The Cosmological Constant 
Ashay Dharwadker 
H501 Palam Vihar District Gurgaon Haryana 122017 India 


We show how to calculate
Einstein's cosmological constant using the Grand Unified Theory. Using
the topological properties of the gauge, we calculate the exact percentages
of ordinary baryonic matter, dark matter and dark energy in the universe.
These values are in perfect agreement with the sevenyear Wilkinson Microwave
Anisotropy Probe (WMAP) observations. Thus dark matter, dark energy and
the cosmological constant are intrinsic properties of the gauge in the
Grand Unified Theory.



Thanks to all the members of the Institute of Mathematics for their support and encouragement. We are pleased to announce that this paper appears as the final chapter of The Grand Unification published by Amazon in 2011.


In the grand unification of the standard
model of particle physics with quantum gravity [1],
Einstein's law of gravity for empty space is written in its simplest form
as
where R_{μν} is the Ricci
tensor [1, §7.4.1]. Einstein [2]
considered a generalization of equation (1) by introducing the cosmological
constant Λ to obtain
where the g_{μν} are given as functions of the spacetime coordinates and define the spacetime metric [1, §7.4.1]. We follow the LambdaCold Dark Matter (ΛCDM) model [1, §8], aka the standard model of big bang cosmology. In the ΛCDM model, the most commonly used solution g_{μν} of equation (2) is known as the FriedmanLemaître metric [3][4]. Since equation (1) is already in good agreement with observation for the solar system (almost flat spacetime), Λ must be small enough not to disturb this agreement. Since the Ricci tensor R_{μν} contains second derivatives of the metric tensor g_{μν}, the cosmological constant Λ must have dimensions (distance)^{2}. For Λ to be small this distance must be very large. It is a cosmological distance, of the order of the radius of the universe. The cosmological constant Λ is equivalent to an intrinsic energy density of the vacuum. A positive vacuum energy density Λ creates a negative pressure which causes an accelerated expansion of the universe. Observational evidence that this is indeed the case was found by astronomers studying supernovae in 1998 [5]. The positive vacuum energy Λ is called dark energy. Calculating the exact percentage of dark energy in the total massenergy of the universe is equivalent to calculating the value of the cosmological constant Λ. In 1934, while measuring the orbital velocities of galaxies in clusters astronomers found a discrepancy between the observed mass and the mass calculated from the equations of motion of the galaxies [6]. The galaxies must contain dark matter which cannot be observed i.e. a form of matter which does not emmit any electromagnetic radiation. Thus, the total energymass of the universe in the ΛCDM model arises from ordinary matter, dark matter and dark energy. The percentages of ordinary matter, dark matter and dark energy in the universe have been experimentally measured with great accuracy by the Wilkinson Microwave Anisotropy Probe (WMAP) [7] in 2010. We shall show how to calculate the percentages of ordinary matter, dark matter and dark energy from the grand unified theory [1] and corroborate the WMAP data. In the grand unified theory [1], the particle frame defines the gauge. Thus, at each point of spacetime there is an associated particle frame. In [1, §7.4.2], we have shown how the particle frame is embedded without selfintersection at each point of spacetime. Each Schrödinger disc of a particle frame carries the curvature and Ricci tensors according to equation (1). We shall show that the particle frame also carries the cosmological constant Λ according to equation (2) and calculate the exact value of the cosmological constant Λ from the topological structure of the gauge. Fix a Planck time interval T during the present epoch in the cosmological timeline [1, §4, §8]. During this Planck time interval T we may observe all the fermions in the universe which constitute the totality of ordinary matter. First consider a particular fermion F. The fermion F is selected according to the Fermion Selection Rule [1, §4.1] from its particle frame. The spin 1/2 fermion F satisfies Dirac's relativistic wave equation [8, §3] and determines exactly one of the four components Ψ_{i} (i = 0, 1, 2 or 3) of the wave function Ψ. Thus, F determines exactly one halfsurface out of the four halfsurfaces that constitute the labeled tRiemann surface of the particle frame, cf. [1, §4.4 The Spin Rule]. This halfsurface consists of 6 Schrödinger discs, one of which is selected for the fermion F. Since the 24 Schrödinger discs of the particle frame are superposed at a spacetime point [1, §7.4.2], the nonzero amplitude of the Schrödinger wave function of F is replicated along the boundaries of all 24 Schrödinger discs of the particle frame. However, exactly one of the discs actually carries an electric charge 0 = 0, 1 =1/3, 2 = 2/3 or 3 = 1 (this is the Schrödinger disc of F). The sign of the electric charge is given by the halfsurface determined by F [1, §4.5 The Electric Charge Rule]. The remaining 23 Schrödinger discs carry a replicated charge 0, 1, 2 or 3, but this charge is now interpreted as a gravitational gauge charge. This explains the reappearance of 0, 1, 2, 3 along with σ as the five parameters of the gravitational gauge group SU(5) in the grand unified theory [1, §7.4.6]. We postulate that the 5 remaining Schrödinger discs of the halfsurface determined by F behave as dark matter. Since each of the 5 Schrödinger discs of dark matter have the same Schrödinger wave function as F, they will exhibit the same massenergy as 5 copies of F. However, since these 5 Schrödinger discs do not have any electric charge defined, they cannot be observed via electromagnetic interactions i.e. will be dark matter. Nevertheless, the 5 Schrödinger discs of dark matter do have a gravitational gauge charge that can be observed over large distances via gravitational interactions. We further postulate that the remaining 18 Schrödinger discs of the other 3 halfsurfaces of the particle frame of F behave as dark energy. By the same argument, their effect can only be observed through gravitational interactions over large distances. Thus, the spin 2 graviton [1, §6.6.1, §6.6.2] must behave in two distinct ways. For the Schrödinger disc of the fermion F and the 5 Schrödinger discs of dark matter on the particle frame, it behaves as an attractive force carrier during gravitational interactions. Whereas, for the 18 Schrödinger discs of dark energy, it behaves as a repulsive force carrier during gravitational interactions. For all spacetime points where there is vacuum, we have a particle frame with no Schrödinger disc selected [1, §4 Figure 4.1]. By the above argument, we shall regard all 24 Schrödinger discs of such a particle frame to represent dark energy. Each fermion F has a positive mass attributed to it by a Higgs particle H via the HiggsKibble mechanism [1][9]. We can pair all fermions with their associated Higgs particles (F, H) during the Planck time interval T. Thus, on the particle frame of F we can superpose the particle frame of H [1, §6.7.1, §6.7.2]. The Higgs particle H consists of 12 Schrödinger discs by the Higgs selection rule [1][9] and only one of the 12 Schrödinger discs will intersect with the Schrödinger disc of F, contributing 1/12 of the mass of F. Now consider all such FermionHiggs pairs in the universe during this Planck time interval T. The total observable mass must be in the ratio (1 + 1/12)/24, since each particle frame has 24 Schrödinger discs. Thus, the percentage of ordinary observable matter in the universe is
Similarly, by the above argument, the percentage of dark matter in the
universe is
In particular, we have calculated the exact value of the cosmological
constant Λ from the grand unified theory [1].
We have shown that ordinary matter, dark matter and dark energy are all
properties of the particle frame which forms the gauge for the grand unified
theory [1]. The percentages (3), (4) and (5)
are in perfect agreement with the SevenYear Wilkinson Microwave Anisotropy
Probe (WMAP) Observations [7]:



References 


Copyright © 2011 by Ashay Dharwadker. All rights reserved. 