**This article is not finished, though it’s in good enough shape to get my ideas across. There may be inaccuracies, misspellings, or grammar errors. There is also much more work to do. This site is a living document. I am an amateur. This is basically just my opinion. I am only hoping to inspire others more capable than myself. The above graph demonstrates the difficulties in testing this sort of thing, so be critical but be fair.
The proposed cosmological model can be summarized with the following postulates.
- All objects are embedded within larger-scale objects.
- Mass density is inversely proportional to any given radius.
- Gravitational binding occurs across all scales.
- H0 is gravitational redshift.
Introduction
At its core, cosmology seeks to accurately describe the distribution of matter and energy in the universe. Typically, this is accomplished by satisfying the Einstein Field Equations, which in turn describe the shape of the overall geometry of the universe.
For nearly a full century now cosmologists have used the FLRW metric, which explores an expanding, homogeneous, and isotropic universe. It’s a simple and intuitive approach that explains the Hubble redshift. The FLRW metric has three possible options – expansion, collapse, or in-between. Expansion is not an inherent feature or direct prediction of general relativity. The expanding solution comes specifically from the assumptions and boundary conditions chosen in the formulation of the FLRW metric
However, in spite of what cosmologists claim, our Universe is not homogeneous. The Universe is not even nearly homogenous. Mass density varies across all scales. Stars, galaxies, clusters of galaxies, super clusters, filaments, veins, walls, and the observable universe all have a different mass density ρ.
The hierarchical structure of the cosmos follows an inverse relationship
ρ(r) = ρ0 (R/r). All objects are embedded within other objects. The most significant difference with this distribution is that it predicts a gravitational redshift that is directly proportional to the distance of light travel (more on this in a moment).
If the Hubble redshift is reinterpreted as gravitational redshift, an entirely new paradigm of cosmology opens up. Many of the mysteries and inconsistencies within ΛCDM become irrelevant.
In addition, this approach is somewhat different than the FLRW in that it is a multi-scale model, meaning the boundary and density are intertwined. The parameters are different for each observation but share a common parameter, gravitational acceleration g ≈ 3.49×10-10 m/s2.
Historical Context
The modern interpretation of the Schwarzschild metric focuses on the fact that it predicts black holes and even horizons. But Schwarzschild’s original 1916 solution does not mention either of the two. Instead, the paper lays out the geometrical structure of spacetime surrounding a non-rotating, spherically symmetric mass.
Schwarzschild’s original 1916 solution was groundbreaking, but Einstein and others in the physics community met his solution with skepticism and confusion, failing to grasp its full implications. The Schwarzschild metric eventually played a crucial role in establishing the fundamental principles of general relativity and advancing the scientific understanding of gravitational physics during that era., but it has never been considered as a cosmological model. Cosmologists have always asserted the universe is homogeneous and isotropic.
On the other hand, the FLRW metric, independently derived by Alexander Friedmann, Georges Lemaître, Howard P. Robertson, and Arthur Geoffrey Walker during the 1920s and 1930s, is a solution to Einstein’s field equations that describes a homogeneous and isotropic expanding (or contracting) universe. It has been the cornerstone of modern cosmology because of its adherence to the cosmological principle, which posits that the universe is homogeneous and isotropic on large scales.
Despite its success, the Lambda-CDM model, under the umbrella of the FLRW metric, still grapples with inherent limitations and unsatisfactory explanations, relying on unobservable constructs such as dark matter, dark energy, and inflation.
Theoretical Framework
“The misconception which has haunted philosophic literature throughout the centuries is the notion of ‘independent existence. ‘ There is no such mode of existence; every entity is to be understood in terms of the way it is interwoven with the rest of the universe.”
Alfred N. Whitehead
The quote above highlights the complications involved in creating an accurate cosmological model. The average galaxy is several orders of magnitude more dense than the observable universe. However, galaxies are typically part of a galaxy cluster, and galaxy clusters reside within superclusters, which reside within larger-scaled structures. All objects are embedded within other objects.
The Schwarzschild metric was chosen to represent this framework because it best represents what one is able to predict when making a cosmological observations. The observable universe is literally a non-rotating, spherically symmetric, massive object. The interior solution was chosen to represent the perspective of being inside the universe.
Gravitational boundaries are fundamentally arbitrary. There is no clear boundary between one star and another. Atoms blend together to form stars. Trillions of stars all blend together to form galaxies, it all blends together to create the universe.
In the context of general relativity, the concept of a gravitational “boundary” refers to something like an event horizon. On the other hand, the surfaces of stars, planets, and galaxies are determined by a complex interplay of various forces (gravitational, electromagnetic, nuclear, etc.), matter distributions, and conditions. These “boundaries” are more fluid and are not defined in the same fundamental way as something like an event horizon.
Mathematical Formulation
The standard Schwarzschild metric is used for the exterior vacuum solution surrounding a spherical mass M:
To derive an interior solution, we specify a variable mass density profile:
Integrating this density yields the enclosed mass within radius r, where M = M(R) is the total mass.
The interior metric is taken as:
The line element provided is in the “conformal time” form of the Schwarzschild metric. It is often used to describe the interior of a spherically symmetric, non-rotating massive object y.
Here f(r) represents the gravitational potential, related to the enclosed mass via:
Smoothly matching f(r) to the Schwarzschild potential at the boundary r = R provides continuity between the interior and exterior regions (observable radius for cosmology). Varying the density profile ρ(r) allows modeling different spherically symmetric mass distributions within this consistent metric framework.
Specifying a radial density profile ρ(r) allows flexibility in modeling the object’s interior composition. The interior metric form with f(r) perturbing the spatial part is justified based on typical forms of GR solutions. Relating f(r) to the enclosed mass M(r) through the -2GM(r)/rc2 potential aligns with gravitational time dilation effects described by GR.
For starlight, this framework predicts a gravitational redshift that is directly proportional to the distance of light travel, r, assuming the observer resides within a region of the universe with no local gravitational effects. For cosmological distances, light must transverse many different scales. The journey begins at the surface of star. Here, the density of mass is the highest the light will ever encounter. As the light transverses outward from the source, it encompasses a larger and larger volume of space. The density of this volume decreases until it reaches the minimum density on the scale of the observable radius, while the gravitational potential increases.
Working backwards from the Hubble parameter, now being interpreted as gravitational redshift, we obtain the following the density profile for the observable universe (4163 Mpc), when H0≈72,000 m/s/Mpc, f(r) ≈ 1.00049193. The observable radius occurs around 1.285e+26 meters with the mass density 9.743e-27 kg/m3or that scale.
This distribution profile reflects the hierarchical structure of our universe. On smaller scales, we find denser structures such as stars and galaxies. As we move to larger scales, the average density decreases, with galaxy clusters, superclusters, and large-scale structures like cosmic filaments and walls having lower average densities, and the observable radius possessing the lowest density.
Dark Energy
Over the past couple of decades, observations of distant Type 1a supernovae have been found to be fainter than expected. “Dark Energy” was coined as the source of the accelerating expansion, but the phenomenon is really just an adjustment to the cosmological constant. Even still the underlying physics behind expansion and acceleration remain a mystery.
In the context of the SIC model, gravitational effects serve as the dominant mechanism for the cosmological redshift, as well as the light curve of distant objects. The difficulty lies in determining an observational difference between the models. The next few graphics demonstrate the relationship between redshift and distance within SIC (interpreted as gravitational redshift) as well as Lambda CDM with and without DE.
The difference between the three models is subtle until the z≈2 region, but the difference would be very difficult to detect from real world observations from any distance less than z≈4. It might be possible to decipher the data if determinations of the luminosity distance function could be trusted from sources beyond 4000 Mpc, or using very large data sets.
Dark Matter
Dark matter is a proposed type of matter used to explain the unexpected rotation curves of galaxies, which can’t be accounted for by just the visible matter. It supposedly doesn’t emit, absorb, or reflect any electromagnetic radiation, making it detectable only through its gravitational effects.
Cosmological models have very little to do with the dynamics of galaxies. However, this model does provide an approximation for the mass and density of objects based on their radius, which could provide insights into the missing mass anomaly.
When the parameters from above are applied to rotational velocity, this model overpredicts the velocity by a factor of ~2 for most galaxies.
The above equation approximates the orbital velocity of a galaxy using the mass enclosed within radius r and the uniform gravitational acceleration that manifests as a result of the inverse mass density relationship in our model.
A comprehensive review of galaxy rotational curves in the context of SIC should be conducted. There may also be additional clues in the formulations of MOND and TeVeS. More work needs to be done in this area, but this is still a dramatic improvement over the FLRW metric, which says nothing about galaxies. the SIC model seems to suggest that there is too much mass in galaxies.
Regardless, this section will serve as a foundation to build upon in the future. Check back in the future for updates.
Gravitational Lensing and Angular-Diameter Distance
Angular-diameter distance (ADDR) was initially formulated to link an object’s physical size to its observed angular size and distance from the observer. The modern interpretation is deeply rooted in the FLRW metric, which provides a framework to calculate ADDR based on the cosmological parameters such as matter density, dark energy, and the curvature of the Universe.
Over the years, various methods have been developed to accurately measure ADDR. These range from using “standard candles” like Type Ia Supernovae, to “standard rulers” like the Baryon Acoustic Oscillations (BAO) scale, to more intricate techniques like gravitational lensing.
This article by F. Hadrović and J. Binney provided an interesting approach to resolving ADDR with gravitational lensing. Their main arguments can be summarized as the following:
- The usual relationship between angular-diameter distance and redshift can be derived using gravitational lensing, incorporating the effects of an inhomogeneous universe.
- Attempts to determine the deceleration parameter q0 from angular-diameter distances would systematically underestimate q0 due to inhomogeneities.
- They provide a statistical model to describe the density field along the telescope beam, assuming a log-normal distribution.
The intent of the article was to develop a more rigorous method for obtaining distance relationships. However, the approach could have fairly dramatic implications within the SIC model.
To incorporate the effects of gravitational lensing into the SIC model, we first need to modify the distance-redshift relationship and, subsequently, the luminosity distance.
In a standard cosmological context, the luminosity distance dL is related to the redshift z by:
Include a correction term ΔdL due to gravitational lensing.
Therefore, the modified luminosity distance becomes:
In this equation, H′(z) is the SIC model’s equivalent of the Hubble parameter, and f(Δρ,μ) needs to be determined either empirically or theoretically to account for the effects.
To formulate f(Δρ,μ) using the following assumptions:
- The correction term f is linearly dependent on Δρ and μ. This is a simplification but provides a starting point.
Here, a and b are constants to be determined, either empirically or theoretically.
- Δρ is a measure of density inhomogeneities along the line of sight, represented as ρ(r) within SIC.
- μ is the gravitational lensing magnification.
To proceed, we need to choose some form for Δρ. A simple approach would be to let Δρ=ρ(r)−ρ0, where ρ0 is the average density and ρ(r) is the density at a particular radial distance r.
The modified luminosity distance would then be:
Cosmic Microwave Background Radiation
Perhaps the most discussed feature of the Schwarzschild solutions is that they predict the the potential existence of black hole singularities. The famous equation to find the Schwarzschild radius for mass M is as follows:
Typically Rs is much smaller than r. For example, a black hole with the same amount of mass as the sun would have a radius around 3 kilometers.
Below the Rs is plotted to show Schwarzschild radius as a function of the mass (green). The yellow line represent the actual distribution of mass throughout most of the universe. The two lines intersect at the observable radius.
Starlight originating from the beyond the observable radius would be redshifted to extremes. However, some non-zero amount of energy would still be able to enter, offering us potential avenues to take for predicting the temperature of the CMBR.
The Unruh Effect is a phenomenon that was first described as a prediction of Quantum Field Theory by Stephen Fulling, Paul Davies, and W.G. Unruh in the 1970’s. In short, the Unruh Effect states that observers in accelerated frames of reference will see a form of radiation, where as observers in inertial frames will not. Hawking radiation comes as a consequence of the equivalence principle and the Unruh Effect.
All objects in the universe reside within non-inertial frames of reference. The entire concept of an inertial frame of reference is part of Newtonian mechanics and not applicable within relativistic physics. The equivalence principle states that gravitational fields are indistinguishable from accelerated frames of reference. The strength of the field is strongest in the space surrounding the object, which reduces with distance and approaches zero at infinity.
For most observations, the frame of reference is not strong enough to produce meaningful change due to the Unruh Effect.
Discussion
We have demonstrated a unique multi-scale model of the universe. The chosen conditions and parameters are consistent with observation and general relativity. The resulting density profile can be directly measured from the Hubble shift.
Summary
Potential cons/concerns:
- The inverse radial density profile may not be physically justified choice for scales <1Mpc.
- The continuity matching conditions need to be verified through the full tensor math.
- Edge cases like ultra-compact objects and objects with high neutron star densities should produce asymmetries.
This framework also needs special consideration within the context of other cosmological phenomenon. There is no clear way of distinguishing between gravitational redshift and redshift caused by the proposed expansion of the universe. If this density profile is accurate, some or all of the cosmological redshift could be interpreted as gravitational redshift. Such a dramatic change in direction would have many implications.
This framework could also be related to MOND. The dimensionless unit expressed above is effectively a minimum gravitational acceleration on the same order as those predicted by MOND. This subject falls outside of the scope of this paper but this is an area that deserves more inquiry.
In addition, if the Universe is not expanding, there should be some explanation for the Cosmic Microwave Background Radiation. While this subject also falls outside of scope, back of the envelope calculations suggest that Harking radiation/the Unruh effect could be the source of the CMB (when using this framework). The Schwarzschild radius of an object is proportional to the mass of that object. When the amount of mass an object possesses happens to be on the scale of the observable universe, the Schwarzschild radius surpasses the actual radius. In short, this metric implies the existence of a different type of black hole (possibly a white hole); one that encompasses the entire universe. [5]
Conclusion
No conclusion should be drawn from this preliminary discussion. However, the presented framework and density profile should be considered by capable parties. The result appears to be an elegant way of representing cosmology within the confounds of general relativity, the cosmological principle, and observational data. Developing a new cosmological model is a monumental task but it has to start somewhere.
References
1.Massimi, M. (2018). Three problems about multi-scale modelling in cosmology. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 64, 26-38. https://doi.org/10.1016/j.shpsb.2018.04.002
2.Tifft, W. G., & Cocke, W. J. (1984). Global redshift quantization. Astrophysical Journal, Part 1, 287, 492-502. https://ui.adsabs.harvard.edu/abs/1984ApJ…287..492T/abstract
3.Willick, J. A., Courteau, S., Faber, S. M., Burstein, D., Dekel, A., & Strauss, M. A. (1997). Homogeneous Velocity-Distance Data for Peculiar Velocity Analysis. III. The Mark III Catalog of Galaxy Peculiar Velocities. The Astrophysical Journal Supplement Series, 109(2). https://doi.org/10.1086/312983
4.Allen, S. W., Evrard, A. E., & Mantz, A. B. (2011). Cosmological Parameters from Observations of Galaxy Clusters. Annual Review of Astronomy and Astrophysics, 49, 409-447. https://doi.org/10.1146/annurev-astro-081710-102514
5.Retter, A., & Heller, S. (2012). The revival of white holes as Small Bangs. New Astronomy, 17(2), 73-75. https://doi.org/10.1016/j.newast.2011.07.003