A universal constant for dark matter-baryon interplay

Recent studies point out that there exists some rough scaling relations for dark matter and some tight connections between dark matter and baryons. However, most of the relations and tight connections can only be found in galaxies, but not in galaxy clusters. In this article, we consider a new expression that can characterize the properties of dark matter-baryon interplay for both galactic and galaxy cluster scales. By using the archival observational data of galaxies and galaxy clusters, we show that the value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{K}}{\boldsymbol{=}}{\bar{{\boldsymbol{n}}}}_{{\boldsymbol{D}}}{\bar{{\boldsymbol{n}}}}_{{\boldsymbol{B}}}{{\boldsymbol{r}}}_{{\boldsymbol{o}}}{\boldsymbol{V}}{\boldsymbol{/}}{{\boldsymbol{v}}}^{{\bf{4}}}$$\end{document}K=n¯Dn¯BroV/v4 is almost a constant and scale independent within the optical radius ro, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{{\boldsymbol{n}}}}_{{\boldsymbol{D}}}$$\end{document}n¯D is the average dark matter number density, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{{\boldsymbol{n}}}}_{{\boldsymbol{B}}}$$\end{document}n¯B is the average baryon number density, v is the characteristic velocity and V is the interacting volume. This would be the first universal relation between dark matter and baryons on both galactic and galaxy cluster scales. We anticipate this result to be a starting point to explain the small-scale problem and the scaling relations for dark matter in galaxies. The constant K discovered may reveal some underlying global interaction between dark matter and baryons.


Method
The effect of dark matter-baryon interplay inside an 'interacting region' would depend on the average dark matter number density n D , average baryon number density n B and the volume of the interacting region V which depends on the optical radius r o . Since most of the galaxies are disk-dominated at r o , the enclosed interacting volume would be a disk-like thin cylinder with radius r o (see Fig. 1). Consider the following product (unit: cm −2 ) , v is the characteristic velocity, m D is the dark matter mass and m B is the average mass of a baryonic particle. Here, we have assumed that all dark matter can interact with baryons. The potential interaction can be gravitational only, or including some other forms of interactions such as scattering.
the constant for galaxies. For 19 , where v′ is the asymptotic circular velocity. Since most of the galactic rotation curves rise to a constant v′ quickly within r o , we assume that the value of v′ is equal to the characteristic velocity v of a galaxy.
Besides, the optical radius r o is closely related to the scale radius of dark matter r s . By using the data of the local volume catalog (LVC) dwarf galaxies 3,30 , a very strong correlation between r s and r o is found (see Fig. 2). Although the relation is not a perfect linear relation, their ratio is close to a constant r s = (2.59 ± 0.55)r o . Since 4 4 4 , the term v in Eq. (1) is canceled. Furthermore, putting r s = (2.59 ± 0.55)r o into Eq. (1) and taking m B = 1.2 m p , where m p is the proton mass, we can calculate the value and the uncertainty of K: On the other hand, the product K can also be understood as follow.
Therefore, K is directly proportional to the ratio of the baryonic mass and the enclosed dynamical mass at radius r o . Note that this ratio is not necessarily equal to the cosmic baryon fraction f = 0.156 ± 0.003 31 . For different sizes and types of galaxies, it is possible to have entirely different ratios at radii r o (see the discussion below). Furthermore, the above relation is obtained by using the Tully-Fisher relation, which is formulated based on the data of spiral galaxies. For elliptical galaxies, many of them satisfy another relation called the Faber-Jackson www.nature.com/scientificreports www.nature.com/scientificreports/ relation 32 . This relation indicates a close relationship between the luminosity L and the stellar velocity dispersion v b of elliptical galaxies: ∝ L v b 4 , which is similar to the form of the Tully-Fisher relation. Recent studies show that the data of some elliptical galaxies also fall on the spiral baryonic Tully-Fisher relation if one assumes a certain value of mass-to-luminosity ratio 33 . Therefore, the above result may also be applicable for elliptical galaxies.
the constant for galaxy clusters. The above deduction method can only be applied for galaxies. It is because there is no Tully-Fisher relation for galaxy clusters. Also, the product ρ r D s is not a constant for galaxy clusters 4 . Fortunately, we can relate the velocity of baryons v′ by the temperature T of hot gas in galaxy clusters, which can be given by Virial relation ≈ v kT m 3 / p . The baryonic component in a large galaxy cluster is dominated by the spherical hot gas halo within the core radius of the hot gas r c . Therefore, we consider the spherical interacting region inside r = r o = r c so that π = V r 4 /3 c 3 . Besides, the hot gas is in hydrostatic equilibrium so that the total mass can be determined 34 : p is the pressure of hot gas, μ = 0.61 is the molecular weight and M(r) is the total enclosed mass. Recent studies show that the x-ray hydrostatic mass measurements are remarkably robust and method-independent 35 . The overall percentage error of mass estimation is about 10-30% if we assume that T is a constant 34,36 . The hot gas surface brightness profile can be determined by x-ray observations. It is usually described by a β-model 34 : where S 0 is the central surface brightness, r c is the core radius and β is a fitted parameter. These parameters can be used to construct the density profile of hot gas 36 : where ρ B0 is the central baryon density. Combining Eqs (4) and (6), we get The total mass density can be obtained by Taking r → 0, we can get the central total mass density ρ t0 . Since dark matter dominates the mass in a galaxy cluster, the central dark matter density is close to the central mass density: www.nature.com/scientificreports www.nature.com/scientificreports/ Note that for most of the galaxy clusters, the density profiles are close to the NFW profile (a cuspy profile). Here, the density ρ D0 represents the average dark matter density for r ≤ r c (i.e. ρ ρ ≈ D D 0 ). Using Eq. (9), the value of ρ D0 for each galaxy cluster can be calculated by the parameters T, β and r c , including their uncertainties.
Putting all the above relations to Eq. (1), the term v and r c would be cancelled naturally. Therefore, we get

Result
For galaxies. In many dark matter-dominated galaxies, the mass density of dark matter for r ≤ r s is close to constant (cored structure: ρ ρ ≈ D D 0 ) 10,37 . Also, recent analyses suggest that the central dark matter column density (the product of the central dark matter mass density and the scale radius) is almost a constant. Early analysis using ∼1000 galaxies gives ρ = 2 (see Fig. 3). Therefore, using this result, we get The narrow range of K (or the ratio of baryonic mass to the enclosed dynamical mass) for galaxies has been known for a decade. The almost constant central dark matter column density for a wide range of galaxies 1-3 and the constant luminous-to-dark matter ratio within one halo scale-length 38 suggest the narrow range of K. Therefore, the almost constant value of K for galaxies is expected.
For galaxy clusters. By using the x-ray data of 64 large galaxy clusters (with r c ≥ 100 kpc) 34 , we get ρ ρ ∝ . ± . is roughly a constant. This correlation has not been discovered and discussed in previous studies. Although the average dark matter density to average baryonic density ratio for a galaxy cluster should be close to ≈ . , the ratio at the central region for all galaxy clusters may not be a constant and equal to f −1 because the density profiles for dark matter and hot gas are entirely different (see the discussion below). This approximately constant ratio of ρ D0 to ρ B0 at r c may reveal some global interplay between dark matter and baryons.
Although different galaxy clusters have different values of β 34 , we approximate the distribution of β by a Gaussian function with an average value β = 0.65 and a dispersion of 0.13 (see Fig. 5). Putting the relation of ρ D0 and ρ B0 to Eq. (10) and using β = 0.65 ± 0.13, we get a scale invariant constant for galaxy clusters:  where ρ ρ ρ ρ ≈ / / D B D B 0 0 . We can see that the values of K for galaxies and galaxy clusters give excellent agreement with each other, within a factor of 2-3. This constant can be regarded as a universal constant for dark matter-baryon interplay because it is independent of scale.
Here, note that the x-ray data used have assumed the hubble parameter h = 0.5 34 . To match the current observed hubble parameter h = 0.68 31 , we have re-scaled the parameters r c and ρ B0 . Besides, using the data of large galaxy clusters can minimize the errors because the uncertainties of the central hot gas temperature significantly affect the mass determination of small galaxy clusters. Therefore, we only choose the large galaxy clusters (with r c ≥ 100 kpc) to perform the analysis. the relation between the constant and the cosmic baryon fraction. As mentioned above, the value of K is directly proportional to the ratio M B /M D (r o ) (for galaxies) or ρ B0/ ρ D0 (for galaxy clusters). One may claim that it is not surprising to have a similar constant value of K for galaxies and galaxy clusters because M B /M D (r o ) or ρ B0/ ρ D0 should be close to the cosmic baryon fraction f = 0.156 ± 0.003, which should be almost a constant for various structures. However, previous studies show that the baryon fractions in various structures are entirely different and the ratios quite depend on the positions of the structures 39,40 . In Fig. 6, we follow the study in 39 and show the corresponding baryon fractions for different structures (from dwarf galaxies to galaxy clusters). We can see that the ratio can be ranging from 10 −3 to 10 −1 , which spans 3 orders of magnitude.
If we assume the NFW profile for dark matter distribution in galaxy clusters and fix the baryon fraction f = M B /(M B + M D ) = 0.156 at a large radius r = 1 Mpc, we can predict the baryon fraction at the characteristic radius r s for various total dynamical mass using hydrostatic equilibrium. In Fig. 7, we can see that the baryon  www.nature.com/scientificreports www.nature.com/scientificreports/ fraction at r s = 100 kpc ranges from 10 −3 to 10 −1 (not close to f = 0.156 ± 0.003) for just a short range of dynamical mass if we consider a fix hot gas temperature (T = 5 keV) and scale radius of dark matter r s = 100 kpc. For various temperature of hot gas and scale radii of different galaxy clusters, the variation of the baryon fraction can be much larger. This large variation can also be seen in galaxies by examining the radial acceleration relation 23 . Therefore, theoretically, the baryon-dark matter density ratio is not necessarily a constant for various structures, but possibly spans at least 2 orders of magnitude. Furthermore, the characteristic radii of the galaxies (∼1-10 kpc) and galaxy clusters considered (∼100-500) kpc are different. There is no reason why the baryon fractions for galaxies and galaxy clusters are close to each other at different characteristic radii. Nevertheless, our results show that this ratio is almost constant at the characteristic radii r o or r c . Based on the result of Fig. 7, it seems that gravitational interaction between dark matter and baryonic matter alone is not enough to provide a satisfactory explanation for the constant value of K. This may reveal the existence of some interplay or self-organizing processes between baryons and dark matter particles within the 'interacting region' .

Discussion
Previous studies have claimed some relations for dark matter. For example, as mentioned above, the central dark matter column density ρ D0 r s is almost a constant for a wide range of galaxies 3 . Some explanations have been given to account for this relation, such as the self-interacting dark matter model 17,41 . However, this relation is not correct for galaxy clusters 4 . Therefore, such scaling relation is not universal at all. Besides, the alleged universal NFW density profile for dark matter only gives good agreement with the data of galaxy clusters 6 and some galaxies 7,8 , but not for dwarf galaxies 9,10 . Therefore, it does not have any universal relation for dark matter which is true for both galactic and galaxy cluster scales. Our result, K is a constant, would be the first universal relation of dark matter for both galactic and galaxy cluster scales. Although this constant is proportional to the ratio of baryon fraction in galaxies and galaxy clusters at the optical radii or core radii, it is not necessarily a constant. As shown in Fig. 6, the baryon fractions can span 3 orders of magnitude variation for different structures. Therefore, our results suggest  Figure 7. The graph of baryon fraction within r s versus total dynamical mass for galaxy clusters, assuming NFW density profile for dark matter and in hydrostatic equilibrium with hot gas. Here, we fix the hot gas temperature and scale radius of dark matter as 5 keV and r s = 100 kpc respectively. www.nature.com/scientificreports www.nature.com/scientificreports/ that some interplay between baryons and dark matter particles may exist so that the constant K or the baryon fraction is almost a constant for galaxies and galaxy clusters within the 'interacting region' .
Note that the choice of the characteristic radius (r o or r c ) in this analysis is not arbitrary. The overlapping regions of dark matter and baryons for galaxies and galaxy clusters are characterized by the optical radii r o and the core radii r c respectively. The size of the interacting volume V also depends on r o or r c . If we consider the scale radius of dark matter r s as the characteristic radius, the volume considered would be much larger, which includes some non-interacting regions. Therefore, it is justified to use r o or r c as the characteristic radius for this analysis.
Although the range of baryon fraction can be very large, in this analysis, we mainly examine the value of K (proportional to the central baryon fraction) within the 'interacting region' (i.e. within r o or r c ). As mentioned above, the narrow range of K in the central regions of galaxies has been known for a decade 38 . However, the narrow range of K in the core regions of galaxy clusters has not been discussed extensively. Many studies only focus on the total baryon fraction for the whole galaxy clusters 40,42 , not for the central regions. Some studies have examined the central baryon fraction of some galaxy clusters 43,44 . However, no extensive discussion about the range of the central baryon fraction has been made. Besides, some discrepancies exist between numerical simulations and observational data for galaxy clusters near the central regions. Previous studies using computer simulations suggest a very wide range of baryon fraction for different radii 45 . However, the simulated baryon fraction is somewhat smaller than that of the observed data [43][44][45] . For r c = 100-500 kpc, the corresponding baryon fraction should be ∼0.02-0.1 based on the simulated results 45 . This gives ρ D0/ ρ B0 ∼ 10-50. Nevertheless, our results suggest a narrower and smaller ratio ρ D0/ ρ B0 = 8.70 ± 3.42 (or a larger value of K) compared with the simulated results, which generally agree with the observed ranges of the central baryon fraction in 43,44 . We explicitly analyze the baryon fraction distribution at core radii of galaxy clusters and quantify the results to get the value of K. It is surprising to see the value of K for galaxy clusters being close to that for galaxies within the 'interacting region' , which has not been discussed and discovered in the early works. Therefore, our analysis gives some new hints to investigate the alleged interplay between dark matter and baryons.
It has been suggested that dark matter-bar yon interaction can explain the obser ved 'mass-discrepancy-acceleration relation' 46 . This relation is closely related to the 'Tully-Fisher relation' 47 . Some recent studies also discover some correlations between dark matter and baryonic component in galaxy clusters 48 . We anticipate that the global properties of dark matter-baryon interaction can give a holistic picture to account for these relations, as well as the 'halo-disk conspiracy problem' 29 . Since the constant K links up the baryon fractions with some of the scaling relations in galaxies, it may provide a more fundamental ground to account for the constant dark matter column density (or ρ ∝ − r D s 0 1 ) and the Tully-Fisher relation. Note that the dimension of the constant K is cm −2 , which is the dimension of 1/cross section. This constant may be closely related to the cross section of the interplay between dark matter and baryons.
It seems that gravitational force only cannot provide a satisfactory account for this interplay. Since baryonic matter is self-interacting while dark matter particles are believed to be collisionless, the density profiles for baryonic matter and dark matter would be different. There is no reason why the values of K (or the baryon fraction) within different characteristic radii are close to each other. Some suggest that baryonic feedback mechanisms can account for the core formation in galaxies 12 . If baryonic feedback can have some self-organizing properties, it may be able to explain the universal nature of K.

Data Availability
The datasets generated during and/or analysed during current study are available from the corresponding author on reasonable request.