Domains Structure of Dark Energy and Dark Matter

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Domains Structure of Dark Energy and Dark Matter

• Feo V Kusmartsev

In collaboration with Franz E. Schunck, and
Eckehard W. Mielke

• We cant see neither Dark Matter nor Dark Energy
• Then why do we talk about it?

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Topics of Discussion

• If Dark Matter exist, is it homogeneously
  distributed ?
• If Dark Energy (?) exist, is the ? has constant
  value over the Universe ?
• What are the content of Dark Energy (?) and Dark
  Matter?
• Einsteinian space

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Energy and dark matter as modified gravity

• The Lagrangian L( R) is singular
• bifurcates into almost Einsteinian spaces,
• with different gravitational strengthand 
  cosmological constant.
• Bifurcation Set a swallow tail or butterfly
  catastrophe
• The coexistance of different Einsteinian domains
• The large scale distribution of darkness

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The dependence of the effective Lagrangian L on
the scalar curvature R

• Not analytical in the (L,R) plane
• several branches of almost linear Lagrangians
  L (R ?eff )/?eff
• distinguished by gravitational constant ?eff and
  cosmological constants ?eff .
• unifying picture of dark matter and energy and
  modified gravity  
• Domains of Darkness? Size of the Domains?

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Legendre map

• the effective nonlinear Lagrangian density
  Leff L (R) g1/2 the determinant g of the
  metric g?? and  the scalar curvature R of
  Riemannian geometry
• The corresponding field momentumP ?L/?RL(R)
• induces the Legendre transformationfrom Leff ?
  H (R ?L/?R L) g1/2
•  

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By conformal mapping to an Einstein frame

• satisfying g1/2 -gt Pn/(n-2) g1/2
• in ngt2 dimensions there arises the effective
  potential
• u (P)   Pn/(2-n) (R P- L)
• The fixed point of conformal mapping gives H0
  and conventional Einstien Gravity L R /(16?G)
• u (P) is a highly nonlinear differential
  equation 

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The parametrization L(R)

• Use Lagrange multipliers or the method of
  Helmholtz, in order to recover  the scalar
  curvature R ?H( P)/?P from H
• where the reparametrized Hamiltonian
  H(P)Pn/(n-2) u( P)
• u(P) plays the role of a generating
  superpotential.
• This yields a parametrization for corresponding
  Lagrangian L P R-H,   convenient for displaying
  the bifurcations of the transformed systems.

A.Benitez, A. Macias, E. Mielke et al, IJMP, A12,
(1977), 2835
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Lagrangian and Curvature

• We relate the dimensionless conformal factor ?
  2?P to a real scalar field ? (?/?)1/2 ln (2?P)
• with self-interacting potential
  U(?)(2?)2/(2-n)u( P)
• ?(n - 1)/(n- 2) is the Brans-Dicke parameter
• ? 8?G is the gravitational constant
• LDM ?g /(2?) R  ? g?? (???) (???) - 2U(
  ?)  
• the following parametric reconstruction of
  curvatureR 2? exp (2??/? ?/ (n-2)
  )n/(n-2)U(?)??/? dU/d?
• the higher--order effective LagrangianLexp (n
  ??/?/(n-2) ?) 2/(n-2)U(?) ??/? dU/d?

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Weighing the Universe ?-f actor

• ? is the density parameter
• Stars visible Universe is made up of stars and
  gas estimate the masses of stars through their
  luminosity.
• Baryons protons and neutrons that make up atomic
  nuclei baryons, estimate though recently the
  cosmic microwave background. Baryonic matter
  includes stars, but not all baryons are
  incorporated into stars.
• Non-baryonic dark matter must be made of some
  exotic stuff non-baryonic matter.
• Dark energy or cosmological constant, to which
  Einstein gave the symbol ?. Now it is used to
  explain the data from supernova explosions, the
  cosmic microwave background and large-scale
  structure.

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The Isotropic Universe
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The Cosmological Principle

• Universe highly isotropic
• CMBR anisotropy ? O(105)
• Unless we occupy the center of the Universe, it
  must also be homogenous
• Isotropy and Homogeneity
• ? maximally symmetric space
• Flat Euclidean space R3
• Closed three-sphere S3SO(4)/SO(3)
• Open three-hyperbola SO(3,1)/SO(3)

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Friedman Equation

• Equation that governs expansion of the Universe,
  k is the enery parameter
• k1 (closed), k1 (open), k0 (flat)
• energy density r
• First law of thermodynamics
• For flat Universe
• Matter-dominated Universe
• Radiation-dominated Universe
• Vacuum-dominated Universe

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Structure Formation

• Jeans instability of self-gravitating system
  causes structure to form
• Needs initial seed density fluctuation
• Density fluctuation grows little in radiation- or
  vacuum-dominated Universe
• Density fluctuation grows linearly in
  matter-dominated Universe
• If only matterbaryons, had only time for 103
  growth from 105 not enough time by now!

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Baryonic dark matter
About 200 seconds after the Big Bang, the
temperature of the universe was similar to that
of the heart of a star, and nuclear fusion
reactions took place similar to those that now
power our Sun. In a few seconds, the neutrons
that were produced in the first moments of the
universe were incorporated into nuclei of the
first few elements of the periodic table
helium-4, a hydrogen-2 (deuterium), helium-3 and
lithium-7 The amounts produced depend on the
density of the universe. The baryon density is
about 0.02/h2, where h is Hubble parameter.
Since h 0.7, we have ? (baryons) 0.04
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Dark Matter hot or cold? The candidates for
non-baryonic dark matter divide into two
categories Hot Dark Matter is made of light,
fast-moving particles. In the early universe,
hot dark matter particles move so fast that they
can be trapped only in the largest concentrations
of matter the largest structures, giant
superclusters of galaxies, form first. The small
picture at bottom right shows a simulated Hot
Dark Matter universe. Massive neutrinos are the
most likely candidate for Hot Dark Matter. Cold
Dark Matter is made of slowly moving particles
that can be trapped in galaxy-sized regions of
the early universe. In a Cold Dark Matter
universe, galaxies form much earlier, and the
great superclusters are less well defined (top
left). WIMPs are cold dark matter. Is dark
matter hot or cold?
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The real universe is shown in the colour picture,
from the Automatic Plate Measurement galaxy
survey. It looks far more like the Cold Dark
Matter simulation, and mathematical tests confirm
this. So the dark matter is cold ?!.
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MAPping the Universe Our best tool for measuring
cosmological parameters is the Cosmic Microwave
Background, relic radiation from about 400000
years after the Big Bang. The tiny temperature
variations in this sea of radiation tell us about
the structure of the Universe at that early era,
and are sensitive to all the cosmological
parameters. The COBE satellite was the first to
discover these temperature variations, but its
angular resolution of 10 was too poor to give
useful results. In the past few years,
ground-based and balloon-borne experiments have
mapped small regions of the microwave sky with
better accuracy, and have provided important
insights. But the most exciting results are
those recently released by NASAs Wilkinson
Microwave Anisotropy Probe, WMAP. The picture
below shows WMAPs map of the Universe 400000
years after the Big Bang. The colour variations
here will eventually evolve into the galaxies,
galaxy clusters and superclusters of the APM
survey above.
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Galactic Dark Matter

• Observe galaxy rotation curve using Doppler
  shifts in 21 cm line from hyperfine splitting

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Galactic Dark Matter

• Luminous matter (stars)
• Wlumh0.0020.006
• Non-luminous matter
• Wgalgt0.020.05
• Only lower bound because we dont quite know how
  far the galaxy halos extend
• Could in principle be baryons
• Jupiters? Brown dwarfs?

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MAssive Compact Halo Objects(MACHOs)

• Search for microlensing towards LMC, SMC
• When a Jupiter passes the line of sight, the
  background star brightens
• MACHO EROS collab.
• Joint limit astro-ph/9803082
• Need non-baryonic dark matter in halo
• Primordial BH of M? ?

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Dark Matter in Galaxy Clusters

• Galaxies form clusters bound in a gravitational
  well
• Hydrogen gas in the well get heated, emit X-ray
• Can determine baryon fraction of the cluster
• fBh3/20.056?0.014
• Combine with the BBN
• Wmatterh1/20.38?0.07

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Cosmic Microwave Background
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Observational evidence for Dark Energy
1)supernova explosions,2) the cosmic microwave
background and 3) large-scale structure
1)Supernovae and Universal Acceleration When a
dying star explodes, it becomes briefly as
luminous as a small galaxy. A certain type of
supernova (Type Ia) always explodes with about
the same brightness these are standard candles.
They are ideal probes for studying the expansion
of the Universe at large distances.
As bright as the host galaxy
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Type-IA Supernovae

• Type-IA Supernovae standard candles
• Brightness not quite standard, but correlated
  with the duration of the brightness curve
• Apparent brightness
• ? how far (time)
• Know redshift
• ? expansion since then

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Collection large samples of Type Ia supernovae in
the 1990s, indicates a strange thing instead of
slowing down over billions of years due to the
action of gravity, the expansion of the Universe
seemed to be speeding up. The only possible
explanation for this is Einsteins infamous
cosmological constant, which is predicted to have
exactly this effect. The picture on the right,
from an analysis of 40 supernovae by the
Supernova Cosmology Project, shows the results
the real Universe is 90 likely to lie within the
green ellipse. The contribution of the
cosmological constant to ? is not zero, and
indeed is greater than the contribution of all
matter (visible or dark, baryonic or
non-baryonic). Unfortunately the supernova data
dont determine the contributions of matter and
energy separately, but only their difference.
The results of the experiment are ?(dark energy)
- ?(matter) 0.4
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Type-IA Supernovae

• Clear indication for cosmological constant
• Can in principle be something else with negative
  pressure
• With wp/r,
• Generically called Dark Energy

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A detailed analysis of WMAPs map of the cosmic
microwave background gives values for all the
main cosmological parameters. WMAPs
conclusions ?(total) 1.02 ? 0.02 ?( matter)
0.27 ? 0.03 The Universe is dominated by dark
energy. ?( baryons) 0.045 ? 0.005 About 85 of
the matter in the Universe is non-baryonic.Over
90 of the baryonic matter is dark (since
O(stars) 0.004). ?( hot dark matter) lt 0.015
(95 confidence) The non-baryonic dark matter is
cold. Thus we can conclude from cosmology that
WIMPs or something like them make up at least 80
of the total matter content of the Universe.
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Cosmic Concordance

• CMBR flat Universe
• W1
• Cluster data etc
• Wmatter0.3
• SNIA
• (WL2Wmatter)0.1
• Good concordance among three

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Constraint on Dark Energy

• Data consistent with cosmological constant w1

• Dark Energy is an energy that doesnt thin much
  as the Universe expands!

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Particle Dark Matter

• Stable, TeV-scale particle, electrically neutral,
  only weakly interacting
• No such candidate in the Standard Model
• Supersymmetry (LSP) Lightest Supersymmetric
  Particle is a superpartner of a gauge boson in
  most models bino a perfect candidate for WIMP
• But there are many other possibilities
  (techni-baryons, gravitino, axino, invisible
  axion, WIMPZILLAS, etc)

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Embarrassment with Dark Energy

• A naïve estimate of the cosmological constant in
  Quantum Field Theory (zero mode energy)
  rLMPl410120 times observation (-gtCatastrophe!)
• The worst prediction in theoretical physics!
• People had argued that there must be some
  mechanism to set it zero
• But now it seems finite???

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Quintessense?

• Assume that there is a mechanism to set the
  cosmological constant exactly zero.
• The reason for a seemingly finite value is that
  we havent gotten there yet
• A scalar field is slowly rolling down the
  potential towards zero energy
• But it has to be extremely light 1042 GeV. Can
  we protect such a small mass against radiative
  corrections? It shouldnt mediate a fifth
  force either.

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Cosmic Coincidence Problem

• Why do we see matter and cosmo-logical constant
  almost equal in amount?
• Why Now problem
• Actually a triple coincidence problem including
  the radiation
• If there is a fundamental reason for
  rL((TeV)2/MPl)4, coincidence natural

Arkani-Hamed, Hall, Kolda, HM
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Minimally coupled dark matter-energy

• dark matter (energy) as a scalar field ? with
  self-interaction U(?) minimally coupled to
  gravity
• 2?LDM ?g R  ? g?? (???) (???) - 2U( ?)
   
• We conformally transformed LDM(R), into an
  effectively higher-order Lagrangian of R
•  

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Model for Dark Energy and Matter

• the potential U m2 ?2 (1 -? ?4)
• m is the  mass of an ultra-light scalar and
• ? the coupling constant of the self-interaction
• u( P)3 m ln (2?P)2 1 -(9 ?/4 ?2 ) ln4 (2?P)
• In n4 dimensions, there is the exact parametric
  solution
• R 6m2P ln(2?P) 1ln(2?P) -27?/(4?2)ln4(2?P)-(9?/
  4?2 )ln5 (2?P) 
• H 3 m2 Pln(2?P)2 1 -(9 ?/4 ?2 ) ln4
  (2?P) 
•  
• L3m2P2 ln(2?P) 2 ln(2?P)-27?/(2?2 )ln4 (2?P) 
  -9?/(4 ?2) ln5 (2?P) 
•   

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Bifurcation of Lagrangians at ?0

• the bifurcation set is the swallow tail
  catastrophe.
• The bifurcation set consists of 3 classes
•        for Rgt0 of two local minima and one
  maximum
•        or Rlt0, we have just one minimum and one
  maximum.
•        Each of the minima merges'' with a
  maximum at the cuspoidal points (A,B)
•        and, then for RgtB or RltA, disappears.

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Classification

• Positive Value of ?
• The minima (the segment A and the segment (B,
  origin) are characterized by, d2H/ dP2 gt0
• The maximim (the segment AB) is characterized
  by, d2H/ dP2 lt0

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Coexistence of the Domains

• These two or three states may coexist with each
  other
• can be described approximately by the same
  effective Lagrangian
• LDM(R) R /?eff - ?eff .
• With different gravitational constant ?eff and
  cosmological constant ?eff .
• The two or three states emerge as a fixed point
  of the conformal transformation
• They are approximate Einstein spaces, but of
  different strength.

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Dark Energy Distribution

• Our bifurcation set indicates that the Universe
  is locally described by Einstein's GR
• each local patch of the Universe may have
  different strength and dark energy')
• In domains (in the maximum), with the positive
  cosmological constant the inflation may be still
  going

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The butterfly catastrophe

•  There are three cuspoidal points A, B and C
  which are associated with the highest
  singularities.
•  In A,B,C points the minimum merges with the
  maximum.
• Each local minimum (the segment AC and the
  semi-infinite segment B) is associated with a
  stable state
• Each local maximum (saddle) is associated with an
  unstable state (segment AB, and the semi-infinite
  segment C).

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An effective Einsteinian gravity

• Domains  with a positive effective cosmological
  constant are still in an unstable inflationary
  phase.
• The strong' gravity state (the semi-infinite
  segment B) with a negative cosmological constant
  corresponds to a stable but collapsing deflation
• The Universe  splits into regions with different
  gravity while locally only one of these phases  
  arises (the first order phase transition)
• The unstable solutions  are only in
  quasi-equilibrium with the stable ones
• At very long time scales (gtgt Planck time) when
  inflation will be ceased, only stable phases
  remain

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Outlook

• For dilute dark matter the Lagrangian is LHE
  R/(2?).
• For dense dark matter, the resulting effective
  Lagrangian is LHE R/(2?eff ) with the
  gravitational coupling ?eff gt? larger than
  Newton's.
• This reminds Milgrom's suggestion of MOND
  (Modified Newtonian Dynamics)
• Depending on its sign, the cosmological constant
  ?eff in a bifurcation can also model dark energy
  or an accelerating phase of the present epoch of
  Universe (like anti-gravity').
• There may arise ?eff lt? - the gravitational
  screening"
• MACHOS can be now described by some specific
  non-linear modifications of the Hilbert-Einstein
  action

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Energy and dark matter as modified gravity

• The effective Lagrangian L( R) is singular
• bifurcates into several almost Einsteinian
  spaces,
• with different gravitational strengthand 
  cosmological constant.
• Bifurc.Seta swallow tail or butterfly
  catastrophe
• The coexistance of different Einsteinian domains
• The large scale distribution of darkness