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 fourdimensional hypersphere
• Analogous to two-dimensional
surface of three-dimensional ball
• Scale factor a = Radius of fourdimensional 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
Relativistic
(RadiationDominated)
Nonrelativistic
(Matter-Dominated)
p-𝜖 Relationship
Second Friedmann
Equation Model
Second Friedmann
Equation Result
Revised First
Friedmann Equation
Model
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
𝑑𝑡
𝑑𝜃
• Notice how the 𝑎 = 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
2𝐴
• The torsion factor approaches 0 rapidly after the Big Bounce, so the 2 term
𝑎
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