Rotation curves and orbits in the scalar field dark matter halo spacetime
aa r X i v : . [ g r- q c ] M a r Rotation curves and orbits in the scalar field darkmatter halo spacetime
Yen-Kheng Lim ∗ Department of Mathematics, Xiamen University Malaysia,43900 Sepang, Malaysia
March 6, 2019
Abstract
The spacetime satisfying the flat curve condition for galaxies is shown to be thezero mass limit of the dilaton blackhole with an exponential potential. We derivethe geodesic equations and by studying rotational curves and light deflection, wefind that the presence of the dilaton increases the gravitational effects overall butdiminishes the contribution from the central mass when the distances are closeto the black hole. Also, the dilaton reduces the size of innermost stable circulartime-like orbits, while the radius of the photon sphere may be larger or smallerthan its Schwrazschild counterpart, depending on the strength of the dilaton field.We also show that a generalisation of the flat-curve-condition spacetime consideredelsewhere in the literature does not solve the Einstein equation.
The discrepancy between the observed galactic rotation curves and the expectation basedon luminous matter [1, 2] has long been an oustanding problem in gravitational physics.Attempts to resolve this issue can be broadly split into two approaches, namely, (i) tomodify the theory of gravity, and (ii) postulating the presence of dark matter in thegalactic halo.In the latter approach, observations lead one to conclude that the dark matter is likelyto take a spherical distribution, and takes up a large proportion of the total galactic massover the luminous matter [3]. Coupled with the observation that the measured rotationalspeeds of the stars appear to tend to a constant against distance, this provides some clues ∗ E-mail: [email protected]
1o determine the possible type of geometry which should describe spacetime around thegalactic halo [4–9].In the present work, we will frequently make reference to a particular spacetime pro-posed by Matos, Guzm´an, and N´u˜nez (MGN) [10], wherein the flat curve condition isimposed to fix the form of a static, spherically-symmetric ansatz for the spacetime. Themetric obtained by MGN has the form g tt ∝ r l , (1)where r is the radial coordinate and l is a constant that depends on the tangential speed ofthe flat curve. If the metric were to describe the spacetime of a galaxy with a flat rotationcurve, it is required that l ∼ − to be consistent with the observed rotational speeds.Having Eq. (1), MG considered a simple scalar field model which would produce sucha spacetime, and concluded that the required ingredients are a scalar field dark matter(SFDM) minimally coupled to gravity with a Liouville-type potential. In this solution,the g rr part obtained upon solving the Einstein equations contains an integration constantwhich was set to zero.Subsequently, Nandi et al. [11] considered a generalisation of the MGN solution wherethe aforementioned constant is not set to zero. With this non-zero integration constant,the g rr is now a non-trivial function. It is then shown that this configuration providesa more realistic energy and pressure density for the dark matter distribution. However,as we will show in the Appendix, this solution cannot be exact, as it does not solve theEinstein’s equation exactly. Nevertheless the constant required in [11] is small, to theorder ∼ − and hence may perhaps be considered an approximate solution.An Einstein gravity theory minimally coupled with a scalar field plus a Liouvillepotential has been well studied in the literature, and many exact solutions have beenobtained [12–17]. A spacetime of relevance to this paper is the dilaton black hole solutiongiven by Chan et al. [12], which reduces to the MGN solution in its zero mass limit.Therefore, as an extension of the results of MGN, we shall consider the geodesics of thedilaton black hole spacetime, including both time-like and null particles.The rest of this paper is organised as follows. In Sec. 2 we review the action forEinstein-dilaton gravity with a Liouville-type potential along with the dilaton black-holesolution. Then the geodesic equations for this spacetime is derived in Sec. 3, followedby Sec. 4 where we explore circular orbits and some examples of rotation curves fortime-like particles. Null orbits and gravitational lensing is briefly considered in Sec. 5.Some concluding remarks are given in Sec. 6. In Appendix A, we rederive the equations ofmotion of MGN and verify that the MGN solution is only possible when the g rr componentis a constant. 2 Equations of motion and the dilaton black hole so-lution
We will be working with Einstein gravity minimally coupled to a scalar field described bythe action (in units where c = 1) I = 116 πG Z d p x p | g | (cid:0) R − ( ∇ ϕ ) − V ( ϕ ) (cid:1) , (2)where p is the number of spacetime dimensions. We will keep p arbitrary in most of ourequations here, though our explicit examples will be motivated by astrophysical contexts,in which case p = 4.Extremising the action gives the Einstein-dilaton equations R µν = ∇ µ ϕ ∇ ν ϕ + 1 p − V g µν , (3) ∇ ϕ = 12 d V d ϕ . (4)We will consider the following dilaton potential: V = 2 V e bϕ . (5)Such a potential allows the Einstein-dilaton equation to be solved exactly. One solutionof our present interest is the black hole [12]d s = − f d t + h d r + r dΣ p − , (6a) f = r a h = k ( p − p − a )(1 − a ) r a − µr − ( p − − a ) , (6b) ϕ = − a p p − r + ϕ , (6c)2 V e bϕ = − ( p − p − ka ( p − a )(1 − a ) , b = 1 √ p − a . (6d)Here, dΣ p − = ˜ γ ij d σ i d σ j denotes a maximally symmetric space with constant curvature k = ±
1, 0. The continuous parameter µ parametrises the mass of the black hole. Inparticular, by applying the appropriate counter-term subtraction on the boundary stresstensor of the spacetime [18], the mass is found to be M = ( p − πG µ, (7)where Σ = R d p − x √ ˜ γ . The scalar field is given up to an arbitrary integration constant ϕ which we can set to zero without loss of generality. Another parameter characterising3he spacetime is a , which determines the strength of the scalar field ϕ . The remainingquantities, b and V are fixed by a via Eq. (6d). Indeed, for a = 0, the solution reducesto the Schwarzschild black hole with f = h = 1 − µr p − .This solution contains the spacetime which satisfies the flat curve condition consideredby MGN [10]. To recover it we simply set µ = 0, p = 4, a = p l/ k = 1, where thesolution reduces to d s = − B r l d t + h d r + r (cid:0) d θ + sin θ d φ (cid:1) , (8a) h = 4 − l , B = 44 − l , (8b) ϕ = −√ l ln r + ϕ . (8c)This is precisely the solution studied in Ref. [10] for a = l/
2. Nandi et al. [11] subsequentyconsidered the metric (8a), but with h = 4 − l C (4 − l ) r − ( l +2) , (9)which is intended as a generalisation of the study by [10] for C = 0. However, Eq. (9)cannot be a solution to the Einstein equations if the scalar field is given by (8c). Wedemonstrate this explicitly in Appendix A.Finally, we write down its corresponding stress-energy tensor,8 πGT µν = ∇ µ ϕ ∇ ν ϕ −
12 ( ∇ ϕ ) g µν − V g µν . (10)Taking the stress-energy tensor to be the form of the perfect fluid, the energy density is − T tt = ρ = − T ij and the pressures is T rr = p , where8 πGρ = 12 (cid:20) − a ( p − f r − a − + k ( p − p − a ( p − a )(1 − a ) r − (cid:21) , (11)8 πGp = 12 (cid:20) a ( p − f r − a − + k ( p − p − a ( p − a )(1 − a ) r − (cid:21) . (12) Having established the background spacetime, we now turn to the problem of particlemotion in the solution. We parametrise the trajectory of the time-like or null particle by x µ ( τ ), where τ is an appropriate affine parametrisation. The trajectory will be a solutionthat extremises the Lagrangian L = g µν ˙ x µ ˙ x ν , where over-dots denote derivatives withrespect to τ . We choose the affine parameter τ such that the invariant g νν ˙ x µ ˙ x ν = ǫ has unit magnitude if it is non-zero. Therefore ǫ encodes the type of geodesic under4onsideration where ǫ = ( − . (13)For simplicity, let us also assume that the motion in dΣ p − = ˜ γ ij d σ i d σ j is suppressed inall but one direction, so that ˙ σ i = ( ˙ φ, , . . . , L = 12 (cid:16) − f ˙ t + h ˙ r + r ˙ φ (cid:17) . (14)Since t and φ are cyclic variables in the Lagrangian, we have the first integrals˙ t = Ef , ˙ φ = Lr , (15)where E and L are constants of motion which we may interpret as the particle’s energy andangular momentum, respectively. Substituting these into g µν ˙ x µ ˙ x ν = ǫ provides anotherfirst integral equation ˙ r + U = 0 , (16)where we have defined the effective potential U = − r − a (cid:20) E − (cid:18) L r − ǫ (cid:19) f (cid:21) . (17)The existence of an orbit requires ˙ r to be real, or equivalently, U ≤
0. Therefore, for agiven choice of constants, the condition
U ≤ r for aparticular orbit.Another equation of motion for r can be obtained by applying the Euler-Lagrangeequation, dd τ ∂ L ∂ ˙ r = ∂ L ∂r , giving ¨ r = − h ′ h ˙ r − f ′ E f h + L hr . (18)where primes denote derivatives with respect to r .Our procedure of obtaining numerical solutions is as follows: For a choice of spacetimeparameters ( µ, a ) and orbital constants ( E, L ), Eqs. (18) and (16) are integrated using thefourth-order Runge-Kutta method, with the aid of (17) to serve as a consistency check ofthe results. The initial conditions we typically adopt is ˙ r = 0, with initial r calculated bysubstituting ˙ r = 0 into (17). For concreteness, in all the following examples we considerspherically symmetric dilaton black hole in four dimensions where k = 1 and p = 4.5 . − . . . .
06 2 4 6 8 10 12 14 16 18 20 U r − − − −
10 0 10 20 (a) a = 0. − . − . . . .
06 2 4 6 8 10 12 14 16 18 20 U r − − − −
10 0 10 20 (b) a = 0 . Figure 1: (Colour online) Plots of effective potential against r along with their correspond-ing orbits with energy E = 0 . L = 3 .
9, in the four-dimensional sphericallysymmetric black hole spacetime with p = 4, k = 1, and µ = 2. The numerical solutionshown here are integrated up to three periods of φ .As a warm-up example of this procedure, consider the following representative orbitsfor the Schwarzschild solution a = 0, µ = 2 in Fig. 1a and the case a = 0 . µ = 2 inFig. 1b. for the choice of E = 0 . L = 3 .
9. The left panels for each case areplots of U against r . As mentioned above, the condition U ≤ r for a trajectory. For the case a = 0 as shown in Fig. 1a, this is approximately8 . ≤ r ≤ . . ≤ r ≤ .
164 for the case a = 0 . This particular E and L for a = 0 corresponds to a closed Schwarzschild orbit labelled (2 , ,
1) in [19]. Circular orbits and rotation curves of time-like or-bits
In this section we shall consider circular orbits of time-like and null particles. Theseorbits are characterised by the property ˙ r = 0. If r is the radius of the orbit, then it isrequired that U ( r ) = U ′ ( r ) = 0. For these orbits to be stable we also require U ′′ ( r ) > U ′ = − a r U − r − a (cid:18) f − rf ′ r L + ǫf ′ (cid:19) , (19) U ′′ = 2 a (1 − a ) r U − a r U ′ − r − a (cid:18) rf ′ − r f ′′ − fr L + ǫf ′′ (cid:19) . (20)For time-like particles, we have ǫ = −
1. Thus U = U ′ = 0 leads to E = 2 f f − rf ′ , L = r f ′ f − rf ′ , (21)giving the required energy and angular momentum for a circular orbit of radius r .The stability of a circular orbit depends on the sign of U ′′ . Substituting U ′ = U = 0and (21) into (20), we have U ′′ = − r − a (cid:18) (4 rf ′ − r f ′′ − f ) f ′ r (2 f − rf ′ ) − f ′′ (cid:19) . (22)Typically, this is positive down to some lower bound of r , below which we only haveunstable circular orbits with U ′′ <
0. This lower bound r ISCO represents the criticalradius which is called the innermost stable circular orbit (ISCO), and we can determineits value by equating Eq. (22) to zero and solving for r .For the Schwarzschild case ( a = 0), this gives the well-known exact value of r = 3 µ .For other values of dilaton parameter a we obtain r ISCO numerically. As shown in Fig. 2,as a is increased from zero, the value of r ISCO decreases. In other words, the presenceof the scalar field allows smaller stable circular orbits, at least naively in terms of thecoordinate radius r .Turning now to the tangential velocity for these circular orbits, various definitions thatwas used in the literature. Here, let us use the definition where the velocity is extractedfrom the spatial part of g µν ˙ x µ ˙ x ν = ǫ . The expression for tangential velocity is [6] v tan = r √ f d φ d t = s rf ′ f , (23) There is another definition proposed by [9] where v tan = q rf ′ . . . . .
56 0 0 . . . . r I S C O a Figure 2: Plot of r ISCO against a . . .
52 2 4 6 8 10 12 14 16 18 20 v t a n r a = 0 a = 0 . a = 0 . (a) µ = 2, various a . . . . .
81 0 10 20 30 40 50 v t a n r µ = 0 µ = 1 . µ = 2 . (b) a = 0 .
04, various µ . Figure 3: Plots of v tan vs r , for (a) fixed µ and different a , and (b) fixedIt is this definition that was used by [10] to determine the flat curve condition which leadto the solution (8). In that case we have a constant v tan = √ a , which was equivalent to √ l in the notation of Ref. [10]. As long as µ = 0, this result is actually independent of ofthe dimensionality of the spacetime.The tangential velocities for non-zero µ and various a are shown in Fig. 3a, wherethe result is perhaps unsurprising. The presence of µ causes v tan to be higher near theorigin, where the gravitating mass is located. Further away, the velocity approaches √ a ,as expected from the flat curve condition. Nevertheless, for small r , it is interesting tonote that larger a corresponds to lower speeds compared to curves of smaller a , indicatingthat the presence of the scalar field diminishes the gravitational attraction of the centralmass. Also, at first glance these distances, this suggests the possibility of observableconsequences for stars near the galactic core. But the values of a we are using in Fig. 3aare quite large. Based on the observed tangential velocities of galaxies, it is required that v tan ∼ √ a ∼ − (in units of c ). 8 . . . . . . . . . .
48 0 0 . . . . r ph a (a) µ = 0 . . . . . . . .
961 0 0 . . . . r ph a (b) µ = 0 . . . . . . . . . .
56 0 0 . . . . r ph a (c) µ = 1 . . . . .
83 0 0 . . . . r ph a (d) µ = 2 . Figure 4: Plots of photon sphere radii against dilaton parameter a for various black holemasses. We now consider orbits for photon trajectories with ǫ = 0. Looking for circular orbits,the condition U = 0 gives L E = r f , which is the standard result of the impact parameterat the distance of closest approach. On the other hand, the condition U ′ = 0 gives2 f = rf ′ → r ph = (cid:20)
12 (3 − a )(1 + a ) µ (cid:21) a , (24)where r ph is the radius of the photon sphere. Clearly, the case a = 0 reduces to thestandard Schwarzschild result of r ph = µ . The radii for other non-zero values of a areshown in Fig. 4. We see that starting small µ , for instance µ = 0 . a increases the size of the photon sphere. If µ is relatively larger, say, µ = 1 in Fig. 4c,the radii develops a maximum at some intermediate a . The location of the maximumdecreases with increasing µ . At larger µ , the the largest photon sphere is for a = 0,in which case all photon spheres of non-zero a are smaller than the Schwarzschild case.Evaluating the second derivative of U at the photon sphere tells us that U ′′ ( r ph ) < J ~v~e ( r ) S L OL Figure 5: Schematic diagram depicting gravitational lensing in the spacetime of a dilatonblack hole. The trajectory begins at the source S and is deflected by the dilaton blackhole lens L at impact parameter J = r / p f ( r ). Then the photon proceeds towards theobserver at the intersection of the trajectory and the optic axis.are considering is not flat, we are not able to consider the deflection of light which comesfrom r → ∞ . Nevertheless, we can consider photon trajectories that begin and end atfinite r . Since the spacetime is not flat at finite distances from the black hole, we shouldbe careful not to take the changes in the coordinate value φ as the bending angle.However, we can adopt the Rindler-Ishak method [21] to establish a coordinate invari-ant quantity which describes the deflection of the photon trajectory. For simplicity, weshall consider the case where the source S , lens L and observer O are co-aligned on theoptic axis, as shown in Fig. 5.. We shall call the point of closest approach L ′ , which hascoordinate value r .In the Rindler-Ishak method, we establish a spatial ortho-normal frame at the locationof the observer O . In the two-dimensional plane containing the null geodesic, it is givenby ~e ( r ) = 1 √ h ∂ r , ~e ( φ ) = 1 r ∂ φ . (25)We take the spatial part of the photon’s four-velocity ( ˙ t, ~v ), where ~v = g ij ˙ x i ˙ x j , where i, j = t , as ~v = ˙ r∂ r + ˙ φ∂ φ . An invariant angle ψ is given bycos ψ = ~e ( r ) · ~v | ~v | , (26)and is depicted as the angle between the direction of the photon and the optic axis at O in Fig. 5. Using the geodesic equations, we can show thatsin ψ = r r p f ( r ) p f ( r ) , (27)where r = r is the coordinate distance of closest approach. The total deflection of the10 . . . .
81 6 8 10 12 14 α J a = 0 a = 0 . a = 0 . (a) µ = 1. . . .
53 6 8 10 12 14 α J a = 0 a = 0 . a = 0 . (b) µ = 2. Figure 6: Plots of bending angle α vs impact parameter J for µ = 1 ,
2, and various a .trajectory beginning with S and ending at O is twice of this angle,ˆ α = 2 ψ = 2 sin − r r p f ( r ) p f ( r ) ! . (28)We can parametrise each photon trajectory by r with its corresponding impact parameter J = r / p f ( r ). By the spherical symmetry of the problem, it suffices to calculate ˆ α byintegrating Eqs. (15) and (18) starting from point L ′ with initial conditions r = r , ˙ r = 0,and φ = 0. The point O where the trajectory intersects the optic axis is φ = π . Thevalue of r at this point is then substituted into (28). The results of the integration areshown in Fig. 6.Between Fig. 6a and 6b, we see that increasing a raises the value of the bending angleuniformly across the range of impact parameter, while µ , reflecting the central black-holemass, causes a large bending angle only for impact parameter close to the black hole, aswe expect intuitively. However, note that for larger µ (say, µ = 2 as shown in Fig. 6b),the bending angles for J close to the black hole are smaller for larger a , showing us thatthe presence of the scalar field diminishes the gravitational effect of the central mass, justas in the time-like geodesic case. In this paper we have derived and studied the geodesic equations of the dilaton black holewith an exponential potential. Since the zero mass limit of this solution corresponds to theMGN spacetime, we may treat particle motion around the dilaton black hole spacetimeas a toy model generalising the MGN solution to include a central gravitating mass.From the rotation curves of time-like particles and gravitational lensing of photontrajectories, we find that the presence of the dilaton scalar field roughly increases the11verall gravitational effects for all distances. On the other hand, at distances close to theorigin where we expect gravitational attraction to be strong, the presence of the dilatondiminishes is effect, as can be seen from the lower tangential velocities and smaller bendingangle at small distances compared to the Schwarzschild case. These diminishing effects atsmall distances clearly not consistent with observations, since we do not expect deviationsfrom standard gravity at sub-galactic distances or large accelerations. However, the casewhere the mass zero or sufficiently small approximates the metric obtained by MGN whichwas designed to satisfy the flat curve condition.Looking at these ranges, the results of Sec. 5 shows that the bending angle is largercompared to the Schwarzschild case of the same mass, and thus the field ϕ plays therole of dark matter which causes an additional bending of light on top of those frombaryonic mass. However, we shall stop short of making these claims in the context ofactual observations, since, as mentioned above, our spacetime is a highly idealised modelwhich does not capture all the subtleties of galactic and cosmological dynamics. A Equations of motion for the MGN solution
In this Appendix we shall rederive the MGN metric and show that the solution with non-zero integration constant considered by [11] cannot be an exact solution to the Einstein-dilaton equations. We begin by taking the metric ansatz to bed s = − f d t + h d r + r dΣ p − , (29)where f and h are functions that depend only on the coordinate r . Assuming the samefor the scalar field ϕ , the Einstein equations are − √ f h (cid:18) f ′ √ f h (cid:19) ′ − ( p − f ′ rf h = 1 p − V , (30a) − √ f h (cid:18) f ′ √ f h (cid:19) ′ + ( p − h ′ rh = ϕ ′ h + 1 p − V , (30b) p − r h ( kh −
1) + h ′ rh − f ′ rf h = 1 p − V . (30c)In this section, primes denote derivatives with respect to r . For completeness, we writedown the dilaton equation, 1 √ f hr p − (cid:16)p f hr p − h − ϕ ′ (cid:17) ′ = 12 d V d ϕ . (31)Eqs. (30) and (31) may serve as a starting point to derive the solution (6).However, we are presently interested in verifying the solution by MGN [10]. To do12his we first take the difference between (30a) and (30b) to obtain h ′ rh + f ′ rf h = 1 p − ϕ ′ h . (32)Using this to remove f ′ from (30c) gives1 r (cid:18) ( p − − kh ) − r h ′ h (cid:19) = − (cid:18) p − ϕ ′ + 1 p − V h (cid:19) . (33)On the other hand, if we used (32) to remove h ′ from (30c) instead, we would have1 r (cid:18) ( p − kh − − r f ′ f (cid:19) = − (cid:18) p − ϕ ′ − p − V h (cid:19) . (34)Finally, we note that since ϕ ′ h = 1 p − ϕ ′ h + p − p − ϕ ′ h = h ′ rh + f ′ rf h + p − p − ϕ ′ h , (35)where Eq. (32) has been used to take the second equality to the third. Substituting thisinto Eq. (30b) gives14 r (cid:20) − r f ′′ f + r f ′ f − r f ′ f + (cid:18) p −
3) + r f ′ f (cid:19) r h ′ h (cid:21) = p − p − ϕ ′ + 1 p − V h. (36)If we substitute k = 1, p = 4, V = 2 V , and f = r l , Eqs. (33), (34), and (36) becomes r (cid:18) − h − r h ′ h (cid:19) = − (cid:18) ϕ ′ + 12 V h (cid:19) , (37a)1 r ( h − ( l + 1)) = − (cid:18) ϕ ′ − V h (cid:19) , (37b)14 r (cid:18) l + (2 + l ) r h ′ h (cid:19) = − (cid:18) ϕ ′ + 12 V h (cid:19) , (37c)thus recovering Eqs. (11)–(13) of [10] precisely. In Ref. [10], to obtain a spacetime thatsupports orbits consistent with the galactic rotation curves, the potential12 V = − l − l e − ϕ/ √ l (38) Obtaining Eq. (37) is more direct if we had started with the Einstein equation in the ‘Einstein tensorform’, R µν − Rg µν = ∇ µ ϕ ∇ ν ϕ − ( ∇ ϕ ) − V g µν . Then Eqs. (37a)–(37c) are simply the tt , rr , and ij -components. Note that our scalar field ϕ is defined such that the Newton’s constant is absorbed. Namely thescalar field Φ in [10] is related to ϕ via Φ = √ κ ϕ , where κ = 8 πG . Furthermore the potential V in [10]is related to V via V = κ V . h = 4 − l , (39)The scalar field solution is ϕ = √ l ln r + ϕ (40)Subsequently, Ref. [11] considered the more general function h = 4 − l C (4 − l ) r − ( l +2) , (41)along with (40) as the scalar field. Ref. [11] investigated the physical properties of thegalactic halo for a (small, but) non-zero C . However, we will now see that (41), togetherwith (40) cannot be an exact solution to (37) unless C = 0, for which it reduces to thecase (39) originally considered in [10]. This is easily seen by direct substitution. Forinstance, substituting (41) into the left hand side of Eq. (37a) gives1 r (cid:18) − h − r h ′ h (cid:19) = 1 r (cid:18) l − ( l + 1) C (4 − l ) r − ( l +2) C (4 − l ) r − ( l +2) (cid:19) , (42)but substituting (40) into the right hand side of the same equation gives − (cid:18) ϕ ′ + 12 V h (cid:19) = 1 r l − lC (4 − l ) r − ( l +2) C (4 − l ) r − ( l +2) ! . 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