Direct Observational Test of a Proposed Cosmological Model

The Proposed Model

Perhaps, the simplest shape of space-time one can think of is the 4-ball in which the radius represents the cosmological time and the hypersurface  (3-sphere) represents our 3-space ( x, y, z )

This attractive shape is not far from the imagination of physicists but it is excluded for two reasons:

1)    It seems to contradict the results of the global application of Einstein’s field equation because according to this model the global geometry of the universe depends only on the age of the universe and has nothing to do with the average density of the universe as (supposed to be) implied by the field equation.

2)    It contradicts the acceleration of the expansion of the universe which is (supposed to be) supported by the observational data about red-shifts because a model of a spherical space with radial time implies a steady expansion of the universe.

Instead of hurriedly excluding this shape of space-time our proposed cosmological model tries to overcome these difficulties.

We can overcome the first difficulty easily by adopting a good definition for the cosmological constant which eliminates the dependence of the global geometry of the universe on its average density and relates it directly to the age of the universe. The cosmological constant which satisfies these requirements is the quantity that composed of two parts: the geometrical part which is the curvature of the universe determined by the age of the universe and the material part which is the average density of the universe .This proposed definition of the cosmological constant will not only enable us to use the simple model of radial time and spherical space but will also solve all the problems of modern cosmology which are generated from the dependence of the global geometry of the universe on its density like the Cosmological Constant Problem without hypothetical concepts (dark energy, inflation, multi-verse, extra dimensions … etc.).

This will not affect the successful local application of the field equation because of the small value of the average density compared to the density of the source of the gravitation field (the zero-point vacuum energy density is canceled out because it is a part of both the cosmological constant and the stress-energy tensor)

Now, let us turn to the second difficulty with our proposed model of the accelerating expansion of the universe or more precisely the cosmological red-shift which is assumed to be a result of the acceleration of the expansion. Surprisingly, our simple spherical model which denies this acceleration offers another interpretation of this cosmological red-shift. It can be proved mathematically by analyzing the world-line of light as it travels through this shape of space-time between the source and the observer that it is a logarithmic spiral which leads to a red-shift given by:

Where theta is the angle between the world-line of the source and the observer. This agrees with the observational data and reproduces Hobble’s Law in small values of theta .The spiral path of light resolves the Horizon Problem (The problem with the standard cosmological model that different regions of the universe have not contacted to each other  but have the same physical properties) because all the radiations emitted from a source will converge to reach again the same source from all direction which may suggests that the radiations with the same properties which we receive from different directions could be emitted from our own galaxy and led by the spiral path to meet our world line again . This may also offers an answer to the problematic phenomena which was discovered recently that some features of the cosmic radiation is aligned with  the orientation of our solar system which can be thought of as the part of radiation which is emitted from the sun the converged to reach it again from all direction and thus appears as aligned with the ecliptic plane for any observer near the sun.

There is indeed so much in these topics which such an approach cannot but miss, but the intention in this stage of our argument is mainly to show that it is a mistake to suppose that the acceleration of the expansion is the only possible explanation of the observed cosmological red-shift or even the best of all possibilities.

The Observational Test of the Proposed Model

The plan of this test is to identify some of the relations between physical properties obtained from this cosmological model which distinguish it from the Standard Cosmological Model and other existing models and compare these relations with the model-independent observational data.

One of the best of these relations is the Redshift-Flux relation because both redshift and flux are model-independent physical quantities and fortunately some natural phenomena such as type la supernova provides us with the tools needed to construct the relation between these two quantities in different cosmological models because this type of supernova give a certain known amount of light and thus represents a standard candle that can be used to calculate luminosity distance which can be used to link the flux  to the red-shift according to the adopted cosmological model.

Now, let us see the mathematical form of this relation in our proposed cosmological model.

The Solution of the Cosmological Constant Problem without any change in GR or QM

Our aim is to introduce a proposed resolution to the cosmological constant problem which satisfies the following requirements :

1) It does NOT involve any modification in General Relativity or Quantum Mechanics.

2) It does NOT generate any propositional quantities or contrived concepts such as Dark Energy , Inflation , Multiverse ... etc.

3) It suggests solutions to other problems of cosmology such as Horizon Problem , Ecliptic Alignment of CMB Anisotropy ... etc.

4) It fulfills the general criteria of science such as simplicity , falsifiability ... etc.

Our strategy is simple and clear . it is based on two assumptions, each of them seems to contradict the results of General Relativity but , interestingly, they are consistent with this theory when they are taken together:
 
The first assumption is concerned with the cosmological model:

The space-time is a 4-ball in which the 3-dimensional spherical surface is the 3-space of the universe and the radius is the cosmological time.

There must be a part of the cosmological constant associated with this model which is the curvature of the space of this spherical shape which can be determined directly and simply from the age of the universe.

In spite of its simplicity and attractions , it may be thought that this assumption contradicts the result of the global application of Einstein's Field Equation because the assumption implies that the global geometry of the universe depends only on the age of the universe and has nothing to do with the average density of the universe as supposed to be implied by the field equation of General Relativity.
On the other hand, this model of the universe also  contradicts the acceleration of the universe which seems to be supported by observational data about the cosmological redshift, because a model of a spherical space with radial time implies a steady expansion of the universe.

Now, instead of hurriedly excluding this beautiful model let us try to overcome theses two difficulties simply by adding another assumption:

Beside the geometrical part of the cosmological constant mentioned above, there exists also a material part subtracted from the right hand side of the field equation, this material part is the average density of the universe .

This assumption leads to the independence of the global geometry of the universe from its average density which is in agreement with our  dearest  model of spherical space and radial time.

Now, let us turn to the second difficulty with our model which is the acceleration of the expansion of the universe or more precisely the cosmological redshift which is supposed to be a result of the acceleration of the expansion of the universe. Surprisingly, our model of the spherical space and radial time offers to us other interpretation of this cosmological redshift because it can be proven that the geodetic path on this shape of space-time which should be followed by light between any source and observer is a logarithmic spiral tends to straight line in large values of the ages of the universe which is associated with redshifts that agree with the cosmological observations.

More details and arguments are found in some papers by the author, see for example:http://vixra.org/author/mueiz_gafer_kamaleldeen 

10 Reasons why we should stop assuming the dependence of the global geometry of the universe on its average density

also in YouTube  https://www.youtube.com/watch?v=y5P9Hhn_i8c

Using Newtonian Approximation of Einstein’s Field Equation to Determine the Cosmological Constant

Introduction

Newton’s law of gravity was replaced by General Relativity for both theoretical and observational reasons and in spite of being essentially different from General Relativity, Newton’s law of gravity can be recovered from the field equation of General Relativity as an approximation in special cases, this fact played an important role during the construction of the field equation. What I am going to put in front of the reader is another major role for this fact.

Analysis

According to the standard modern cosmology, there are three possibilities of the shape or the large-scale geometry of our universe but whatever is the real shape of the universe, the field equation of General Relativity should be applicable and correct in all these possibility.

Let us fist pay our attention to the fact that every case in Newton’s Gravity can be proved with General Relativity.

Now when we consider the possibly of spherical shape of the universe in which the matter is uniformly and homogenously distributed throughout the space as appears in the large scale (our argument will not be affected even if this is not a precise description of our universe because General Relativity and Newtonian Approximation is applicable and correct also in the ideal case)   we will find according to Newton’s law of gravity that the value of the gravitational field in any region of such a space must be zero because of the total symmetry and similarity of all directions ( there is no preferable direction for a gravitational force that could act on an object) so we arrive at the important fact that in this case the gravitational field is zero regardless of the value of the density of matter in the universe . But can this be explained by General Relativity? In General Relativity, as we are always told, the existence of matter must be associated with the geometrical deformation of space that causes gravitational effects? Is it a contradiction? 

This conflict between general theory of  relativity and its Newtonian Approximation cannot be resolved except if we abandon the unnecessary and unjustified assumption that the large scale geometry of the universe should depend on the average density simply by assuming that this average density of the universe is a part of the cosmological constant and therefore the large-scale geometry of the universe well not be affected by this density while the local application of the field equation ,in which deviations from uniformity come to light, remains with its well-known successful results .

More arguments to support this inevitable idea and more details and results including the resolution of the Cosmological Constant Problem  is found in other papers by the author such as:

- The Independence of the Global Geometry of the Universe from Its Average Density

- The Detestability of the Zero-Point Energy in General Relativity and Quantum Mechanics

- A logical Analysis of the Cosmological Constant Problem and Its Solution

- Another Cosmological Constant to Solve Major Problems of Cosmology

- The Resolution of the Flatness Problem without Inflation

- A Comparison between the Standard Cosmological Model and a Proposed Model with Radial Time and Spherical Space

- Adaptable Cosmological Constant to Solve Major Problems of Modern Cosmology

The Cosmological Constant Problem

https://www.youtube.com/watch?v=n0WxG-A9OHw

A Logical Analysis of the Cosmological Constant Problem and Its Solution

What I am about to put before the reader is an exploration of the logical relations between the propositions which constitute the structure of modern cosmology within which the Cosmological Constant Problem is created in order to avoid the risks inherent in unjustified confidence in absolute validity of some of these relations in hope that the problem will be solved within our well-defined existing concepts by refutation of such false relations instead of invoking metaphysical notions such as dark energy, multi-verse … etc.
The Analysis
Let us first make a list of the basic propositions and statements of modern cosmology associated with the cosmological constant problem:
⦁ Einstein’s Field Equation is correct.
⦁ Zero-Point Energy exists and is very large.
⦁ The global curvature of space is very small.
⦁ The observational data about cosmological red-shifts are correct.
⦁ The accelerating expansion of the universe is true.
⦁ The global geometry of the universe depends on its average density.
Then we turn to the claimed logical relations between these statements:
⦁ The 1st statement implies the 6th statement.
⦁ The 4th statement implies the 5th statement.
⦁ The 6th statement implies that the 2nd statement contradicts the 3rd.
Now let us examine the validity of these relations carefully:
Starting from the first relation the author argues that it is incorrect, a fact which has often been overlooked, because the dependence of the global geometry of the universe in the average density which is thought to be a necessary result of the field equation can be taken away by assuming that the cosmological constant (or part of it) is the average density of the universe.
 In this case any homogeneous distribution of matter and energy throughout the universe cannot affect the geometry of the universe because the contribution of this distribution on the stress-energy tensor in one side of the equation is canceled out by its contribution on the cosmological constant in the other side of the equation and thus the field equation is not affected by such a distribution regardless of its density.
Beside its simplicity and ability to cut the cosmological constant problem at its roots there is nothing to be lost by adopting this assumption because the successful local applications of the field equation will not be affected.
Now let us turn to the second relation , perhaps another advantage is gained if we managed to get rid of more restrictions caused by this relation if proved to be false. The important question to be answered is whether or not the accelerating expanding of the universe is the unique explanation of the observational data of the cosmological red-shifts. The accelerating expanding is a very undesirable idea because it is responsible for the excluding of the best of all cosmological models in terms of physical simplicity and mathematical beauty which is the cosmological model of spherical 3-space and radial time.   
Fortunately and interestingly the second relation is false and the cosmological red-shift observational data attributed to acceleration of the expansion can be explained easily as a result of geodesic path of light associated with the shape of space-time in the cosmological model with radial time and spherical space.
(More arguments which support this resolution and more details are found in other papers by the author.)