Intrinsic Nature of Dark Matter in the Galactic Halo

Ugur Camci Department of Chemistry and Physics, Roger Williams University, One Old Ferry Road, Bristol, RI 02809, USA; ucamci@rwu.edu; ugurcamci@gmail.com Version August 10, 2020 submitted to Entropy; Typeset by LATEX using class file mdpi.cls Abstract: We obtain more straightforwardly the main intrinsic features of dark matter distribution 1 in the halos of galaxies by considering the spherically symmetric space-time, which satisfies the 2 flat rotational curve condition, and the geometric equation of state resulting from the modified 3 gravity theory. In order to measure the equation of state for dark matter in the galactic halo, we 4 provide a general formalism taking into account the modified f (X) gravity theories. Here, f (X) 5 is a general function of X ∈ {R,G, T}, where R,G and T are the Ricci scalar, the Gauss-Bonnet 6 scalar and the torsion scalar, respectively. These theories yield that the flat rotation curves appear 7 as a consequence of the additional geometric structure accommodated by those of modified gravity 8 theories. Constructing a geometric equation of state wX ≡ pX /ρX and inspiring by some values of 9 the equation of state for the ordinary matter, we infer the properties of dark matter in galactic halos 10 of galaxies. 11


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In order to obtain results which are relevant to the galactic dynamics, we assume that the galactic 69 halo has spherical symmetry and that dragging effects on material particles (stars and dust) are 70 inappreciable. Therefore, we restrict our study to the static and spherically symmetric metric. The 71 most general static and spherically symmetric metric can be written as 72 ds 2 = −A(r)dt 2 + B(r)dr 2 + dΩ 2 , where dΩ 2 = C(r) dθ 2 + sin 2 θdϕ 2 and (r,θ,ϕ) are the spherical coordinates. Then, the 73 equations of motion for a test particle in the space-time (1) can be derived from the Lagrangian 74 2L = −A(r)ṫ 2 + B(r)ṙ 2 + C(r) θ 2 + sin θ 2φ2 , where a dot means derivative with respect to the proper time. From the above Lagrangian (2), the generalized momenta become Version August 10, 2020 submitted to Entropy

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where E is the total energy of a test particle and L i are the components of its angular momentum. 75 Using L 2 = L 2 θ + L 2 ϕ / sin 2 θ which is the first integral corresponding to the squared total angular 76 momentum, the norm of the four-velocity (g µν u µ u ν = −1) yields where V(r) ≡ A(1 + L 2 /C) is the effective potential. Thus, the conditions for circular orbits 78ṙ = 0 and ∂ r V(r) = 0 lead to The definition of tangential velocity of a test particle is given by [12] Then, using the constants of motion (5), it follows that the tangential velocity of this test particle which has the form v 2 tg = rA 2A for C(r) = r 2 , where a prime represents derivative with respect to 83 r. In the flat rotation curves region, where v tg ≈ constant, integration of Eq. (7) gives which takes the form A(r) = A 0 r for C(r) = r 2 (A 0 is an integration constant and = 2v 2 tg ). Thus,

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The generic action that we will consider for modified gravity theories reads where κ 2 = 8πG N is the standard gravitational coupling, g = det(g µν ), L m is the ordinary matter 90 Lagrangian and the function f depends on X ∈ {R, G, T, ...}. Hereafter, in order to obtain effects of the 91 DM in the halos of galaxies, we have assumed that there exists no the luminous matter, i.e. L m = 0.

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Before obtaining the field equations by varying the Lagrangian according to the coefficients of metric

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(1), we note the fact that the metric variable B does not contributes to dynamics by using a point-like where F i are the functions of the metric coefficients q i = {A, B, C} and their first and/or second 100 derivatives, ρ X (q i , q i , X, X , X ) and p X (q i , q i , X, X , X ) are the density and pressure for the modified 101 gravity theory, respectively. Then, inspiring by some values for the EoS parameter used in the 102 ordinary matter, the geometric EoS defined by w X ≡ p X /ρ X has capacity to inform us about the kind After varying (13) with respect to A, B and C, the field equations have the form of (10)-(12), where ρ R and p R are defined by where For an exact solution of the power law gravity 114 f (R) = f 0 R n , the metric coefficients are given by . Therefore, the tangential velocity for this case becomes due to the relation = 2v 2 tg . Here, the density and pressure of R n gravity have the form where ρ 0 and p 0 are found as The relation (19) and the form of ρ R and p R explicitly represent that the constant tangential 119 velocity v tg , the density ρ R and pressure p R vanish as n = 1, which is the GR case, that is, it 120 can be concluded that the GR does not give any information about the DM dominated region of 121 galaxies. Furthermore, the geometric EoS w R = p R /ρ R is constant, and found to be where L f (R) is as given by (13). Then, solving the field equations obtained from the above where f 1 = −2 f 0 2 ( − 2)/( 2 − 2 + 8), and the density and pressure are with the geometric EoS w RG = −1 − 4/ which reduces to w RG = −1 − 2v −2 tg since the relation 130 = 2v 2 tg .

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• f (T) gravity: In this gravity theory, the gravitational contributions to the metric tensor 132 become a source of torsion T rather than curvature R, i.e. the connection considered in this 133 theory of gravity is distinct from the regular Levi-Civita connection which is replaced with its 134 Weitzenböck analog. In order to be consistent with the study [16], we will use the signature 135 (+, −, −, −) for spherically symmetric metric which has the form ds 2 = A(r) 2 dt 2 − B(r) 2 dr 2 − 136 C(r) 2 (dθ 2 + sin θ 2 dϕ 2 ). The Lagrangian L f (T) for the latter metric is given by After varying the above Lagrangian with respect to A, B and C, the field equations for power-law f (T) = f 0 T n gravity can be written in the form of (10)-(12) as follows Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 12 August 2020 doi:10.20944/preprints202008.0274.v1 where ρ T and p T are defined as which are the torsion contributions to energy density and pressure. If n = 1, then the theory 141 becomes the Teleparallel Equivalent of General Relativity (TEGR), and the energy density and 142 pressure given by (29) vanish, just as expected. Using the field equations for f (T) = f 0 T n 143 gravity, where n = 0, 1, 1 2 , 5 6 , 5 4 , 3 2 , it is reported the following exact solution in Ref. [16] where = 4n(n − 1)(2n − 3)/(4n 2 − 8n + 5). Here it is found the relation = v 2 tg because of 145 the form of metric coefficients in this case, which yields that the tangential velocity has the form For this solution, the density and pressure of T n gravity have the form where ρ 1 and p 1 are obtained as which gives a constant geometric EoS w T = p 1 /ρ 1 as (34)  All test particles in stable circular motion move at the speed of light when v tg = 1, but this 161 gives rise to a contradiction by observations at the galactic scale. Furthermore, the tangential velocity 162 v tg tends to zero in the limit of large r. Thus, the tangential velocity v tg has to be at the interval 163 0 < v tg < 1. During early epochs of the universe, the DM velocity is not so small than the speed of 164 light, for example the relative velocity is v rel ∝ 0.3c at freeze-out epoch [17]. While at later times such 165 as the DM halos today and during the recombination epoch, the DM velocity is very smaller. For

Discussions and conclusions
we can exactly calculate n for some specific tangential velocities. For instance, if v tg = 10 −3 , 174 which is the rotational velocity of spiral galaxies, then we find n = 1.000001 or n = 0.4999985.

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Defining the polynomial discriminant D = Q 2 + P 3 , we can solve algebraically the latter cubic 204 equation. If D > 0, one of the roots is real and the other two roots are complex conjugates.