Understanding the Formation of Galaxies with Warm Dark Matter

The formation of galaxies with warm dark matter is approximately adiabatic. The cold dark matter limit is singular and requires relaxation. In these lecture notes we develop, step-by-step, the physics of galaxies with warm dark matter, and their formation. The theory is validated with observed spiral galaxy rotation curves. These observations constrain the properties of the dark matter particles.


Introduction
The formation of galaxies is qualitatively different if dark matter is warm instead of cold.The cold dark matter limit is singular.It turns out that understanding galaxies, and the formation of galaxies, is less difficult if dark matter is warm.So, we add to the cold dark matter ΛCDM cosmology one more parameter, namely the temperature-to-mass ratio of dark matter, and let observations decide whether dark matter is warm or cold.These lecture notes do bring new understanding of dark matter, and constrain the properties of the dark matter particles.

Warm dark matter
Let us consider warm dark matter as a non-relativistic classical noble gas of particles of mass m, density ρ and temperature T .By "classical" we mean that the velocity distribution of the non-relativistic particles of dark matter in the early universe is assumed to be the Maxwell distribution, i.e. is not degenerate.For a discussion on how dark matter might have acquired the Maxwell distribution of velocities, see [2].By "noble" we mean that collisions, if any, do not excite internal states of the particles.Recall that for a nonrelativistic gas, 1  2 m v 2 = 3 2 kT .v 2 is the root-mean-square thermal velocity of the dark matter particles.Recall that for adiabatic expansion of a noble gas, T V γ−1 = T V 2/3 = constant (here V is the volume), so is an adiabatic invariant.We will review these concepts in more detail in the next lecture.We want to stress that equation ( 1) is valid for a collisionless or collisional gas, and is valid whether, or not, the particles are bouncing off the walls of an expanding box of volume V .
To verify these statements, let us now consider a free particle in a homogeneous universe with expansion parameter a(t), normalized to a(t 0 ) = 1 at the present time t 0 .The particle has velocity v at the space point r.In time dt the particle advances vdt and arrives at the space point r ′ .Due to the expansion of the universe, r ′ moves away from r with velocity So the velocity of the free particle relative to r ′ , when it coincides with r ′ , is reduced by −dv = (da/a)v, so Note that this expression is in agreement with equation (1).As the universe expands, velocities decrease in proportion to a −1 , and temperatures decrease in proportion to a −2 .It is not necessary to invoke General Relativity.The preceding arguments are valid also in the non-relativistic physics of Newton.During galaxy formation, the dark matter particles have a velocity field (to be shown in Figure 2 below) in addition to the thermal velocity.Whenever the velocity field diverges, dark matter cools as described in the preceding paragraph.
The density ρ(a) of matter is proportional to a −3 .We define the "adiabatic invariant" of warm dark matter as the comoving root-mean-square thermal velocity of the dark matter particles: Ω c ρ crit is the mean dark matter density of the universe at the present time (throughout we use the notation and the values of parameters of [1]).The sub-index h stands for the dark matter halo.We will use the sub-index b for "baryons", mostly hydrogen and helium.These sub-indices will be used only when needed.Consider a free observer in a density peak in the early universe.This observer "sees" warm dark matter expand adiabatically, reach maximum expansion, and then contract adiabatically into the core of a galaxy.The adiabatic invariant (4) in the early universe remains constant in the core of the galaxy throughout the formation of the galaxy.This statement is nontrivial, so we will study it in detail in the following lectures, and finally will validate it with observations.
To the cold dark matter cosmology ΛCDM, that has six parameters, we add one more parameter, namely the adiabatic invariant v hrms (1), and obtain the warm dark matter cosmology ΛWDM.As we shall learn in the following lectures, we are able to measure v hrms and ρ h in the core of spiral galaxies, and will therefore be able to obtain v hrms (1), and then decide whether dark matter is warm or cold.

The exponential isothermal atmosphere
This lecture is included to remind the reader of results we will be using later on.(For more background, I recommend studying the awe inspiring Feynman Lectures on "The exponential atmosphere".)James Clerk Maxwell presented several clever arguments to obtain the distribution of velocities of the particles in a gas in thermal equilibrium.The number of particles per unit phase space volume d 3 rd 3 p ≡ dxdydzdp x dp y dp z is proportional to exp [−p 2 /(2mkT )]: where p = mv is the particle momentum.The momenta p are assumed isotropic (later on we will lift this assumption when needed).This Maxwell distribution has been validated experimentally (which settles the issues of the clever arguments), and is our point of departure in these lectures.Using the definite integrals the following well known results are readily obtained: where the pressure P is defined as twice the momenta p z of particles with p z > 0 traversing unit area in unit time.
Let N/V be the number of particles per unit volume of the gas.Then the normalized Maxwell distribution is The number of particles with momenta in the interval p z to p z +dp z , traversing unit area in unit time, obtained from (8), is Now consider gas in equilibrium in a column in a uniform gravitational field g = g z e z with g z < 0. The potential energy of a particle at altitude z = h is Φ = m(−g z )h.We wish to obtain the distribution of momenta at altitude h.We consider particles with p z > 0 and p 2 z /(2m) > Φ, with momenta in the interval p z to p z + dp z .These particles reach altitude h with momenta in the interval p ′ z to p ′ z + dp ′ z , where Subtracting these two equations, and keeping first order terms, obtains Multiplying by dt obtains This equation is Liouville's theorem in one dimension: the volume of phase space occupied by particles moving in a conservative field is a constant of the motion.
In equilibrium, the flux dF ′ of particles at altitude h with momenta in the interval p ′ z to p ′ z +dp ′ z is the same as the flux dF of particles at altitude 0 with momenta in the interval p z to p z + dp z , since these are the same particles.Therefore, expressing dF in terms of variables at altitude h, we obtain: where So, the density varies with altitude as Comparing ( 13) with ( 9), we obtain the distribution of particles at any altitude: where the energy of a particle is Equation ( 16) is the Boltzmann distribution.The proportionality constant in ( 16) is independent of altitude.Note that T in ( 16) is the same T as in (8), so the column of gas is isothermal!These equations are valid for general potential energies Φ, whether, or not, proportional to h.An amazing result of these calculations is that the root-mean-square velocity v 2 of the particles of the gas is independent of altitude!This is because only the more energetic particles reach a higher altitude.Another amazing result is that the column of gas is isothermal because of the Maxwell distribution of momenta, not because of thermal contact, or not, with the walls (if any) of the gas column.
We have not considered particle collisions.It turns out that results are unchanged because collisions conserve energy.
If the particles are collisionless, the thermal velocities may not be isotropic, in which case we will consider separately each component of the thermal velocity.
One more equation that we will be using in the sequel is in agreement with (15).This equation expresses that the difference of pressure P (z) − P (z + dz) supports the weight of the gas between z and z + dz.This result may again be surprising since there are no membranes at z and z + dz on which the particles can bounce off.Equation (18) expresses conservation of momentum.

The cored isothermal sphere
Let us repeat the arguments of the preceding lecture, but this time we consider a self-gravitating gas with spherical symmetry.The equations to be solved are The rotation velocity V (r) of a test particle in a circular orbit of radius r is given by First we seek a particular solution of the form ρ(r) ∝ r n , with v 2 r independent of r, i.e. we limit the scope of the present lectures to the isothermal case.There is a single solution of this form (with n = −2): The potential energy difference between particles at r ′ and r is Φ = 2kT ln (r ′ /r), so (15) is valid.There are dn particles in phase space volume 4πr 2 drdp r , that will later occupy phase space volume 4πr ′2 dr ′ dp ′ r .Then so dF ′ = (r/r ′ ) 2 dF .The total mass of the halo can be defined as M(r max ) with ρ(r max ) = Ω c ρ crit .Then M(r max ) ∝ T 3/2 ∝ V 3 , which is the Tully-Fisher relation if M(r max ) is proportional to the absolute luminosity.Equations ( 19) can be solved numerically in radial steps dr, starting from r min .To start the numerical integration we need to provide two boundary conditions, e.g.g r (r min ), and ρ(r min ).For solutions with no black hole at r = 0 we can take g r (r min ) ≈ −G4πr min ρ(r min )/3.For these isothermal spheres with a core the solution for r ≫ r c is (21), while the solution for r ≪ r c is The two asymptotes meet at a core radius Since the cored isothermal sphere has formed from a density perturbation in the early universe, the two parameters ρ 0 and v 2 r are related by the adiabatic invariant (4): Therefore the cored isothermal sphere is defined by a single independent parameter, ρ 0 or v 2 r .Equation ( 26) assumes isotropic velocities in the core, which is valid due to the spherical symmetry.Further justification of (26), and observational validation, will be given in following lectures.
We note that a measurement of V (r ≫ r c ) obtains v 2 r , and a measurement of dV (r ≪ r c )/dr obtains ρ 0 , and together they obtain the adiabatic invariant v hrms (1).

An example
As an example with extremely large ρ 0 , let us consider the spiral galaxy UGC11914.The rotation curves of this galaxy, observed by the SPARC collaboration [3], are presented in Figure 1.We fit these rotation curves by solving the equations: We have included κ b and κ h to account for rotation.These parameters are quite uncertain, tho a rough estimate is κ b ≈ 0.98 and κ h ≈ 0.15 [4].We can eliminate κ b and κ h from the numerical integration by replacing We start the numerical integration at the first measured point at r min , and end at the last measured point at r max , in steps dr.To start the numerical integration we need six boundary conditions: Top: Observed rotation curve V tot (r) (dots) and the baryon contribution V b (r) (triangles) of galaxy UGC11914 [3].The solid lines are obtained by numerical integration as explained in the text.Bottom: Mass densities of baryons and dark matter obtained by the numerical integration.to include the possibility of a black hole at the center.Good fits are obtained assuming v 2 rb /(1 − κ b ) and v 2 rh /(1 − κ h ) are independent of r.We vary the six boundary conditions to minimize the χ 2 between the observed and calculated rotation velocities, and obtain: Uncertainties are statistical from the fit.The core radius is 0.70 kpc.The core dark matter density ρ h (r min ) is 2.6 × 10 8 times the mean dark matter density Ω c ρ crit .We note that this galaxy indeed has a black hole at its center.From v 2 rh and ρ h (r min ) we obtain the adiabatic invariant Let us mention that similar results are obtained from galaxies spanning 3.5 orders of magnitude in absolute luminosity [5], validating that the adiabatic invariant in the core of galaxies is conserved, confirming that v hrms (1) is of cosmological origin, and that dark matter is indeed warm!If we add an r-dependence to v 2 rh we generally obtain a higher χ 2 of the fit, so indeed, within uncertainties, the dark matter halo is isothermal, at least out to the observed rotation curves, indicating that the particles have the Maxwell-Boltzmann momentum distribution.We note that v 2 rb /(1 − κ b ) is of the same order of magnitude as v 2 rh /(1 − κ h ), because dark matter and baryons fall into the same potential well.Since thermal equilibrium im- ), dark matter is not in thermal equilibrium with baryons (mostly hydrogen atoms).So, on galactic scales, we can neglect dark matter-baryon interactions.If dark matter feels only the gravitational interaction, it can be shown that deviations of its trajectory, or interchange of energy with baryons, can be neglected on galactic scales.We also note that the ratio of dark matter to baryon densities in the galaxy is of the order of the universe average.
The galaxy density (21) reaches Ω c ρ crit at The age of the universe at, say, redshift z = 6, is 3 × 10 16 s.In this time a particle with constant velocity 198 km/s reaches 0.2 Mpc, less than r max , and much farther than the last observed rotation velocity.
From (36), neglecting κ h , we estimate that dark matter becomes nonrelativistic at expansion parameter i.e. after e + e − annihilation, and while the universe is still dominated by radiation.Now let us do the following back-of-the-envelope approximate calculations.An ultra-relativistic gas with zero chemical potential has a number density of particles n(T ) = 0.1218 • (kT /( c)) 3 (N b + 3N f /4) [1].At expansion parameter a hNR the temperature is T h (a hNR ) ≈ m h c 2 /(3k).The present number density is n(1) = n(a hNR )a 3 hNR = Ω c ρ crit /m h , if dark matter particles do not decay or annihilate at ≈ a hNR , i.e. if there is no "freeze-out".From these equations we obtain For scalar dark matter, i.e.N b = 1 and N f = 0, and neglecting κ h , we estimate the mass of the dark matter particles from (36): m h c 2 /e ≈ 140 eV.At expansion parameter a hNR the photon temperature is T γ (a hNR ) = T 0 /a hNR .From the preceding equations we obtain or, for our example, T h (a hNR )/T γ (a hNR ) ≈ 0.354.That this ratio is of order 1, given that the dark matter mass is uncertain over 89 orders of magnitude [1], is surely telling us something!Furthermore, note that dark matter is sufficiently cooler than photons to (marginally) evade the "thermal relic" mass limits obtained from the Lyman-α forest [6], and sufficiently cool to not spoil the success of Big-Bang Nucleosynthesis [4].Finally, within experimental uncertainties, dark matter is in thermal and diffusive equilibrium with the Standard Model sector at T somewhere between the top quark mass m t and the temperature T C of the deconfinement-confinement transition from quarks to hadrons (decoupling at a lower temperature compromises the agreement with Big Bang Nucleosynthesis [4]).For a proper treatment of the preceding estimates see [2] [7]: equations (40) and (41) change by less than 1%.For a summary of measurements and their interpretation, see [8].For details of each measurement see the citations in [6].
6 Adding particles to a self-gravitating isothermal gas Consider a self-gravitating isothermal gas in equilibrium with density ρ(r).
We try to find a family of self-similar distributions by adding more particles to the gas.Distances scale as r → αr, densities scale as ρ(r) → βρ(αr), and the particle velocities (including thermal velocities) scale as v → γv.
The temperature scales as T → γ 2 T .The gravitation field scales as g(r) = V 2 /r → (γ 2 /α)g(αr) or as g(r) = −GM/r 2 → αβg(αr), and the mass of the gas scales as M(r) ∝ r 3 ρ → α 3 βM(αr) or as M(r) ∝ T r → γ 2 αM(αr), so These relations are in agreement with equations ( 27)-(30).The kinetic energy of the gas scales as E K → α 3 βγ 2 E K , and the potential energy of the gas scales as E P → α 5 β 2 E P , so the virial theorem E K = −E P /2 for the scaled gas is satisfied.
To complete the description of the problem at hand, we still need to specify the equation of state of the gas, namely T ρ(αr) −2/3 = constant, or γ 2 = β 2/3 .We are then left with a 1-dimensional family of similar solutions with γ = All these similar solutions have the same adiabatic invariant T ρ(αr) −2/3 .During the accretion of new particles, the self-gravitating gas remains in thermal and mechanical equilibrium to a good approximation, and the adiabatic invariant remains constant.

Galaxy formation
Consider a cored isothermal sphere.According to (43), this cored isothermal sphere is defined by a single parameter, e.g. the core density ρ 0 .As the universe expands, new particles, if available, fall into the halo potential well.The halo contracts as r → αr.The converging particles increase the temperature of the halo in proportion to γ 2 , and the densities scale as ρ(r) → βρ(αr), where α, β and γ satisfy (43).To a good approximation, thermal and mechanical equilibrium is maintained throughout this galaxy evolution.The adiabatic invariant in the core of the galaxy remains constant, i.e. γ = β 1/3 .As an example, assume that the core density ρ 0 of the cored isothermal sphere increases by a factor 8. According to the adiabatic invariant, or the noble gas equation of state, the thermal velocity v 2 r in the core increases g(r, t) is the gravitation field, r is the proper (not comoving) coordinate vector, ρ h (r, t) and ρ b (r, t) are the mass densities, v h (r, t) and v b (r, t) are the velocity fields, and v 2 rh (t) and v 2 rb (t) are the radial (1-dimensional) velocity dispersions, i.e. thermal velocities, of dark matter and baryons (mostly hydrogen), respectively.Equations ( 47) and (48) express the conservation of momentum.The static limits of equations ( 44)-(48) are equations ( 27)-(30).These equations need to be supplemented by the equation of state of the gas.The Maxwell distributions of baryons and dark matter have different temperatures kT b = m p v 2 rb and kT h = m h v 2 rh , respectively.As an example we assume no rotation, i.e. κ h (t) = κ b (t) = 0, and set the adiabatic invariant to v hrms (1) = 490 m/s.For baryons we take v brms (1) = 21 m/s, corresponding to hydrogen decoupling from photons at z ≈ 150 [9].We set the initial ρ h (r) and ρ b (r) as shown in Figure 2. We integrate the equations in steps dt, and for each t, in steps dr, starting at r min , to calculate the new ρ b (r, t + dt) and ρ h (r, t + dt).The above equations are supplemented by the adiabatic conditions, so, for each step of t, we set, only once, i.e. for all r: This isothermal prescription, valid at least out to the last observed rotation velocity, can not be correct beyond r max where the universe is expanding homogeneously.However, this is not a problem since, beyond r max , ∇( v 2 rh ρ h ) and ∇( v 2 rb ρ b ) are zero.While the hydrogen and helium gas remains adiabatic, i.e. until excitations, radiation, shocks and star formation become significant, we require (50) in the core of the galaxy.An example of the formation of a galaxy is shown in Figure 2.

Conclusions
We have studied galaxies, and galaxy formation, assuming that dark matter is warm.This scenario requires the addition of one parameter to the cold dark matter ΛCDM cosmology, namely the adiabatic invariant v hrms (1).We find that the formation of galaxies, all the way from linear perturbations in the early universe, until the galaxies run out of new particles to accrete, is adiabatic to a good approximation, and that the adiabatic invariant in the core of the galaxy is conserved.The run out of new matter to accrete determines the final density and radius of the galaxy core (note that it is not necessary to invoke dark matter self interactions or baryonic feedback to justify a core).The observed spiral galaxy rotation curves allow measurements of v hrms (1) [5].These measurements are consistent for galaxies with absolute luminosities spanning 3.5 orders of magnitude [5], so the analysis is validated by observations, and the interpretation that v hrms (1) is of cosmological origin is confirmed.Independent measurements of v hrms (1) are obtained by studying the consequences of the warm dark matter free-streaming suppression factor τ 2 (k) of the cold dark matter power spectrum of density perturbations, i.e. galaxy stellar mass distributions, galaxy rest-frame ultra-violet luminosity distributions, first galaxies, and their effect on the reionization optical depth [10].All of these measurements of v hrms (1) are consistent, and constrain the warm dark matter particle properties [8] [6] [11].
Figure 1:Top: Observed rotation curve V tot (r) (dots) and the baryon contribution V b (r) (triangles) of galaxy UGC11914[3].The solid lines are obtained by numerical integration as explained in the text.Bottom: Mass densities of baryons and dark matter obtained by the numerical integration.

Figure 2 :
Figure 2: The formation of a warm dark matter plus baryon galaxy with zero angular momentum is shown.The densities ρ h (r) and ρ b (r), and the velocity fields v rh (r) and v rb (r), are presented at time-steps that increase by factors 1.4086 (or √ 1.4086 for the dot-dashed lines).The initial perturbation is Gaussian, with parameters listed in the figure.Dark matter is warm with v hrms (1) = 490 m/s.