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28 October, 2014

Cosmological Parameters and Expansion History of our
Universe
Alec Wright
Abstract: The publication of the theory of relativity has since provided us with the knowledge to determine
certain cosmological parameters of our Universe
...
Derivations of different cosmological parameters are exhibited in this paper, and evaluated to
theoretically model the evolution of the Universe under different circumstances
...
Modelling the expansion history requires distance parameters, Hubble parameters,
and scale factors to be derived
...
Data obtained from multiple studies of 255
Supernovae revealed the Hubble parameter at current time to be 45
...
3 kilometres per second per
mega parsec
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5 % of the critical density, with the Cosmological Constant density
remaining ambiguous given an unknown curvature of the Universe
...


1
...
What is perhaps the most significant aspect about the nature of the Universe is its
evolution through time, requiring the famous theory of relativity to derive cosmological parameters
to explain how it changes with time
...
e
...
The introduction of the Cosmological Constant can represent this repulsive
force in Cosmological equations in order to provide the viability of a static finite Universe as
described
...

The subject of this paper is to discuss the expansion history of a spatially finite Universe
under the consideration of the Cosmological Constant acting as an extra density parameter
...
The redshift of light also plays an
important role in understanding the expansion history of such a Universe since it is required to
analyse distances to objects
...
The study of Type 1a Supernovae, the Cosmic
Microwave Background Radiation, and Baryonic Acoustic Oscillations require theoretical distance
models such as proper distances and luminosity distances as functions of redshift, before analysis
and comparisons are made with retrieved data
...


2
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The Friedmann equation takes place as one of the three essential
equations used to study the evolution of a finite homogenous isotropic Universe that is either static
or undergoing expansion/contraction
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The derivation of the corresponding density reveals a Cosmological Constant density
unchanging over time (i
...
constant) due to the negative pressure the CC (Cosmological Constant)
enforces
...


(1)

Corresponding to flat curvature of space (k=0), such a property of the Cosmological Constant
allows the solution of the Friedmann equation for a CC-dominated Universe to be exponential
...
We can compare this with scale factors
corresponding to matter-dominated and radiation-dominated Universes with flat curvature
...
Time corresponding to matter-dominated, radiation-dominated,
and Cosmological Constant Dominated Universes with Flat Curvature
...

The expansion of such
Universes decelerate over time but at a decelerating rate, meaning the rate of expansion slows
down over time but will never reach a stopping point
...
For example, consider a finite Universe with a mixture of the
Cosmological Constant and matter densities
...
A similar idea applies to a
mixture of radiation and Cosmological Constant densities for a finite Universe with flat curvature
...

When considering a non-zero curvature (i
...
positive/negative) for a finite Universe, the derivation
of a scale factor becomes mathematically difficult, rather a proportionality between scale factor and
conformal time [2] will be as effective to evaluate by considering a substitution of conformal time
with regular time [3]
...
In the case of matter-dominated and radiationdominated Universes, scale factors follow the path of an oscillating system in terms of conformal
time
...
In regards to a CC-dominated Universe, similarly to the flat Universe scenario, an

Figure 2: Scale Factor Models vs
...


exponential proportionality between scale factor and conformal time, derived from equation 1,
revealed an increasing expansion rate for a closed finite Universe with positive curvature
...
However, for
a CC-dominated Universe where such density is constant and using similar logic as for a flat
Universe, options of expansion rates are limited to only the acceleration of the Universe
...

Although it is still possible for a spatially finite Universe with negative curvature, the fact that it is
open implies a constant expansion rate
...
Physically, this means an accelerating expansion proportional to conformal time,
however, converting to normal time indicates a similar rate of expansion to the flat Universe
scenario
...

In the case of a CC-dominated Universe with negative curvature, it is again revealed an increasing
expansion rate, from the proportionality of scale factor to the exponential power of time
...
However, depending on curvature depends on the rate at which the
expansion is accelerating, as observed in Figure 2
...
The Hubble parameter of a finite Universe undergoing expansion
or contraction describes the rate at which space changes at any point in time
...
The mathematical definition implies the Hubble parameter is different for a finite
Universe depending on factors such as curvature and the dominating density
...
In the case of a
flat finite Universe, matter-domination and radiation-domination scenarios permit the Hubble
parameter to be inversely proportional to time
...
e
...
For a CC-dominated Universe, the Hubble parameter remains
constant over time, and its magnitude is only dependant on what the value of the Cosmological
Constant is
...
These three dominations of
matter, radiation, and CC density are related by the significant observation that the Hubble
constant will never reach zero, regardless of the variation with time
...

The determination of the Hubble Parameter corresponding to positive or negative curvature
requires similar derivations in terms of conformal time
...
Models
of the Hubble parameter for both matter domination and radiation domination exhibit decreases in
the Hubble constant over time
...
In regards to a CC-dominated Universe with positive curvature, the Hubble
parameter remains constant over time
...
In the case of matter-dominated and radiation-dominated Universes with negative
curvature, the Hubble parameter theoretically decreases over time but will never be equal to zero
...


Figure 3: Hubble Parameter Models vs
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Observe the
constants for a CC-dominated Universe with different curvature directly above the time-axis
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Proper and Luminosity Distance Derivations
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The simplest
possible scenario of a flat static homogenous isotropic Universe requires very few calculations to
measure distances
...
However,
in the case of an expanding or contracting Universe regardless of curvature, the redshift of
luminous objects becomes paramount in measuring such distances
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The
redshift of a luminous object can be used to determine the rate at which the observed object is

moving away from the observer, so it is logical to model proper distances and luminosity distances
as functions of redshift in order to observe the variation in distance of such objects
...


(3)

Considering dominations of different densities and curvature is necessary once more in order to
model proper distances
...
Presuming a flat Universe, matter-domination and radiation-domination scenarios reveal
proper distances to be inversely proportional to the 1+z quantity, and with the addition of a
constant, means proper distance increases with redshift
...
These correlations related to each
density domination agree with corresponding expansion rates for each Universe in the sense that,
proportionality-wise, proper distance decreases similarly as the Hubble Parameter decreases with
time
...


Figure 4: Proper Distance Models vs
...
Observe that the Proper Distance
Model corresponding to radiation-domination does not differ for positive or negative curvature of
space
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Considering a matter-dominated Universe, in the case of positive
curvature, proper distance increases with increasing redshift and eventually becomes constant as
redshift becomes arbitrarily large
...

These models are reasonable theoretical estimates given the corresponding spheroidal and
hyperbolic shapes of the Universe to positive and negative curvature respectively
...
The case for a radiation-dominated Universe however is
different in the sense that proper distance increases with increasing redshift regardless of
curvature
...
For negative curvature of space,
however, this model corresponds to the imaginary component of proper distance whereas the
model for positive curvature corresponds to the real component of the same quantity
...

Derivation of equation 3 for a CC-dominated Universe revealed a logarithmic proportionality
between the proper distance and redshift, similar to that of the radiation-dominated scenario, in the
case of positive curvature
...

Luminosity Distance derivations require a significant value procured from the Friedmann-LemaitreRobertson-Walker metric
...


(4)

If an object located at a very large distance from an observer emits radiation corresponding to the
visible spectrum of light, the luminosity distance can be determined from the redshift of the emitted
radiation in collaboration with equation 4, with a dependence on the curvature of space
...
However, the distance increases with different proportionality to redshift, as exhibited in
Figure 5
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These models
for luminosity distance are similar to the case of negative curvature when considering the fact that
luminosity distance increases with redshift
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It is shown that a matter-dominated Universe with
negative curvature allows a constant luminosity distance as a function of redshift (along the x-axis)
...
Redshift corresponding to each density domination and
curvature of space (flat=blue, positive=green, negative=red)
...
Also, observe
the case for a CC-dominated Universe with positive curvature
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It is difficult to correlate such a relation in the case of a matter-dominated
Universe with negative curvature, however trust in the mathematics of the derivation is entailed
...

It is also observed that the case of a CC-dominated Universe allows the luminosity distance to
reach a maximum at some redshift value then decreases back to zero as redshift increases
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This is in the sense that the luminosity distances for each density
domination fail to increase at rates compatible with the corresponding expansion rates
...
Generally,
however, all other models for luminosity distance exhibit an increase in distance as redshift
increases
...
Observational Results
Type 1a supernovae are considered preeminent “standard candles” in determining properties of
objects such as luminosity, distance, absolute and apparent magnitudes, and redshift
...
The main aim of the SDSS 2 study is to determine

density parameters for matter and the Cosmological Constant
...
The observation of 255 type 1a supernovae allowed us to obtain data on the
redshift and distance modulus of each object, as well as the relative error, which can confirm the
validity of the density parameters
...
It is still unclear as to
which density is currently dominating the Universe, but an analysis of the Hubble constant can
provide details for the density-domination as well as the curvature of space
...
If the proper distance is equal to the luminosity distance multiplied by the
(1+z) factor for corresponding redshift values for each supernovae measured, then the Universe is
flat, meaning the sum of the density parameters of matter and the Cosmological Constant is 1
...


Recession Velocity vs
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Luminosity Distance for 255 Type 1a Supernovae with error bars
corresponding to approximately 10% of magnitude
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9452,
reinforcing the viability of the linear trend line
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95 plus/minus 2
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The chi-squared value allows this to be a reasonable
estimate of the Hubble constant of the Universe, which can lead to significant calculations of
density parameters
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58 plus/minus 0
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The viability of this value must also be reasonable since it was directly
derived from the Hubble constant
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(5)

Considering a spatial region inside the Universe sizeable enough to be homogenous, the density of
matter in that region is theoretically equal to the total matter density if the Universe
...
The use of the combination of the Virial Theorem and mean radial velocities of the 255
supernovae provides a matter density inside this region to be 1
...
However, this does not imply that the
Universe is dominated by the Cosmological Constant as the curvature of space is unknown
...
In the case of positive curvature or flat
curvature, the Cosmological Constant density would dominate the Universe since by definition the
density parameter must be equal to or greater than 54%
...
It is simpler to leave the parameter of the density
at 54% of the critical density, which is a density of 1
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In the case of positive curvature, the density is greater than this value, but must be less
than the critical density in order to imply a “Big Bang” beginning of the Universe
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4
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A Hubble parameter value of 45
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3 kilometres per second per mega parsec
can suggest an accelerating expansion if constant over time, regardless of the curvature of space
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This Hubble constant gives a current age
of the Universe to be approximately 2
...
The evolution of the Universe strongly
depends on the density parameters of both matter and the Cosmological Constant
...
Since matter density decreases as scale factor
increases, as time becomes arbitrarily large, the Cosmological Constant will become more
dominant until it reaches some density value, at which point the sum of the densities of matter and
the Cosmological Constant will amount to greater than the critical density, and alter the curvature
of the Universe to become positive and closed
...

In the case of positive curvature, the density of the Cosmological Constant must also dominate, as
the sum of this and the density of matter must amount to be greater than the critical density
...
In regards to negative curvature of space,
considering the fact that the density of both matter and the Cosmological Constant must sum to
less than the critical density means that the CC density parameter must be less than 54%
...


5
...

Such a
dependence is clearly evident in the models of scale factor, Hubble parameter, proper distance,
and luminosity distance, as well as the density parameters of matter and the Cosmological
Constant
...
However,
considering the different scenarios for curvature, it is very likely the Cosmological Constant
contributes approximately 54% of the total density of the Universe
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95 +/- 2
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This is the Hubble parameter at the current
time, corresponding to the current age of the Universe to be approximately 2 billion years old
...
This will inevitably lead the Universe
to live an infinite lifetime, regardless of any Curvature
...
References
[1] PHYS3080
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Expansion History Overview
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[2] Longair Lectures
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Fundamentals of Cosmology
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[3] C, Hirata
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Redshift and Conformal Time
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[4] F, Abdalla
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Observational Cosmology: The Friedmann Equations 2
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[5] A, Liddle
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An Introduction to Modern Cosmology
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