Duality and Lambda in Cosmology
Peter Wilson
Abstract: An expression for the cosmological constant using cosmic parameters is proposed: lambda=k/R^2. Here, R is a distance, characteristic of the universe; k is a dimensionless ratio, also characteristic. R is defined as the average distance between local clumps of matter (e.g. galaxy clusters) that do not participate in the Hubble expansion; k is the coupling efficiency (0<k<1) of local contraction to cosmic expansion. In practical terms, R is the distance beyond which the Hubble relation (v=Hr) is valid; k is the fraction of radiant energy converted to gravitational energy in the universe at-large. Dimensional analysis is used to argue that the model must include R & k, and that the two appear as lambda in the above relation. With lambda having an observed magnitude 10^-52 /m^2, and R an estimated 10 Mly (10^23 m), the proposal requires a k of magnitude 10^-6, or about 1 part per million.
Background
General Relativity (GR) is famous for being exact in all circumstances to the limit of model and measurement precision. Cosmology presents a special challenge, however, because there is no Newtonian model with which to compare it. This is because the universe appears infinite, and the Newtonian math “blows up” when one uses it to model the infinite. Therefore, cosmology was new territory when GR came along, and it came as a complete surprise in 1998 when it was discovered that the standard model was deficient. Specifically, the expansion of the universe was observed to be accelerating, not decelerating, as had been predicted (1, 2). Hence, the addition of “dark energy” to the model, which remains enigmatic 12 years later.
Traditionally, solutions to Einstein’s field equations of GR have been couched in terms of either/or: the universe either expands or contracts. Complex gravitational systems, however, are dualistic in nature: some parts of the system contract, while other parts expand. For example, massive stars tend to sink to the center of star clusters, while low-mass stars tend to get flung out as a consequence. A complete description must include both processes. Inevitably, some fraction of the energy lost by the contracting part(s) is coupled to expansion of the larger. Duality postulates that the “cosmological constant” represents the quantity of such coupling in the cosmos, and that the standard model, sans lambda, either ignores the effect, or fails to incorporate it correctly.
Abstract: An expression for the cosmological constant using cosmic parameters is proposed: lambda=k/R^2. Here, R is a distance, characteristic of the universe; k is a dimensionless ratio, also characteristic. R is defined as the average distance between local clumps of matter (e.g. galaxy clusters) that do not participate in the Hubble expansion; k is the coupling efficiency (0<k<1) of local contraction to cosmic expansion. In practical terms, R is the distance beyond which the Hubble relation (v=Hr) is valid; k is the fraction of radiant energy converted to gravitational energy in the universe at-large. Dimensional analysis is used to argue that the model must include R & k, and that the two appear as lambda in the above relation. With lambda having an observed magnitude 10^-52 /m^2, and R an estimated 10 Mly (10^23 m), the proposal requires a k of magnitude 10^-6, or about 1 part per million.
Background
General Relativity (GR) is famous for being exact in all circumstances to the limit of model and measurement precision. Cosmology presents a special challenge, however, because there is no Newtonian model with which to compare it. This is because the universe appears infinite, and the Newtonian math “blows up” when one uses it to model the infinite. Therefore, cosmology was new territory when GR came along, and it came as a complete surprise in 1998 when it was discovered that the standard model was deficient. Specifically, the expansion of the universe was observed to be accelerating, not decelerating, as had been predicted (1, 2). Hence, the addition of “dark energy” to the model, which remains enigmatic 12 years later.
Traditionally, solutions to Einstein’s field equations of GR have been couched in terms of either/or: the universe either expands or contracts. Complex gravitational systems, however, are dualistic in nature: some parts of the system contract, while other parts expand. For example, massive stars tend to sink to the center of star clusters, while low-mass stars tend to get flung out as a consequence. A complete description must include both processes. Inevitably, some fraction of the energy lost by the contracting part(s) is coupled to expansion of the larger. Duality postulates that the “cosmological constant” represents the quantity of such coupling in the cosmos, and that the standard model, sans lambda, either ignores the effect, or fails to incorporate it correctly.
Since the dimensions of a problem are theory invariant, dimensional analysis is used here to argue that the above two parameters, k & R, are required for a model to be minimally complete, but one or both are missing from the pre-1998 standard model without lambda. This is done by examining dualistic systems, and concluding that four is the minimum number of parameters which must be specified in order to have a unique solution to a given problem. The “standard model” comes in varying degrees of sophistication, but in the simplest case, the GR model appears at least one parameter short: R is missing entirely, and the inclusion of k, as defined here, is unclear.
Einstein included the thermodynamic behavior of radiation in his equations (as a gas of pressure p), but apparently not the interaction of radiation with matter (k). Lambda appears as a catch-all term, and has the same effect in GR as k and R have in a finite system. That is, k and R typically combine in such a way as to cause the system to expand at an accelerating rate. This will be illustrated below.
[Note: the “R” and “k” used here are not the same R and k used to express Einstein’s General Relativity. The “R” that appears in GR is the, “Ricci curvature tensor…which represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in a Euclidean space." Here, “R” is simply a distance. Likewise, the k in GR is a dependent variable, and is a whole number: 0, 1, or -1. Here, k is an independent variable, and is a real number between 0 and 1.]
Dimensions in a Finite Dualistic System
Newton’s law of gravity expresses the static force between two objects using three variables: the masses of the two objects, and the distance between them; or m1, m2 & r. Employing the Cosmological Principle, essentially a set of simplifying assumptions, we consider only symmetric and uniform problems, and set m1=m2=M. That is, we reduce the system to two variables, M & r. To include dynamics, we must also specify a rate-of-change, which we will call h, expressed in terms proportional to distance, that is: dr/dt=hr.
Einstein included the thermodynamic behavior of radiation in his equations (as a gas of pressure p), but apparently not the interaction of radiation with matter (k). Lambda appears as a catch-all term, and has the same effect in GR as k and R have in a finite system. That is, k and R typically combine in such a way as to cause the system to expand at an accelerating rate. This will be illustrated below.
[Note: the “R” and “k” used here are not the same R and k used to express Einstein’s General Relativity. The “R” that appears in GR is the, “Ricci curvature tensor…which represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in a Euclidean space." Here, “R” is simply a distance. Likewise, the k in GR is a dependent variable, and is a whole number: 0, 1, or -1. Here, k is an independent variable, and is a real number between 0 and 1.]
Dimensions in a Finite Dualistic System
Newton’s law of gravity expresses the static force between two objects using three variables: the masses of the two objects, and the distance between them; or m1, m2 & r. Employing the Cosmological Principle, essentially a set of simplifying assumptions, we consider only symmetric and uniform problems, and set m1=m2=M. That is, we reduce the system to two variables, M & r. To include dynamics, we must also specify a rate-of-change, which we will call h, expressed in terms proportional to distance, that is: dr/dt=hr.
Figure 1
Figure 1 illustrates the simplest dualistic system that is uniform. Two pairs of bodies orbit a common center-of-gravity, each pair in turn orbiting a moving center. Because the close pairs are orbiting in the same direction as the wide pair, tidal interaction causes transfer of angular momentum from the close pairs to the wide pair. Hence, the close pairs spiral in, and the wide pair spiral out: dualism. The dense parts are contracting, while the less dense part is expanding. More to the point, the contraction drives, or powers, the expansion. The bottom panel shows the system some time later.
What is the minimum number of parameters required to model such a system? Consider each body to have a mass of 0.5M; call the distance between the wide pair R, and that between the close pairs r; call the expansion-rate of the larger system H, and the contraction-rate of the close pairs h. In the idealized case, conservation of energy can be used to tie the expansion rate to the contraction rate, which in turn can be used to determine r. Therefore, only three parameters completely specify a solution: M, R & H.
In all non-ideal systems, however, energy coupling is not 100% efficient. Due to frictional losses and weak interaction with distant bodies, the energy lost by the close pairs spiraling in will be greater than the energy gained by the wide pair spiraling out. Therefore, in the “real world,” we must include a 4th parameter, k, that specifies the coupling efficiency of energy lost to energy gained. So the simplest real-world dualistic system requires a minimum of four parameters to completely specify it: M, R, H & k.
Behavior of finite dualistic systems
While the system pictured in Figure 1 would get “hotter” with time--the close pairs would spiral together faster and faster--we imagine the general case that may be limited in its rate of decay for some reason, such that the rate of energy loss is constant. If k is also constant, then the wider parts will gain energy at a constant rate. Because work in a gravitational system is inversely proportional to distance, if energy is input into the wider part at a constant rate, as R increases, dR/dt increases as well, that is, the rate of expansion will accelerate.
Such behavior is exhibited by the Earth-Moon system, although the source of energy is rotational, not gravitational, as in Figure 1. Earth’s rotation is coupled to the moon through tidal interactions. Because Earth is so much more massive than the Moon, and spins considerably faster than the Moon orbits, rotational energy is converted to gravitational at an approximately constant rate. Therefore, the rate of expansion of the system--about 3 cm/yr--is accelerating (3). Billions of years ago, when the Moon was in a steeper part of Earth’s gravity well, the expansion was slower.
What is the minimum number of parameters required to model such a system? Consider each body to have a mass of 0.5M; call the distance between the wide pair R, and that between the close pairs r; call the expansion-rate of the larger system H, and the contraction-rate of the close pairs h. In the idealized case, conservation of energy can be used to tie the expansion rate to the contraction rate, which in turn can be used to determine r. Therefore, only three parameters completely specify a solution: M, R & H.
In all non-ideal systems, however, energy coupling is not 100% efficient. Due to frictional losses and weak interaction with distant bodies, the energy lost by the close pairs spiraling in will be greater than the energy gained by the wide pair spiraling out. Therefore, in the “real world,” we must include a 4th parameter, k, that specifies the coupling efficiency of energy lost to energy gained. So the simplest real-world dualistic system requires a minimum of four parameters to completely specify it: M, R, H & k.
Behavior of finite dualistic systems
While the system pictured in Figure 1 would get “hotter” with time--the close pairs would spiral together faster and faster--we imagine the general case that may be limited in its rate of decay for some reason, such that the rate of energy loss is constant. If k is also constant, then the wider parts will gain energy at a constant rate. Because work in a gravitational system is inversely proportional to distance, if energy is input into the wider part at a constant rate, as R increases, dR/dt increases as well, that is, the rate of expansion will accelerate.
Such behavior is exhibited by the Earth-Moon system, although the source of energy is rotational, not gravitational, as in Figure 1. Earth’s rotation is coupled to the moon through tidal interactions. Because Earth is so much more massive than the Moon, and spins considerably faster than the Moon orbits, rotational energy is converted to gravitational at an approximately constant rate. Therefore, the rate of expansion of the system--about 3 cm/yr--is accelerating (3). Billions of years ago, when the Moon was in a steeper part of Earth’s gravity well, the expansion was slower.
Dimensions in an Infinite Dualistic System (the Universe)
In the Friedmann solution or “standard model,” there are two independent equations for modeling a homogeneous, isotropic universe (http://en.wikipedia.org/wiki/Friedmann_equations). These model the universe using five parameters: expansion rate; density; pressure, k, and lambda. However, pressure is implicitly set to zero in Figure 1, the simplest symmetric dualistic system, so we must explicitly set it to zero in the cosmological model under consideration in order to compare apples-to-apples. That leaves four variables.
At first glance, the number of variables is consistent with a finite dualistic model. However, the k in the Friedmann equations is not the coupling factor, k, used here; it represents a short-hand for three sets of solutions, and it can be 0, 1 or -1. In the solution for a flat universe, it is zero. Our universe appears flat, so the term with k disappears, and there are just three parameters: expansion rate, density and lambda. Prior to 1998, however, lambda had been assumed to have a value of zero, and the standard model was expressed using only two parameters: expansion rate and density.
As we have seen, a finite gravitational system that is dynamic and dualistic requires a minimum of three parameters to specify it, ideally, and four in the real world. Since the universe is dynamic, dualistic and real, we assume modeling it requires a minimum of four parameters, as well.
In the Friedmann solution or “standard model,” there are two independent equations for modeling a homogeneous, isotropic universe (http://en.wikipedia.org/wiki/Friedmann_equations). These model the universe using five parameters: expansion rate; density; pressure, k, and lambda. However, pressure is implicitly set to zero in Figure 1, the simplest symmetric dualistic system, so we must explicitly set it to zero in the cosmological model under consideration in order to compare apples-to-apples. That leaves four variables.
At first glance, the number of variables is consistent with a finite dualistic model. However, the k in the Friedmann equations is not the coupling factor, k, used here; it represents a short-hand for three sets of solutions, and it can be 0, 1 or -1. In the solution for a flat universe, it is zero. Our universe appears flat, so the term with k disappears, and there are just three parameters: expansion rate, density and lambda. Prior to 1998, however, lambda had been assumed to have a value of zero, and the standard model was expressed using only two parameters: expansion rate and density.
As we have seen, a finite gravitational system that is dynamic and dualistic requires a minimum of three parameters to specify it, ideally, and four in the real world. Since the universe is dynamic, dualistic and real, we assume modeling it requires a minimum of four parameters, as well.
Figure 2
Such a minimally specified model is illustrated at left. Suggested is an infinite number of particles, by imagining the cube repeats in all directions (Figure 3, Artist‘s Impression of Infinity). Top and bottom panels in Figure 2 represent early and later times; the larger system is expanding at a rate of H. Each cube has a dimension of R, as defined in the abstract: the distance beyond which all sources are red-shifted. Within each cube is a large number of bodies that sum to mass M. Gas pressure within each clump is always non-zero, and can be arbitrarily high in local regions up to black hole formation. Pressure between clumps is explicitly zero, however, in that the model presupposes divisions between regions, where non-relativistic particles fall towards one clump or the other, rendering gas pressure effectively zero between clumps.
As in Figure 1, the close particles are getting closer. For completeness, we consider all forms of energy in which “closer” implies lower energy: gravitational, electromagnetic and nuclear. Conservation of angular momentum and electromagnetic forces conspire to prevent the clump from collapsing all at once. Nonetheless, the total radiant power of the clump, or its luminosity density (LD), represents the net rate of local contraction in terms of energy, and this power is almost constant. A fraction of this radiant energy, k, is coupled to the expansion of the larger system. Thus, like the finite system in Figure 1, this model is completely specified with four parameters: mass of clumps (M); distance between clumps (R); rate of contraction of clumps (their radiant power, or luminosity density, LD); and the coupling of local-contraction (LD) to non-local expansion, k. Since k is not observable, an unspecified relationship is presumed to exist that yields H, that is: H=f(M, R, LD, k). Thus, the model is causally specified with M, R, LD & k, but H, M, R, & k are sufficient to uniquely characterize it.
As in Figure 1, the close particles are getting closer. For completeness, we consider all forms of energy in which “closer” implies lower energy: gravitational, electromagnetic and nuclear. Conservation of angular momentum and electromagnetic forces conspire to prevent the clump from collapsing all at once. Nonetheless, the total radiant power of the clump, or its luminosity density (LD), represents the net rate of local contraction in terms of energy, and this power is almost constant. A fraction of this radiant energy, k, is coupled to the expansion of the larger system. Thus, like the finite system in Figure 1, this model is completely specified with four parameters: mass of clumps (M); distance between clumps (R); rate of contraction of clumps (their radiant power, or luminosity density, LD); and the coupling of local-contraction (LD) to non-local expansion, k. Since k is not observable, an unspecified relationship is presumed to exist that yields H, that is: H=f(M, R, LD, k). Thus, the model is causally specified with M, R, LD & k, but H, M, R, & k are sufficient to uniquely characterize it.
Figure 3, Artist's Impression of Infinity
Since the Friedmann solution with lambda “works,” we ask how the minimum of four parameters in the dimensional analysis transpose into the accepted model using only three. Density is M/R^3, but this ratio alone is insufficient to uniquely specify either one, so R or M must appear again. That leaves k and a second use of R or M. Since energy-coupling (k) is a multiplicative factor, and distance (R) is divisive, it is plausible that the two independent variables can be combined into one, the former in the numerator, the latter in the denominator.
Or, since k & R appear nowhere else in the equations, lambda must represent them. Table 1, below, lists this proposed conversion explicitly.
Table 1
Standard Model Inputs Dualistic Inputs (H, M, R, k)
Expansion rate, H H
Density M/R^3
Lambda k/R^2
Or, since k & R appear nowhere else in the equations, lambda must represent them. Table 1, below, lists this proposed conversion explicitly.
Table 1
Standard Model Inputs Dualistic Inputs (H, M, R, k)
Expansion rate, H H
Density M/R^3
Lambda k/R^2
Testing the Duality Hypothesis
Referring to Figure 1: if we call the energy lost by the close pairs the system’s luminosity density, or LD, then clearly a causal relation exists, H=f(M, R, LD, k). Duality proposes that the expansion rate of the cosmos, or Hubble’s constant, is likewise dependent on the same four variables, as suggested by Figure 2.
Confirming this requires an independent expression with k in it. Such a relationship is suggested by Friedmann’s second equation, which includes pressure. As argued above, in the simplest model, matter-pressure is effectively zero between clumps, so only radiation-pressure is important. Radiation pressure is related to luminosity density, so LD enters the Friedmann solution obtusely. However, mass and distance are wrapped into one as density in the standard model, confounding the issue. Presently, the author cannot suggest an independent equation based on M, R, LD & k, so the proposal is unproven.
Summary
The simplest possible gravitational model of the universe would seem to require a minimum of four independent variables to specify it uniquely (Figure 2). The pre-1998 standard model with only two parameters, density and expansion rate, appears fatally underspecified. The addition of the third term, lambda, can be sufficient to uniquely specify the problem only if two of the three terms are combinations of four independent variables. Thus, it is proposed that the three inputs in today’s standard model represent four, as per Table 1.
Table 2 summarizes the published and estimated values of the parameters discussed herein.
Table 2, Cosmic Parameter Summary
Cosmic Parameter Published Value Estimated Value
Average clump mass, M 10^43 kg
Distance between
clumps, R 10^23 m
Matter density 3 x 10^-27 kg/m^3
Luminosity Density or
contraction rate, LD 2.5 x 10^-33 W/m^3
Coupling of contraction
to expansion, k 10^-6
Expansion rate, H 2.3 x 10^-18 /s
Dark energy/lambda 10^-52 /m^2
Mainstream cosmology stands by its use of only three parameters to characterize the cosmos gravitationally: expansion rate, density and lambda. While the third input suffers for physical explanation, dimensional analysis demands the appearance of k and R in the model, and suggests that lambda represents these two as k/R^2. A solution to Einstein’s field equations of GR, expressed explicitly in terms of M, R, LD & k, could confirm the proposal, and resolve the paradox of dark energy.
References
(1)S. Perlmutter et al. (1999). "Measurements of Omega and Lambda from 42 high redshift supernovae". Astrophysical J. 517: 565–86.
(2) Adam G. Riess et al. (1998). "Observational evidence from supernovae for an accelerating universe and a cosmological constant". Astronomical J. 116: 1009–38.
3) Linda T. Elkins-Tanton, The Earth & Moon, 2006, p. 127
Appendix: http://en.wikipedia.org/wiki/Friedmann_equations
