Emilio Casuso Romate and John Beckman- Dark Matter and Dark Energy: Breaking the Continuum Hypothesis?

8/3/2019 Emilio Casuso Romate and John Beckman- Dark Matter and Dark Energy: Breaking the Continuum Hypothesis?

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Volume 3 PROGRESS IN PHYSICS July, 2006

Dark Matter and Dark Energy: Breaking the Continuum Hypothesis?

Emilio Casuso Romate and John Beckman

Instituto de Astrosica de Canarias, E-38200 La Laguna, Tenerie, SpainE-mail: [email protected]

In the present paper an attempt is made to develop a ractional integral and dieren-

tial, deterministic and projective method based on the assumption o the essential

discontinuity observed in real systems (note that more than 99% o the volume occupied

by an atom in real space has no matter). The dierential treatment assumes continuous

behaviour (in the orm o averaging over the recent past o the system) to predict the

uture time evolution, such that the real history o the system is orgotten. So it

is easy to understand how problems such as unpredictability (chaos) arise or many

dynamical systems, as well as the great difculty to connecting Quantum Mechanics

(a probabilistic dierential theory) with General Relativity (a deterministic dierential

theory). I ocus here on showing how the present theory can throw light on crucial

astrophysical problems like dark matter and dark energy.

1 Introduction

In 1999 I published [1] the preliminaries o a new theory: the

General Interactivity. It was a sketched presentation o the

mathematical basis o the theory, i. e. the ractional integral

treatment o time evolution. In the present paper we extend

the ideas o General Interactivity to the ractional derivatives,

and so we can explain the outer atness o rotation curves,

last measures o SN Ia at high redshits, the uctuations in

the CMB radiation and the classical cosmology theory.

In 1933 Zwicky [2] ound that the Coma cluster o

galaxies ought to contain more matter than is inerred romoptical observations: many o the thousands o galaxies in

the cluster move at speeds aster than the escape velocity

expected rom the amount o visible matter and rom the

Newton theory o gravitation. In the 1970s, many authors

discovered that the speed o stars and clouds o hydrogen

atoms rotating in a galactic disk is nearly constant all the

way out to the edge o the galaxy [3, 4]. Using Newtons

law o gravitation, this implied that the amount o matter at

increasing radius is not alling away, against the observed

star-light suggests. Over past two decades, the measured

deection o light rom a distant star by a massive object

like a galaxy (gravitational lens) points to a mass-to-light

ratio or the lensing galaxies o about 150, and yet i galaxies

contained only observed stars the expected value would be

between 5 and 10 [5]. From the observed cosmic micro-

wave background (CMB, the relic radiation o the Big Bang

that flls the Universe) uctuations, we need that 23% o

the Universe is dark matter, and 73% is dark energy

[6, 7, 8, 9, 10]. Recent observations o SN Ia brightness show

that the expansion o the Universe has been speeding up.

This unexpected acceleration is also ascribed to an amount

o dark energy that is very similar than 73% o the Uni-

verse [11].

In Section 2 we show a review o the theory, in Section 3

we apply the theory to account or the observed dark matter

and dark energy, and in Section 4 we develop the conclusions.

2 The model

I start rom two hypothesis: (1) the irreversibility in time

o natural systems and (2) the interactivity among all the

systems in the Universe. These hypotheses imply an intricate,

unsettled and discontinuous (and hence non-dierentiable)

space-time. The dierential treatment projects a variable

X(t), whose value is known at a time t, to a successive

time, t + t, through the assumption o a knowledge otheir time derivative, X(t), as ollows: X(t + t) = X(t) ++ X(t)t. In many cases, to a good approximation, thereis proportionality between X(t) and X(t) so that X(t) X(t+t). Here I extend this projection, but with two crucialmodifcations: (a) I project a complete distribution o real

values (a set o measured values ordered in time) instead

o individual values at one time, and (b) I generalize the

derivative to the Liouville ractional derivative (to take into

account the possibility o the discontinuous space-time o

the system under study). This then gives the undamental

equation o the new dynamics:

dFRACdt

X(tpast) X(tfuture) . (1)

X(tpast) being a table o values o the variable X untilthe present time, X(tfuture) the same number o values oX but rom the present time to the uture (a projection), and a value between 0 and 1 that includes the key inormationabout the history o the system.

But or more physical sense, one must take the inverse

o equation (1), i. e.

X(tpast) 1

()

Ttpast

X(tfuture)

(tpast tfuture)1dtfuture (2)

82 E. Casuso Romate and J. Beckman. Dark Matter and Dark Energy: Breaking the Continuum Hypothesis?


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July, 2006 PROGRESS IN PHYSICS Volume 3

which is the ractional integration (or the Riemann-Liouville

integral) o X(tfuture), T being a time-period characteristico each system. The frst hypothesis, irreversibility, suggests

the necessity o projecting the values o X(t), weighted

by a unction o time that must be similar to the unctioncharacteristic o critical points, such as observed in the well

known irreversible phase transitions in Thermodynamics; or

example, the orm (TE TEC)0.64 or the time correlation

length o an infnite set o spins with a temperature TE nearthe critical temperature TEC [12]. Compare this with the term(tpast tfuture)

1 in equation (2). I call this weighting

generalized inertia; it is characteristic o each system in

the sense o incorporating into the exponent the history oall the interactions suered by the system, including those

interactions avoided by the dierential approximation (high

order terms in Taylor expansions) due to its small values.

To use the undamental equation (1) with maximum ef-

ciency, I invert equation (2) because this is an Abel integral

transorm, and there is a technique developed by Simmoneau

et al. [13] to optimize the inversion o Abel transorms. This

technique consists in making a spectral expansion using a

special kind o polynomials whose coefcients are obtained

by means o numerical integration, thus avoiding the basic

problem o amplifcation o the errors, a problem inherent

in numerical dierentiation; in the technique o Simmoneau

et al., measurement errors are incorporated into the coef-

cients o the spectral expansion and then propagate with time

without being amplifed.

In the present context one can see the time as a critical

variable, each present being an origin o time coordinates,with two time dimensions: the past and the uture. We should

note that in Quantum Mechanics two independent wave unc-

tions are needed (the real part and the imaginary part o the

total wave unction) to describe the state o a system at each

moment in time.

One can view General Interactivity as a third approxima-

tion to reality: the frst was the conception o continuous and

at space-time by Newton, the second was that o continuous

and curved space-time by Einstein. Here I see a discontinuous

space-time whose degree o intricacy measures the essential

cause o changing. As in Newtonian Dynamics, where the

orces are the causes o changing, and in General Relativity,

where modifcations o the metric o space-time are the

cause o changes in the motion o all massive systems, in

General Interactivity the exponent gives us a measureo the intricacy o the space-time seen by each system

through a given variable X. But how can we see Gravity

rom the new point o view o General Interactivity? From

(dierential) Potential Theory we know that the modulus o

the gravity orce per unit mass is the ollowing unction o

mass distribution, (x), in space:

FG(x) = G

(x)

|x x|2d3x (3)

and, comparing with the three-dimensional ractional integra-

tion o (x) we have:

R(x)

=

2

32

2

(x)

|x x|3d3x; (4)

FG(x) can be identifed with the 1-integral o (x) in three-dimensional space (= 1) except or a constant. So in the

present context the gravity orce can be interpreted as a one-

dimensional projection o the three-dimensional continuous

distribution o matter. It is not, then, a complete integral (this

would be = 3) and so the sum (integral) or obtaining thegravity is more intricate than the mass distribution (contin-

uous by defnition), i. e. the real discontinuity o mass distrib-

utions is transerred to the ractional integral instead o

working with a discontinuous (x). Gravity, like the electro-static orce, whose expression is very similar to FG(x),is seen as an inertial reaction o space-time, which would

tend to its initial (less intricate,i. e. simpler) state, towards astructure in which the masses were all held together without

relative motions; both orces are seen as reactions against the

action o progressive intricacy in the general expansion o

the Universe ollowing the Big Bang.

We take the total mass-energy o the Universe as the

observable magnitude X(t) to evolve in time using Eq. (2).The constancy o this variable gives 1 = 1

2(T2 t2past)

(where I take squared variables or simplicity in the use o

Simmoneau et al.s inversion technique). The greater past-

time variable, tpast, less indicating that the space-time ismore intricate with time; this is the reason or integrating

more ractionally (less ). So the parameter can also beconsidered as a measure o the entropy o the Universe.

Another key to understand General Interactivity comes

rom the classical Gaussian and Planckian distribution unc-

tions, to which real systems in equilibrium tend. The equi-

librium distribution unction or systems o particles, or col-

lisional and or collisionless systems (in the non-degenerate

limit [14]) is Gaussian; classical Brownian motion is an ex-

ample [15]; the equilibrium distribution unction or systems

o waves is a Planckian, and a key example is blackbody

radiation. I both distribution unctions evolve in time, then,

using the inversion o equation (2), we have the same fnal

result: the Planckian distribution. This tells us that whatever

the initial distribution at the beginning o the Big Bang (per-

haps both the Planckian characteristic o interacting waves

and the Gaussian characteristic o interacting particles co-

existed), their time evolution leads to a Planckian distribu-

tion, thereby connecting with the actual observed spectrum

o the Cosmic Microwave Background, which appears to be

almost perectly Planckian.

But, why is the Planckian more stable than the Gaussian

with the passing o time? The answer I propose is that suc-

cessive critical transitions (at each time), due to the complex-

ity caused by interactions at large distances, tend to ampliy

the Gaussian distribution to all range o energies, making it

E. Casuso Romate and J. Beckman. Dark Matter and Dark Energy: Breaking the Continuum Hypothesis? 83


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atter. This breaks the thermal homogeneity because o the

very dierent time evolution o many regions, due to the

delay in the transmission o inormation rom any one zone

to others that are ar away (note that the speed o the light

is a constant). This amplifcation goes preerentially to highenergies because there is no limit, in contrast with lower

energies, or which the limit is the vacuum energy.

In this context, then, the Universe is seen as an expansion

o objects that emitting inormation (electromagnetic waves)

in all directions, and one can dierentiate between two basic

kinds o interactions: (a) at small distances (the distance

travelled by light during a time that is characteristic o

each system) orming coupled systems showing macroscopic

(ensemble) characteristics, such as temperature or density,

well dierentiated rom those o their surroundings; and

(b) at large distances, interering one system rom another in a

complex manner due to the permanent change in the relative

distances due to the constancy o the speed o light, the

huge number o interactions and the internal variation o the

sources themselves. Note that this distinction between small

and large distances can be extended relative to each physical

system. For instance, a cloud o water vapour (as in the

Earths atmosphere) constitutes a system o water molecules

interacting over short distances, while the interaction between

one cloud and another is considered to take place over a large

distance. Inside a galaxy, the stars in a cluster are considered

to interact over short distances, while the interactions between

that cluster and the remaining stars and gas clouds in the

galaxy are considered as interactions over large distances.

In General Relativity there are no point objects; instead,all the objects in Nature are considered as systems o other

objects, even subatomic particles appearing to be composed

o others yet smaller.

I now ocus on one o the most puzzling interpretations o

Quantum Mechanics: the wave-particle duality. In Quantum

Mechanics the objects under study show a double behaviour

depending on what type o experiment one makes. An elect-

ron behaves as a particle in collisions with other electrons,

but the same electron passing through two gaps (enough

small and enough near each other) behaves as a wave in

that the outcoming electrons orm an intererence pattern.

In General Interactivity each particle is considered as a

system, and we know that the equilibrium distribution o

random particles is Gaussian, and that ater the time evolution

given by Eq. (2) the distribution transorms into a Planckian

(the interaction with the other systems drives the random

set o particles) which is the distribution to which a set o

interacting waves in a cavity naturally tend. Furthermore,

the Planckian can be decomposed into a set o Gaussians,

so that the double nature o matter/energy is ensured. The

act that a Planckian can be the result o the addition o

Gaussians o dierent centres and amplitudes is interpreted

as the Planckian representing an ensemble o random motions

in turn represented by Gaussians, which fnd a series o walls

to which resulting in certain reexion and certain absorption.

As already demonstrated [15], both processes, reexion and

absorption by a barrier, are equivalent to the addition and the

subtraction, respectively, o two Gaussians: the main Gaus-

sian and that which emerge as a consequence o the barrier(by displacing its centre to the other side o the barrier). A

Gaussian, then, converts into a set o several other Gaussians

at progressively smaller amplitudes as a consequence o the

existence o barriers, and the envelope is a Planckian. There

is a partial reection at each barrier in the direction o higher

energies, while the reection is total to the lower energies and

the absorption o unreected part must be added to the let

o the barrier. This argument can be applied to explain the

Planckian distribution observed in the Cosmic Microwave

Background Radiation: the energy barriers can be thought o

as the consequence o the existence o wrinkles in space-time,

caused by the fniteness o the Universe (closed box) and the

uncoupled expansion o the content with respect to the box,

or by breaking o the expansion because o the collision o

the outer parts with another medium, or by the succession o

several bangs at the beginning, instead o only one bang.

3 Dark Matter and Dark Energy

Another example o application o this theory is the generali-

zation o one o the most important theorems in Field Theory,

Gausss theorem, leading to a possible solution (as a kind o

Modifed Newtonian Dynamics theory) o the well known

problem o the lost mass o the Universe and its associated

problem o dark matter [16]. Assuming the well knownobservation o the infnitesimal volume occupied by matter

relative to holes in Nature (the nucleus o an atom occupy less

than 1% o the atoms volume, and gas clouds in the inter-

stellar medium have densities o 1 atom per cubic centimeter

or less), one must consider the possibility o relaxing the con-

tinuum hypothesis. The Gausss theorem can be expressed,

simplifed and or the gravitational feld, asS

gNdS = 4GM , (5)

where gN is the intensity o the gravity feld over a closedsurace, S, which contains the mass, M, which is the origin

o the feld, on the assumption o continuity, and G is the uni-versal gravity constant. So, integrating Eq. (5) on the assump-

tion o gN constant overS, we get gN=4GM/S, withS =

S

dS. I we take as the starting point the dierentialorm o Gauss theorem, and then we take in Eq. (5) the rac-

tional, instead o the ull, integral and also assume g const,we have

g(r) =4GM

1( 22 )(2 )

SdX

|SX|2

. (6)

Because is less than 2, g is greater than gN, and thisresult could explain the observational act o gN being very



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July, 2006 PROGRESS IN PHYSICS Volume 3

050

100

150

200

r(pc)

Fig. 1: Rotation curve (kms1) or NGC3198. Crosses are observ-

ational data points taken rom van Albada et al. [17]. Full line is the

prediction o the present theory, and dashed line is the prediction

by the Newton law o gravitation.

small in explaining, or instance, many galactic rotation

curves ar rom the central regions. For, assuming spherical

simetry one has:

g(r) = gN4r2

1(22 )

( 2 )

20

2r2 sin dr2

. (7)

And taking = 1 one has g(r) = gNr, which introduced

in the classical centriugal equilibrium identity V2

r = g leads

to the amazingly V = (GM)1/2 constant as is observed orat rotation curves that needs dark matter (see Fig. 1).

More complicated treatment can be made: in the integral

treatment one can consider the basic constituents o matter

(the atoms) and the infnitesimal size o the volume occupied

by the atomic mass (the nucleus) with respect to the size

o the atom, and then one fnds the necessity o take into

account that ubiquitous nature o the big holes existing inside

the matter (the atoms and molecules inside the very low

density galactic gas clouds ampliy the hole eect respect

to the whole cloud and then ampliy the inuence in the

macroscopic gravitational (massive) behaviour). So one can

consider the hypothesis o continuity as a frst approximation,

and one can re-examine the Gauss theoremS

gdS = 4G

V

dV , (8)

where g is the gravitational feld over the surace S, S isany closed surace containing the massive object which is

the source o the feld, G is the gravitational constant,

is the density, and V is the volume contained within thesurace S. And one can generalize Eq. (5) in the sense o take

both integrals as ractional integrals ( and respectively)which leads to normal integrals or some especial case. I

one assumes, or simplicity, spherical symmetry or the gasmass distribution in the galaxy, one has:

= 0 e(rr0)

r (9)

and assuming g constant over the now non-necessary conti-nue surace (the ractional integration takes this into account)

one has:

g = 162G0C2()

f()r2C1()r2

r2r3e

rr0r dr , (10)

where f() is some unction o ,

C2() = 3/2

32

2

, (11)

C1() = 1

22

2

, (12)while

gN = 4G0

r2

r2e

rr0r dr . (13)

So, in the especial case when = 3 and = 2 we haveg = gN. Then, expanding r

3 1+(3)log r + . . . as 3 and r2 1(2)log r + . . . as 2, and in-

cluding the expansions into Eq. (13) one has

g 8C2()

f()C1()

1 ( 2)log r

gN

4G0r2

( 3)(log r)r2e

rr0r dr

.

(14)

And integrating by parts and taking very large values or

r, we have

g gN8C2()

f()C1()

1 + ( 2)(3)log 2r

. (15)

And or typical values o observed at rotation curves

(5kpc r 20kpc) we have that g gNr represents a goodapproximation. So, or certain values o and ( less than2 and greater than 3) one has that outer rotation curves can

be at as observed.

But the most puzzling problem up-to-date in cosmology

is the necessity o adding ad hoc a dark energy or negative

pressure (the so called by Einstein cosmological constant)

to the main equation o General Relativity to account or

the last measures on supernovae Ia and the uctuations in

the cosmic microwave background radiation which implies

a at accelerating expanding universe. The feld equation

o General Relativity was ormulated by Einstein as the

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generalization o the classic Poisson equation which relates

the second derivative o the potential associated to thegravitational feld with the assumed continuous mass distrib-

ution represented by the volume density :

2 + 4G = 0 . (16)

For comparison, the similar equation in General Rela-

tivity, which relates the mass and energy distribution with

the dierential changes in the geometry o the continuum

space-time, is (see e. g. Einstein [18]):

R

1

2gR

+ T= 0 . (17)

But or the last equation be coherent with the last inde-

pendent measures o SN Ia and uctuations o CMB radia-

tion, we need to add a term g to the let side o equationwhich represents near 73% o all the other terms. This prob-

lem is avoided naturally i we consider a discontinuous

space-time, and then we re-ormulate the equations by using

the ractional derivative instead the ull derivative. In that

case, the second derivative is less than the ull derivative,

and then the cosmological constant is not needed at all to

equilibrate the equations. In act the -ractional derivativeo the unction r is given by [19]:

Dr =( + 1)

( + 1)r (18)

or greater than 1, greater than 0. But as 1,r being the gravitational potential. And as one cansee, taken a fxed value o , as increase, the -derivativedecrease. Or to be more precise, i we assume that the con-

stant to be added to the let side o Eq. (17) represents the

73% o all the matter and energy in the Universe, one has:

lim1

( + 1)R2

( 2 + 1)R 1.73 , (19)

where R is a characteristic scale-length o the Universe.And the relation (19) works or values o greater butvery near 2, being 2 the value corresponding to the usual

second derivative. So we conclude that taking a value, or

the derivatives in the feld equations, slightly greater than

the usual 2, we are able o include the cosmological constant

inside the new ractional derivative o the classical feld

equations.

4 Conclusions

The new theory o the General Interactivity can be applied

to many felds o natural science and constitutes a new step

orward in the approximation to the real behaviour o Nature.

It assumes the necessity o explicitly taking into account

the real history o a system and projecting to the uture.

However, it also takes into account the non-uniormity o the

distribution o holes in the Nature and is thereore a theory

o discontinuity. The new theory can account or naturally

the needed amounts o dark matter and dark energy as a light

modifcation o classical feld equations.

Acknowledgements

This work was supported by project AYA2004-08251-CO2-

01 o the Spanish Ministry o Education and Science, and

P3/86 o the Instituto de Astrosica de Canarias.

References

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Emilio Casuso Romate and John Beckman- Dark Matter and Dark Energy: Breaking the Continuum Hypothesis?