# Feebly Self-Interacting Cold Dark Matter: New theory for the Core-Halo structure in GLSB Galaxies

## Abstract

We explore the low energy cosmological dynamics of feebly self-interacting cold dark matter and propose a new simple explanation for the rotation curves of the core-halo model in massive LSB (Low Surface brightness)galaxies. We argue in favor of the truly collisionless nature of cold dark matter,which is feebly,self-interacting at small scales between epochs of equality and recombination.For this,we assume a model, wherein strongly coupled baryon-radiation plasma ejects out of small regions of concentrated cold dark matter without losing its equilibrium. We use the Merscerskii equation i.e. the variable mass formalism of classical dynamics.We obtain new results relating the oscillations in the CMB anisotropy to the ejection velocity of the baryon-radiation plasma,which can be useful tool for numerical work for exploring the second peak of CMB. Based on this model, we discuss the growth of perturbations in such a feebly self-interacting,cold dark matter both in the Jeans theory and in the expanding universe using Newton’s theory.We obtain an expression for the growth of fractional perturbations in cold dark matter,which reduce to the standard result of perturbation theory for late recombination epochs. We see the effect of the average of the perturbations in the cold dark matter potential on the cosmic microwave background temperature anisotropy that originated at redshifts between equality and recombination i.e. . Also we obtain an expression for the Sachs-Wolfe effect,i.e. the CMB temperature anisotropy at decoupling in terms of the average of the perturbations in cold dark matter potential.

Self-interacting,Cold Dark matter,Core-Cusp,Dark matter halos, Collisionless,WIMP,Rotation Curves,LSB Galaxies, Baryon-Radiation plasma,Dark Energy,Equality,Recombination,Decoupling, ,Perturbations,Jeans Theory, CMB, Sachs-Wolfe effect,Merscerskii equation

## I Introduction

Flat cosmological models with a mixture of ordinary baryonic matter, cold dark matter, and cosmological constant(or quintessence) and a nearly scale-invariant, adiabatic spectrum of density fluctuations are consistent with standard inflationary cosmology. They provide an excellent fit to current observations on large scales( Mpc). Currently, the constitution of the universe is 4% baryons, 23% dark matter and 73% dark energy (1); (2); (3); (4); (5); (6).

In the standard hot Big Bang model, the universe
is initially hot and the energy density is dominated by radiation. The
transition to matter domination occurs at . In the
epochs after equality and before recombination, the universe remains
hot enough. Thus the gas is ionized, and the electron-photon scattering
effectively couples the matter and radiation (7). At
, the temperature drops below K. The
protons and electrons now recombine to form neutral hydrogen
and neutral helium. This event is usually known as recombination
(8); (9); (10). The
photons then decouple and travel freely. These photons which
keep on travelling till present times are observed as the cosmic
microwave background(CMB). The cold dark matter theory including
cosmic inflation is the basis of standard modern cosmology.

This model ,which we shall call ”Standard”CDM,or sCDM for short is consistent with many fundamental cosmological observations:the age of the universe as measured from the oldest stars (11)),the extragalactic distance scale as
measured by distant Cepheids (12)); the primordial abundance of the light
elements (13)), the baryonic mass fraction of galaxy clusters (14), the amplitude of the Cosmic Microwave Background fluctuations measured by COBE
((15)(16), the present-day abundance of massive galaxy clusters (17) ), the shape and amplitude of galaxy clustering patterns (18)), the magnitude of large-scale coherent motions of galaxy systems (19), (20) ), and the world geometry inferred from observations of distant type Ia supernovae (21) , (22)),large scale structure data (23) among others.Moreover,this model assumes the dominance of non-baryonic matter (24); (25),which is massive,weakly interactive and collisionless (26) and has negligibly small primeval velocity dispersion together with electromagnetic neutrality (27); (28).The word Cold means that the thermal motions of DM
particles were essentially negligible at the time of matter-radiation
equality i.e.
the ratio ,where is the gravitation potential and T and M
represent the temperature and mass of the dark matter particle respectively.Though,sCDM
predicts and matches many available data fairly well (29); (30) and predicts a lot about structure formation (31); (32),but at small scales,the core-cusp issue is one of the major problems which cannot be explained by sCDM.The origin of this problem lies in the fact that there is no mechanism in sCDM which can explain a very steeply rising rotation curves from CDM simulations as compared to slowly rising ones from observational data
(33),(34).There is a need to explain the positive slopes in the outer parts of rotation curves of LSB galaxies,upto the outermost measured point,especially more positive than the HSB galaxies (34).Either the picture suggested by CDM simulations is wrong or we need new physics under the CDM paradigm to resolve the core-cusp issue i.e. to try to explain the observed rotation curves of DM dominated galaxies (LSBs)(35)(36).

The existence of clusters( mpc) and groups of galaxies suggests
that the galaxy formation is due to the gravitational instability
of a spatially homogenous and isotropic expanding universe. The
Perturbations of such a model have been first investigated by
(62). Then within the framework
of general relativity by (40); (41). Also
it was studied in a newtonian model(42). The results
obtained from relativistic theory were similar to the standard
Jeans theory. In the early universe, radiation fixes the expansion
rate. This is due to low density and self-gravity of dark matter
(43); (44).

We know that the dark matter reveals itself only through its gravitational interactions.Except some recent astrophysical observations,there are no observational facts that indicate existence of significant interactions between the ordinary and dark matter particles.The observations of the Bullet cluster,more formally known as 1E0656-56 (45) and other merging clusters CL 0152-1357 (46) and MS 1054-0321 (47) prove that the dark matter components of the colliding clusters do not perturb each other much during the collision, allowing themselves to pass right through.Independent results on the dark matter cross-section supporting the collisionless nature have been reported from the studies of A2744 (48),MACS J0025. 4-1222 (49).There are several other merging clusters that provide results similar to the Bullet cluster A1758 (50),(51),A2163 (52).That is these observations support the WIMP based dark matter models

We know that the self-interacting nature of cold dark matter has been supported by some recent observational data.The merging cluster of galaxies have the power to directly probe and place limilts on the self-interaction cross-section of dark matter (53) .This has been used to place upper limits on the dark matter particle self-interaction cross-section of the order of 1 .If the dark matter has a large self-interaction cross-section,then it would not be located in the vicinity of the cluster galaxies.Williams and Saha have suggested that a kpc scale separation between stellar and dark matter components in the cluster A3827 may be evidence for the dark matter with a non-negligible self-interaction cross-section (54).A large self-interaction cross-section for darkmatter much beyond the upper limit on the cross-section derived from the Bullet cluster has been derived by Randal et al (55).A higher limit on the DM self-interacting cross-section has been derived in (56).The galaxies outrunning the dark matter in the Musket Ball cluster DLSCL J0916.2+2951 suggests that dark matter particles do interact and slow down like a gas.The self-ineracting nature of dark matter has been proposed as a possible solution for the shallow core of the halos (57).This excludes a very strong self-interacting DM, as this could lead to gravothermal catastrophe with a very steep central density profile in galaxies (58).

In this paper,we strongly argue in favor of the collisionless nature of cold dark matter particles,which are feebly self-interacting at very small scales.This we do by an assumption inherent in the model, which ensures that the dark matter is collisionless for all epochs upto recombination.This is an improvement over the standard perturbation theory in the static and the Newtonian expanding scenario.We incorporate the feebly,self-interacting nature by allowing the cold dark matter to feel the pressure of the baryon-radiation fluid,at very small scales.This is because of the very heavy nature of cold dark matter particles and due to minor fluctuations in it’s velocity as a result of self-interactions.

After equality, the main contribution
to the gravitational potential is due to the cold dark matter.Therefore in the metric of the perturbed FRW universe, the main contribution to
the potential comes from an imperfect fluid, i. e. the dark
matter. We consider that the difference is very small as
compared to .The ratio of to is proportional to the ratio of the photon mean free path to
the perturbation scale. Here and are the newtonian potential
and the perturbations to the spatial curvature in a conformal Newtonian
gauge respectively(see eq. 48 sec II). We neglect the contribution
of the baryon density in realistic models, where baryon contributes only
a small fraction of the total matter density.The potential of cold dark matter is time-independent both
for long wavelength and short wavelength perturbations.This is because it is highly non-relativistic. There is a strong coupling between the
baryon and the radiation in the plasma.So, we treat it as a single perfect fluid for low baryon densities. We neglect the non-diagonal components in the energy- momentum tensor of dark matter. This is because for very small photon mean free paths is same as . Therefore, we treat it as a single perfect fluid for many epochs after equality
upto the recombination. It is only after recombination, that the baryon
density starts increasing. This is after the primordial nucleosynthesis
of hydrogen and helium is complete. The radiation after decoupling from
matter at very late recombination epochs evolves separately. The CMB
anisotropy of epochs between equality and recombination gives important
information about the perturbations in the cold dark matter i. e. the
modes that enter the horizon before recombination.

We assume a model,based on the variable mass formalism,inspired from the Merscerskii equation of classical dynamics.Here,the strongly coupled baryon-radiation plasma ejects out
of a concentrated region of cold dark matter.In the classical Merscerskii problem it is the relative velocity of escaping
(or incident)mass with respect to the center of mass of the body (which is at rest…).We can use this model here for a small volume element filled in by the baryon-photon fluid,whose skeletal structure is composed of dark matter .This volume element is of the order of kpc.The outermost skeletal region extending for kpc has exponentially decreasing density profile.Due to very heavy and ultra non-relativistic nature of CDM, the center of mass of the volume element will be at rest for an anisotropic ejection of baryon-photon plasma.In the standard Einstein static case of jeans theory or in the expanding Newtonian one,we cannot appreciate the feeble self-interactions triggered in the cold dark matter particles,at very small scales,due to the flow of strongly coupled baryon-photon plasma.The investigation of the perturbation theory using the variable mass formalism gives an opportunity to see the relative transfer of energy and momentum between cold dark matter particles,especially if they can self-interact.Also,in the standard Newtonian approach or in the Jeans theory,we do not have a method to preserve the collisionless nature of cold dark matter,throughout the evolution of perturbations in the matter,with its various substituents.Here,we can preserve the collisionless nature of cold dark matter particles throughout and still recover the standard CMB.Moreover,we have a new physics to explain the core-halo structure of massive,DM dominated (LSB) galaxies.

The picture that the dark matter forms the skeletal structure,which is filled in by other matter is supported from latest cosmological observations(59)
(60).Also,we know that the baryonic matter collapses in the gravitational potential of dark matter.This implies that that there will be regions of dark matter across which there will be a flow of baryonic matter and radiation.Therefore,we need a model,which can emphasize the flow of strongly coupled baryon-radiation plasma through any volume element formed by the skeletal structure of dark matter.This could help us in understanding the growth of velocities and internal masses of galaxies with distance from the galactic centre and thus try to resolve the core-cusp issue in LSB galaxies.

We know that there are only two major components of universe between epochs of equality and recombination,i.e. the strongly coupled baryon-radiation fluid under the influence of the gravitational potential of cold dark matter.Since,we assume a picture of a volume element with its skeletal structure formed from cold dark matter and the strongly coupled baryon-radiation fluid flowing through it,we are inspired to study this two-component system from variable mass dynamics.This is a formalism of classical dynamics which is used to study the evolution of the skeletal structure of the rocket,because of the gases ejecting out of it. Based on this model,we
study the dynamics of the cold dark matter after equality and upto many
epochs near recombination. Note that this sort of flow assumes that the
baryon -radiation plasma ejects out without losing its equilibrium in
the presence of cold dark matter. This is possible only if we assume
that the cold dark matter is truly collisionless,but it can be feebly self-interacting.This we do by allowing the cold dark matter to feel feeble pressure,triggered by its own self-interactions,and also because of its very heavy nature,at very small scales.This we do by ensuring that the baryon -radiation plasma obeys an equilibrium equation for all epochs between equality and recombination. This assumption can also
accomodate a feebly,self-interacting dark matter at very small scales, which can
transfer energy and momentum to the outer core (57).

We discuss the dynamics of feebly,self-interacting DM in the variable mass formalism with a purpose to explore a new physical explanation under the CDM paradigm for the rotation curves of massive, DM dominated galaxies (LSBs).Also,we explore this formalism,to bring in more parameters sensitive to the baryon-to-photon ratio in the CMB , e.g. : the ejection velocity of the baryon-radiation fluid with respect to the concentrated region of cold dark matter and :the square of the velocity of sound.This we know can be used to get more information about the second and higher peaks of CMB,through numerical analysis.The
strongly-coupled baryon -radiation plasma is only under the influence
of the potential of cold dark matter at these epochs. We assume that the
entropy of photons relative to Cold dark matter is initially spatially uniform
on supercurvature scales(wavelengths greater than ). During
this time, the matter and radiation densities vary in space. In other
words, we consider adiabatic perturbations. As the universe expands, the
inhomogeneity scale becomes smaller than the curvature scale. Thus the
components move with respect to one another and the entropy of photon
relative to cold dark matter particle starts varying spatially. In contrast, the entropy of photons relative to
baryons remains spatially uniform on all scales. This will continue until the
baryons decouple from radiation.

We use Merscerskii equation to study the dynamics of the strongly
coupled baryon- radiation plasma, after the cold dark matter starts
to dominate. This occurs after epochs of equality.Here we
imagine a flow of the baryon - radiation plasma across regions of highly
concentrated non -relativistic cold dark matter.We accomodate the self-interactions of ultra non-relativistic cold dark matter by letting it feel some pressure due to its heavy nature and minor fluctuations,though it does not itself contribute to pressure.The term in (Eq. 2)
sec. II represents the minor fluctuations in the velocity of cold dark matter particles due to self-interactions. The baryon -radiation
plasma can be assumed to have an ejection velocity with respect to the
cold dark matter after equality. We do not disturb the equilibrium of
the ejecting baryon-radiation plasma, even in the presence of cold dark
matter. This we do to argue in favour of truly collisionless nature of
cold dark matter. We ensure this by a specific assumption(see eq. 27
sec. II)which is valid for all epochs between equality and recombination,as an improvement over the standard perturbation theory. This is the time, when the radiation densities
rapidly start to decrease. This is due to the fact that its scaling is proportional to
.The scaling of the cold dark matter density is proportional to . On the basis
of this model, we study the adiabatic perturbations between epochs of
equality and recombination. This we do, both in the Jean’s theory and in
an expanding universe in Newtonian theory. We then find out the effect
of perturbations in cold dark matter potential on the anisotropy in
temperature of radiation at these epochs. This we do for modes greater
than the curvature scales. These are the modes which enter horizon
before recombination.Finally,we use certain assumptions inherent in the model (see eq.36 sec. II and eq.110 sec. III to resolve the core-cusp problem in small-scale sCDM cosmology using DM dominated galaxies as ideal test cases.In particular,we try to explain the rotation curves of LSB galaxies.

The paper proceeds as follows. In section II, we discuss the Dynamics
of Cold dark matter in baryon-radiation plasma using the Merscerskii
equation in the Jeans theory.We ensure the collisionless nature of cold dark matter throughout, by an assumption inherent in the model.Then we discuss the adiabatic perturbations
in the scope of this model.We suggest the growth of dark energy with negative pressure at recombination epochs,which cancels the force due to pressure of baryon-radiation fluid (see eq.(32)
sec. II). We also deduce an equation which represents
how the anisotropy in the temperature of radiation is affected by the
perturbations in the cold dark matter in the epochs between equality
and recombination. We derive an expression for the Sachs-Wolfe effect. In
Section III, in the scope of this model, we discuss the dynamics of the above
mentioned scenario in the expanding universe in the Newtonian theory.We keep the cold dark matter collisionless throughout by an inherent assumption.We
show how the anisotropy in the temperature of radiation is affected
by the perturbations in Cold dark matter potential, in an expanding
universe.This depends on ,i.e. the ejection velocity of the baryon-photon fluid,a parameter sensitive to baryon-to-photon ratio,in addition to ,in the traditional approach (see eq.(128),eq. (119) eq.(133)
sec. II) . Also we evaluate an expression for the Sachs-Wolfe effect in
the expanding universe. We then write the equation for the evolution of
cold dark matter perturbations with time.Finally,in sec. IV,based on one of our assumptions (eq.36 sec. II and eq. 110 sec. III)which are important in deriving the CMB in our model,we propose a new physical explanation under the paradigm of sCDM to resolve the core-cusp controversy for galaxies in general,with regard to massive,LSB(DM dominated)galaxies.In particular,we try to explain the outer rotation curves of the massive,LSB galaxies and the linearly increasing mass of galaxies upto the last measured point (61).

## Ii Gravitational Instability: Jeans theory Representation

In Jeans theory we consider a static, non-expanding universe. Also we assume a homogeneous, isotropic background with constant time -independent matter density (62). This assumption is in obvious contradiction to the hydrodynamical equations. In fact, the energy density remains unchanged only if the matter is at rest and the gravitational force, vanishes. This inconsistency can in principle be avoided if we consider a static Einstein universe, where the gravitational force of the matter is compensated by the antigravitational force of an appropriately chosen cosmological constant. We consider the fundamental equation of dynamics of a mass point with variable mass. This is also referred to as the Merscerskii equation.

(1) |

It should be pointed out that in an inertial frame, is interpreted as the force of interaction of a given body with surrounding bodies. The last term is referred to as the reactive force . This force appears as a result of the action that the added or separated mass exerts on a given body. If mass is added , then the coincides with the vector . If mass is separated, ,the vector is oppositely directed to the vector .

We consider a fixed volume element in Euler (non-co-moving) co-ordinates . After equality, when the cold dark matter starts to dominate in small regions of space, we write:

(2) |

Here the first term on L. H. S represents the acceleration in the mass of
cold dark matter of mass and is the velocity
of dark matter element.We can write this term as we allow the dark matter to feel the pressure of the baryon-radiation fluid at small scales.This we assume due to very heavy nature of dark matter and due to minor fluctuations in its velocity because of self-interactions. The first term on the R. H. S. represents the
gravitational force on the cold dark matter. The second term on right
is for the force due to the pressure of the baryon -radiation plasma
(63). The last term on R. H. S. is the reactive force
on the cold dark matter due to the ejection of baryon - radiation plasma
from regions dominated by cold dark matter.This ejection is not isotropic.This ejection has time and spatial dependence,so that the reactive force does not vanish. This last force is only due to the model which we assume here. Here is the pressure
of the baryon-radiation plasma,and is the ejection velocity of
the Baryon - radiation plasma with respect to the concentrated region of
cold dark matter. Note that in (eq.2), we do not take the pressure of
cold dark matter into account. This is because the cold dark matter
is highly non - relativistic. Therefore, we neglect its pressure i.e.( term).
We assume that after equality, the strongly coupled Baryon - radiation
plasma starts to decouple from the matter. The matter at these
epochs is predominantly Cold Dark matter.

After Equality, in regions dominated by highly non - relativistic Cold dark matter, we write the continuity equation for the ejection of baryon- radiation plasma complex :

(3) |

where is the ejecting mass of baryon-radiation plasma and is the energy density of baryon radiation plasma in the concentrated region of cold dark matter, from where it is ejecting out. In this model, we assume that the ejection of baryon -radiation plasma out of a concentrated region of heavy Cold dark matter does not disturb the equilibrium of the dark matter. This we can assume only because of the truly collisionless nature of Cold dark matter. This is the fundamental assumption of this model. We neglect the flux of the Cold dark matter out of a region of volume in the time that the baryon -radiation plasma flows out of this region.The cold dark matter acts as a skeletal structure for the small volume element .This explains the absence of time derivative of in eq. (4).The rate of flow is entirely determined by the flux of the baryon - radiation plasma and we write :

(4) |

We write (Eq. 2) as:

(5) |

where is the energy density of cold dark matter, and is the gravitational potential of cold dark matter.

(6) |

Now using the Jeans theory, we introduce small perturbations about the equilibrium values of variables, W. Bonnor(42):

(7) |

(8) |

(9) |

(10) |

where , the speed of light, and

(11) |

(12) |

where S is the entropy of cold dark matter element. Also we write:

(13) |

where is the speed of sound. Neglecting dissipation, we write:

(14) |

(15) |

From (eq. 4) we write:

(16) |

We substitute the values from (eq. 7)-(eq. 15)in (eq. 16)and (eq. 6) to get:

(17) |

(18) |

(19) |

(20) |

We now take the divergence of (eq. 18)to get:

(21) |

Now using (eq. 17) and (eq. 19) we get :

(22) |

In a static universe the total energy remains constant. So for small inhomogeneities we write:

(23) |

Thus (eq. 22) becomes:

(24) |

For strongly -coupled baryon - radiation plasma before recombination, with low baryon densities we write :

(25) |

Thus we get:

(26) |

The baryon - radiation plasma is only affected by the gravitational potential of the cold dark matter in these epochs. We argue that the baryon-radiation plasma ejecting out of concentrated regions of cold dark matter is always in equilibrium, even in the presence of cold dark matter. This is possible only if the cold dark matter is truly collisionless. So for epochs between equality and recombination, we write:

(27) |

The above equation is of profound importance in this model. This is because, it ensures that the equilibrium of the ejecting baryon -radiation plasma is not disturbed, even when it flows out of concentrated regions of cold dark matter. This we can write, because the distribution function of pressure of the baryon-radiation plasma depends only on . We assume here that the chemical potential is much smaller than the temperature. So from small perturbations about the equilibrium values of variables and in the (eq. 27), about a fixed temperature T, we get,

(28) |

Thus we write:

(29) |

Here , which represents the anisotropy in temperature of radiation for small baryon densities after equality. For small perturbations in cold dark matter, using (eq. 29) in (eq. 24), we write:

(30) |

For small perturbations in Cold dark matter, we neglect the term . This is due to cold dark matter being highly non -relativistic. Also we neglect . This is due to the fact, that in this model, we assume that the ejecting velocity of baryon - radiation plasma does not suffer interactions and collisions. This is because of dark matter being truly collisionless. So, we further write:

(31) |

In the above equation, we see the contribution of divergence in the anisotropy of radiation at epochs between equality and recombination. At epochs near recombination, the strongly coupled baryon-radiation plasma starts rapidly to decouple from the cold dark matter. During these epochs, therefore the actual velocity of the baryon -radiation plasma does not change appreciably due to the gravitational force of cold dark matter. During these epochs, the rate of increase of ejection velocity is only due to gravitation potential of cold dark matter. There is no force due to the pressure of the baryon -radiation plasma. Therefore we write:

(32) |

We can argue that due to high baryon densities, the pressure of the baryon-radiation plasma vanishes at these epochs. But the baryon densities can never be very high. This is because it will hinder hydrogen nucleosynthesis after recombination. Thus it has to be assumed that the pressure of the baryon - radiation plasma vanishes not due to very high baryon densities, but due to a strange form of energy(dark energy)with negative pressure, which has started to dominate near recombination. Therefore, the acceleration in the ejection velocity will be primarily due to the gravitational forces So, we write:

(33) |

(34) |

(35) |

The above (eq. 35) shows that the actual velocity of the baryon-radiation plasma stops to increase at epochs, where it is almost about to decouple from cold dark matter. Therefore, at these epochs the acceleration of cold dark matter reverses its sign and we write:

(36) |

With these Assumptions the (eq. 31) reduces to:

(37) |

Considering adiabatic perturbations:

(38) |

We use:

(39) |

Then we write:

(40) |

The (eq. 37) now reduces to:

(41) |

or

(42) |

The above equation has two solutions:, where

(43) |

Here the Jeans length is :

(44) |

Thus is real for or , where

(45) |

when

(46) |

Where is the total equilibrium value of energy density and is the equilibrium energy density of baryon-radiation plasma, at these epochs. We interpret from above, that all the modes near recombination are real. This is because the value of starts increasing rapidly near recombination. This is due to the decoupling of baryon - radiation plasma from cold dark matter. This explains the maximum frequency of fluctuations in the CMB temperature anisotropy at red -shifts of recombination. For the solutions are:

(47) |

For , where stands for the mean free path of free- streaming photons after equality, free-streaming becomes important. Free-streaming refers to the propagation of photons without scattering. We write the equation for free-streaming photons in the conformal Newtonian gauge (64). The gravitational potential is primarily due to the cold dark matter. We argue that the photons can still be described by the equilibrium distribution functions. This is because the cold Dark matter is collisionless. Therefore, their mutual interactions will not disturb the equilibrium of the ejecting baryon-radiation plasma. Also, we treat the baryon-radiation plasma as free-streaming photons for low baryon-densities. We use the metric below:

(48) |

where corresponds to the Newtonian potential and is the perturbation to the spatial curvature. The equation for free- streaming photons is:

(49) |

(50) |

We can make the above assumption, for epochs between equality and recombination. This is because after Equality, and before recombination, the dominant contribution to the potential is due to cold dark matter. So, we neglect the non-diagonal components of the energy-momentum tensor of the baryon-radiation plasma. Recall that the difference is very small compared to . Thus we assume it to be a single perfect fluid. In a static Einstein universe, we neglect the terms due to the Hubble parameter . At scales of the order of mean-free path of the free-streaming photons in the baryon-radiation plasma, i. e. of the order of , we neglect the spatial inhomogeneities . We therefore write:

(51) |

Using the assumption of (eq. 50), we write:

(52) |

Recall that the Newtonian potential corresponds to the dark matter potential . This remains constant between epochs of equality and recombination. Thus we write:

(53) |

(54) |

The momentum per unit volume of radiation at epochs of decoupling is equivalent to radiation pressure. Also, if the photons, just at the epochs of decoupling are in thermal equilibrium, and that it has same average energy associated with each independent degree of freedom, we write:

(55) |

where is the increment in radiation pressure associated with each independent degree of freedom at decoupling.For a unit volume ,decoupling photons’ energy density is same as radiation pressure per degree of freedom, we therefore write:

(56) |

We use (eq. 40 and (eq. 47) to write:

(57) |

where

(58) |

Therefore, for epochs of decoupling, we write:

(59) |

or

(60) |

From the above equation we can write:

(61) |

Therefore, we write:

(62) |

The above result is the same as that predicted by Sachs-Wolfe.Here it is important to note that in the (eq.60),the problem of divergences in the temperature anisotropy does not arise as the term represents the average perturbation in the gravitational potential of the cold dark matter between the epochs of equality and recombination.We know this can’t take zero value. Three
types of effects(due to fluctuations in density, velocities
and potential)simultaneously contribute to the CMB temperature
anisotropy. The fluctuations that matter at scales beyond
are those in the gravitational potential (Sachs-Wolfe
effect)(65).

The above equation represents the anisotropy in the CMB temperature, for
epochs near recombination, for regions where sufficient primordial helium
synthesis takes place, even before recombination, or the dark energy has
started to dominate. Recall that in assumption of(eq. 32), we argue that
the force due to pressure of baryon-radiation plasma vanishes near
recombination epochs, and that it can be only due to the fact that the
dark energy with negative pressure has started to dominate. The value of
is constant. This is because the potential of cold dark matter
remains constant for many epochs between equality and recombination. Thus
in a static universe, with this model, we see that the dominant component
in CMB temperature anisotropy fluctuations is near recombination. This
is because, at epochs near recombination, the perturbations in the cold
dark matter potential are very small. This is because in this model, the
decoupling of the baryon - radiation plasma from concentrated regions
of cold dark matter, near epochs of recombination is almost near
completion.

We now write the CMB fluctuations for supercurvature modes i. e modes with
, and for regions where sufficient primordial
helium synthesis takes place, even before recombination, i. e. the
baryon-densities are high. We neglect the effect of gravity at these
epochs. We can do so because at late recombination epochs, the pressure of
baryon-radiation plasma is low, and the negative pressure of a strange form
of energy(dark energy), which starts to dominate at these epochs, cancels
the effect of forces due to low pressure baryon-radiation plasma and
that of gravity. we therefore write:

(63) |

Also for , free-streaming of photons is no longer relevant. Therefore the scattering of photons will dilute the anisotropy to a large extent. For late recombination epochs, when the radiation has almost decoupled from matter, we write:

(64) |

We have neglected gravity here again, because at these epochs, the effect of dark energy, which had started to dominate from some earlier epochs is to cancel the forces due to gravity and the pressure of baryon-radiation plasma. This is possible even in regions of lower baryon densities, which has higher pressure than the regions of higher baryon densities. This is only because the dark energy with negative pressure had been dominating from some earlier recombination epochs. The above equation shows that the frequency of fluctuations in CMB temperature anisotropy spectrum of supercurvature modes(modes which enter the horizon very early near recombination epochs), at late recombination, remain constant till today. This is valid both for regions of low or higher baryon densities(where sufficient primordial helium nucleosynthesis takes place before recombination). This is because at very late recombination epochs when the radiation has almost decoupled fully from matter, the speed of sound approaches a constant value of . The effect of scattering of photons will dilute this anisotropy in supercurvature modes. So, it is of not much cosmological significance.

## Iii Instability in Expanding Universe: Newtonian theory

Using our model (see Sec. I), we study the same scenario, i. e. of cold dark matter in the presence of strongly coupled baryon -radiation plasma at epochs between equality and recombination. Here we use Newtonian theory of expanding universe (42). We treat the dynamics between cold dark matter and the baryon - radiation plasma, again in the framework of the Merscerskii equation.We ensure the collisionless nature of cold dark matter by the same inherent assumption as in (sec. II).We assume that the strongly coupled baryon -radiation plasma is in the presence of a gravitational potential. This potential, is only due to cold dark matter, at epochs after equality and before recombination. We neglect the non-diagonal components of the energy-momentum tensor of the cold dark matter. This is because, the difference is very small as compared to . We consider dark matter as highly non -relativistic fluid compared to the baryon - radiation plasma. We consider epochs when the Dark matter has already started clustering. We neglect the rate of flow of cold dark matter out of a given region of space, as compared to the baryon-radiation plasma. Therefore we write:

(65) |

where is the relative velocity of the baryon - radiation plasma with respect to the cold dark matter. We write the Merscerskii equation for cold dark matter with baryon radiation plasma, ejecting out of concentrated regions of space dominated by cold dark matter:

(66) |

In an expanding flat, Isotropic and homogenous universe, we write:

(67) |

Therefore we write:

(68) |

We first discuss for scales i. e the curvature scale. In this case, we neglect the velocities of Cold dark matter particles. This is because, there is not enough time to move highly non -relativistic cold dark matter upto distances greater than the Hubble scale. Therefore the entropy per cold dark matter particle remains constant on supercurvature scales(). So we write:

(69) |

The (eq. 66) gives:

(70) |

We take the divergence of the above equation to write:

(71) |

To get the above equation, for near recombination epochs, we use:

(72) |

We use the above assumption because for epochs near recombination, the baryon-radiation plasma starts to decouple fast. Therefore, only the gravitational force determines the acceleration of dark matter. Now for Adiabatic perturbations, we write:

(73) |

Also, we write the following perturbations for other variables:

(74) |

Where the variables have their usual meanings (see Sec. II). We use (eq. 71) in (eq. 66) and (eq. 68) to write :

(75) |

and

(76) |

We use the Langragian co-ordinates and write:

(77) |

where

(78) |

We write:

(79) |

where is the fractional amplitude of perturbations. We write (eq. 75) in the Co-moving coordinates, using (eq. 65) and (eq. 77) to get:

(80) |

also we write the (eq. 76) in the co-moving coordinates to get:

(81) |

We write (eq. 69)in co -moving coordinates:

(82) |

from (eq. 71), we get:

(83) |

We take the divergence of (eq. 76), and use (eqn’s. 68-77-79) and (eq. 80) to get:

(84) |

We Neglect the fourth term i. e. . This is because, it is proportional to , which is very small. This is due to very small divergence in the perturbations to the ejection velocity of baryon -radiation plasma. With this assumption, again we argue that the cold dark matter is truly collisionless. So, in this model, it does not disturb the equilibrium of the ejecting baryon-radiation plasma. From (eq. 71), we get:

(85) |

Using the above, we write (eq. 84) as:

(86) |

Therefore, neglecting the term containing , we write:

(87) |

We use:

(88) |

Thus, we write the equation for the evolution of fractional amplitudes of perturbations in the baryon -radiation plasma as:

(89) |

For very late recombination epochs, the radiation starts to decouple rapidly from matter. Therefore, baryon densities are very low. So we write:

(90) |

where is the speed of sound.

The (eq. 89) shows, that for low baryon densities, fractional amplitudes of perturbations in the radiation, which originate at late recombination epochs grow at the fastest rate. This is because the value of is maximum at late recombination epochs. It is because of the lowest baryon densities, in the coupled baryon -radiation plasma at these epochs. Also we can see that the second term in the exponent in (eq. 89)vanishes for very late recombination epochs, as ratio . We thus write the equation for evolution of the fractional amplitude of perturbations in the Cold dark matter, which originate after equality, when the clustering of dark matter had already started.

(91) |

The above equation represents the growth of the fractional amplitudes of perturbations in the cold dark matter, which originate after equality, for scales .It is important to note that for late recombination epochs,it reduces to the standard result . If baryons contribute a significant factor of the total matter density, CDM growth rate will be slowed down between equality and the recombination epochs (66). Also we see that the perturbation growth rate will slow with scale factor (67). The CDM density fluctuations will dominate the density perturbations of baryon-radiation plasma (66). This is because for scales , the density perturbations of baryon-radiation plasma are washed out by the scattering of photons at scales , which is the mean free path of photons. The perturbations in the cold dark matter will cease to grow when the dark energy starts to dominate (68). We can interpret this from (eq. 91). This is because with the growth of dark energy, the value of will decrease. This will then lead to ceasing of growth of CDM perturbations. There is existence of non-linear structures today. This implies that the growth of fluctuations must have been driven by non-baryonic dark matter, which was not relativistic at recombination. Also, we see that the perturbations at supercurvature scales grow slowly. Recall that these are the modes which enter the horizon very early, well before the recombination epochs. The slow growth of such modes is because, it is only at late recombination epochs that the second term in the exponent in (eq. 91) will vanish due to . Also, it is only at late recombination epochs, that the value of reaches its maximum value of , just before decoupling. The amplitude of the fractional density perturbations in the cold dark matter, in (eq. 91) will be maximum when the ratio. This will occur when

(92) |

In writing the above equation, we use the result of (eq. 71). So for epochs when the density of baryon-radiation plasma is equal to the density of the cold dark matter, (eq. 91) is:

(93) |

Now we discuss the gravitational instability for scales . This originates after equality,with dark matter dominance in the presence of strongly coupled baryon - radiation plasma in an expanding universe. We discuss it in the Newtonian theory. The decoupling of the strongly coupled baryon - radiation plasma from the non -relativistic cold dark matter starts after equality. Let us assume that the separation of this plasma from the cold dark matter gives a relative velocity of to the baryon - radiation plasma. At small scales with low baryon densities, we neglect the contribution of non-diagonal components in the energy- momentum tensor of dark matter. Therefore, we treat it as a perfect fluid for many epochs between equality and recombination. We treat the strongly coupled baryon - radiation plasma as a perfect fluid for epochs between equality and recombination. This is because of low baryon -densities at these epochs. Recall that the baryon densities only starts increasing substantially after recombination. This is when the primordial nucleosynthesis of hydrogen and helium will start. However, as an exception in certain regions, the primordial nucleosynthesis may start at epochs before recombination. We now conclude that there is sufficient time for the Cold dark matter to flow through distances at scales . This is because from (eq. 91), we see that the perturbations in the cold dark matter, at small scales, grows at the fastest rate. Therefore we write the Merscerskii equation as below:

(94) |

where

(95) |

At small scales, we assume that in the time that the baryon -radiation plasma flows out of a given region of space, the inhomogeneity in Cold dark matter in that time duration is negligible. So we assume that the baryon-radiation plasma can flow out of a concentrated region of dark matter without generating the collision terms. This is because the cold dark matter is highly non-relativistic and collisionless. Also, because the dark matter has already started to cluster. Therefore we write:

(96) |

where is the eulerian cordinate. We write the continuity equation for flow of the strongly coupled baryon - radiation plasma as :

(97) |

Here is equal to

(98) |

We take the divergence of (eq. 94) and neglect the spatial dependence of . This is due to the collisionless nature of dark matter. We also assume that is constant for small scales. We thus write the Friedmann equation:

(99) |

For adiabatic perturbations we write from :

(100) |

where the variables have their usual meanings (see Sec. II). Then we use the perturbed values of variables in (eq. 94), to write: