Dec 27, 2015
1.Introduction1.IntroductionThe standard cosmology is a successful framework for
interpreting observations. In spite of this fact there were certain questions which remained unsolved until 1980s.
For many years it was assumed that any solution of these problems would have to await a theory of quantum gravity.
The great success of cosmology in 1980s was the realization that an explanation of some of these puzzles might involve physics at lower energies: “only” 1015 Gev, vs 1019 Gev of quantum gravity.
THE CONCEPT OF INFLATION WAS BORN.
What follows is an outline of the main features of inflation in his “classical” form;
The reader will find more than one model of inflation in scientific literature; here we will refer to the standard inflation which involves a first order cosmological phase transition.
2.Classical problems of standard 2.Classical problems of standard isotropic cosmologyisotropic cosmology
2.1 The horizon problemFrom CBR observations we know that:
510T
TOn angular scales >> 1°.
Sandard cosmology contains a particle horizon of radius:
*
0
*2)(
*)(*
t
po tdtta
tatR In the radiation dominated era,when a(t) ~t 1/2. (We will use natural
units, c=1).
In the matter dominated era (a(t) ~t 2/3) :
*
0
*3)(
*)(*
t
po tdtta
tatR
R po(t0)=3t0 ~6000 Mpc h-1
At t= tls (last scattering) Rpo(tls)=3tls
Because of the expansion of the universe the universe at last scattering is now:
Subtending an angle of about 1°
The microwave sky shows us homogeneity and isotropy on angular scales >>1°
MpcMpch
zzt
t
tt
t
tt
ta
tat
lsls
ls
lsls
lsls 100
1
6000
1
1333
)(
)(3 1
210
31
00
32
00
Why do we live in a nearly homogeneous universe even though some parts of the universe are not (or not yet ) causally connected???
2.2 The flatness problem
From the first Friedmann equation:
22
3
8
a
kGH
We have (see appendix 1):2
001
0
1
1
1
a
a
1
11
)(
1
0
2
00
a
a
z
)(zf
At the Plank epoch:
2
0
43
0
2
00)(
a
a
a
a
a
a
a
azf
eq
eqeq
eq
Remembering that
1 Ta
6128
4240 10
102.1
1073.21038.2)(
eV
eVh
T
T
T
Tzf beq
We have:
1
1101
1
0
61
To get Ω0 1 today requires a FINE TUNNING of Ω in the past.
At the Plank epoch which is the natural initial time, this requires a deviation of only
1 part in 101 part in 1061 61 !!!!!!
However , if Ω =1 from the beginning Ω =1 forever
But a mechanism is still required to set up such an initial state
To solve the horizon problem and allow causal contact over the whole of the region observed at last scattering requires a universe that expands more than linearly (yellow in the previous figure)
1)( tta
In the figure we have
ACCELARATED EXPANSION
This is the most general features of what become known as the INFLATIONARY UNIVERSE.
Equation of state (from the second Friedmann equation):
tta exp)(
pGa
a3
3
4
We want 0a 3
1p
The general concept of inflation rests on being able to achieve a negative-pressure equation of state.
This can be realized in a natural way using quantum field theory.
4.Basic concepts of quantum 4.Basic concepts of quantum field theoryfield theory
4.1 The Lagrangian density
)(2
1 VL Real scalar field
)(V Potential of the real scalar field , usually in the form
where m is the mass of the field in natural units
22
2
1)( mV
The restriction to scalar field is not simply for reasons of simplicity but because is expected in many theory of unification that additional scalar field such as the Higgs field will exist.
The scalar field is in general complex. We will use a real one only for simplicity.
4.2 Energy momentum tensor and equation of state
The Lagrangian density written above is obviously invariant under space-time translations of the origin of the reference system.
The existence of a global symmetry leads directly to a CONSERVATION LAW, according to the Noethern’s theorem.(See appendix 2 for details).
The conserved energy-momentum tensor is
From this we read off the energy density and pressure, since:
With the conventions that
qqq
LLT
44T pT 11
t
z
y
x
ic
4
3
2
1
1
111
LLT
4
444
LLT
If we add the requirement of homogeneity of the scalar field:
0321
)(2
)(2
2
2
Vp
V
If )(2
2
V
The equation of state is : pThis is of the type we need in order to solve the horizon problem! (p< -1/3 ρ).
4.3 Dynamics of the fieldFrom the Euler –Lagrange equation of motion:
We now derive the equation of motion for the scalar field.
In order to be correct in general relativity the lagrangian density L needs do take the form of an invariant scalar times the jacobian
In a Friedmann-Walker-Robertson model:
The Euler-Lagrange equation than becomes:
From which it’s not difficult to obtain:
With the requirements of homogeneity of the field:
0
LL
ijgg
g
det
)(3 tag
0)()( 33
LaLa
03 2
V
a
a
03
V
a
a
5.Cosmological implications5.Cosmological implications5.1 Evolution of the energy densityIf: The universe is dominated by the scalar field Φ with Lagrangian
and p= -ρ , that’s to say
The scalar field is not coupled with anything
From the relation
Adding the equation of state for the field (p= -ρ) and solving we have:
and since
with
From the first of the Friedmann equation:
)(2
1 VL
)(2
2
V
33 adt
dpa
dt
d
const
)(2
2
V
)(2
2
V constV )(
constGVGH )(3
8
3
82 constH
5.2 Exponential expansion
From the first Friedmann equation:
More then linear expansion: this is what we need in order to solve the horizon problem
constHa
a
2
2
aHa
Htea
5.3 Necessity of Cosmological Phase Transition
The discussion so far indicates a possible solution of the problems of standard cosmology, but has a critical, missing ingredient.
In the period of inflation the dynamics of the universe is dominated by the scalar field Φ, which has as equation of state.
There remains the difficulty of returning to a “normal” equation of state:
THE UNIVERSE IS REQUIRED TO
UNDERGO A COSMOLOGICAL
PHASE TRANSITION
p
5.4 Necessity of Reheating
The exponential expansion produces a universe that is essentially devoid of normal matter and radiation;
Because of this the temperature of the universe becomes <<T, if T was the temperature at the beginning.
We know that at the end of the inflation the temperature has to be high enough in order to allow the violation of the barion number and nucleosynthesis.
A phase transition to a state of 0 vacuum energy, if istantaneous, would transfer the energy of the field to matter and radiation as latent heat.
THE UNIVERSE WOULD THEREFORE BE REHEATED
6.The potential of the scalar field 6.The potential of the scalar field and the SRD approximationand the SRD approximation
In order to solve the equation of motion of Φ we have to specify a particular form of the potential.
Different forms of V(Φ) have been explored during the years and each of them produces a different type of expansion of the universe.
Requirements on V(Φ) :
1.In order to have negative perssure:
From this system we derive a(t)
V2
GVa
a 3
82
03 ' Va
a
2. THE SRD (SLOW-ROLLING-DOWN) APPROXIMATION:
The solution of the equation of motion become tractable if we make the socalled SRD approximation:
From the equation of motion we have: The condition than becomes a condition on Φ:
(using the first of Friedmann equations)
H3
'V
'3 VH V
2
)(9
'2
22
VH
V )(33
'2
2
VH
V
VV
mV p 83
' 22
24'
22pm
V
V
'V
V
24pm
3. (From Friedmann equation).
We will use a potential of the form:
const)(V
constV )(
)('V
2234 aTbVeff
In the figure we can see the temperature dependent potential of the form written above, illustrated at various temperatures:
At T>T1 only false vacuum is available;
At T<T2, once the barrier is small enough, quantum tunneling can
take place and free the scalar field to move: we have a first order transition to the vacuum state.
It’s important to remark that the energy density difference between the two vacuum states is
4TV
7.The Inflation solution of 7.The Inflation solution of standard cosmology problemsstandard cosmology problems
7.1 The horizon problemIn order to solve the horizon problem we need the horizon of the inflationary epoch to be
now bigger than ours:
Horizon during inflation: constHe
dte
ta
dttatOE
tHt
Ht
t
1'
)'(
')()(
'
Our horizon (matter dominated expansion)
Growth of inflationary horizon from the end of inflation up to nowExpansion of the horizon
during inflation
If ti<<te
00)( 3
1t
a
ae
H e
ttH ie
eei HtttH ee )(
Ha
ate eHte
003
If the comoving entropy is conserved, then: a3T3=const
(This is non true when p=p(T,Θ) , that’s to say: when pressure is not only function of the temperature.This is what happens for example during phase transition at a temperature different from the critical one)
Remembering that in natural units:
30
30
33 TaTa ee
00 T
T
a
a ee e
Ht
T
HTte e
003
2891944317 109101010310103 pp mt
1pptm
From SRD
(1 st Fried.equat.)
If we are dealing with a quantum field at temperature μ, then en energy density is expected in the form of vacuum energy.
Where μ 10 15-16 Gev (From GUT theories
V
22
pm
VH
4
epe TmT
H
We define:
Te = Temperature at the end of inflation
Its value is strongly dependent on
reheating
1
2
f
f
T
H
e
431 1010
pmf
eTf
2
A phase transition to a state of zero vacuum energy , if instantaneous, would transfer the energy
To normal matter and radiation (case of perfect reheating)
the universe would therefore be reheated.
In approximation of “perfect” reheating:
It will be proved below that this is also exactly the number needed to solve the flatness problem
4
eT 12 f
32810109 eHte 60 eHt
foldingseNHte
7.2 The flatness problemAs we have already seen, from the first of Friedmann equations we have (see appendix 1 for
details):
We take: t*=ti and t=te
Remembering that ρ is nearly constant
during inflation, we have:
2
1
1
)(
*)(
)(
*)(
1*)(
1)(
ta
ta
t
t
t
t
2
11 11
e
iie a
a
Exponential expansion: Hteta )(
NHtttH
e
i eeea
aeei 22)(
60N
Nie e 211 11
1e
We deduce:
because of the factor
We would like to have an estimate of the parameter Ω(t) at the present epoch Ω(t0) Ω0
again the relation
with tt0
te
1202 ee N
2
1
1
)(
*)(
)(
*)(
1*)(
1)(
ta
ta
t
t
t
t
2
001
10
1
1
a
aee
e
2
0
2
0
2110 11
a
a
a
ae eq
eq
eeq
eq
eNi
2
0
23
0
4
2110 11
a
a
a
a
a
a
a
ae eq
eq
e
eqe
eqNi
eq
eNi
e
eqNi TT
Te
a
aae
0
221
2
02110 111
110107
100010103
1011 15253
294
21512011
0
iiGev
Geve
If we have perfect reheating:
7.3 Number of e-foldings: criteria for inflationAs we have already seen, successful inflation in any model requires more than 60 e-foldings of the expansion.The implications of this fact are easily calculated using the SRD equation:
HH
V
H
V23
)('
3
)('
Vm
Hp
22 1
83
Using the first of Friedmann equations:
H
m
V
VH
m pp
8'8
22
Hdtm
d p
8
2
222
48e
p
t
t p md
mHdtN
e
i
e
i
eie HtttH )(
ppep
e mmHtm
24
60
4
2
N > if V’<
A model in which the potential is sufficiently flat (V’<<) that slow-rolling down can begin will probably achieve the critical 60 e-foldings.
The criterion for successful inflation is thus that the initial value of the field exceeds the Plank scale (mp)
8.Ending of inflation8.Ending of inflation
The relative importance of time derivatives of Φ increases as Φ rolls down the potential and V approaches zero.
The inflationary phase will cease!
The field will oscillate about the
bottom of the potential, with
oscillation becoming damped
because of the
friction term.H3
If the equation of motion remains the one written above (absence of coupling), then:
1. We will have a stationary field that continues to inflate without end, if V(Φ=0)>0.
2. We will have a stationary field with 0 energy density.
BUT
If we introduce in the equation the couplings of the scalar field to matter field:
this thing will cause the rapid oscillatory phase to produce particles, leading to reheating
0)('3 VH
8.1 Absence of couplingFrom the relation:
It’s not difficult to derive:
And in presence of the scalar field and radiation:
Remembering that:
Equation of motion 0
33 )( adt
dpa
dt
d
0)(3 pH
04)(3 rr HpH rrp 3
1
)(2
)(2
2
2
Vp
V
)('V
0
0
r
r
04)('3 rr HVH
8.2 Adding a term of coupling:It’s the same thing as varying the equation of motion of the scalar field
We have in this way:
and also (harmonic oscillations)
0)('3 VH 0)('3 VH
This extra term is often added empirically to represent the effect of particle creation;
The effect of this term is to remove energy from the motion of Φ and damping it in the form of a radiation background;
Φ undergoes oscillations of declining amplitude after the end of inflation and Γ only changes the rate of damping.
For more detailed models of reheating see Linde (1989) and Kofman , Linde & Starobinsky (1997).
24 rr H 2
42
2
30 RhgT
= Temperature of reheating
G=degree of freedom
4RhT Energy density for relativistic
particles in the case of perfect reheating
Because of the factor even in the case of perfect reheating is << of the initial one
g30
2
RhT GUTT
A plot of the exact solution for the scalar field in a model with a potential.
The top panel shows how the absolute value of the field falls smoothly with time during the inflationary phase, and then starts to oscillate when inflation ends.
The bottom panel shows the evolution of the scale factor a(t). We see the initial exponential behavior flattening as the vacuum ceases to dominate
The two models shown have different starting conditions: the former (upper lines in each panel) gives about 380 e-foldings of inflation; the latter only 150.
(From Peacock,1999).
2)( V
9. Relic fluctuations from Inflation9. Relic fluctuations from Inflation
9.1 Fluctuation spectrum During inflation there is a true event horizon, of proper size 1/H This fact suggest that there will be thermal fluctuations present, in analogy with black
holes for which the Hawking temperature is:
The analogy is close but imperfect, and the characteristic temperature here is:
The inflationary prediction is of a horizon scale amplitude fluctuation
The main effect of these fluctuations is to make different parts of the universe have fields that are perturbed by an amount δΦ with:
22
2
8 c
GMR
R
hckT s
sH
24hcH
kT
2
2HH
2
H
We are dealing with various copies of the same rolling behavior Φ(t) but viewed at different times, with:
The universe will then finish inflation at different times, leading to a spread in energy density.
The horizon scale amplitude is given by the different amounts that the universe have expanded following the end of inflation:
t
2
2HHtHH
2
H (Indetermination on the scalar field, from
quantum theory of fields. See Peacock, 1999 for details)
This plot shows how fluctuations in the scalar field transform themselves into density fluctuations at the end of inflation.
Inflation finishes at times separated by in time for the two different points, inducing a density fluctuation
t
tH
9.2 Inflation couplingFrom the SRD equation, we know that the number of e-foldings of inflation is:
If
Since N ≈ 60 and the observed value of fluctuations
(Really weak coupling!!!)
'3 2
V
dH
dHHdtN
4V
2
2
H
N
2321
3
332
'
3N
H
V
HHH
510 H
1510
If
From the first of Friedmann equations:
And since is needed for inflation,
22mV
2
332
2
3
'
3
m
H
V
HHH
pp m
VH
m
VH
22
53
232
23
102
3
pp
H m
m
mm
V From CBR observations
pm
pmm 510
This constraints appear to suggest a defect in inflation, in that we should be able to use the theory to explain why , rather than using observations to constrain the theory510 H
9.3 Gravity WavesInflationary models predict a background of gravitational waves of expected
rms amplitude:
It’s not easy to show from a mathematical point of view how such a prediction arises.
Here is enough to say that everything comes from the fact that in linear theory any quantum field is expanded into a sum of oscillators with the usual creation and annihilation operators.
The fluctuations of the scalar field are transmuted into density fluctuations, but gravity waves will survive to the present day.
prms m
Hh
10. Conclusion10. Conclusion
To summarize, inflation: Is able to give a satisfactory explanation to the
horizon and flatness problem; Is able to predict a scale invariant spectrum, but
problems arises with the amplitude of the fluctuations predicted (or alternatively with the coupling constant λ );
Is strongly linked with quantum field theory.
11.References11.References
• Kofman, Linde, Starobinsky,1997:hep-ph/9704452 • Linde,1989:Inflation and quantum cosmology,
Academic Press.• Lucchin,1990:Introduzione alla cosmologia, Zanichelli.• Peacock,1999:Cosmological physics, Cambridge
University Press.• Ramond:Quantum field theory.• Weinberg,1972:Gravitation and cosmology, John Wiley
and sons.
Appendix 1Appendix 12
00,
00,
2
3
8
a
kGH
crcr
20
020
2
a
kHH
20
020
20 a
kHH
At t=t0
20
020 )(
)1(ta
kH
Substituting this result in the first equation:
2
00
20
00
20
2 1
a
aHHH
And remembering that
0
020,0
,0
0
0
2
3
8
3
8
HGGH crcr
crcr
It’s not difficult to get the following equation:
2
001
0
1
1
1
a
a
Appendix 2Appendix 2Given a lagrangian density L for the field and the transformations: xa
xxxx '
Def:
aaaa xxx ''
qaqa
aaa
x
xx
~
)(~ '
A
If L is invariant for “A”:
aqa
LLJ
And
0 qJ This is the Noethern’s theorem