Universal dark matter and expansion age
from an electromagnetical cosmology
S. J. M. STOELINGA
Centre for High-Energy Astrophysics (CHEAF), University of Amsterdam Kruislaan 403, NL-1098 SJ Amsterdam, The Netherlands
(ricevuto il 17 Aprile 1996; approvato il 23 Luglio 1996)
Summary. — From our simple electromagnetical cosmology which is consistent with recent observational results, it is suggested that the dark matter mainly consists of roughly equal proportions of nucleons and neutrinos, and that for long-distance scales an apparent paradoxical small expansion age applies as observed.
PACS 98.80 – Cosmology.
PACS 04.50 – Gravity in more than four dimensions, Kaluza-Klein theory, unified field theories; alternative theories of gravity.
1. – X-ray examination of hot ionized gas in galaxy clusters using the Einstein Observatory satellite, the ROSAT All-Sky Survey and the EXOSAT, led to an estimate of the universal density parameter of baryonic matter up to VbB 0 .3 [1-3]. It thus
concerns dark matter for a great part. Re-examination of systematic errors in the determination of the primordial He-4 abundance in galaxies led to a value up to VbB
0 .2 [4]. These values agree with that from the baryonic-matter model universe of Cen et
al. [5], where Vb4 0 .2 6 0 .1 . As another dark-matter candidate the neutrino with a
rest mass in the 10–100 eV range is often proposed, providing a contribution to the mass density which has at least the above value [6-8].
Concerning the Hubble constant H0, the inferred values are roughly in the range
50–100 km s21Mpc21 where, however, those from long-distance scales such as from
the Virgo cluster of galaxies, strongly correlate with the reported values for the distance. Recently, to obtain an absolute reference to the extragalactic distance scale, from advanced observations of Cepheid variables Pierce et al. [9] determined the distance to the Virgo galaxy cluster (B15 Mpc), leading to a Hubble constant
H04 ( 87 6 7 ) km s21Mpc21. The corresponding expansion ages of the universe are
( 11 .2 60.1) Gy and (7.3 60.6) Gy for V40 and 1, respectively. The authors conclude that the estimated ages of galactic globular clusters of ( 16 .5 62) Gy [10] thus raise a problem regarding the universal age, where the origin of this paradox does not appear attributable to the observational data. New Hubble Space Telescope (HST) observa-tions of Cepheids by Freedman et al. [11, 12] led to H04 ( 80 6 17 ) km s21Mpc21from a
distance to the Virgo cluster of about 17 Mpc. Using the HST for observations of Cepheids in more nearby galaxies at 12 Mpc, a value of ( 69 68) kms21Mpc21 is
obtained by Tanvir et al. [13] and ( 52 68) kms21Mpc21 at 4 Mpc by Saha et al. [14].
A small value such as the latter is also obtained by Sandage and Tammann [15] from globular star clusters at 21 Mpc in the Virgo cluster and by Nugent et al. [16] from the physics of Ia supernovae at roughly 10 Mpc, where Riess et al. [17] derived ( 67 67) kms21Mpc21 from such objects at about 3 Mpc. For the value of
50 km s21Mpc21expansion ages of 20 Gy and 13 Gy for V 40 and 1 respectively apply,
so that not necessarily an inconsistency with the ages of the oldest stars is involved [16, 18].
In the following, together with dark matter we will discuss the above problem from our simple electromagnetical cosmology which is consistent with recent observational results [19-21]. Then, the universal geometry is defined by the original static Einstein universe where, however, equivalence of cosmic space and time is introduced involving a continuous creation of mass energy which is in favour of the universality of the physical laws (see below). The matter of the homogeneous model universe is described by electrons and protons (nucleons). The universal mass M B2p2c2ROG, where radius R 4cT, with c the velocity of light and T4H021B 14 Gy , H0B 72 km s21Mpc21, and
where G is the gravitational constant, while in a comparable state of the standard model Vn4 (rnOrc) B0.4, with nucleon density rnB H02OG and critical density rcB
3 pH2
0O4 G . Furthermore, the classical electric field of the electron is connected with a
gravitational part
(
eq. (2.1))
, which is magnetic if electrons are electromagnetic (Lorentz), so that their field corresponds to the field of a point-like magnetic dipole. Then, we deal with an electromagnetical cosmology where increasing units from varying fundamental atomic quantities (Lemaître, Hoyle) in proportion to T are found to apply such as the charge e and the masses meand mp of the above particles. Thisleaves the numerical values of the main cosmological and atomic quantities invariant. However, taking the local units as invariant allows for an increase of the values of parameters such as R and T. Apart from a dimensional factor of about an order of unity, the values can be roughly expressed (MKSA units) in simple powers of c, such as
R Ac3, G Ac21, k
cA c4A ˇ21, with kccircular wave number of a cut-off frequency from
the relation of general relativity to quantum fluctuations of the cosmic vacuum
(
Sakharov (sect. 3))
, where 2 pˇ is Planck’s constant, and e2A ac24, with a the fine-structure constant, while we have (eOmec2)2A G . Also, general principles are
found to hold such as the universality of the physical laws [22], Newtonian as well as Einsteinian, which is apparently paradoxical, and the similarity with respect to different scales, the so-called self-similar cosmology [23].
From the recession in our model universe the momentum mc of a rest mass equals the relativistic momentum of m which, on the space-like hypersurface, is related to a positional uncertainty defined by the space parameter as given by R. Thus, in the case of fundamental relation, a volume Rl2 is connected with a particle, where l is the particle dimension, roughly the particle Compton wavelength, respectively. Then, the universal mass is given by MOmpB (ROlp)2 as an upper limit of protons (electrons),
radius, and which is one of the large-dimensionless-numbers coincidences (Dirac) that are included. This leads to the Newtonian relationship [19] of our simple cosmological theory [20], where in Machian equations e and me play a main role in the universal
electromagnetic mass energy (sect. 2).
In the homogeneous universal state, from the quantum energy of lower wave-length kc21, the volumes RlO2e connected with electromagnetic field momentum
mec as produced by a point-like charge allow quanta up to a wavelength A c (sect. 2).
From the above positional uncertainty as given by R, this wavelength is related to a lower rest mass energy Aˇc2
B 102eV , where kc21is related to roughly Mc2. From an
upper limit connected with the universal space, a volume ARˇ related to this rest mass results which gives a particle dimension Aˇ1 O2. Taking the point-like electron
charge into account, in our electromagnetical-model world of electrons and protons, this mass energy corresponds to the field energy of an e1e2 dipole, where the
involved dipole length turns out to be correctly predicted by the above particle dimension. Also, in agreement with a universal mass built from these dipoles, using
rpB re (cf. above) the dipoles correctly predict the value of the proton mass. Then,
applying the above-mentioned self-similarity with regard to the dipole-to-proton ratios of proton and universe, we roughly obtain an equal contribution from dipoles and protons to the universal mass (dark matter) which, according to the above, yields V B8O3p40.9. The extra contribution from the dipoles to the universal mass density does not alter the rough approximations that led to our cosmological relationship [19] as denoted at the beginning of this section. Furthermore, our simple consideration about the cosmic background radiation (CBR) delivering the 3 K temperature ([21], sect. 2) leads to a photon-to-proton ratio in the universal space of roughly 3O2 times the dipole-to-proton ratio mentioned above. All these values suggest the dipoles are neutrinos, from recent experimental, observational and theoretical results (sect. 3).
In the case of inhomogeneity such as in the present universal state, regarding regions of low density between surrounding concentrations, a reduced value for the proportionally related space parameter may be taken into account. For, according to the above, the dipoles constituting the universal mass have a lower value of mass energy related to the upper wavelength and the fundamental connection with the space parameter as given by R, and provide in relation to their dimension a measure for the homogeneous universal space. Then, taking everywhere the same numerical values of atomic quantities into account
(
cf., e.g., ˇ and the dimensionless number a ([24], sect. 2))
, the decrease of the densities of the dipoles and resultant nucleons lead to an increase of the unit of space in the regions concerned, i.e. to an increase of the by-definition-numerically-invariant velocity of light. As for the dipoles in the nucleons and the universal space argued above, applying the self-similarity to the distribution of particles over the clusters and the surrounding regions, for long-distance scales we obtain an apparent paradoxical small expansion age.2. – In our electromagnetical cosmology, for the stationary Lorentz-type electron (sect. 1) we have the picture that curvature of space-time connects radial electric energy with magnetic gravitational energy in a Newtonian way such that
mec24
1
2]fe(re) re1 Fme(Rme) Rme(, (2.1)
where fe(re) 4e2O4 pe0re2, with e0 the dielectric constant of the vacuum and where mec2 Rme 4 Fme(Rme) B G 2 p2
g
meM R2 meh
. (2.2)In (2.2) the magnetic gravitational force Fme(Rme), representing the electron mass from
its interaction with the universal mass (Mach), balances the electric cosmological repulsion [20] and from M B (ROre)2mp (sect. 1), this gives
mec2 Rme B G 2 p2
g
memp re2h
. (2.3)From eqs. (2.2) and (2.3), a Newtonian interaction of me with the universal mass
roughly corresponds to the Newtonian interaction of mewith a proton mass regarding
a relative distance of about re. The proton mass roughly corresponds to the uncertainty
in mass which is connected to this interval, for we have re4 alOe, and
a B2pmeOmp, where a 4e2O4 pe0ˇc , and a21B ln (ROre). The repulsional electric
force fe(re) 4mec2Ore, i.e. it balances a magnetic centrifugal force [21].
In the above equations, e and meplay a main role in the universal electromagnetic
mass energy. Using the expressions in simple powers of c (sect. 1), from the lower wavelength kc21, the volumes RlO2e connected with momentum mec (sect. 1) allow
quanta up to a wavelength B (lO2
ekc) Ac. Thus, being the rest mass momentum
represented by an equal distribution of the mass over a distance as given by R, this quantum wavelength is related to a lower rest mass energy ARˇOcAˇc2
B 102eV while, since M ARc3 (sect. 1), kc21 defining an energy A c is roughly related to the
upper mass energy ARkcc ARcOˇBMc2. An upper limit of lower rest masses
connected with the universal space is AMOˇAR3
ORˇ , so that the particle dimension A ˇ1 O2B rd (sect. 1, see below).
Concerning this lower mass energy, for the field of the Lorentz-type electron as mentioned before we have written a magnetic-dipole field of the form
(
E(r), H(r))
4 q(e 21 0 r , ced3 r) 4 pr3k
1 2v2Oc2 , (2.4)where r, NrN4rFre, is the position vector from the origin O of a stationary reference
frame to any point P, charge q 4e
k
1 2v2Oc2at O with velocity v » r , NvNCc, and E(r) is the classical field of the electron while H(r) is a magnetic gravitational part on a circlet about P that has infinitesimal radius d with edany unit radius vector » r . Then
(sect. 1), the above lower mass energy corresponds to the field energy of an e1e2
dipole as given by the volume integral of e0E2d(r), where Ed(r) 4
(
3 r(p Q r) Or52pOr3
)
O4 pe0, is the electric field of the dipole with p 4erd, NrdN 4 rdB 10217m , the
vectors of the dipole moment and dipole length, respectively, and where the middle of rd is situated at an origin O with r, r re, as denoted above, while the magnetic
gravitational part provides a factor 2. The dipole agrees with rdA ˇ1 O2 (argued above)
and correctly predicts the value of the proton mass from (rpOrd)3ˇ A mp, where rpB re
(sect. 1), so that mpcorresponds to about 2 Q 107dipole masses. Thus (sect. 1), conceiving
the universal mass as built from dipoles we obtain that this value roughly represents the dipole-to-proton or -nucleon ratio in the universe by taking self-similarity into
account. The density of the free electromagnetic energy from the charges in our model universe equals about (e2
O4 pe0lOe)2 rOmp, where r is the average mass density, which
from the Stefan-Boltzmann law leads to a temperature T0B 3 K of the CBR [21]. Then,
we have about ( 2 e2
O4 pe0lOekT0) B3Q107photons per proton, where k is the constant of
Boltzmann, which from the above gives a photon-to-dipole ratio in the universal space of roughly 3O2.
According to the self-similarity, in an inhomogeneous state such as in the present universe the mass in the clusters (cf. end of sect. 3) roughly equals the mass in the surrounding regions (cf. end of sect. 1). Then, the energy from the space in these regions may be conceived as proportionally represented by roughly half the total number of particles in the universe (cf. Vb 0 .3 as measured from galaxy clusters while
VnB 0 .4 ). In the expression for the total number, (ROl)2, this corresponds to an
increase of l by roughly a factor k2 from an increase of the unit of space (sect. 1). Consequently, in the regions concerned an apparent reduction of the Hubble time by a factor of about 0.7 applies.
3. – The above photon-to-dipole ratio roughly equals 3O2, where from standard big-band cosmology the photon-to-neutrino ratio of each kind of neutrino (left- and right-handed) is expected to be 11O6 [25], which suggests that the dipoles are neutrinos. Also, the rest mass energy of an order of 102eV agrees with theoretical
results regarding the energy of relevant neutrinos with respect to the dark matter (sect. 1). Besides, rdB 10217m agrees with upper limits on neutrino dimensions as
obtained from experimental and observational results, among others on the basis of a proposed neutrino production by e1e2 annihilation (cf. above suggestion) in the
supernova [26]. The correct value for mp refers to the concept of lepton and quark
compositeness on the scale of electrons as denoted in [20], where a universal mass built from such neutrinos refers to the cosmogony according to a hot-dark-matter model based on neutrinos with a rest mass of a few tens of electron volts [7]. Values of VnB VnB 0 .4 roughly agree with theoretical and observational results, respectively (sect.
1), where V 4Vn1 VnB 0 .9 (sect. 1) corresponds to V 4 1 as predicted by
naturalness arguments and by inflationary-universe models. As for the electron [21], the angular momentum connected with the proton charge results from integration of r 3 (E3H)Oc2 over the visible universal space-like hypersurface where agreement (0.7ˇ) with the quantum value (ˇO2)k3 refers to the universality of the laws (sect. 1). In the same way we have a universal angular momentum RMc
(
cf. self-similarity (sect. 1))
. For the dipole an angular momentum obtained as in the above, where no quantum effects have been taken into account, has negligible value. Regarding its stability also magnetic centrifugal forces play a role (sect. 2). The source of these inertial forces is (equivalence principle) the photon spin which also governs the electromagnetic potential as the source of gravitation and as represented by the curvature of space-time, i.e. the curvature is described to connect the radial electric energy of the photons with an opposing magnetic gravitational part (sect. 2), so that the latter may be conceived as dragging energy (e.g., [27]) from the spin. The H-field in eq. (2.4) concerns the magnetic spin-1 part of the energy and stands for the spin-2 gravitational field, so that the graviton [20] represents two parallel photon spins. From the Sakharov approach to gravitation [28], kc, which is of an order of magnitude of the reciprocalPlanck length, is supposed to determine the limit of definiteness of space which agrees with the connection between kc and M (sect. 2).
A clumping on cluster scale of nucleons and dipoles or neutrinos (sect. 2, [5]) leads in the surrounding regions, as depending on the reduced mass density, to an increase of the by-definition-numerically-invariant velocity of light by a factor of roughly 1.4. From a consequent apparently reduced Hubble time by a factor of about 0.7 in these regions we have that for long-distance scales an apparent paradoxical small expansion age applies, as observed (sect. 1). Concerning the reported values of the expansion age for V 40, as in our cosmology, there is no expansional deceleration.
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