The Paleoclimatic evidence for Strongly Interacting Dark Matter Present in the Galactic Disk
TThe Paleoclimatic evidence for Strongly Interacting Dark MatterPresent in the Galactic Disk
Nir J. Shaviv Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
Using a recent geochemical reconstruction of the Phanerozoic climate which exhibits a 32 Maoscillation with a phase and the secondary modulation expected from the vertical the motion ofthe solar system perpendicular to the galactic plane [1], we show that a kinematically cold stronglyinteracting disk dark matter (dDM) component is necessarily present in the disk. It has a localdensity ρ dDM = 0 . ± .
03 M (cid:12) / pc . It is also consistent with the observed constraints on thetotal gravitating mass and the baryonic components, and it is the natural value borne from theToomre stability criterion. It also has surface density Σ dDM = 15 ± (cid:12) / pc and a vertical velocitydispersion of σ W = 8 . ± . ρ hDM (cid:46) .
01 M (cid:12) / pc . If thedDM component follows the baryons, its average density parameter is Ω dDM = 1 . ± .
5% and itcomprises about 1/8 to 1/4 of Milky Way (MW) mass within the solar circle.
PACS numbers: 98.35.Ce, 98.35.Df, 98.35.Hj, 95.35.+d
I. INTRODUCTION
A standard astronomical method to indirectly detectdark matter in the MW disk is to find a difference be-tween kinematic determinations of the total density ofgravitating mass and estimates for the baryonic massdensity. This type of evidence for missing mass can betraced to Oort [2, 3]. The local mass density measure-ment itself, known as the “Oort limit”, was found byhim to be ρ ∼ .
15 M (cid:12) / pc , more than the observedbaryon density. Subsequent analyses were contradictory.Whereas some recovered Oort’s result [4, 5], others foundno conclusive evidence for any missing matter [6–8]. Thedebate was mostly considered resolved with the analy-sis of the Hipparcos data [9], giving ρ total ∼ . ± .
01 M (cid:12) / pc , compared with ρ baryon ∼ .
095 M (cid:12) / pc .The difference is consistent with the small amountof dark matter expected from the “halo” dark matter(hDM) component. Extrapolating the dark matter den-sity from z = 1 − ρ hDM = 0 . ± .
003 M (cid:12) / pc [10]. Similarly, a standard sphericallysymmetric NFW profile that would fit the rotation curveat the solar galactic radius gives ρ hDM = 0 . (cid:12) / pc [11]. Thus, measurements of the different densities atthe plane leave little room for an appreciable “disk” darkmatter (dDM) component.Measurements of the column densities leave more roomto hide dDM, but are still consistent with no dDM atall. Typical results for the total column density includeΣ . = 74 ± (cid:12) / pc [12], Σ . = 74 +25 − M (cid:12) / pc [13] and Σ . = 71 ± (cid:12) / pc [8]. On the other hand,different estimates for the total baryon column densityrange between 50 to 60 M (cid:12) / pc [10, 12, 14, 15]. Since ρ hDM ∼ .
008 M (cid:12) / pc corresponds to Σ hDM , . ∼
18 M (cid:12) / pc , there is little room for additional dDM.There are however two caveats. First, it was shownthat kinematic determinations of the density at the MWplane suffer from systematic uncertainties due to the ex-pected perturbation by spiral arm passages [16]. Because the density increase associated with the interstellar gasis abrupt, stars with a relatively small vertical oscillation( (cid:46)
100 pc) cannot adjust “adiabatically” to the changedpotential such that the whole stellar distribution devel-ops “ringing” motion which can systematically distortthe inferred mass density. The apparent contraction ofthe stars in the solar vicinity towards the plane is a sig-nature of this effect [16]. Without the constraint of ref.[9], a local disk of dark matter cannot be ruled.Moreover, estimates for the total baryonic column den-sity is obtained using a vertical potential which neglectsthe existence of excess dark matter in the disk. How-ever, by introducing dark matter, the vertical potentialis deeper such that the total baryonic column density in-ferred from observations and modeling is smaller, leavingmore room for Dark Matter, as recently pointed out [17],and as borne also in the analysis below.With the above caveats considered, there is significantroom for excess dark matter at the MW disk, with a col-umn density of (cid:46)
20 M (cid:12) / pc , as also pointed out by [17].It does not prove that a disk exists, since without reli-able density measurement at the plane and sufficientlylarge uncertainties in the column densities, a no dDMsolution is still possible. However, it becomes inconsis-tent once the paleoclimate data [1] is considered. Be-low, we also show that other Massive Compact Disk
Ob-jects (“Macdos”) are inconsistent implying that it cannotbe an unseen baryonic component or gravitationally col-lapsed Dark Matter.We begin in § II with building a self-consistent modelof the vertical structure of the MW. We continue in § IIIwith a discussion of the disk stability to self gravity andin § IV with the implications. In § V we show that alterna-tive explanations to the paleoclimatic data and the dDMare not plausible, and then end with a discussion on theimplications to dark matter and a summary in § VI. a r X i v : . [ a s t r o - ph . GA ] J un II. MODEL FOR VERTICAL STRUCTURE
We follow the standard methodology and approxima-tion to solve for the vertical dependence of the vari-ous mass components and the gravitational potential,e.g., refs. [9, 15]. We assume that each componentfollows a thermal equilibrium distribution of the form ρ i = ρ ,i exp (cid:0) − Φ( z ) /σ z,i (cid:1) . We note however, like ref.[17], that some of the components have their local den-sity determined from observations, while others, in par-ticular the interstellar gas, have their column densitydetermined. The values themselves are taken from ref.[15]. We also add to the model a standard hDM com-ponent with a constant background density of 0 . ± .
005 M (cid:12) / pc [10] (i.e., we solve for 0.005, 0.008 and0 .
011 M (cid:12) / pc ). We also add a dDM component with agiven ρ , dDM at the plane and vertical dispersion σ z, dDM .For the gravitational potential we neglect the rotationcurve term [10], in which case we have K z = − ∂ Φ ∂z , and Σ( z ) = (cid:90) z − z ρ ( z (cid:48) ) dz (cid:48) = | K z | πG . (1)To solve a model, the density at the MW plane isfirst guessed for the components with observational con-straints on the column densities. The vertical galacticpotential can then be integrated, giving the total columndensities of all the different components, including thosewith a fixed column density. The densities of the lat-ter can then be iterated for until their integrated columndensity agrees with the observational constraints.With a given model, we can plot the total baryonicand gravitating column densities (e.g., up to 1.1 kpc), thecolumn density and dDM density at the Galactic plane,as depicted in fig. 1. III. DISK STABILITY
In addition to the above observational considerationson the densities and column densities, it is interestingto analyze the stability of the dark matter disk to theeffects of self gravity. Toomre found a criterion to theinstability of local axisymmetric disturbances of a thindisk of collisionless particles [18], such as stars, whichshould pertain to dark matter particles as well. If wedefine Q ≡ σ R κ/ . G Σ, with σ U being the radial veloc-ity dispersion and κ the radial epicyclic frequency, then Q < Q crit = 1 is a necessary condition for instability.The disk can be unstable to non-axisymmetric pertur-bations (i.e., to bars and spirals) for somewhat smallerdensities, i.e., to Q crit ∼ . − .
5. Disks which are un-stable, with
Q <
1, will generate sufficient waves to kine-matically heat up the disk, thus increasing Q to “stable”values. We therefore expect Q (cid:38) Q for the solar neighborhood. First, the local mass is het-erogeneous. Not only is there an important contributionfrom gas, the stellar component can be described as a combination of different populations with different kine-matic characteristics. The stability criterion can then bewritten as [19]2 πGk Σ g κ + k c g + 2 πGkκ n (cid:88) j =1 Σ j Ψ j > , (2)with c g being the sound speed of the gas. Also,Ψ j = 1 − exp( − k σ j /κ ) I ( k σ j /κ ) k σ j /κ , (3)and I is the zeroth Bessel function.The second modification is the effects of thick disks.This is important because one components’ most unsta-ble wavelength could be small compared with the scaleheight of another component. Although there is no exactsolution to this problem, we can apply the useful reduc-tion factor ansatz of ref. [20], giving the criterion:2 πGk Σ g κ + k c g + 2 πGkκ n (cid:88) j =1 Σ j Ψ j kh j > , (4)with h j = Σ / (2 ρ ,j ) being the effective scale height ofcomponent j . Thus, given a model solution, we cancalculate the value of Q . This is plotted in red infig. 1. Evidently, dDM disks which are denser than about0 . (cid:12) / pc at the plane are unstable. Note that we as-sume the radial and vertical dispersion of the dDM arethe same. For stars, σ W ∼ σ U / IV. RESULTS AND IMPLICATIONS TO DDM
The model results are plotted in fig. 1. The differentconstraints are denoted by the shaded regions. For the to-tal mass we take a range of 71 ± (cid:12) / pc as determinedby ref. [12]. For the effective density we take the paleo-climate determinations [1]. Several consequences can bereached. First, a no dDM solution is permissible only ifthe background density of hDM is on the large side andthe paleoclimate data is discarded. If it is accepted, how-ever, then only hDM densities of (cid:46) . (cid:12) / pc can pro-vide solutions satisfying all the constraints, which span ρ dDM = 0 . ± .
03 M (cid:12) / pc . It also has a surface densityΣ dDM = 15 ± (cid:12) / pc and a vertical velocity dispersionof σ W, dDM = 8 . ± . V. ALTERNATIVE EXPLANATIONS?
The conclusion that dDM exists rests on several as-sumptions. As mentioned above, the first is that the ρ eff = M ⊙ / pc ρ eff = M ⊙ / pc Σ tot = M ⊙ / pc Σ tot = M ⊙ / pc Σ dDM = M ⊙ / pc Σ dDM = M ⊙ / pc Σ b = M ⊙ / pc Σ b = M ⊙ / pc = = ρ hDM = M ⊙ / pc - - - - ( ρ dDM /[ M ⊙ / pc ]) Log ( σ d D M /[ k m / s ] ) ρ hDM = M ⊙ / pc - - - - ( ρ dDM /[ M ⊙ / pc ]) Log ( σ d D M /[ k m / s ] ) ρ hDM = M ⊙ / pc - - - - ( ρ dDM /[ M ⊙ / pc ]) Log ( σ d D M /[ k m / s ] ) FIG. 1: Model solutions assuming ρ hDM = 0 .
008 M (cid:12) / pc (large panel), ρ hDM = 0 .
005 M (cid:12) / pc (top right) and ρ hDM =0 .
011 M (cid:12) / pc (bottom right). Large panel includes contour levels of Σ b (dashed gray), Σ tot (solid green, with the green shadedregion denoting observational constraints), Σ dDM (dashed blue) and the equivalent constant density ρ eff needed to give theobserved 32 Mr oscillation seen in the geological data (solid gray, and shaded region denoting the paleoclimatic constraint) aregiven as a function of the ρ , dDM and σ dDM . The red contours denote the Toomre Q value. A region near ρ , dDM ∼ . (cid:12) / pc and σ dDM ∼ / s satisfies all constraints. The small plots are abridged and only include observationally constrained regions. paleoclimate data is due to the vertical motion of the so-lar system. It requires that the discrepancy between thelocally measured baryon density and the effective totaldensity measured over 550 Ma is not due to large densityvariations. Last, it assumes that the unexplained com-ponent is non-baryonic and not, for example, massivecompact disk objects of baryonic origin.The statistical significance of the paleoclimate signalwas discussed in ref. [1]. It is clear beyond any doubtthat the periodic signal exists (at 17 σ ). If it is not dueto the vertical motion it must be some other regular sig-nal (e.g., due to some unknown very long interaction inthe planetary interactions), which coincidentally has thecorrect phase to be the vertical motion (1 in 6 probabil-ity) and a secondary frequency modulation with a correctphase and period to mimic the radial epicyclic motion ofthe solar system (1 in 60 probability).If the baryon / effective density inconsistency is dueto density variations, then the average ISM gas densityhas to be large by a factor of a few more than the localdensity. However, such large variations in the density (oforder a factor of 2) should leave a fingerprint in the pale-oclimate data in the form of cycle to cycle jumps that areof order 1 / √
2, or about 20 Ma. Fig. 2 replots fig. 4 of ref.[1], which is the detrended paleotemperature proxy datafolded over the 32 Ma period. The difference is that timeis here distorted to remove the radial epicyclic motion bydefining a “distorted” time ˜ t as d ˜ t = dt/ (cid:112) ρ ( R ( t )) /ρ , with ρ ( R ( t )) being the modeled density in the plane atthe modeled R ( t ), due to the radial epicyclic motion.Since some of the apparent variations can be due to ag-ing errors (whether measurement or additional climatevariations), some of the typically (cid:46) δρ/ρ (cid:46) .
3, which cannotexplain the large discrepancy between baryon and pale-oclimate measurements.Although measurement of the missing gravitationalcomponent does not provide any direct indication to itsnature, the fact that it can keep itself kinematically coldimplies that it can self interact. This is apparent from thesmall vertical velocity dispersion, of 8 . ± . VI. DISCUSSION AND SUMMARY
The main argument against the existence of a dDMcomponent is the small difference between the measuredbaryonic density and kinematic determination of the lo-cal mass density. However, spiral arm passages distortthe inferred mass density by O (1). On the other hand,paleoclimate data indicates that the local mass density × ×× × ×× × × × × × × × ×× × ×× × ×× × × × × × × × ×× δ O ⩽ - ⩾ [ t,32 Ma ] +
32 Ma shifted copy t [ M a ] FIG. 2: The same as fig. 4 of ref. [1], except that time isdistorted to remove the radial epicyclic motion described inthe text. The vertical axis spans the Phanerozoic. The hori-zontal axis is the modified time folded over a 32 Ma period.For convenience, the horizontal axis shows two 32 Ma peri-ods. The blue and red circles (connected by dashed lines) arethe modeled plane crossings (blue) and the maximal excur-sions from the plane (red), respectively. The disk radii andcolor correspond to the detrended and high pass filtered δ Osignal, as given by the scale on the right (in (cid:104) ). is about twice larger than the local baryonic matter, im-plying that a cooling dDM component should be present,since it cannot be consistently explained otherwise. Thefact that the dDM cools down to form a disk is relevantfor two major reasons.First, if a cooling dDM component exists and can over-come the heating from viscous stirring, then the predic-tion should be that it cooled down to a velocity disper-sion which is marginally stable, i.e., Toomre’s Q (cid:38) Q ∼ ρ ∼ . . Namely, the paleoclimaticmeasurement recovers the theoretical prediction.Allowing the dDM to cool requires that the cooling re-action rate is several time faster than the Hubble rate. For example, we can consider cooling through a reac-tion 2 d → d + (cid:96) , where d is a dDM particle and (cid:96) isa light DM particle required to take the kinetic energy,then nσ dd(cid:96) v (cid:38) N H where n is the number density ofdDM before the disk cools down (assuming it is formed“puffed”), v is the typical Keplerian velocity which char-acterizes the typical random component that a puffed updDM halo would have. N is the typical number of in-teractions required for the cooling to take place. Taking v ≈ (cid:112) GM MW (cid:12) /r (cid:12) with M MW (cid:12) the amount of masswithin our galactic radius r (cid:12) , then one finds that dDMat r (cid:12) can cool if σ dd(cid:96) m d (cid:38) π N α Hr / (cid:12) G / M / MW (cid:12) ≈ N α . cm gr . (5)Here we assumed that a fraction α of the DM mass is inthe cooling component. For α ∼ . N ∼
10 we findthat the cross section should satisfy σ dd/m H (cid:38) / g.Another cooling reaction could be inverse-Comptonlike cooling, through d + (cid:96) → d + (cid:96) . However it mayrequire a relic (cid:96) background which on one hand has to becold enough as to not leave a Baryon-like Acoustic Oscil-lation in the cosmic microwave background [22], but nottoo cold to leave a negligible background on which the dDM cannot cool.Last, we note that a kinematically cold and dense disk,which could periodically perturb the Oort cloud (andcause mass extinctions) is also ruled out as it would bekinematically unstable. It would develop horizontal per-turbations which would quickly heat the disk. It cannotform collapsed objects (“dark stars”), as those will thenhave a Hubble time to heat to (cid:38)
25 km/s.Many thanks Yoram Lithwick, Erik Kuflik, YonitHochberg, Kris Sigurdson, James Owen, and ScottTremaine for fruitful discussions. This research projectwas supported by the I-CORE Program of the Planningand Budgeting Committee and the Israel Science Foun-dation (center 1829/12) and by ISF grant no. 1423/15. [1] N. J. Shaviv, A. Prokoph, and J. Veizer, Scientific Re-ports , 6150 (2014).[2] J. H. Oort, Bull. Astron. Inst. Netherlands , 249 (1932).[3] J. H. Oort, Bull. Astron. Inst. Netherlands , 45 (1960).[4] J. N. Bahcall, Astrophys. J. , 926 (1984).[5] J. N. Bahcall, C. Flynn, and A. Gould, Astrophys. J. , 234 (1992).[6] O. Bienayme, A. C. Robin, and M. Creze, Astron. Astro-phys. , 94 (1987).[7] K. Kuijken and G. Gilmore, MNRAS , 651 (1989).[8] K. Kuijken and G. Gilmore, ApJ , L9 (1991).[9] J. Holmberg and C. Flynn, Mon. Not. Roy. Astro. Soc. , 209 (2000).[10] J. Bovy and S. Tremaine, Astrophys. J. , 89 (2012).[11] R. P. Olling and M. R. Merrifield, MNRAS , 164(2001), astro-ph/0104465. [12] J. Holmberg and C. Flynn, MNRAS , 440 (2004),arXiv:astro-ph/0405155.[13] A. Siebert, O. Bienaym´e, and C. Soubiran, Astron. As-trophys. , 531 (2003), astro-ph/0211328.[14] C. Flynn et al., MNRAS , 1149 (2006).[15] C. F. McKee, A. Parravano, and D. J. Hollenbach, As-trophys. J. , 13 (2015), 1509.05334.[16] N. J. Shaviv, ArXiv e-prints (2016), 1606.02595.[17] E. D. Kramer and L. Randall, ArXiv e-prints (2016),1604.01407.[18] A. Toomre, Astrophys. J. , 1217 (1964).[19] R. R. Rafikov, MNRAS , 445 (2001).[20] A. B. Romeo, MNRAS , 307 (1992).[21] B. Nordstr¨om, Proc. IAU , 31 (2008).[22] F.-Y. Cyr-Racine et al., Phys. Rev. D89