Abstract

A family of potential–density pairs has been found for spherical haloes and bulges of galaxies in the Newtonian limit of scalar–tensor theories of gravity. The scalar field is described by a Klein–Gordon equation with a source that is coupled to the standard Poisson equation of Newtonian gravity. The net gravitational force is given by two contributions: the standard Newtonian potential plus a term stemming from massive scalar fields. General solutions have been found for spherical systems. In particular, we compute potential–density pairs of spherical, galactic systems, and some other astrophysical quantities that are relevant to generate initial conditions for spherical galaxy simulations.

1 INTRODUCTION

In recent years important progress has been achieved in understanding the dynamics that led to the formation of galaxies. Two- and three-dimensional N-body simulations of galaxies and protogalaxies have been computed using millions of particles, giving a more realistic view of how galaxies, quasars and black holes could have formed (Barnes & Hernquist 1992; Barnes 1998).

The Universe's composition at the time galaxies formed could be, theoretically, very varied, including visible and dark baryonic matter, non-baryonic dark matter, neutrinos and many cosmological relics stemming from symmetry-breaking processes predicted by high-energy physics (Kolb & Turner 1990). All these particles and fields, if present, should have played a role in structure formation. Accordingly, recent independent observational data measured in the cosmic microwave background radiation at various angular scales (de Bernardis et al. 2000; Bennett et al. 2003), in Type Ia supernovae (Riess et al. 1998; Perlmutter et al. 1999), as well as in the 2dF Galaxy Redshift survey (Peacock et al. 2002; Efstathiou 2002), suggest that Ω=ΩΛm≈ 1, or ΩΛ≈ 0.7 and Ωm≈ 0.3, implying the existence of dark energy and dark matter, respectively. One particular candidate for dark energy is a scalar field potential usually called ‘quintessence’ (Caldwell, Dave & Steinhardt 1998). In this way galaxies are expected to possess dark components and, in accordance with the rotation curves of stars and gas around the centres of spirals, they might be in the form of haloes (Ostriker & Peebles 1973) and must contribute to at least 3–10 times the mass of the visible matter (Kolb & Turner 1990).

In order to construct numerical galaxies in a consistent way using gravity and particle physics components, one must find for the correct physical scheme. The theoretical framework to explain the existence of dark components finds its origin in theories of elementary particle physics, with the addition of the action of gravity. There are plenty of theories (grand unification schemes, string theories, brane worlds, etc.) that involve such physics, but scalar–tensor theories (STT) of gravity are typically found to represent classical effective descriptions of such original theories (Green, Schwarz & Witten 1988). In this way, the scalar fields of these theories are the natural candidates to be the quintessence field (Caldwell et al. 1998; Boisseau et al. 2000; Amendola 2001), as a remnant of some cosmological function that contributes with ΩΛ≈ 0.7 today. It has even been suggested that the quintessence field is the scalar field that also acts on local planetary scales (Fujii 2000) or on galactic scales (Matos & Guzman 2001; Mota & Van de Bruck 2004). Moreover, massive scalar fields might account for the dark matter components of galaxies in the form of haloes.

Motivated by the above arguments, in the present report we study some STT effects in galactic systems. Specifically, we compute potential–density pairs coming from such theories in their Newtonian approximation. The results presented here will be useful in constructing numerical galaxies that are consistent with grand unification schemes.

This paper is organized as follows: In Section 2 we present the Newtonian approximation of a general scalar–tensor theory of gravity. Solutions are presented in terms of integrals of Green's functions, and some expressions for the velocities and dispersions of stars in galaxies are given. In Section 3, solutions for a family of potential–density pairs (Dehnen 1993) and the NFW model (Navarro, Frenk & White 1996) are presented, and some relevant observational quantities are computed. In Section 4 we present conclusions.

2 SCALAR FIELDS AND THE NEWTONIAN APPROXIMATION

Let us consider a typical scalar–tensor theory given by the following Lagrangian: 

1
formula
from which we get the gravitational equations 
2
formula
and the scalar field equation 
3
formula
where ()′≡∂/∂φ. In the past we have studied (Cervantes-Cota & Dehnen 1995a, b), among others, this type of non-minimal coupling between gravity and matter fields in a cosmological context for the very early Universe, where all relativistic effects have to be taken into account. In the present study, however, we want to consider the influence of scalar fields within galaxies, and therefore we need to describe the theory in its Newtonian approximation, that is, gravity and the scalar fields are weak and velocities of stars are non-relativistic. We expect to have small deviations of the scalar field around the background field, defined here as 〈φ〉= 1. If one defines the perturbation graphic, the Newtonian approximation gives (Helbig 1991; Salgado 2002) 
4
formula
 
5
formula
where we have introduced 
formula
and α≡ 1/(3 + 2ω). In the above expansion we have set the cosmological constant term equal to zero, since on galactic scales its influence should be negligible. We only consider the influence of dark matter due to the boson field of mass m governed by equation (5), that is a Klein–Gordon equation with a source.

Note that equation (4) can be cast as a Poisson equation for graphic: 

6
formula

The new Newtonian potential is now given by 

7
formula
The next step is to find solutions for this new Newtonian potential given a density profile, that is, to find the so-called potential–density pairs. We present the solutions first for point-like masses and secondly for general density distributions.

2.1 Point-like masses

The solution of these equations is, in this case, known: 

8
formula
where 
9
formula
where ms is a source mass, and the total gravitational force on a particle of mass mi is 
10
formula
Here graphic is the Compton wavelength of the effective mass (m) of some elementary particle (boson) given through the potential V(φ) and ω(φ). In what follows we will use λ instead of m−1. This mass can have a range of values depending on particular particle physics models. The potential u is the Newtonian part and uλ is the dark matter contribution, which is of the Yukawa type. There are two limits. On the one hand, if r≫λ (or λ→ 0) one recovers the Newtonian theory of gravity. On the other hand, if r≪λ (or λ→∞) one again obtains the Newtonian theory, but now with a rescaled Newtonian constant, GGN (1 +α). There are stringent constraints on the possible λ–α values determined by measurements on local scales (Fischbach & Talmadge 1999).

In the past the above solutions have been used to solve the missing mass problem in spirals (Sanders 1984; Eckhardt 1993) as an alternative to considering a distribution of dark matter. This was done by thinking that most of the galactic mass is located in the galactic centre, and then considering it as a point-like source. In our present investigation we do not avoid having dark matter, since our model predicts that bosonic dark matter produces, through a scalar field associated to it, a modification of Newtonian gravity theory. This dark matter is presumably clumped in the form of dark haloes. Therefore we will consider in what follows that a dark halo is spherically distributed along an observable spiral and beyond, having some density profile (Section 3). Next, we compute the potentials, and some astrophysical quantities, for general halo density distributions.

2.2 General density distributions

General solutions to equations (5) and (6) can be found in terms of the corresponding Green functions 

11
formula
 
12
formula
(BC = boundary conditions) and the new Newtonian potential is 
13
formula
The first term of equation (13), given by ψ, is the contribution of the usual Newtonian gravitation (without scalar fields), while information about the scalar field is contained in the second term, that is, arising from the influence function determined by the Klein–Gordon Green's function, where the coupling ω (α) enters as part of a source factor.

Given that we are interested in spherical galaxies and bulges in what follows, we will only consider the case of spherical symmetry. Additionally, we use flatness boundary conditions (BC) at infinity, such that the boundary terms in equations (11) and (12) are zero. Moreover, regularity conditions must be applied to spatial points where the potentials are singular. For spherical systems these conditions mean that graphic at the origin. Accordingly, performing the integrals in these equations we obtain ψ for a system of radius R, 

14
formula
and for graphic 
15
formula
where we have defined the functions 
16
formula
and we have written the density as 
17
formula
Here M is a reference or scaling mass and a is a scaling length that we use to define the following quantities: rar/a, RaR/a, λa≡λ/a and η≡α/λa. Then, the new Newtonian potential can be written as 
18
formula
where 
19
formula

2.3 Intrinsic and projected properties

We now want to present the intrinsic and observable quantities of interest that characterize the steady state of spherical galaxies and bulges. These include the circular velocity, the velocity dispersion, the distribution function, the projected velocity dispersion and the projected density.

The circular velocity for a spherical galactic systems is, from the above equations, 

20
formula
For ra < Ra one obtains straightforwardly the circular velocities for stars inside a dark halo of size R. For raRa one gets 
21
formula
where 
22
formula
which gives the circular velocities outside the halo (if there are stars there). From equations (19) and (20) one obtains, after redefinition of the multiplicative constant, the expression found in the past for rotational velocities due to point masses (Sanders 1984).

A full description of stars in a galaxy is given by the collisionless Boltzmann equation, which after being integrated over all velocities yields the Jeans equations relating the potential and density of a system to its velocity dispersion (Binney & Tremaine 1994). In particular, the velocity dispersion profile is important in determining the stationary state of the system. For equilibrium stationary states, we found the velocity dispersion for a spherically symmetric stellar system (using equation 20) to be 

23
formula
where we have assumed that graphic and graphic. We have used similar expressions to generate initial conditions of protogalaxies to study the influence of scalar fields on the transfer of angular momentum during collisions (Rodríguez-Meza et al. 2001).

The distribution function of the isotropic models can be calculated using the Abel transform (Binney & Tremaine 1994). First, we define graphic being the total energy, and Ψ(ra) =−(a/MN(ra), and we obtain r as a function of Ψ by inverting. The mass density is 

24
formula
The inversion of this equation is done using the Eddington formula and we obtain for the distribution function 
25
formula

The projected velocity dispersion for an isotropic system is given by (Binney & Tremaine 1994) 

26
formula
where graphic is the projected density. This also gives us the cumulative surface density 
27
formula

3 POTENTIAL–DENSITY PAIRS

In the past some potential–density pairs have been studied within the framework of Newtonian gravity. The Jaffe density model (Jaffe 1983) and Hernquist model (Hernquist 1990) have been generalized by Dehnen (1993 by introducing a free parameter (γ) that determines particular density models for galaxy description. Another density profile of interest, obtained from N-body cosmological simulations, has been proposed by Navarro et al. (1996, NFW). In this section we will consider both models in order to study the influence of scalar fields on spherical systems.

3.1 Dehnen's profile

We use the family of density profiles for spherical haloes and bulges of galaxies proposed by Dehnen (1993, 

28
formula
where a is a scaling radius and M denotes the total mass. The Hernquist profile corresponds to γ= 1 and Jaffe's to γ= 2.

Solving the Poisson equation without a scalar field, i.e. for Newtonian gravity, the potential that corresponds to the density of equation (28) is (Dehnen 1993) 

29
formula
Owing to the influence of the scalar field, however, ψ(r) does not represent the total gravitational potential of the system. The effective Newtonian potential is now graphic determined by equation (13). Thus, using equation (28) for the density, we obtain for ra < Ra 
30
formula
and for raRa 
31
formula

Using equation (28) one finds explicitly the functions m, p and q: 

32
formula
 
33
formula
 
34
formula
where 2F1(a, b; c, z) is a hypergeometric function.

For ra < Ra the integral in the first term of equation (30) can be evaluated, giving for the standard Newtonian potential 

35
formula
which for Ra→∞ reduces to equation (29).

The circular velocity is, for ra < Ra, 

36
formula
and for raRa 
37
formula
The first term on the right-hand side in each of equations (36) and (37) is the contribution from ψ and the second term is the contribution from the scalar field. The latter term depends on Ra, λa and η=α/λa, rather than on α alone, which is typical for forces exerted on point systems (see Fischbach & Talmadge 1999). Therefore, the stringent constraints on α found for point masses at local scales are less severe here, since the above terms can still yield a large amplitude, even when α is small. One sees that for rR, λ, the scalar field influence decays exponentially, recovering the standard Newtonian result.

The velocity dispersion can be computed by substituting equations (36) and (37) into equation (23), and this determines completely the initial conditions for spherical galaxies. Fig. 1 shows the circular velocity and the velocity dispersion curves, graphic, for the Hernquist model (γ= 1) for η= 1 (upper curves) and η= 0 (lower curves), i.e. upper curves are computed with the scalar field taken into account and the lower curves without a scalar field. In the limit λ→∞ one recovers the Newtonian result as expected. The computations were done with R→∞.

1

(a) Rotation curves and (b) velocity dispersion curves for the Hernquist model. The upper curves were computed with λa= 1 and η= 1, while the lower curves were obtained for λa= 1 and η= 0, i.e. without a scalar field.

1

(a) Rotation curves and (b) velocity dispersion curves for the Hernquist model. The upper curves were computed with λa= 1 and η= 1, while the lower curves were obtained for λa= 1 and η= 0, i.e. without a scalar field.

3.2 NFW profile

The N-body cosmological simulations (Navarro et al. 1996) suggest a universal density profile for spherical dark haloes of the following type: 

38
formula
where rs is a scale radius, δc is a dimensionless constant, and ρcrit is the critical density (for closure) of the Universe. This profile is the same as Dehnen's with γ= 1 (Hernquist model) for rrs, but it differs at other radii.

This halo density profile was obtained in cosmological simulations in which the Newtonian theory of gravity was used. Because we are considering λ≪H−1, then all predictions of such cosmological simulations must be taken into account; this is because in the present formalism for r≫λ one gets the Newtonian theory.

Solving equations (5) and (6) with the density profile (38) we get for ra < Ra, 

39
formula
and for raRa 
40
formula
where the reference length a=rs. The circular velocity is given by equations (36) and (37) where the functions m(ra), o(ra), p(ra) and q(ra) are given by 
41
formula
 
42
formula
 
43
formula
 
44
formula
where Ei(z), Shi(z) and Chi(z) are the exponential, hyperbolic sine and hyperbolic cosine integrals, respectively.

In Fig. 2 we plot the circular velocity and its dispersion for the NFW profile. The values of λa and η are the same as in Fig. 1. The upper curves correspond to the case in which we take into account the scalar field, while the lower curves were computed for standard Newtonian gravity. We took the size of the distribution of mass to be R= 20 rs.

2

(a) Rotation curves and (b) velocity dispersion curves for the NFW density profile. Curves were computed with λa= 1 and upper (lower) curve with η= 1(η= 0).

2

(a) Rotation curves and (b) velocity dispersion curves for the NFW density profile. Curves were computed with λa= 1 and upper (lower) curve with η= 1(η= 0).

The influence of the scalar field is to enhance, especially the inner regions, both the velocity curves and dispersion. These effects are a little more pronounced in the Hernquist model than in the NFW model.

4 DISCUSSION AND CONCLUSIONS

We have found potential–density pairs of spherical galactic systems within the context of linearized scalar-tensor theories of gravitation. The influence of massive scalar fields is given by graphic, determined by equation (15). General expressions have been given for circular velocities and dispersions of stars in the spherical system; by stars we mean probe particles that follow the dark halo potential. Specifically, these results were used to find potential–density pairs for the generic Dehnen density parametrization, as well as for the NFW profile. In general the contribution due to massive scalar fields is non-trivial (see e.g. equations 36 and 37), and, interestingly, forces on circular orbits of stars depend on the amplitude terms Ra, λa and η=α/λa. This means that, even when local experiments force α to be a very small number (Fischbach & Talmadge 1999), the amplitudes of forces exerted on stars are not necessarily very small and may contribute significantly to the dynamics of stars. Alternatively, one may interpret the local Newtonian constant as being given by (1 +α) 〈φ〉−1, instead of by 〈φ〉−1 (1 in our convention). In this case, the local measurement constraints are automatically satisfied, and at scales larger than λ one sees a reduction of 1/(1 +α) in the Newtonian constant. If this were the case, then the upper curves in Figs 1 and 2 would have to be multiplied by 1/(1 +α).

In the past, different authors have used point solutions, equations (8) and (9), to solve the missing mass problem encountered in the rotation velocities of spirals and in galaxy cluster dynamics (Sanders 1984; Eckhardt 1993). These models were used as an alternative to avoid dark matter. Indeed, for single galaxies one can adjust the parameters (α, λ) to solve these problems without the need for dark matter. However, these models do not provide a good description of the systematics of galaxy rotational curves because they predict the scale λ to be independent of the galactic luminosity, and this conflicts with observations for different galactic sizes (Aguirre et al. 2001) unless one assumes various λ values, and hence various fundamental masses, m, one for each galaxy size. Such a particle spectrum is not expected from theoretical arguments; it represents a considerable fine-tuning of masses. This criticism would also apply to our models. The purpose of the present investigation was, however, not to present a model alternative to dark matter to solve the missing mass problem, but to compute the influence of the scalar-field dark matter distributed in the form of a dark halo. This contribution, together with those from bulge and disc, will give rise to a flat velocity curve, as in Newtonian mechanics. Yet, the influence of the scalar field from STT yields some profile modification, especially in inner regions, as shown in Figs 1 and 2.

Acknowledgments

This work was supported in part by the DAAD and CONACYT grant number 33278-E.

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