The Elastic Universe Model

Learning about General Relativity and the expansion of the universe inspired me to do this project. Isaac Newton described gravity as an attractive force between masses, but he didn’t understand what caused it. Einstein’s General Relativity suggests that space-time is a fabric that is warped and curved by mass and energy, and this curvature causes objects to move toward one another, which we perceive as gravity. As the universe expands, this fabric stretches over time (Einstein, 2001). However, I wondered what would happen if this fabric resisted being stretched, like a rubber band?

A rubber band is perfectly relaxed when unstressed, but when it is stretched, it develops increasing resistance (Khan Academy, n.d). What would the universe be like if the fabric of space-time behaved in a similar way? In more detail, how would it affect its evolution and expansion?

Several possible endings have been suggested for our universe, including the Big Crunch, the Big Freeze, the Big Rip, and the Vacuum Decay (Mack, 2024). Based on our current assumptions, given that matter and radiation continue to dilute, and dark energy remains a constant, the Big Freeze seems to be the most likely ending (NASA, 2023). How would our predictions change if we included resistance in our assumptions and modified our models accordingly?

This is why I developed the “Elastic Universe Model” to design and test this what-if scenario, make testable predictions based on it, and explore how similar those predictions are to our recent findings.

Design process:

Since this is a theoretical model, it cannot be tested through experiments. The only way to achieve quantitative results in this project is to determine how the increasing stiffness of space-time would affect the universe using math. This includes its evolution over time and its fate.

Defining the restoring term:

Inspired by Hooke’s law (Figure 1), I developed the “space-time’s elastic restoring force” (Figure 2).

Since in Hooke’s law, the constant stiffness of the spring is represented by k, I chose the Latin letter Ƙ to represent the constant stiffness of space-time.

In cosmology, the size of the universe is represented by the scale factor a(t) (Carroll, 1997). In this model, the scale factor plays a role analogous to displacement in Hooke’s Law. Therefore, changes in "a" represent the stretching of space-time.

Modification:

Since I was trying to predict how space-time’s resistance would affect the evolution of the universe, I needed to modify equations that govern the cosmic expansion.

These equations are called the Friedmann equations (Figure 3), which determine the rate of expansion and acceleration of the universe by accounting for factors such as the density of matter and radiation, and most importantly, dark energy (Weinberg, n.d).

To allow the model to account for a resistance to expansion, I placed my restoring term in the second Friedmann equation (Figure 4).

The value of Ƙ and units:

In cosmology, constants are determined using observations- but since this is a thought experiment, Ƙ can not be derived from observations, meaning it needs to be assigned a value.

I assigned 3 different values to Ƙ (Ƙ1, Ƙ2, and Ƙ3) to determine how different values of stiffness influence the rate of acceleration.

To ensure consistency with the acceleration terms in the second Friedmann equation, I assigned Ƙ units of s⁻².

The turning point analysis

The ratio(r):

I set up the ratio r by manipulating the modified Friedmann (Figure 1). This number determines the ratio of the strength of the restoring force (-Ƙa) to dark energy (Λ). Accordingly, the value of r determines the dominating force and the sign of acceleration:

r < 1 → Ƙa < Λ, dark energy dominates, expansion accelerating

r = 1 → Ƙa = Λ, equilibrium, acceleration stops (the “turning point”)

r > 1 → Ƙa > Λ, restoring force dominates, expansion decelerates

Since a, the scale factor, represents the size of the universe in cosmology, using r and the value of Ƙ, we can determine the universe’s size at different force dominance stages.

The pie charts:

Based on my ratio formula (Figure 1), I designed pie charts with different values of r to visually demonstrate the restoring force vs. dark energy domination (Figure 2).

Since I assigned 3 different values to Ƙ, I performed this process three times with fixed r values and percentages to demonstrate how Ƙ influences the turning point.

The graphs:

I used Python with NumPy and Matplotlib libraries to graph the dynamics. My code calculates acceleration (ä) with respect to the scale factor (a) using my modified equation.

Since my variable is Ƙ, I created three different graphs using the three assigned Ƙ values.

Ultimately, all three of my graphs revealed a parabolic relationship with acceleration peaking at different "a" values based on the assigned value of Ƙ (Figure 3).

The conservation of energy

The elastic potential energy term:

By integrating the restoring term from the modified equation, I developed the elastic potential energy term.

The equation:

Dark energy is what currently drives the expansion of the universe (NASA, 2025). According to this equation, the work done by dark energy gets stored as elastic potential energy in the fabric of space-time (Figure 4). Since this energy has a different source and origin than other energy density components, I refer to it as a kind of “exotic" energy.

Big Crunch: the ultimate fate

According to the developed graphs, the value of Ƙ has a great influence on the acceleration turning point:

Ƙ1 = 2 * 10-36 s-2: Acceleration peaks at a=0.83 and changes sign at a=1.67.

Ƙ2 = 2 * 10-37 s-2: Acceleration peaks at a=8.24 and changes sign at a=16.7.

Ƙ3 = 2 * 10-39 s-2: Acceleration peaks at a=828 and changes sign at a=1670.

This shows that all three chosen Ƙ values result in a deceleration in the cosmic expansion. According to the parabolas, before the deceleration begins, the universe goes through a phase of more rapid acceleration, followed by a less rapid acceleration. After the acceleration changes sign, the expansion begins to slow down until it stops.

At this point, since dark energy doesn’t do any more work, the stored elastic potential energy that has been accumulating converts into inward kinetic energy.

As a result, the universe contracts and snaps back into a singularity, similar to the predictions of the Big Crunch theory.

Based on the predictions, the fate of an elastic universe depends heavily on the value of Ƙ. If the Ƙ has an approximately similar value to the term Λ, a turning point in acceleration will occur when the universe is 1.67 times its current size, and will lead to a “Big Crunch” fate. A similar thing would happen if the Ƙ value is only slightly smaller, with the turning point being at a=16.7 instead. If the Ƙ value is significantly smaller, resorting force won't dominate until the universe is 1670 times its current size.

In our Big Bang model, we assume that the universe started with a singularity, which we define as a point where curvature becomes infinite (Einstein, 2001). In my model simulation, however, the universe starts with a relaxed rubber band. Therefore, in an elastic universe, singularities are relaxed regions of space-time. This changes how we define singularities in an elastic universe.

Assuming that the value of Ƙ1, with its strength similar to dark energy, is the most reasonable assigned value, if our own universe did have a restoring force, we would detect slight changes in supernovae and galaxy recession speeds, indicating that the acceleration is slowing. If we could observe supernovae when the universe is about 3 times its current size, we would find them too bright, indicating that expansion is decelerating.

Final note:

Recent DESI results suggest cosmic acceleration may be slowing (DESI collaboration, 2025), which is independently consistent with what the Ƙ1 scenario in my model predicts.

Comparison: Since I made some testable predictions about the cosmic expansion, I am excited to compare my model’s predictions more closely with DESI findings to see if there are more similarities.

Singularity predictions: In an elastic universe, what we define as “absolute singularities” cannot form. I am curious to explore how this would affect our understanding of black holes and the way we define them.

Tensor equation modification: Possibly the most ambitious step will be modifying Einstein’s tensor equations so I can define the restoring force as an original, built-in property of space-time.

I would like to thank:

My parents for their support. This project wouldn't be possible without them.

Mr. Sean Mahedy at Bear Creek Secondary School for introducing me to the concept of Hooke's law and giving me feedback on my work.

Mr. Martin Lancaster at Bear Creek Secondary School for reviewing my work, guidance in developing testable predictions for this model, and offering advice on my physical demonstration.

Those who reviewed my work and provided feedback to improve clarity and presentation.

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