Baryon-Interacting Dark Matter
A Core-Forming Model with Baryon Heated Dark MatterWessel Krah
August 18, 2020
Abstract
Populaire abstract: Dat er in schijfstelsels specifieke relaties te vinden zijn tussen de ver-snelling van materie en de afstand van deze materie tot de kern van het stelsel is al langer bekend. Deze relatie wordt omschreven door de "Mass-Discrepancy Acceleration Relation" (MDAR) en im-plicieert het bestaan van extra, onzichtbare massa. Deze extra massa wordt donkere materie (DM) genoemd. Het "donker" heeft als oorsprong dat deze materie geen interactie heeft met licht en dus niet gezien kan worden maar wel zwaartekracht uitoefend. Dat deze donkere materie bestaat is voor een groot deel zeker maar wat deze donkere materie nu precies is en hoe deze donkere materie omgaat met normale materie is nog niet helemaal zeker. In deze scriptie wordt er naar een hypo-thetisch model van donkere materie gekeken die zowel zwaartekracht uitoefent alsook een andere, nog onbekende, interactie heeft met normale materie. Bij deze onbekende extra interactie wordt de donkere materie dan opgewarmd. Dit zorgt er vervolgens voor dat de donkere materie die normaal naar het midden zou vallen van het sterrenstelsel dan weer naar buiten zal worden geaccelereerd. Gebaseerd op de hydrodynamische vergelijkingen van Boltzmann en de zwaartekracht vergelijking van Poisson is het mogenlijk om de opwarmingssnelheid, de vergelijking voor de druk, de ontspan-ningstijd en de vergelijking van de warmtetransport te achterhalen. Daar dan weer op gebaseerd en als de aannamen wordt gemaakt dat het profiel van de DM een pseudoisothermisch profiel heeft is het dan mogelijk om te laten zien dat het model meerdere bekende observaties voorspeld zoals "Central Surface Density Relation"(CSDR) en de "Baryonic Tully-Fisher Relation" (BTFR). Ook is het mogelijk de EDGES anomalie te verklaren en enkele voorspellingen te doen.
Scientific abstract: Multiple observational scaling relations about disk galaxies are known. Like the Mass Discrepancy-Acceleration Relation (MDAR) which encodes both the Baryonic Tully-Fisher Relation (BTFR), which tells about the rotational velocity in the flat part of rotation curves, and the Central Surface Density Relation (CSDR), which describes the observed diversity in galaxy rotation curves. Making a working model of galaxy formation which encodes these observational facts has so far been challenging though. The proposal which [1] gives is that the MDAR is the result of baryons interacting with Baryon-Interacting Dark Matter (BIDM), which heats up said dark matter. Using a hydrodynamical approach it becomes clear that the MDAR follows if; the BIDM equation of state is the ideal gas law; the relaxation time is the Jeans time and if the heating rate is inversely proportional to the BIDM density. An added benefit of using these assumptions and the hydrodynamical equations is that they enjoy an approximate anisotropic scaling symme-try in equilibrium. This symmesymme-try in the BIDM dominated regime gives the way the asymptotic rotational velocity of two galaxies should relate according to the BTFR. When assuming a cored pseudo-isothermal profile at equilibrium the parametric dependence of the BTFR follows. Lastly in equilibrium in the central region of high surface brightness galaxies the CSDR follows. When looking at the time dependent case a combination of the BTFR and the CSDR follows where if one is assumed the other one will follow. When just looking at the heating rate some predictions based on BIDM can be made. One of these is that the density profile should look like a core instead of a cusp. Another is that BIDM gives a possible explanation to the EDGES anomaly.
1
Introduction
1.1 The history of dark matter and its related problems
The idea that there could be something like dark matter, matter which does not interact with light but does have a gravitational effect, originates from 1933 [2]. In that year a Swiss astronomer named Fritz Zwicky published a paper on galaxy clusters [3] where he noted that the cluster that he observed should be unstable. The different galaxies were moving so fast that either the cluster should rip itself apart or there should be a much higher mass than could be deduced from all of the luminous matter. Zwicky back then said that there had to be dark matter that made sure that the cluster stayed together. His findings of unstable clusters were later also found by multiple other astronomers who also came to the conclusion of dark matter.
Only in the late 1950’s was this problem examined further. Bigger and bigger catalogues of stars were becoming available and the data much more precise. The problem remained however and a solution was needed. Multiple solutions were proposed including the existence of multiple regions of ionised hydrogen, large amounts of dwarf galaxies, dense gravitational radiation and cosmological black holes. Some even proposed that it was all due to observational errors and then others proposed that the clusters were, in fact, flying apart and that it was more of a coincidence that we happened to find so many intact clusters. In the end it turned out to be difficult to find a good answer to this problem.
This changed with the introduction of rotation curves. Rotation curves are curves of galaxies where on the x-axis is the radius of the galaxy and on the y-axis is the rotational velocity of matter in the galaxy. These rotation curves of galaxies suddenly became easier to observe in the 1950’s with the discovery of the 21 cm emission line of otherwise hard to detect neutral hydrogen. According to both Newtonian and Einstein’s laws the rotational velocity should taper off when the radius is beyond the bulk of the mass which could be measured by measuring the amount light emitted by stars. As was found out in the 1970’s [4] The rotational velocity however remained flat all the way until the edge of what could be measured. This either implied that the working theory of gravity was wrong or again there was a case of missing mass. However again not much urgency was put on finding a correct theory. A new, theoretical, problem arose which once more made dark matter an attractive candidate. In the early 1970’s a resurgence of general relativity came with the increase in physics schooled astronomers. One of the problems these astronomers wanted to solve was wether the universe is closed, flat or open. According to the Friedmann equations an isotropic homogenous universe could either be closed, flat, or open. The Friedmann equations describe the conflict between the push of the expansion of the universe and the pull of gravity where a closed universe would eventually collapse back into itself and a flat or open universe would expand forever. Two important parameters existed which determined the end of the universe. One was the Hubble constant H0 and the other is the deceleration parameter q0, which, if you believe that Einstein’s constant is zero (∆ = 0), completely depends on the universal
density ρ. Both of these parameters are observables and thus a lot of research went into finding out what they were. There was however also a preference for the parameters as most astronomers thought the universe had to be closed based on “aesthetic” or “religious” reasons. The Hubble constant was quickly found however this posed a problem. The required density to make the universe closed was ρc 10−29g/cm3 while the observed density was ρ 10−31g/cm3 based on stellar observations. This caused a lot of people to look for mass not in the stars in galaxies but in other sources of matter. Only then did people start taking the theory of dark matter seriously and see it as a solution to all three problems. There was still the problem of what exactly this dark matter is. In general most theories fall under three different brackets. There are the MACHO’s or MAssive Compact Halo Objects, the WIMP’s or Weakly Interacting Massive Particles and lastly the theory that there is no dark matter at all, MOND or MOdified Newtonian Dynamics. All three of these theories have their own problems. MACHO’s usually refer to primordial black holes which are massive and compact but also do not emit any observable light. It has however been difficult to make a working theory which accounts for all of the missing mass while also taking into account that none of these black holes have ever been observed either directly or indirectly through gravitational lensing. MOND usually gives great theories for the rotational velocity of galaxies problem but then struggles to also explain the behaviour
of galaxy clusters [5] and explain the results of the Cosmic Microwave Backround (CMB) map, which contains indirect evidence of both dark matter and dark energy when analysed. Lastly WIMPs have been the most successful in describing the universe aside from the fact that they too have not yet been observed either in CERN or through telescopes.
Another problem that a lot of WIMPS have is the “cusp versus core” problem. Most of the types of dark matter particles that have been modeled and simulated only interact through gravity. The density distribution then tends to form a cusp where the density starts to go towards infinity as the radius goes to zero. This is not supported through observations however as such high densities would have an observable effect on the surrounding stars. What is more in line with observations is a core where there is a roughly constant density until a certain radius after which the density lowers. Such a density distribution does not happen when the dark matter particles only interact through gravity, but if they are heated up by the baryons through some alternative interaction then a core might be formed [1]. One such a core forming model with extra interactions that cause dark matter heating will be explored in this paper.
1.2 MDAR and the BTFR
The Mass Discrepancy Acceleration Relation is a relation between the baryonic rotational acceleration of matter in a galaxy and the mass of that galaxy and encodes the flat part of the rotation curves which are a problem for the Newtonian analysis. It also gives an indication that there is an acceleration scale in the rotation curves of galaxies. To begin it is important to realise that radial velocity and acceleration are linked through Vr2 = a so V2V2
N ewton
= a a
N ewton which means a difference in measured
radial velocity and the radial velocity expected from the Newtonian equations can be explained through an acceleration difference. One interesting thing to note is that this effect of missing acceleration is only present below the characteristic acceleration a0 ≈ 10−10m/s2 as can be seen in figure 1. MDAR [6] is
typically described as a relation between a baryonic gravitational acceleration and a total gravitational acceleration: g = ( gb gb a0 √ gba0 gb a0, (1) Where gb is the baryonic gravitational acceleration and g is the total gravitational acceleration. The BTFR or Baryonic Tully-Fisher Relation [7] is a relation that follows from the MDAR. It is the relation between the mass of a galaxy and the asymptotic rotational velocity of that galaxy. This is easy to derive if the edge of said galaxy is in the dark matter (DM) dominated regime. First near the edge of galaxies the gravitational acceleration can be simplified to gb = GMb
r2 where G is the gravitational
constant and Mb is the total baryonic mass. Next in the DM dominated regime g =
√
a0gb. Combining
these two pieces with the centripetal force Fcp= Vf lat2 m
r and F = ma gives
Vf lat4 = a0GMb, (2)
which is just the effect of the MDAR in the flat part of the rotation curve.
1.3 CSDR and LSB scaling
The Central Surface Density Relation or CSDR is yet another consequence of the MDAR. It is related to the diversity of shapes and behaves like:
Σ(0) = ( Σb(0) Σb(0) aG0 q 2a0 πGΣb(0) Σb(0) a0 G , (3) where Σ(0) = 2R∞ −∞dzρ(−
→r ) is the central dynamic surface density with z being the coordinate transverse to the galactic disk. The baryonic surface density is then Σb(0) =R∞
−∞dzρ(−
→r ). The central dynamic surface density Σ(0) is then the sum of the baryonic surface density Σb(0) and the dark matter
Figure 1: The mass discrepancy acceleration relation[6]. It shows that there is a missing accel-eration which becomes more apparent as the Newtonian accelaccel-eration from baryons drops below a0≈ 10−10m/s2.
ΣDM = 2
Z ∞
0
drρ(r). (4)
In the CSDR High Surface Brightness or HSB galaxies refer to galaxies where Σb(0) a0
G is the
case. Low Surface Brightness or LSB galaxies refer to the opposite so where Σb(0) a0
G
LSB’s are particularly interesting because they have an extra scaling symmetry according to MOND. According to MOND in LSB’s below the gravitational acceleration scale a0 the dynamics are invariant under spacetime transformations so −→x → λ−→x and t → λt. This implies that two LSB’s with similar total mass Mb but different scale lengths L1 and L2 have similar rotation curves with just a scale
difference. Combining this with the BTFR however indicates that these galaxies should be related by
V2(R) =
Mb,2
Mb,1
14
· V1(RL1/L2), (5)
where R is the axisymmetric radius and where V (R) is the rotational velocity and dependent on R.
2
Baryon-Interacting Dark Matter
In order to make a model for baryon-interacting dark matter one can start off with the Boltzmann equation which is usually collisionless and add in a collisional term to turn it into the Boltzmann transport equation. The first term of this Boltzmann transport equation or the zeroth velocity moment is the mass equation which basically says that DM mass must be conserved:
∂ρ ∂t +
− →
∇ · (ρ−→u ) = 0, (6)
where ρ is the DM density and −→u is the DM bulk velocity and is defined as −→u ≡ h−→v i where −→v is the velocity dispersion.
The second term of the Boltzmann transport equation or the first velocity moment deals with conser-vation of momentum. It says that all the DM momentum, which is balanced on the left side of the equation, can only change due to the influence of gravity, which can either come from DM sources or baryonic sources. It looks like:
∂ ∂t+ − →u ·−→∇ ! ui+ 1 ρ∂iP ij = gi, (7)
here Pij ≡ ρh(vi− ui)(vj− uj)i is the pressure tensor and gi is the gravitational attraction. The
total gravitational attraction is determined by the Poisson equation as −→∇ · −→g = −4πG(ρb + ρ). For
most of the evaluations the ρb will be an input specified by observations.
The third term of the Boltzmann transport equation or the second velocity moment deals with the conservation of energy. Again all the energy contained is balanced on the left hand side and on the right hand of the equation it tells us that the total energy can only be changed due to the energy exchange between the baryons and the dark matter. It looks like
3 2 ∂ ∂t+ − →u ·−→∇ ! T m + 1 ρP ij∂ iuj+ 1 ρ − → ∇ · −→q = ε˙ m, (8)
where T ≡ m3h|−→v − −→u |2i is the local DM temperature, −→q ≡ 1 2ρh(−
→v − −→u )|−→v − −→u |2i the heatflux
of the DM and mε˙ is the local heating rate which happens purely due to interactions with baryons. An illuminating way to view this last equation is to multiply both sides with the density ρ = mDM
volume3
which on the left hand side gets rid of all of the ρ1 and on the right hand side gives us a heating rate per unit volume. For the rest of the paper an isotropic case will be assumed which means that Pij ≈ P δij
which is valid for |−→u | v where v is the one dimensional velocity dispersion according to v =pT /m.
2.1 Scaling Symmetries
Looking closely at equations 6, 7, 8 and the Poisson equation it becomes clear that these equations are invariant under isotropic spacetime scaling symmetries or:
−
→x → λ−→x ; t → λzt, (9)
which is valid for an arbitrary z. Following this the various other quantities then transform as v → λ1−zv; − →u → λ1−z−→u ; − →g → λ1−2z−→g ; ρ → λ−2zρ; ρb → λ−2zρb; Pij → λ2−4zPij; − →q → λ3−5z−→q , (10)
With these transformation rules the heating rate has to transform as ˙
ε m → λ
2−3z ε˙
m. (11)
These transformation rules might seem non-physical and just an emergent property of the equations but they are powerful tools that will show their worth later on. When one demands that these scaling symmetries are emergent from the DM for a particular value of z then this allows us to put constraints on the DM equation of state, heat flux and heating rate.
2.2 DM equation of state
In order to solve equations 6, 7 and 8 a equation of state for the DM is needed. If one looks at the scalings in equation 10 it becomes possible to identify an equation of state if one assumes that the equation of state has the form P = P (ρv). The equation of state would then look like:
P = ρv2, (12)
which is an obvious and recognisable choice for an equation of state and which would make the equation of state invariant under the isotropic spacetime scalings. It is also the obvious choice if the assumption is made that the DM is sufficiently dilute. Which means that nλwavelength 1 where n is
2.3 Heatflux and Relaxation Time
A natural expression for the heatflux −→q can be found in Fourier’s law [8]: −
→q ≈ −κm−→∇v2, (13)
where κ is the thermal conductivity. Dark matter cannot radiate energy away and it is also assumed that the thermal differences from the equilibrium state are small, to be precise |dlogvdlogr2| 1. The thermal conductivity κ has the form of
κ = O(1)ρv
2t relax
m , (14)
where trelax is the relaxation time of the dark matter. The relaxation time can be seen as the time it takes for the dark matter to reach equilibrium through self-interactions and or interactions with baryons. The scaling symmetries in the equations 10 say that the relaxation time has to scale as trelax→ λztrelax. This is important as a generic relaxation mechanism will not necessarily give rise to
such a scaling for an arbitrary z.
If, for example, the dark matter reaches equilibrium due to self-interactions then the relaxation time will follow from these self-interactions. Self interactions generally have a cross-section that looks like σ = σ0(c/v)α for a fixed α and where σ0 is a constant. The relaxation time is then the inverse of the
self-interaction rate trelax = σnv1 which would result in trelax = σm0ρv(v/c)α which would give trelax a
scaling of trelax→ λ(3−α)z−1+αt
relax which would only agree with scalings in equation 10 if z = 1−α2−α.
While there might be self-interacting dark matter where the self-interactions have an associated re-laxation time, this could be a rather long time period. Especially considering that no evidence of self-interacting DM has been found, this self-interacting relaxation time could be longer than the dy-namical time of galaxies. At this point another relaxation mechanism might be more relevant, namely the Jeans time [9] or the gravitational free falling time:
trelax=
O(1) √
Gρ, (15)
which is the time it takes for a cloud of density ρ to collapse due to gravity. The reason why the Jeans time makes sense is discussed in [10] but it can be argued that the dark matter will follow the mechanics of the baryons since they interact and will therefore have the same relaxation time. Two other reasons for choosing the Jeans time as the relaxation time is because it helps in the reproduction of the MDAR and also has the scaling properties required by the scalings in equation 10. The Jeans time being the relaxation time means that the thermal conductivity as shown in equation 14 becomes
mκ = Ccond
r ρ Gv
2, (16)
where Ccond is a O(1) constant.
2.4 Heating Rate
Up until this point scale invariance for any arbitrary z could be preserved. For the heating rate this is no longer possible however. In order to find an expression for the heating rate it is nescessary to look at the BTFR. The BTFR states that Vf lat/Mb is a universal constant for rotationally supported
galaxies. It is imperative that this does not transform under the scaling symmetry. The only way this is possible is if z becomes
z = 1
2, (17)
and from this it follows that the heating rate will transform as ˙
ε m → λ
1/2 ε˙
which will fix the parametric dependence. It is also logical that the heating rate is dependent on ρ and ρb as well as the velocities of the baryons and the dark matter. Both of the densities transform as ρ → λ−1ρ and ρb → λ−1ρb. For the velocity dependence of the dark matter it is important to
realise that for rotationally supported galaxies the DM bulk velocity is negligible to the DM velocity dispersion |−→u | v. This means that in the rest of the analysis the spin of the DM halo will be ignored. For the baryonic case it is the other way around. As in the baryonic velocity dispersion is negligible compared to the bulk velocity vb |−→Vb|. Then lastly we have to compare the DM velocity dispersion
and the baryonic bulk velocity. In the outer edges of the galaxy in the flat part of the rotation curve these two values are roughly comparable while in the centre |−→Vb| v. So while in general the heating
rate would depend on the baryonic bulk velocity in this analysis the heating rate will only depend on the DM velocity dispersion. The transformation law of the DM velocity dispersion is v → λ1/2v so this means that the heating rate will have the general form of
˙ ε m = vF ρb ρ , v2 ρ , (19) where F ρb ρ, v2 ρ
is a general function that depends on ρb
ρ and
v2
ρ. In order to get a more precise
function for the heating rate it is relevant that in this analysis the baryons are heating up the dark matter and thus that the heating rate is an extensive quantity as a function of the number of baryons. Thus the heating rate should be linear with ρb
˙ ε m = v ρb ρf v2 ρ , (20) where f v2 ρ
is a general function which is dependent on vρ2. From a model building perspective this dependency is desirable and it makes sense in the DM dominated regime where ρb
ρ 1. There
the heating rate should drop to zero as there are no baryons to heat the DM and this is also what our equation tells us. It is however assumed that equation 20 holds generally and not just specifically in the DM dominated regime.
The next assumption made is based on the fact that f
v2
ρ
has the dimensions of acceleration. The assumption is that f
v2
ρ
is roughly constant and has the value of a0 which is the characteristic
acceleration found in the MDAR. This then fixes the heating rate to ˙
ε
m = Cheata0v ρb
ρ , (21)
where for concreteness Cheat was added which is Cheat ≈ O(0.1) which in the cooling case as found in [10] offered a good fit to rotation curves. The assumption that f reduces to a0 is nice from a phenomenological standpoint as it helps in describing the MDAR which has this characteristic acceleration scale. It however does not explain why it numerically has this acceleration scale. An idea is that it gets resolved by some other mechanism that operates over cosmological timescales which are significantly longer than the timescales in this analysis. This makes sure that a0 can be treated as a
constant in our timescales while in longer timescales it might change. The inverse density dependence in the heating rate is also nice phenomenologically as it makes sure the energy exchange between the baryons and dark matter gets lowered in high density environments. One last word about the sign of Cheat. In [10] the sign was negative meaning that the DM was heating up the baryons. This leads
to multiple problems like the formation of cusps and dark disks, both of which are not supported by observations. Dark disks form if there is an initial DM spin during the galaxy formation. If the sign is positive this becomes less of a problem as cusps are turned into cores which are supported by observations. This happens because the heated up DM will move faster and thus move further outward, causing a cusp to be turned into a core.
2.5 Deep MOND scaling
Now if we combine the equation of state 12, the heatflux 13, and the heating rate 21 together with the boltzmann equations 6, 7 and 8 we get
∂ρ ∂t − → ∇ · (ρ−→u ) = 0; (22) ∂ ∂t+ − →u ·−→∇−→u + 1 ρ − → ∇(ρv2) = −→g ; (23) 3 2 ∂ ∂t+ − →u ·−→∇v2+ v2−→∇ · −→u − 1 ρ − → ∇ · Ccond r ρ Gv 2−→∇v2 = Cheata0v ρb ρ ; (24) − → ∇ · −→g = −4πG(ρ + ρb), (25)
where 25 is the Poisson equation. These equations are still invariant under the scaling transforma-tions of 9 and 10 if z = 1/2. One interesting thing to note is that in the DM dominated regime is that the ρb can be neglected compared to ρ in the Poisson equation. This gives rise to a larger approximate
symmetry − →x → λ−→x ; t → λyt; v → λ1−yv; − →u → λ1−y−→u ; − →g → λ1−2y−→g ; ρ → λ−2yρ; ρb→ λ−2yρb, (26)
which reduces to the original z = 1/2 scale symmetry for y = 1/2 but where y for the rest is arbitrary. This new scale symmetry is only approximately valid in DM dominated regions. Even though it is only approximately valid it still has important consequences. Namely it results in the scaling found in equation 5. Again if we have a galaxy which has scale length L1, total baryonic mass Mb,1 and rotation curve velocity
− →
V1 which also is a solution to our equations. It then follows that the
equations must also have a solution with L2, Mb,2 and
− →
V2 which are related to the 1 case through
L2 = λL1; Mb,2 = λ4−4yMb,1; − → V2(λ−→x ) = λ1−y − → V1(−→x ). (27)
This is the same as saying that the rotation curves of two different galaxies must be related according to5.
3
Testing of the Theory using a Cored Profile
As previously stated in the DM dominated regime the velocity scaling as found in 27 should result in the scaling found in 5 which is what the BTFR says, namely Mb ∝ Vf lat4 . The normalisation of the
BTFR is a difficult task to do from a theoretical standpoint. If one starts from abundance matching (AM), which is the process by which one checks how much mass is missing based on the rotational velocity observed and how much mass that implies versus the amount of visible mass, with Navarro-Frenk-White (NFW) [11] halos one gets the correct starting point of the relation in the baryonic mass range but with a scatter that is too high. When following the curvature of the predicted BTFR it then gives a Vf latthat is too large or a DM mass that is too large at the LSB end. Still with a scatter that
is too large. If we assume that the normalisation that AM gives us is correct in the intermediate-mass regime, the heated DM in the LSB galaxies should be expelled due to this heating. Which in turn brings the enclosed DM mass down and thus also Vf lat.
From this point on to make more analytic predictions the assumption is made that, through self-interactions and through baryon-DM self-interactions, the DM halos will form a cored pseudo-isothermal profile at the baryonic disk. It will follow that equation 22 to 25 are consistent with this profile and will reproduce the MDAR.
3.1 Reproducing the MDAR
In order to reproduce the MDAR using the cored pseudo-isothermal profile the parameters of said profile should take on specific values. The profile has the form
ρ(r) = ρ0 1 + r rc 2, (28)
where ρ0 and rcare as of yet unknown parameters and where spherical symmetry is assumed. rho0 is the central density and rc is the radius of the core. Another way to write down rc is through the
asymptotic velocity dispersion, denoted by v∞,
rc=
v∞
√ 2πGρ0
. (29)
v∞ is the velocity dispersion at radius infinity because the profile is not strictly isothermal and
a higher temperature means a higher velocity dispersion. At the large distance regime (r rc) the
density profile becomes
ρ(r rc) = r2cρ0 r2 c+ r2 ≈ ρ0r 2 c r2 = v∞2 2πGr2, (30)
which implies the flat rotation curve where Vf lat=√2v∞due to the results of abundance matching.
A flat rotation curve means that the dark matter dominates which also justifies the spherical symmetry. The BTFR 2 then tells us that the velocity dispersion must be related to the baryonic mass through
v∞4 = 1
4a0GMb, (31)
which through equation 30 fixes one of the parameters and simplifies 28 to
ρ(r) = 1 4πG √ a0GMb r2 c + r2 . (32)
The second parameter is fixed thanks to the CSDR 3 where the pseudo-isothermal profile inside equation 4 gives
ΣDM = πρ0rc, (33)
due to the fact that the profile is spherically symmetrical. From this point onwards it is important to make a distinction between LSB’s where DM dominates everywhere or Σb a0/G and with HSB’s
where DM only dominates on the outside and baryons dominate in the centre. For LSB’s the CSDR 3 implies ρ0rc= 2 π r a0 2πGΣb(0), (34)
which gives us the second constraint but only for LSB’s because there most of the surface density is given by the dark matter and thus the total surface density can be approximated as the dark matter surface density. The constraint is given through equation 29 and the previous constraint 31
rc= 1 4 s πMb 2Σb(0) , (35)
for HSB galaxies where in the centre Σb a0/G, the CSDR does not constrain the ΣDM because
the baryons dominate and cannot be ignored. The constraint however depends on which functional form for the MDAR is assumed.
From the scaling perspective which is a prevalent subject in this analysis, the cored pseudo-isothermal profile breaks the z = 1/2 scaling symmetry by introducing an explicit scale rc. This symmetry is only restored at the flat part of the rotation curve where r rc as seen in 30 which has the scaling
3.2 Testing the Parametrically fixed Profile
Now that we have a cored pseudo-isothermal profile with the parameters fixed through the MDAR and the CSDR it is important to test it against the equations 22 till 25. This test is only for the LSB galaxy case or the DM dominated case and where the DM halo has negligible spin. If the DM halo has no real spin then the DM bulk veloity is zero so −→u = 0 and the mass continuity equation 22 is already satisfied. The other equations then reduce to
− → ∇(ρv2) = ρ−→g ; (36) − → ∇ ·r ρ Gv 2−→∇v2 = −Cheat Ccond va0ρb; (37) − → ∇ · −→g = −4πG(ρ + ρb), (38)
where equation for conservation of momentum 23 becomes the Jeans equation 36. Since we are working in the DM dominated case spherical symmetry is a good approximation. The equation Jeans equation 36 and the Poisson equation 38 are solved approximately by
ρ(r) ≈ v
2(r)
2πGr2, (39)
where it is assumed and later on will be verified that v(r) is a slowly-varying function [12] meaning that the behaviour near infinity is similar to the behaviour at infinity. v(r) is determined by the heat equation or energy balance equation 37 which using spherical symmetry and equation 39 becomes
1 r2 d dr v4rdv dr = −r π 2 Cheat Ccond va0Gρb, (40)
which, if you approximate the v on the right hand side as nearly constant, can be integrated once to give rdv 4 dr = − 1 √ 2π Cheat Ccond a0GMb, (41)
which in turn can be integrated again to give v4(r) = √1 2π Cheat Ccond a0GMbln R0 r , (42)
where R0 is an arbitrary length scale. Since v depends on r logarithmically the assumption that v
is slowly varying is justified.
The downside of this analysis is that the logarithmic dependence of v(r) means that scale invariance is not restored for r rc completely. However using the approximate relation V ≈
√
2v as used before the rotation curve is nearly flat with
Vf lat4 ≈ a0GMbln
R0
r , (43)
which has the upside of matching the parametric dependence of the BTFR 2 namely a0GMb. R0 cannot be fixed using the equilibrium treatment nor can its scatter be determined. For this the
dynamical evolution towards equilibrium will have to be analysed.
3.3 CSDR in HSB Galaxies
In the centre of galaxies where r rc the DM density is approximately constant, so ρ ≈ ρ0. From this the Jeans equation 36 becomes
− →
where the solution is v2 = φ + αv∞2 where φ is the gravitational potential and where α is an O(1)
constant where the precise value is irrelevant. What is relevant is that v2 approaches v∞2 near the centre and the gradient is fixed by the gravitational field.
The next assumption that is necessary is that we are working sufficiently close to the centre that the baryon distribution stretches on infinitely so r L, where L is the disk scale length, but that we we are working sufficiently far that the baryon disk looks perfectly flat so r Lz, where Lz is the disk scale height. The result of these assumptions is that the baryon distribution is approximated by the surface density, so
ρb ≈ Σbδ(z), (45)
and since the distance is L the surface density can be taken as homogeneous and thus is Σb a
constant and given by the central value Σb(0).
The heat equation or energy balance equation 37 becomes discontinuous in the normal direction which fixes the magnitude to
r ρ0
Gv∞|∇⊥v
2| = Cheat
2Ccond
a0Σb(0), (46)
which if combined with equation 29, equation 33 and 44 gives
ΣDMg⊥= r π 2 Cheat 2Ccond a0Σb(0), (47)
where the transverse component of the gravitational field is solved by integrating Poisson’s equation 38. For HSB’s this gives
g⊥=≈ 2πGΣb(0), (48)
which if filled into equation 47 gives
ΣDM = r π 2 Cheat 2Ccond a0 2πG, (49)
which implies ΣDM ≈ a0/G which matches the behaviour as seen in the CSDR.
3.4 Approach to Equilibrium
The focus up until this point was on the static equilibrium situation. This has proven to be useful since the BTFR was reproduced parametrically in equation 43, with only a logarithm of scale R0 whose scatter and magnitude is unknown. As well as the CSDR for HSB galaxies in equation 49. The equilibrium analysis can only bring us so far however. The proper dynamical analysis requires numerical simulations which is beyond the scope of this analysis. A simpler analysis that is time dependent but ignores density and velocity gradients and treats the mas average quantities can still reproduce the CSDR in the DM-dominated regime if one treats the BTFR as a given.
In order to this dynamical analysis it is best to start with the equations 22 till 25 and add into that the entropy density per DM particle which is given by the Sackur-Tetrode [13] equation
s = ln (2π)3/2m 4v3 ρ +5 2, (50)
which allows us, if we so choose, to get rid of v, since it now is an implicit function of ρ and s, and express the boltzmann equations and Poisson equation in terms of ρ, −→u and s. An equation for the entropy density follows if one combines the continuity equation 22 and the heat equation 24:
∂ ∂t+ − →u ·−→∇s + 1 ρv2 − → ∇ · −→q = ε˙ mv2, (51)
− →q = −2 3Ccond r ρ Gv 4−→∇(s + lnρ), (52)
In order to simplify the equations we now also approximate the mass and entropy densities as nearly uniform. This means that the gradients of the entropy and mass approaches zero−→∇s,−→∇ρ ≈ 0. This also means that the heat flux becomes zero due to equation 52 which means that equation 51 simplifies to ∂s ∂t = ˙ ε mv2, (53)
which is what one would expect. Namely that the entropy per DM particle becomes higher as they get heated by the baryons. If you assume that the initial DM entropy is negligible then one can integrate this equation over the relaxation time to get
˙ ε
mv2trelax≈ 1, (54)
which expresses the condition for equilibrium. With equations 33 and 29 it is possible to estimate that the DM surface density is ΣDM =
pπ 2 q ρv2 G ≈ q ρv2
G which combined with 15, 21 and 54 tells us
that
Σ3DM v2 ≈
a0ρb
G2 , (55)
which gives us an expression for the DM surface density in the dynamical case. If the assumption is made that there is an approximate linear relation between disk scale length and disk scale height Lz ≈ L/8 then one can calculate the mean baryon density through ρb ≈ πLM2bL
z ≈
Mb
L3 which if combined
with the central baryonic surface density of an exponential disk of scale length L which is Σb(0) = Mb
2πL2
than this gives us an approximate relation between the mean baryon density and the baryon surface density ρb≈
Σ3/2b (0) √
Mb which if substituted into equation 55 we get
Σ3 DM v2 ≈ a0Σb(0) G 3/2 √ a0GMb , (56)
which if combined with the BTFR which was also taken as an assumption v2 ≈√a0GMb gives
ΣDM ∝
r
a0Σb(0)
G , (57)
which is the desired CSDR for DM dominated or LSB galaxies. On average will the dark matter have more mass than the baryonic matter since there is roughly 5 times more dark matter than baryonic matter so it is not surprising that the resut matches the DM dominated CSDR. This is good however since the baryonic dominated regime was already explored in the equilibrium treatment.
4
Implications and Constraints
Even though there is no specific model for the by baryons heated dark matter discussed in this analysis there are still some things which can be said and constraints and implications which can be discussed. One assumption made is that the details that are discussed in this analysis, like the heat transport due to collective excitations of a DM medium, are valid at all times, even when the dark matter is highly condensed. The only equation needed to derive the very general results will be the heat equation 21 where Cheat= 101 and a0= 10−8cm/s2 so that the heating rates becomes
˙ ε m = 10 −9ρb ρv cm s2 . (58)
4.1 Cusp versus Core
One issue that most variants of dark matter runs into is the cusp versus core issue [14]. All indirect observations of dark matter tell us that the dark matter density in the cores of galaxies is roughly constant. Meaning that it is a constant dark matter density core. While most simulations of dark matter which only interacts through gravity turns into a density profile like the NFW profile, which is a profile that behaves like a power law and approaches infinity near the centre. This is then called a cusp. The process by which dark matter usually forms into such a cusp is called violent relaxation and has a characteristic time in the order of the dynamic time or Jeans time. The dark matter relaxation method as described in this analysis and in [10] also has a characteristic time in the order of the Jeans time and thus competes with this cusp forming process. It is expected that the relaxation mechanism kicks in before the NFW profile is reached. In fact since the dark matter will be heated up by the baryons it is expected that such a cuspy profile is never reached because the heated up dark matter will move further out in the galaxy due to the associated increase in kinetic energy and will therefore form a core. Baryon heated dark matter therefore could be a solution to the cusp versus core problem.
4.2 Early Universe constraint
In the early universe the baryons were in thermal equilibrium with photons by Compton scattering until z ≈ 200. This process cooled the photons and left visible spectral distortions in the CMB map. If the baryons were heating up the dark matter at the same time then this might have a visible effect on these distortions. The dominant constraint comes from the redshift range 104 . z . 106. [15] the constraints were studied in the case of light DM (m mb) scattering elastically with baryons and/or
photons. The steps are to first take the energy exchange rate per baryon εnn˙
b where n is the DM number
density and nb is the baryon number density and then compare this to the thermal energy per baryon ET ≈ mbvb2 where vb is the baryon velocity dispersion. Then one has to compare this to the Hubble
rate. If the thermal energy is significantly bigger than the energy exchange rate then the amount of energy lost will not be noticeable. So first define
≡ εn/n˙ b Hmbvb2
, (59)
which if combined with equation 21 becomes
= Cheata0v 6Hv2
b
, (60)
where also has been used that n = mρ. has to increase in time in this range since vb will lower due to the expansion of the universe and it is the dominant term. This means that the constraint at z ≈ 104 is the most important. The baryons at that time are in thermal equilibrium with radiation so vb2 = Tγ/mb where Tγ is the CMB temperature. The CMB temperature at z ≈ 104 is Tγ ≈ 2eV and
H ≈ 10−27eV which if combined with the values used in equation 58 means that |z=104 ≈ 10
v
c, (61)
where the DM particles in this analysis are assumed non-relativistic, expecially at that time since they are only allowed to heat up and thus go faster, so the spectral distortions caused by the heating of the DM are negligible.
4.3 Merging Clusters
While this analysis has mostly focused on DM-baryon interactions, DM self-interactions are not dis-missed. One way to contrain these DM self-interactions is through the merging of galaxy clusters [16]. The constraint looks like
σ m .
cm2
where σ is the self-interaction cross section and the precise constraint depends on the assumptions made. Although the constraint 62 was made with DM self-interactions in mind it also applies to the heating rate obtained from DM-baryon scattering. The cross section constraint can be translated into a heating rate constraint by using mε˙ ≈ ρmσv3 where the characteristic energy exchanged per collision of DM particles is mv2 and the interaction rate ψ translates to a cross section through ψ ≈ ρσv. The constraint then becomes:
˙ ε m .
cm2
s3 . (63)
The DM density to baryon density is roughly ρ ≈ 5ρb [17] and the relative velocity between two
clusters is usually v ≈ 108cm/s from which we obtain a rough value for the baryon heated DM ˙
ε
m ≈ 5 × 10
−2cm2
s3 , (64)
which satisfies the constraint 63. What is even better is that only a couple orders of magnitude of improvement on the observational bound 62 would come close to the predicted heating rate. This means that merging clusters could help us detect DM-baryon interaction.
4.4 EDGES anomaly
There was a recent measurement of the 21-cm absorption spectrum from the Cosmic Dawn epoch by the EDGES collaboration to study the first stars and their behaviour [18]. One of the things they found is an excess which implies a cooler hydrogen gas at z ≈ 17 than predicted by the ΛCDM model. The baryon heated dark matter could provide a solution for this as it could have cooled the hydrogen gas prior to the Cosmic Dawn. If, for example, the DM particles are sub-GeV in mass then if they scatter elastically with the baryons their velocity-dependent cross section
σ(v) = σ v 1km/s −4 , (65)
would explain the signal if σ & 10−20cm2. However if mε˙ ≈ nbσ(v)v3 is used, which is a similar equation as used in the merging cluster case but this time with DM-baryon interactions in mind, where nb is the baryon number density which is estimated by nb = 2x10−7(1 + z)3cm−3and the characteristic
velocity is v = 1km/s the bound then translates to ˙ ε m & 10
−8cm2
s3 , (66)
where z ≈ 17 is used. The baryon heated DM used in this analysis gives us the value ˙
ε
m & 5 × 10
−5cm2
s3 , (67)
where again ρ ≈ 5ρb has been used. This means that the heating mechanism as described in this analysis can explain the EDGES excess. In fact it overshoots the excess which could be a slight problem depending on how the explicit particle model looks.
5
Conclusions
The ΛCDM model is rather accurate when describing the universe on cosmological scales, but it can sometimes struggle to describe the physics in astronomical scales like the rotation curves of galaxies which are encoded in the MDAR. The MDAR is a relation between the total gravitational field and the Newtonian acceleration generated by the baryons alone at every radius in disk galaxies. Even though it has little scatter it is still difficult to give an explanation for it, even after all the history of trying to find and describe dark matter. The current method of describing the MDAR is based on feedback processes, and while promising since it has been found out that stellar feedback, which is
the effect of the massive amounts of energy released during supernova explosions, has a characteristic acceleration of order a0, it is not sufficient yet to explain the diversity found in the rotation curves in galaxies which are encoded in the MDAR. Nor does cold dark matter tell us about the core-cusp transformation process since regular cold dark matter, which does not interact with baryons aside through gravity, should form cusps according to numerous simulations while observations imply that dark matter should form cores.
That is why it might be a worthwhile investment to look at the possibility that there may be a new, non-gravitational, interaction between the dark matter and the baryons. Most of the time research is done in long-range interactions which act like MOND and are either fundamentally new forces or are emergent from the DM, like superfluid DM [19]. This analyis, however, looks at direct interactions between baryons and dark matter at any length scale instead of an effective modification of gravity or just through feedback processes. This interaction is only done to heat up the DM-fluid but through a "bottom-up" approach. It has given insights in how the DM-baryon interactions should look like to reproduce the MDAR, BTFR and CSDR.
This analysis has found that the DM microphysics can be encoded in three physical quantities and that if these quantities have their conditions satisfied that the MDAR will follow from them. These three quantities and there conditions are;
• The equation of state P = P (ρ, v), which has to approximately look like an ideal gas and thus follow the ideal gas law P = ρv2. This will generally be the case in the dilute limit where the inter-particle separation is large compared to the mean free path.
• The relaxation time trelax, which falls out of the heat conductivity, which is set by the Jeans time trelax. The mechanism on how this could be achieved can be found in [10] and is if the DM is in
the Knudsen regime. • The heating rate ε˙
m, which has the relation ˙ ε
m ≈ Cheata0v ρb
ρ. This relation is the most important
as it has the most constraints and tells the most about the microphysics of the DM.
These are neither unique nor necessary conditions to obtain the MDAR, just sufficient. One inter-esting thing to note is that these assumptions combined with the hydrodynamical equations and the Poisson equation give rise to an anisotropic scaling symmetry. This gives a guide for model building. In the DM dominated regime this scaling symetry is enhanced to a family of scalings of one-parameter which implies 5.
Specifically these quantities combined with a cored pseudo-isothermal profile give rise to the paramet-ric dependence of the BTFR, up to a logarithm of r. In the central regions of HSB galaxies the DM reproduces the CSDR. When studying the time dependent problem a constraint on the combination of DM velocity dispersion and surface density, which matches the combination of the BTFR and the CSDR, comes to light, which means that if the BTFR is taken as a given the CSDR follows naturally. The model, being as general as it is, can handle the constraints given through multiple observations and can even give predictions for various astrophysical observables like the prediction of DM-baryon interactions becoming visible with a couple of orders of magnitude improvement on merging clusters and being able to explain the EDGES excess. The assumption is made that the effective DM-baryon theory is still valid in these environments. An explicit particle physics model will still be a challenge but the heating rate of the DM in this analysis appears to be observationally viable. Further development is almost always possible. Examples of possible future research are:
• The effect on the baryons: In this analysis the baryons were treated as an external energy source without any specific dynamics. This is approximately true if the typical energy lost by the baryons will not change the total energy of the baryons by a significant amount in the timescales studied. One can estimate the energy lost by a baryon per unit length by dEb
dl &
Cheatmba0v
Vb which could
become significant, despite the fact that Cheat ≈ O(0.1), in DM dominated regions like LSB
galaxies.
• Numerical simulations: A numerical simulation would help confirm some of the predictions made in this analysis like whether or not the equilibrium solution is stable like whether the furthest reaches of the galactic disks are not too perturbed by the heat exchange. It would give confirmation
whether or not the relaxation time is as predicted. Most importantly it would quantify the expected scatter for the BTFR and give a value for the characteristic scale R0 appearing in the logarithm.
• A particle physics model: While this rather general physics model based purely on the effective hydrodynamical desription of the DM has given multiple predictions, it would also be interesting to see what the constraints on the heating rate 58 would mean for the microphysics.
References
[1] Benoit Famaey, Justin Khoury, Riccardo Penco, and Anushrut Sharma. Baryon-Interacting Dark Matter: heating dark matter and the emergence of galaxy scaling relations. Journal of Cosmology and Astroparticle Physics, 2020(06):025, 2020.
[2] JG De Swart, Gianfranco Bertone, and Jeroen van Dongen. How dark matter came to matter. Nature Astronomy, 1(3):1–9, 2017.
[3] Adam Riess. Dark matter, Dec 2017. URL https://www.britannica.com/science/ dark-matter.
[4] Harold Irving Ewen and Edward Mills Purcell. Observation of a line in the galactic radio spectrum. In Classics in Radio Astronomy, pages 328–330. Springer, 1951.
[5] Theodorus Maria Nieuwenhuizen. How Zwicky already ruled out modified gravity theories without dark matter. Fortschritte der Physik, 65(6-8):1600050, 2017.
[6] Stacy S McGaugh. The mass discrepancy-acceleration relation: disk mass and the dark matter distribution. The Astrophysical Journal, 609(2):652, 2004.
[7] R Brent Tully and J Richard Fisher. A new method of determining distances to galaxies. Astron-omy and Astrophysics, 54:661–673, 1977.
[8] Edward A. Mason. Transport properties, Apr 2020. URL https://www.britannica.com/ science/gas-state-of-matter/Transport-properties#ref507001.
[9] Gilberto M Kremer, Martín G Richarte, and Elberth M Schiefer. Using kinetic theory to examine a self-gravitating system composed of baryons and cold dark matter. The European Physical Journal C, 79(6):492, 2019.
[10] Benoit Famaey, Justin Khoury, and Riccardo Penco. Emergence of the mass discrepancy-acceleration relation from dark matter-baryon interactions. Journal of Cosmology and Astropar-ticle Physics, 2018(03):038, 2018.
[11] Julio F. Navarro, Carlos S. Frenk, and Simon D. M. White. The Structure of Cold Dark Matter Halos. The Astrophysical Journal, 462:563, May 1996. ISSN 1538-4357. doi: 10.1086/177173. URL http://dx.doi.org/10.1086/177173.
[12] Nicholas H Bingham, Charles M Goldie, and Jef L Teugels. Regular variation, volume 27. Cam-bridge university press, 1989.
[13] Sackur-tetrode equation. URL https://www.oxfordreference.com/view/10.1093/oi/ authority.20110803100435896.
[14] W. J. G. de Blok. The Core-Cusp Problem. Advances in Astronomy, 2010:1–14, 2010. ISSN 1687-7977. doi: 10.1155/2010/789293. URL http://dx.doi.org/10.1155/2010/789293.
[15] Yacine Ali-Haïmoud, Jens Chluba, and Marc Kamionkowski. Constraints on Dark Matter Inter-actions with Standard Model Particles from Cosmic Microwave Background Spectral Distortions. Physical Review Letters, 115(7), Aug 2015. ISSN 1079-7114. doi: 10.1103/physrevlett.115.071304. URL http://dx.doi.org/10.1103/PhysRevLett.115.071304.
[16] Maxim Markevitch, AH Gonzalez, D Clowe, A Vikhlinin, L David, W Forman, C Jones, S Murray, and W Tucker. Direct constraints on the dark matter self - interaction cross - section from the merging cluster 1e0657-56. Arxiv preprint astro-ph/0309303.
[17] Masataka Fukugita and P. J. E. Peebles. The Cosmic Energy Inventory. The Astrophysical Journal, 616(2):643–668, Dec 2004. ISSN 1538-4357. doi: 10.1086/425155. URL http://dx.doi.org/10. 1086/425155.
[18] Judd D. Bowman, Alan E. E. Rogers, Raul A. Monsalve, Thomas J. Mozdzen, and Nivedita Mahesh. An absorption profile centred at 78 megahertz in the sky-averaged spectrum. Nature, 555(7694):67–70, 2018. doi: 10.1038/nature25792.
[19] Lasha Berezhiani, Benoit Famaey, and Justin Khoury. Phenomenological consequences of super-fluid dark matter with baryon-phonon coupling. Journal of Cosmology and Astroparticle Physics, 2018(09):021, 2018.
