Large number hypothesis and the matter-dominated universe

Dirac’s large number hypothesis (LNH), in the formG/G ₀=HH ₀ ⁻¹ , is applied to the matter-dominated cosmological era, using the framework of the scale covariant theory (Canuto et al., 1977). We obtain explicit expressions forR andβ _a as functions ofR _E , whereR andR _E are the scale factors of the cosmological Robertson-Walker metric, expressed in atomic and gravitational units, respectively, andβ _a is the ratio between the rates of gravitational and atomic clocks. The parameters in these expressions are , the deceleration parameter in gravitational units, and (t ₀)H ₀ ⁻¹ where (t ₀) is the present epoch value of the derivative ofβ _a with respect to atomic time. We find that a necessary condition for the LNH to be compatible with a Robertson-Walker model is that (t ₀)H ₀ ⁻¹ ≥ ₂ ¹ . The only experimental values for (t ₀) available at present are those based on the lengthening of the Moon’s period of revolution around the Earth, suggesting 0.86≥ (t ₀)H ₀ ⁻¹ ≥0.21; the more promising technique of radar ranging to the inner planets has not yet produced a value for (t ₀). Using the lunar data, it follows that 0≤ ≲0.42 corresponding to an open universe (k=−1). Closed models (k=1, >1/2) are not compatible with the LNH since the required values of (t ₀)H ₀ ⁻¹ are more than an order of magnitude above the observational upper limit. Presented at the Dirac Symposium, Loyola University, New Orleans, 1981.     

Modified large number hypothesis

In the Brans-Dicke theory, a certain large number hypothesis is equivalent implicitly to an equation of state. The equation of state corresponding to Dirac's large number hypothesis, however, is not reasonable. The Whitrow-Randall relation is regarded as a modification of Dirac's large number hypothesis, but it is not in fact in keeping with Dirac's original intention to relate only a single cosmological parameter to the gravitational constant. In view of those facts, an alternative modification of Dirac's large number hypothesis is proposed.     

Generalized large number hypothesis and cosmological constant

By generalizing Dirac's large number hypothesis we infer that the cosmological constant varies witht ^–2, as expected from earlier studies.     

Large numbers hypothesis. I. Classical formalism

A self-consistent formulation of physics at the classical level embodying Dirac’s large numbers hypothesis (LNH) is developed based on units covariance. A scalar “field”ϕ (x) is introduced and some fundamental results are derived from the resultant equations. Some unusual properties ofϕ are noted such as the fact thatϕ cannot be the correspondence limit of a normal quantum scalar field. Presented at the Dirac Symposium, Loyola University, New Orleans, May 1981. NAS-NRC Senior Research Fellow 1981–1982.     

Gravitational vacuum hypothesis and cosmology with variable particle number

We discuss a heurisitic model of gravitational vacuum as a set of virtual, radiating planckeons, particles with Planck size (L) and mass (/cL). Combined with Dirac's large number hypothesis, this gives the minimum universe scale factor valuea _min10^–13 cm, the strong interaction length (a = L just whena=a _min ). Taking this state as an initial one using standard quantum techniques, we consider particle creation by planckeons. Under some reasonable assumptions we obtain the present number of particles with nucleon mass close to observations,N 10⁸⁰. A criterion for gravitational stability of particles is formulated and some applications of the corresponding mass formula are considered. In particular, Fermi's weak interaction constant is expressed in terms ofa andL and a finite value for the neutrino mass is obtained.     

Large numbers hypothesis. II. Electromagnetic radiation

This paper develops the theory of electromagnetic radiation in the units covariant formalism incorporating Dirac's large numbers hypothesis (LNH). A direct field-to-particle technique is used to obtain the photon propagation equation which explicitly involves the photon replication rate. This replication rate is fixed uniquely by requiring that the form of a free-photon distribution function be preserved, as required by the 2.7 K cosmic radiation. One finds that with this particular photon replication rate the units covariant formalism developed in Paper I actually predicts that the ratio of photon number to proton number in the Universe varies ast ^1/4 precisely in accord with LNH. The cosmological red-shift law is also derived and it is shown to differ considerably from the standard form ofνR = const.     

Large numbers hypothesis and Whitrow-Randall-Sciama relation in Brans-Dicke theory

It has recently been suggested in Brans-Dicke theory that the so-called Whitrow-Randall-Sciama relation plus the assumption of a constant deceleration parameter implies the large numbers hypothesis. It is shown that this claim is not always true.     

Large numbers hypothesis: Reply to a paper by Beesham

We discuss a recent paper by Beesham, showing that his criticism of an article by Berman is based on a contradiction, though his arguments are interesting.     

Large numbers hypothesis. IV. The cosmological constant and quantum physics

In standard physics quantum field theory is based on a flat vacuum space-time. This quantum field theory predicts a nonzero cosmological constant. Hence the gravitational field equations do not admit a flat vacuum space-time. This dilemma is resolved using the units covariant gravitational field equations. This paper shows that the field equations admit a flat vacuum space-time with nonzero cosmological constant if and only if the canonical LNH is valid. This allows an interpretation of the LNH phenomena in terms of a time-dependent vacuum state. If this is correct then the cosmological constant must be positive.     

Large numbers hypothesis. III. Kinetic theory,statistical physics,and thermodynamics

Dirac's large numbers hypothesis (LNH) is incorporated into kinetic theory, statistical physics, and thermodynamics using the self-consistent formalism of units covariance. The ingeodesic equation and matter creation introduce modifications of the most fundamental laws of the subject. Liouville's theorem no longer holds, the Boltzmann equation is modified, as is theH-theorem. This affects the second law of thermodynamics in that for canonical LNH neither reversible nor adiabatic processes are possible (as expected). A significant result is that the collision terms have the same form as in standard physics. This means that equilibrium distribution functions are identical to those of standard physics, as required for self-consistency with the precepts of LNH. The net effect of LNH is as though all matter in our Universe were weakly coupled to a large heat bath.NAS-NRC Senior Research Associate 1981–1983.     

Republication of: Large number coincidences and the anthropic principle in cosmology

This is a reprinting of the paper by Brandon Carter, first published in a little-known volume of conference proceedings in 1974, that moved the anthropic principle from the realm of philosophical speculations to the subject of theoretical physics. The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by George Ellis.     

Large boson number IBM calculations and their relationship to the Bohr model

Recently, the SO(5) Clebsch-Gordan (CG) coefficients up to the seniority v _max = 40 were computed in floating point arithmetic (T.A. Welsh, unpublished (2008)); and, in exact arithmetic, as square roots of rational numbers (M.A. Caprio et al., to be published in Comput. Phys. Commun.). It is shown in this paper that extending the QQQ model calculations set up in the work by D.J. Rowe and G. Thiamova (Nucl. Phys. A 760, 59 (2005)) to N = v _max = 40 is sufficient to obtain the IBM results converged to its Bohr contraction limit. This will be done by comparing some important matrix elements in both models, by looking at the seniority decomposition of low-lying states and at the behavior of the energy and B(E2) transition strengths ratios with increasing seniority.     

Large atom number dual-species magneto-optical trap for fermionic 6Li and 40K atoms

We present the design, implementation and characterization of a dual-species magneto-optical trap (MOT) for fermionic ⁶Li and ⁴⁰K atoms with large atom numbers. The MOT simultaneously contains 5.2 × 10^{9 6}Li-atoms and 8.0 × 10^{9 40}K-atoms, which are continuously loaded by a Zeeman slower for ⁶Li and a 2D-MOT for ⁴⁰K. The atom sources induce capture rates of 1.2 × 10^{9 6}Li-atoms/s and 1.4 × 10^{9 40}K-atoms/s. Trap losses due to light-induced interspecies collisions of ～65% were observed and could be minimized to ～10% by using low magnetic field gradients and low light powers in the repumping light of both atomic species. The described system represents the starting point for the production of a large-atom number quantum degenerate Fermi-Fermi mixture.     

Large negative numbers in number theory,thermodynamics, information theory,and human thermodynamics

We show how the abstract analytic number theory of Maier, Postnikov, and others can be extended to include negative numbers and apply this to thermodynamics, information theory, and human thermodynamics. In particular, we introduce a certain large number N ₀ on the “zero level” with a high multiplicity number q _i ? 1 related to the physical concept of gap in the spectrum. We introduce a general notion of “hole,” similar to the Dirac hole in physics, in the theory. We also consider analogs of thermodynamical notions in human thermodynamics, in particular, in connection with the role of the individual in history.     

Asymptotic oscillation hypothesis

The observed persistence of threshold effects into the Regge asymptotic region of single-particle inclusive cross sections can be accommodated in a natural fashion by complex Regge poles which, if factorizable, will produce asymptotic damped oscillations in a wide variety of other observable quantities. It is shown that a single pair of such poles, the imaginary part of whose position in the complex J plane is of the order ± 1, can qualitatively explain various anomalies that have been observed in high energy phenomena. Predictions are made for future measurements.     

A hierarchy hypothesis

Why attraction and repulsion between likes should not enjoy equal status in nature is considered. By postulating a hierarchy of isolated systems of finite radii whose associated charges form a geometric series with enormous imaginary common ratio, and by identifying a universe (the content of an infinite cosmos within a Hubble radius of an observer), an electron, and a neutrino as three consecutive members of the hierarchy (in fact the only three observable because of the uncertainty principle), it is possible to treat gravitational and electromagnetic phenomena as perfectly analogous and complementary for the overall structure of the cosmos. An isolated system behaves, from an external viewpoint, as an elementary particle, and from an internal viewpoint, as a universe. Remarkable relationships between physical constants emerge.