A Study of Three Galaxy Types, Galaxy Formation, and Warm Dark Matter

We try to bridge the gap between the theory of linear density-velocity-gravi-tational perturbations in the early universe, and the relaxed galaxies we observe today. We succeed quantitatively for dark matter if dark matter is warm. The density runs of baryons and of dark matter of relaxed galaxies are well described by hydro-static equations. The evolution from initial linear perturbations to final relaxed galaxies is well described by hydro-dynamical equations. These equations necessarily include dark matter velocity dispersion. If the initial perturbation is large enough, the halo becomes self-gravitating. The adiabatic compression of the dark matter core determines the final core density, and provides a negative stabilizing feedback. The relaxed galaxy halo may form adiabatically if dark matter is warm. The galaxy halo radius continues to increase indefinitely, so has an ill-defined mass.


Introduction and Overview
How does a particular linear density perturbation in the early universe evolve to become a relaxed galaxy that we observe today? We approach this question in (arguably) reverse chronological order. We begin with a study of relaxed elliptical galaxies with a cusp dominated by baryons. We find that the density runs The purpose of the present study is to find out how nature obtains these parameters starting from a particular linear density-velocity-gravitational perturbation in the early universe. In order to simplify the problem at hand, we study spiral galaxies with a core dominated by dark matter. Then we can, to a first approximation, neglect baryons, and focus the study on the two parameters Starting with given initial perturbations, we integrate the equations numerically to study the formation of the dark matter halo of a relaxed galaxy. We can indeed obtain the parameter 2 rh v′ from these integrations. However, the equations discussed so far apparently do not fix We use the standard notation in cosmology as in [3].

The Stationary Galactic Halo
We consider relaxed galaxies, i.e. galaxies with no sign of recent collisions or mergers, or other extraneous features. We model the galaxy as two self-gravitating,  (2) and the equations of conservation of the r-component of momentum of particles with velocity 0 r v > (or, separately, 0 r v < ), valid for collision-less gases, or for gases with elastic collisions [4]: The gravitation field is  κ , appear in the following combinations: Good fits to observed rotation curves are generally obtained assuming 2 rh v′ and 2 rb v′ are independent of the radial coordinate r (in some galaxies 2 rh v′ decreases at large r, so the present analysis is valid up to that radius). We solve these equations in spherical coordinates, assuming spherical symmetry. These equations need four boundary conditions, e.g. ρ . We integrate the equations numerically from the first observed radius min r to the last one max r , in the disk of the galaxy, and vary the four boundary conditions to minimize a 2 χ between the calculated and observed rotation curves. Note that we do not use density templates.
The observed independence of 2 rh v on r in these relaxed galaxies implies that the particles of the gas have a 1D phase space density that follows the non-relativistic Boltzmann distribution, i.e. the number of dark matter particles in dr and d rh v is proportional to [6] ( )( ) where ( )

Elliptical Galaxies with a Cusp Dominated by Baryons
Numerical integrations of (2) and (3) for the elliptical galaxies M87, NGC 5846,   The mass-to-light ratio is fixed at dominating the enclosed mass, the asymptotic solutions of (2) and (3) are Examples of galaxies that obtain good fits, with the assumption that 2 rh v′ is independent of r, are presented in Table 1. For galaxy Mrk1216, the assumption that 2 rh v′ is independent of r obtains a good fit for 50 kpc r < , while 2 rh v′ declines for larger r.
, the baryonic mass of the galaxy is finite: Replacing b M by L ϒ , i.e. by the mass-to-light ratio ϒ times the absolute luminosity L, and taking the logarithm of (8), obtains the equation of a plane in the space ( ) 2 eq ln , ln , ln . This plane is known in the literature as the "fundamental plane" of elliptical galaxies.

Spiral Galaxies with a Core Dominated by Baryons
, and its contribution from baryons ( ) b v r , at small r, determine the dark matter density in the core of the galaxy: Figure 5 presents the rotation curves and density runs of the spiral galaxy NGC 0024 with a core (arguably) dominated by dark matter. Additional examples can be found in [2] and [13]. First galaxies (arguably) have a core dominated by dark matter. For simplicity, in the following we will study the formation of galaxies dominated by dark matter, since then the density run ( ) h r ρ of the relaxed galaxy is described with just two parameters: the dark matter reduced velocity dispersion

Formation of the Galactic Halo
Let us consider a universe with zero spatial curvature, dominated by non-relativistic warm dark matter. The mean density of the early universe is where t is the age of the universe, and ( ) a t is the expansion parameter (normalized to 1 a = today). The velocity of expansion is r v Hr = , with Hubble parameter ( ) If dark matter is warm, the root-mean-square of the r-component of the dark matter particles dispersion velocity (with respect to a comoving observer) is This velocity dispersion scales as ( ) is an adiabatic invariant. Note that the velocity v of a free particle (with respect to a comoving observer momentarily at the position of the particle) scales due to the expansion of the universe.
For simplicity, we consider only dark matter. The formation of the galactic halo can be illustrated by integrating numerically Newton's equation

,
h Gρ π ∇ ⋅ = − g (12) International Journal of Astronomy and Astrophysics the continuity equation and Euler's equation Newton's equation, and the last two terms in (14), obtain the steady state dark matter halo of Section 2. Again we take 2 rh v′ independent of r for this steady state solution. This approximation is valid out to a radius where the rotation velocity ( ) v r remains flat, often beyond the range of observations, and the present analysis is limited to that radius.
Linear perturbation theory is obtained for Equations (12), (13), and (14), with , one scalar mode that decays as 3 2 a − ∝ , and one scalar mode that grows as a ∝ due to gravitational collapse, and will survive [1]. φ is the gravitational potential. r is the proper coordinate, and a r is the co-moving coordinate. We assume dark matter is collision-less. If are damped and collapse due to free streaming. (For collisional gases, , This equation will be discussed in Section 4, and its importance, to fix and stabilize the galaxy core density and radius, will be illustrated by the simulations. min r is the smallest r in the numerical integration, and is chosen much smaller than the galaxy core radius c r . To understand the formation of the galactic halo we need to include velocity dispersion, i.e. the last term in (14), since it is needed to obtain agreement with the observed rotation curves of relaxed galaxies as shown in Section 2. Once we include velocity dispersion, Equations (12), (13) and (14) become incomplete since 2 rh v′ remains unspecified, and we can not, consistently, omit the adiabatic invariant constraint (16).

B. Hoeneisen International Journal of Astronomy and Astrophysics
The simple implementation (16) of adiabatic expansion is justified as follows.
To integrate Equations (12), (13), and (14) we need to assign a value to 2 rh v′ , possibly rand t-dependent. From numerical studies described below, we find that the solutions are insensitive to 2 rh v′ for t less than turn-around, or for large r (beyond the pivot point discussed below). Thus, for the problem at hand, it is sufficient to assign an r-independent value (16) to Equations (12), (13), (14), and (16) are approximate: they are hydro-dynamical equations that treat dark matter as a continuous medium, not as a superposition of particle orbits, and do not include relaxation mechanisms to damp oscillations to attain a relaxed final state. Relaxation is beyond the scope of the present study. For a review of cosmological simulations of galaxy formation see [14].
As an example we consider a galaxy with stellar mass  The initial dark matter velocity is the Hubble flow, and the dark matter adiabatic invariant, in this simulation, is ( ) . We note, in Figure 6, that the central density decreases with time, turns around, increases, and approaches the final steady-state galaxy halo with asymptotes marked with lines C D E. The increase of the core density is due to negative dark matter velocity rh v after turn-around shown in Figure 7. This dark matter is falling back to the core.
Let us recall that the final steady-state galaxy halo is determined by (12), (13), and (14) with 0 t ∂ ∂ → and 0 rh v = , i.e. by Newton's Equation (12), and the momentum conservation equation (17) These static equations require two boundary conditions, namely the dark matter reduced velocity dispersion as in (6). Therefore the pivot point P determines the final reduced velocity dispersion 2 rh v′ . The galaxy core radius is defined by  The condition to obtain a self-gravitating core is that the core radius at turn-around be less than the would be pivot point radius, so that a pivot point can form. Passing this bottle-neck, the core collapses, see Figure 6 and Figure 8.
What phenomena determines the second parameter c ρ ? In the cold ΛCDM scenario, the central density increases to infinity, i.e. the dark matter particles fall to 0 r ≈ , overshoot, splash-back, overshoot again, etc., and various relaxation mechanisms, and the virial theorem (assuming a well-defined halo mass h M ), are invoked to finally attain a relaxed galaxy halo [16]. These phenomena are not captured by Equations (12), (13), and (14), and the numerical integration breaks down at core collapse, see the first panel of Figure 8.
In the case of warm dark matter we complement Equations (12), (13), and (14) with the equation of adiabatic expansion (16). Note, in Figure 10, that the value  independent of time is un-natural, but helps understand galaxy halo formation. Note that the value of 2 rh v′ does not change the pivot point P, and affects (does not affect) the halo formation for r less than (greater than) r of the pivot point.
we use the core density ( ) min , r t ρ to update 2 rh v′ at each time step using (16). After core turn-around, the core density c ρ increases, and hence 2 rh v′ increases. This is a negative feedback on the further growth of c ρ , see Figure   10. In summary, (16) not only determines the steady-state core density (once the pivot point P, and the final 2 rh v′ are determined), but also provides a negative feedback that tends to stabilize the core density. With fixed 2 rh v′ the collapse is run-away.
Note in Figure 6 and Figure 7 that dark matter keeps falling onto the steady state galaxy halo, so the radius of this halo continues to grow. Equating the halo density (18) with the density (9) of the homogeneous universe, we obtain the constant velocity with which the halo grows: . Note that the aver-International Journal of Astronomy and Astrophysics age velocity of a dark matter particle in the halo times the age of the universe t, is of the order of the radius of the halo at time t. Note also that the halo mass keeps growing so is an ill defined concept. Let us mention that dark matter particles with orbits well within the boundary of the halo at time t have completed many orbits in time t. So 2 rh v′ in the core and in the inner asymptotic region are connected, and attain thermal equilibrium. Finally, let us mention that, as the galaxy halo radius grows, new dark matter particles acquire orbits bound to the galaxy, populating the tail of the non-relativistic Boltzmann distribution.
As the central density c ρ in Figure 6 increases, so does the core temperature due to adiabatic compression. This compression brings rh v to its equilibrium value 0 rh v = out to a radius r that increases with time, see Figure 7. Numerical integration breakdown occurs when 0 rh v = approaches the pivot point P. The negative rh v beyond P indicates that the dark matter halo radius keeps growing.
If we artificially "jump" the pivot point P, i.e. we stop the numerical integration before 0 rh v ≈ reaches P, and resume the integration after passing P, we obtain the result shown in Figure 11 (left panel). In nature we do not see any significant extraordinary phenomena at P, see, for example, the figures in Section 2.
Neither do we see any instability in the integration of the static equations, see Figure 11 (right panel).   , see (16). Figure 14 illustrates the hierarchical formation of galaxies. Reversing the sign of the perturbation of Figure 12 we can study the expansion of voids, see Figure 15.
To the preceeding examples we may add a velocity perturbation, e.g.

( )
for a growing initial mode. Figure 14. Illustration of the hierarchical formation of galaxies (with two pivot points). The initial density perturbation is a double top-hat. Figure 15. Same as Figure 12, except that the sign of the perturbation has been reversed in order to study the expansion of voids. International Journal of Astronomy and Astrophysics at which dark matter particles become non-relativistic (uncorrected for dark matter halo rotation). Each measurement was obtained by fitting the rotation curves of a spiral galaxy in the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample [12] with the indicated total luminosity at 3.6 μm. Figure from [19]. References to the original rotation velocity measurements can be found in [12]. Two of these measurements were given in Figure 4 and Figure 5. Full details of each measurement are presented in [13].

Conclusions
Let us consider a relaxed galaxy dominated by dark matter. is obtained by numerical integration, starting from the linear perturbation, see Section 3. This asymptotic halo is obtained adiabatically, beyond the pivot point P, even for the top-hat perturbation, see Figure 13.
The second parameter, i.e. the core density c ρ , is a mystery in the cold dark matter ΛCDM model. In that model, the core expands, reaches turn-around, collapses, overshoots, followed by splash-back, overshoots again, etc., and several relaxation processes are invoked to reach a virialized state. How can the observations in Figure 16 be explained in the cold ΛCDM scenario?
On the other hand, we are able to obtain c ρ if dark matter is warm, for then International Journal of Astronomy and Astrophysics Equations (12), (13), and (14)  The velocity dispersion term in (14) is needed to understand the density run ( ) h r ρ of relaxed galaxies, see Section 2. Once this velocity dispersion term is included, Equations (12), (13), and (14) become incomplete since it is then necessary to specify 2 rh v , i.e. we need to add the equation of adiabatic expansion (16) for both theoretical and experimental reasons, see Section 4, and Figure 16.
The observed dark matter core is evidence that dark matter is warm, see (19).
To understand the baryon density run properties, see [2] and references therein.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.