should be interpreted in accordance with the sense of its notation and with regard to the underlying assumptions (usually not reflected in notation). In particular, weaker constraints should not be treated as "worse" than the stronger ones but, for the most part, as speaking in favor of variability.

9.3 Dependence on the employed theory and assumptions

Many factors involved in determination of the bounds put on G variation are theory or assumption dependent. For example, stringent constraints derived from BBN (Accetta et al. [1], Rothman & Matzner [68]) are valid only for Brans-Dicke theory; likewise, the constraint derived by Guenthner et al. [32] bases on the Brans-Dicke type theory with varying G. Most of constraints, even when not visibly shown in their notation, base on observations testing Dirac's LNH, i.e. are focused on the possible decrease of G. This in particular concerns the results derived from geophysical and paleontological data: impact of the Earth surface temperature on ancient organisms, expansion of Earth and the relevant difference in paleolatitudes between two sites of known separation (allowing to deduce the paleoradius), spin-down of the Earth due to its expansion, recession of the Moon and its impact on tides reflected in fossils. The respective data depend on too many conditions to repose excessive trust in their precision, and thus to consider them as fully reliable assumptions. In his extensive review study, Uzan [74] pays attention on these other sources of uncertainty connected with particular constraints.

9.4 Variation of Newton's constant and the age of universe

Assuming that increase of G extends the age of universe, the rate of G variation would be smaller than the here quoted value 7.25 x [10.sup.-11] [yr.sup.-1] thus better fitting observations. However, according to the Friedmann equation

[H.sup.2] = [([??]/a).sup.2] = 8[pi]G/3 [rho] - [kc.sup.2]/[a.sup.2] + [LAMBDA][c.sup.2]/3, (49)

variation of G has a negligible impact on the age of universe. For k = 0 (flat universe) and [LAMBDA] = 0, density becomes critical ([[rho].sub.c] = 3[H.sup.2][(8[pi]G).sup.-1], and thus Friedmann equation reduces itself to identity [H.sup.2] = [H.sup.2] becoming insensible to the change of G. In such a case, the age of universe simply equals the inverse of Hubble's constant (t = [H.sup.-1]). However, for [LAMBDA] [not equal to] 0, currently estimated for [[LAMBDA].sub.(const)] [approximately equal to] [10.sup.-52] m, dark energy (in a form of cosmological constant) predominates from a certain moment, so that t and [H.sup.-1] more and more diverge. In an accelerating universe driven by dark energy, the rate of increase of G determined by G [varies] [R.sub.u] also accelerates, which means that its declining in the unit time gradually slows down. Hence, in the far future, G [varies] [R.sub.u] will translate to G [varies] [H.sup.-1] rather than to G [varies] t.

9.5 Equivalence of gravity and inertia

As is known, there are (currently) four notions of mass: 1) active gravitational mass--measure of ability to create gravitational field or curvature, 2) passive gravitational mass--measure of "sensing" the gravitational field by a body (in the Newtonian depiction, respectively: measure of the force exerted by a body and measure of the force experienced by a body), 3) inertial mass--measure of resistance against the force accelerating a body (including the force of gravity), and 4) mass as a measure of energy according to E = [mc.sup.2]. Numerous experiments performed over a long time up to present days have shown with increasing precision that inertial mass and passive gravitational mass are proportional to each other: [m.sub.(inert)] ~ [m.sub.(pass)] (week equivalence principle). In turn, since active and passive gravitational masses are interchangeable according to the Newton's third law, so also active gravitational mass and inertial mass are proportional to each other:

[m.sub.(act)] ~ [m.sub.(inert)]. (50)

The active gravitational mass is proportional to the Newton's constant: [m.sub.(act)] ~ G. In fact [m.sub.(act)] is inseparable from G, which means that any change in the active mass should be interpreted as the change in the Newton's constant. Consequently, inertial mass is thought to follow the putative variation of G:

[DELTA]G [right arrow] [DELTA][m.sub.(inert)]. (51)

This would make the so-called "inertial reaction force" always (i.e. also in the time-slice experiments) equivalent to the gravitational force. At the same time, variation of the Newton's constant would not affect the mass interpreted as the source of positive energy. Accordingly, the tests on G variation derived from celestial mechanics (e.g. LLR) would be basically ineffective, while the other ones (e.g. based on stellar physics) would still remain valid.

10 Quantitative predictions of the varying Chandrasekhar limit hypothesis, based on G [varies] [R.sub.u], c = const

We shall now consider the VCM hypothesis in the form related to the BHC specified as G [varies] [R.sub.u] i.e. G [varies] t. On the assumption that the rate of Hubble expansion is approximately uniform, the Chandrasekhar limit depends on cosmic time as [M.sub.Ch] [varies] [t.sup.-3/2]. This determines characteristic "delay time" for a single white dwarf, defined as the time needed to reach the WD's mass by the decreasing [M.sub.Ch]. It makes thereby a basis for the quantitative predictions of VCM as to the rate of supernovae events, interpreted as a function of cosmic time. While, in general, the anticipated by VCM ability of a single WD to become the supernova meets the problem of the paucity of SNe Ia progenitors, the detailed predictions obviously demand more circumstantial investigation. One has to regard: 1) the number of single WDs within a given area (in particular, the number of their representative sample); 2) the mean/median mass of this sample; 3) the respective "delay time" for the median mass, determined by [M.sub.Ch] [varies] [t.sup.-3/2]. Besides, in predicting the rate of distant SNe Ia one should also regard the related to distance intrinsic time of the observed events, and a corresponding value of Chandrasekhar limit. Once a distance is well defined, the respective limit should be treated as constant, considering the negligible (compared with the assumed rate of change in [M.sub.Ch]) time devoted to observation. Instead, for the nearby SNe Ia one may fairly assume [M.sub.Ch] [approximately equal to] 1.4M[dot encircle].

Let us apply the above to our Galaxy. For the sake of simplicity (an also taking into account the uncertainty in all data), we shall not regard the contribution of SNe Ia originated in binaries. We aim to estimate the present rate of SNe Ia, deriving it from accessible data, according to the above quoted three points. As is known, the Galaxy contains roughly 100-400 billion stars, above 97% of them supposed to end as white dwarfs, which however includes both actual WDs and the potential ones. According to the estimations based on SPY project, the space density of WDs within the radius of 20 pc is (4.8 [+ or -] 0.5) x [10.sup.-3] [pc.sup.-3] while the corresponding mass density amounts to (3.2 [+ or -] 0.3) x [10.sup.-3] M[dot encircle] [pc.suip.-3], which gives the overall mean mass [(M).sub.WD] [approximately equal to] 0.665 M[dot encircle] (Holberg et al. [35]). Instead Kepler et al. [39], basing on catalog elaborated by Eisenstein et al. [24] from the SDSS Data Release 4, found significant difference in the WD's mean mass between DA and DB stars (hydrogen and helium layers, respectively); namely [(M).sub.DA] [approximately equal to] 0.593M[dot encircle] and [(M).sub.DB] [approximately equal to] 0.711M[dot encircle]. Considering the number of DA and DB in the sample (7167 and 507, respectively), one gets the [(M).sub.WD] [approximately equal to] 0.6M[dot encircle]. We shall us this value in the further calculations.

In order to estimate the total number of white dwarfs in the Milky Way, we have to multiply the WD's space density by the Galaxy volume. Certainly, such an extrapolation is encumbered by significant uncertainty, as it is doubtful whether the sample obtained from the relatively close neighborhood (thin disc, in general) is typical for the whole Galaxy including thick disc, halo and the galactic bulge. Different parts of Galaxy vary in age, so WD's population is likely inhomogeneous in age and density. Evaluating the radius for 15,000 pc and the mean thickness for 5,000 pc and multiplying this by WDs' local density, one obtains: (3.5 x [10.sup.12] [pc.sup.3]) x (5 x [10.sup.-3]) [approximately equal to] 1.7 x [10.sup.10]. This gives an insight into the actual number of WDs, consistent with a list brought by the Research Consortium on Nearby Stars (RECONS). According to the latter, 8 of the nearest 100 stars are the white dwarfs, which, provided this to be the representative ratio, gives the total number between 0.8 x [10.sup.10] to 3.2 x [10.sup.10], dependently on the assumed total number of stars (100-400 billion).

The next step is to derive the "mean delay time" [(T).sub.del] for the WD's mean mass [(M).sub.WD]. The respective algorithm reads

[(T).sub.del] = [(t/[T.sub.u] + 1).sup.2/3] x [T.sub.u] - [T.sub.u] (52)

[T.sub.u]-age of universe, t--an auxiliary delay time not regarding the power index, yielding

t = [T.sub.u]/([M.sub.Ch]/[DELTA][M.sub.WD]) - 1, (53)

where [DELTA][M.sub.WD] = [M.sub.Ch] - [(M).sub.WD]. After conversion, one has

[(T).sub.del] = [([M.sub.Ch]/[(M).sub.WD]).sup.2/3] X [T.sub.u] - [T.sub.u]. (54)

Inserting [M.sub.Ch] - 1.4M[dot encircle], [(M).sub.WD] = 0.6M[dot encircle] and [T.sub.u] = 13.8 Gyr, one obtains [(T).sub.del] [approximately equal to] 10 Gyr. Dividing the number of white dwarfs in Galaxy by that time gives the rate of roughly 1-3 events per year, a frequency exceeding the observed rate by a factor > [10.sup.2]. However, this prediction does not concern the present rate but a hypothetic rate averaged over the above calculated [(T).sub.del]. One should not identify (or confuse) "averaged" with "uniform" mainly because WD's masses subject, in general, to the Gaussian distribution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (55)

([sigma]-standard deviation, [mu]-mean of the distribution) and the respective probability function:

Prob[a [less than or equal to] x [less than or equal to] b] = [[integral].sup.b.sub.a] f(x) dx. (56)

The observed standard deviation is significantly smaller than one ([[sigma].sup.2] << 1) yielding substantial peak around the median mass 0.6M[dot encircle]. Obviously, only WDs of the mass close to 1.4M[dot encircle], corresponding with the relatively short mean delay time, contribute to the present rate of SNe Ia. We assume that any single white dwarf of the mass close to [M.sub.Ch] is, dependently on specific conditions (rotation, chemical composition), a potential SN Ia at any moment during the slated delay time. Admittedly, the most massive known WD only slightly exceeds 1.3M[dot encircle]; this however should be associated with the fact that less than one-millionth of the whole population of WDs in Galaxy are identified so far. A similar difficulty concerns specifying the expression "close to 1.4M[dot encircle]". Bearing in mind an inevitable uncertainty, let us determine the respective range for [b-a] [approximately equal to] 0.1M[dot encircle], assuming that, dependently on detailed conditions, any WD of the mass between 1.3-1.4M[dot encircle] may become the SNe-Ia. For that mass range, the unit normal distribution yields less than 0.1% of the entire population, say, [approximately equal to] [10.sup.7]. The mean mass of this "representative sample" is 1.35M[dot encircle]. It follows:

[T.sub.del] [approximately equal to] [(1.4/1.35).sup.2/3] x 13.8 - 13.8 [approximately equal to] 0.34 (Gyr). (57)

The respective rate is then

[10.sup.7]/3 X [10.sup.8] = 3 x [10.sup.-2] ([yr.sup.-1]). (58)

This still slightly exceeds the observed rate, provided the latter is [less than or equal to] 1 events per 100 years. However, considering the mentioned above reservations, it would not be reasonable to attach excessive importance to this or that particular number. The real number of single WDs from the representative sample may prove to be much smaller than [10.sup.7]. The mass-range of potential progenitors may appear slightly narrower or wider. In general, more accurate data may support or falsify our hypothesis.

11 Conclusion

We have considered the SNe Ia progenitor problem in the context of general problem of the constancy of fundamental constants. Basing on arguments derived from the black-hole cosmology, we have singled out the Newton's constant as the most probable candidate for "inconstant constant". Since the increase of G involves the decrease in the value of Chandrasekhar limit [M.sub.Ch], both questions meet together yielding a hypothesis according to which a single white dwarf can alone become the progenitor of SN Ia.

Admittedly, the ongoing progress in observational techniques together with an improvement in stellar physics may bring solution to the progenitor problem dispensed with violating the constancy of Chandrasekhar limit. A tacit heuristic strategy connected with searching for the SNe Ia progenitors consists in attempts of making the SD and DD models flexible enough to eliminate the observed discrepancies. For the time being however the problem still exists, which makes solutions going beyond the binary paradigm justifiable and noteworthy.

The unbiased estimations seem to support the main thesis of this article, i.e. that [M.sub.Ch] decreasing according to G [varies] [R.sub.u] may explain the paucity of SNe Ia progenitors. It is to be noted that, predicted by G [varies] [R.sub.u] immense growth of the Newton's constant from the initial to present value (G/[G.sub.0] = [[delta].sup.2] [approximately equal to] [10.sup.124]) almost completely applies to the very early and early universe, preceding structure formation. Since the oldest SNe Ia detected so far: SN UDS1 0Wil (Wilson) and SN 1997ff reach about 11 Gyr the part of increase of the Newton's constant shaping the Chandrasekhar limit does not exceed the one order of magnitude, being much smaller in the case of overwhelming majority of the observed events.

Submitted on June 11, 2014 / Accepted on October 22, 2015

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Maciej Rybicki

as-Zubrzyckiego 8/27, 30-611 Krakow, Poland

E-mail: maciej.rybicki@icloud.com
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