thesis_andyl.pdf

Effects of Massless Conformally-Coupled Fields on

Future Classical Singularities - A Semiclassical

Approach

Presented before the faculty of the Wake Forest University

Department of Physics as a requirement for graduating with

the distinction of Honors in Physics

Andrew J. Lundeen

Summer 2015

Approved by

Eric D. Carlson

Paul R. Anderson

Gregory B. Cook

2Contents

I. Introduction 3

II. Background 4

III. The Order-Reduced Method 7

IV. A Significant Feature of All < 0 Cases 8

V. The = 0 Case: Nearly All Models Evolve to a Type II Singularity 9

VI. The Type I (Big Rip) Singularity and the Little Rip Singularity 11

A. The < 0 Case 12

B. The > 0 Case 13

1. Proof That the Big Rip and Little Rip Singularities Are Avoided for > 0 14

VII. The Type III Singularity 16

A. 1 < B 32

16

B. 32< B 6= 2 18

C. B = 2 19

VIII. The Type II and Type IV Singularities 20

IX. Discussion of Results and Significance of Findings 23

X. Acknowledgments 23

References 24

3I. INTRODUCTION

In conflict with previously commonly-held expectations, the universe is not slowing down

in its expansion, but instead is currently accelerating in its expansion according to recent

observations [1]. Normally, one would expect the universe to be slowing in its expansion

due to gravity, but the evidence seems to indicate that the contrary is occurring. The

problem of understanding the mysterious cause that must be driving the universes outward

acceleration, and the goal to better understand the implications of such an effect on the future

of the universe are both focuses of current research. The latter concern will be considered

in this paper. First, though, it is helpful to consider some history of research in cosmology

to better place this work in context.

In 1929, Edwin Hubble showed that objects in the universe seem to expand at a rate that

is proportional to their separation distance [2]. Hubble observed the brightness of stars called

Cepheid variables and compared this to their known luminosity. Such a comparison allowed

him to measure their distance from Earth. Hubble also observed the rate of recession (or

approach) of the Cepheids host galaxies by measuring the galaxies redshift (or blueshift).

Combining the data, Hubble was able to plot the recession rate (or rate of approach) of

the galaxies as a function of their distance from Earth. He found the relationship to be

approximately linear. The slope of this line is now known as Hubbles Constant, H0. This

constant measures the current expansion rate of the universe. Cosmologists believe, however,

that the universes rate of expansion is not constant, and in fact there is convincing evidence

to suggest that it is currently increasing. In other words, the universe is presently accelerating

its expansion.

The aforementioned evidence comes from observations of stellar eruptions known as type

Ia supernovae [3]. These supernovae are similar to Cepheid variables in that both are stan-

dard candles (their peak luminosity is known to a relatively high degree of precision), but

these supernovae are much brighter. In fact, type Ia supernovae are visible at distances of

billions of light years, since their luminosity is comparable to the luminosity of entire galax-

ies. Extending Hubbles method to these supernovae, one finds that the recession velocity

is slower for large distances (the brightness of the supernovae are larger than they a lin-

ear Hubble relationship would predict). In other words, observations of type Ia supernovae

indicate that the universe is currently accelerating outward in its expansion. This is odd,

4because all baryonic matter (ordinary matter) and so-called dark matter (whose compo-

sition is unknown) should cause the expansion rate of the universe to slow due to gravity.

This suggests that something besides ordinary and dark matter exists in the universe that

causes the universe to accelerate its expansion, and this so-called dark energy is currently

the dominant component of the universe.

More evidence exists that indicates the existence of dark energy [1]. Observations of the

Cosmic Microwave Background Radiation (CMBR) from the Big Bang show that the CMBR

is nearly uniform in every direction. This, along with the evidence that the universe on a large

scale is flat, requires that the ratio crit

be very close to one, where is the energy density

of the content of the universe, and crit is the critical density (H20 8pi3 crit). However,

current measurements of using only the energy content of ordinary and dark matter yields

.27, or 27% of critical density. This suggests that the remaining 73% is somethingbesides ordinary or dark matterthe aforementioned currently dominant component of the

universe, dark energy.

What dark energy is is not known. The universe accelerates its expansion if the equation

of state relating the pressure P to the energy density satisfies w P< 1/3. The central

aim of this paper is to investigate equations of state of particular interest, namely ones that

lead to future singularities (divergences in the size of the universe a, the Hubble rateH a/a,the energy density , and others). We investigate whether or not including a quantum term

in the Friedmann equation removes or softens these singularities. In particular, we examine

the effects of massless, conformally-coupled fields when the universe undergoes rapid change,

such as rapid change in its scale factor a or its time derivatives. Similar work has been

explored in detail in the literature [46], but this paper is novel in that the conclusions are

more general, and we attempt to examine every possible interesting case.

Throughout, we use units such that ~ = c = G = 1, and we employ the conventions of

Misner, Thorne, and Wheeler [7].

II. BACKGROUND

In order to model the expansion of the universe, an equation of state relating pressure to

energy density is needed. However, since the nature of the dark energy is unknown, several

viable models exist that could accurately describe the future expansion of space. Let the

5equation of state be Pc = Pc(c), where Pc is the pressure and c is the energy density of

dark energy. The type of function used to model Pc determines the dependence of c on the

scale factor of the universe a through the equation

c = 3H [c + Pc(c)] , (2.1)

where a dot denotes a derivative with respect to time, and H is the Hubble expansion rate,

defined as

H aa. (2.2)

Notice that in Eq. (2.1), if c + Pc(c) < 0 for all values of c, then despite the fact that

the universe is expanding and the matter in the universe is separating with it, the dark

energy density increases with time. If the dark energy density increases without bound it

can result in a future singularity. Assuming the dark energy is the dominant contribution,

one can then solve for the scale factor a by solving the classical Friedmann equation for a

flat universe,

H2 =8pi

3c, (2.3)

to determine a(t).

Many forms of Pc(c) exist that could each accurately describe the dark energy. However,

many models cause divergences in a, , P and/or P in finite (or in one category, infinite)

time. These equations of state are said to contain future singularities. Table 1 classifies the

main types of classical cosmological singularities. Note that these model equations are simply

modelsthere are many viable models that can construct the future classical singularities

we investigated. Note that we are interested in the behavior of the universe only near the

singularity. Our model equations of state should not be taken literally except near the

singularity.

For the Big Rip singularity model with B = 1, we can write Pc = wc, where w =

1 A < 1. This is a significant parameter value for B, since Pc = wc is one of themost common models currently in the literature, with w being experimentally measured to

be very close to 1 [1].In all cases except for the Little Rip singularity, the singularity is reached within a finite

time. Singularities that occur only in higher time derivatives can easily be created by using

6Type of Singularity Divergence When Model Equation Parameters

Type I/Big Rip a, , P FinitePc = c ABc

12 0 B > 1

Type II P Finite Pc = cA|sc|B B < 0Type IV P Finite A > 0 0 < B 0 it is unstable and highly sensitive to the

initial conditions. Because of this, there is a problem in figuring out how to choose initial

conditions so that the appropriate solution may be found. To resolve this issue, we first

solved the Friedmann equation using a method that I will refer to as the order-reduced

approach, but is also commonly known as the Parker-Simon method [8, 9].

III. THE ORDER-REDUCED METHOD

The order-reduced approach is a perturbative expansion where we treat the quantum

terms as corrections to the classical energy density. In this approach, one solves Eq. (2.3) for

H and then substitutes this expression in each of the terms of Eq. (2.4), so that q f(Hc),where Hc is dependent only on the form of the classical energy density contribution. In other

words, the quantum energy density contribution is now written in terms that depend only

on a and a. Next, using this expression for q, one can solve Eq. (2.8) by specifying only one

initial condition. The solution is stable, and the initial condition should simply match the

classical solutions initial condition.

The following equations summarize how to get the order-reduced solution to the Fried-

mann equation. First, rewrite Eq. (2.1) as

dcda

= 3a

[c + Pc(c)] (3.1)

8and solve to find c(a). Next, insert this into the classical Friedmann equation to find an

expression for H

Hc =

8pic

3. (3.2)

Now write the quantum contribution in terms of this Hubble parameter

q = (2HcHc Hc2 + 6HcHc2) + Hc4. (3.3)

Lastly, solve the Friedmann equation using this expression for the quantum contribution

H2 =8pi

3[c + q(c)] . (3.4)

Matching the initial value of a to the initial value of a in the classical solution specifies the

solution.

The order-reduced method works well while the perturbations are small. Since the solution

is stable and reliable until the quantum effects become significant, the order-reduced solution

is able to guide the initial condition choices in the third-order full Friedmann equation before

the singularity is approached.

IV. A SIGNIFICANT FEATURE OF ALL < 0 CASES

Well now consider a simple proof by contradiction that the universes size increases

without bound when < 0. Begin by assuming that the universe will reach a maximum

size. When the universe reaches a maximum size, H = 0. Therefore, Eq. (2.8) with the

quantum energy density Eq. (2.4) simplifies to

0 =8pi

3(c + ||H2). (4.1)

Since both terms on the right hand side of this equation are positive, it must not be possible

for H to equal 0. It follows that if H > 0 initially, H will always be strictly positive.

This proves that a(t) is an increasing function in all cases when < 0. Since a is always

increasing, quantum effects will not allow us to evade the singularity.

9V. THE = 0 CASE: NEARLY ALL MODELS EVOLVE TO A TYPE II

SINGULARITY

When the parameter in Eq. (2.4) vanishes, Eq. (2.8) simplifies to

H2 =8pi

3(c + H

4). (5.1)

Solving it, we find

H2 =39 256pi2c

16pi. (5.2)

We have chosen the minus sign because this gives Eq. (2.3) in the limit that c is small.

Clearly, there is a value max 9256pi2 which makes the argument in the square root 0; if were to exceed this value, the value of H2 would become complex.

If the equation of state is that of a Type II or Type IV singularity and if s < max,

then the singularity at c = s occurs before max is reached. In this case, H2 is a smooth

function of c in the neighborhood of s, so that any singularities resulting from the classical

equation H2 = 8pi3c will be manifested in the quantum equation as well.

However, if s > max (or if c diverges), then c will attain the value max; it follows

from Eq. (5.2) that H2|max H2max = 316pi . However, since

cH2 =

8pi9 256pi2c

, (5.3)

then

cH2|max =. (5.4)

Since H is finite when c = max, this implies that H = here. However, if H diverges,this means that P and its derivatives diverge. This is precisely what characterizes the Type

II singularity. Hence, for = 0, the only equation of state that does not cause the universe

to evolve to a Type II singularity is the Type IV singularity model with s < max.1

Next, we seek to derive a function a(t) that describes the scale factor of the universe in

the neighborhood of max when = 0 model-independently. Starting with Eq. (5.2), we

1 Even for the Type IV singularity model with s < max, the universe experiences a Type II singularity at

= max after it first experiences the Type IV singularity at = s.

10

take the time derivatives of both sides, and we make a substitution for the c term using

Eq. (2.1) to get the expression

H =12pi

9 256pi2c(c Pc). (5.5)

Next, we approximate the value of c by using a Taylor expansion of c(t) around the

point t = tmax to first order:

c max + c(t tmax) = max 3Hmax(max + Pmax)(t tmax), (5.6)

where Pmax Pc(max). Approximating c+Pc = max+Pmax in Eq. (5.5), and substitutingEq. (5.6), we find

H 3(max + Pmax)

16Hmax(tmax t) (5.7)

As expected, the analytical model predicts that H diverges as t approaches tmax. Inte-

grating this equation, we find that H approaches a finite value, namely

H Hmax 3(max + Pmax)(tmax t)

4Hmax. (5.8)

Solving this for a, we find that

a amax exp[Hmax(tmax t) +

(max + Pmax)

3Hmax(tmax t) 32

]. (5.9)

This describes what happens to the universe when quantum effects are taken into account

when = 0 for all classical equations of state that have future singularities. However, for

the cases > 0 and < 0, the analysis is far more difficult and must be examined separately

and type-by-type. First we will examine the Type I singularity and the closely-related Little

Rip singularity.

11

VI. THE TYPE I (BIG RIP) SINGULARITY AND THE LITTLE RIP

SINGULARITY

As described in Table 1, the Type I (Big Rip) and Little Rip singularities are modeled by

the equation of state

Pc = c ABc , (6.1)

where A > 0, 12 0. Substituting this into Eq. (2.3) yields, for B 6= 12 ,

a = a0 exp

{(3BA

) 112B 1

1B[(2B1)

2pi(tst)

] 2B22B1

}, (6.3)

c =[A(2B1)

6pi(tst)

] 22B1

. (6.4)

When B = 1, we again have the Type I singularity, but the equation for the energy density

instead is described by

c = ca3A , (6.5)

where c > 0, and the corresponding equations in terms of time are

a =[A

6pic(ts t)]23A. (6.6)

c =1

6piA2(ts t)2 . (6.7)

Fig. 1 illustrates the classical behavior of a for the Big Rip and Little Rip singularities.

(The Little Rip is considered a singularity because the energy density diverges.) To

investigate what impact the quantum contribution has on the universes evolution for these

types, we first need to choose the sign of . First we will look at the < 0 case.

12

FIG. 1: In the Big Rip singularity (left), the scale factor diverges in finite time, while inthe Little Rip singularity (right), it diverges in infinite time. Since each time derivative of

the scale factor also diverges, it follows that , P , and P also diverge. In the Little Rip, thescale factor diverges faster than exponentially, as shown.

FIG. 2: Numerical results from our Little Rip model with < 0. The blue line is theclassical result, the red line is the order-reduced result, and the black lines are the

full-quantum computation with three choices of initial conditions. For the full-quantumresults, the scale factor diverges in finite time independent of initial conditions, which

means that the Little Rip is converted to a Big Rip for < 0.

A. The < 0 Case

The solutions to the Friedmann equation for < 0 are not sensitive to the initial condi-

tions. We used the order-reduced method to help set the initial conditions, but explored a

range of such conditions. For these cases we found that the singularities remain or become

more singular. For the Little Rip, numerical computations indicated that the quantum ef-

fects cause the universe to evolve to a Type I singularity instead for a wide range of initial

conditions.

13

With the numerical results as a guide, we attempted to understand these models analyt-

ically. We conjectured that near the singularity, the scale factor behaves as

a c(ts t). (6.8)

Here, c > 0 is a constant, ts is the time where the singularity occurs, and we postulated

that > 0.

To solve for the value of , we first wrote the Friedmann equation, substituting Eq. (6.8)

for each a. Next, we looked for the terms of the lowest powers on (ts t), i.e. the leadingorder terms, in the q contribution using our postulate that > 0. We discovered that the

quantum terms typically overtake and surpass the classical contribution so that to leading

order the quantum terms must cancel each other. We found that, indeed, > 0 and for

B < 1 it is given by

=3||+92 + 3||

. (6.9)

For B = 1, the equation for c is different and the situation is more complicated. For

(34A + 1)2 < 1 +

3|| , the quantum effects still dominate the classical contribution, and we

found that the value of is still given by Eq. (6.9). For (34A + 1)2 > 1 +

3|| , the classical

contribution can no longer be neglected, but we still find Eq. (6.8) is valid, but with = 43A

.

For the special case (34A+ 1)2 = 1 +

3|| , it is necessary to modify Eq. (6.8) to

a c(ts t)43A [ ln(ts t)]13A . (6.10)

The end result is that, once again, we have a Type I divergence.

In all cases of the Type I and Little Rip singularities for < 0, we found that the universe

evolved towards a Type I singularity.

B. The > 0 Case

For > 0, numerical solutions exhibited interesting results, both for the Big Rip and the

Little Rip. Independently of initial conditions, the universe eventually reaches a maximum

size. Fig. 3 shows typical results. We did not think this was a coincidence, so we looked for

a way to prove this seemingly-generic feature.

14

FIG. 3: Classical (blue), order-reduced (red), and full-quantum (black) for Big Rip (left)and Little Rip (right) cosmologies with > 0. Note that the full-quantum cosmology always

reaches a maximum size, evading the singularity.

1. Proof That the Big Rip and Little Rip Singularities Are Avoided for > 0

We claim that for massless, conformally-coupled fields, self-consistent quantum backre-

action effects avoid the Big Rip and Little Rip singularities when > 0. Well use a proof

by contradiction, assuming a increases without bound, ultimately resulting in conflicting

statements.

The Friedmann equation, Eq. (2.8), can be written as

1

4

(2HH H2 + 6HH2

)=(H2 3

16pi)2 + (A c)

4, (6.11)

where

A 9256pi2

. (6.12)

Now, one can prove that

H3/2d

dt

[1

a

d

dt(aH1/2)

]=

1

4(2HH H2 + 6HH2) , (6.13)

and show that

H3/2d

dt

[1

a

d

dt(aH1/2)

]= H5/2

d

d ln a

[H3/2 +

1

3

d

d ln aH3/2

]. (6.14)

15

By direct substitution, we have that

d

d ln a

[H3/2 +

1

3

d

d ln aH3/2

]=(H2 3

16pi)2 + (A c)

4H5/2. (6.15)

It is important to note that, in Eq. (6.15), no assumptions were made. Let a` be the scale

where c reaches the value A, then since c is an increasing function of a, c > A for a > a`.

Thend

d ln a

[H3/2 +

1

3

d

d ln aH3/2

]< X for a > a` , (6.16)

where X > 0.

Now, let[H3/2 + d

d ln aH3/2

]a=a`

= R. Then integrating Eq. (6.16), we see that

[H3/2 +

1

3

d

d ln aH3/2

]aa`

< [X ln a]aa` . (6.17)

Since H3/2 is positive, this implies

1

3

d

d ln aH3/2 < RX ln (a/a`) . (6.18)

Integrating both sides yields

1

3H3/2 1, we have a Type III singularity. This is

what we shall examine next.

VII. THE TYPE III SINGULARITY

The equation of state that leads to a Type III singularity is given by Eq. (6.1) with 1 0. Substituting this into Eq. (2.1) gives the relation

c =[3A(B1) ln

(asa

)] 11B

(7.1)

where as > 0 is the value of a that marks when the singularity is reached.

Fig. 4 shows the classical behaviors of the scale factor and energy density of the Type III

singularity. The effects of quantum fields on Type III singularities depend on both the sign

of and the value of B.

FIG. 4: The classical Type III singularity scale factor (left) and energy density (right). Thedashed green line denotes the singular value as = 10.

A. 1 < B 32

For < 0 we know that the universe must always increase in size. However, the classical

Type III singularity occurs at a finite value of a, a = as. Therefore, the singular value of a is

always reached. Our numerical results, shown in Fig. 5, suggest that the singularity remains

Type III.

17

FIG. 5: Classical (blue), order-reduced (red), and full-quantum (black) for a Type IIIsingularity with B = 5/4 for < 0 (left) and > 0 (right). Note that for the full-quantum

result, the singularity is inevitably reached for < 0 but inevitably avoided for > 0.

We analytically modeled the divergence to leading order with the expression

a as exp[c

(ts t)]. (7.2)

We found the value of by substituting this in Eq. (2.8), keeping only the most divergent

terms, and found that

=4(B 1)2B 1 , (7.3)

so that the leading order classical energy density term cancels the leading order term in the

quantum energy density term. (Since the order of H2 is not of leading order, the leading

order terms present on the right hand side of the equation must cancel.) By matching the

coefficients of the leading order terms, we found that

c = [(1)(3)]1B2B1

[3A

4(2B1)

] 12B1

. (7.4)

For 1 0 for < 0. The behavior must be

slightly modified if B = 32, but it is still a Type III singularity [10].

However, for > 0, expression (7.4) yields nonsensical results. There is good reason that

this model fails for > 0. E. Carlson found a proof which shows that for this case a reaches

a maximum value instead of reaching as [10]. This is also backed by numerical evidence,

which is shown in Fig. 5.

18

B. 32 0 the singularity can be reached or avoided depending on initial conditions. For both

signs of we found that H no longer diverged at as. However, H remained divergent, as

illustrated in Fig. 6.

FIG. 6: The scale factor (left) and its second time derivative (right) for classical (blue),order-reduced (red), full-quantum (black), and singular value as = 10 (green) for a Type IIIsingularity with B = 7/4 with < 0. Note that for the full-quantum neither the scale factor

nor its first time derivative diverge at the singularity, but the second derivative does,indicating the singularity has been softened to a Type II singularity.

FIG. 7: Classical (blue), order-reduced (red), full-quantum (black), and singular valueas = 10 (green) for a Type III singularity with > 0 and B = 7/4. Note that the

singularity can be reached or avoided depending on initial conditions.

Therefore, we suspected that for values of B in this range, the universe should exhibit a

Type II singularity. Indeed, when B > 32, the contribution of the c term to H is no longer

19

divergent, and hence H is generally dominated by non-divergent terms. To study this, we

modified our expression for a and postulated that instead

a as exp[c1(ts t) c2

2(ts t)2 c

(ts t)

], (7.5)

where c1 and c2 are arbitrary parameters that depend on initial conditions.

Solving this equation by similar methods as before for the value of , we found that

=3B 4B 1 . (7.6)

This equation implies that the value of lies in the range of 1 < < 3, due to the range of

values available for B.

As before, we substituted this value for to find the value of the c term and found that

c =(B 1)3 [c1(B 1)]

B1B

2(B 2)(2B 3) . (7.7)

For 32< B < 2, we find 1 < < 2, and, using Eq. (7.5), it is evident that at t = ts

there is no divergence in a, but there is a divergence in a, which is characteristic of a Type

II singularity. For 2 < B, we have 2 < < 3, which causes a and a to be finite while a(3)

diverges, indicating a Type IV singularity. Eq. (7.7), however, breaks down for B = 2.

C. B = 2

For this case, we modified our model for a to

a as exp[c1(ts t) c2

2(ts t)2 + c

2(ts t)2 ln(ts t)

]. (7.8)

We found the leading order divergent terms all on the right hand side of Eq. (2.8), so we

found an expression for c by matching coefficients. We found that

c =1

6Ac12. (7.9)

Note that Eq. (7.8) implies a still diverges at t = ts, so it is a Type II singularity.

20

A summary of our findings of the effects of quantum fields on the classical Type III

singularity is given below.

Value of B Sign of Quantum Effect

1 < B 32

+ Always avoided Always Type III

32< B 2 + If not avoided, Type II Always Type II2 < B

+ If not avoided, Type IV Always Type IV

This concludes our findings about the Type III singularity. The final singularities that

we investigated are the Types II and IV singularities.

VIII. THE TYPE II AND TYPE IV SINGULARITIES

The Types II and IV singularities share similar characteristics. We modeled them with

the equation of state

Pc = c A|s c|B, (8.1)

where A, s > 0. This yields the energy density equation

c = s sgn(as a)[3A(1B)

ln(asa

)] 11B . (8.2)For the Type II singularity, B < 0, and for the Type IV singularity, 0 < B < 1/2. (If B = 0,

then the equation of state transforms into Pc = c A, which is a particular Little Ripcase.)

Fig. 8 shows the classical behavior of the Type II singularity. The Type II singularity

is characterized by the divergence of P . P diverges when the second time derivative of a

diverges.

Now, as before, we investigate what happens when quantum contributions are included.

Our numerical results indicate that the Type II singularity is softened. Fig. 9 shows these

results for < 0, and Fig. 10 shows comparable results for > 0. For < 0, a must reach

the singularity at as since a cannot reach a maximum. However, our results indicate that

there is no longer a divergence in the second or third time derivatives of a. The singularity

has been softened to a divergence in higher time derivatives of a. For > 0, if as is reached

21

FIG. 8: The Classical Type II singularity. Note that the scale factor (blue) passes throughthe singular value (green).

the singularity is softened in the same wayhowever, initial conditions can be chosen such

that a never reaches as, in which case the singularity is avoided.

Similarly, our numerical results of the Type IV singularity indicate that it is softened. We

do not provide figures for these results because they look similar to Figs. 9 and 10. Again,

for < 0, a must reach as, so the singularity is softened, and for > 0 it is also softened at

as if initial conditions are not chosen such that a reaches a maximum before then.

For both these cases, the singularity (if reached) is softened so the divergence depends

on time derivatives of the scale factor that do not appear in Eq. (2.8), so we consider the

singularities to be effectively removed.

FIG. 9: The scale factor (left) and its second time derivative (right) for classical (blue),order-reduced (red), full-quantum (black), and singular value as = 10 (green) for a Type II

singularity with B = 2 with < 0. Note that for the full-quantum the second derivative ofthe scale factor is continuous, indicating that the singularity has been softened.

22

FIG. 10: The scale factor (left) and its second time derivative (right) for classical (blue),order-reduced (red), full-quantum (black), and singular value as = 10 (green) for a Type II

singularity with B = 2 with > 0.

As before, we examined these singularities analytically, postulating that the expansion

can be approximated by

a exp[c1(t ts) + c2

2(t ts)2 + c3

3(t ts)3 + c

|t ts|

]. (8.3)

Here, c1 and c2 are free parameters that depend on initial conditions. Using this model in

the Friedmann equation and looking to cancel the highest-order terms in the classical and

quantum energy densities, we found that

c3 =c2

2 + 6c12( 1

16pi c2) c14 s

4c1. (8.4)

We also found that

= 3 +1

1B (8.5)

and

c = [3c1A(1B)]1

1B

2c1(1)(2) . (8.6)

For Type II, 3 < < 4, and for Type IV, 4 < < 5. One can show with these ranges

that the divergence is now in a(4) and a(5), respectivelyhence the singularities are effectively

removed.

23

IX. DISCUSSION OF RESULTS AND SIGNIFICANCE OF FINDINGS

With the nature of dark energy unknown, the future of the universe is uncertain. Some

viable models contain future singularities, points where quantities like energy density and

pressure diverge. We found, however, that quantum fields can have a significant effect on

the universe when it experiences a rapid growth. Table II summarizes our findings.

Equation of State Classical Classification ConstraintQuantum Classification

> 0 < 0 = 0

Pc = c ABc ,

Little Rip B 12

AvoidedType I

Type IIBig Rip/Type I 1

2 0 32 0 Type IV 0 < B < 1

2Type IV

TABLE II: Effects of massless conformally-coupled fields on future classical singularities.

We note that by Effectively Removed we mean that the singularities have been softened

to divergences that are likely not physically significant.

Although model equations are given in Table II, these models were only used to create

numerical results which guided the more general analysis. Most of our findings are model-

independent.

To conclude, in many cases massless conformally-coupled fields have a significant impact

on future singularities. Sometimes the singularities are even completely avoided. Future

research could investigate whether non-conformal fields have effects on singularities that

differ qualitatively from conformal fields.

X. ACKNOWLEDGMENTS

Id like to thank Dr. Paul Anderson for the guidance he provided me in this research since

it began in 2013. Id also like to thank the Wake Forest physics department for funding my

efforts in this research in the summer of 2013. Further, Id like to thank the Wake Forest

Research Foundation for funding my work this project in the summer of 2014.

Lastly, I would like to personally thank Dr. Eric Carlson for his support, constructive

feedback, patience, insights, and contributions throughout the work on this project. His

24

mentorship has been valuable to me.

[1] P. A. R. Ade, et al., arXiv eprints arXiv:1502.01589 (2015).

[2] E. Hubble PNAS 15 (1929)

[3] A. G. Riess, A. V. Filippenko, P. Challis, et al., Ap. J. 116, 1009 (1998).

[4] J. D. Bates and P. R. Anderson, Phys. Rev. D 82, 024018 (2010).

[5] S. J. M. Houndjo EPL 92, 10004 (2010).

[6] S. Nojiri, S.D. Odintsov, and S. Tsujikawa, Phys. Rev. D 71, 063004 (2005).

[7] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

[8] J. Z. Simon Phys. Rev. D 43, 3308 (1991).

[9] L. Parker and J. Z. Simon Phys. Rev. D 47, 1339 (1993).

[10] E. D. Carlson, private communication

thesis_andyl.pdf

Documents