universe

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 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  0 ) . This satisfies



the equation of state  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 , 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 ) .