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					Undergraduate Research Opportunity Programme in Science

                        (UROPS)




     2 ways for the Universe to end


                       Submitted by

                    Teo Choon Hoong




                       Supervised by

                   A/P Brett McInnes




               Department of Mathematics

             National University of Singapore

          Academic Year 2001/2002 Semester 2
                               Contents


Introduction                               1


Origin of the Universe                    1


Equations and their meanings              3


Changes in the Universe                   10


Big Crunch theory                         11


Big Smash theory                          19


Conclusion                                23


References                                24
                                     Introduction

 It was once thought by many including Albert Einstein that the Universe is static and

unchanging in time . However , there will be no redshifts in such a Universe and it is also

unstable : It would either collapse or expand forever should any density fluctuation occur

. Thus , it is not a model that can explain the real Universe .

 In the mid-twenties , Edwin Hubble observed that the light reaching us from distant

galaxies are „red-shifted‟ . This means that the Universe is actually expanding . When

light travels from one place to another , it is passing through the intervening space-time .

When space-time is expanding , the light passing through it will be „stretched‟ and this

results in a longer wavelength for the light and thus , it is termed „redshift‟ .

 Since a static Universe in no longer possible , this leads to the conclusion that the

Universe must end some day .

 First , we start by giving a brief introduction to how the Universe began and we move

on to examine the different equations we will be using in the discussion and next , onto

the various models that describes how the Universe will end . Finally , we decide on the

model that can best describe the ultimate fate of the Universe .


                             Origin of the Universe

 About 15 billion years ago a tremendous explosion started the expansion of the

universe. This explosion is known as the „Big Bang‟ . At the point of this event , all of the

matter and energy of space was contained at one point .
 The origin of the Big Bang theory can be credited to Edwin Hubble . When he was

observing the galaxies , Hubble found that that a galaxy‟s velocity is proportional to its

distance . That means , galaxies that are twice as far from us move twice as fast . This

also implies that the Universe is expanding in all directions and that every galaxy took

the same amount of time to reach its current position from a common starting point .

 After the Big Bang , the temperature of the Universe was extremely hot owing to the

intense motion of particles which were made up of both matter and anti-matter . Around

10^-43 seconds after the event , an almost equal amount of particles and anti-particles

existed . These particles and anti-particles collide and annihilate with each other ,

producing energy as a result . As the Universe favors particles over anti-particles , one

particle was spared for every billion of such collisions and our Universe today is made up

of those which were spared from the collisions .

 During this period of time , the Universe expands from a size of an atom nucleus to over

1000 meters in width in less than one thousandth of a second .

 The universe at this point was ionized plasma where matter and radiation were

inseparable . When the universe aged to one hundredth of a second old , neutrons begin to

decay on a massive scale . This allows for free electrons and protons to combine with

other particles . At this period of time , the radiation is so dense (1014g/cm3) that no light

is visible .

 Photons , neutrinos , electrons and quarks began to form as the Universe expands and

cooled . After the universe had cooled to about 3000 billion degrees Kelvins , composite

particles such as proton and neutrons become the common state of matter . After three

minutes and a temperature of one billion degrees , protons and neutrons were slowing
down enough in order to allow the formation of light elements( light-element

nucleosynthesis ) .

 For the next 300000 years , the Universe expands and cools to a temperature of 10000

Kelvins and this allow the helium nuclei to absorb free-floating electrons and form

helium atoms . Meanwhile hydrogen atoms were bonding together and forming lithium .

It is here that the density of the universe has expanded to the point where light can be

perceived .

 A brief summary of the most significant events :

      10-4 seconds: Baryogenesis occurs , quarks condense under strong interaction to
       form nucleons (e.g., Protons and Neutrons) .
      1 second: Nucleosynthesis occurs , universe cools enough (photon energies ~ 1
       MeV) for light nuclei to form (e.g., deuterons, alpha particles) .
      104 years: Radiation density becomes equal to matter density , since the radiation
       density has extra factor of a-1 due to red-shifting . Matter density is the dominant
       energy density after this epoch .
      105 years: Recombination occurs and electrons are combined with nucleons to
       form atoms . This time also coincides with the decoupling of photons from matter
       , giving rise to a surface of last scattering of the cosmic background radiation .
      1010 years: The present .

 After an introduction to the origin of the Universe , we will move on to derive some of

the equations used in this discussion and what they are trying to explain .


                      Equations and their meanings

 In this section , we will be looking at the equations we will be using in the later sections

and try to explain their meanings .

Friedmann equation

 The Friedmann equation describe the expansion of the Universe . Friedmann developed

it as a relativistic equation in the framework of general relativity but here , we will
derive it using Newtonian mechanics and try to explain the meaning of the equation with

the help of relativity .

For simplicity sake , we consider a sphere with a uniform density ,  ,that is expanding

uniformly in all directions . Assume that there is a particle of mass , m , at a distance r

from the center of the sphere . We know from Newtonian mechanics that the potential

energy is given by

                                   P.E = - GMm/r                                          (1)

 Where M is the mass of the sphere that has an influence on the particle and it has a

value of 4/3r^3 and G is the gravitational constant . We have the kinetic energy of the

particle as

                                 K.E = ½ m( dr/dt )^2                                     (2)

 Using ( 1 ) and ( 2 ) together , we have the total energy

                             T.E = ½ m( dr/dt )^2 – 4/3Gr^2m                            (3)

 We can generalize this to a particle in the Universe as the Universe looks the same

everywhere , so we can consider any place to be its center . ( 3 ) gives evolution of the

separation , r , between any 2 particles since the Universe is homogeneous and isotropic .

 We let the relation between r and the comoving distance x be

                                         r = a(t)x                                        (4)

 Where a(t) is a function of time . The galaxies are in fixed positions in a comoving

coordinate system( comoving coordinates are coordinates with respect to which

comoving observers are at rest ) . a(t) is also known as the scale factor of the Universe

and it measure how the physical separations , r , varies with time .

 Substituting ( 4 ) into ( 3 ) , we can make T.E to be in terms of a(t) . We get
                    T.E = ½ m( da/dt )^2 .x^2 - 4/3Gm( da/dt )^2.x^2

 Next , we multiply both sides by 2( ma^2.x^2 )^-1 . Rearranging the terms , we have

                     ( da/dt )^2/( a^2 ) = 8/3G + 2T.E/( ma^2.x^2 )

 Let kc^2 = -2T.E/( mx^2 ) , we get

                          ( da/dt )^2/( a^2 ) = 8/3G - 2kc^2/( a^2 )                       (5)

 ( 5 ) is the Friedmann equation . Here , a and d are constants and since x , the comoving

separation is a constant by definition , this concludes that k is a constant . For a given

Universe , the value of k determines it‟s geometry and stays constant throughout .

 If we use general relativity to derive the Friedmann equation , we can see that the

constant k tells us that gravity is a result of the curvature of space-time . It can only take 3

values :

k =0            flat Universe

k =1            closed Universe

k = -1          open Universe( deSitter Universe )


 A graph below shows how the Universe will progress for different values of k . Here ,

we take the cosmological constant to be 0 for simplicity sake .
Geometry            q0     Fate of Universe         Name

Flat            =1 1/2     Open Universe            Einstein-DeSitter Model
Hyperbolic      <1 <1/2    Open Universe            Open Model
Spherical       >1 >1/2    Closed Universe          Closed



 Shown below is a 3 dimensional view of how each model will look like .
 For the flat Universe , it follows the Euclidean geometry that is :

 1) the shortest distance between 2 points is a straight line

 2) the sum of interior angles in a triangle is 1800

 3) the circumference of a circle is 2R , where R is the radius

 4) 2 parallel lines remain at a fixed distance apart

 For the flat Universe , it is static and it has no edges( it is infinite ) .

 For the spherical Universe , it has the following properties :

 1) the shortest distance between 2 points is a curve

 2) the sum of interior angles in a triangle is greater than 1800

 3) 2 parallel lines converge

 4) the circumference of a circle is less than 2R

 The spherical Universe is a finite Universe with no edge .

 For the hyperbolic Universe :

 1) the shortest distance between 2 points is a curve

 2) the sum of interior angles in a triangle is less than 1800

 3) 2 parallel lines diverge

 4) the circumference of a circle is greater than 2R

 The hyperbolic Universe is the opposite of the spherical Universe and it will expand

forever .

The conservation equation

 To solve the Friedmann equation , we need to relate the density of the materials in the

Universe ,  , to the pressure of the materials , p . The equation that relates this two

quantities together is the conservation equation .
 We will use the first law of thermodynamics to derive the conservation equation .

Assuming that the expansion of the Universe is adiabatic( that means no heat is loss or

gained in the process ) , we have from the first law of thermodynamics :

                                    dQ = dE + pdV = 0                                      (6)

 Here , dQ is the rate of change of heat , dE is the rate of change of energy , p is the

pressure and dV is the rate of change of volume . We have from ( 6 ) ,

                                    dE/dt = -pdV/dt                                        (7)

 Assuming that the volume has a radius b and a uniform density ,  . Using these and

E = mc^2 , we have the following ,

                                      E = 4/3b^3c^2

 Differentiating with respect to time , we have the rate of change of energy to be

                     dE/dt = 4/3b^2c^2d.db/dt + 4/3b^3c^2.d/dt

 and the rate of change of volume is

                                   dV/dt = 4/3b^2.db/dt

 The scale factor R(t)( at time t ) can be related to the scale factor R(0)( at time t=0 ) by

an expansion parameter , a :

                                        R(t) = aR(0)

 Differentiating with respect to time , we have the rate of expansion :

                            dR/dt = da/dt.R(0) = da/dt.( R(t)/a )

 Rearranging the terms , we have ,

                               dR/dt.( 1/R(t) ) = da/dt.( 1/a )

 We recognize this as the Hubble law with the Hubble constant ,

                                       H = da/dt( 1/a )
 da/dt( 1/a ) gives us the rate of expansion of the Universe . After knowing the

significance of a , let us get back to the conservation equation . Since a(t) = a(0).b , we

have

                            da/dt( 1/a ) = a(0)db/dt/( a(0).b ) = db/dt( 1/b )              (8)

 Substituting the expression for dE/dt and dV/dt together with ( 8 ) into ( 7 ) , we get the

conservation equation :

                                 d/dt + 3da/dt(  + p/( c^2 ) )/a = 0                      (9)

 The conservation equation tells us that the density changes as the Universe expands are

due to 2 factors : 1) the dilution in density as the volume increases and 2) loss of energy

as work has been done as the volume increases .

The acceleration equation

 We can combine ( 5 ) and ( 9 ) together to form the acceleration equation . This equation

describes the rate of expansion of the Universe( For an expansion parameter a , the rate of

expansion of the Universe is given by da/dt.1/a ) . Differentiating ( 5 ) with respect to

time , we get :

            2( da/dt )/a^3.( aä – ( da/dt )^2 ) = 8/3Gd/dt + 2kc^2( da/dt )/a^2

 Substitute for d/dt from the conservation equation , we obtain :

                  ä/a = ( da/dt.1/a )^2 = -4G(  + p/( c^2 ) ) + kc^2/( a^2 )

 Referring back to the Friedmann equation again , we have :

                            ( da/dt.1/a )^2 = -4G(  + p/( c^2 ) )                     ( 10 )

The Dominant Energy Condition

 The Weak Energy condition( WEC ) states that the energy density( energy per unit

volume ) measured by any observer must be greater than zero . The Dominant Energy
condition( DEC ) is the WEC with an extra condition : The pressure p must not exceed

the energy density  . We have the equation of the state , p/ . We will look at this in

greater details when we discuss about the Big Smash .


                           Changes in the Universe

 Since the Big Bang , the Universe has been expanding . The meaning that the Universe

is expanding means that galaxies are moving further apart from each other and it‟s not

that the galaxies are moving but the space between them that is expanding . We can use

the Friedmann equation to describe this and to be able to make a comparison between

different models , we have to choose one as benchmark . This will be the flat Universe .

 Now , we move on to find the density of the flat Universe which is also known as the

critical density ,  . This density corresponds to a Universe with zero curvature( k=0 )

and for this type of Universe , its expansion is slowing down and approaches 0 as t

approaches infinity . This is the Einstein-DeSitter model and it will never collapse . The

reason for finding  is that we can use it to compare against the different values of

density we have for each type of Universe . Any value lower than  will result in a

Universe that expands forever and any value higher than it will have the Universe

suffering a collapse in the future .

 Substituting k=0 into ( 5 ) and we define the Hubble constant , H(t) = da/dt.1/a and the

deceleration parameter q(t) = -ä(t)/( a(t).H(t)^2 ) . Substituting into ( 5 ) and ( 10 ) , we

have

                   0 = 2 ä/a + ( da/dt )^2/( a^2 ) = -2q(t)H(t)^2 + H(t)^2

                            ( da/dt )^2/( a^2 ) = H(t)^2 = 8/3G
 After solving for  , we get the critical density .

                                      = 3H(0)^2/( 8G )                                   ( 11 )

 Now , we have to find out if the density of the Universe allows it to expand forever or

will the Universe end up collapsing some day . In the next section , we will take a look at

the Big Crunch theory which describes a Universe that will collapse and argue that a

model described by the Big Crunch theory will portray the fate of our Universe .


                                Big Crunch theory

 The Big Crunch theory assumes that there is enough matter in the Universe to slow

down it‟s expansion and finally , causes the Universe to contracts . This type of Universe

has a positive curvature( k=1 ) and it has a density that is greater than the critical density ,

 .

 It is analogous to a bullet fired from Earth that has a velocity that is less than the escape

velocity . Clearly , the bullet will not be able to escape from the Earth and will attracted

back to the Earth by its gravity in the finite future . Similarly , a Universe that has a

density greater than  will be forced into contraction by its gravity in the future .

  Now , we have to find out if the Universe contains enough matter to achieve the density

needed .

Dark matter

 When we look at the sky , we can see lots of stars with our unaided eyes . Astronomers ,

with the help of instruments can detect other things like dusts , radiation from stars ,

particles as well as other galaxies and whatever is visible to them . However , recent

evidence shows that there could be other matter that lies undetected in our Universe . We
look at some evidence that suggests this .

 A spiral galaxy is constantly rotating and it is held together by the gravity produced by

the amount of matter it contained . We would expect the force of gravity to be the

strongest near the center of the galaxy as that is where the matter appears to concentrate

and deceases with distance . Using Newtonian mechanics , we have the force of attraction

of a star with mass m , at a distance r , from the center of the galaxy to be :

                                F = -GMm/( r^2 ) = -mr^2

 Here , F is the force of attraction , G is the Gravitaional constant , M is the mass of the

matter within the radius r and  is the angular velocity of the star . Rearranging the

equation , we have :

                                      ^2 = GM/( r^3 )

 This tells us that  is proportional to the total mass within the radius r and inversely

proportional to r . Near the center of the galaxy , r is small and M is large as the matter

concentrates at that region . As we move towards the edge of the galaxy , r increases

linearly but there is not much increase in M as the matter thins out along the spiral arms .

This means that  should decrease as r increases .

 However , that is not the case . Astronomers found that the stars at the edge of the

galaxy are moving at the same speed as the ones near the center . If that is the case , the

gravity of the galaxy will not be enough to hold these stars in their orbit and they should

fly out of the galaxy .

 This led to the conclusion that there must be matter that are undetected in the galaxy and

this matter provides the extra gravitational force that helps to prevent the stars , dust and

gas on the edge of the galaxy from being flung into deep space .
 Dark matter is not confined to spiral galaxies . Clouds of hot gas are found to surround

elliptically shaped galaxies and the gravitational force produced by the visible matter in

the galaxies is not sufficient to keep the gas from evaporating . Compare this to the Earth

and moon . The gravitational force provided by the moon is not sufficient to retain

whatever atmosphere there might be on it. On the other hand , Earth is much more

massive and this allows it to have an atmosphere .This suggests that dark matter exists in

these galaxies .

 Dwarf galaxies are galaxies that are smaller than spiral galaxies . They are believed to

be the most common type of galaxy in the Universe . Observations show that there are

seven such galaxies surrounding the Milky Way and two of these are pretty close .

 Since they are small in size , they have a small gravitational field compared to Milky

Way and they should be torn apart by the gravitational force exerted by our galaxy long

ago . The fact that they did not suggests that they must contain much more mass than can

be seen and this too suggests that dark matter exists in such galaxies .

Candidates for Dark matter

 In this section , we move on to discuss about the possible candidates for dark matter .

Stars produce radiation through the process of nuclear fusion . Since nuclear fusion works

under extreme conditions , the more gigantic the star is , the more efficient the process

will be . This means that astronomers will have to search for stars that are small in size

and other matter which does not produce electromagnetic radiation efficiently . A few

options are being ruled out .

 Dust seems attractive but if it is the dark matter we are searching for , it would have to

fill up a large portion of a galaxy to produce the desired gravitational effect and it means
that optical astronomy would not be possible as it will block the visible light passing

through it . Hot gas could be detected by ultraviolet and X-ray telescope and cold gas , by

radio telescope so they are not suitable candidates either . We will proceed to discuss

about the more suitable candidates for the dark matter .

White dwarf

 A white dwarf is a small and dense star that was formed by the collapse of a star

which has the size of our sun . When the star exhausts it fuel , it collapses to form a lump

of matter which has a size similar to Earth . Due to their size , they are luminosity is

pathetic compared to that of the sun but a teaspoonful of it would weigh about ten tons .

If a large number of white dwarfs are formed when a galaxy is still young and over the

period of time , exhaust whatever fuel they have and cooled to invisibility , they could be

used to account for the dark matter .

Neutron stars

 When stars that are five or more times more massive than our sun exhaust their fuel

and collapse , their diameter dropped from about a hundred thousand miles to about ten

miles . This indicates that the density of a neutron star is extremely dense and in fact , it‟s

density is about ten million times that of a white dwarf .

 To have neutron stars as an ideal candidate , we have to assume that a large number of

neutron stars were produced when the galaxy is young and forming but the problem is

that when gigantic stars collapse to form neutron stars , they release a lot of energy and

this results in a catastrophic explosion( supernova ) . These explosions could interfere

with the formation of the galaxy and the galaxy might not even be formed .
Black holes

 When a star with the mass about twenty times greater than that of our sun collapses and

explodes , the gravitational force produced will be so intense that even light could not

escape from it . In the surrounding area of a black hole , nothing escapes from it and it is

truly dark . The only way to detect its presence is through its gravitational field and its

effect on neighboring stars .

 However , the number of stars which has the mass that is massive enough to form black

holes are observed to be far less than the ones that would become neutron stars .

Low mass stars

 Red and brown dwarfs are called low mass stars as they have mass that is a tenth that of

the sun or less . One example of a red dwarf star is the Proxima Centauri , the star that is

nearest to us . Since they have small mass , the nuclear reactions in these stars do not

function as effectively as they do in the sun so they are very dim .

 Observations show that there are many red dwarfs around . About two third of the

ninety or more stars near the sun are red dwarfs .

 Brown dwarfs have mass that is even less than that of red dwarfs . Due to their mass ,

nuclear fusion cannot occur in them and the only way to detect their presence is by the

heat generated by their slow gravitational contraction .

 Some astrophysicist suggest that the conditions at the time when galaxies are still

forming were just right for large number of brown dwarfs to be formed . However ,

brown dwarfs emit most of their energy at infrared wavelengths as they are cool . That

means that we have to search for them using space based equipment as radiation of such

wavelength will be absorbed by the atmosphere .
 Searches have been carried out and over the years , astronomers found that brown

dwarfs could be as common as stars .

Cosmions

 Cosmions are particles that are believed to be formed under the extreme conditions that

are present when the Universe is very young . They interact very weakly with matter and

neither produce nor absorb electromagnetic radiation . Some types of cosmions are

discussed below .

Neutrinos

 Neutrinos are particles with little or no mass . The Big Bang theory predicts that there

are roughly a billion times more neutrinos than protons in the Universe and since

neutrinos interact very weakly with matter , they could be the dark matter we are looking

for .

 However , neutrinos are moving too fast and a Universe dominated by it will not be the

same as ours . To trap neutrinos , the galaxies must be very massive and this means that

the dwarf galaxies could not have enough neutrinos to account for the dark matter . This

is because neutrinos obey the Pauli exclusion principle and since lots of neutrinos will be

packed into a small space , all the low energy quantum levels will be filled up and those

occupying the higher energy levels will escape from the galaxies due to their weak

gravitational field .

Photinos

 Photinos are theoretical particles also known as Cold Dark Matter( CDM ) that are

moving at a speed that is much slower than neutrinos . They interact very weakly with

matter and they could have been produced in a large amount when the Universe is very
young and have sufficient mass to account for the dark matter .

 Experiments to verify the existence photinos like detecting the neutrinos produced by

photino interactions in the Earth and using space telescopes to search for particles such as

cosmic ray antiprotons and positrons produced in the galaxy halo by photino interactions

are being carried out . However , they still remain theoretical .

Will the Universe ever collapse
 The existence of dark matter suggests that the density of the Universe is higher than

previously thought and this leads us to wonder if the density is high enough to cause the

Universe to collapse . The answer is NO ! We‟ll look at the evidence that backs up this

claim .

Density of the Universe

 To know the total mass contained in the Universe , we can make use of several methods

. We shall start with the mass-to-light ratio method . In this method , we first measure the

average mass to light ratio( M/L ) of the largest possible galaxy and multiply this result

by the total luminosity density( L/V ) of the Universe . This gives us the density of the

Universe . The density measured using this method is around 1/3 of the critical density .

 Next , we move on to using the Baryon fraction method . This method measures the

density of the Universe by measuring the ratio of the baryonic mass to the total mass in

the galaxies . The ratio is found to be roughly 0.15 and from the big bang model , we

have the baryon density to be 0.045 ± 0.0025 . With these information , we have the

value of the density of the Universe to be 0.3 ± 0.1 .

 We get the density of the Universe to be around 30% of the critical density from 2

independent measurements thus , we claim that there is not enough mass in the Universe
to allow it to collapse .

Expansion of the Universe

 In this section , we will refute the Big Crunch theory by showing that the expansion of

the Universe is accelerating . From the second derivative test , the Universe has to be

decelerating now if Big Crunch is going to take place .

 We can measure the expansion rate of the Universe using the relation between distance

and redshift . If a body is found to be further than expected for a flat Universe , it means

that the expansion rate is increasing . Likewise , if it is nearer , it shows the opposite .

 Objects of known luminosity are used and we use the inverse square law , F = L/( 4d^2

) . Here , F is the flux observed at Earth , L is the luminosity of the object and d is the

distance apart .

 The Supernova Cosmology Project and the High-Z Supernova Search are carried out to

collect the relevant data of supernova . Different analysis techniques and different

samples of supernovae are used and the results show that the supernovae are more

distant than expected for a decelerating Universe thus , it appear that the Universe is

accelerating .

Curvature of the Universe

 The curvature of the Universe can be measured from the cosmic microwave

background( CMB ) . The CMB spectrum gives a measure of the inhomogeneity in

matter and energy at very high red-shifts which corresponds to a few hundred thousand

years after the Big Bang .

 When radiation and hot gases of baryons oscillates , on the scales corresponding to the

sound horizon( the maximum distance pressure wave can travel from the creation of the
Universe to the time the CMB is emitted ) , it has time to undergo maximum collapse but

not rebound . Thus , a peak in the power spectrum corresponding to the sound horizon is

expected on the angular scale .

 The physical length corresponding to the sound horizon is affected by the curvature of

the Universe only . For a flat Universe , it has an angular diameter of about 0.50( this

gives a power spectrum peak near l  200 , here , l is the integer multipole moment and a

given value of l corresponds to an angle of /l radians ) and it is larger and smaller for

close and open Universe respectively .

 Recent measurements show that the value of l  215 and this indicates that the Universe

is flat .

Conclusion

 From the discussion above , we conclude that the Universe will not collapse as any one

of the above argument will refute the Big Crunch theory . Since a closed Universe is not

possible and a static Universe is also being ruled out , we will proceed to the next model :

the deSitter Universe .


                                   Big Smash theory

 Latest observations revealed that we live in a spatially flat Universe that has a

density that is lower than the critical density and is accelerating . Models that are used to

describe this situation focus on the cosmic energy having negative pressure and with the

equation of state , p/  -1 .

 This is to satisfy the Dominant Energy Condition( DEC ). In this section , we will be

looking at something different . We are going to discuss what happens when  < -1 .
 We assume that the Universe is dominated by stuffs with a phantom energy component .

This energy component has positive energy density  , but negative pressure p , such that

p +  < -1( this violates the DEC which requires that p +   0 and  > 0 ) . This satisfies

the equation of state  < -1 .

 From ( 9 ) , we have the conservation equation . We let c=1 , we have :

                                  d/dt + 3da/dt(  + p )/a = 0

 We substitute  < -1 into the equation above , we end up with :

                                      d/dt > da/dt.( 1/a )

 We can rewrite this as :

                                           d/da > 1/a

 Solving for this , we obtain :

                                             > ln( a )                                 ( 11 )

 Since the Universe is expanding , da/dt is positive so we can see that the rate

of change of density of the Universe is an increasing function.

 Now , we have to show that the density and pressure approaches infinity at finite time

 With the constancy of , we can integrate the conservation equation and get ( 6 ) of [7]

                                   = K[ - + ( 1 +  )( t/T ) ]^2                      ( 12 )

 Here , K is a positive constant and we can see that when t approaches the value

                                  t = ( /( 1 +  ) )/T                               ( 13 )

we have the density approaching infinity at a finite time and we use the relationship

between pressure and density to conclude that the pressure will also have an infinite value

in a finite time in a Universe where the DEC is violated and this results in the Universe

being shattered into disconnected pieces in the future and this is termed the Big Smash .
 Taking (U) = M/V( density = mass/volume ) , we can see that the mass of the Universe

increases as the volume increases and at a faster rate . This implies that energy is created(

E = mc^2 ) as the Universe expands and this violates the law of conservation of energy

which states that the total energy of an isolated system is always conserved . This

seems to suggest that the law of conservation of energy is violated in such a Universe .

 The model we have discussed above is the one proposed by Caldwell . We have from

classical mechanics , the speed of sound , v = ( dp/d )^1/2.c = ( dp/d )^1/2( here , we

take c=1 ) . In the Caldwell Universe , DEC is violated thus , the pressure p exceeds the

density  and we have  < -1 .

 Since p =  ,  dp/d  =    > 1 , this implies that ( dp/d )^1/2 > 1 and we have

shown that the speed of sound will exceed that of light in such a Universe .

 Indeed , Big Smash is something that is very bizarre and here , we will compare 2

cosmological models with  < -1 . R. R. Caldwell proposed one that will end up in a

Big Smash while the other is by Brett McInnes that has no Big Smash .

Smash free model
  In this section , we will give a verbal description of the Smash free Universe and

compare it with Caldwell Universe .

 The size of the Smash free Universe is a power of cosh( kt ) which is a monotonically

increasing function with k being a constant , t is the time and cosh( kt ) = ( e^kt + e^-kt

)/2 . Taking the derivative of cosh( kt ) with respect to t , we have :

              d/dt( cosh( kt ) ) = d/dt(( e^kt + e^-kt )/2 ) = ( k( e^kt – e^-kt )/2 )    ( 14 )

 We obtain the minimum value of the function by setting ( 12 ) to be equals to zero .

                                    k( e^kt – e^-kt )/2 = 0
 implies

                                            t=0.

 Thus , the size of the Universe obtains a minimum value when t = 0 and the minimum

value can be very close to zero and this makes the start of the Universe indistinguishable

from the Big Bang . However , it will not have a Big Bang as the size of the Smash free

Universe will never be zero though it will be close to it .

 For Caldwell Universe , the value of  is less than –1 and it is a constant while for the

Smash free Universe ,  is not a constant though it is less than –1 , this is very important

as it is only with a non constant  that a Smash can be prevented . As  is a function of

time , we can make the value of  approach –1 as time approaches infinity . The proof is

as follows , using ( 19 ) , ( 20 ) and ( 21 ) from [7] , we have

                               = 3/( 8L^2 ) - 3/( 8L^2 )a^-

 and

                        p = -3/( 8L^2 ) – ( /3 – 1 ) 3/( 8L^2 )a^-

 Here ,  = 3( 1 +  ) and

                                 a = cosh( t/( 2L ) )^( 2/ )

  = p/ = ( -3/( 8L^2 ) – ( /3 – 1 ) 3/( 8L^2 )a^- )/( 3/( 8L^2 ) - 3/( 8L^2 )a^- )

 We can further simplify the equation above to obtain

                                = -1 – ( /3 )cosech( t/2L )^2                       ( 15 )

 As t approaches infinity , a approaches infinity and since cosh( x )^2 – sinh( x )^2 = 1 ,

we have sinh( x )^2 approaching infinity . Cosech( x )^2 = 1/( sinh( x )^2 ) implies that

cosech( x )^2 approaches zero as t approaches infinity . From this , we can see that the

value of  approaches a limiting value which is –1 .
 This gives us something that looks like a cosmological constant that describes an open

Universe( deSitter Universe ) . To get a Big Smash , we must have the scale factor a(t) to

be infinite in a finite time but if a cosmological constant is present , a(t) is proportional to

cosh( t/L ) and cosh( t/L ) only approaches infinity as time approaches infinity . In this

case , a(t) is a power of cosh( t/L ) and thus , a Big Smash is prevented .

 For the Smash free Universe discussed above , it was very large in the past , contracts to

a minimum size and expands again . We are currently living in the period of time where

it is undergoing an expansion .


                                       Conclusion

 After discussing about the different ways in which the Universe can end , we will end

off the discussion by deciding on a model that best describes the fate of the Universe .

One question we must answer however , is that , will our prediction of the fate of the

Universe be accurate ?

 Since we do not know exactly the present composition of matter in the Universe and

future transformations between different kinds of matter and the initial conditions of the

Universe , any prediction we make about the Universe now will not be accurate in the

future . This implies that a model that we believe will best describe the Universe now

may not be so in the future .

 Nevertheless , we can try our best to describe the fit a model to the Universe now and

improve on it when the need arises .

 Using the data we have at hand and assuming that the conservation laws holds , the

simplest model that can be used will be the deSitter Universe . In this Universe , the rate
of expansion will increase while the amount of energy in it remains constant . As time

approaches infinity , the Universe will become flat and its density goes to zero .

 We could be wrong in our choice of model , as there could be other more bizarre ways

for the Universe to end and this , only time can tell .


References

[1] A.A. Starobinsky , “ Future and Origin of our Universe: Modern View ” , astro-
    ph/9912054 ( 1999 )

[2] Brett McInnes , “ The DS/CFT correspondence and the Big Smash ” , hep-th/0112066
   ( 2001 )

[3] Gary Scott Watson “ An exposition on inflationary cosmology ” , astro-ph/0005003
   ( 2000 )

[4] George F.R Ellis and Ruth M. Williams “ Flat and curved space-times ” . Clarendon
   Press ; New York : Oxford University Press , ( 1988 )

[5] Neta A. Bahcall , Jeremiah P. Ostriker , Saul Perlmutter and Paul J. Steinhardt
    “ The Cosmic Triangle: Assessing the state of the Universe ” ,
    astro-ph/9906463 ( 1999 )

[6] P.James E.Peebles , David N.Schramm , Edwin L. Turner and Ricard G. Kron .
     The evolution of the Universe .
     http://130.94.24.217/specialissues/0398cosmos/0398peebles.html

[7] Peter Coles and George F.R Ellis “ Is the Universe Open or Closed ? ” New York :
   Cambridge University Press, ( 1997 )

[8] R.R Caldwell , “ A Phantom Menace ? ” , astro-ph/9908168

[9] S.W.Hawking and G.F.R.Ellis “ The large scale structure of space-time ” .
Cambridge University Press , England ( 1973 )

[10] Wallace Tucker and Karen Tucker “ The Dark Matter : contemporary science's quest
for the mass hidden in our universe ”. New York : Morrow , ( 1988 ) .

				
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