Derivation of parametric equations of the expansion of a

closed universe with torsion Prastik Mohanraj

Big Bang Cosmology We observe redshift, where galaxies become redder in color over time. This means that the galaxies are speeding away from us. So, we can see that the universe must be expanding. If the universe is expanding, it must have been smaller, denser, and hotter in the past. Thus, it is likely that the universe started from something extremely small, that being the Big Bang. Source: BBC iWonder

Models of the Universe

The 3D plane of the universe is curved around the 4D hyperspace of the universe in one of these three different shapes. This is a result of matter-energy density on the largest scales in the universe.

These images are of the Cosmic Microwave Background Radiation, which is a footprint left by the hot universe around 380,000 years after the Big Bang. The observed CMB fluctuations in temperature give information about the earliest moments of the universe.

Closed Universe

• Constant positive curvature (k = 1)

• A three-dimensional hypersurface of a four-dimensional hypersphere

• Analogous to two-dimensional surface of three-dimensional ball

• Scale factor a = Radius of four-dimensional hypersphere

Closed Universe Expansion

• The further two galaxies are from each other, the faster they recede from one another in an expanding universe: Hubble’s Law

• If the universe is expanding, then it would be impossible to go around the universe since one would need to exceed the speed of light – some parts of the universe appear to expand faster than other parts from an observer’s view

• Source: One-Minute Astronomer

Friedmann Equations

• a = scale factor of universe

• k = curvature factor (-1, 0, or +1)

• 𝜖 = energy density of universe

• 𝜅 = constant value

• p = pressure

• 𝑓(𝑎, 𝜖) = 0 except for the immediate beginning of the universe

Friedmann Equations Model

Period p-𝜖 Relationship Second Friedmann Equation Model

Second Friedmann Equation Result

Revised First Friedmann Equation Model

Relativistic (Radiation-Dominated)

Nonrelativistic (Matter-Dominated)

Cycloid

• When a circle rolls along the plane, the curve connecting the position of one distinct point on the circle at different times produces a cycloid

• Geometric derivation of 𝑥, 𝑦 coordinates yields parametric equations

Cycloid with the Nonrelativistic Friedmann Equations

The result is equivalent to the nonrelativistic Friedmann Equation model, thus the cycloid function is a solution to the nonrelativistic universe expansion.

Dilemma

• In the diagram, there are points where the scale factor = a on the vertical axis is equal to 0

• This means that the universe has become a point, with infinite density: a singularity

• This can be avoided with Einstein-Cartan Theory, which uses a torsion tensor to eliminate gravitational singularities

Trochoid

• The coefficients of the parametric functions for 𝑥, 𝑦 are changed to:

• As long as 𝐴 > 𝐵, the scale factor = a will always be a positive value

• Notice how the 𝑎 =𝑑𝑡

𝑑𝜃 result from the cycloid function is preserved

here

• Another format for writing the scale factor = a in the trochoid function is

Revised Friedmann Equation Model with the Torsion Factor • The Friedmann Equation will be revised to incorporate the added torsion factor,

which is only present in the beginning of the universe, thus, the relativistic model will be revised

• The revised Friedmann Equation model is:

• Notice how the coefficient 𝐴2 − 𝐵2 will be close to 0 with 𝐴 >≈ 𝐵, and only make an impact on the function when 𝑎 is very small

• Instead of a singularity with the Big Bang, the non-singular bounce of the universe at the minimum scale factor, creating an alternate Big Bounce

• The torsion factor approaches 0 rapidly after the Big Bounce, so the 2𝐴

𝑎2 term

makes the only impact

• Thus, the torsion factor can be removed from the model after the Big Bounce

Novel Solution to Revised Model

• A solution to the model is:

• Here,

• The 𝐸(𝜃, 𝑘) function an elliptic integral of the second kind

Proof of Solution

• The

The result is equivalent to the Friedmann Equation with the added torsion factor, thus this is a working solution to the model.

As 𝐵 → 𝐴, torsion disappears

• Torsion fully disappears when torsion factor = 0 : This is when 𝐴 = 𝐵

• This result gets substituted into the parametric equations to obtain:

Substituting these functions a and t into the relativistic model without the torsion factor yields a true result. Thus, this solution is true.

Dark Energy Universe

• Dark energy is present only in the nonrelativistic, matter-dominated era

• The dark energy factor is proportional to 𝑎2, and will be added to the nonrelativistic model of the Friedmann equation, resulting in:

• Further into the universe’s evolution, where a becomes extremely high, this model reaches the limit of:

• This is the exponential increase behavior of scale factor that we expect in a dark-energy dominated era, thus the model is valid

Conclusion

• Here, we have presented models of the Friedmann Equations incorporating different eras of the evolution of the universe

• The novel solution to the Big Bounce time period of a positively curved universe is presented here:

• and with

• These parametric equations are solutions to the Friedmann equations incorporating the Einstein-Cartan Theory torsion factor

Acknowledgements and References