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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?

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

Energy and dark matter as modified gravity

- The Lagrangian L( R) is singular
- bifurcates into almost Einsteinian spaces,
- with different gravitational strength and

cosmological constant. - Bifurcation Set a swallow tail or butterfly

catastrophe - The coexistance of different Einsteinian domains
- The large scale distribution of darkness

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?

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 momentum P ?L/?RL(R)
- induces the Legendre transformation from Leff ?

H (R ?L/?R L) g1/2

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

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

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

curvature R 2? exp (2??/? ?/ (n-2)

)n/(n-2)U(?)??/? dU/d? - the higher--order effective Lagrangian Lexp (n

??/?/(n-2) ?) 2/(n-2)U(?) ??/? dU/d?

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.

The Isotropic Universe

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)

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

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!

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

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?

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 ?!.

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.

Galactic Dark Matter

- Observe galaxy rotation curve using Doppler

shifts in 21 cm line from hyperfine splitting

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?

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? ?

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

Cosmic Microwave Background

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

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

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

Type-IA Supernovae

- Clear indication for cosmological constant
- Can in principle be something else with negative

pressure - With wp/r,
- Generically called Dark Energy

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.

Cosmic Concordance

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

Constraint on Dark Energy

- Data consistent with cosmological constant w1

- Dark Energy is an energy that doesnt thin much

as the Universe expands!

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)

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???

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.

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

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

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)

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.

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

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.

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

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).

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

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

Energy and dark matter as modified gravity

- The effective Lagrangian L( R) is singular
- bifurcates into several almost Einsteinian

spaces, - with different gravitational strength and

cosmological constant. - Bifurc.Seta swallow tail or butterfly

catastrophe - The coexistance of different Einsteinian domains
- The large scale distribution of darkness