The Age of Multiple Universes by Michael Robinson and Chris Bezerra-Clark

PHYS205 Term Project Submitted Tuesday, December 05, 2006

Instructor: Dr. Dimitrios Psaltis

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Introduction

The age of the universe is a perplexing problem that has be considered for thousands of years. Ever since our first human ancestors have been able to look up in to the night sky they have wondered why we are here, how we came to be, and what our universe really is. Man has burned with a desire to find the answers the natural world. One question that carries many implications is the origin and age of the universe. By attempting to quantify how long the universe has been, scientists could deduce how our universe came to be and what its nature has been all along. From some studies we can determine that the universe began as a singularity. Theoretically this means that every single atom and bit of energy that makes up all the galaxies and stars in the universe arose from a single point. Since this point held all the energy and matter of our universe it was extremely dense. At some point this singularity became unstable, and it began to expand rapidly. One might try to quantify the speed at which the newly born universe was expanding, during the cosmic inflation the universe’s radius would have increased by scales of a factor 26

10 in 3310

! seconds. Using basic kinematics one could simply find the radius of today’s universe and see how long it would take the universe to reach that radius using the value of cosmic inflation previously stated. But this value would be grossly incorrect. In order to correctly approximate the age of the universe one would need to incorporate general relativity with the cosmological parameters. In 1922 Alexander Friedman did this, creating what is known as the Friedman equations.

Using these equations in a computer program that solves differential equations, time will be run backwards to study the age of the universe. The Friedman equations require several quantities to be known in order to solve the equation. This report will consider values corresponding to the most recent and accurate data of the known universe, as will as vary these parameters to consider other possible universes. The ages and origins of these universes will be compared and analyzed to find trends and other interesting factors that may arise. The age found for the actual universe will be compared to what is considered (presently) as the actual age of the universe, which is 13.7 billion years old. The Problem Assumptions

The primary assumption of this report is that Friedman’s equation accurately models the changing size of the universe, since the computer program will be using Friedman’s equations to run time backwards until the universe collapses. The type of Friedman equation used makes several assumptions itself, namely that the three considerable densities in the universe are radiation, matter (both luminous and dark), and energy in a vacuum (which will be referred to as dark energy for this report). If other types of densities exist or are significant, this model will not take them into account. The Friedman equation assumes that the universe is uniform and isotropic. Uniform refers to the universe being of a constant density and is often referred to as homogenous. Isotropic means that change in the universe is independent of direction, or that what is true for something in one direction is true in another. When calculating the age of the universe using parameters that best match the experimental data, several assumptions will have to be made for this constrained universe. It will be assumed that the universe began as a single point in a flat universe. The other parameters

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used for the normalized densities of radiation, matter, and dark energy (denoted as Ωr,m, or d) are 5

103!

"=#r

, 27.0=!m

, 73.0=!d

, and will be assumed that they best model the known universe. Equations The Friedman Equation was presented for this problem as:

( )22

0

22

23

81

!"""

#!

! R

kcG

dt

ddmr$++=%

&

'()

*%&

'()

* .

(1) where “α” is a scale factor, referring to the ratio between the distance of an object from another at time “t” compared to the initial distance between the two (R0). The three values of “ρ” refer to the densities of radiation, matter, and dark energy (denoted r, m, and d). “G” is the gravitational constant of the universe. The term “k” is a cosmological parameter which can have the value of 0 for a flat universe, 1 for an open universe, and -1 for a closed universe, and “c” is the speed of light.

As the universe expands, the available space for matter, radiation, and dark energy to occupy and therefore the densities of each change as a function of time. The densities relate in the following way:

( )( )

( )( )

( ) ( )0

0

0

3

4

==

==

==

tt

tt

tt

dd

m

m

r

r

!!

"

!!

"

!!

.

(2-4) Note: t = 0 represents the value of the density today.

These densities can be expressed in terms of the critical density:

G

H

c

!"

8

32

= .

(5) where “H” is the Hubble parameter defined as the normalized derivative of the scale factor. To normalize each density, divide the density by the critical density:

c

d

d

c

m

m

c

r

r

!

!

!

!

!

!

="

="

="

4

(6-8) Add the normalized densities to obtain the value Ω, the density parameter:

!=!+!+!dmr

. (9)

If Ω is equal to one then the universe will be closed but expand indefinitely, if it is less than one then the universe is ‘saddle’ shaped and will also expand indefinitely, and if it is more than one the universe can have multiple possibilities.

The following relationship is known and is useful in solving the Friedman equation into a form usable for this model:

2

0

22

0 )1(

R

c

k

H=

!" .

(10) This equation is taken from Pg. 23 in Modern Cosmology.

Multiplying t by 0

H will give us a dimensionless value of t:

ttH !=0

(11)

Solving the Friedman Equation

The following shows the method used to solve the Friedman equation into a form useable in a program based on the Rutta-Kruge method of differential equation solving. (See Appendix for the actual codes used)

Substitute equations 2-4 into 1:

22

0

2

34

2

23

81

!"

!

"

!

"#!

! R

kcG

dt

dd

mr $%&

'()

*++=%

&

'()

*%&

'()

*

(12) Substitute equations 6-8 into 12:

22

0

2

34

2

23

81

!!!"

#!

! R

kcG

dt

dd

mr

c$%&

'()

*++

++

+=%

&

'()

*%&

'()

*

(13) Substitute equation 5 into 13:

22

0

2

34

2

0

2

28

3

3

81

!!!"

"!

! R

kc

G

HG

dt

dd

mr #$%

&'(

)*+

*+

*=$

%

&'(

)$%

&'(

)

(14) Simplifying:

5

22

0

2

34

2

0

2

2

1

!!!

!

! R

kcH

dt

dd

mr "#$

%&'

()+

)+

)=#

$

%&'

(#$

%&'

(

(15) Multiply both sides of the equations by 2! :

22

0

22

2

3

2

4

2

2

0

2

2

2

1

!

!!

!

!

!

!!

!

! R

kcH

dt

dd

mr•

"##$

%&&'

(•)+

•)+

•)=•#

$

%&'

(#$

%&'

(

(16) Simplifying:

2

0

2

2

2

2

0

2

R

kcH

dt

dd

mr !"#

$%&

'(+

(+

(="

#

$%&

')

))

)

(17) Substitute equation 10 into 17:

k

kHH

dt

dd

mr)1(202

2

2

0

2!"

!#$

%&'

("+

"+

"=#

$

%&'

()

))

)

(18) Simplify. Substitute equation 11 into 18:

)1(202

2

2

0

2

0

2

!"!#$

%&'

("+

"+

"=#

$

%&'

(

)HHH

td

dd

mr ***

*

(19) Simplify and take the square root of each side:

)1(2

2

2

!"!#$

%&'

("+

"+

"=#

$

%&'

(

)*

**

*d

mr

td

d

(20) What is left is a solution to the Friedman equation. This solution is in dimensionless form that the program will run this equation through the Runge-Kutta method.

( )12

2!"!#

$

%&'

("+

"+

"=

)*

**

*d

mr

td

d

(21) The Friedman equation is valid only if the following inequalities remain true:

11

11

01

+=!>"+"+"

#=!<"+"+"

=!="+"+"

k

k

k

dmr

dmr

dmr

(22-24) Equation 21 does not violate these boundary conditions.

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Numerical Method To integrate the dimensionless solution to the Friedman equation we found, we will use the 4th Order Runge-Kutta method for solving ordinary differential equations. The method is given by these equations:

[ ]

[ ]

[ ]

6336

,

2,

2

2,

2

,

)(,

4321

34

23

12

1

kkkkff

kfttgtk

kf

ttgtk

kf

ttgtk

ftgtk

tftgdt

df

itt

ii

ii

ii

ii

++++=

+!+"!=

#$

%&'

(+

!+"!=

#$

%&'

(+

!+"!=

"!=

=

!+

(25-31) Note: These variables do not correspond to any variables used in the Friedman equation. They are simply variables used to illustrate the 4th Order Runge-Kutta method.

The preceding is run in a loop until ttt =!+ )( . This method is very good in giving precise values for the solutions to the equations put in. Because it is of the fourth order the error in the solution is on the magnitude of 4

t! , so if the time-step is small enough, the error will not affect the answer as much as a larger time-step. This is an effective method for us to use because it does not have very many steps, it has a small magnitude of error, and can be easily produced and modified to show many different types of universes. Results The age of the universe was considered for three different primary universes, with multiple possibilities or sub-universes for the distribution of densities. One additional universe (which would fall under the Universe 1 category) was considered using the values of experimental data which are presently considered to best describe the universe. Universe 1 This universe is defined by the k parameter equaling 1. For this universe the total value of the normalized density (Ω) is equal to 1, as defined by the Friedman equation itself. The values of the normalized densities for radiation, matter, and dark energy were varied within this universe, and its corresponding age was considered for each case. The seven sub-universes were defined by their different values for the normalized densities of each kind, and were the only variables. The individual values were chosen to provide a range of valid possibilities; allowing matter or dark energy to be dominant in the present universe or allowing a balance between them.

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Allowing radiation to presently dominate or to even have a significant value (greater than Ωr = 0.02) created interesting and improbable results in this universe. As time ran backwards in the program, normally the value of the scale factor α became smaller and smaller until almost reaching zero. At this point the remainder of the reported values read as “nan” or not a number. This is most likely due to the program’s inability to evaluate the square root of a negative number. Since the values approach zero, the age of the universe was determined by the value of time between the last numerical value given and the first appearance of “nan.” When Ωr was given a higher value than approximately 0.02, the program did not approach zero and then “break”, but continued on giving negative values. A negative scale factor is not valid, as it is defined as the ratio of two positive distances. Only one large value of normalized radiation density was considered for this universe, because of the strange results. For the other six possibilities considered, Ωr was left at 3.00x10-5, which at present considered to be the most likely value today.

Graph 1 shows the data collected:

Graph 1

For this and all following graphs, “R”, “M” and “D” refer to the normalized density value of radiation, matter, and dark energy respectively when time equals zero, or today. These values used to build the equations. The seven possibilities shown allow for the domination of matter or dark energy (factors of approximately 2,10, and 100) and a balanced scenario. The scenario of

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the dark energy dominating matter by a factor of approximately 2 is not included as this case is considered likely to be the actual universe, which will be considered separately. The graph proposes several interesting trends. If matter is dominate in today’s universe, the universe is younger, decreasing its age with increasing factors. Similarly, if dark energy is dominate in today’s universe, the universe is older, increasing its age with increasing factors. When radiation is dominate, the program begins to produce negative values for the scale factor alpha, and this is shown on the graph. However, if the equation is accurate for this scenario until α<0, then the universe is younger here than the scenarios when matter dominates. Table 1 displays the seven different Ω distributions for Universe 1, and their corresponding age of that possible universe.

Ages of the Universe for Varying Values of Ω (Ω= 1 and k= 0)

Ωr Ωm Ωd age (billions

of years) uncertainty in data (±)

3.00x10-5 0.99 0.01 8.904 0.067 3.00x10-5 0.90 0.10 9.306 0.067 3.00x10-5 0.70 0.30 9.977 0.067 3.00x10-5 0.50 0.50 11.18 0.067 3.00x10-5 0.10 0.90 17.07 0.067 3.00x10-5 0.01 0.99 26.71 0.067

0.80 0.10 0.10 6.759 0.067 Table 1

The age reported here was determined by taking the average time corresponding to the last numerical value provided by the program and the next time. The uncertainty was determined to be ±0.067 when considering that the actual time when alpha becomes zero could be any time between those two ending times. The time step used was 134 million years, and the error is one half of this. (Ex: For the first possible sub-universe listed on the table, the age of the universe is between 8.837 and 8.971 billion years ago). Universe 2 This universe is defined by its k value equaling -1. This has a corresponding Ω value less than 1. Unlike Universe 1, Ω is not a specific, and many different values of Ω could be analyzed ranging anywhere between 0 and 1, providing for a large number of sets of distributions for the various normalized density values of radiation, matter, and dark energy which would be valid of considering. For the purposes of this report, only one value of Ω (Ω= 0.5) will be considered, with six varying density distribution. When testing the program used for Universe 2, the same problem was found as in Universe 1 was found; when the normalized density of radiation was significant, the program began to produce negative results. One such scenario was left for consideration, but as with Universe 1, the other values of Ωr were left at 3.00x10-5. The remaining distribution of Ω values considered for radiation and matter were chosen to show the domination of each and a balance between them.

Graph 2 shows the data collected:

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Graph 2

The six possible normalized density distribution here allow for the domination of either matter or dark energy (factors of approximately 50 and 5) and for a balanced scenario. Also shown is the possibilities (high radiation) that produce negative results in the scale factor alpha. Trends in Universe 2 are similar to those of Universe 1. A high present radiation density seems to be improbable as well as indicates a relatively young universe. A large ratio of dark energy density to matter density creates an older universe and a small ratio creates a younger universe. Though when given a significant value strange data is produced, it seems possible that a radiation dominant present universe would create the youngest universe of all. Table 2 displays the six different Ω distributions for Universe 2, and their corresponding age of that possible universe.

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Ages of the Universe for Varying Values of Ω (Ω= 0.5 and k= -1)

Ωr Ωm Ωd age (billions

of years) uncertainty in data (±)

3.00x10-5 0.49 0.01 9.975 0.067 3.00x10-5 0.40 0.10 10.51 0.067 3.00x10-5 0.25 0.25 11.58 0.067 3.00x10-5 0.10 0.40 13.46 0.067 3.00x10-5 0.01 0.49 15.87 0.067

0.30 0.10 0.10 8.502 0.067 Table 2

The age and uncertainty was evaluated the same way as in Universe 1. When compared to Universe 1, it appears that a k= -1 and Ω=0.5 universe is slightly younger than a k=0 universe when comparing the ratios between matter and dark energy versus age. Universe 3 This universe is defined by its k value equaling positive one, and therefore its Ω value equaling more than 1. The number of valid Ω values for this universe is infinite, as the range for Ω is from zero to infinity. For each valid Ω value, there would a set of valid distribution of radiation, matter and dark energy normalized density values to consider, increasing the length of analysis even more. For the purposes of this report, one value of Ω (Ω=1.5) will be considered. Again, seven distributions of the various normalized densities will be evaluated and the age of the universe for each scenario will be considered. When testing Universe 3 for results, several strange occurrences were discovered. In this universe, negative values of alpha did not always appear when the radiation was set to a significant value. Other distributions did not even approach zero. Because of the nature of Universe 3, radiation in this case was not held at a very small value. The trials considered here allowed radiation, matter and dark energy to dominate over the others (factors of 150 and 4) and tried a scenario when the three were balanced. Graph 3 was created based on the data points created by the Universe 3 program.

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Graph 3

In this graph two of the density distributions behave as expected, starting at α=1 and ending at α=0. Four of trials continue past α=0 and become negative, which does not follow logically with the definition of the scale factor. When the normalized radiation density is at a value of 0.25, the alpha function stops a zero when matter has a normalized density of 1.0, and dark energy of .25, but passes zero and moves to negative numbers when radiation is kept the same and the other two values are switched. This did not occur in Universe 1 or 2. The final trial (red line) decreases from α =1 as expected, but then levels off and does not approach or cross α =0, as had all other data either considered in testing or displayed in this report. At 13.46 billion years before present the leveling function stops and the program produces “nan” results, as had the functions which approached zero. Some of the reasons for this inconsistent result were discovered through further research. According to Modern Cosmology and a source online from Manchester University (see Sources section) the origin and fate of the universe falls into three possible categories. The universe could have originated from a singularity and expand to infinity; the universe could have originated from a singularity, expand for a period of time, and contract back into a singularity; the universe could have originated from an infinite size, contract to a finite size, and expand back to infinity. Both sources do not consider radiation significant at present, but show what possibility occurs for varying values of the normalized density of matter and dark energy. This is shown in figure 1:

Time (years)

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Figure 1

The two normalized densities in the figure are for present day values which are used to solve the Friedman equation in a similar fashion as shown in this report. The green line shows the boundary for expanding and contracting universe. The red line shows the boundary for the infinite-infinite universe, or no big bang scenario. Any universe in between will have a big bang and expand forever. The blue line shows the k=0 universe. Below would be the k= -1 universe and above the k=+1 universe. The values found for the red line (R=0.01, M=0.01. D=1.48) in graph 3 lie within the “no big bang boundary”. This is why it does not approach zero, as it should actually become greater and approach infinity. It is unclear why it apparently stops at its apex and does not start increasing as time continues backwards. The remainder of the values in all 19 other trials fall between the green and red line, and would all have a value for time when alpha equals zero. Table 3 displays the seven different Ω distributions for Universe 2, and their corresponding age of that possible universe.

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Ages of the Universe for Varying Values of Ω (Ω= 1.5 and k= +1)

Ωr Ωm Ωd age (billions

of years) uncertainty in data (±)

0.01 1.48 0.01 7.967 0.067 0.01 0.01 1.48 infinite ------- 1.48 0.01 0.01 5.958 0.067 0.50 0.50 0.50 7.699 0.067 0.25 1.00 0.25 7.843 0.067 0.25 0.25 1.00 9.975 0.067 1.00 0.25 0.25 6.628 0.067

Table 3 The age of the differing possibilities of this universe are smaller than both Universe 1 and 2. High levels of dark energy also produce the oldest ages within this model, and high levels of radiation the youngest. The Likely Universe Given in the problem were values of each present normalized density, the present value of the Hubble constant, and value of k. These values are based on experimental data from several sources, including the Hubble telescope and the WMAP satellite. The Hubble constant was used in the program to solve the Friedman equation, though it eventually dropped out of the equation, making it valid for all possible universes and various densities. Using the best experimental data for the universe, it is possible to calculate the actual age of the universe as accurately as possible, at least within the scope of this report. For this model, Ωr =3.00x10-5, Ωm =0.27, and Ωd =0.73. Running these values through the program produced an age of the universe of 13.394 billion years. Graph 4 shows the function alpha for the best experimental data.

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Graph 4

The curve is fairly linear from present to 10 billion years ago, and then begins to decrease rapidly. This indicates a rapid expansion following the big bang for approximately three billion years, and since then a constant growth. This universe falls within the big bang to infinite expansion universe shown in figure 1, as do all of the k=0 universe possibilities. As with Universe 1 the likely universe is older than other possibilities due to the larger amount of dark energy density compared with matter density. At present, many scientists believe that the universe is 13.7 billion years old, compared to the 13.4 billion years found using the method in this project. The 2.2% discrepancy could be accounted for by program due either to rounding errors or by a mistake in the model used. Conclusion The method of determining the age of the universe found the likely universe to be 13.394 billion years old with an uncertainty of 67 million years. This value is has a difference of 2.2% from what is presently considered the actual value for the age of the universe, which would increase the error of the data to 313 million years. When considering the open, closed, and flat universes (k=1,0,-1) and manipulating the densities many interesting possibilities arose. Though all models, except for one, shared an origin and fate, a wide range of possible ages occurred ranging from 5.968 billion years to 26.71

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billion years old. The data further indicated that the larger the value of dark energy density, the older the universe, and the larger the value of matter density, the younger the universe. It also hints that a large radiation density at present would shorten the age of the universe even more. Manipulating the value of radiation density for present caused many unusual trends to develop in the Friedman model, causing the universe to take on a negative dimension. Most likely this indicates impossibility in the chosen values, meaning that the value of the radiation is presently quite small compared to that of matter or dark energy, especially for Universe 1 and 2. Universe 3 could, however, handle a high radiation density scenario, and contains all three possibilities for the origin and fate of the universe, making it the most “open” universe of all. By using the Friedman model, one can consider an infinite number of possible universes, with an infinite number of possibilities within each one. Though this report only considers a handful of those with a rather simplistic scope, this report surely points toward the well of knowledge that can be discovered through application and experimentation of this model.

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Sources Gorini, Vittorio and Ugo Moschella. Modern Cosmology. Institute of Physics Publishing,

Philadelphia, 2002. pg 23.

Leahy, J.P.. http://www.jb.man.ac.uk/~jpl/cosmo/friedman.html. Maintained by the University of Manchester, Jodrell Bank Observatory. Site last visited at 10:05pm, 12/4/06.

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Appendix The following pages are the four codes used to solve the problem. The first code was used for Universe 1, the second for Universe 2, the third for Universe 3, and the fourth for the likely Universe.