Transcript Diapositiva 1

Instituto Avanzado de Cosmología
Connections between Dark Energy
and Particle Physics
Axel de la Macorra
Instituto de Física, UNAM
Instituto Avanzado de Cosmologia
Cosmology on the beach, Playa del Carmen, Jaunuary 2010
El Universo Invisible
1er Congreso de Cosmología IAC
Outline
• Brief Introduction
• Properties of Dark Energy
• Theoretical Considerations
• Dynamics of Dark Energy (Scalar Fields)
• Specific models:
pNGB
Condensates
Coupled (interacting) DE Models
Late time generation of DE
• Conclusions
Cosmology on the Beach, Playa del Carmen,
January 2010
Dark Energy
• Dark Energy has been established in the last 10 years
• It is one of the most interesting and open question in physics
• But the nature and dynamics of Dark Energy “DE” is still not well know
How detectable is DE on earth?
A volume of the earth size filled
with DE weights less than 0.001 gr
It was a big surprise for most scientist even though there were some hints from
• age of globular cluster (age ~ 13.5 billion years)
• structure formation
both required a larger age of the universe
Without DE 8-10 billion years
with DE 13.7 billion years
Cosmology on the Beach, Playa del Carmen,
January 2010
General approach to Dark Energy
M2pl=1/8p G = 1
Can introduce Dark Energy via
1)
New Particles (modification of Tun )
2)
Scalar-Tensor (non-minimal coupling f(f) R )
3)
f(R) modification (e.g. MOND)
4)
Inhomogeneities (live in a huge Bubble)
for reviews see
• Dynamics of dark energy. E. Copeland et al Int.J.Mod.Phys.D15:1753,2006. hep-th/0603057
• Dark Energy and the Accelerating Universe. J. Frieman et al, Ann.Rev.Astron.Astrophys.46:385,
2008, arXiv:0803.0982
• The Dynamics of Quintessence, The Quintessence of Dynamics. E. Linder Gen.Rel.Grav.40:329356,2008, arXiv:0704.2064
Cosmology on the Beach, Playa del Carmen,
January 2010
Basics
Einstein eqs. in
a FRW metric
eq. of state
w = 1/3 radiation
w = 0 matter
w = -1 cosmo. cte
acceleration requires w < -1/3
Cosmology on the Beach, Playa del Carmen,
January 2010
5 year WMAP, Komatsu et al
w = cte
Cosmology on the Beach, Playa del Carmen,
January 2010
5 year WMAP, Komatsu et al
w = cte
-1.11 < w < 0.86 (95% CL)
w = wo + w’ z/(1+z)
w’=dw/dz ( at z = 0 )
- 1.32 < w < 0.86 (95% CL)
Cosmology on the Beach, Playa del Carmen,
January 2010
Improved DarkVector
Energy Constraints
from ~100 New CfA Supernova Type Ia Light Curves.
Fields
M.Hicken et al Astrophys.J.700:1097-1140,2009.
i.e.
w = - 0.87 +/- 0.06
Cosmology on the Beach, Playa del Carmen,
January 2010
Reconstruction from Hubble diagram V, w(z)
Reconstruction:
i)
Model independent
or
i)
Piece wise w(z) for z+Dz
ii)
Choose a parametrization of w(z)
But
• Involves an integration and is not precise enough to extract w(z)
• Results depend on the priors used
• require extra data sets (LSS, BAO, WL)
see Tegmark, Takada, Zaldariaga courses
and Bean, Crawford, Roe, Suntzeff talks
or all “Cosmologia en la Playa”
Cosmology on the Beach, Playa del Carmen,
January 2010
There is a strong degeneracy in w(z) and Wm on the expansion history
due to the integration on the luminosity distance
Steinhardt et al ‘02
Cosmology on the Beach, Playa del Carmen,
January 2010
Improved Cosmological Constraints from New, Old and Combined Supernova Datasets.
Supernova Cosmology Project (M. Kowalski et al.). Astrophys.J.686:749-778,2008.
Cosmology on the Beach, Playa del Carmen,
January 2010
Parametrization of w
i)
w = wo constant
ii)
w = wo + w1 z
iii) w = wo + w1 z/(1+z)
The values and evolution of w(z) depend heavily on the parametrization used
4 free parameters
Yellow = 95% C.L.
Yellow no cross over the
w = -1 line
Corasaniti et al PRD’04
How to solve the nature of Dark Energy ?
Need two fundamental ingredients
Inspiration
(go to the top of a pyramid and
recieve the “energy”)
e.g. Sun pyramid or Tulum
careful thinking …
A. Riess et al ‘06
Dark Energy Properties
Generic Properties DE models must satisfy:
• Amount of Dark Energy
WDE = 0.72 +/- 0.03
• Present mass density Wm = 0.28 +/- 0.03
• Constraint from NS
Bean et al Wf < 0.045
• Distance to last scattering, z=1089: RCMB = 1.70 +/- 0.03
• SDSS luminous red galaxy, baryon acoustic oscillation (BAO)
distance parameter z = 0.35 gives A with n = 0.95
• Distance ratio from z = 0.35 to z =1089 gives R0.35 = 0.0979 +/- 0.0036
Cosmology on the Beach, Playa del Carmen,
January 2010
Dark Energy Properties for scalar fields
•
Slow roll conditions must be satisfied
1)
|V’/V|
2)
|V’’/V| << 1
•
Weakly coupled to SM particles
•
Scalar field light mass (induces a long range force)
<< 1
Present values of mass and dark energy
M2pl=1/8p G = 1
Cosmology on the Beach, Playa del Carmen,
January 2010
Particle Phsyics and Dark Energy
• What is the Nature of Dark Energy?
• Is it a cosmological constant w= -1 or a particle w(z) ?
• Why is DE relevant today ?
“Coincidence Problem”
What do we expect from a good theoretical model ?
1)
Derive the potential V(f )
2)
Small number of free parameters
3)
Reasonable choice of values for the free parameters
(i.e no fine tuning of parameters)
4)
Initial condition of scalar field f and energy density r(f)
5)
Account for the long period of radiation and matter domination
6)
and of course have a good fit to the data
e.g. Scalar potential
V (f) = L4 f (f /M)
need to derive the functional form f (f /M) and explain the parameters L, M
A. de la Macorra, Inst. de Física, UNAM, IAC
Ultra violet Vacuum Energy
Vacuum Energy
L = 0.003 eV
Quantum field vacuum corrections
k = Planck mass 1019 GeV?
k = Electroweak Scale 100-1000 GeV?
r is too large !
The Standard Model “SM” has no cutoff k --> Planck mass
• The mass of the Higgs is expected to be O(100-1000) GeV
(quntum corrections give m =O(mpl) or to the scale of SM validity)
• need new physics beyond TeV
• e.g. Supersymmetry (scalar + fermion loops cancel)
scalar loop
fermion loop
=0
+
j=spin, susy ameliorates the UV problem
but it is still too large
Cosmology on the Beach, Playa del Carmen,
January 2010
Naturalness
We measure a parameter A(m) at a scale m << L (e.g. Mpl)
we should be able to determine it from A(L) with L >> m
• We do not want a fine tuning between A(L) delta A
• We would like to have A(L) ~ d A ~ A(m)
• For mass m with V ~ f4 one has dm ~ L
• For gauge coupling constant g one has d g ~ Log[ L/m ]
Potential
one loop
effective potential
Potential
it is not enough to derive Vo to give Dark Energy
and but we should ensure that the radiative corrections do not spoil the DE behavior
Cosmology on the Beach, Playa del Carmen,
January 2010
Particle Phsyics and Dark Energy
A) Scalar fields f (spin cero particles) present at high energies (after inflation)
Mpl > L >>TeV
•
Fundamental scalar fields f
(e.g. tracker behavior of scalar fields)
B) Produce DE scalar field at a late time and low energy scale L << TeV
a) Fundamental field generated dynamically at small scale L
e.g. produced by the decay of other particles
b) Composite scalar field f generated at a phase transition scale L
( e.g. can have mpl >> L )
i) fermion condensate f = <YY>
ii) vector condensate A = <Vm Vm>
•
L = L gut e
8p 2 / g g2 u t b
L QCD  200 MeV
since L is closer to present scale of DE we have
less fine tuning of the parameters in the DE potential
•
help to explain the coincidence problem
Phase transition we expect V = O(L4 ) with L the scale of sym. breaking
A
e.g.
Vi/VDE
Cosmology on the Beach, Playa del Carmen,
January 2010
Radiation
log[Energy]
120 orders of magnitude
Evolution of Energy Densities
w = p/r
DE
1
r rad  a , wr =
3
r mat  a 3 , wm = 0
4
Matter
Cosmological Constant
initial size of universe
log[a]
COINCIDENCE PROBLEM
1)
Cosmologial constant w = -1
2)
Quintessence (scalar field) w = w(a)
r L  a 0 = cte, wL = 1,
today
dynamics, e.g.
scalar field
rf  a 3(1 w) , wf (a)
Dark Energy Models
Generic Properties: Scalar fields with weak coupling to the SM
• Quintessence
Scalar field with standard (canonical) kinetic term,
a slow roll potential V and w > -1
• K-essence (include Tachyons)
Scalar field with non standard kinetic term
• Phantom
Scalar field with negative kinetic term, can have w < -1
Mixture of any of the above
Model buliding:
• Tracking Models
• Pseudo Nambu-Godstone Bosons, pNGB
• Condensate Models
• Assisted Inflation
• Interacting (coupled) Models
• Chameleon Models
• Late Generation of DE
• Oscillating Models
• Mocker Models
• Quartessence and Chaplygin gas models
• Skating Models,
• Wet fluid
• Leveling Models
• Quintom Models
many others....
•Tracking Models
• •Quartessence
•Wet
•Quintom
•Condensate
•Interaction
•Chameleon
Pseudo
Oscillating
•Mocker
•Skating
fluid
Nambu-Godstone
Models
Models
Models,
Models
Models
Models
Models
Bosons,
pNGB
•Late
scalar
Generation
fields
and
that
ofChaplygin
DE
redshift
gas
(track)
models
as the dominate energy component,
•Leveling
Models
•Assisted
Inflation
Dynamics
Equivalent
Scalar
effective
Interaction
DE
Transition
Go
potential
from
fields
scalar
free
corresponding
to
between
from
that
the
and
field
field
acquiere
cross
sum
matter
mass
behavior
DE
produced
of
over
depends
like
and
ascalar
to
a
constant
a
small
the
behavior
other
w
circle
=by
w+1
on
mass
=
afluid,
w
to
in
-1
late
the
to
component
phase
cosmological
line
trough
e.g.
cosmological
time
environment
dark
space
phase
non-perturbative
matter
and
transition
constant
aconstant
cosmological
or
neutrinos
like
like
symmetry
behavior
behavior
constant
along
Attempt
Late
they
time
to
are
unify
insensitive
dark
matter
of
to
initial
and
field
dark
conditions
(F.Briscese,
energy
buy
A.M.)
w
>-0.7
et
al)
Approach
aproduction
cosmological
constant
as
the
density
nears
a (Steinhardt
limiting
value
and
Slow
rollcurves
of
the
DE
depends
having
multiple
breaking
(Barenboim
(Holman
(Vikman,
(Bienetruy,
(Amendola,
(Khoury,
along
the
curve
&
Odintosov
(mass
Brax
Naidu,
dw/da
A.M.)
&
van
Lykken
et
of
is
de
al
04)
=protected
dw/da
04)
-3(1Bruck)
et&06;
al,
=
w^2)
Hu
Barenboim,
C on
from
w(1
et
[physically
al)
+quantum
w)
(Linder
Mena
corresponding
corrections)
Requejo,
06)fields
& Quigg
but
to aneed
field
06)dw/da
fine-tuning
moving
across
of
a
(Makler,
de
Oliveira,
Waga
03
for
an
overview)
have
parabolic
tracks,
respectively
dw/da
=-3(1
+ w)(wwa)
and
= -3(1
+
(Liddle
et conditions
al,
Coley et(Frieman
al)
the
constant
initial
potential]
(Linder
05;
et al,
Liddle
Choi)
et al, 05)]
w)(w + w_a). (Linder, 06)
Quintessence
large number of DE models, e.g.
acceleration
acceleration if
Cosmology on the Beach, Playa del Carmen,
January 2010
Evolution of Scalar Fields
e =1 quintessence e = -1 phantom
Friedmann eq.
Autonomous evolution eqs.
Classify the models by the limit of l = - V’/V ,
e = 1 canonical
e = -1 phantom
g = 1 + w, for e =1
Cosmology on the Beach, Playa del Carmen,
January 2010
Stability issues
perturbations around the solution
the perturbations have an eq. of motion
Cosmology on the Beach, Playa del Carmen,
January 2010
Tracker Fields
Tracker behavior = scalar field evolves as the dominant fluid
eq.motion
eq. of state
Tracker condition
Cosmology on the Beach, Playa del Carmen,
January 2010
Tracker Fields
However, tracking behavior may be reach later than present time, e.g.
For IPL tracker needs n > 5
and has (n=5) w > - 0.75
trackers have w > - 0.7
For V =
Cosmology on the Beach, Playa del Carmen,
January 2010
V = m2 f2
The field oscillates around the minimum with w = 0
1
1
0.8
0.8
0.6
0.4
0.5
0
w
0.6
V=x^2
0.2
0.4
0.5
0
0.2
0.2
1
0
1
2
t
3
4
0
0.5
V = L9/f5
1
1.5
t
2
2.5
0
3
0.5
1
1.5
t
2
2.5
Runaway. n = 5, wtr = 0.3, L = 10 TeV, w = - 0.75, W = 0.72
1
1
0.5
0.8
0.5
0.4
w
0.6
0.3
0
0.4
0.5
0.2
0.2
0.1
1
0
0
0
0
1
2
3
4
2
6
8
10
12
0
14
2
4
6
5
n = 1, wtr = 2/3 L = keV,
L5/f
8
10
12
14
t
t
t
V=
4
1
w = - 0.87,
W = 0.72
1
2
0.8
0.5
1.5
1
w
0.6
0.4
0.5
0.5
0.2
0
0
1
2
3
t
4
5
0
0
1
0
1
2
3
t
4
5
0
1
2
3
t
4
5
Tachyons
Tachyon: the lowest string excitation in
D-brane or D-antiD brane systems
Tachyons were motivated by String. They
represent the lowest energy state in Dbranes and V has a form
both give a w = 0
Cosmology on the Beach, Playa del Carmen,
January 2010
Copeland et al PRD 05
Tachyon Potentials
n = 2, acceleration depends on Vo
0 < n < 2, gives acceleration (1)
cases (2) and (3)
2 < n, in case (1), (4) and (5) with
w = 0 at late times
Cosmology on the Beach, Playa del Carmen,
January 2010
K-essence: Scalar fields with non canonical kinetic terms
string motivation: at weak coupling g
field, conformal transformation
obtain
acceleration w < -1/3 for X < 2/3
Cosmology on the Beach, Playa del Carmen,
January 2010
Phantoms: Fields with negative kinetic term
acceleration
Big Rip:
as t => ts H and r go to infinity at finite time
(but avoided if V has a maximum)
p-Nambu-Goldstone Bosons pNGB
• A global continuos symmetry has massless Nambu-Godlstone bosons
• Non-perturbative effects may break the symmetry and give a small mass
• The mass is protected from loop corrections by the global symmetry
• e.g. axion fields
• Typical potential is:
Slow roll V’/V << 1
fa > Mpl helps inflation but V will have corrections from instanton contributions
expand around the extrema
duration of inflation
Cosmology on the Beach, Playa del Carmen,
L.Sorbo et al ‘05
January 2010
60
1
3
2.5
2
1.5
1
0.5
0
50
V V'
w
0.5
0
40
30
20
0.5
10
0
1
0
2
4
6
8
t
10
12 14
0
2
4
6
t
8
10
12
0
2
4
6
At the maximum the pGNB is tachyonic so instabilities arise
Cosmology on the Beach, Playa del Carmen,
8
t
January 2010
10
12
14
• pNGB are scalar fields with mass protected by the symmetry
however,
• Models with fa < 0.1 Mpl are extremely fine-tuned
• Models with fa > Mpl have a V with instanton contributions
Cosmology on the Beach, Playa del Carmen,
January 2010
Possibe way out
1)
fa > mpl not good from strings or GR
2)
Many pNGB
3)
Two pNGB (mixing with QCD type hidden sector)
v = 10 eV, M = 1019 GeV
m = 0.001 eV
pNGB may work if the scale m can be brought close to
DE scale, i.e. late time phase transition.
Cosmology on the Beach, Playa del Carmen,
January 2010
Interacting Dark Energy General Analysis
Define effective equations of state
which fluid dominates depends on
the sign of Dweff
e.g. for d = c H r dm , c constant
a late time attractor
Cosmology
on la
the
Beach, Inst.
Playa
Carmen,
A. de
Macorra
de del
Física,
UNAM January 2010
Observational constraints on an interacting dark
energy model, R. Maartens et al, arXiv:0907.4987
Cosmology on the Beach, Playa del Carmen,
January 2010
Interacting Dark Energy f
e.g. Scalar Field f and Fermions y
mass
Fermi-Dirac distribution with a field dependent mass
M
gs degrees of freedom
Density
Pressure
A. de la Macorra,
on la
the
Beach, Inst.
Playa
Carmen,
Inst. Cosmology
de Física, UNAM
A. de
Macorra
de del
Física,
UNAM January 2010
“Cosmology of mass-varying neutrinos driven by quintessence …”
A, W. Brookfield, et al Phys.Rev.D73:083515,2006,
CDM
f
astro-ph/0512367
n
n
Cosmology on the Beach, Playa del Carmen,
January 2010
How to
How to obtain w < -1 ?
1) For the interacting fluids
2) For the non-interacting fluids
using
get
Cosmology on the Beach, Playa del Carmen,
January 2010
= - 1.06
w < -1 can be an “optical effect”
Describe the universe with
i)
non-interacting DE and DM
wDE
and wm = 0
ii) Interacting DE and DM
Non Interacting
Interacting
wIDE = wf and wm = 0
wDE : apparent eq. of state as seen for the non-interaction DE
wDE can be < -1
if x > 0
wDE
i) For x = 0
wap = wf
ii) For x > 0
wap < wf we can have wDE < - 1 !
<-1
even though wIDE = wf > -1
(for a growing function f(f) i.e. f (a<1) /fo(ao=1) < 1
Cosmology on the Beach, Playa del Carmen,
January 2010
Neutrinos in Cosmology
Neutrinos density
From HM experiment
Implications to Dark Energy
With out HM:
-0.94 < w < -1.28
95%CL
with HM:
-1.09 < w < -1.67
95%CL
w is more negative !
A cosmological constant is not within
the 95 % CL
A.Melchiorri, P.Serra, R.Bean A.M.
Astropart.Phys. ‘07.
Condensate Model
A.M. PRL ‘01, JHEP ’03,PRD‘05
Evolution of coupling constants vs energy
SU(3) QCD
Dark Energy
SU(Nc=3),
Nf = 6, b = 3
SU(2) Weak interact.
SU(1) E.M. interact.
one loop evolution
1
g 2 (L)
=
1
g 2 (L gut )
b = 3N c  N f
L gut  10 GeV ,
16
g 2 gut  1 / 2

b
L
Log
[
]
2
L gut
8p
Condensation or
phase transition scale
L = L gut e
8p 2 / g g2 u t b
L QCD  200 MeV
L DE  40 eV
What happens to elementary particles when the coupling becomes strong ?
The particles form neutral bound states.
Quarks form:
p =< d d 
pions,
p =< uud , n =< udd 
protons, neutrons
L QCD  200MeV
Dark Energy is a bound states made out of fundamental particles
f = (< YY )
1/ 3
L DE  40 eV
Dark Energy
Scalar field
Initial Conditions for V and f
fi = (< YY )1/ 3 = L DE
V (fi ) = L DE 4 2 / 3fi 2 / 3 = L DE 4
mf (fi ) =
2
 2V
f 2
= L DE 4 2 / 3fi  2 2 / 3 = L DE 2
Cosmology on the Beach, Playa del Carmen,
January 2010
Dark Energy Model
Dark Group:
SU(Nc=3), Nf=6
using supersymmetry y non perturbative and exact results we determine the
potential V (Affleck-Dine-Seiberg):
1 /( N c  N f )
W = ( N c  N f )( LbDE / det < YY )
V=|
f = (< YY )1/ 3
dW 2
4 n
| = L DE f  n
df
L DE  40 eV
Phase Transition or condensation scale
The potential V is generated below the energy scale
L DE
when the coupling constant g becomes large
A.Cosmology
de la Macorra,
Física, UNAM
onInst.
the de
Beach,
Playa del Carmen,
January 2010
r r  a 4
radiation
DE
r m  a 3
rf  a 3(1 w)
matter
i) For E > LDE fundamental particles are massless
and we have w = 1/3
ii)
At E = LDE, phase transition !
Effective scalar field and potential V are generated
w is dynamical for E < LDE.
V(f )
Effective potential
Effective potentital
f
Dark Energy evolution (after phase transition)
wo = - 0.92
Having extra particles coupled at high energies with the standard model
gives a smaller energy density for our Dark Group
Late time generation of Dark Energy
F.Briscese, A.M. 08, 09
Overview
1) The universe contains no dark energy field f
2) At late time the field f is generated by a relativistic field j, via a quantum
transition
4) The scale of the re-generation is dynamically obtained
given in terms of the coupling “g” between
G/H > 1
f, j
5) we can unify inflation with dark energy with inflation
6) We can use the same interaction for the inflaton f decay and its the late time
re-generation
Cosmology on
A.the
de la
Beach,
Macorra,
Playa
IFUNAM,
del Carmen,
IAC
January 2010
Inflation – Dark Energy Unification
i)
Inflation (accelerates univ.) => Flat at high energy
ii)
Dark Energy (accelerates univ.) => Flat al low
energy
iii)
but we require a long period of deceleration
dominated by radiation and later by matter
(nucleosynthesis, formation of galaxies, stars etc)
Are the 2 inflation periods connected ?
Can we have a single field producing inflation?
iii)
and
Require a V:
|V’/V|<1 , |V’’/V| < 1
dr/r =105 and
V(fo)=Vo
f coupling Vint (j relativistic field c , y SM particles)
Cosmology
on la
the
Beach, Inst.
Playa
Carmen,
A. de
Macorra
de del
Física,
UNAM January 2010
V(f)
Inflaton- Dark Energy Unification
f coupling Vint (j relativistic field c , y SM particles)
The process takes place
when
1)
G/H > 1
Inflaton Decay
f --> j  j  j
2) Reheating with
Standard model
with SM particles
3) Dark Energy
Re-generation
f and j relativistic
if we take
Cosmology on
A.the
de la
Beach,
Macorra,
Playa
IFUNAM,
del Carmen,
IAC
January 2010
Re-generation process:
• Start with No f particles
• E > Egen with n = V = r = 0
• Only for E < Egen
f particles are produced
• Potential V is generated
Cosmology on the Beach, Playa del Carmen,
January 2010
Linear evolution of perturbations
• DE Homogenous via Equation of state w(t) (Adiabatic sound speed ca(t) )
• DE Perturbations via Sound speed cs(t)
• Dark Matter
General fluid (e.g. Dark Energy)
with ca2 = w
^ = rest frame of the D. E. fluid
q = fluid velocity perturbation
x = Dark Energy
• DE perturbations are crucial to distinguish between different DE models
• sound speed cs
• adiabatic sound speed ca
A. deBeach,
la Macorra Inst.
de Física,
Cosmology on the
Playa
delUNAM
Carmen,
January 2010
Linear growth cs2 = w and w = wo + wa z/(1+z)
Nonlinear growth w = wo + wa z/(1+z)
Collapsed halos of mass M + dM
Total number of halos with M > Minf
L. Abramo et al astro-ph/0707.2882
Conclusiones
• Our universe is dominated today by Dark Energy
• Do not know the nature of Dark Energy
• Cosmological Constant or dynamical DE ?
• Determination of the cosmological parameters depend on the priors
used (e.g. parametrization of dark energy w(z) )
• Many models of Dark Energy in the market
• Perhaps the best (simplest) candidates are scalar fields but need to
derive the potential V = L f (f) and explain the smallness of the scale L
and the functional form f (f)
• DE perturbations are important to break degeneracy
(Cosmological constant has no perturbations or interaction)
• scalar fields produced at late time have less fine tuning on the
parameters, could explain the coincidence problem (e.g. condensates
models)
• or protected by symmetries (e.g. pNGB)
• or ….
Cosmology on the Beach, Playa del Carmen,
January 2010
Cosmology on the Beach, Playa del Carmen,
January 2010
Lagrangian
Eq. of motion
perturbations around the classical background
account for the quantum states (particles)
Eqs. of motion and Boltzmann equations
Cosmology on
A.the
de la
Beach,
Macorra,
Playa
IFUNAM,
del Carmen,
IAC
January 2010
Initial Energy Densities
Initial Conditions
gi =
r DE
p2
=
g D TD 4 ,
30
r rad
p2
=
g r Tg 4
30

(bos. deg 
7
ferm. deg)
8
g rad = 228 for MSSM E  1 TeV
g rad = 3.36 for E < 0.1 MeV
g D = 97.5 for Dark Group E  L DE = 42 eV
i) Initial = Unification Scale E = Lgut
W rad (L gut ) =
we have Tg = TDE
and all particles are massless, i.e. radiation.
ii)
At phase transition
E ~ LDE = 40 eV
g rad
= 0.7
g rad  g D
gD
W D (L gut ) =
= 0.3
g rad  g D
W rad (L DE ) =
W DE (L DE ) =
g rad
g rad  g D (TD / Tg )
4
g DE (TDE / Tg ) 4
g rad  g D (TD / Tg )
4
= 0.9
= 0.1
Cosmology on the Beach, Playa del Carmen,
January 2010
Cosmology on the Beach, Playa del Carmen,
January 2010