Revisiting perfect fluid dark matter: Observational constraints from our galaxy

We revisit certain features of an assumed spherically symmetric perfect fluid dark matter halo in the light of the observed data of our galaxy


Introduction
A few years ago, in Ref.
[1], a perfect fluid dark matter model was developed that was shown to have many attractive theoretical aspects. The solution may be thought of as a dark matter induced spacetime embedded in a static cosmological Friedmann-Lemaître-Robertson-Walker (FLRW) background 1 . The motivation for developing an isotropic perfect fluid model (we leave open the question of particle identity of dark matter) came from the fact that predictions from such model at stellar and cosmic scales have been observationally well corroborated so far. More recently, Harko and Lobo [2] investigated dark matter as a mixture of two non-interacting perfect fluids, with different four-velocities and thermodynamic parameters. González-Morales and Nuñez [3] compared two different dark matter models, one is a perfect fluid and the other is a scalar field [3]. See also [4].
The model considered here assumes that a spherical dark matter distribution is the only gravitating source. This assumption is of course an oversimplification since, although the bulge is quite spherical and is dominated by old stars, the Milky Way has a strongly flattened stellar distribution. However, we know from the vertical velocity dispersion of stars as a function of distance from the disk plane that the local disk mass density is almost identical to the sum of the densities that can be attributed to stars, gas and stellar remnants. Therefore, there is practically little dark matter hidden in the disk. Hence, to explain the rotation curve measurements, we are forced to assume that dark matter resides in the halo region dominating its mass, is spherically distributed and, if it is non-baryonic, would not be expected to collapse into a disk-like structure.
Specifically, the hypothesis of dark matter arose because the Newtonian circular velocity v 2 c = GM(r) r of circularly moving probe particles caused by the luminous mass distribution M (r) is not supported by observations [5,6]. The circular velocity becomes nearly flat, v 2 c ≃ constant, at distances far away from the center (halo region), which is possible only if M (r) ∝ r. Therefore, almost every galaxy is assumed to host a large amount of non-luminous matter, the so called gravitational dark matter, consisting of unknown particles not included in the particle standard model, forming a halo around the galaxy. Naturally, dark matter is at the core of modern astrophysics. Many well known theoretical models for dark matter exist in the literature, for instance, see  (the list is by no means exhaustive). Some models that do not hypothesize dark matter appear in [30][31][32][33][34][35][36][37][38]. Well known density profiles originated in [39][40][41]. Excellent reviews are to be found in [42][43][44][45].
In this paper, we shall revisit the model of perfect fluid dark matter, developed in Ref.
[1], in the light of the observed/inferred data of our galaxy. Our analysis would require three ingredients: (i) A method, viz., the Faber-Visser 1 The reason for the appearance of static FLRW background around the imbedded perfect fluid dark matter is already explained in Ref.
[1]. The Einstein field equations are solved with perfect fluid stress tensor in both the cases but we sought a static solution from the start. While working on a local problem (flat rotation curve), the scale factor is usually fixed to R 0 = 1 today.
[46] method of combined post-Newtonian measurements of rotation curves and gravitational lensing for measuring the equation of state ω(r) of the dark matter and determining the rotation curve mass (m RC ), the lens mass (m Lens ) and the post-Newtonian mass (M pN ). However, for the perfect fluid solution we consider here, it suffices to consider only the rotation curve as a constraint, while the lens mass will be a result of computation. (ii) Two observed inputs, viz., the circular velocity ℓ = 2v 2 c /c 2 0 of probe particles, where c 0 is the speed of light in vacuum, and the radius R MW of our galactic halo. (iii) An observational constraint, viz., the one imposed by the attractive nature of dark matter so that p/ρ = ω(r) ≥ 0. The nature is attractive because the very existence of dark matter is speculated from observations of the Doppler shifted light emanating from neutral hydrogen clouds moving on stable circular orbits in the galactic halo [20,32,47,48]. Using these ingredients, we shall analyze how choices of the adjustable parameter D appearing in the dark matter metric lead to different types of scenarios.
The following are our new results: Depending on the values of D, we show that (i) The observable masses m RC , m Lens and M pN compare well with the masses inferred by other independent means. (ii) There could appear an intriguing negative pressure matter sector (ω < 0) beyond the halo radius 2 . (iii) The Faber-Visser χ− factor has values near unity so that pressure contribution to the post-Newtonian mass M pN is negligible. Hence the perfect fluid dark matter resembles dust (ω ∼ 0) akin to CDM model. (iv) There is flexibility in the halo radius in the sense that our model can accommodate extended radii. All these imply that, fundamentally, the perfect fluid model stands up to actual observations on mass, equation of state and in addition predicts marginal quitessence matter at asymptotic distances, all within a single formulation.
In Sec.2. we briefly outline the perfect fluid dark matter and in Sec.3, display the Faber-Visser post-Newtonian observables. In Sec.4, we apply the galaxy inputs to those observables and deduce the most suitable range of D that agrees with the observational constraints from our galaxy. In Sec.5, we conclude the paper. We take G = 1, c 0 = 1, unless specifically mentioned.

Perfect fluid dark matter
The general static spherically symmetric space-time is represented by the following metric ds 2 = −e ν(r) dt 2 + e λ(r) dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) , where the functions ν(r) and λ(r) are the metric potentials. For the perfect fluid, the matter energy momentum tensor T α β is given by T t t = ρ(r), T r r = T θ θ = T ϕ ϕ = p(r), where ρ(r) is the energy density, p(r) is the isotropic pressure. Considering flat rotation curve as an input, an exact solution of Einstein field equations is derived in [1]: where B 0 > 0, D are integration constants and v c is the circular velocity of stable circular hydrogen gas orbits treated as proble particles. The exact energy density and pressure are The free adjustable parameter D, having the dimension of (length) −2 , in the solution is extremely sensitive and its value can be decided only by observed physical constraints. In the present case, the constraint is that the galactic fluid be non-exotic and attractive, i.e., the equation of state parameter ω(r) = p(r) ρ(r) ≥ 0 must hold within the halo radius. With this information at hand, an interesting aspect of the solution can be found from Eqs. (7) and (8).
It can be seen that the integrated quantity, call it M 0 = 4π r 0 ρ(r)r 2 dr derived from exact ρ(r) given by Eq. (7), is identical with the Newtonian mass M N derived in Eq.(23) below. One could as well call M N the post-Newtonian counterpart of M 0 since ρ(r) in Eq.(23) is expressed as derivatives of post-Newtonian masses. The question then we ask: What quantity derived from the exact solution differs from its measurable post-Newtonian counterpart? One such quantity is the total mass within a radius r with pressure contribution, which is defined by, using Eqs.(7) and (8) We shall see in the next section that its post-Newtonian counterpart is just M pN (r) = ℓr 2 . Hence the theoretical and observable masses in principle differ depending on arbitrary values of D, even when D = 0. Therefore, let us proceed to define the post-Newtonian observables.

Faber-Visser post-Newtonian observables
We shall only quote the relevant expressions here. For details, see Faber-Visser [46]. They considered the metric in the form which is completely determined by the two metric functions Φ(r) and m(r).
Comparing it with the metric (1), we have The potentials Φ RC (r) and Φ Lens (r), obtained respectively from the rotation curve data and gravitational lensing observations, are derived to be .
The lensing potential Φ Lens (r) is a fundamental observable quantity. When the pressures and matter fluxes are small compared to the mass-energy density then Φ RC (r) = Φ Lens (r), otherwise they may not be equal. One pseudo-mass, inferred from rotation curve measurements, is given by Another pseudo-mass m Lens (r), obtained from lensing measurements, is defined as For the equation of state parameter for perfect fluid, we should evaluate ω and impose the constraint that up to r = R MW , which will provide a limit on D. From the first order approximations of Einstein's equations, one obtains [46] Then Eq. (18) yields We have intentionally kept in the left hand side of the above Eq.(21) the transverse pressure component p t for transparency, remembering that for perfect fluid p r = p t , an exact equality that was used to derive the metric (1). It is to be emphasized that, observationally, such exact equalities as p r = p t are impossible to attain. It follows that the difference in dimensionless pressures is not zero but [46] which is just the post-Newtonian version of isotropicity of the perfect fluid. However, this value of the right hand side for our galaxy (and presumably for all other samples as well) is exceedingly small but not exactly zero! The next issue is whether the model is Newonian or not, that is, how much of pressure contribution to mass is there. For this, we need to compare the two integrals, one is the Newtonian mass M N (r) given by, using Eqs. (19) and (20), and the other is the mass in the first post-Newtonian approximation [46] M pN (r) = 4π r 0 (ρ + p r + 2p t )r 2 dr = ℓr 2 .

Application to our galaxy
Zaritsky [45] collated the published older results (till 1998, see e.g., [49][50][51][52][53]) and demonstrated that they are all consistent with a galactic halo that is nearly isothermal with a characteristic circular velocity oscillating between v c ≃ 180 to 220 km sec −1 at 15 kpc. There are however more recent works on constraining the mass and extent of the Milky Way's halo (see e.g., [54,55,56]). We shall use these data in our computations below. Dehnen et al [54] suggested a virial radius R vir ∼ 200 kpc, enclosing a virial mass M vir ∼ 1.5 × 10 12 M ⊙ . We adopt them as the halo radius R MW and mass M MW of our galaxy. Xue et al [55] observed that the Milky Way's circular velocity curve at ∼ 60 kpc gently falls from the adopted value of v c ≃ 220 km sec −1 at the Sun's location to v c ≃ 175 km sec −1 and implies M (< 60 kpc) = (4.0±0.7)×10 11 M ⊙ . Deason et al [56] infer that the mass within 150 kpc probably lies in the range (5 − 10) × 10 11 M ⊙ . The measured fall in v c is not considered serious because the implied mass ratio between the two extremes is only (175/220) 2 = 0.63.
Our strategy is to first find ω(r) from the Faber-Visser Eq.(21) using the input of v c (that is, ℓ) at some radius r. Next, within the halo boundary R MW ∼ 200 kpc, we impose the constraint ω(r < 200 kpc) > 0, which means attractive dark matter halo. At the boundary itself, we impose that ω(R MW ) = 0, thereby allowing for a change of sign in ω beyond the halo boundary. We then analyze in detail the numerical limits on ω(r) using the observed value of ℓ and different signs of the adjustable parameter D.
We now consider three cases of signs of D and omit mentioning its dimension in what follows. Case (1): D = 0. This value suggests itself. Then, from Eq.(26), we have ω(r) = 2.8 × 10 −7 and χ(r) = 1 ∀r, which imply that the perfect fluid approximates to dust dark matter. Because of negligible pressure, as evidenced by the Faber-Visser function χ(r) = 1, this case is more consistent with the Cold Dark Matter (CDM) paradigm for galactic fluid [57]. We now use ℓ = 6.80 × 10 −7 , D = 0 in the expressions for the masses m RC , m Lens , M pN (r) and find that all have nearly the same values 3 within r = 60 kpc, viz., m RC = m Lens = M N = M pN = 4.25 × 10 11 M ⊙ (Fig.1). The last two equalities suggest that the model is Newtonian, that is, pressure contribution is negligible [see Eqs. (23,24)]. Within the current level of uncertainties in the values of observed mass, it is evident that our common value is quite comparable with the value M (< 60 kpc) = (4.0 ± 0.7) × 10 11 M ⊙ inferred by Xue et al [55]. Assuming no further significant fall-off in the circular velocity from v c ≃ 175 km sec −1 (distinct from the radial velocity dispersion via Jean's law), we find that at r = 150 kpc, m RC = m Lens = M pN = 1.0 × 10 12 M ⊙ . This mass value is reasonably consistent with the range (5 − 10) × 10 11 M ⊙ suggested by Deason et al [56].
Case (2) (Fig.3). This case has several interesting features and Fig.3 is the most eloquent illustration of how the constraint Eq. (28) can eventually determine the behavior of matter in all distance sectors. First, the sector having ω(r) < 0 has positive energy density ρ(r) > 0 and negative pressure p r (r) + 2p t (r) < 0 (Fig.4), but the matter is not exotic as it still does not violate the Null Energy Condition (NEC) 4 (Fig.5). Interestingly, the matter does not resemble either the cosmological phantom (ω < −1) or quintessence matter (ω < −1/3) because, as is evident from Fig.3, ω(r) ∼ −10 −7 at any finite r > 200 kpc. However, at r → +∞, we find from Eq. (26) that ω(∞) → (−1/3)+, which therefore marginally corresponds to quintessence matter, and there appears no singularity in ω(r) at any radius. Second, Fig.6 shows χ(r) ≃ 1 for r ∈ [0, 400 kpc], indicating that the pressure contribution is quite insignificant, thereby once again supporting the CDM paradigm. Third, it can be easily seen from Eq.(24) that the measure M pN (r) gives a comparable galactic mass ∼ 1.4 × 10 12 M ⊙ enclosed within the radius R MW , while other mass estimates are also very close to it. Fourth, from Eq. (22), it follows that 4πr 2 [p r (r) − p t (r)] ∼ 10 −14 , a very minute difference as expected of perfect fluid from the observational point of view. Finally, note that the value of D actually determines the terminating radius where ω(r) = 0. For example, for D > −4.84 × 10 −18 , the radius can be arbitrarily shifted away from R MW ∼ 200 kpc (for an illustration, see Fig.7). This means that D can be adjusted to the possibility of having a larger Milky Way halo than considered here (viz., 200 kpc).
In view of these merits, we can say that the range −4.84 × 10 −18 ≤ D ≤ 0, which in turn leads to 0 ≤ ω(r) ≤ 2.8 × 10 −7 for the perfect fluid dark matter, can reasonably describe our galactic halo. The simple physical requirement of an attractive halo thus leads to a strong constraint indicating a dust-like dark matter (ω ∼ 0 or p ∼ 0).
We wish to point out here that so far we focused only on the constraint from the Milky Way, but there must be constraints from, for example, the Cosmic Microwave Background (CMB) on deviations from ω = 0 for dark matter. Using CMB, supernovae Ia and large scale structure data in a fluid dark matter model, Müller [59] found constraints on ω as follows: −1.50 × 10 −6 < ω < 1.13 × 10 −6 if the dark matter produces no entropy and −8.78 × 10 −3 < ω < 1.86 × 10 −3 if the adiabatic sound speed vanishes, both at 3σ confidence level. Clearly, we see that both the ranges in [58] concentrate around the value ω ∼ 0, which is in very good agreement with our result of a dust-like halo. By observing effects of perturbations on CMB and matter power spectra, Kumar and Lu [60] conclude that the current observational data favor the CDM scenario with the cosmological constant type dark energy at the present epoch. This is the most recent result on the CMB constraint.
Case (3): D > 0. Eq.(28) then suggests 0 < D < 5.67 × 10 −12 giving ω(200 kpc) > 0. Let us take a concrete value, for example, D = 5 × 10 −12 to see what that means. We find from Eq.(26) that ω(200 kpc) > 0 but ω → ∞ occurs at r sing = 212 kpc. Fig.8 shows the occurrence of cosmological quintessence matter (ω < −1/3) immediately beyond 212 kpc. If D > 5×10 −12 , then ω(r) < 0 inside the halo boundary, both contrary to our assumption that ω(200 kpc) = 0. Fig.9 then shows that M pN (r) is larger than M N (r), meaning that there is substantial pressure in the halo, as confirmed also by the χ-factor that shows values away from unity, signalling the presence of non-negligible pressure as opposed to the CDM paradigm. All these features could make the case truly intriguing if one is ready to live with a singularity in ω(r). One might consider any other allowed value respecting D < 5.67 × 10 −12 , say D = 2.83 × 10 −12 (This particular value corresponds to ω(200 kpc) = 1 3 ), then we find from Eq.(26) that the singularity just shifts to a larger radius r > 200 kpc, and quintessence matter begins to appear from that radius onwards, as shown in Fig.10. However, as we see the unphysical singularity can only be arbitrarily shifted at will by choosing D but not removed.

Conclusions
We revisited the perfect fluid dark matter model in the light of the Faber-Visser post-Newtonian formalism that requires simultaneous measurement of pseudo-mass profiles from rotation curve and gravitational lensing observations. However, for the model considered here, no constraints from lensing were used. The lensing mass together with other masses were computed using recent data. The formalism provides information of the equation of state of the galactic fluid, especially the pressure component in it. We saw above how the variation of a crucial parameter D, that has dimension of the cosmological constant (kpc) −2 , can lead to different scenarios. We deduced the post-Newtonian version of isotropicity in Eq. (22) and computed the equation of state parameter ω(r), the observables such as the post-Newtonian mass M pN , the rotation curve and lens pseudo-masses from Eqs. (21,24,16,17) respectively in terms of the metric functions 5 .
The computation of the above observables for our galaxy was done taking into account the data on rotational velocity ℓ = 6.80 × 10 −7 from Xue et al [55] and requiring an attractive halo (ω(r) ≥ 0) at least within the halo radius R MW ∼ 200 kpc [54]. The choice of the values of D obtained from Eqs. (27,28), when used in Eq. (26), led to the profiles of ω(r). Salient features of our analysis are summarized below: The case D = 0 in Eq. (27) immediately led to ω(r) = 2.8 × 10 −7 and χ(r) = 1 ∀r, which imply that the perfect fluid approximates to dust dark matter. The masses within r = 60 kpc, viz., m RC = m Lens = M N = M pN = 4.25 × 10 11 M ⊙ are found to be in excellent agreement with the value M (< 60 kpc) = (4.0 ± 0.7)× 10 11 M ⊙ inferred by Xue et al [55]. The mass within r = 150 kpc obtained by Deason et al [55] as well as the virial mass M vir from Dehnen et al [54] are also found to be quite comparable (Fig.1).
The case D > 0 signals the presence of non-negligible pressure in the halo as opposed to the CDM paradigm but also leads to a singularity in ω(r) that can only be arbitrarily shifted at will by choosing D but not removed.
In view of the consistency with the recent galactic data and flexibility as above, we suggest an overall range −4.84 × 10 −18 ≤ D ≤ 0, which in turn leads to 0 ≤ ω(r) ≤ 2.8 × 10 −7 for the perfect fluid singularity-free equation of state of dark matter. As we see, the values are concentrated around D ∼ 0 leading to a strong constraint of dust-like dark matter (ω ∼ 0), which is supported also by CMB constraints [59,60]. This is the main result of our paper.
We recall that an acceptable practical, working definition of a galactic halo is still debatable [45]. One theoretically sound definition is that the halo is the volume enclosing all of the mass that has already decoupled from the Hubble flow. Another definition is the virial radius enclosing gravitationally bound halo mass. All of them are problematic for practical measurements. The only viable solution is to altogether avoid defining the halo as a discrete entity. Instead, one should focus on the mass profile or on the mass within a selected, fixed radius. However, all these arguments do not rule out a future observation of a terminated discrete halo, even though it might extend to hundreds of kpcs farther than the adopted R MW = 200 kpc. Within this ideology, the interval 0 ≤ ω(r) ≤ 2.8 × 10 −7 allowing flexibility in the halo radius does make good sense.

Acknowledgment
The authors wish to thank an anonymous referee for his/her insightful comments that led to significant improvements over the initial version.