#  A Model For Dark Matter Haloes.

H. L. Helfer<EMAIL_ADDRESS>Department of Physics and
Astronomy, The University of Rochester,  Rochester, NY 14627

# II. A GR Source Term $T^{\mu\nu}$ For Dark Matter.

H. L. Helfer<EMAIL_ADDRESS>Department of Physics and
Astronomy, The University of Rochester,  Rochester, NY 14627

###### Abstract

Two forms are suggested for the source term $T^{\mu\nu}$ associated with an
aggregate of dark matter (with the properties described in Paper I). Both have
large pressure-like components which dominate the density terms. Using one
form a simple model of the spiral galaxy halos is developed which can match
the observed ‘flat’ outer rotation curves of some galaxies,including the Milky
Way. It can also represent the ascending outer rotational curves of small
spiral galaxies such as M33. See Figs. 1, 2.

The analysis of the Milky Way rotation curve gives $p_{a}/c^{2}\sim 6\times
10^{-27}\ {\rm g\ cm}^{-3}$ at $R\sim 60-80$ kpc near the outer edge of the
halo. The surrounding dark matter (DM) cloud has $p/c^{2}\sim 0.1p_{a}/c^{2}$.
The dark matter cannot come closer to the center than $R\sim 4$ kpc. It
reaches a maximum $p_{b}/c^{2}\sim 10^{2}p_{a}/c^{2}$ at $R_{h}\simeq 8$ kpc
and then falls rapidly $p/c^{2}\propto 1/r^{2}$. It is hoped that by analyzing
other galactic rotation curves these models can be used to infer properties of
the intergalactic DM matter.

The other form for $T^{\mu\nu}$ for DM is useful in cosmology. Then a (non-
constant) cosmological ‘constant’ term $\Lambda$ needs to be added to
Einstein’s equation in order to allow use of the standard $p\ -\rho$ relation
for DM. We suggest that at least some of the ‘dark energy’ present may have
been misinterpreted and not be real, resulting from adopting an improper form
of $T^{\mu\nu}$ for DM.

Keywords: galaxies:halos, galaxies;rotation curves, dark matter, dark energy,
cosmology: cosmological constant

Keywords:, galaxy:haloes, dark matter, dark energy, gravitation,
cosmology:theory

##  1 Introduction

###  1.1 Preliminary Concepts.

In this paper, dark matter (DM) refers to the unknown material responsible for
the extended gravitational fields needed in constructing models for
gravitational lensing, for modeling the motions of ordinary matter (OM) in
clusters of galaxies and in the halos of many spiral galaxies, including the
Milky Way.

For these environments, a model for DM was proposed which assumed it is
comprised of particles that follow time-like world-lines similar to those of
non-relativistic OM particles. However, the DM particles have a different
kinematics than OM particles have, being influenced by their internal motions,
so that their total momenta is space-like. See Paper I (Helfer (2019)).

[This model of DM may also represent the ‘cold’ DM encountered in the $\Lambda
CDM$ cosmology. We do not (and do not need to) speculate upon the particle
physics nature of the constituents of DM. We also noted that speculation about
the classification of internal motions within fundamental particles is
appropriate; a recent calculationattributes $\sim 68\%$ of the mass of the
proton to internal motions. ]

As emphasized in Helfer (2019), an important observational property of dark
matter (DM) is that it does not appear in substantial amounts in ordinary
stars. As a consequence, a suggestion was made that DM objects were
characterized by having their total momenta being space-like, reflecting the
presence of two different types of internal momentum flows: (1) a time-like
component $\hat{U}$ which indirectly specifies its motion through space-time;
and (2), a space-like component $\hat{S}$ which is transported with the
object. The particle actually moves along a complicated path whose tangent is
the vector $U$. The path is a tight helix whose axis is the particle’s world-
line, described by a tangent vector $\langle\hat{U}\rangle$; it satisfies
$\langle\hat{U}^{\nu}\rangle\langle\hat{U}^{\mu}\rangle_{;\nu}=0$, the usual
geodesic equation. It was shown that part of $\hat{U}$ appears as an internal
spin-like motion which can couple to the object’s orbital angular momentum;
this coupling creates a centrifugal barrier which prevents the DM object from
penetrating into deep gravitational potential wells such as those around
proto-stars. This explains its absence in stars.

One component of the internal field $\hat{S}$ is needed to provide strict
local conservation of momentum for balancing the helical component of
$\hat{U}$. The other part111It is referred to in text and in Paper I as
$\hat{S}^{z}$. of $\hat{S}$ would be carried along and contributes to the
local DM energy-momentum tensor $T^{\mu\nu}$, a source term for Einstein’s
equation of GR. It is relatively large, because the total momentum is taken to
be space-like. This paper deals with the form of this contribution when
dealing with aggregates of these DM particles. Like rest-mass, this DM
aggregate momentum flux contributes significantly to the gravitational field.

A very simple DM model of the halo region of a galaxy is constructed by
studying the effects of angular momentum conservation on determining the
distribution of the halo DM. The model’s utility is tested by requiring it to
explain the outer part of galactic rotation curves, commonly attributed to DM
halos. Successful models for the large spiral galaxy, the Milky Way, and the
small spiral M33, are presented. [See Figs. 1 & 2].

Appendix A summarizes the observations, primarily for the large spirals,
exhibiting ‘flat’ rotation curves. The notation used in this paper is that of
Paper I.222 One has $c=1$ normally and the local Lorentz metric
$g_{\mu\nu}$has the signature $(+1,-1,-1,-1)$. Also the time-like momentum
vector is written as $\hat{U}=m_{0}U$ and similarly for the space-like
momentum vector $\hat{S}=m_{1}S$.

###  1.2 Outline.

####  1.2.1  The DM Source Term $T^{\mu\nu}$

In many GR problems the energy-momentum tensor $T^{\mu\nu}$, used as a source
term, is taken to be that of a classical fluid. First, we start with examining
some problems of using a kinetic theory representation of matter to form a
fluid energy-momentum tensor primarily in order to understand what the symbol
$p$ means; there is a distinction between thermodynamic pressure terms and
other forms of momentum flux. In general in this paper $p$ is not a
thermodynamic pressure term. This kinetic theory approach can used to
construct $T^{\mu\nu}$ for ordinary matter (OM) when the geometry is not
simple. This construction of $T^{\mu\nu}$ for OM involves using averaged
equations of motion as well as equations of continuity for ensembles of
particles; it is given in Section IIA (and in Appendix B).

Next, in Sections IIB & IIC one focuses on how to construct $T^{\mu\nu}$ for
DM, incorporating contributions from both $\hat{U}$ and $\hat{S}$ using the
model of DM particles of Paper I. Two different forms of $T^{\mu\nu}$ are
suggested. A criterion for choosing when either form is applicable is
developed in Section IIE333 One of these forms may actually be that used in
cosmology; see Section IID. It can be written more conventionally in the form
representing a standard thermal fluid, provided one adds a cosmological
‘constant’ term $\Lambda$; see Kolb & Turner (1990),Weinberg (1972). In this
approach one finds $\Lambda$ needs to be variable.

Finally, the representation of a DM halo as an intrusion in a larger DM
structure of cosmological size is discussed in Section III and in Appendix C,
using one of these two forms of $T^{\mu\nu}$. The outer rotation curves of the
large Milky Way galaxy and of the small spiral M33 are successfully modeled by
this representation and parameters characterizing the DM halos derived. See
Figs (1) & (2).

Most of this paper is ‘technical’, i.e. it deals with the many details of
setting up a usable physical description of the properties of the DM gas
comprising a halo. A more casual read would focus on the overall description
provided in the next subsection, 1.2.2, and the ease which with many observed
rotation curves can be interpreted, given in Section III. Possible
cosmological implications are discussed in Section 2.4, in which it is pointed
out that the inclusion of a ‘$\Lambda$’ term may be required if one adopts an
inproper equation of state for DM.

####  1.2.2  Comments About The DM Halo Structure.

The DM galactic halos, being extended, are modeled in Section III using an
interior Schwarzschild metric for the halo region. [See eq.(14)] A typical DM
halo is characterized by a very large momentum flux; this flux formally
appears as a ‘pressure’ term when $T^{\mu\nu}$ is written in the conventional
fluid form, but it is not to be regarded as a conventional thermodynamic
pressure. The DM density term is relatively small. The halos are not in
hydrostatic equilibrium. The structure of a halo may be regarded as if it were
a very slowly flowing interior region of a large intergalactic DM cloud in
which the galaxy was presently embedded; details of the flow are determined by
conservation of angular momentum.

It was noted in Paper I that using this metric, the motions of non-
relativistic particles (not near a singularity) are really controlled by
variations in the metric component, $B$, not by those in $A$. Accordingly the
circular rotational velocity is given by eq.(45) and a large momentum flux
$p$, attributable to DM, can mimic ‘missing mass’ when the more conventional
interpretation, which disregards the possibility of a momentum flux
contribution, is used. For our simple spherical model of the halo region, the
variation of $p(r)$ is discussed in Appendix B, Section C and details are
provided about how one fits observed rotation curves with models, using as
representative the large galaxy, the Milky Way, and for a much smaller galaxy,
M33. [See Section III.]

In general, the halo is divided into two adjoining parts. In the outer halo,
some DM particles are not severely constrained by their angular momenta and
can ‘free-fall’ towards the central OM galaxy; the pressure and density are
given by eqs. (33) In the small inner halo the DM particles’ motions are
severely constrained by their angular momenta; the pressure and density are
given by eqs. (34). [The boundary between the two regions is given by eq.
(31).] For large galaxies, the outer portion dominates and results in ‘flat’
rotation curves. For small galaxies, only the inner halo structure is
relevant; this structure results in the ‘linearly’ increasing rotation curves.

##  2  The Construction Of The Energy-Momentum Tensor.

“…about the dread right-hand side of the Einstein’s equations…” 444Kolb &
Turner 1990, pg 48.

In most GR models, for reasons of homogeneity and isotropy, the source term is
assumed to be of the form of the hydrodynamic tensor:

$T^{\mu\nu}_{fluid}=(\rho(x)+p(x))\bar{U}^{\mu}\bar{U}^{\nu}-p(x)g^{\mu\nu}.$
(1)

Einstein Einstein (1950) adopted this, regarding the ‘pressure’ term as
representing an averaged momentum flux of the kinetic motions of the OM. In
this spirit we emphasize that the requirement $T^{\mu\nu}_{\ ;\nu}=0$ need not
be regarded as the equivalent of a thermodynamic equation-of state. In order
to connect to standard GR models, we shall adopt this ideal fluid form, eq.
(1), for representing the volume average of $T^{\mu\nu}$ for simple DM models
as well, when the local distribution of DM momentum flux is isotropic.

The difficulty with eq.(1) is that only in the simplest of ideal situations
are the quantities $\rho,\bar{U},p$ well defined. At the very least, one may
always add a term $\lambda g^{\mu\nu}$ ,where $\lambda=$ constant, to eq.(1)
effectively modifying the definitions of $\rho\ \&\ p$. If one specifies
$T^{\mu\nu}_{fluid}$ accurately, then ‘in principle’ the geometric structure
of the LHS of Einstein’s can be determined. The real problem is to reverse the
procedure: to ‘observe’ or adopt a geometric structure and from this to
determine the tensor source term. Then one has the problem of interpreting
what $\rho,p$ (and the other components of the stress-tensor) mean physically.

To interpret $T^{\mu\nu}$ for classical OM one must start with aggregates of
discrete entities. A kinetic theory representation (e.g.  Weinberg (1972)) of
them must first be developed and then averages performed in an enclosure to
get a fluid representation.555The definition of a local enclosure may not be
easy; consider the problem of extending the Kerr solution to to include
rotating clouds of particles. Only in cases involving the simplest time
dependences, can co-moving coordinate differences be used to define local
enclosures. Such a kinetic treatment for one constituent of OM is given in
Appendix B. It emphasizes that the condition $T^{\mu\nu}_{\ ;\mu}=0$ is
equivalent to a strong form of the fluid equation of mass conservation which
effectively includes particle number conservation.

Normally we regard $T^{\mu\nu}$ as a sum of terms, one for each constituent.
Since the LHS of Einstein’s equation is non-linear in the affine connections,
$T^{\mu\nu}$ must be inclusive; in effect the sum of all contributions from
individual constituents need to be considered together. Since one component
can contribute most of the mass and another most of the momentum flux, $\rho$
and $p$ for the composite need not be simply related.

In practice approximations are introduced. Local mass motions are ignored and
$p$ is assumed to be a thermodynamic momentum. This excludes considering many
astronomical problems in which local mass motions transport a momentum flux
greater than that associated with a thermodynamic pressure term and one must
evaluate the spacial components of $T^{\mu\nu}$. This is the case for the DM
halo models discussed below where the thermodynamic pressure terms are
ignored.

In the bold experiment of constructing a cosmology for the Universe, we came
face-to-face with unknown source terms, normally referred to as ‘cold dark
matter” and ‘dark energy’. These together do not allow a comfortable
interpretation of $\rho,p$ as a conventional fluid. The variables
$\rho,p,\bar{U}$ are not really well-defined in physical terms for the unknown
source material. So it should be as no surprise that the energy momentum
tensor for the component representing DM, discussed below, requires changes in
the definitions of $\rho$ and $p$, because the energy and momentum
distributions and the equations of motion for DM are quite different from
those of OM.

###  2.1  The Actual Pressure Term In $T^{\mu\nu}$.

First we consider when $p$ is a thermodynamic pressure and when it simply
represents a momentum flux for OM objects. For large aggregates of streams of
similar particles in a very small region, ${\cal R}$, not near a gravitational
singularity, it is useful to introduce ‘number’ densities $n_{s}$, each of
which contains a mass factor $m_{s}$ . Put $U_{s}^{\mu}={\bar{U}}^{\mu}+\delta
U_{s}^{\mu}$, where

${\bar{N}}{\bar{U}}^{\mu}=\Sigma_{s}n_{s}U_{s}^{\mu},\ \ {\rm with}\ \
{\bar{N}}=\Sigma_{s}n_{s}\ {\rm and}\ \Sigma_{s}n_{s}\delta U_{s}^{\mu}=0.$
(2)

Here, ${\bar{U}}$ and ${\bar{N}}\equiv\surd(-g)\rho$ are fields characterizing
the region ${\cal R}$, with $({\bar{N}}{\bar{U}}^{\nu})_{;\nu}=0$, giving the
ensemble’s energy density conservation; set

$T^{\mu\nu}\equiv\Sigma_{s}T^{\mu\nu}_{s}=F^{\mu\nu}+P^{\mu\nu},\ \ {\rm
where}$ (3) $F^{\mu\nu}={\bar{N}}{\bar{U}}^{\mu}{\bar{U}}^{\nu}\ \ {\rm and}\
P^{\mu\nu}=\Sigma_{s}n_{s}\ \delta U_{s}^{\mu}\delta U_{s}^{\nu}$ (4)

with $T^{\mu\nu}_{\ \ ;\nu}=0$. The trace is
$T=g_{\mu\nu}T^{\mu\nu}={\bar{N}}$, the rest-mass density. The structure
equations $T^{\mu\nu}_{\ ;\nu}=0$ are derived in Appendix B, using the
individual particles’ equations of motion and the equation of continuity for
each stream of particles.

When ${\cal R}$ is small enough (so that the $\Gamma^{\xi}_{\mu\nu}$ are
slowly varying in it) we assume the geodesics are nearly constant:
$U^{\xi}_{s;\mu}\cong 0.$ Now, it is quite remarkable that:

$P^{\mu\nu}_{\ \ ;\xi}=\Sigma_{s}\delta U_{s}^{\mu}\delta U_{s}^{\nu}\
n_{s,\xi}\ .$ (5)

(This follows by differentiation by parts and noting that because
$U^{\mu}_{s;\xi}=0$, one has $(\delta
U_{s}^{\mu})_{;\xi}=-{\bar{U}}^{\mu}_{;\xi}$ ) This means that the local
change of $P^{\mu\nu}$ with position is independent of the affine connections,
both frame forces and gravitational forces, the mean velocity
${\bar{U}}^{\mu}$, and of details of the variations of each $\delta
U_{s}^{\mu}$ with position. It is a function only of the stream densities; for
this reason we regard it as determined by thermodynamic considerations. If
${\cal R}$ is embedded in a warmer thermodynamic enclosure, only $P^{\mu\nu}$
is expected to change.

$P^{\mu\nu}$ is similar to a conventional kinetic-theory stress tensor. Under
the assumption of stationarity, statistical thermodynamic averaging would give
$F^{\mu\nu}\to\rho\delta^{\mu}_{0}\delta^{\nu}_{0}$, and under the assumption
of isotropy, for (non-relativistic) streams $P^{\mu\nu}\to
p\delta^{\mu}_{i}\delta^{\nu}_{i},{\rm for}\ i=1,2,3$. In this simple case the
structure equations $T^{\mu\nu}_{\ ;\nu}=0$ have a thermodynamic
interpretation. However, suppose the region contains part of a rotating
stellar disk population; the spacial components of $F^{\mu\nu}$ would not be
zero and the structure equations would not have a thermodynamic meaning.

In the simple cold DM halo models developed in this paper, the variance terms,
$P^{\mu\nu}$, are assumed very small and ignored.

###  2.2  The Local $T^{\mu\nu}$ For One Dark Matter Object.

There are two contributions to $T^{\mu\nu}$ from DM objects; see eq (8). One
is similar to that just noted for ideal OM objects; it corresponds to
averaging the particles’ transport of mass and translational momentum along
their world lines, $\langle\hat{U}\rangle$. The other, which we now discuss,
results from the transport of the particles’ internal structure motions. In
particular, we focus on the $\hat{S}^{z}$ momentum carried internally by DM
particles.

As discussed in Paper I, Section II, the motions of a DM object may be
described at any point by two vectors $U,S$. Using local momentum
conservation, the only large internal components contributing for cold DM to
$T^{\mu\nu}$ are $\hat{U}^{0}\ \&\ \hat{S}^{z}$, since, in the local frame of
reference, $\hat{S}^{0}=0$. We follow the notation of that paper.

[ One defines $\hat{U}_{s}^{\mu}=m_{0}U_{s}^{\mu}\ ,\
\hat{S}_{s}^{\mu}=m_{1}S_{s}^{\mu}$ with $U_{s}^{\mu}U_{s\mu}=1,$ time-like,
and $S_{s}^{\mu}S_{s\mu}=-1,$ space-like. The index $s$ labels the particular
object. Also for halo DM, $m_{0}=k\Gamma m,\ m_{1}=\Gamma m$ where $m$ is
arbitrary. In the local Lorentz frame of an object used, $\mu=z$ defines the
direction of translation and $S_{s}^{0}=0$. We use multiplicative constants
$a,b$ satisfying $a^{2}m_{0}\equiv b^{2}m_{1}$. For DM, $a^{2}=\Gamma,\
b^{2}=k\Gamma$, with $k^{2}<1$ and $\Gamma^{-2}=1-k^{2}$. The important
parameters $k,\Gamma$ characterize the total space-like internal velocity of a
DM object]

As before in dealing with ordinary matter, we consider only objects that do
not terminate in the volume under study.666 This implies
$S^{\nu}_{s;\nu}=U^{\nu}_{s;\nu}=0$. See Appendix B, Sections A,B. To insure
$T^{\mu\nu}_{s;\nu}=0$ , one needs to modify the procedure used in Appendix B
for OM, because the equations for $\hat{U}^{\mu},\hat{S}^{\mu}$ are not simple
geodesics but describe helices. The two equations of motion, determining the
development of ${\hat{U}},{\hat{S}}$ in a very small region ${\cal R}$
containing their separate paths, effectively differ only by multiplicative
constants. (See eqs. (14,15 & 18) in Paper I. ) Since $T^{\mu\nu}$ is found by
summing all streams within a small volume, the distance between segments of
the actual two close-by paths of the momentum components of DM object is not
relevant. Multiplying these equations by constants to correct for this scale
difference, the difference between the two equations vanishes. Therefore, we
consider as the first part of the internal energy momentum tensor for a DM
object, comprised of the momentum pair $(\hat{U}_{s},\hat{S}_{s})$,

$\epsilon(a^{2}+b^{2})a^{2}m_{0}{\tilde{T}}_{s}^{\mu\nu}\equiv a^{2}\
{\hat{U}}_{s}^{\mu}{\hat{U}}_{s}^{\nu}-b^{2}\
{\hat{S}}_{s}^{\mu}{\hat{S}}_{s}^{\nu}\Rightarrow a^{2}m_{0}\
[m_{0}U^{\mu}_{s}U^{\nu}_{s}-m_{1}S^{\mu}_{s}S^{\nu}_{s}],$ (6)

where the $\epsilon=\pm 1$; both signs are of interest. The normalization used
here is chosen so that for $\epsilon=+1$, one has $\tilde{T}_{s}\equiv m>0$;
this leads to a form useful in cosmology. By construction
${\tilde{T}}^{\mu\nu}_{s\ ;\nu}\equiv 0$. 777 Using the local Lorentz
coordinate system used in discussing the guiding center solution in Paper 1,
one finds ${\tilde{T}}_{s}^{xx}={\tilde{T}}_{s}^{yy}=0$ so that the ‘spinning’
motion discussed in that paper in the $xy-$plane does not directly contribute
to this approximate source term. With this form of $T^{\mu\nu}$ no external
conservation laws are associated with $\tilde{T}^{\mu\nu}_{\ ;\nu}=0$.

For DM one has $m_{0}/m_{1}=k<1$ and to good approximation we can exclude the
$U^{\mu}U^{\nu}$ term when $k$ is sufficiently small. The dominant term is the
one containing $(S^{z})^{2}$.[ We note that for OM carrying small amounts of
transverse momentum, one would use $m_{1}/m_{0}=v\ll 1$; in the limiting case
we could exclude the $S^{\mu}S^{\nu}$ terms, recovering the form given in
equation (3).] We assume the internal fields causing the internal motions
represented by the $U\ \&\ S$ vector fields are included in the mass factors
$m_{0},m_{1},m_{2}$. The transported components of $\hat{S}$ are specified at
any point by eq.(1) of Paper I; their point-to point modification by the
presence of very strong gravitational curvature is not needed or included
here.

###  2.3  An Averaged Local DM Energy-Momentum Tensor.

Again, for aggregates of DM particles the momenta transport associated with
their averaged motion along their world-lines is specified by tensors similar
to those for ordinary matter, $P^{\mu\nu},F^{\mu\nu}$ previously discussed;
these may be incorporated later in eq.(8).

For representing the internal motions, we note that In the local Lorentz
frames used in Paper I the only significant stream components contributing to
${\tilde{T}}_{s}^{\mu\nu}$ are: $U_{s}^{0},U_{s}^{z},S_{s}^{0},S_{s}^{z}$,
using the notation of that paper. For ‘adding’ isotropic ensembles of similar
DM objects in a small region to form volume averages for the fluid
representation we can follow the same procedures used before888We can set a
particular $S^{0}_{s}\equiv 0$ only in one rest frame. So for an ensemble of
DM streams one really get an extra term
$\delta\tilde{T}^{00}=-\bar{m}_{1}{\cal N}\langle(S^{0})^{2}\rangle$ which we
will assume is small. From the variance in the distributions of
${U^{\mu}_{s}}\ \&\ S^{\mu}_{s}$, we get the equivalent of conventional
pressure terms expressed by the tensor $P^{\mu\nu}$. There is only one volume
averaging really involved. That should include components of the motion
associated with the particles’ motions around their world-lines. So we may
include these in the definitions of $F^{\mu\nu}\ \&\ P^{\mu\nu}$. in Appendix
B for ordinary matter. For $\epsilon=+1$ one finds the leading terms are:

$\langle\tilde{T}^{00}\rangle=+\bar{m}_{0}{\cal N}\langle(U^{0})^{2}\rangle,\
\
\langle\tilde{T}^{xx}\rangle=\langle\tilde{T}^{yy}\rangle=\langle\tilde{T}^{zz}\rangle\cong-\bar{m}_{1}{\cal
N}\langle(S^{z})^{2}\rangle/3,$ (7)

with $\langle\tilde{T}^{\mu\nu}\rangle=0$ otherwise. Here
${\bar{m}}_{0},\bar{m}_{1}$ are average stream mass densities999including the
normalization factor $1/(a^{2}+b^{2})$ in the averages.; ${\cal N}$ is the
average number of DM objects - each represented by a $(U_{s},S_{s})$ pair- in
a unit volume; and $\langle(S^{z})^{2}\rangle$ is an average of the squared
value of that component of $S$ which is normal to the spin-plane in the
guiding center solution. [ To simplify we shall consider
$\bar{m}_{0},\bar{m}_{1}$ as constants with only the number density ${\cal N}$
satisfying an equation of continuity; see eq.(21). ] Since
${\tilde{T}}^{\mu\nu}_{s\ ;\nu}=0$ for each object, one may adopt
$\langle\tilde{T}^{\mu\nu}\rangle_{;\nu}=0$ for the average.101010 One may
consider having many different ensembles of DM objects in a given volume, each
specified by certain characteristics e.g. $m_{0}$, energy, etc. In this case,
the quantities listed in eq.(7), including ${\cal N}$, should be labeled by a
name; for simplicity, we omit these labels now.

An ensemble average for the internal motions may be represented by the usual
ideal fluid representation
$T^{\mu\nu}_{fluid}=(\bar{\rho}+\bar{p})\bar{U}^{\mu}\bar{U}^{\nu}-\bar{p}g^{\mu\nu}$,
but now the dominant term $\bar{p}\equiv\langle\tilde{T}^{xx}\rangle$ is
negative. This is acceptable because $\bar{p}$ represents an averaged momentum
flux, not a thermodynamic pressure term and other terms contribute to the
total energy-momentum tensor.

For the cold DM halo we are interested in only in the case $U^{0},S^{z}\cong
1$. (See Paper I.) Consequently at any two positions $x_{1},x_{2}$ one has
$\bar{p}(x_{1})/\bar{p}(x_{2})={\cal N}(x_{1})/{\cal N}(x_{2})$; the similar
statement for the $\bar{\rho}$ ratio needs a slight modification. [See
eq.(27).] These relations determine the variation of the source terms,
$\langle\tilde{T}^{\mu\nu}\rangle$ , expressed as functions of the fluid
variables $\bar{\rho}$ and $\bar{p}$. Because $\bar{\rho},\bar{p}\propto{\cal
N}$, one has $\bar{p}\cong-\bar{c}^{2}\bar{\rho}/3$ where
$\bar{c}^{2}\cong(\bar{m}_{1}/\bar{m}_{0})\langle(S^{z})^{2}\rangle/\langle(U^{0})^{2}\rangle>1$
is to be assigned or determined for each ensemble of DM objects; so each
ensemble, so defined, is the equivalent of an ‘isothermal’ gas.

One may try to use this internal energy- momentum tensor as the dominant term
in the expression for the total fluid tensor $T^{\mu\nu}$. [Note this form
results from the choice $\epsilon=+1$.] Using it we see that the DM fluid
would act as to assist compression and resist expansion because of the
negative ‘pressure’ term; this physics may be compensated for by including an
additional source term such as $\lambda g^{\mu\nu}$.

The alternate form for ${\tilde{T}}_{s}^{\mu\nu}$, resulting from the choice
$\epsilon=-1$, turns out to be very useful for constructing halo models; in
this case the components of $\langle\tilde{T}^{\mu\nu}\rangle$ are given by
eqns.(7) with the signs reversed and similarly for $\bar{\rho}\ \&\ \bar{p}$.
In both cases, $\bar{\rho}\ \&\ \bar{p}$ are specified once the equation of
continuity for ${\cal N}$ is solved; with the assumption
$\bar{m}_{0},\bar{m}_{1}$ are constants, this is equivalent to solving
$\langle\tilde{T}^{\mu\nu}\rangle_{;\nu}=0.$

####  2.3.1  The Complete DM Source Term.

Finally, we recall we should add in a conventional term representing the
averaged mean time-like momenta fluxes transported through space time along
the world-line time-like trajectories, just as we represented the contribution
to $T^{\mu\nu}$ for ordinary matter:

$T^{\mu\nu}_{alt}\equiv\langle\tilde{T}^{\mu\nu}\rangle+q_{a}\langle
V^{\mu}V^{\nu}\rangle,$ (8)

where, if $V_{s}$ represents an averaged $U_{s}$ motion for a particular DM
object, $\langle V\rangle$ represents the local volume averaged velocity for
many objects.111111Paper I introduced $V_{s}$ as a ‘guiding center’ velocity,
a mass-weighted averaged component of $U_{s}$ (without ‘spin’ ) normalized so
that $V^{\alpha}V_{\alpha}=1$ and satisfying $V^{\nu}V^{\mu}_{;\nu}=0$. (In a
small local volume one may average motion along the individual guiding path,
determining mean motions and the averages of conserved quantities. We shall
assume $q_{a}={\cal N}\breve{m}_{0}/\langle(V^{0})^{2}\rangle$ where
$\breve{m}_{0}=\alpha m_{0}$ and $\alpha\leq 1$. When $\langle V^{\mu}\rangle$
is non-relativistic, it is only the term $T^{00}_{q}\equiv q_{a}\langle
V^{0}V^{0}\rangle=\alpha{\cal N}m_{0}$ that contributes to $T^{\mu\nu}_{alt}$.
Both tensors in $T^{\mu\nu}_{alt}$ have components that vary
thermodynamically; in this treatment of the cold DM halos we ignore these
terms.121212Also,because we don’t know how the DM particles are formed, it is
possible that the ratio of the two tensors may not be fixed for any DM
aggregate; we ignore this possibility.

We note that the form of $T^{\mu\nu}$ for one particle usually adopted for OM
is also composite, consisting both of rest-mass terms and internal motions.
See Yang & Liu (2018) .

[ A much simpler approximation results when one can re-interpret the extra
$q_{a}\langle V^{\mu}V^{\nu}\rangle$ term as a small OM component of the
source term associated with the DM component given by eqs(7). In cosmological
models, we have both DM and OM occurring together and the precise ratio of one
to the other is not known. This extra term can be included in the uncertain
specification of the OM contribution, allowing us to ignore the problem of
specifying $q_{a}$.]

###  2.4  The Form Of $T^{\mu\nu}$ For DM Used In Conventional Cosmology.

The form, characterized by $\epsilon=+1$, may be put into the standard fluid
representation of a cold DM fluid used in cosmology by adding an extra term of
the form of the ‘cosmological constant’ term to this source term. This results
in an effective (small) positive pressure term $\bar{p}_{eff}=\delta p$ in the
final fluid representation. Let’s define

$T^{\mu\nu}_{dark}\equiv\ T^{\mu\nu}_{alt}-\lambda(p)g^{\mu\nu}.$ (9)

One has that
$T^{00}_{dark}=\bar{\rho}+q_{a}\langle(V^{0})^{2}\rangle-\lambda\equiv\hat{\rho}$
and for the space-like components e.g.
$T^{zz}_{dark}\cong\bar{p}+q_{a}\langle(V^{z})^{2}\rangle+\lambda\equiv\hat{p}$.
For cold matter, $\langle(V^{0})^{2}\rangle\cong 1,\
\langle(V^{z})^{2}\rangle\ll 1.$ Choose $\lambda=-p+\delta p>0$ where $\delta
p>0$, representing a classical fluid’s positive pressure term, is small; Then
$T^{00}_{dark}=\hat{\rho}\cong\bar{\rho}-|\bar{p}|>0$ and e.g.
$T^{zz}_{dark}=\hat{p}\cong q_{a}\langle(V^{z})^{2}\rangle+\delta p=0+\dots$ .
[We may now include the usual pressure and internal energy terms associated
with the neglected term $P^{\mu\nu}$, expressing the variances in the velocity
distributions.] In the local rest frame all terms of $T^{\mu\nu}_{dark}$ are
diagonal and positive and we can therefore write

$T^{\mu\nu}_{dark}=(\hat{\rho}+\hat{p})U^{\mu\nu}-\hat{p}g^{\mu\nu}$ (10)

as for a normal fluid. But then we must rewrite Einstein’s equation as

$R^{\mu\nu}-g^{\mu\nu}/2=8\pi G\ T^{\mu\nu}_{dark}+\Lambda g^{\mu\nu},$ (11)

where we have put $\Lambda=8\pi G\lambda$. It is necessary to add the extra
term because $\lambda$ is not a constant but really varies as $p$ does. Its
variability makes no difference because in eq.(11) we have added and
subtracted the same term to the RHS of the equation. So we have a ‘normal’
ordinary matter fluid representation of the dark matter providing we also
introduce a variable cosmological term into Einstein’s field equations. Since
$T^{\mu\nu}_{dark\ ;\nu}=-g^{\mu\nu}\lambda_{,\nu}$ there is no strict local
conservation of energy and momentum for DM with this source term. We suggest
that this is the form implicitly used for representing DM in cosmological
models.131313 In the standard cosmological model $\Lambda$ should be primarily
determined by the needed rate of cooling and expansion during the short period
of nucleo-genesis; for this reason $\Lambda$ probably may be replaced with an
average value corresponding to this period. [Even for OM with some spin terms,
$p$ is small and not rapidly varying, so a a small constant value for
$\Lambda$ is an acceptable assumption for energies $\leq\sim 100$ Mev.]

With this interpretation, setting $\Lambda=\lambda\cong|p|$ and using eq.(7),
one estimates from observational cosmological parameters a value, $\langle
k\rangle$, of the parameter describing the space-like momentum component of a
typical DM object:

$\Omega_{DM}/\Omega_{\Lambda}\cong(\rho-|p|)/|p|=[\bar{m}_{0}\langle(U^{0})^{2}\rangle]/\
[\bar{m}_{1}\langle(S^{z})^{2}\rangle/3]-1=3\langle k\rangle-1;$ (12)

using $\Omega_{DM}/\Omega_{\Lambda}\sim 1/3$, one finds $\langle k\rangle\sim
0.4$ for cosmological models.

###  2.5  A Local Criterion For $T_{s}^{\mu\nu}$, Allowing The Choice
$\epsilon<0$ for DM.

The augmentation $T^{\mu\nu}_{alt}\to T^{\mu\nu}_{dark}$ is cosmetic. It was
done in order to make $T^{\mu\nu}_{dark}$ look like a familiar fluid stress
tensor with positive density and small positive pressure terms; the price we
pay for this is that we must introduce a ‘cosmological constant’ term in
Einstein’s equation. One could use have used the peculiar looking term
$T^{\mu\nu}_{alt}$ alone. If we were to drop this cosmetic requirement, the
question then arises: ‘What additional restraints must a chosen form for
$T^{\mu\nu}$ satisfy in order for it to be acceptable as a source term
(without requiring augmentation)?’

So, specify a replacement for eq.(6). Perform a thought experiment. Consider
an local ensemble of similar DM objects in a local Lorentz frame in which
their guiding center motions appear isotropic. [Equivalently, consider a
representative $(\hat{U},\hat{S})$ averaged over all possible spacial
orientations of the local Lorentz frame.] Then
$\langle\tilde{T}^{\mu\nu}\rangle$ is diagonal with equal space components in
a small region. Add $\lambda g^{\mu\nu}$ to it where
$\lambda=\langle\tilde{T}^{zz}\rangle$. The resulting sum $T_{E}^{\mu\nu}$
then has only one component, $T_{E}^{00}$. This construct uses Einstein’s
original focus on the one term $T^{00}=\rho_{eff}$ as the source of the
attractive gravitational field with the other components of $T^{\mu\nu}$
playing a minor role. For example, suppose we used eq.(6) itself; then one
finds

$T_{E}^{00}=\epsilon{\cal N}(\bar{m}_{0}-\bar{m}_{1}/3)\equiv\rho_{eff}.$ (13)

In our case an attractive gravitational field results with $\epsilon=-1$ for
DM if $\bar{m}_{1}>3\ \bar{m}_{0}$ or $k<1/3$. After matter decouples from
radiation in the standard cosmological model, one would expect for DM that
$k_{s}$ would decrease with time. So it is possible to adopt $\epsilon=-1$ for
the halo models, changing the signs of the averaged components given in eqs.
(7) when $k$ is small. Halo cold DM will then be charactered by dominant
positive pressure terms rather than by the positive density terms
characterizing OM.

[ There is a formal difficulty that $T\equiv g_{\mu\nu}T^{\mu\nu}$ itself has
the wrong sign when $\epsilon=-1$; one would like to match that the sign of
the Riemann constant on the LHS of Einstein’s equation . But the procedure
outlined determines a value of $\lambda$ and $\lambda g^{\mu\nu}$ may be added
to the source terms as a remedy when Einstein’s equation is actually being
solved.] But, it is always possible to use $\epsilon=+1$ if we introduce the
cosmetic $\lambda$ term, so that both $\rho\ \&\ p$ appear positive. In this
case the physical interpretation of the source terms is not direct, but can be
done using eqs.(11,10,9).

##  3 Description Of A Simple DM Halo Structure.

We regard a galactic halo as the reorganization of the local intergalactic
medium due to the presence of a concentrated OM gravitational source, similar
to the Debye sphere surrounding an ion in a plasma. Because the gravitational
field is small, we can split up the DM into two separate parts and
correspondingly consider two separate gravitational fields represented by two
different metrics. The DM halo part is superimposed upon a background large
intergalactic DM cloud which will represented locally by a fluid with a
uniform density $\rho_{00}$ and pressure $p_{00}$. So, for example,
$p_{halo}=p-p_{00}$.

The DM halo objects move under the attraction of a central galaxy. The local
DM objects follow orbits modified by their spin terms. Assuming spherical
symmetry for the galactic potential, the DM orbits satisfy the integrals of
motion given in Paper I. We therefore have conservation of orbital angular
momentum for each DM object. Also, the structure equations $T^{\mu\nu}_{\
;\nu}=0$ are satisfied by conserving the number flux of DM objects. (See
Section IIC.)

Our DM halo $T^{\mu\nu}$ is constructed with $\epsilon=-1$, so that the
pressure and density are given by eqs.(7) with the signs reversed. For
convenience we assume little OM in the halo region141414 This has the drawback
of ignoring OM galactic disk contributions and should be remedied in future
work. and only one type of low temperature DM, that for which the energetics
at $r\geq r_{q}$ are given by $K:\Gamma(k,0,0,1)$ in a local Cartesian frame.[
See eq.(1), Paper I]. As seen from a point in the halo, the attracting central
field is weak, $\Psi$, is small. So $(S^{z})^{2}$ is very much larger than
$\Psi$ in the halo and its variation can be ignored.

####  3.0.1 The Adopted Structure Metrics.

To represent the local intergalactic DM medium for $r>r_{q}$ one uses a
Robertson-Walker metric and cosmology, adopting a non-zero cosmological
constant to insure approximate stationarity. For simplicity, for the galaxy
the source contributions of the inner stellar disc population and of the
central bulge to the galactic mass are lumped together at the origin. For
$r<\sim r_{p}$ we assume the OM Milky Way galaxy can be represented by an
exterior Schwarzschild metric. For defining the structure of the halo we also
assume spherical symmetry. The halo, the intermediate region, $r_{p}\leq r\leq
r_{q}$, is represented by a interior Schwarzschild metric One has for the
galactic metrics:

$d\tau^{2}=B(r)dt^{2}-A(r)dr^{2}-r^{2}d\theta^{2}-r^{2}\sin\theta^{2}d\phi^{2},$
(14)

where $A^{-1}\equiv 1-2G{\cal M}(r)/r\equiv 1-2\Phi$. Write $B\equiv 1-2\Psi$;
its precise form is to be specified by the halo model . For the exterior
Schwarzschild solution, in vacuo, one has $AB=1$ and $\Psi=\Phi$. Details of
the representations used are given in Appendix C.

####  3.0.2 The Local Source Terms

For the halo material consider only low-velocity DM, for which $V^{0}\approx
1$. Adopt eqs.(6,7) with $\epsilon=-1$ for representing the basic DM halo. In
the remainder of this paper, the DM is assumed collisionless and locally
isotropic at large $r$. The halo source term151515For cold DM our estimates
use $\langle U^{0}\rangle,\langle V^{0}\rangle,\langle(S^{z})^{2}\rangle\simeq
1$. may be written as
$\bar{T}^{\mu\nu}=(\bar{\rho}+\bar{p})\bar{U}^{\mu}\bar{U}^{\nu}-\bar{p}g^{\mu\nu}$.
The signs of the components given by eq.(7) are reversed for $\epsilon=-1$.
Contributions from the variances of the various velocity distributions are
neglected. Note that the pressure term $\bar{p}\cong\bar{m}_{1}{\cal N}/3$ is
large and positive with $\bar{p}\gg|\bar{\rho}|$.

The effective halo density $\bar{\rho}\cong-\bar{m}_{0}{\cal N}+q_{a}$ is
small and and can be of either sign. One has a negative contribution from
eq(7) (since $\epsilon=-1$) and a positive contribution coming from the term
$q_{a}\langle(V^{0})^{2}\rangle\sim q_{a}$ which is hard to estimate without
detailed models or constraints on $q_{a}\bar{m}_{0}$ from observations. To
lowest order, since we are already neglecting halo ordinary matter’s
contributions we shall assume $\bar{\rho}\approx 0$.

The evaluation of how density and pressure in a static halo vary is done in
Appendix B, using particle flux conservation and angular momentum
conservation. The Appendix is quite detailed because the usual assumption of
“hydrostatic equilibrium” used for OM does not play any direct role here.The
halo structure is given by eqs.(33, 34). The DM pressure terms appearing in
$T^{\mu\nu}$ dominate in determining the potential $\Psi(r)$ with variations
in the potential $\Phi(r)$, dictated by the average density, playing a very
minor role. It is easy to replicate the observed ‘flat’ rotation curves; the
rotation velocity is given by eq.(45). The formal halo structure, expressed in
terms of the variations of the metric components,$A,B$ is outlined in Appendix
C.

###  3.1 Two Quite Different Halo Models.

Henceforth, we write $p,\rho$ for $\bar{p},\bar{\rho}$ neglecting halo
contributions from OM. There are two types of physical halo situations
considered. The first, (a), is for a galaxy at ‘rest’ with respect to the
local DM intergalactic medium, the halo structure extending out to $r=r_{q}$.
The second, (b), represents the halo around a rapidly moving galaxy; in this
case we expect a bow-shock discontinuity161616 At such a boundary the values
of $\rho_{a},\ p_{a}$ would be enhanced over their intergalactic values by the
factor ${\cal N}_{a}/{\cal N}_{q}$; using $V_{gal}\sim\ 200\ {\rm km\ s}^{-1}$
and $\hat{v}_{a}\sim\ 2-20\ {\rm km\ s}^{-1}$ for the stagnation velocity as
estimates one sees an enhancement factor of $\sim 10\ -\ 100$ may occur. to be
traveling in front of the galaxy. Behind the discontinuity we expect a
stagnation zone whose radius $r_{a}<r_{q}$ sets the outer halo limit; in this
case $p(r_{a})=\zeta\ p(r_{q})$ where $\zeta\gg 1$.

In Appendix B, we consider a flux of incoming DM objects with low angular
momenta and show that conservation of angular momentum flux in the outer halo
regions requires

$p_{\ halo}\equiv p-p_{00}=\delta p=[(r_{q}/r)^{2}-1]p_{00}\ \ {\rm for\ type\
(a)},$ (15) $p=(r_{a}/r)^{2}p_{a}\ \ {\rm for\ type\ (b)}.$ (16)

For type (a), there exists an $r_{a}<r_{q}$ such that for $r\leq r_{a}$ the
subtractive term term $p_{00}$ can be ignored; then the two equations have the
same form for $r\leq r_{a}$.

The Einstein field equations for $A\ \&\ B$ (see eq.(45)) then directly
require that the rotational velocity $v^{2}_{cir}\propto pr^{2}$ for the outer
DM halo since there $\Phi(r)$ is small. Since $p\propto 1/r^{2}$, these
pressure estimates force $v_{cir}\cong$ constant for most of the outer halo
where $r_{b}^{\star}<r<r_{a}$.

In the very outermost halo region, $r_{q}\geq r>r_{a}$ , which we call the
‘rim’, the radial velocity terminates gradually $v_{cir}^{2}\propto 4\pi
Gp_{00}(r_{q}^{2}-r^{2})$ for type (a) but more abruptly for type (b),
depending upon the width and structure of the putative shock discontinuity. We
suggest adopting $r_{q}\cong 3r_{a}$ for type (a) halos. As one goes inward
the DM pressure rises to a maximum value and then falls rapidly to zero at
$r_{p}/2$ in a small inner annular zone whose size is determined by the
streams’ conserved initial angular velocity ($r_{a}v_{a}$) and the mass of the
central galaxy. The rather sharp boundary between the zone of freely falling
streams, where eqs.(15,16) hold, and the inner zone is given by eq.(31). We
emphasize that eqs. (15,16 ) do not assume an equation of hydrostatic
equilibrium.

###  3.2  Models For The Milky Way Galaxy and M33.

####  3.2.1 The Observed Rotation Curve.

The observational rotation curve for the inner part of the Milky Way, for
$R\leq\sim 8$ kpc is discussed by Clemens (1985); we neglect this region. Mróz
et al (2019) gives Cepheid observations determining the outer part, $5\leq
R\leq\sim 15$ kpc. The observed outer rotation velocity curve is quite flat,
allowing from noise, perhaps falling from $\sim 230\pm 15\ {\rm km\ s}^{-1}$
at $R=6$ kpc to $\sim 215\pm 15\ {\rm km\ s}^{-1}$ at $R=14$ kpc. We now
construct a DM halo model which matches these observations.

####  3.2.2  Choosing Parameters For A Model.

First, one may predict a rotation curve starting initially with theoretical
model parameters. One may calculate $p(r),\ \rho(r)$ in the halo - see
Appendix B - and the metric coefficients $A(r),B(r)$; this gives a detailed
solution for $v_{circ}(r)$, the rotation curve. See eq.(45). The mass of the
central galaxy ${\cal M}_{0}$ is needed. Often it is not well estimated, but
again this can be refined by observations. So only two really arbitrary
parameters, $p_{a},v_{a}$ need to be specified in constructing a particular DM
halo model. These two parameters also describe the internal state of the
intergalactic DM cloud enclosing the halo. The choice of $v_{a}$ is severely
constrained by eq.(37).

The important input data for such a model halo prediction are the main
‘outside’ parameters $r_{a},p_{a}$ and an estimate of the average DM peculiar
velocity, $v_{a}$, at $r=r_{a}$. With this the average intrinsic angular
momentum each DM particle carries at $r=r_{a}$ can be estimated. In practice
this quantity can be replaced by the parameter171717 It is defined in the
‘Approximations’ section of Appendix B. $\eta$ which , with $R_{h}$, determine
the boundaries of the inner parts of the halo. Both $\eta$ and $R_{h}$, the
‘inside’ parameters, can be estimated from observations near the ‘knee’ of the
rotation curve. Less accurately, from the ‘end’ of the ‘flat’ rotation curve
$r_{a}$ can be estimated.181818For incomplete rotation curves, the
approximation $r_{a}\simeq 10R_{h}$ is reasonable. Finally the halo density
$\rho_{a}$ at $r=r_{a}$ needs to be specified; because it is small, it should
not significantly affect the structure of the halo. Put
$|\rho_{a}|\equiv\tilde{k}_{obs}p_{a}$ and to first approximation one may put
$\tilde{k}_{obs}(r)\approx 0$.

It is not difficult to reverse the procedure and estimate the model parameters
from a given rotation curve by successive approximations. From the ‘knee’
region one estimates $R_{0}$ and sets the potential due to central OM to be
$\Phi(r)=G{\cal M}_{0}/r\equiv V_{circ}^{2}(R_{0})\ {\rm at}\ r=R_{0}$ for
$r\geq R_{0}$; to be consistent ,one provisionally adopts
$\tilde{k}_{obs}\approx 0$. At the far end of the rotation curve where $r$ is
large one has $p\propto 1/r^{2}$; therefore choose a value for $r_{a}$: then
since then $V_{circ}^{2}(r)\cong 4\pi Gp(r)r^{2}\to v_{\infty}^{2}$, a
constant, one has $p_{a}$ determined. Now construct a first approximation to
the rotation curve, $V^{2}_{first}\approx\Phi(r)+v_{\infty}^{2}$, calculating
from the outside inwards $r\to R_{0}$. This theoretical approximation will
depart from the observations for $r\leq\hat{r}_{h}$; we may take this to
determine $\hat{r}_{h}\approx r_{h}$ the maximum compression of the DM; then
the observations in the interval $[R_{0},r_{H}]$ determine the value of $\eta$
(and by establishing the boundary $r_{b}^{\star}$, a value for
$v^{2}_{a}r_{a}^{2}$).

A further refinement would include a contribution to $\Phi(r)$ from the OM
disk population for $R_{0}\leq r\leq\hat{r}_{h}$; this could be determined for
the Milky Way from the Gaia observations. With this known, non-zero values of
$\tilde{k}_{obs}(r)$ may be investigated

####  3.2.3  Parameters For The Milky Way.

As an example, we constructed a representation for the Milky Way galaxy’s
rotation curve for $r\geq 6$ kpc, using a type $(b)$ model. For the OM galaxy,
we adopt $V(r)^{2}=G{\cal M}_{0}/r$, with $V(r=6\ {\rm kpc})=220\ {\rm km\
s}^{-1}$. This model uses $\eta=0.5$ and $R_{h}=8$ kpc as the effective inner
edge of the dense part of the DM halo; no DM can go below $\sim 4$ kpc and the
boundary between the inner and outer halo zones is at $r_{b}^{\star}=12$ kpc.
Eq. (45) gives $v_{cir}(r)$.

We use the approximations191919These greatly simplify the the detailed halo
calculations in the range $r=4-7$ kpc where the halo contributions are very
small; they are good approximations in the range $r=9-12$ kpc.  discussed in
Appendix B; in this case one sets $v_{cir}^{2}\to v_{\infty}^{2}=4\pi
G(p_{a}+\rho_{a})r_{a}^{2}$ for $r\gg R_{h}$ for the asymptotic halo solution.
We chose $v_{\infty}=175\ {\rm km\ s}^{-1}$ and $\rho_{a}\approx 0$. The outer
‘edge’ of the halo was chosen to be $r_{a}\approx 60-80$ kpc; this corresponds
this results in estimating $v_{a}\approx 30-25\ {\rm km\ s}^{-1}$. using
eq.(37). The results are given in Table 1.

Table 1:  Model $V_{cir}$ vs. r For Milky Way 6 kpc | 237 km/s | 14 kpc | 227 km/s
---|---|---|---
7 | 230 | 16 | 221
8 | 227 | 20 | 212
9 | 227 | 40 | 195
10 | 228 | 60 | 188
12 kpc | 234 km/s | 80 kpc | 185 km/s
Figure 1: Fig.1 The observed rotation curve for the Milky Way, given by Mróz
et al (2019) . The calculated model points of Table 1, represented by gold
stars, are superimposed on their results.

This is in very good agreement with the results of Mróz et al (2019); see
their Fig 2 and their mean solid line. [Our tabulated values are perhaps $\sim
5\ {\rm km\ s}^{-1}$ too low.] For data covering from $6-16$ kpc , they find—
202020Another of their models gives $V_{cir}=\approx 222\ {\rm km\ s}^{-1}$ at
$r=7.6$ kpc .$V_{cir}\approx 234\ {\rm km\ s}^{-1}$ at $r=8$ kpc and a linear
fall-off, $dV_{cir}/dr=-1.34\pm 0.20\ {\rm km\ s}^{-1}/{\rm kpc}$. This
Cepheid data is in good agreement with the maser data Read et al (2014). The
observations of $v_{cir}$ for $r>14$ kpc still have large error.

[Caveats: For the inner Milky Way , a spatially averaged rotation curve is not
available and it is known that some inner regions on opposite sides of the
galaxy exhibit differing rotation curves Mihalas & Binney (1981). Also,
estimates of the luminous mass ${\cal M}_{0}(r)$ differ by a factor of two
Trimble (2000); Bahcall et al (1983); Bahcall (1986); Gilmore (1984),
reflecting differences in reduction procedures. Another fit to the data used
in the rotation curve of Clemens Clemens (1985) would allow a lower value
$R_{h}\approx 6$ kpc; this probably the lowest value acceptable. Within the
uncertainties of interpretation of the observations, somewhat larger values of
$R_{h}$ up to $9-9.5$ kpc are not really excluded. Small variations in
$\tilde{k}$ are permissible; We used $\tilde{k}\equiv|\rho|/p=0$. For
$\tilde{k}=0.1$ , $V_{cir}$ is reduced by $\sim 2\ {\rm km\ s}^{-1}$ at $r=12$
kpc.]

####  3.2.4 The Rotation Curve for M33

If the central galaxy hasn’t enough mass, or the angular momentum of the dark
matter in the halo is sufficiently high, then there may not be a “free-fall”
outer zone of the halo . Then the dark matter concentration in the vicinity of
the galaxy is severely limited by its conservation of angular momentum, the
pressure being given by its “inner zone” value, see eq. (34). But because the
small spiral M33 has only $\sim 1/10$ the mass of the Milky Way galaxy in its
interior, even a small contribution from a dark matter accumulation in its
vicinity leads to a rise in the rotation curve. We assume the observed region
corresponds only to the inner DM halo region.

As with the Milky Way , the motions in the interior of the small spiral, M33
are quite complicated Kam, et al (2015). In the region beyond $\sim 15$ arc
minutes, $\sim 3$ kpc, the rotation curve seems reasonably represented by the
observations of Corbelli & Palucci (2000). We adopt $v_{cir}=95-100\ {\rm km\
s}^{-1}$ at $r=15$ arc minutes as characterizing the $1/r$ potential
representing the OM central galaxy. Also take the outermost points of the
rotation curve as determining $r_{h}\cong 80$ arc minutes. with
$v_{cir}(r_{h})=145\ {\rm k\ s}^{-1}$. Then the only unknown model parameter
is $\eta$; its choice is specified by matching the observations in the entire
region $15\leq r\leq 140$ arc minutes. One finds the assignment $\eta=0.15\ -\
0.18$ reasonable.

Figure 2: The observed rotation curve for the outer portion of M33, as given
by Corbelli & Palucci (2000). The model points, represented by gold stars, are
superimposed upon the observations in (a), and compared to the linear
representation some people prefer, in (b). The model shown used $\eta=0.15$.
Here, 5 arcmin $\approx 1$ kpc.

####  3.2.5  The Model Parameters Seem Acceptable.

In addition to the requirement that the model predicts observed rotation
curves, one requires that the model parameters are reasonable. For the Milky
Way, arbitrarily assuming $r_{a}=10R_{h}\sim 80$ kpc, one finds
$p_{a}/c^{2}\cong 6.0\pm 0.3\times 10^{-27}{\rm g\ cm}^{-3},$ the ‘errors’
reflecting uncertainty in the choice of $\tilde{k}_{obs}$. Alternatively,
using212121 The mean peculiar speed of DM objects in originating regions is
designated by $v(r_{a})\equiv v_{a}$. In paper I it is suggested that the mean
intrinsic rotational speed of a DM object is in equilibrium with its peculiar
translational velocities. This numerical estimate assumes the DM objects have
baryonic masses. $v_{a}\sim 30\ {\rm km\ s}^{-1}$ and
$r_{a}^{2}v_{a}^{2}\equiv\ G{\cal M}_{0}R_{h}$ with ${\cal M}_{0}\approx
10^{11}{\cal M}_{\odot}$ Mróz et al (2019), one estimates $r_{a}=63\pm 15\
{\rm kpc}$; the uncertainty reflecting the uncertainties in $R_{h}$ and ${\cal
M}_{0}$. The probable range for the DM pressure is $p_{a}/c^{2}\sim 1.5\times
10^{-26}\ {\rm g\ cm}^{-3}$ at $r_{a}\sim 50\ {\rm kpc}$ to $p_{a}/c^{2}\sim
0.5\times 10^{-26}\ {\rm g\ cm}^{-3}$ at $r_{a}\sim 80\ {\rm kpc}$ .

For the large spirals, the extent of the observed haloes sets a lower limit to
the value of $r_{a}$. Using $r_{a}\sim 50-100$ kpc, this range for
$p_{a}/c^{2}$ needed looks reasonable. For both types (a),(b) of halos it is
reasonable to adopt $p_{a}\sim 10\ p_{q}$ where $p_{q}$ is the effective
density of DM in intergalactic space. These should be compared to other
appropriate densities (See Bahcall (2000); Scott et al (2000).)The limiting
cosmological critical density is $\rho_{c}\sim 10^{-29}\ {\rm g\ cm}^{-3}$.
The mean densities of luminous matter in clusters of galaxies are $\sim
10^{-26}-10^{-28}\ {\rm g\ cm}^{-3}$; this is on the order of our estimate for
$p_{q}/c^{2}$.

In the MW inner halo the DM pressure is $p_{b}=p_{a}(r_{a}/R_{0})^{2}\approx
50-100\ p_{a}$ corresponding to $p_{b}/c^{2}\sim 10^{-24}-10^{-25}\ {\rm g\
cm}^{-3}$ which is slightly smaller than the OM mean density of the MW galaxy
distributed inside a sphere of $10$ kpc radius, $\langle\rho\rangle\sim
2\times 10^{-24}\ {\rm g\ cm}^{-3}$. The local mean density of OM is estimated
to be $\sim 1.4\pm 0.6\times 10^{-23}\ {\rm g\ cm}^{-3}$ Mróz et al (2019).

For the representative small galaxy M33, we may estimate $p_{b}$ at
$r_{b}\approx 16$ kpc from $v_{circ}^{2}\cong 4\pi Gp_{b}r_{b}^{2}$. Because
of the observed linearity of the rotation curve the value of $p_{b}$ is
reasonably determined even though the value of $r_{b}\cong r_{h}$ is
uncertain. One finds $p_{b}/c^{2}\approx 1.1\times 10^{-25}\ {\rm g\
cm}^{-3}$. The agreement of this value of $p_{b}$ with the value found for the
large Milky Way galaxy suggests one ‘infall’ mechanism is sufficient to
explain both types of rotation curves.

####  3.2.6 Improvement Of The Modelling Procedure.

To significantly improve these models one needs observations to estimate the
amounts of ordinary matter entrapped within the DM haloes and flowing through
the inner edge of the DM halo. Also, the mass of the OM disc population in the
outer regions, $r\geq\eta R_{h}$ is needed. It is possible that outer OM disk
populations may cause ‘wiggles’ in the rotation curve. With a combined OM and
DM model one could determine the density contribution $q_{a}$ of eq.(8).

This is important because the only model parameter not yet well determined is
$|\rho_{a}|\ \equiv\tilde{k}_{obs}p_{a}$. Trials in fitting indicate
$\tilde{k}_{obs}$ is small, so to lowest approximation we used
$\tilde{k}_{obs}\approx 0$. To determine it from the rotation curves requires
assumptions about the outer OM galactic disk density and the subsequent
modification of $\Phi(r)$. We do not have a theoretical estimate of this
parameter. [See the discussion of eq.(8).]

###  3.3  Observational Challenges.

A main observational challenge is to determine the interface between the outer
halo and the intergalactic medium $r_{a}\leq r\leq r_{q}$ and the value of
$r_{a}$. Observations of DM lensing Mandelbaum et al (2006) in earlier epochs
are modeled with $r_{q}>100$ kpc. In the case of the MW galaxy,, which
exhibits a flat rotation curve at $\sim 200\ {\rm km\ s}^{-1}$, one has limits
set by disturbances at $\sim 60$ kpc by the dwarf galaxies LMC & SMC, and at
$\sim 700$ kpc by M31; evidently the assumption of a spherical gravitation
potential would would be crude and useful only in selected directions. A study
of MW ‘halo objects’ suggests the halo changes character at $\sim 30$ kpc and
might extend to $\sim 120$ kpc [Battagia et al (2005)]. Also very low surface
brightness observations of emissions from ordinary matter trapped in the
extended DM regions around galaxies would be rewarding in possibly mapping out
motions in the halo.

It should be remarkably easy to use this procedure to represent the observed
halo radial velocity curves of many galaxies. The derived model parameters may
be used as a first probe of the variations in the intergalactic DM medium.
There really is an interesting unexplored observational field here. There is
no reason to assume that since the epoch of decoupling from ordinary matter,
most of the DM (- $\sim 3/4$ of the total matter -) should remain anywhere
near homogenously distributed in the Universe.

Not all galaxies enveloped in a DM intergalactic medium need have haloes such
as those calculated. Our model requires that the peculiar velocities of a
significant portion of the local DM be low. The components of DM carries
intrinsic angular momentum which, combined with orbital angular momentum,
forms a centrifugal barrier to prevent close approach to the center of
attraction. (See Paper I.) The DM halo’s inner edge effectively starts at
$r_{p}\cong R_{h}\equiv r_{a}^{2}v_{a}^{2}/G{\cal M}_{0}$. A low mass galaxy
may have its DM halo starting well beyond its visible structure. A high mass
galaxy may have its halo structure begin inside its visible structure. The
Milky Way probably is such a galaxy. So far, observations of other large
spirals suggest values $R_{h}\sim 4-8$ kpc. Any OM entrained with the DM at
large distances, having only orbital angular momentum will fall through the DM
barrier at $r_{p}$. However one guesses it should be a minor halo component;
one expects OM to constitute $\sim\Omega_{b}/\Omega_{d}\sim 0.2$ Tegmark et al
(2004); Longair (2005) of the halo density at large $r$.

####  3.3.1 Other Considerations.

We also note that DM with high angular momentum may provide the external
pressure boundary condition needed to construct classic Bonner-Ebert models of
protogalaxies. Ebert (1955); Bonnor (1956) Because of past galaxy-galaxy
collisions, it would be impossible to rule out the presence of DM inside
galactic disks. DM entrapped within galaxies by galaxy-galaxy collisions may
thermalize and loose almost all its orbital angular momentum, thereby
providing a confining pressure to aid form very massive star models.

DM with larger peculiar velocities than those assumed ($\sim 30\ {\rm km\
s}^{-1}$) could form haloes with similar structures but at much larger
distances; this may provide models of galaxy lenses. We have not allowed for
the presence of OM in the haloes. The presence of the Sunaev-Zeldovich effect,
indicating the presence of OM , precludes direct application of these models
to the haloes of galaxy clusters.

##  4  Summary.

A model of the dark matter (DM) surrounding galaxies has been developed which
represents the observed rotation curves and can explain the absence of DM in
small potential wells such as the solar neighborhood. The model is based on
conservation of angular momentum in the flow of DM around a galaxy. In it (see
Paper I) the DM is treated as a new state of matter in which matter while
traveling along time-like paths also carries considerable transverse momentum.
This transverse momentum results in intrinsic angular momentum terms. These,
coupled to the usual orbital angular momentum, form centrifugal barriers,
preventing DM objects from passing close to centers of gravitational
attraction. In addition, the DM objects can transport an internal form of
momentum. For aggregates of moving DM objects, the local sum of the internally
transported momentum fluxes appears as a new form of ‘pressure’, not a
thermodynamic pressure. For DM this term is large and must be included in
specifying the energy-momentum tensor, the local source of the gravitational
field and in calculating rotational velocities in the outer parts of galaxies.
The halo cannot be primarily OM: (1) it is not seen; (2) there is no reason
why OM halo material would be diminished in the region $r_{p}\leq r\leq
r_{h}$.

###  4.1 Halo Structure

Two alternate forms of the energy-momentum tensor representing DM are derived
in this paper. One features a large positive momentum flux appearing as a
dominant pressure-like term. Using this the structure equations for the halo
are developed and examined; these show that the pressure increases
considerably as one approaches the central attractor but suffers an abruptly
termination because of the centrifugal barrier. The DM matter forms an annulus
around the central galaxy, a ‘halo’, and does not enter the central region.

In this halo region, the circular velocity is given by one of Einstein’s
equations eq.(45). Here $\Phi$ represents (mainly) the central OM galaxy’s
gravitational potential and is small and slowly varying. In the outer halo
regions $p\propto 1/r^{2}$, so $v_{cir}\cong$ constant. The variation of $p$
is determined by consideration of particle flux and angular momentum
conservation.

A model for the Milky Way galaxy uses an inner edge of the halo at $R_{h}=8\
{\rm kpc}$ and an outer edge at $r_{a}\sim 80\ {\rm kpc}$; the pressure there
is $p_{a}/c^{2}\sim 6\times 10^{-27}\ {\rm g\ cm}^{-3}$, with the
intergalactic DM pressure being perhaps a factor of 10 lower. The local
intergalactic DM contributing to the MW halo seems to have low peculiar
velocities $v_{pec}\sim 30\ {\rm km\ s}^{-1}$. A model fitting the rotation
curve of the small spiral M33 is constructed using only the inner structure of
the halo; in it the centrifugal barrier is important.

Aggregates of such DM objects could comprise the main component of the
intergalactic medium. Because of the gravitational attraction of an (ordinary
matter) galaxy, the intergalactic DM medium locally compresses to form a
galactic halo around a galaxy. One expects, because of variations in the local
peculiar velocities of the DM objects, that the size of the halos would vary.
Models of galactic haloes can be calibrated from observations with sufficient
accuracy to provide a tool for investigating variations in density, pressure
and the peculiar velocities of the intergalactic DM medium.

###  4.2 Cosmology

The other form of $T^{\mu\nu}$ for DM found can be made to resemble the
conventional form of $T^{\mu\nu}$ used in cosmology to represent ordinary
matter (OM), providing a cosmological ‘constant’ term $\Lambda$, slightly
varying, is also introduced. This raises the possibility that at least some
part of the ‘dark energy’ is not a separate entity but due to the
representation of DM used.

If the alternate representation $T^{\mu\nu}_{dark}$ is adopted for cosmology,
then $\Lambda\sim 8\pi Gp$ is not constant and the Standard Cosmological Model
would need some re-discussion. We do not claim this is the only reason for the
$\Lambda$ term’s presence in Einstein’s equation, but this simple DM proposal
provides a rationale for its presence. It is needed in modern cosmology to get
the observed baryon acoustic oscillation spectrum and was based on the
assumption that the DM was simply like an undetected form of ordinary matter
with an ordinary fluid $T^{\mu\nu}$ featuring a conventional $p-\ \rho$
relation. However, this spectrum represents only the epochs when the Universe
was becoming transparent , when the optical depth is less than $\sim 3$ and
does not justify the assumption that $\Lambda$ was constant at earlier epochs.

It is possible that $k=|\rho|/p$ varies on cosmological time scales. The
observed cosmological parameters Longair (2005)
$\Omega_{d}/\Omega_{\Lambda}\approx 0.15/h^{2}\sim 0.3$ would then represent
an averaged $k$; it would very likely represent the value of $k$ when particle
formation was frozen, i.e. when the temperature was $\sim 1-5$ Mev.

[(An earlier version of some of these ideas is available on-line Helfer
(2017).]

##  5 Appendices.

##  6  Appendix A: Astronomy.

###  6.1 Inferred Properties Of Dark Matter.

There exist many large spiral galaxies (including the Milky Way) with ‘dark
matter’ haloes. The observational characteristic of these spiral galaxy halos
are galaxy rotation curves, $v_{rot}(r)$ consisting of: (1) a central
spherical contribution followed by a (roughly) linear part :
$v_{rot}(r)=V_{0}(r/r_{00})$ for $r\leq r_{00}\sim 2\ -\ 4$ kpc; (2) a
(fairly) ‘smooth’ transition zone $r_{00}\leq r\leq r_{0}\sim 4-8$ kpc; and
(3) a roughly “flat” part $v_{rot}(r)=V_{0}$ for $r_{0}<r\leq r_{1}\sim 16\ -\
50+$ kpc. This outer limit is hard to estimate and in a few cases may be $\geq
100$ kpc. A value $V_{0}\sim 200\ {\rm km\ s}^{-1}$ characterizes large
spirals. Rubin (1980, 1983); Fich & Tremaine (1991); de Zeeuw & Franz (1991);
Kulessa & Lynden-Bell (1992); Sofue & Rubin (2001). For the Milky Way see Mróz
et al (2019).Read et al (2014).These rotation velocities222222These are
defined by the observed line-of-sight motions of extreme Population I objects
such as very luminous HII regions or HI gas and molecular clouds, known to
depart by less than $\sim 5\%$ from circular orbital velocities $v_{cir}(r)$
in galactic disks outside the very central region. The absence of these
objects in the outer parts of galaxies limits our present knowledge of the
rotation curves at large $r$.  are normally interpreted as circular
velocities, setting $v_{cir}^{2}=G{\cal M}_{obs}/r$ where ${\cal
M}_{obs}\propto r$ when $r_{1}\geq r\geq r_{0}$ and regarding ${\cal M}_{obs}$
as a true mass distribution. No galactic mass distribution compatible with the
observed stellar distributions predicts this behavior in the outer zone
Mihalas & Binney (1981). An unseen (‘dark’) matter component is usually
hypothesized, probably amounting to several times the mass contribution
inferred for the stellar and interstellar matter contribution. Because the
halos are strongly concentrated to the center, the halo DM must be moving at
non-relativistic velocities; this is also true for DM in clusters of galaxies:
$v<\ \sim 400\ {\rm to}\ <\sim 10^{3}\ {\rm km\ s}^{-1}$ (in clusters) are
conservative upper limits.

There is also a class of smaller spiral galaxies showing smaller halos whose
outer rotation curves have an approximately constant positive slope. The small
galaxy M33 studied in this paper -see Fig.2- may be considered a prototype.

Presently, there is no direct evidence that this ‘dark’ matter (DM) interacts
with ordinary matter (OM) through electromagnetic interactions or collisions.
However, in the standard cosmological model both types of matter remained
coupled to radiation until $T\sim 1$eV. To be coupled the DM must have an
interaction cross-section with $n(t)\sigma cR(t)/\dot{R}(t)>1$. Allowing for
the model’s uncertainties we note that for DM to remain coupled for energies
down to $T\sim 10$ eV (or $1$ KeV) one must have an interaction cross-section
$\sigma>10^{-29}\ ({\rm or}\ 10^{-33})\ {\rm cm}^{2}$, if DM is represented by
particles of baryonic mass. [If they have less than an electron’s mass, one
has the additional problem of explaining their low velocities in galactic
halos.] Also, such particles would be entrapped within a typical star like the
sun if $\sigma>2\times 10^{-37}\ {\rm cm}^{2}$.

##  7  Appendix B: Supplementary Physical Arguments.

###  7.1  The Vanishing Of The ‘Line’ Divergence $U^{\nu}_{s;\ \nu}$

The technical point here is that we are are considering functions defined only
along a line so we cannot directly use the Gauss divergence theorem with no
sources or sinks present. However $U^{\nu}_{s;\nu}$ is a scalar, independent
of a coordinate system choice. At any point, choose a geodesic coordinate
system and rotate the coordinate axes so that $\tau$ is the coordinate along
one of the axes. Then $U_{s}^{\mu}$ has only component, $\mu=\tau$, and
$dU_{s}^{\mu}/d\tau=0$ because $U_{s}^{\tau}$ is a geodesic’s tangent vector.
For DM when we consider confluences of streams such as $S_{s}$ in a small
region, we shall also assume $S^{\mu}_{s;\mu}=0$ to prevent local creation or
destruction of streams; rigorously, in the presence of spin one only has
$\langle S_{s}^{\mu}\rangle_{;\mu}=0$ established in the guiding center
approximation.

###  7.2  $T^{\mu\nu}$ For Ordinary Matter.

We start with a method for constructing $T^{\mu\nu}$ for ordinary matter (OM)
when mass motions are significant and their associated momentum flux exceeds
any assignable pressure term. An example is the rotating stellar disk in a
spiral galaxy. It may be useful for numerical solutions to Einstein’s
equations

The only formal requirements in GR on the fluid energy-momentum tensor are
that it is symmetric $T^{\mu\nu}=T^{\nu\mu}$ and divergence-free
$T^{\mu\nu}_{\ ;\nu}=0$; consequently, a term $\lambda g^{\mu\nu}$, where
$\lambda$ is constant, can always be added to $T^{\mu\nu}$.

For each OM object $s$ with constant mass $m_{s}$ following the time-like path
$x^{\mu}_{s}(\tau)$ we define an energy momentum tensor by using the geodesic
equation $U^{\nu}_{s}U^{\mu}_{;\nu}=0$ and requiring mass conservation. Set

$T_{s}^{\mu\nu}\equiv m_{s}f_{s}U^{\mu}_{s}U^{\nu}_{s}$ (17)

along the path and determine the variation of $f_{s}$ by requiring

$(m_{s}f_{s}U^{\nu})_{;\nu}\equiv 0=[m_{s}f_{s}\surd(-g)U^{\nu}]_{,\nu}\ \ \
{\rm or}$ (18) $d\hat{m}_{s}/d\tau_{s}+\hat{m}_{s}U^{\nu}_{s,\nu}=0,\ \ {\rm
where}\ \ \hat{m}_{s}\equiv m_{s}\surd(-g)f_{s},$ (19)

then $T^{\mu\nu}_{s\ ;\nu}=0$ and $T^{\mu\nu}_{s}$ can be used as a fluid’s
source term for determining the curvature tensor. So, for satisfactorily
representing even one typical object as a source, one must introduce a
‘density’ distribution function. Then the distributed object may be thought of
as equivalent to a localized stream with a finite cross-section and mass
density $\hat{m}_{s}$.

One may show (see Appendix A) that $U^{\nu}_{s;\ \nu}=0$ if the objects are
neither created nor destroyed in the region under study.232323This divergence-
free condition is equivalent to choosing the radiation gauge in EM theory.
Henceforth we drop the accent, writing $m$ for $\hat{m}_{s}$ when speaking of
a ‘typical’ object. Then eq.(18) has three useful forms depending upon the
functional form of $m$ available: (1) steady state one-dimensional pipe
flow242424 i.e. For an aggregate of close-by objects all moving similarly, the
argument of Appendix A1 holds and we can regard them collectively as a narrow
confined stream of objects.:

$mv={\rm constant};$ (20)

(2) given the fluid form $m_{s}(\tau)=m_{s}(x(\tau))$, the equation of mass
continuity results,

$\partial_{t}\ m+{\vec{v}}\cdot\nabla m=0,$ (21)

where $U^{\mu}_{s}=\Gamma(1,{\vec{v}})$; and (3), given the kinetic form
$m_{s}(\tau)=m(\ x(\tau),U^{\mu}(x(\tau))\ )$, Liouville’s equation results,

$[U^{\mu}\partial_{\mu}-\Gamma^{\mu}_{\alpha\beta}U^{\alpha}U^{\beta}(\partial/\partial
U^{\mu}]m=0,$ (22)

which plays an important role in interpreting some stellar kinematics.
Chandrasekhar (1942); Battagia et al (2005) Each of these equations hold only
on a time-like path $x^{\mu}(\tau)$.

Eqs.(21,22) use an additional constraint, $\nabla\cdot\bar{U}=0$, for
individual particles and for simple252525 We may need to allow for the
exchange of energy and momentum among different components of the source
terms. particle ensembles. It is needed when the total number of particles is
conserved. For a fluid this concept has no meaning, so it must be imposed on
fluid representations to enforce the particulate nature of matter.

[Depending upon the interval of space-time being studied, this condition can
be somewhat relaxed if the total source term $T^{\mu\nu}_{fluid}$ is broken
down into a sum of constituent fluids. For example, consider radioactive decay
of a nucleus $A\to B+C+\gamma$. The number densities $n_{A},n_{B},n_{C}$ are
modified, also their associated momenta are changed and 4-momentum is
transferred to the radiation field tensor.. The fluid $T^{\mu\nu}$,
representing each species, is changed, but their sum in the source term in GR
is not. So, e.g. $T^{\mu\nu}_{A;\nu}\neq 0$. But because the source term
$T^{\mu\nu}_{fluid}$ is inclusive we can use the ideal fluid form
$T^{\mu\nu}_{A;\nu}=0,$ and ignore interactions such as particle destruction.
Similarly for considering star densities and the formation of black holes and
proto-stars. ]

Now, suppose we have a finite number of streams in a small region ${\cal R}$.
Conventionally we treat ${\cal R}$ as an energy-momentum reservoir and
effectively define local fields  $\bar{N}\bar{m}_{s}$, $\bar{U}$ and
$\bar{T}^{\mu\nu}$ extending throughout ${\cal R}$ as averaged values of the
included streams. [The averaging is detailed in text.] Then
$\bar{T}^{\mu\nu}=F^{\mu\nu}+P^{\mu\nu}$ where only $P^{\mu\nu}$, expressing
contributions from the variances of the velocity distributions, need require a
thermodynamic equation-of-state to evaluate; its contribution is usually small
for non-relativistic OM.

We emphasize that the spacial components of $F^{\mu\nu}$ represent the
streams’ average momentum flux Landau & Lifshitz (1959), and not really
thermodynamic ‘pressure’ terms. The assumption of isotropy in cosmology allows
confusion between the components of $F^{\mu\nu}$ and of $P^{\mu\nu}$; but in
modeling individual galaxies it is the streaming motions of various galactic
populations contributing to $F^{\mu\nu}$ that are the source of the spacial
terms in the stress tensor.

###  7.3  The Halo Structure.

####  7.3.1  Representation Of The Averaged $T^{\mu\nu}$ In The DM Halo Model.

We use only DM with $\epsilon=-1$, so that the components of the basic energy
momentum tensor are given by eq. (7), with the signs reversed to represent the
source function. From eq.(8) we must add a contribution from the other source
term $q_{a}\langle V^{\mu}V^{\nu}\rangle$; for cold DM only the (positive)
density term contributes. So, for the sum of the two contributions to the halo
density we write $\rho=-\tilde{k}p$, where $\tilde{k}$ is expected to be
small.

First we derive eqs.(6, 7), slightly generalizing the notation to allow
inclusion of anisotropic distributions of DM. In the halo annular region (e.g.
$r_{\hat{p}}\leq r\leq r_{q}$ in the spherically symmetric case) specify the
halo $T^{\mu\nu}$ by the change of $T^{\mu\nu}$ from its value in the
background model (at e.g. $r=r_{q}$). The halo material responds to the
gravitational attraction of the central attractor, and the DM with
sufficiently small angular momentum is compressed as it falls in. In type (b)
halos DM falls inward from a stagnation zone on time-like trajectories at
nearly parabolic262626 In the integral of energy conservation along a typical
guiding center path, eq.(20) of Paper I, ${\cal E}\cong 0$.speeds as judged by
an observer at $r_{\hat{p}}$; i.e.
$v(r_{a})\ll\surd(2\Phi(r_{p})\approx\surd(2G{\cal M}/r_{p})$. In general we
consider low-temperature halos in which at $r\geq r_{a}$, the same is true and
most of the DM objects have small peculiar velocities at distance and are
isotropic.

We write for the change in a typical spacial diagonal component of $\delta
T^{\mu\nu}$ at the halo point $r\geq r_{p}$,

$\delta p={\cal N}mP-{\cal N}_{q}m_{q}P_{q}=\left[{{\cal N}\over{\cal
N}_{a}}{m\over m_{a}}{P\over P_{a}}-\zeta\right]p_{aa}.$ (23)

Here ${m}$ is an average mass density, ${\cal N}$ is the number of DM objects
in a unit volume and ${P}$ represents a local value of one spacial component
of $\langle T^{\mu\nu}\rangle$. We consider only the major velocity
contributors to these averages. For the isotropic case $P$ represents the
square of the ‘speed’272727 Alternatively, this corresponds to
$s^{2}\equiv(S^{r})^{2}+J_{s}^{2}/r^{2}$ transported locally along the mean
‘guiding center ’paths $x^{\mu}_{s}(\tau_{s})$ discussed in Paper I, and
refers to that component of $S^{\mu}$ not involved in the internal ‘spin’
motion. $\langle(S^{z})^{2}\rangle/3$ of eq.(7). Here, $p_{aa}\equiv\ {\cal
N}_{a}\ m_{a}P_{a}$ and $\zeta p_{aa}={\cal N}_{q}m_{q}P_{q}$, so that $\zeta$
is small for $r\ll r_{a}$ and one may write $\delta p\cong p.$ [A similar
expression holds for $\delta T^{00}$ in terms of the average value of
$(U^{0})^{2}$.] Again, ${\cal N}$ need not specify all the local DM, but just
a particular subset.

Using the steady state one dimensional pipe flow, (see Sec. II) one has
$m(r)/m_{a}=s_{a}/s(r)$ where $s$ is the speed along the fluid’s streamline .
By eq(1) of Paper I, $S^{z}\sim 1$ is very large compared to $\Phi$, and we
may take it not very much affected by the gravitational field in the halo and
we can set $m/m_{a}=S^{z}_{a}/S^{z}\simeq 1-\Phi_{a}+\Phi\sim 1\ \ {\rm and}\
\ P/Pa=(S^{z})^{2}/(S_{a}^{z})^{2}\simeq 1-2\Phi+2\Phi_{a}\sim 1$. Hence for
halo DM

$p/p_{a}={\cal N}/{\cal N}_{a}.$ (24)

The density term, $T^{00}$, can be expressed in terms of $p/p_{a}$ and the
potential $\Phi$ in eqs.(27, 28).

For non-spherically symmetric geometries, the rule to follow is that the
vector component $S^{z}_{s}$ is transported along a particle’s world-line with
the velocity $u^{z}_{s}$. So, if in a local spacial Cartesian coordinate
system, $\grave{x},\grave{y},\grave{z}$ one has for the ensemble’s mass motion
tensor $F^{\mu\nu}$ only the diagonal elements $a_{i}^{2}(\grave{U}^{i})^{2},\
i=1,2,3$, one should use $P\to T^{ii}=a_{i}^{2}(S^{z})^{2}/3,\ i=1,2,3.$

####  7.3.2 Momentum Flux.

The pressure terms in eq. (7) represent a type of momentum flux. Statistical
treatment of large numbers of streams of DM carrying momentum flux can be
modeled after the standard treatment of radiative flux to introduce a momentum
density. (See e.g. Chandrasekhar (1950); Rybicki & Lightman (1979).) Normally
one writes for the energy (or momentum) crossing a surface element $dA$ in a
solid angle $d\Omega$ oriented at an angle $\hat{\theta}$ to the normal of
$dA$ in a time interval $dt$,

$dF=(I/v\pi)\ v\cos\hat{\theta}\ d\Omega dAdt,$ (25)

defining $I\equiv I(\theta,\phi)$, the local specific intensity; it may be
labeled by other parameters, such as the energy. Here $v$ is a velocity along
the axis of the solid angle. The standard procedure is to focus on the
development of $I$, rather than $F$, since it represents a ‘density’. One has
e.g. $(I/v\pi)={\cal N}\langle h\nu\rangle$, the energy density in isotropic
radiation flows. For a steady-state when $I$ is independent of $\theta,\phi$,
the total transfer of flux crossing $dA$ in one direction in unit time is
$dF=IdA$. For us, dealing with momentum flux, $I\propto{\cal N}$ of eqs. (23,
24), the local density of streams. For this momentum flux, $I\propto{\cal
N}\langle(mS^{z})S^{z}\rangle=p$, using $v=S^{z}$. So to determine $p$ we
focus on determining the flux of DM streams in a halo. For isotropic flows
through concentric spherical annuli in free space one requires for flux
conservation:

$4\pi r_{a}^{2}I(r_{a})=4\pi r^{2}I(r)\ \ \ {\rm or}\ \ r_{a}^{2}/r^{2}={\cal
N}(r)/{\cal N}(r_{a})=p(r)/p(r_{a}),$ (26)

For the mass carried by the incoming streams one takes
$I=\langle\bar{m}_{0}\rangle{\cal N}\cong\langle m_{0}U^{z}\rangle{\cal
N}/\langle U^{z}\rangle$; since conservation of mass gives $\langle
m_{0}U^{z}\rangle=$constant along a stream-line, one has,

$\rho(r)/\rho(r_{a})\cong\langle U^{z}(r_{a})/U^{z}(r)\rangle\ p(r)/p(r_{a});$
(27)

here $U^{z}$ is that component of $U^{\mu}$ perpendicular to the ‘spin’ plane,
$(U^{z})^{2}\equiv\langle U^{r}\rangle^{2}+\bar{L}_{v}^{2}/r^{2}$. (See eq.
(20) of Part I.) In the outer part of the halo $\langle
U^{z}(r_{a})/U^{z}(r)\rangle\cong 1$. In the inner part, one has $\langle
U^{z}\rangle$ determined by the gravitational potential (see Appendix C):

$\rho(r)/\rho(r_{a})\cong[\Psi(r_{a})/\Psi(r)]^{0.5}p(r)/p(r_{a}).$ (28)

####  7.3.3  Angular Momentum Conservation.

Now examine momentum conservation for a particular aggregate, with number
density ${\cal N}$, of incoming DM objects which at large $r$ have specified
low angular momenta; ${\cal N}$, considered as a function of $r$, is subject
to the constraint that each object in the aggregate must conserve its
associated angular momentum in a spherical potential; the gas is
collisionless.

Consider flow through a portions of a sphere; the normals of the area elements
are the radius vectors. For convenience suppose at some large $r=r_{a}$, in a
volume element all the streams of objects considered have the same stream
density $\langle m_{a}\rangle$ and the same (small) average speed $v_{a}$
(where $U_{s}^{\mu}=\gamma(1,\vec{v}_{a,s})$).

For each stream of objects the orbital angular velocity
$L_{s}=r_{a}\times(r_{a}U^{\phi}_{s})=v_{a}\ r_{a}\sin\theta_{s}\equiv
v_{a}b_{a,s}$ is a constant and is determined just by the value of
$\theta_{s}$ ( the angle between the 3-vectors $r_{a}$ and $\vec{v}_{a,s}$).
The total number of inflowing streams at a shell of radius $r_{a}$ is then
$F_{a}={1\over 2}\cdot 4\pi r_{a}^{2}\cdot\int I_{a}\cos\theta\
\psi(\theta,\phi)\sin\theta d\theta d\phi$, where $\psi$ is a distribution
function for the ingoing streams. For large $r$ we adopt a distribution
$\psi=1$, corresponding to velocities being isotropic282828The specification
of $\psi$ is part of the specification of the angular velocity distribution;
in principal it could be a function of $v_{a}r_{a}$. In near equilibrium the
symmetries of the velocity distribution reflect the symmetries of the
effective gravitational potential; see Chandrasekhar (1942) . so
$F_{a}={1\over 4}\cdot 4\pi r_{a}^{2}I_{a}$; the collection of all these
streams carries orbital angular momenta up to $L_{a}=r_{a}v_{a}$.

Let us follow such an aggregate moving in towards the center, assuming for
convenience that $v_{a}$ is small enough that each stream follows a nearly
parabolic orbit, i.e. $v_{a}^{2}\ll V_{0}^{2}(R_{0})$, where $V_{0}(R_{0})$ is
close to the galaxy’s maximum rotation speed. For any large
$r=\tilde{r}<r_{a}$, a similar argument gives
${\tilde{F}}=\pi\tilde{r}^{2}\tilde{I}$ for the inflowing streams. There is a
region of ‘free-fall’ in which the streams are not appreciably deflected; in
it $\tilde{F}=F_{a}$ so eqn. (26) is satisfied.

####  7.3.4 The Inner Halo

Inside this region streams with large orbital angular velocity $L_{s}$ will be
deflected. We calculate the loss to the inflow. In the background space
suppose we regard each sphere represented by a (fictitious) collection of
‘bound’ DM circular orbits, each with an associated orbital angular velocity
given by $L_{cir}^{2}=r_{cir}^{2}v_{cir}^{2}(r)$. For a falling stream to pass
through a sphere of radius $r_{cir}$ it must have its distance of closest
approach to the center be less than $r_{cir}$.

For example, let $\Phi=G{\cal M}/r\equiv V^{2}(r)$ with ${\cal M}$ a constant,
representing the (ordinary matter) inner part of a galaxy for $r\geq R_{0}$.
For a stream of DM particles following a parabolic orbit (${\cal E}\cong 0$)
one requires $L^{2}_{s}\leq 2L_{cir}^{2}+W_{u}^{2}\equiv L_{\ast}^{2}(r)$
where $W_{u}$ represents the intrinsic angular momentum associated with the DM
object (See Paper 1.) One has $L_{cir}^{2}=G{\cal M}[r_{cir}-r_{p}]$ where
$r_{p}\equiv W^{2}_{u}/(G{\cal M})$. [ Bound circular orbits cannot be defined
for DM for $r_{cir}\leq r_{p}.$] With parabolic infall velocities, DM objects
have their turning point at $r=r_{p}/2$.

Consider a particular sphere $r_{b}\leq r_{a}.$ Only those streams with low
$L_{s}$ reach $r_{b}$; they have the same (invariant) values of $L_{s}$ as
they had when they were at $r=r_{a}$. We calculate the fraction $f$ of
inflowing streams at $r_{a}$ that can reach this particular $r_{b}$; one has
$f=(r_{a}v_{a})^{-2}\int_{0}^{L_{\ast}}LdL$, using $L=r_{a}v_{a}\sin\theta$ as
the variable of integration instead of $\theta$:

$f\cong r_{b}(r_{b}-r_{p}/2)V^{2}(r_{b})/(r_{a}^{2}v_{a}^{2}).$ (29)

This loss must be compensated for by an decrease in $I_{b}$ since the total
number of streams with assigned $L_{s}$ which can travel from $r_{a}$ to
$r_{b}$ is conserved. Consequently292929 Note we have not assumed that at our
starting point, $r=r_{a}$, $v_{a}r_{a}$ is the maximum orbital angular
velocity possible, $v_{a}^{2}\leq v_{cir}^{2}(r=r_{a})$. since $\pi
r_{b}^{2}I_{b}=f\cdot\pi r_{a}^{2}I_{a}$, one has, for an inner zone

$I_{b}/I_{a}={\cal N}_{b}/{\cal
N}_{a}=p_{b}/p_{a}=\tilde{v}_{b}^{2}/v_{a}^{2}\equiv
V^{2}(r_{b})[1-r_{p}/(2r_{b})]/v_{a}^{2};$ (30)

this definies $\tilde{v}(r_{b})$. This holds303030 More general potentials can
be used. We really assume that in the outer halo $rv_{cir}$ is an increasing
function of $r$, so that there is a limiting radius $r_{b}^{\ast}$ at which
‘free-fall’ ends and for which $r<r_{b}^{\ast}$ equation (30) holds. for all
$r_{b}\leq r_{b}^{\star}$ for which $\tilde{v}_{b}r_{b}\leq r_{a}v_{a}$.

####  7.3.5 The Boundary Between The Two Halo Parts.

The boundary between the two regions, $f=1$, occurs at $r=r_{b}^{\star}$ where

$r_{b}^{\star}-{r_{p}\over 2}\cong v_{a}^{2}r_{a}^{2}/(G{\cal M}_{0});$ (31)

the RHS assumes $r_{b}^{\star}$ is close to the inner halo edge where $G{\cal
M}(r)\simeq G{\cal M}_{0}\ =r_{b}^{\star}V^{2}(r^{\star}_{b})$. This is true
for only high mass galaxies and DM with low initial angular velocities,
$r_{a}v_{a}$; this is the case of most interest. Then the DM generally is in
the free-fall zone where $p\propto r^{-2}$. But, for low-mass galaxies
$r_{b}^{\star}$ may be close to $r_{a}$, severely limiting any ‘free-fall’
zone.

In this inner zone the change in the gravitational potential can be
appreciable because of contributions from OM galactic disk populations and
$\rho$ actually has a complicated behavior; by itself eq. (28) gives

$\rho(r)|\propto p(r)/\surd\Psi(r).$ (32)

####  7.3.6  The Imput Parameter $v_{a}$.

From the discussion in Section 3.2 in text one has for the Milky Way galavy,
$r_{b}^{\star}\simeq 12$ kpc. Using the approximation eq.(31) and
$r_{a}\approx 10r_{h}\cong 80$ kpc one estimates $v_{a}\sim 25\ {\rm km\
s}^{-1}$. Considering the uncertainty in $r_{a}$, this is in good agreement
with its estimate in text, based on the observed value of $v_{\infty}$.

For the small galaxy M33, the estimate is more uncertain because less is
observed and $v_{\infty}$ is not known. The text uses $r_{h}\cong 16$ kpc and
$r_{b}^{\star}\cong 18$ kpc for representing the rotation curve. For
theoretical reasons, one expects $r_{b}^{\star}$ to be ‘close’ to $r_{a}$. If
we use $r_{a}\approx r_{b}^{\star}$ then eq.(31) inplies $v_{a}\sim 40{\rm km\
s}^{-1}$. If $r_{a}\approx 2r_{b}^{\star}$, then $v_{a}\sim 20{\rm km\
s}^{-1}$. Also, because of the natural width of the distribution of $p$ DM
near its peak, $r_{h}$, it is possible that a lower value for $r_{h}$ should
have been inferred: $r_{h}\cong 12-14$ kpc , resulting in $\sim 20\%$ increase
in $v_{a}$.

It is interesting, then, that both galaxies can be characterized by the same
imput velocity, $v_{a}\sim 20-40{\rm km\ s}^{-1}$ despite their differences in
size and OM mass.

####  7.3.7  Some Caveats.

The infall argument holds for ordinary matter as well as dark matter since it
involves conservation of a flux, which may be a mass flux as well as a
momentum flux. If some of the in-falling streams do not come from a real
stagnant region and actually have appreciable mean stream velocities at large
distances, the treatment would need modification, since the expression for $f$
holds only in the stream’s rest frame. In this paper, using our expression for
$f$, we are actually considering a restricted class of incoming matter and our
estimates for densities, etc. at large distances refer only to them. Because
of collisions, ordinary matter particles can exchange angular momentum and
some part of the initial flow will always reach the central galaxy with very
low angular momentum and fall in. But, as postulated, the DM particles always
possess a non-zero total orbital angular momentum and cannot penetrate deeply
into the central galaxy.

####  7.3.8  Approximations For The Halo Pressure & Density.

For the region $r\leq r_{a},$ use as unit of length $R_{h}\equiv
r_{a}^{2}v_{a}^{2}/(G{\cal M}_{0})$ ( which shall turn out to be close to the
maximum of the rotation curve $R_{0}$ for many cases of interest). Represent
the intrinsic angular velocity by $W_{u}^{2}=r_{a}^{2}(\xi v_{a})^{2}$ with
the maximum orbital angular velocity of the incoming DM halo streams given by
$L_{u}^{2}=r_{a}^{2}v_{a}^{2}$. For the form of $T^{\mu\nu}$ adopted,
$\rho\leq 0$. f One writes $\rho_{a}=-\tilde{k}p_{a}$ and estimates
$\tilde{k}\leq 0.1$ but in this section we allow for the possibility of a much
larger value. The formal inner edge of the halo, defined in terms of the
lowest bound ‘circular’ path is $r_{p}=\xi^{2}R_{h}$ and the furthest inward
DM parabolic orbits can go is $r_{p}/2=(\xi^{2}/2)R_{h}\equiv\eta R_{h}$; we
will use $\eta\approx{1\over 2}$ as representative in the approximations
because we are interested only in infall with low angular momentum. In these
units the boundaries of the inner zone are given by $\eta R_{h}\leq
r\leq(1+\eta)R_{h}=r_{b}^{\star}$. We used $\eta=1/2$ in our models.

In the outer region one has, from eqs.(26,27):

$p(r)=p_{a}(r_{a}/r)^{2};\ \ \rho(r)/\rho_{a}\simeq p(r)/p_{a},$ (33)

ignoring the variation of the small quantity $\Psi$ in the outer zone in
calculating $\rho$.

In the inner zone, using eqs. (27,30) we allow for the variation of $\Psi$ in
calculating the density. Using the notation $p_{b}\equiv
p_{a}(r_{a}/r_{b}^{\star})^{2};\
\rho_{b}\equiv\rho_{a}(r_{a}/r_{b}^{\star})^{2}$ and $r=xR_{h}$ (so that
$\eta\leq x\leq 1+\eta$ in the inner zone ),

$p=(1+\eta)^{2}p_{b}(x^{-1}-\eta x^{-2});\ \
\rho=(1+\eta)^{2}\rho_{b}[(x/(1+\eta)]^{0.5}(x^{-1}-\eta x^{-2}).$ (34)

For the DM mass in the inner zone, using $M_{1}=(4\pi/3)\rho_{b}R_{h}^{3}<0$,
one has

$\delta{\cal M}_{1}(r)\equiv\int_{\eta R_{h}}^{r}4\pi\rho
r^{2}dr=M_{1}(1+\eta)^{3/2}\ [6x^{5/2}-10\eta x^{3/2}+4\eta^{5/2}]/5,$ (35)

so that, using $\eta=0.5$ one has $\delta{\cal M}_{1}(r_{b}^{\star})\cong
3M_{1}$. Also, for the additional DM mass above the boundary, $r_{b}^{\star}$,
in the outer zone, one finds

$\delta{\cal M}_{2}(r)\equiv\int_{r_{b}^{\star}}^{r}4\pi\rho
r^{2}dr=3(1+\eta)^{2}M_{1}[x-(1+\eta)].$ (36)

In the inner zone the pressure reaches a maximum at $r=R_{h}$ and then, going
inwards falls rapidly to zero at $r=\eta R_{h}$. One finds the halo structure
effectively starts at $r\cong R_{h}$, i.e. for $r_{p}/2\leq r<R_{h}$, $p,\rho$
may be taken as small. For $r_{b}^{\star}\geq r\geq R_{h}$, one may consider
the pressure and density as approximately constant; they then fall $\propto
1/r^{2}$ as we go further out. For either type (a) or type (b) halos the size
of the halo, using $R_{h}R_{0}\approx R_{0}^{2}$, is constrained by the
angular momentum condition

$r_{a}^{2}/R_{0}^{2}\cong V_{0}^{2}/v_{a}^{2},$ (37)

where $V_{0}^{2}\equiv G{\cal M}_{0}/R_{0}$; this imposes an important
constraint on the choice of $v_{a}$. With these expressions for $p,\rho$ the
structure of the halo can be developed and the functions $A(r),B(r)$, used in
defining the metric, can be calculated. See Appendix C.

####  7.3.9  The ‘Shielding’ Distance $r_{s}$

For type (a) halos, which may be quite extended, there is an additional
feature because the model DM halo density may be negative if $q_{a}$ is small.
In the limiting case one may define a ‘shielding’ distance,
$r_{s}>r_{b}^{\star}$, such that

$\delta{\cal M}_{1}(r_{b}^{\star})+\delta{\cal M}_{2}(r_{s})+{\cal M}_{0}=0,$
(38)

since $\rho<0$ for DM. This corresponds to $\Phi(r_{s})=0$ and in Appendix C
is used to define the edge of the halo $r_{q}$. Since $r_{q}$ is For $r<r_{q}$
Using $\eta=0.5$ and $\rho(r_{b}^{\star})\equiv-\tilde{k}p_{b}$, and for the
outer zone $p_{b}(r_{b}^{\star})^{2}=p_{a}r_{a}^{2}$. one finds using equation
(45) for the observed asymptotic circular velocity, $v_{\infty}$, that

$r_{s}/R_{h}\cong(1.5\tilde{k})V^{2}(R_{h})/v_{\infty}^{2},$ (39)

where $V^{2}(R_{h})R_{h}\equiv G{\cal M}_{0}$. For the MW galaxy
$r_{s}/R_{h}\approx 2.2/\tilde{k}$. The significance of $r_{s}$ is that for
$r>r_{s}$ the details of the potential representing the central attractor are
no longer relevant and the metric coefficients are purely determined by the
background intergalactic DM cloud. It is reasonable to consider $r_{s}\geq
r_{q}>r_{a}$ for type (a) halos and this the proper definition of $r_{q}$.
Again, this value of $r_{s}$ is crude because the value of $q_{a}$ has been
assumed low and we have neglected positive density contributions from OM
trapped in the halo.

##  8  Appendix C: Metric Parameters For A Galaxy’s DM Halo.

The DM halo is represented by the standard fluid energy momentum tensor
$T^{\mu\nu}=(p+\rho)U^{\mu}U^{\nu}-pg^{\mu\nu}$ (with $c^{2}=1$). When
$r_{p}\leq r\leq r_{a}\leq r_{q}$ one has $p$ as a large pressure term and
$\rho<0$ with$|\rho|\ll p$. For $r$ outside this range,
$T^{\mu\nu}_{halo}\cong 0.$

###  8.1 Outside

The intergalactic medium is represented by the Robertson-Walker metric

$d\tau^{2}=d\bar{t}^{2}-R(\bar{t})[(1-\bar{k}r^{2})^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}]$ (40)

where the structure equations are:

$3\ddot{R}=-4\pi G(\rho+3p-2\lambda)R,\ \ \dot{R}^{2}+\bar{k}=8\pi
G(\rho+\lambda)R^{2})/3$ (41)

and the ‘energy’ conservation constraint

$3(p+\rho)d\ln R/d\bar{t}=-d(\rho+\lambda)/d\bar{t}$ (42)

hold for $r>r_{q}$. Here, we have added $\lambda g^{\mu\nu}$ to the usual
fluid source term. One can get a static model (with $R=1$) by choosing
$\lambda$ appropiately. For our case of the surrounding intergalactic cloud
being purely DM, $p\to p_{00}$ is the dominant term with $\rho\to\rho_{00}$
being small and negative. For a quasi-static model take $\rho+3p-2\lambda=0$
so that $\lambda\cong 3p_{00}/2$ . Also $\rho+\lambda\cong\lambda$ so that one
may take $\dot{R}\cong 0.$ Then $\bar{k}\cong 4\pi Gp_{00}$, having the
dimensions of an inverse length squared, sets the scale of the curvature
effects. For example, for $p_{00}/c^{2}=10^{-27}{\rm g\ cm}^{-3}$ one has
$1/\surd\bar{k}\equiv L\simeq 10^{9}$ l.y. Consequently, one may take
$\bar{k}\cong 0$ for our halo models.

subsection The Form Of $A(r)$ In The Halo.

In the Schwarzschild model (equation (6) of Paper I)) for the entire halo
$r_{p}/2\leq r\leq r_{q}$, we have $A=1/(1-2G{\cal M}(r)/r)\equiv 1/(1-2\Phi)$
where ${\cal M}(r)={\cal M}_{0}+\delta{\cal M}$; ${\cal M}_{0}$ is the mass of
the central attractor at $r\approx 0$ and $\delta{\cal M}$ is given by eqns.
(35, 36); For DM , $\delta{\cal M}<0.$ The halo begins at $\eta R_{h}$ with
$\eta\sim 1/2$; there one has $\rho=p=0$. Eqs.(33, 34) give approximations for
$\rho,p$ for the halo. Using these approximations for the mass variations one
has

$\Phi(r)=G{\cal M}_{0}/r+4\pi G\rho_{b}r^{2}(1-R_{h}^{3}/r^{3})/3,\ {\rm for}\
\ R_{h}\leq r\leq r_{b}^{\star},$ (43)

where $r_{b}^{\star}=R_{h}(1+\eta)$. This matches There is a similar
expression for $\Phi(r)$ for $r_{b}^{\star}\leq r\leq r_{a}$, resulting in
e.g.

$\Phi(r_{a})=G({\cal M}_{0}+{\cal M}_{1})/r_{a}+4\pi
G\rho_{a}r_{a}^{2}[1-r_{b}^{\star}/r_{a}],$ (44)

where $\rho_{a}r_{a}^{2}=\rho_{b}(r_{b}^{\star})^{2}$ has been used. One has
$v_{a}^{2}r_{a}^{2}=G{\cal M}_{0}R_{h}$ and, for massive galaxies
$[1-r_{b}^{\star}/r_{a}]\approx 1$. [See eq. (31.] As noted in Appendix B,
there is a limit to $r_{a}$, provided by a shielding length $r_{s}$, see
eq.(39), if the (negative) halo density provides significant shielding. To
join onto the exterior solution, theRobinson-Walker metric, at $r=r_{a}$ one
choose the parameter $\bar{k}$ appropriately.

###  8.2 The Form of $B(r)$ In The Halo

The restriction $T^{\mu\nu}_{\ ;\nu}=0$ requires
$\Psi^{\prime}\cong{p}^{\prime}/({\rho}+p)$. But this standard relation is not
directly useful because: (1) $p,\rho$ have different dependences on $\Psi$;
(2) there are other neglected contributions, see eq.(8); and (3)$\Psi$ has an
additional contribution from the OM central galaxy. Also, the quantity
$k\equiv-\rho/p$ may not be constant in the inner halo. In the outer parts of
DM halos , the variation of $p$ is predictable and $\Psi$, mainly due to DM,
can be directly determined from observations (since
$v^{2}_{cir}\equiv-r\Psi^{\prime}$) assuming strict circular motion. Generally
other combinations of Einstein’s equations are more useful for determining
approximations for the three quantities, $p,\rho,\Psi$. Because $k$ is small
and its variations should have minor effects in the outer halo, we initially
assume it constant.

For the Schwarzschild metric, we have $A^{\prime}/A+B^{\prime}/B=8\pi
G(p+\rho)rA$. Again using $B=1-2\Psi$, where $\Psi$ is small in a halo model,
one may rewrite this Einstein equation in a more useful form:

$v_{cir}^{2}\equiv-rd\Psi/dr=4\pi Gpr^{2}+\Phi(r),$ (45)

where $\Phi$ is known -see the previous section- and for most of the halo is
small. For our simplified version of $T^{\mu\nu}$ for DM one has $\delta{\cal
M}<0.$ Both $p$ and $\rho$ are continuous at the joining points $r=r_{p}/2$
and $r=r_{b}^{\star}$. Using our approximations for the density, eqns.(33, 34)
for $r_{p}/2\leq r\leq R_{h}$, one has $\Psi(r)\cong\Phi(r)=G{\cal M}_{0}/r$
in the innermost interior halo zone. Next one has

$v_{cir}^{2}=4\pi Gp_{b}r^{2}+G{\cal M}_{0}/r+4\pi
G\rho_{b}r^{2}[1-R_{h}^{3}/r^{3}]/3\ \ {\rm for}\ R_{h}\leq r<r_{b}^{\star}.$
(46)

For the outer zone, since $p\propto 1/r^{2}$, one has

$v_{cir}^{2}=4\pi Gp_{a}r_{a}^{2}+4\pi
G\rho_{a}r_{a}^{2}[1-r_{b}^{\star}/r]+G[{\cal M}_{0}+{\cal M}_{1}]/r$ (47)

for $r_{b}^{\star}\leq r<r_{a}$. Again
$\rho_{a}r_{a}^{2}\cong\rho_{b}(r_{b}^{\star})^{2}$ , and
$\rho_{a}=-kp_{a},{\cal M}_{1}<0$ [if there are no contributions from OM]. For
large $r$, one has $v_{cir}\cong$ constant.

####  8.2.1  The Outer Rim of the Halo.

In the outermost halo region, $r_{q}\geq r>r_{a}$ where $r_{a}$ is large and
$\delta\rho=-k\delta p$ (see eq.(15)) one has

$\Phi(r)=G{\cal M}_{T}/r+4\pi G(-kp_{00})\
[r_{q}^{2}(r-r_{a})-(r^{3}-r_{a}^{3})/3]/r\ $ (48)

with

$\Phi(r_{a})=G{\cal M}_{T}/r_{a}\ {\rm and}\ \Phi(r_{q})=G{\cal
M}_{T}/r_{q}-8\pi Gkp_{00}r_{q}^{2}/3.$ (49)

Here ${\cal M}_{T}={\cal M}_{0}+\int_{R_{h}}^{r_{a}}4\pi\rho(r)r^{2}dr$. In
practice we use $r_{q}\sim 3r_{a}$. Then, to satisfy the boundary condition
$A\to 1$ at $r=r_{q}$ we suggest adding the small constant
$\Phi_{0}\equiv-\Phi(r_{q})$ to $\Phi(r)$ throughout the entire region $r\leq
r_{q}$.313131 For the usual exterior Schwarzschild solution representing an
isolated source , one chooses $\Phi_{0}=0$ because the solution is supposed to
hold as $r\to\infty$. Here we impose another boundary condition. We shall
ignore it for $r\leq\sim r_{a}$ although it may affect estimates of the
shielding distance $r_{s}$. Using eqs.(45, 15), one has

$v_{cir}^{2}=4\pi Gp_{00}[r_{q}^{2}-r^{2}]+\Phi(r)-\Phi_{0}$ (50)

This shows both $v_{cir},\ \Psi^{\prime}\to 0$ as $r\to r_{q}$. Since we only
determine $\Psi^{\prime}(r)$, see eq.(45), a constant $\Psi_{0}$ may be added
to it. We suggest requiring $\Psi_{0}=-\Psi(r_{q})$ to satisfy the boundary
condition $B\to 1$. Then $d\bar{t}^{2}\to dt^{2}[1-2\Psi(r)]$ establishes the
clock rate comparison inside and outside; this is to be expected because the
halo is in a gravitational potential well with the halo model sitting on top
of the background DM cloud model.

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