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4 where M 2 = c 2 1018 [GeV ], R is the Ricci scalar, f (φ) is the Gauss-Bonnet coupling function have dimensions p 8πG ≈ × of [length]2, that represent ultraviolet (UV) corrections to Einstein theory. In the above equation (µ,ν)=(0, 1, 2, 3). We define the Gauss-Bonnet invariant as

= R2 4R Rµν + R Rµνρσ. (2) G − µν µνρσ The variation with respect to the field φ gives us the equation of motion for the scalaron field

φ ∂ V (φ)+ ∂ f (φ)=0. (3) − φ G φ

The variation of the action over the metric gµν simplified by the Bianchi identity gives

1 1 1 1 0 = M 2 Rµν gµν R + ∂µφ∂ν φ gµν ∂ φ∂ρφ + gµν ( V (φ)+ f (φ) ) (4) p − 2 2 − 4 ρ 2 − G   f (φ) 2RRµν + Rµ Rνρ 2Rµρστ Rν +4Rµρστ Rν − ρ − ρστ ρστ + 2R µ ν 2gµν R 2 4Rνρ µ 4Rµρ ν f (φ) ∇ ∇ − ∇ − ∇ρ∇ − ∇ρ∇  +4 2f (φ) Rµν +4gµν ( f (φ)) Rρσ 4 ( f (φ)) Rµρνσ. ∇ ∇ρ∇σ − ∇ρ∇σ The metric of a spatially flat homogeneous and isotropic universe in FLRW model is given by:

3 2 ds2 = dt2 + a2(t) dxi , (5) − i=1 X  where a(t) is a dimensionless scale factor, from which we define the Ricci scalar R and the GB invariant in FLRW geometry as G

R =6 2H2 + H˙ = 24H2 H˙ + H2 . (6) G     We start by considering φ = φ (t). On the other hand, Eq.(4) is written as

1 2 ′ φ˙ + 24H3f (φ) φ˙ + V (φ)=3M 2H2, (7) 2 p ′ where φ˙ = ∂tφ, f (φ)= ∂φf (φ) and H =a/a ˙ is the Hubble parameter. The scalar term vanishes in φ˙ = 0, leads to 2 2 V (φ = constant)=3Mp H (Cosmological density). The solution Eq.(7) is then evaluated as in the form of an energy ˙ 2 2 2 2 2 equation. There are dark matter sectors: ΩDM = φ /6Mp H and the dark-energy sectors ΩDM = V (φ) /3Mp H . ′ ˙ 2 For an interaction between dark energy and dark matter sectors we have ΩI = 24Hf (φ) φ/3Mp . The fraction of dark matter is ΩDM = 1 ΩDE ΩI . It is possible, for appropriate choices of the potential and the coupling function, that competition between− these− terms leads to a minimum of the effective potential. We consider in this work the dilatonic-type:

−kφ +kφ V (φ)= V0e and f (φ)= f0e , (8) in this case we refer to the scalaron is a dilaton field. Indeed, the coupling function remain invariant under the simultaneous sign change (k, φ) ( k, φ). The equations of motion Eq.(3) is invariant under the transformation V (φ) f (φ). In what follows,→ − we shall− assume, the term f (φ) represents a second term of effective potential. In that←→ case −G the harmonic term dominates in the potential, so−G one can approximate the equation of motion to find the damping harmonic oscillation [19].

CONDITION FOR SCALARON MASS

Next, we study the mass of the scalar field which will describe the mass of dark matter. In order to find an explicit expression for scalaron mass we need to consider an effective potential in the equation of motion Eq.(3), which can be written as the Klein Gordon equation in FLRW metric as

φ¨ +3Hφ˙ + ∂ V (φ) ∂ f (φ)=0. (9) φ − G φ 3

Let us now use the above expressions in order to examine the evolution of the scalaron. Observe that the term V (φ) f (φ) acts as the effective potential for the perturbations. The effective potential [21, 22] that includes the GB term− G can be written as

V = V (φ) f (φ) . (10) eff − G We notice that the effective potential of the scalaron includes the Gauss-Bonnet coupling and the Gauss-Bonnet invariant. In other words, the Gauss-Bonnet term affect the potential structure of the scalaron; thus, the scalaron mass depends on the matter contribution. The particles of the field φ comes from the fluctuation around the minimum of the effective potential Veff (φ). Usign (8), the second derivative of the effective potential with respect to φ is

2 ∂ 2 −kφ 2 +kφ Veff = k V0e k f0e . (11) ∂φ2 − G As before the existence of a minimum in the effective potential requires = 0. More precisely, we define the effective mass as a function of the field φ G 6

2 2 ∂ m = Veff (R, φ) , (12) eff ∂φ2 φmin

where φmin as a minimum value of the scalaron φ. The effective mass is then given by

− m = k V e kφmin f e+kφmin , (13) eff 0 − G 0 p when the minimum of the scalaron is very small, the solution could be given by meff = k√V0 f0. Domain of validity of fluid description of dark matter is for m be real. One finds that has the bound − G eff G V − 0 e 2kφmin . (14) G ≤ f0 If the condition is satisfied everywhere in spacetime, then the adiabatic approximation will be good everywhere. In particular, considering the maximum value of as G

V0 −2kφmin max = e . (15) G f0

At this point we should mention that for = max, the scalaron mass will be zero. The inequality above can be written as G G

V0 −2kφmin q − 4 e , (16) ≥ 24H f0 where q = 1 H/H˙ 2 is the deceleration parameter. We mention that according to Eq.(16), even if = , the − − G Gmax asymptotic value of q depends on the model parameters V0 and f0. Furthermore the isolated points where q = qmin constitute a set of measure zero and do contribute to find meff .

DARK MATTER STABILITY VIEWPOINT

Let us describe the effective potential in an environment that surrounds the matter. The GB invariant can be greatly simplified to the matter energy density ρ [21], giving

= ρ. (17) G

Therefore, in order to satisfy Eq.(14) one must set that V0 and f0 have the same sign. We take into account this proposition, we finally obtain

− V = f e kφ ρ e2kφmin ρe2kφ , (18) eff 0 max −  4 which was previously discussed in the context of the models [21]. At the low kφ, the effective potential can be approximated by V0eff . In the high-kφ, Veff approaching to V0 exp(kφ). Our discussion of the stability in the 2 following sections will be valid for meff (φ) > 0. We can then express the mass meff as a function of the matter energy density − m2 = k2f e kφmin (ρ ρ) , (19) eff 0 max − V0 −2kφmin where we have taken into account that ρmax = f0 e . The mass of small fluctuations being an increasing function of the matter energy density. At the low matter energy density, the effective mass can be approximated 2 −3kφmin by k V0e . In the high-density region, meff approaching to 0, which corresponds to the dark matter halo around galaxies. To be more precise, the dark matter is very important in the extremities of the when ρ 0. Apart from the evolution of the scalaron that was extracted above, the other important observable is when∼ the matter energy density becomes maximum, the mass of this field will be zero. The effective pressure and density are peff = V (φ)+ f (φ) ρ and ρeff = V (φ), respectively. Then, the effective equation of state for this system is given by −

peff ρ 2k(φ−φmin) ωeff = = 1+ e , (20) ρeff − ρmax we mention that one can have parameter regions in which ωeff lies in the matter energy density and the scalaron. If V0 > 0, the pressure for this system will be negative. The system will be stable if

−2 dρeff ceff = > 0. (21) dpeff

Here ceff is the effective sound speed. This equation gives a simple prescription for computing when a given theory 2 will be stable. We calculate ceff for a fixed density,and we find ωeff = 1/ceff . We mention that for ωeff = 0, one − find ρ = ρe2k(φ φmin). Even if φ φ , one obtains the asymptotic value ρ ρ . Additionally, to assess max ∼ min ≃ max possible behaviours of the sound speed squared and ωeff , we remark that the stable regime arises in the case

2k(φ−φmin) ρmax < ρe . (22)

In this expression, k should be positive so that the theory will be stable (since φ > φmin). In case that k is negative, the mass meff correspond to the tachyonic instability. Hence, inserting ρ < ρmax into Eq.(22) we ρmax 2k(φ−φmin) result to 1 < ρ < e , it is necessary that k > 0. In the regime of large densities, we have φ > φmin. 2 If φ φmin, the effective sound speed can be greatly simplified, giving ceff > 0. In summary, this model provide≫ that the Einstein-Gauss-Bonnet dark matter corresponding to the stable regime. Now, since the right hand side involves exponential factors, we expect that the field φ evolved by the logarithmic function of the density of matter.

ROTATION CURVES FROM MOND-EGB GRAVITY

It’s well known that the orbital velocities v of planets in planetary systems and moons orbiting planets decline with 2 GM −11 3 −1 −2 distance according to Kepler’s third law [23] by this relationship v = r , with G 6, 67 10 m Kg s . In contrast, the revolve around their galaxy’s centre at equal or increasing speed over≈ a large× range of distances [24]. The rotational speeds of stars inside the galaxies do not follow the rules found in other orbital systems. A solution to this conundrum is to suppose the existence of dark matter and to assume its distribution from the galaxy’s center out to its halo [25]. The effective potential be used to determine the orbits of planets [26], also in the cosmological evolution analysis of the chameleon field, for the detection of dark energy in orbit [27]. According to the MOND theory [15], the rotational velocity of stars along the length of a galaxy is: 4 v0 = GMa0. (23) −10 −2 where a0 1, 2 10 ms and M is the mass of a crude approximation for a in the outer regions of a galaxy. The Eq.(23≈) predicts× that the rotational velocity is constant out to an infinite range and that the rotational velocity doesn’t depend on a distance scale, but on the magnitude of the acceleration a0. We suppose that the constant a0 ni more constant, after seeing the form of galaxy rotation curves in MOND [16]. We start by the changing GM v4 GM , (24) 0 → f  0  5

2 where f0 have dimensions of [length] . In this case we can write that the rotational velocity of a galaxy by

GMk − kφmin 2 2 v (r)= e ρmax ρ (r). (25) meff − p Here, ρ is is the local matter energy density and ρmax is the maximum density of the galaxy and M is the mass of the galaxy. We consider a disk of radius r and its center is the galactic center. This modification based on EGB gravity in- stead of Newton’s gravity as in the case of MOND theory, which will later introduce a relatistic term in this potential. The choice of this potential has several advantages, since it generalizes the Newtonian potential and it has a relation with the potential of MOND theory [15]. The potential above has a relativistic aspect which is related to parameters of the EGB gravity. The term 1 ρ(r) represents the relativistic part of this potential. The description of the rotation − ρmax of galaxies in the relativisticq EGB gravity is better compared to the Newtonian frame, and gives a complete dynamic in space-time. On the other hand, the MOND theory remains incomplete since it has a lack of relativistic treatment of the rotation of galaxies. Note that in the edges of the galaxy we take a low matter energy density ρ (r ) 0, yields max ∼

V (GM)1/2 (ρ )1/4 θ , (26) max ∼ max 0 2 − kφmin where θ0 = ke 2 /meff . Note that the parameter θ0 strongly affects the behavior of the velocity Vmax. The galaxy’s rotation curves remains almost constant or increasing in the edges of the galaxies [24, 28, 29]. To describe the maximum rotational velocity of a galaxy, we use the speed Vmax Eq.(26). which exactly corresponds with an almost constant rotation if θ0 is fixed. Or an increase in rotation relative to the center, if θ0 increases. In what follows, we will draw the curve of θ0 according to the observation data. To determine the value of θ0, we study the maximum values of rotational velocity Vmax of galaxies according to the density ρ and the masse Min of galaxies. Using the observation results, we can determine the values of θ0 [30–33]:

− − 1 − 1 6 4 2 θ 8.167 10 V ρmax M , (27) 0 ≈ × × max × × Annexe. to find the value of θ0 we have to calculate (M,ρmax, Vmax) for some galaxies in the tables of

FIG. 1: These points are obtained by the observation data [24]. Begins with spiral galaxies to the irregular galaxies. We notice that that the values of θ0 are almost constant (θ0 ≈ 0.12).

According to the figure (1), the parameter θ0 permit us to determine the rotational velocity of galaxies. The values of θ0 are very close under normal conditions, but there are exceptions like dwarf spheroidal galaxy. 6

SUMMARY

We have presented a study that is designed to describe dark matter Einstein-Gauss-Bonnet gravity coupled with scalar fields. We also analyzed the effects of the GB invariant on the evolution of effective potential. Also, the stability of dark matter was discussed in the context of this model. We have presented an explicit procedure to construct the effective equation of state describing the scalar fields. The potential describes the rotation of galaxies by two parameters k and meff introduced by EGB gravity. Indeed, these two parameters describe the dark matter hidden in galaxies. We have compared our results with the ones obtained by the observation results. We have used the EGB gravity and MOND theory, in relationship with the field φ to provide a qualitative description of modified gravity. We found an expression of the velocity curve of the galaxies depends on the parameter θ0. We obtained a graph figure (1) of θ0 using the observation data (Annexe) which corresponds exactly to the evolution of the rotation of galaxies. We have shown that θ0 is an essential parameter, which is constant for spirals, lenticular and elliptical galaxies. But is no longer a constant for irregular and spheroidal . This shows that the parameter θ0 plays an important role in describing both the rotation and the type of galaxies. The future work will have to test if this model also corresponds to other observations, such as the CMB.

ANNEXE

−22 2 10 Galaxy ρ(×10 Kg/m ) Min(×10 M⊙) Vmax(Km/s) θβ Milky Way 13, 10 12, 10 220 0, 086 NGC 7331 8, 70 14, 70 268.1 0, 105 NGC 4826 45, 00 1, 90 180.2 0, 130 NGC 6503 18, 40 0, 958 121 0, 154 NGC 7793 12, 00 0, 88 117.9 0, 174 UGC 2885 15, 30 11, 70 300 0, 114 NGC 253 10, 00 4, 30 229 0, 160 NGC 925 8, 60 2, 00 113 0, 120 NGC 2403 5, 20 2, 90 143.9 0, 144 NGC 2841 8, 02 17 326 0, 121 NGC 2903 6, 30 6, 70 215.5 0, 135 NGC 3198 3, 26 6, 00 160 0, 125 NGC 5585 10, 1 0, 59 92 0, 173 NGC 4321 8, 80 16, 8 270 0, 098

TABLE I: Large spirals galaxies

−22 2 10 Galaxy ρ(×10 Kg/m ) Min(×10 M⊙) Vmax(Km/s) θβ NGC 4303 6, 81 3, 68 150 0, 124 NGC 5055 5, 21 7, 07 215 0, 138 NGC 4736 14, 00 1, 77 198.3 0, 198 NGC 5194 1, 00 4, 00 232 0, 299 NGC 4548 3, 20 3, 80 290 0, 287

TABLE II: Messier Spirals 7

−22 2 10 Galaxy ρ(×10 Kg/m ) Min(×10 M⊙) Vmax(Km/s) θβ UGC 3993 3, 10 17.8 300 0, 138 NGC 7286 4, 60 0.59 98 0, 224 NGC 2768 10, 00 1.98 260 0, 268 NGC 3379 0, 90 1.10 60 0, 151 NGC 2434 1, 00 5.00 231 0, 266 NGC 4431 13, 00 0.30 78 0, 193

TABLE III: Lenticular and Elliptical Galaxies

−22 2 10 Galaxy ρ(×10 Kg/m ) Min(×10 M⊙) Vmax(Km/s) θβ WLM(DDO 221) 0, 92 0, 00863 19 0, 539 M81dWb 5, 00 0, 007 28, 5 0, 588 Holmberg II 3, 64 0, 0428 34 0, 307 NGC 3109 8, 00 0, 0299 67 0, 605 NGC 4789a 93, 00 0, 0188 50 0, 303 NGC 3034 22, 00 1, 00 137 0, 163

TABLE IV: irregular dwarf galaxies

−22 2 Galaxy ρ(×10 Kg/m ) Min(M⊙) Vmax(Km/s) θβ Carina 6, 50 3.38 × 106 8, 5 0, 007 Leo I 13, 60 7.74 × 106 12, 5 0, 006 7, 40 3.40 × 106 12 0, 01 Fornax 0, 373 12.40 × 106 11, 5 0, 01

TABLE V: dwarf spheroidal galaxy (dSph)

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