Dark Energy demystified. The preliminary mathematical equations on the two forces driving the expansion and contraction of the universe.
Max Gaofei Yan, Alexander Zhiqiu Yan
Abstract
To date, no theory can completely explain the observations of the universe expanding at an accelerated rate. Dark Energy currently represents the theoretical repulsive force that counteracts gravity leading to this expansion, but no evidence of this energy exists. The initial premise of this study is that the Universe has an expanding volume consisting of only photons outside the baryonic regions. Building on this argument, the paper will explain how the Ideal Gas Law can be utilized to forecast the Universe’s expansion by applying the pressure force exerted by the galaxies within a hypothetical spherical universal container. Derivations produce an equation of pressure force 
. The paper will also apply new gravity equations based on the understanding of gravity exerted by mass and EM Waves. Derivations produce an equation of total gravity force 
which shows that gravitational forces on the matter within the universe will decline faster than predicted in mainstream gravity theories. Equating the pressure and gravity forces provides a relationship for gravity and pressure at the edge of the Universe with mass: 
with r being the radius of the universe with mass, R being the radius of the universe which EM waves have reached, and T being the measure of kinetic energy in the universe. This paper demonstrates how the Universe can expand or contract, and at what rate, based on the variables in these two forces’ equations. This paper concludes that the faster decline in gravity force with a steady pressure force can describe the phenomena that Dark Energy tries to explain. These conclusions match observations of the Universe’s past and current expansion trends. With this equation, we can also predict how the Universe will evolve in the long-term future.
Table of content
Abstract
Section 1
1.1 A current cosmology problem
1.2 Our approach to this problem
1.3 Two basic assumptions of our Universe
The edge of the Universe with Mass and beyond
Section 3
Pressure force at the edge of the Universe with Mass
Section 4
4.1 The gravity force of the Universe
4.2 Comparison gravity force on the Universe between today’s mainstream and new gravity theory.
Section 5
5.1 Equation for gravity and pressure equilibrium at the edge of the Universe with Mass
5.2 How the Universe expands or contracts based on pressure and gravity
5.3 Recap of how the universe evolved in the past
5.4 What Dark Energy really is
5.5 Future of the universe: Big Freeze, Big Rip, Big Crunch or Big Bounce
Section 6
Further research
Conclusion
Section 1
1.1 A current cosmology problem
Recent observations and measurements have determined the size of the Universe relative to its age [1]. This has produced a curve for the expansion of the Universe over time shown in Figure 1. The Universe expanded rapidly following the so-called Big Bang. Following this rapid expansion came a period of deceleration, but recent observations indicate that the Universe is now undergoing accelerated expansion this period of stagnation. This acceleration has perplexed the scientific community as well-established theories predicted the Universe expanding at a constant rate or even contracting.
Figure 1 – a graph showing the cosmic distance change as a function of time [2]
Recent studies and theories on the accelerating expansion of the universe have attributed this acceleration to Dark Energy. This theory has generated questions, many of which stem from the inconclusive evidence that such energy exists and the inability to explain how this energy led to the varying expansion rates of the Universe. In current theories, Dark Energy represents any unknown “repulsive” or “anti-gravity” force or source of energy, and the term is unfortunately used as a catch-all for any force or energy source the scientific community does not understand. As of now, no research has been able to directly detect Dark Energy [3].
1.2 Our approach to this problem
This paper will outline a theory that avoids the problems Dark Energy poses when explaining the accelerating expansion of the Universe, and it will also provide cosmologists a new path to analyzing the expansion, and possibly contraction, rate of the Universe.
The basis of this paper’s approach acknowledges the existence of a photon-filled space outside the Universe with Mass. By utilizing the principles of the Ideal Gas Law, this paper can determine a pressure force which expels mass at the boundary of the Universe with Mass into the photon-filled space. As explained in the following sections, this pressure force from the Ideal Gas Law can help model the expansion of the Universe. In addition, a new field theory of gravity can demonstrate how gravity on matter in the Universe can weaken much faster than previously thought. Equations of the opposing pressure and gravitational forces will be derived in preliminary math equations, compared, and provide a conclusion to how the Universe expands or contracts.
1.3 Two basic assumptions of our universe
Before deriving the equations of the two forces, the paper makes the following two assumptions:
- The Universe expands in all directions at the same speed so that the Universe maintains a spherical shape with the mass evenly distributed in cosmological large scale.
- In cosmological large scale, each galaxy randomly moves to all possible directions like gas molecules in a volume.
Important notes:
- Presently, the Universe is perceived to possess a spherical volume, extending in all directions, and its expansion is observed as an expansion of the space itself [4].
- Unless it is specifically stated, the distance measure in this paper is proper distance.
- The radius of the Universe with Mass is unknown and represented with r. The observed universe is believed to be a mere fraction of the actual Universe with Mass [5].
The intricacies of the cosmos continue to be a subject of ongoing debate among researchers. This paper seeks to employ a simplified model of the universe to arrive at preliminary mathematical equations that depict the underlying mechanism behind its expansion or contraction. Further data, as outlined in Section 6, acquired through continued research may provide a more exact and refined mathematical correlation of this mechanism.
Section 2
The edge of the Universe with Mass and beyond
This paper defines the edge of the Universe as the outer surface or skin of the spherical Universe with Mass. Within the edge exists all the matter with mass in the Universe. The edge defines the boundary of the Universe with Mass. Current mainstream discussions about the expansion or contraction of the Universe only consider the expansion of this edge. This paper will also refer to this edge when discussing the Universe’s expansion or contraction.
Since EM waves/photons emitted within the Universe with Mass travel at light speed and the Universe with Mass’s edge expands at a slower speed, the space beyond this edge is filled solely with photons. This creates a space outside the Universe with Mass containing EM waves/photons that constantly expands outward from the Universe with Mass at the speed of light.
This new region of space has not been considered by those researching the expansion of the Universe. This space beyond the edge of the Universe with Mass, and together with the Universe with mass, represents the whole Universe. This means that the actual space of the Universe is much larger than that usually discussed.
In Figure 2, the blue circle represents the expanding universe containing matter with mass. The space between the blue circle and the black ring represents the space where EM waves/photons have traveled. This shows how the actual Universe is larger than the Universe with Mass. This idea provides the foundation for the use of a pressure force derived from the Ideal Gas Law in the following section.
Figure 2 – 2D Diagram showing the Universe with mass and the space beyond consisting of EM waves/photons [6]
Pressure force at the edge of the Universe with Mass
This theory uses the Ideal Gas Law where the Universe is a container of galaxies acting as air molecules and where a pressure force replaces Dark Energy as the driver for the Universe’s increasing expansion rate.
Most research today does not assume that pressure acts on the cosmological scale comparable to the Ideal Gas Law. Two main reasons lead to this.
First, all measurable points inside the Universe with mass have the same pressure in all directions, therefore canceling out the pressure force.
Secondly, and most importantly, there persists the assumption that no space lies beyond the Universe with Mass, and hence, there would be no edge as defined in this paper and nothing for the pressure to push outward towards.
These explanations only hold when analyzing pressure within a Universe with Mass without a space beyond its edge. As explained in Section 2, these conditions do not hold following this paper’s assumptions, and at this edge, the pressure could have a surface to push on assuming:
1. There does exist a space beyond the Universe with mass, and
2. Phenomenon on either side of the edge do not cancel out the pressure.
The regions of space with mass and that filled with only EM waves/photons create two different regions of pressure. The Universe with Mass has a pressure created by galaxies that move randomly in all directions like gas molecules. The space beyond the edge does not have matter with mass to create pressure as it only has EM waves/photons moving away from the Universe with Mass. This creates a pressure gradient at the edge of the Universe with Mass that results in an outward-pushing force. Figure 3 depicts how the galaxies move randomly within the Universe with Mass and how the EM waves/photons move away from the Universe with Mass.
Figure 3 – 2D Diagram showing the direction of movement of galaxies within the Universe with mass and photons in the space beyond the edge
Universe with mass having an edge and space filled by EM waves/photons beyond this edge creates a pressure force that has an outward direction into the space created from the EM waves/photons. Recognizing the existence of a region beyond the Universe with Mass invalidates the two primary reasons why the Ideal Gas Law cannot be applied at the edge of the Universe with Mass. This means the Ideal Gas Law can model the pressure force along the edge of the Universe with Mass, which solves part of the Dark Energy problem.
Since this new theory relies on the Ideal Gas Law, this means it must follow the assumptions of the Ideal Gas Law:
- Collisions between air molecules are elastic, and their motion is frictionless.
- The total volume of the air molecules is at least a magnitude smaller than the volume it occupies.
- There are no intermolecular forces acting between molecules and their surroundings.
- Air molecules are in constant motion and distances between them are significantly larger than the molecules themselves.
The characteristics of galaxies in the Universe mostly meet these four assumptions. The Universe is magnitudes larger than the galaxies that occupy it, and galaxies are in constant motion while generally remaining extremely distant from each other. While collisions between galaxies are not elastic and galaxies do have a gravitational force acting on each other, the same can be said of real gases that still follow the Ideal Gas Law’s trends. This allows the Ideal Gas Law in Equation 1 to predict the trend of the pressure force that drives the Universe’s expansion in this new theory.
(Equation 1)
Equation 1 represents the Ideal Gas Law with P being pressure, V being volume, n being moles of ideal gas, R being the gas constant, and T being the absolute temperature.
To apply the Ideal Gas Law in describing the pressure force on the Universe with Mass’s edge, a few of the variables will need redefining and further research. n may represent the moles of galaxies. R for the Universe is unknown and will be designated as Ru. Since the Ideal Gas Law uses temperature as an indicator of molecular kinetic energy, the absolute temperature for the Universe may not be applicable as it does not indicate kinetic energy of the galaxies.
The variables from the Ideal Gas Law may not directly apply to galaxies and phenomena on a cosmologically large scale; however, this preliminary paper aims to describe the mechanism for the Universe’s expansion or contraction. The pressure determined by the Ideal Gas Law affects the Universe’s expansion and contraction, but the details on how the Ideal Gas Law’s variables should be redefined requires further discussion and research.
Pressure acting on the Universe with Mass’s edge can be defined as a force over an area. The pressure acting on the edge of the Universe with radius r is the pressure force acting on the inner surface of the spherical Universe. This is shown in Equation 2.
(Equation 2)
The volume of the spherical Universe is shown in Equation 3.
(Equation 3)
Substituting Equations 2 and 3 into the Real Gas Law Equation and solving for pressure force FP provides Equation 4. This is the pressure force at the edge of the Universe with Mass, and it helps to drive the Universe’s expansion.
(Equation 4)
Section 4
4.1 The gravity force of the universe
While pressure acting on the edge of the Universe allows the Universe to expand, gravity causes the Universe to contract.
According to a new field theory, gravity exists between two EM waves and between EM waves and objects with mass [7].
The equations for gravity between photons and between photons and matter with mass are shown in Equations 5 and 6 respectively.
(Equation 5)
(Equation 6)
Where:
F – gravity force between two photons or photon and an object with mass
Y1 = 3.61 × 10−111𝑚3𝑘g
Y2 = 4.91 x 10-61 m3 s-1
f – photon frequency
m – mass of object
r – distance between two photons or object with mass and photon
If EM wave/photons create a gravitational force, then this would affect the expansion of the Universe. Photons within and exiting the Universe with Mass would produce a force acting on the edge of the Universe with Mass.
To understand the gravity force acting on the edge of the Universe with Mass, Figure 4 will be used to denote the radii used in the Equations 7 through 10.
Figure 4 – 2D Diagram showing radius of the Universe with mass as r and the entire Universe as R
Gravity force from mass:
Following the assumption of a spherical Universe, Equation 5 shows the force of gravity at the edge of the Universe with Mass due to mass located within the Universe. G is the gravitational constant. m is the mass of the galaxy at the edge of the Universe with Mass, and M is the mass of the whole Universe. r is the radius of the Universe with Mass.
(Equation 7)
Gravity force from EM waves:
With the understanding of the new theory on gravity mentioned above, the equation for gravity force between photon and matter with mass is:
(Equation 6)
Where:
F – gravity force between photon and an object with mass
Y2 = 4.91 x 10-61 m3 s-1
f – photon frequency
m – mass of object
r – distance between object with mass and photon
As discussed above, the Universe with Mass is located at the center of the entire Universe and has a galaxy with mass m at r distance from its center. The total number of photons is 
, with k being the number of photons and f being the frequency of each photon. Equation 6 shows the gravity force from photons/EM waves on this galaxy at the edge of the Universe with Mass:
(Equation 8)
Since R is the region of photons, that means R is increasing at the speed of light.
This paper outlines the mechanisms behind the two forces responsible for the expansion or contraction of the Universe. In Equation 6, the density of the photons in the entire Universe is simplified as being uniform. The preliminary mathematical equations could be adjusted to align with data on the propagation of EM waves in the Universe while retaining the same mathematical relationships describing this mechanism.
Total gravitational force:
Summing these gravitational forces provides the total gravitational force at the edge of the Universe with Mass. Equation 7 shows this sum.
This summation can be simplified by grouping constants into single variables. While the number of photons does vary with time, taking these values at a single point in time makes this value constant. We have A = 
, B = 
This allows for the force of gravity at the edge of the Universe to be more easily calculated at a point in time. Equation 10 shows the mass, photon, and gravitational constants grouped into single variables.
(Equation 10)
4.2 Comparison gravity force on the universe between today’s mainstream and new gravity theory.
Current theories on the Universe’s accelerating expansion only use the conventional force of gravity of matter with mass, which is shown in Equation 7.
(Equation 7)
Current observations indicate that dark matter generates approximately five times the gravitational force in the Universe over regular matter [8]. With this information and an understanding of the conventional gravitational force, the total gravitational force can be calculated as follows:
(Equation 7-1)
As previously discussed, the total gravitational force, considering the presence of gravitational force from EM waves, can be calculated as follows:
(Equation 9)
The comparison of the total gravitational force in Equation 7-1 and Equation 9 is as follows:
In Equation 7-1, if the value of r remains constant, the force of gravity, Fg, will not vary.
If r remains constant and R increases at the speed of light in Equation 9, the force of gravity, Fg, will decrease.
If r does not increase at the speed of light, comparing 
Equation 7-1 with 
in Equation 9 shows that the inclusion of EM waves in the force of gravity at the edge of the Universe with mass would lead to a faster decline in gravitational force as the Universe expands.
By only using this component of gravity at the edge of the Universe with Mass, the expected force of gravity does not decline as quickly as it would when assuming EM waves also produce a gravitational force. This faster decline in gravitational force as EM waves propagate away over time, increasing R at the speed of light, provides one aspect of why the Universe with Mass experiences accelerating expansion.
Section 5
5.1 Equation for gravity and pressure equilibrium at the edge of the Universe with mass
At the edge of the universe with mass, a galaxy will be affected by two forces: the pressure which pushes it outward and the gravity which pulls it inward. When these two forces are equal, we have Fg = Fp.
Placing the gravitational and pressure forces in an equation allows for further simplification and easier analysis of how the different variables affect the force acting on the edge of the Universe with Mass. Equation 11 sets the gravitational force in Equation 10 equal to the pressure force in Equation 4 and further simplifies the two expressions. The left-hand side of the equation represents Fg with two variables r and R while the right-hand side represents Fp with one variable, T:
= 
(Equation 11)
(Equation 11-1)
If we group constant 3n
(Equation 12)
In Equation 12, A 
, B 
and C (as 3n
and the right side represents outward pressure 
Whichever side is greater dictates a positive or negative acceleration, and the difference in magnitude dictates the magnitude of acceleration.
5.2 How the universe expands or contracts based on pressure and gravity
Comparing the gravity and pressure components from Equation 12 can explain the accelerating, decelerating, and stagnant expansion of the Universe shown in Figure 1. These different scenarios are shown in Equation 13, and Inequalities 1 and 2.
(Equation 13)
(Inequality 1)
(Inequality 2)
Equation 13 shows Fgequal to Fp. This means that galaxies at the edge of the Universe are in equilibrium, and that the edge of the Universe with mass experiences an equal outward push and inward pull. In this scenario, the Universe with mass does not expand or contract with any acceleration. This represents the period of stagnation in Figure 1.
Inequality 1 shows Fgless than Fp. In this scenario, the galaxies at the edge of the Universe exert an overall outward force, expanding the Universe at an accelerated rate. This is what cosmologists are currently observing.
Inequality 2 shows Fggreater than Fp. In this scenario, the galaxies at the edge of the Universe exert an overall inward force, contracting the Universe at an accelerated rate. This represents the period of decelerating expansion in Figure 1.
5.3 Recap of how the universe evolved in the past
Figure 5 – Graph showing change in radius of the universe over time [9]
Figure 5 is the observed change in radius of the Universe since the theorized Big Bang. From roughly 2 to 9 billion years from the beginning, the size and temperature of the Universe remained relatively steady on the cosmological scale. During this time, the expansion of the universe was decelerating. After roughly 9 billion years, an unknown force started to expand the universe with mass, and the universe has expanded at an accelerated rate ever since.
By applying Equation 12 and the two derived Inequalities, we can explain the curve shown in Figure 5.
After the theorized Big Bang, the Universe experienced a period of extremely high temperatures that led to a high-pressure force and therefore rapid expansion. After a very hot beginning, the universe started to cool down rapidly, which means T declined significantly. The amount of force 
on the left side of Equation 9 would have been greater than the right side 
, hence we have Inequality 2-1:
(Inequality 2-1)
At this period time of the universe’s evolution, the expansion of the universe decelerated since Fg > Fp.
T eventually became relatively stable. R and r kept increasing, with R increasing at the speed of light and much faster than r. The left side of Equation 12 decreased, hence:
(Equation 13)
During this period of the Universe’s evolution, Fg= Fp, the expansion of the universe continued at a steady rate.
While R and r increased and T remained constant steady, the left side of the Equation 12 declined and led to Inequality 1-1:
(Inequality 1-1)
Now Fg < Fp, and the universe started to expand at an accelerated rate, as what we observe today. This increasing acceleration has puzzled physicists and cosmologists; however, with this understanding of the universe and a new field theory, the acceleration can be explained with gravity declining much faster than believed in mainstream gravity theories, as discussed in Section 4.2.
The above provides a logical explanation for the process by which the universe changes its size at various points in time as a result of the balancing of both sides of Equation 12.
5.4 What Dark Energy really is
After decades of observations on the expansion of the universe, there has been no valid explanation on why the gravity force as described by the mainstream gravity theories is not enough to slow the expansion of the universe. It is particularly perplexing that the slowing down of the universe's expansion occurred closer to the so-called Big Bang, and the speedup resumed afterwards. Researchers termed Dark Energy to explain this phenomenon. This Dark Energy is so pervasive now that some theories about 75% of the universe is made of Dark Energy [8, 10]. However, the idea of Dark Energy has raised more questions than it has answered and there is no evidence to back up the theory, other than the observed acceleration of the Universe's expansion.
This paper argues that Dark Energy is a result of the rapid decline in gravitational force (as shown in Equation 9) and a relatively consistent outward pressure force (as depicted in Equation 4), working in conjunction as described in Inequality 1.
5.5 Future of the universe: Big Rip, Big Freeze, Big Crunch or Big Bounce,
(Equation 12)
Equation 12, which involves the variables r, R, and T, not only provides a logical explanation for current observations of the evolution of the Universe, but it can also predict its future state in the long term.
Current mainstream theories predict various outcomes for the universe, including the Big Freeze, Big Rip, Big Crunch, or Big Bounce [11].
The Big Freeze (also known as the Big Chill) is a hypothetical scenario in which the ongoing expansion of the universe results in it approaching absolute zero temperature asymptotically.
The Big Rip is a scenario in which a constant and increasing rate of the Hubble constant ultimately results in the complete disintegration of all material objects in the universe, beginning with galaxies and culminating in a finite amount of time with the breakdown of even the smallest forms, into unbound elementary particles, radiation, and beyond. As the energy density, scale factor, and expansion rate reach infinity, the universe terminates in an effectively singular state. Under the Big Rip scenario, dark energy not only accelerates the expansion of the universe at a consistent rate, but does so in an exponential manner, ultimately causing the fabric of reality to be torn apart.
The Big Crunch is another theoretical scenario for the ultimate fate of the universe, in which the Universe would stop expanding and eventually collapse back into itself, resulting in a reversal of the Big Bang.
The "Big Bounce" is another hypothetical scientific scenario which suggests that the Universe underwent a phase of contraction before the Big Bang. The concept is based on the idea that the universe has no beginning or end, and that it has gone through an infinite cycle of expansion and contraction. According to this theory, the Universe experiences a Big Bounce instead of a Big Bang, and it may eventually experience another Big Bounce in the future.
Figure 6 – Diagram showing the expansion of the Universe over time and its potential future expansion or contraction [12]
At present, the Big Rip is the most widely accepted prediction based on mainstream theories of gravity and dark energy.
Alternatively, Equation 12 may lead to outcomes different from the Big Rip. If the T in the equation approaches zero, even with declining gravity, there remains a non-zero gravitational force, potentially making the left side (Fg) of the equation greater than the right side (Fp). This would cause the universe with mass to contract. The universe with mass eventually contracts to a certain size, triggering a strong chemical or nuclear reaction that causes it to "explode" again. This cycle of contraction and expansion is referred to as the Big Bounce, and Equation 12 can explain its mechanism.
Varying the values of r, R, and T in Equation 12 will result in varied outcomes regarding the expansion or contraction of the universe. Utilizing this equation and its principles, combined with additional data from future observations, we can enhance our prediction of the universe's future.
Section 6
Further research
From the assumptions made in this paper and the preliminary equations used to derive Equation 12, this paper outlines the mechanism of a novel explanation for the observed and predicted expansion of the Universe with Mass.
Given the preliminary nature of this theory, it has several aspects that need further investigation in order to make accurate and precise predictions about the expansion or contraction of the Universe. These include: determining the exact radii of the Universe with Mass r and the entire Universe R; determining if the Universe is spherical beyond our current observational scale; understanding how the mass M of the entire universe changes, if at all; finding a way to equate the absolute temperature T in the Ideal Gas Law to another cosmologically equivalent value; exploring the potential changes in the density of EM waves outside the Universe with mass; and refining the definitions of n and Ru in Equation 4.
Future studies and additional data can help to refine the preliminary equations that dictate the process of expansion and contraction outlined in this paper.
With this understanding of the mechanism and its new equations on how our universe changes its size, we can also enhance the precision of calculations, analysis, and predictions in other areas of cosmology research.
Conclusion
By defining the edge of the Universe with Mass as the point of interest when discussing the expansion of the Universe, implementing a pressure force at the edge of the Universe with Mass, and using a novel system of gravity, this paper outlines a potential mechanism that could explain the observed and predicted expansion and contraction of the Universe. Equation 12, 
shows the effective gravity and pressure that acts on the edge of the Universe with Mass, and by adjusting the three variables, the observed expansion trends of the Universe can be explained. Through more study, these variables can be determined, and this mechanism, given accurate variables, matches the observed and predicted expansion or contraction of the Universe better than current dark energy theories.
References
- NASA Goddard Space Flight Center, "Mystery of the Universe’s Expansion Rate Widens with New Hubble Data," NASA, January 8, 2019. https://www.nasa.gov/feature/goddard/2019/mystery-of-the-universe-s-expansion-rate-widens-with-new-hubble-data
- Grøn Ø. The Discovery of the Expansion of the Universe. Galaxies. 2018; 6(4):132. https://doi.org/10.3390/galaxies6040132
- Sunny Vagnozzi, Luca Visinelli, Philippe Brax, Anne-Christine Davis, and Jeremy Sakstein, “Direct detection of dark energy: The XENON1T excess and future prospects,” Phys. Rev. D 104, September 15, 2021. https://www.cam.ac.uk/research/news/have-we-detected-dark-energy-cambridge-scientists-say-its-a-possibility
- Bennett C. L., et al. "Shape of the Universe," NASA, updated May 6, 2020. https://map.gsfc.nasa.gov/universe/uni_shape.html
- Vardanyan M., Trotta R., and Silk J., “Applications of Bayesian model averaging to the curvature and size of the Universe” arXiv, Feb 28, 2011. https://doi.org/10.1111/j.1745-3933.2011.01040.x
- Yan, Max. Gaofei. "Outline on 'The dark energy is the gravitational force between the photons outside the universe and the universe'," Zenodo, December 4, 2020. https://zenodo.org/record/4306961#.ZB48WPbMK5c
- Yan, Max. Gaofei. "Unified Field Theory for Gravity and Electromagnetism and Its Explanation for Dark Matter Observations," Zenodo, March 27, 2020. https://zenodo.org/record/4057270#.ZB48-fbMK5c
- NASA, “Dark Energy, Dark Matter,” NASA, April 30, 2023. https://science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy
- Brainerd, J. "Q&A: How Accelerating Expansion Affects Red Shift," Sky & Telescope, June 22, 2020. https://sky-lights.org/2020/06/22/qa-how-accelerating-expansion-affects-red-shift/
- Keller, J. "The Expanding Universe: A Receding Mirage," Solar Center, Stanford University, 1997. http://solar-center.stanford.edu/activities/jeff/Solution.html
- Chien, A. (2021, January 20). The beginning to the end of the universe: The Big Crunch vs. the Big Freeze. Astronomy Magazine. https://astronomy.com/news/magazine/2021/01/the-beginning-to-the-end-of-the-universe-the-big-crunch-vs-the-big-freeze
- Siegel, E. (2018, September 29). Ask Ethan: Could the Big Rip lead to another Big Bang? Forbes. https://www.forbes.com/sites/startswithabang/2018/09/29/ask-ethan-could-the-big-rip-lead-to-another-big-bang/?sh=67704e733189