Sup ernatural InationInation from

Sup ersymmetry with No Very Small

Parameters

by

Marin Soljacic

SUBMITTED TO THE DEPARTMENT OF PHYSICS IN PARTIAL

FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

BACHELOR OF SCIENCE IN PHYSICS

AT THE

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JUNE

c

Marin Soljacic All rights reserved

The author hereby grants to MIT p ermission to repro duce

and to distribute publicly pap er and electronic

copies of this thesis do cument in whole or in part

Signature of Author

DepartmentofPhysics

May

Certied by

Lisa J Randall

Asso ciate Professor of Physics

Thesis Sup ervisor

Approved by

June L Matthews

Professor of Physics

DepartmentofPhysics Undergraduate Thesis Co ordinator

Sup ernatural InationInation from

Sup ersymmetry with No Very Small

Parameters

by

Marin Soljacic

May

Submitted to the Department of Physics

on May in Partial Fulllment of the

Requirements for the Degree of Bachelor of Science in

Physics

Abstract

Most mo dels of ination incorp orate small inexplainable parame

ters that do not exist normally in particle physics theories We de

signed a mo del that avoids this it implements ination using only

the parameters that already exist in the particle physics theories We

succeeded this by using two coupled quantum elds instead of using

only one as it is usually done

Using more elds than just one makes the mo del more complicated

and the analysis is harder The prop er analysis of the predictions of

our mo del would therefore b e purely numerical However as we show

the approximate analytical expressions we obtain are in an excellent

agreement with the exact numerical solutions Therefore weusethem

to analyze and calculate its predictions

Thesis Sup ervisor Lisa J Randall

Title Asso ciate Professor of Physics

Contents

BigBang

Justication of BigBang mo del

The problems of the old BigBang mo del

The problem of the domain walls

The monop ole problem

The baryon asymmetry

The horizon problem

The atness of space

Other problems and the solution

Ination

How do es ination do it

Smallscale inhomogeneity

Existing mo dels of ination and our goal

Our mo dels

Where do our mo dels come from

Density p erturbations

Mathematical analysis of the number of efolds

Stage

Stage

Stage

Total number of efolds

Computer conrmation of the number of efolds

Stage

Stage

Stage

The spike in the density p erturbations

Creating the plots of masses

Description of the program

Conclusion

Acknowledgements

List of Figures

The numerical simulation for Stage The example wecho ose

is for M mo del We pick some typical values of the parame

ters M GeV andM

Dierent parameters or mo del dier only quantitatively

The numerical simulation for the development of and the

envelop e of in Stage All the parameters are as for Figure

This particular example is for the mo del with

l

The numerical simulation for the development of E in Stage

corresp onding to the development of and shown in

e

Figure

The numerical simulation for the development of in Stage

All the parameters are as for Figure This is just a

continuation of Stage here

The numerical simulation for the development of E in Stage

corresp onding to the development of shown in Figure

This Figure presents the result of optimizing our M mo del

with M at the Planck scale given the constraints of the den

sity p erturbations and the constraining spike in the density

p erturbations

This Figure presents the result of optimizing our M mo del

with M at the GUT scale given the constraints of the den

sity p erturbations and the constraining spike in the density

p erturbations

This Figure presents the result of optimizing our M mo del

with M at the intermediate scale given the constraints of

the density p erturbations and the constraining spike in the

densit y p erturbations

This Figure presents the result of optimizing our mo del with

the Yukawa coupling with given the constraints of

the density p erturbations and the constraining spike in the

density p erturbations

Note Fol lowing a long tradition in Cosmology we set al l the

constants k c and h to throughout this paper They are easy

B

to reintroduce later on the basis of dimensional analysis

For a general review of the subject of inationary cosmology

please refer to

BigBang

Starting in the b eginning of this centurymanycharacteristics of the observ

able universe lead scientists to a conclusion that universe was created in a big

explosion some billion years ago Therefore all the energy of the uni

verse was once concentrated in a v ery small volume of enormous temp erature

and energy density This is the essence of the socalled the Hot BigBang

mo del of the creation of the universe

An imp ortant thing to realize ab out the idea that the whole universe

evolved from a small volume is that we are not talking ab out a purely three

dimensional explosion like the explosion of a b omb A feature of an explosion

which happ ens in a three dimensional space is that it has a well dened origin

at which the explosion happ ened In contrast the explosion of the universe

do es not have such a three dimensional origin

Instead the explosion happ ens in what we can picture to be four dimen

sions The universe is like a three dimensional surface of a four dimensional

ballo on that is b eing blown As the volume of the ballo on increases anytwo

points on its surface moveaw ay from each other the p oints that are further

away move apart faster In reality this picture is slightly more complicated

since we are not sure yet whether the universe is really like a ballo on or

like a four dimensional hyp erb ola nevertheless the essence of the description

is the same

Justication of BigBang mo del

In it was discovered that all observable stars are moving away from the

solar system This fact was detected by measuring the radiation sp ectrum

of the stars and observing that the further away a star was from us the

shift its sp ectrum had The red doppler shift meant of bigger red doppler

course that the star was moving away from us Of the galaxy sp ectra

measured by the present day a red shift was observed in all except a very

few closest galaxies This universal expansion was in a direct accord with

the BigBang mo del

If we are to b elieve the Hot BigBang mo del we should exp ect that at

the earliest stages of the development of the universe all the particle sp ecies

are in a thermal equilibrium with each other that is the whole universe was

in a thermal equilibrium To solve for the energy in each particle sp ecies

and the energy distribution of particles of each particle sp ecies we can use

the black body mo del but with many more degrees of freedom than the

black body we usually talk ab out each particle sp ecies represents a degree of

freedom in which the energy can b e stored Since all particles are relativistic

at temp eratures this high the energy distribution of particles in each particle

sp ecies will b e the same as for a black body

However as the universe adiabatically expands it co ols down the density

of each sp ecies decreases and the collisions b etween the particles needed to

keep the universe in a thermal equilibrium b ecome rare Dep ending on the

scattering cross section of a particular sp ecies one after the other particle

sp ecies decouple from the thermal equilibrium Nevertheless at the moment

of the decoupling each sp ecies still has the energy distribution which follows

from the black body mo del of the temp erature at the time of the decou

pling After the decoupling this energy redshifts due to the expansion of the

universe It turns out that the redshift do es not destroy the shap e of the

energy distribution the only thing it do es is to decrease the eective tem

p erature the distribution is at Consequently we exp ect to observe some

background photon radiation still present from the time of the creation of the

universe And indeed a cosmic microwave background radiation CMBR

with the energy distribution of a black body at temp erature of K is

readily observed

There is many other evidence that supp orts the Hot BigBang mo del

dif among the most imp ortant is lightelement abundances which are very

cult to explain without employing the Hot BigBang mo del Consequently

the Hot BigBang mo del is universally accepted to day and it is used to make

predictions ab out the state of the universe all the way to sec after the

bang

The problems of the old BigBang mo del

In the recent decades scientists b egan to realize signicant disadvantages of

the old Big Bang mo del In fact Collins and Hawking showed that the

set of initial data needed for the universe to evolveintoauniverse similar to

ours is of measure zero I summarize in this section many of the problems of

the old the BigBang mo del For a comprehensive discussion please see

The problem of the domain walls

In the b eginning the universe is at a very high temp erature Consequently

 

the term T which app ears in the p otentials of all elds dominates the

potentials and all elds except the ones which are very weakly coupled so

temp erature coupling is small are conned to the origin the

However as the temp erature decreases this term vanishes and the elds

are free to move to their true minimum This can happ en either through

rolling towards the minimum if there are no barriers b etween the origin and

the true minimum or through quantum tunneling if the barriers exist But

there are many elds in particle physics theories which have more than one

true minimum For example this could be a scalar eld which has a cos

potential In this particular case the eld can roll either to the minimum

on the left or to the minimum on the right In any case the symmetry is

broken

There is no reason for a eld to move to the same minimum in two causally

disconnected regions of the universe However if it do es not a domain wall

is created between the two regions Many regions which were causally dis

connected a long time in the past are within the observable universe to day

So we exp ect that there should b e many domain walls present in the observ

able universe Unfortunately it turns out that a single domain wall in the

entire universe would have energy density to o high and lead to unacceptable

cosmological consequences

The monop ole problem

Besides the domain walls which can conceivably be created as a result of

sp ontaneous symmetry breakings most grand unied theories predict that

magnetic monop oles could also be created These monop oles are extremely

heavy and they decay very slowly Zeldovich and Khlop ov showed that

magnetic monop ole density in the observable universe should therefore be

comparable to the baryon density the energy density of the universe would



be times higher than it is to day and the universe would have collapsed

a long time ago

The baryon asymmetry

The most antimatter existing to dayistheonewhich is created in the high

energy colliders There is no reason in principle that the whole universe has

to b e comp osed from matter instead from antimatter in fact we could not

tell the dierence at all and would probably be calling antimatter to be

matter to day and vice versa Due to the symmetry between the two it is

hard to conceivewhythetwo are not present in same abundances And while

some clever philosophical attempts have been made to solve this problem

none of them explains why the ratio of baryons to photons is exactly as small



as

The horizon problem

As we mentioned the cosmic microwave background radiation has a tem

p erature of K This temp erature is universal across the whole universe

with only small deviations of K The observable universe to day encom



passes ab out regions whichwere causally disconnected at the time of the

CMBR decoupling The fact that the CMBR is the same across the whole

observable universe is therefore hardly explainable within the framework of

the old the BigBang theory

The atness of space

This is one of the biggest problems of the old theory It can b e formulated in

several ways but the essence of the problem is that the only natural length



scale in general relativity is the Planck length l m The fact that

P

the universe is at on all observable scales instead of b eing curved on the

order of l is therefore a complete mystery to the old theory

P

Other problems and the solution

There are some other problems with the old Big Bang mo del including

the problem of largescale homogeneity the galaxy formation problem the

gravitino problem the problem of Polonyi elds and the vacuum energy

problem However in Professor Alan Guth prop osed the concept

of ination Ination if implemented correctly solves all of these problems

except the vacuum energy problem Consequently ination is a universally

accepted theory to day

Ination

The old Big Bang mo del provides a reliable account for the development of

the universe from sec from the Big Bang till to day Its only problem is

that the set of initial parameters at t sec needed for the universe to

evolve to a universe qualitatively similar to ours is extremely unlikely So it

is desirable to develop an upgrade on the theory which would explain how

come the initial conditions were so sp ecial

Avery interesting philosophical idea was prop osed in order to solve some

of the problems It is called the anthropic principle Unless the Universe

and the laws of physics were as they are intelligent life could not have de

velop ed to discover and discuss them And while it is true that some of

the initial conditions indeed had to be sp ecial so that life could develop

the anthropic principle do es not solve many of the problems for example it

explains nothing ab out the uniformity of CMBR across the sky

On the other hand the concept of ination a purely scientic concept

solves all the problems except the vacuum energy problem starting from

almost any initial conditions provided they allow for ination to happ en

many mo dels of ination exist that naturally predict a universe similar to

ours

How do es ination do it

The basic idea of ination is that there was a time interval in a region of

the universe when the dominant comp onent of the energy was the vacuum

energy According to the Friedmann equation the scale factor R then grows

exp onentially

To see this we write the Friedmann equation



k G R

 

R R

where k corresp onds to a closed universe k to a at universe and

k to an op en universe Now in the case when is constant this is

solvable exactly but we can save ourselves some eort It is clear that R

grows faster and faster As it grows the term prop ortional to k b ecomes

vanishingly small and we are left with a pure exp onential expansion of R

is called the number of efolds The lnR R

f inal initial

Consequently a patch whichwas very small b efore ination and therefore

very smo oth can grow to b e bigger than the region which will evolveinto the

observable universe to day Furthermore the energy density of the relics

existing b efore ination drops to almost zero However during ination the

entropy in the patch is xed and the temp erature drops to zero We now

need a mechanism to reheat the universe ie since we assumed the patch

was small the initial entropy in the patch was small also As the patch

encompasses the whole observable universe to day we want to increase the

entropy in the patch For this to happ en we allow the vacuum energy to

decay into radiation once ination ends But since the initial conditions

for the decay are uniform across the patch this decay will not cause new

inhomogeneities across the patch

If implemented correctly ination dilutes the magnetic monop oles and

inates the domain walls created b efore ination well out of the observable

universe to day Furthermore it explains the uniformit y of CMBR In ad

dition the radius of curvature of space which is prop ortional to R grows

enormously thereby solving the atness of space problem Ination also

solves all the problems we did not elab orate on except the vacuum energy

problem However describ ed the way we did it it is not clear how ina

tion solves the smallscale inhomogeneity problem or the baryon asymmetry

problem but it turns out that the smallscale inhomogeneitycan be solved

through a correct implementation of ination while baryongenesis can be

solved without con tradicting ination We will elab orate on the issue of the

smallscale inhomogeneity b elow and concerning the issue of baryongenesis

we refer the reader to Section of our pap er

Smallscale inhomogeneity

Every workable mo del of ination has a mechanism that at a certain mo

ment transforms the vacuum energy resp onsible for inating the universe

into matter or radiation energy thereby stopping ination and reheating

the universe We argued ab ove that the inated patch stays smo oth during

ination b ecause it was smo oth in the b eginning But the quantum uctu

ations of the elds resp onsible for ination cause ination to end at dierent

erse this causes inhomogeneities across times in dierent parts of the univ

the universe

These inhomogeneities manifest themselves in nonuniformities of the

CMBR temp erature across the sky whichwas observed by the Cosmic Back

ground Explorer COBE satellite It is b elieved that precisely these initial

inhomogeneities evolved into the smallscale anisotropywe see to day indeed

according to computer simulations exactly the magnitude of the CMBR

temp erature anisotropies observed to day corresp onds to the magnitude of

the initial p erturbations needed to result in the observed structure of the

universe to day

Existing mo dels of ination and our goal

In his revolutionary pap er in which he intro duced the concept of ination

Prof Guth also prop osed the rst mo del of ination It turned out later

that the particular mo del he prop osed did not work Consequently many

cosmologists came up with their own mo dels of ination and there are many

workable mo dels of ination to day Unfortunately there is not enough data

yet so we can not decide on the correct mo del

Nevertheless most mo dels of ination include very small parameters this

is necessary to get enough efolds while at the same time pro ducing the den

sity p erturbations of the correct magnitude There are already existing small

inexplainable parameters in particle physics theories at many places For ex

scale to the Planck scale and the ample we mention the ratio of the weak

magnitude of the Yukawa coupling of the electron However intro ducing

new inexplainable parameters whose only motivation is to implement ina

tion correctly weakens the credibility of a theory Our goal in this pro ject

was therefore to develop a mo del of ination which follows naturally from

particle physics theories using only the scales already present in the particle

physics theories and without including any new inexplainable parameters

Our mo dels

The essential idea b ehind our mo dels is that we use two quantum elds

instead of only one to implement ination This gives us more freedom in

designing the mo del We used this opp ortunitywell and succeeded to design

a mo del that do es not require any new small parameters

The rst eld which we call has a lo cal maximum at the origin and

its true minimum is far away from the origin it is on the order of M The

P

value of the p otential in when provides the vacuum energy density

needed for ination The p otential of the other eld has only a term

 

m Since m is small compared to H the p otential in is naturally

at and slowrolls towards the origin It is the slow roll of this eld that

provides the required number of efolds

However there is an additional coupling term between and in the

potential which causes to have ahuge mass at the origin when is large

So we start of with and large After many efolds when b ecomes

small enough the coupling term is not dominant for m t any more and

is free to drop into its true minimum Now as b ecomes large the mass

of grows enormously due to the coupling term and gets to the origin

very quickly b eginning its coherent oscillations Here a third eld which

is coupled to enters the scenario all the energy leaks from into very

quickly and we can forget ab out from this pointonwards

So on gets to its true minimum starting its coherent oscillations this

energy in eventually reheats the universe since it turns out that the en

ergy stored in at the moment of the reheating is many orders of magnitude

smaller than the energy in

We actually havetwo mo dels with two dierent coupling terms and the

equations of their p otentials are as follows






m




p

V M cos



M

f

from now on referred as to M mo del and



  

m




p

V M cos

f

referred to as mo del

Since is the factor that holds at its false vacuum state for many efolds

we avoided the need to tune the p otential in acts like a switch to end

ination so the shap e of the p otential in is almost irrelevant for obtaining

enough efolds while the chosen shap e of the p otential in is actually the

most natural p otential to exp ect As we will see later the required density

p erturbations are formed in a natural way also

Where do our mo dels come from

Our mo dels follow from sup ersymmetric theories So it is to be exp ected

that the scale M app earing in the mo dels should be of the order M for an

I

explanation why this is so please see

As mentioned ab ove the particular shap e of the p otential in is almost

irrelevant This can b e seen from the Taylor expansion of the p otential only

at the very late stages of ination are the terms higher than the constant

and the mass term relevant so their particular choice do es not make much



dierence in the mo del at all Wechose the particular cos shap e b ecause of

its simplicity to think ab out

By Taylor expansion of the p otential in we see that the imaginary m



M f at the origin We exp ect that f M or equivalently m m

P W

According to the rules of sup ersymmetric theories the shap e of the p otential

in is as natural as they get The most natural value to exp ect for m is

m also so this is the value we pick for m

W

The only terms in the p otentials and which we did not justify

yet are the coupling terms In fact the coupling terms are the only terms

where the two p otentials dier So we justify the coupling terms now

The coupling term in the M p otential is derived from a sup erp otential

 

W

M

Our mo del is motivated by the prop erties of mo duli elds or at di

rections of the standard mo del For the standard mo del at directions

M could conceivably be equal to M M or M We explore all these

P G I

p ossibilities later To derive the p otential from W we write



X

W

V

i

i

where s are all the elds app earing in W

i

Since we are taking the scalar elds to b e real we put an additional factor

of to nally get the coupling term






M

In the second typ e of the coupling we explore we assume a sup erp otential

of the form

W

So after using and adding an additional factor of to accountfor

the fact that wetake the scalar elds to b e real we get the nal form of our

coupling term to b e

  



This is so b ecause we take h i so the other terms in V evaluate

to zero The coupling in is a typical Yukawa coupling Therefore we

assume

Density p erturbations

As discussed b efore ination predicts an extremely isotropic universe all

anisotropies are inated away Now fortunately for us although the universe

seems very isotropic at the large scales it is not p erfectly isotropic at small

scales Unless ination had a way to make up for this fact we would have

to abandon inationary theory

Ination in fact do es have a mechanism to predict some density p ertur

bations The basic idea is that although the evolution of the h i can be

describ ed with the classical equations of motion the developmentof is not

p erfectly classical there are some small quantum uctuations of around

h i These uctuations cause ination to end at slightly dierent times in

dierent parts of the universe therebyproviding for the initial energy density

p erturbations

As the time evolves these initially small p erturbations are amplied

through the gravitational and some other eects to result in the small scale

anisotropies observed to day The magnitude of these initial p erturbations

can be readily calculated for any particular inationary mo del once it is

calculated it has to be compared with the density p erturbations calculated

from the CMBR anisotropies as explained ab ove Finally the mo del has to

be designed in such a manner as to predict the same density p erturbations

as the ones calculated from the CMBR

Now we calculate what density p erturbations are predicted byourmod

els The equation derived from the CMBR anisotropies is given in for

the slowroll regime



V

p



M V

P

After a few lines of algebra this gives us the following constraint for the

M mo del

s



M f

N 



dens

e





m M M

p

while the mo del has the constraint

N 

dens

e



In the equations ab ove we dened m H and m H N

dens

is the number of efolds between the time at which the scales relevant for

determining the density p erturbations cross the horizon and the time when

m t at the origin We will see in the next section Eq how

big exactly is this parameter N for our mo dels these results will b e more

dens

clear after the next section

determines the scale dep endence The exp onential in Eqs and

of the density p erturbations



n

s

This concludes the discussion ab out the scalar density p erturbations

Conceivably there could also exist tensor density p erturbations that are due

to the fact that the graviton eld mo des are conceivably excited in de Sitter

space in a similar manner as the scalar p erturbations are created How

ever it turns out that our mo dels predict that the total energy stored in

the graviton degrees of freedom will b e negligible to the energy stored in the

scalar p erturbations so we skip the derivation of the tensor p erturbations

and refer the interested reader to for this derivation

Mathematical analysis of the number of efolds

In this section weprovide the analysis for the M mo del the analysis for the

second mo del is quite analogous so we just state the result

However b efore going in to the analysis of the numb er of efolds wehave

to say a few words ab out the decayof into which is a eld of small mass

coupled with Once the mass of b ecomes large the energy in decays

into relativistic particles and decouples from the eld

Weusetwo dierent mo dels for the decayof into Each of these two

decay mechanisms can be applied to either M mo del or mo del For the

explanation where the decay rates come from please see In the rst

mo del the decay rate is

m t

b

While is small t is small also and we can neglect the decay of

into Although we did take it into account in the computer simulations

of the mo del development In the other mo del



m t

q

l



M

P

So essentially when b ecomes large enough t for the decayof into

b ecomes signicant compared to H while t is still growing enormously

This makes the energy stored in decay rapidly and since for the parameters

we are using the oscillations in are always bigger than t the fact that

t is large do es not slow down the propagation of In short after the

time t b ecomes comparable to H we can forget ab out very so on the

fact that is decreasing in turn makes propagate much faster thereby

further increasing m t and hence t

Stage

We dene Stage as the period during which is slowrolling towards the

origin and is small enough not to inuence the propagation of signi

cantly

Consequently in this stage the equation of motion of is



H m

In our mo dels the terms other than the constant make a negligible con

tribution to energy during the relevant last efolds of ination Therefore

H changes negligibly during ination and we can make the substitution

N Ht where N is the number of efolds and

s



M

H

M

P

Furthermore we dene N at the moment when the mass squared

of at the origin equals exactly zero and N to be negative b efore that

Consequently Eq b ecomes

d



dN

and therefore

N

N e

c

q

where M m

c

At the same time the equation of motion for is

d





dN M

In this equation we assumed that is slowrolling in Stage which ac

tually is not true but is still a fairly go o d approximation it gives the correct

solution to within efolds We checked this numerically as explained in

the next section Also as will be clear from the denition of below and

c



from the fact that throughout Stage the term is negligible

c

in Stage Consequently we did not include this term in Eq

Anyway combining Eq and Eq and also using Taylor expan

sion in the exp onential we get

 

N

d

dN

According to this equation

N
N  w

N e

i

where w is some initial value of and N is the number of e

i

folds when These two values are calculated from the fact that when

i

the mass of at origin is small the quantum uctuations in dominate

its classical evolution This quantum development is describ ed in The

results are that essential

s

r

e



H w

i

and

s

N w

Stage ends when b ecomes large enough to start inuencing the de

p

velopment of This happ ens roughly when M H So we

c

calculate the total number of efolds N in Stage from Eq by setting

N From this we get our nal answer for the number of efolds in

c

Stage to b e

v

s

u

u

c

t

ln N w

i

Stage

As explained ab ove at the end of Stage starts to oscillate rapidlywhile

its amplitude decreases b oth b ecause of H and Because the oscillations

in are very rapid compared to the development of for the parameters of

interest to us we can approximate the developmentof by the development

of its envelop e The equation of development of amplitude of is

H t

e e

Stageisthe stage of development of from the moment that begins

its rapid oscillations till the momentwhen decays into Stage therefore

ends when t H

According to our denition of for the rst mo del of decayof into

b

and Stage ends almost immediately after Stage ends Consequently

c

b

the total number of efolds in Stage for the mo del N

b



For the mo del of decay with we get that the Stage ends when

l



 

HM M

P

l

l

To calculate N we note that it turns out that during Stage the term





is negligible and that essentially keeps with the minimum of its

potential which moves right as redshifts away The potential we are

talking ab out is therefore

 



m

V



M

and from this p otential

p

q

m M

l

N  l

m M e N

l



l

N



from which we conclude using Eq and the fact that t is negligible

during Stage that



M

P

l

N ln





H

We can also calculate the energy in at the end of Stage We assume

that all the energy that was stored in particles at the end of Stage is

transfered into E Consequently for

b

 

H

c

b

E



and for

l

l 

N

c

l 

l

E





M

Stage

Stage is the p erio d b etween the momentwhen till the momentwhen

b egins its coherent oscillations around its true minimum

Since is gone it do es not inuence the development of anymore

Furthermore it turns out that all the terms of the p otential in other

than the mass and the constant term are negligible all the way till the end

Therefore the equation of motion of is approximately



d d





dN dN

From the ab ove equation it is clear that

rN

N f ge

c l

dep ending on which mo del of decay we implement We dened

s



r

Using the ab ove and the fact that Stage ends approximately when

reaches its true minimum we get

f

N ln



r f g

c l

where is the nal value of that is the p osition of the true minimum of

f

p

Consequently f

f

Furthermore we can calculate E at the end of Stage since we know

that during Stage the energy in the relativistic particles is redshifting

N



away with H atthe end of Stage E E e



Total number of efolds

From the ab ove we conclude that from the time the mass of at the origin

equals exactly zero till the time starts its rapid oscillations around its

true minimum there are total of

N N N N N

tot  

efolds where N and N dep end slightly on which mo del of decay we im

 

plement However it turns out that in the end N for the two mo dels of

tot

decay diers by only efolds

The derivation for the mo del with Yukawa coupling is quite analogous

so we just state the results here N through N are dened as for the rst



mo del but the s at which the stages end are slightly dierent

p

H

c

and

p





M H

P

l



Some useful facts to have in mind when trying to develop the intuition for

the development of the mo dels is that N when This is so b ecause a

tot

smaller presents a atter p otential for As we can see from the expression

for N N In addition N decreases with an increase of as

tot tot

we can see from the expressions for N and N However this dep endence is

much smaller

This concludes our analysis of the numb er of efolds after the time N

We need this N to determine which N go es into Eq and Eq to

tot

determine the density p erturbations for the two mo dels The time when the

relevant scales enter the horizon is given by

M T

RH

ln ln N

Mpc

GeV GeV

which for our mo dels evaluates to

M m

ln ln N

Mpc

GeV GeV

Consequently the N which go es into the equations for determining the

density p erturbations is given by

N N N

dens tot Mpc

N determined this way is efolds bigger than the true N de

dens dens

termined numerically as in the next section Nevertheless this is a small

deviation and we will use N determined analytical way in all future cal

dens

culations

Unfortunately the predictions for E are not as great as the ones for

N the mistake is anywhere between and However it turns out

tot

that b ecause of a big redshift any initial E at is typically many orders of

magnitude smaller than the energy in particles at the time when these

particles reheat the universe Consequently E is irrelevant for the entire

discussion The only thing that we should worry ab out is that E could

mayb e cause over pro duction of gravitinos but it turns out that we are safe

from this danger also All these facts are describ ed in much more detail in

Computer conrmation of the number of efolds

The numerical simulations of the propagation of the mo dels were done in

MATLAB Every stage ab ove was develop ed separately and the ending val

ues of and of one stage were taken to be the initial values for the next

stage We explored a wide range of values of dierent parameters that could

conceivably be of interest to us This convinced us that the analytical solu

tion given in the previous section typically diered from the numerical one

by efolds for all parameters of interest to us Since this was a small devi

ation wewere allowed to use Eq ab ove for all further calculations since

this saved us enormous amount of time compared to using the numerical

simulation for every set of parameters

In this section we describ e the whole metho d and give the co des together

with the results of propagation for one representative set of parameters which

is illustrativehow the metho d works Again the numerical calculation is used

only as a check of the analytical expressions for N s of dierent stages and

the expressions for E that are given in the previous section

Stage

The program in MATLAB that solves numerically for the evolution of Stage

essentially solves the system of two coupled dierential equation of second

order in twovariables and The initial conditions are

i

N 

e and The equations are given by

c

V

H

and

V

H

We propagate the ab ove equations until the moment when b egins its

rapid oscillations around its minimum At this p oint Stage starts

A typical development of this stage is given in Figure

Once starts oscillating we take its amplitude as the initial condition

for Stage We also take the values of and as the initial conditions for

Stage

Stage

This simulation is also done in MATLAB The equation for propagation of

is the same as Eq but this time is replaced b y the envelop e of its

amplitude It is oscillating to o rapidly for MATLAB to b e able to propagate

it long enough On the other hand since it is oscillating so rapidly compared

to the evolution of the eld eectively sees only the envelop e of the

amplitude of The evolution of envelop e is given by

e

t H

e e

Furthermore it is trivial to calculate the total energy stored in eld




E



M

Or in the other mo del 9 10

8 10 ψ

7 10

6 10 GeV 5 φ 10

4 10

3 10

2 10 0 1 2 3 4 5 6 7 8 9

e−folds

Figure The numerical simulation for Stage The example we cho ose is

for M mo del We pick some typical values of the parameters

M GeV and M Dierent parameters or

mo del dier only quantitatively

  

E

We need to calculate E b ecause we want to calculate the total energy

stored in as a function of time eventually The equation for E is

E H t E

The factor H is included b ecause particles are relativistic so they

redshift with H A continuation of the propagation of the parameters of

Figure is shown in Figure 14 10

12 10 φ

10 10

8 10

6 10 GeV ψ

4 10

2 10

0 10

−2 10 0 1 2 3 4 5 6

e−folds

Figure The numerical simulation for the development of and the enve

lop e of in Stage All the parameters are as for Figure This particular

example is for the mo del with

l

As we said we also use MATLAB to calculate the propagation of E the

result corresp onding to the development of shown in Figure is shown in

Figure

This concludes our simulation for Stage The continuation of the de

velopmentofthe mo del is done in the next section We take the nal values

of and E as the initial values for Stage

Stage

Stage is the numerical simulation of the development of the mo del once E

is to o small to inuence the developmentof or E in anyw ay Consequently

we set and use the equations of the previous section to propagate

e 30 x 10 10

9

8

7

6

5 GeV^4 4

3

2

1

0 0 1 2 3 4 5 6

e−folds

Figure The numerical simulation for the development of E in Stage

corresp onding to the development of and shown in Figure

e

the mo del further Obviously in these equations E now The result of

the simulation is shown in Figures and

This concludes the whole development till the point when starts to

oscillate The numerical simulations of many examples of dierent parame

ters prove that our analytical expressions were rather go o d and that we are

justied in using them for the further exploring of the mo dels

The spike in the densit y p erturbations

As we said b efore when m the quantum uctuations dominate the

development of It turns out that this feature of our mo dels results in a

large energy density p erturbations which should conceivably be observable

at the scales that enter the horizon while m We calculate the exact

19 10

18 10

17 10

φ 16 10 GeV 15 10

14 10

13 10

12 10 0 5 10 15

e−folds

Figure The numerical simulation for the developmentof in Stage All

the parameters are as for Figure This is just a continuation of Stage

here

shap e of these p erturbations in They lo ok approximately like a gaussian

of width w and maximum height

This spikeisacharacteristic of our mo dels that should conceivably distin

guish it from the predictions of the other mo dels one day the relevant scales

b ecome observable However the scales observable to day through CMBR

have density p erturbations much smaller than as we said b efore Conse

quently wehave to make sure to put this spike in the scales unobservable

to day We sho wed in that if the peak of the spike is at more than w e

folds smaller scales than Mp c its contribution to the density p erturbations

are negligible and we can use the formulas from Section to determine

the density p erturbations

Consequently in our calculations we have to make sure that the scales 32 10

30 10

28 10

26 10

24 10

GeV^4 22 10

20 10

18 10

16 10

14 10 0 5 10 15

e−folds

Figure The numerical simulation for the development of E in Stage

corresp onding to the development of shown in Figure

that determine the density p erturbations are at least w away from the p oint

when b ecause that p oint is the p eak in the spike In other words we

c

imp ose the requirement

N w

dens

Creating the plots of masses

Our main goal in this pro ject was to develop a mo del which will not incorp o

rate any new inexplainable small parameters it will p erform ination using

only the natural already existing parameters

In the case of our mo dels the requirement of naturalness means that all

weak mass scales should be of order H meaning and In

addition the mass M should be of order M GeV and the fact that

I

m and m are at weak scale means that we would like them to be of the



order GeV

The most natural values of m and m we were able to obtain are pre

sented as functions of given M in Figures The reader can pickavalue of

M they consider to b e most natural and the most natural other parameters

obtainable given the constraints of our mo dels and the picked value of M

can be read o the plots All the plots are for

b

Figure This Figure presents the result of optimizing our M mo del with

M at the Planck scale given the constraints of the density p erturbations

and the constraining spike in the density p erturbations

As we said N Consequently we want to have N as large

tot tot

Figure This Figure presents the result of optimizing our M mo del with

M at the GUT scale given the constraints of the density p erturbations and

the constraining spike in the density p erturbations

as p ossible But we have to ob ey the constraint given by Eq and

Eq Therefore the best we can do is if we imp ose the requirement on

our parameters such that they result in

N w

dens

The other constraints that we have to ob ey are given by Eq or

The fact that we are determining once was alternatively Eq

imp osed to be as small as p ossible is a go o d strategy b ecause the density

p erturbations are determined more strongly with than with while

Figure This Figure presents the result of optimizing our M mo del with M

at the intermediate scale given the constraints of the density p erturbations

and the constraining spike in the density p erturbations

N dep ends more strongly on So the two variables are virtually de

dens

coupled and we can optimize them almost separately

This pro cedure yields the b est values and as a function of the

other parameters that are inputs to our mo dels

We wrote a program to implement this pro cedure a typical example of

of M the program the program is given in the app endix For every value

nds the optimal values of and So for a vector of variables M it

returns a vector of the corresp onding optimal and These vectors are

then used to create the plots of masses in Figures The plots are a result

Figure This Figure presents the result of optimizing our mo del with the

Yukawa coupling with given the constraints of the density p ertur

bations and the constraining spike in the density p erturbations

of the optimization of our mo del they are the most natural parameters our

mo dels can feasibly use

As explained ab ove all the plots and the program are for We men

b

tioned ab ove the the mo dels with typically propagate for efolds more

l

so their parameters would lo ok a bit worse but not much more so Never

qualitatively theless they would be quite the same

Description of the program

The program in the app endix is for the mo del with the Yukawa coupling The

program used to calculate the parameters for M mo del is quite analogous

The idea of the program is to search iteratively for the values of and

till b oth parameters are obtained with the sp ecied precision In this

section we give a brief description of each pro cedure The reader should then

have no problem to analyze and understand the co de

Pro cedure main denes the main parameters In addition it calls the

pro cedures that calculate the optimal parameters for every M required to

calculate it for Furthermore it sets up the initial values for the iteration

Finally it outputs the vectors of M and the corresp onding calculated values

of the masses

Pro cedure ndh just returns the value of H given M

Pro cedure densityp ert takes as inputs and and returns the

value of the density p erturbations that would result from such parameters

Pro cedure nefoldanal takes as inputs and and returns analyt

ical value of N w this is the value we would like to constrain to zero

dens

according to Eq

The rst pro cedure that do es some minimizing work is ndmupsi

which takes as given and nds the value of to satisfy the density

p erturbations constraint given M and

Similarly the pro cedure ndmuphi takes as given and nds the

value of to satisfy Eq given M and

The nal pro cedure that do es all the work of nding and for each

M it is called with is called itterate It takes an initial value of Given

this it nds the to satisfy the density constraint It do es so using

calculated this way to nd needed to ndmupsi Next it uses the

satisfy Eq given It do es so using ndmuphi Given this it

nds the to satisfy the density constraint Etc till the required precision

is satised after many iterative steps

Conclusion

As the reader can see from Figures the resulting values of the masses are

excellent We have achieved our initial goal our mo dels do not incorp orate

any new small parameters Instead they implement ination correctly using

only the parameters already existing in the particle physics theories

One of the distinguishing predictions of our mo dels is that the scalar

index n is very close to but in general slightly bigger this is clear from

Eq and from Figures Furthermore there is a characteristic spike

in the density p erturbations at so far unobservable scales In addition our

mo dels predicts that the tensor p erturbations are negligible For further

implications of our mo dels please see

Acknowledgements

The work was done under the guidance and sup ervision of Professor Lisa

Randall and Professor Alan Guth I am extremely grateful to them for

making this pro ject p ossible and for all of their help and guidance

We are very grateful to Andrew Liddle David Lyth and Ewan Stewart

for discussions and the Asp en Center for Physics where these discussions to ok

place We also thank Sean Carroll Csaba Csaki Arthur Kosowsky Andrei

Linde and Bharat Ratra for their comments We thank Bernard Carr Jim

Lidsey Avi Lo eb Paul Schechter and Paul Steinhardt for discussions and

corresp ondence ab out blackholes We also thank Krishna Ra jagopal for his

comments on the manuscript

The work of LR was supp orted in part by the Department of Energy un

der co op erative agreement DEFCER NSF grant PHY

NSF Young Investigator Award Alfred Sloan Foundation Fellowship and

DOE Outstanding Junior Investigator Award The work of MS was sup

p orted in part by MIT UROP The work of AHG was supp orted in

part by the Department of Energy under co op erative agreement DEFC

ER

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