A Scalar Field Dark Matter Model and Its Role in the Large-Scale Structure Formation in the Universe

Abstract

We present a model of dark matter based on scalar-tensor theory of gravity. With this scalar field dark matter model we study the non-linear evolution of the large-scale structures in the universe. The equations that govern the evolution of the scale factor of the universe are derived together with the appropriate Newtonian equations to follow the nonlinear evolution of the structures. Results are given in terms of the power spectrum that gives quantitative information on the large-scale structure formation. The initial conditions we have used are consistent with the so-called concordance ΛCDM model.

1. Introduction

The standard model of cosmology is supported by three main astronomical observations: the surveys of supernovae Ia, the cosmic microwave background radiation (CMB), and the primordial nucleosynthesis. These observations together with other modern cosmological observations, like galaxies surveys (SDSS, 2dF), galaxy rotation curves, the Bullet Cluster observation, studies of clusters of galaxies, establish that the universe behaves as dominated by dark matter (DM) and dark energy. However, the direct evidence for the existence of these invisible components remains lacking. Several theories that would modify our understanding of gravity have been proposed in order to explain the large-scale structure formation in the universe and the galactic dynamics. The best model we have to explain the observations is the ΛCDM model, that is, the model of cold dark matter (CDM)—nonrelativistic particles of unknown origin—with cosmological constant (Λ). In particular, this model explains very well the universe on scales of galaxy clusters and up [1].

The ΛCDM model has become the theoretical paradigm leading the models of the universe to explain the large-scale structure (LSS) formation and several other observations, where “large” means scales larger than 1 Mpc—about the size of the group of galaxies that our Milky Way belongs to. Together with the cosmic inflation theory, this model makes a clear prediction about the necessary initial conditions that the universe has to have in order to have the structures we observe and that those structures build hierarchically due to a gravitational instability. One of its main predictions is that the density profile of galaxies, clusters of galaxies, and so on is of the form [2, 3]𝜌_NFW(𝑟)=𝜌₀𝑟/𝑟₀1+𝑟/𝑟₀², (1)

a density profile known as Navarro-Frenk-White profile (NFW). Parameters 𝜌₀ and 𝑟₀ must be fitted, for example, using rotation curves of galaxies.

The ΛCDM model and its success in explaining several observations—this is why this model is also known as the concordance model—yields the following conclusions: on large-scales, the universe is homogeneous and isotropic, as described by the Friedmann-Lamaître-Robertson-Walker (FLRW) metric. The geometry of the universe is flat, as predicted by inflation. The dark matter is cold (nonrelativistic at decoupling epoch). The initial density fluctuations were small and described by a Gaussian random field. The initial power spectrum of the density fluctuations was approximately the Harrison-Zeldovich spectrum (𝑃(𝑘)∝𝑘^𝑛, 𝑛=1) [4, 5].

In terms of the composition of the universe, the above conclusions can be summarized as follows: Hubble’s constant (Expansion rate of the universe at the present epoch): 𝐻₀=73.2±3.1km/s/Mpc. Density parameter (combined mass density of all kind of mass and energy in the universe, divided by the critical density): Ω₀=1.02±0.02. Matter density parameter (combined mass density of all forms of matter in the universe, divided by the critical density): Ω_𝑚=0.241±0.034. Ordinary matter parameter density (density of mass of ordinary atomic matter in the universe divided by the critical density): Ω_𝑏=0.0416±0.001. Density parameter of dark energy (energy density of dark energy in the universe divided by the critical density): Ω_Λ=0.759±0.034 [6].

Even though of all successes of the ΛCDM, this model has several problems. Some of them are as follow: The exotic weakly interacting particles proposed as dark matter particles candidates are still undetected in the laboratory. The number of satellites in a galaxy such as the Milky Way is predicted to be an order of magnitude larger than is observed. The cuspy halo density profiles problem. The lack of evidence in the Milky Way for a major merger is hard to reconcile with the amount of accretion predicted by ΛCDM. With respect to the inclusion of the cosmological constant, the ratio of the vacuum energy density to the radiation energy density after inflation is 1 part in 10¹⁰⁰, a fine tuning coincidence. The cosmological constant has the wrong sign according to the string theorists who prefer a negative Λ instead of a positive Λ. The ΛCDM model predicts that large structures should form last and therefore should be young whereas observations tell us that the largest galaxies and clusters appear old. Several candidates have been proposed in the past that pretend to substitute the role that the cosmological constant plays to accelerate the expansion of the universe. That the universe is expanding is supported mainly by the observations of the supernova project SNIa [7, 8]. And we have led to conclude that the universe is now dominated by an energy density with negative pressure and occupies about 70% of the universe in the present epoch. This energy is called generically the dark energy. Several models to explain this dark component have been proposed, and they may be classified according to their equations of state as the follow: quintessence dark energy [9, 10], phantom energy [11, 12], or the quintom cosmology paradigm [13] (see also the review [14] for even more details). Cosmological constant has as its equation of state one in which the pressure is the negative of the density.

From the 𝑁-body numerical simulations point of view we have two works, in which a detailed analysis of two scalar field possibilities has been explored. One is the coupled quintessence models [15] and the other is the extended quintessence models [16]. The former considers a scalar field coupled minimally to the Ricci scalar and is coupled to the Lagrangian matter contribution. The latter is of the type of a scalar-tensor theory in which the scalar field is introduced to model dark energy with a potential that is an inverse power law. The model we present in this paper pretends to model the dark matter contribution in the large-scale structure formation with a scalar field model that stems from the newtonian limit of a general scalar-tensor theory [17].

Another problem that the ΛCDM problem is facing is the following. Almost a decade ago a bow shock in the merging cluster 1E0657-56, known as the Bullet Cluster, observed by satellite Chandra indicates that the subcluster—found by [18]—moving through this massive (10¹⁵ℎ⁻¹𝑀_⊙) main cluster creates a shock with a velocity as high as 4700 km s⁻¹ [19, 20]. A significant offset between the distribution of X-ray emission and the mass distribution has been observed [21, 22], also indicating a high-velocity merger with gas stripped by ram pressure. Several authors have done detailed numerical noncosmological simulations [23–27]. One of the key input parameters for the simulations is to set the initial velocity of the subcluster, which is usually given at somewhere near the virial radius of the main cluster. Authors of [28] have run cosmological N-body simulation using a large box (27 ℎ⁻³Gpc³) to calculate the distribution of infall velocities of subclusters around massive main clusters. The infall velocity distribution was given at 1–3 𝑅₂₀₀—similar to the virial radius—and thus it gives the distribution of realistic initial velocities of subclusters just before collision. This distribution of infall velocities must be compared with the best initial velocity used by [23] of 3000 km s⁻¹ at about 2𝑅₂₀₀ to be in agreement with observations. Authors of [28] have found that such a high infall velocity is incompatible with the prediction of the ΛCDM.

Therefore, there are plenty of problems that the concordance model has to solve and finally, it does not tell us what are dark matter and dark energy.

The program to study the large scale structure formation should be to start with primordial initial conditions which means to give the initial relevant fields, such as, for example, density and velocity fields at the epoch of last scattering (~z = 1100, the value of the redshift at that epoch, i.e., a photon emitted at that epoch is redshifted as 1+𝑧=𝜆_𝑟/𝜆_𝑒, with 𝜆_𝑒 the wavelength of the photon when emitted, and 𝜆_𝑟 is the wavelength of the same photon when observed; the expansion factor for a universe with a flat geometry is related to the redshift as 1+𝑧=1/𝑎). Then, evolve this initial condition using an 𝑁-body scheme up to the present epoch (𝑧=0).

Some questions we have to answer are as follows What is the distribution of the LSS sizes? What is the amount of mass and its distribution at large scales? How are the voids distributed through the space? Are these voids devoid of any matter? How dose the LSS evolve with time? What is the DM equation of state? What is its role in the LSS formation processes and galactic dynamics? What are the implications of the observed LSS on the cosmological model of our universe and on the structure formation? And of course, what is the nature of the dark energy and matter?

During the last decades, there have been several proposals to explain DM, for example, Massive Compact Halo Objects (Machos), Weakly Interacting Massive Particles (WIMPs) such as supersymmetric particle like the neutralino. Other models propose that there is no dark matter and use general relativity with an appropriate equation of state. Or we can use scalar fields, minimally or nonminimally coupled to the geometry.

In this work we are mainly concerned with the problem of dark matter and its consequences on the large-scale structure formation process. Our DM model is based on using a scalar field (SF) that is coupled nonminimally to the metric through the Ricci scalar in the Einstein field equations. A scalar field is the most simple field of nature. Nordstrom proposed a gravity theory by 1912, before Einstein [29]. Scalar fields have been around for so many years since the pioneering work of Jordan, Brans, and Dicke. Nowadays, they are considered as (a) inflation mechanism (b) the dark matter component of galaxies (c) the quintessence field to explain dark energy and so on. Therefore, is natural to consider dark matter models based on modifications of Einstein’s general relativity that include scalar fields. In this paper we will show results about the role this scalar field plays on the nonlinear large-scale structure formation of the universe. In particular, we will show how the power spectrums predicted by this model compare with the power spectrums predicted by ΛCDM and the ones that come from observations.

So we organize our work in the following form In Section 2 we present the general theory of a typical scalar-tensor theory (STT), that is, a theory that generalizes Einstein’s general relativity by including the contribution of a scalar field that couples nonminimally to the Ricci scalar. In Section 3 we show how the Friedmann equations become within an STT and present our model for the evolution of the universe expansion factor 𝑎(𝑡). In Section 4 we present the 𝑁-body method which we will use to obtain the evolution of the large-scale structures. Our results for an initial condition of the fields that is consistent with the observations are given in Section 5. Finally, our conclusions are given in Section 6.

2. General Scalar-Tensor Theory and Its Newtonian Limit

The Lagrangian that gives us the Einstein equations of general relativity is√ℒ=−𝑔𝑅16𝜋𝐺. (2) The Einstein field equations that are obtained from the above Lagrangian, in the limit of small velocities as compared with the speed of light and small forces, limit known as the Newtonian limit, give us the standard Newtonian potential due to a point particle of mass 𝑚Φ_𝑁𝑚=−𝐺𝑟, (3) where 𝐺 is the gravitational constant. What we intend to do in this work is to obtain the consequences in the LSS formation processes when we rise the constant 𝐺 to a scalar field, 1/𝐺→𝜙. But we go beyond this approach and include in the Lagrangian two additional terms that depend on this field, kinetic and potential terms. We will show, in particular, that the Newtonian limit of this theory gives for the Newtonian potential due to a mass 𝑚 [30],Φ_𝑁𝑚=−𝐺𝑟1+𝛼𝑒^−𝑟/𝜆. (4) That is, the standard Newtonian potential is modified by an additional term that has the form of a Yukawa potential.

Then, we start with the Lagrangian of a general scalar-tensor theory√ℒ=−𝑔𝜔𝜙16𝜋−𝜙𝑅+𝜙𝜕𝜙²−𝑉𝜙+ℒ_𝑀𝑔_𝜇𝜈. (5) Here 𝑔_𝜇𝜈 is the metric, ℒ_𝑀(𝑔_𝜇𝜈) is the matter Lagrangian, and 𝜔(𝜙) and 𝑉(𝜙) are arbitrary functions of the scalar field. The fact that we have a potential term 𝑉(𝜙) tells us that we are dealing with a massive scalar field. Also, the first term in the brackets, 𝜙𝑅, is the one that gives the name of nonminimally coupled scalar field.

When we make the variations of the action, 𝑆=∫𝑑⁴𝑥ℒ, with respect to the metric and the scalar field we obtain the Einstein field equations [29]𝑅_𝜇𝜈−12𝑔_𝜇𝜈1𝑅=𝜙8𝜋𝑇_𝜇𝜈12𝑉_𝑔𝜇𝜈𝜔𝜙𝜕_𝜇𝜙𝜕_𝑣𝜙−12𝜔𝜙𝜕𝜙²𝑔_𝜇𝜈+𝜙_;𝜇𝜈−𝑔_𝜇𝜈□𝜙, (6) for the metric 𝑔_𝜇𝜈 and for the massive SF 𝜙 we have1□𝜙=3+2𝜔8𝜋𝑇−𝜔′𝜕𝜙²+𝜙𝑉′−2𝑉, (7) where ()^′≡𝜕/𝜕𝜙. Here 𝑇_𝜇𝜈 is the energy-momentum tensor with trace 𝑇, and 𝜔(𝜙) and 𝑉(𝜙) are in general arbitrary functions that govern kinetic and potential contribution of the SF. If in Lagrangian (5) we set 𝑉(𝜙)=0 we get the Bergmann-Wagoner theory. If we further set 𝜔(𝜙)=constant the Jordan-Brans-Dicke theory is recovered. The gravitational constant is now contained in 𝜙. Also, the potential contribution, 𝑉(𝜙), provides mass to the SF, denoted here by 𝑚_SF.

2.1. Newtonian Limit of an STT

The study of large-scale structure formation in the universe is greatly simplified by the fact that a limiting approximation of general relativity, the Newtonian mechanics, applies in a region small compared to the Hubble length 𝑐𝐻⁻¹ (𝑐𝐻₀⁻¹≈3000ℎ⁻¹ Mpc, where 𝑐 is the speed of light, 𝐻₀=100ℎ km/s/Mpc, is Hubble’s constant, and ℎ≈(0.65−0.75)) and large compared to the Schwarzschild radii of any collapsed objects. The rest of the universe affects the region only through a tidal field. The length scale 𝑐𝐻₀⁻¹ is of the order of the largest scales currently accessible in cosmological observations and 𝐻₀⁻¹≈10¹⁰ℎ⁻¹ yr characterizes the evolutionary time scale of the universe [31].

Therefore, in the present study, we need to consider the influence of SF in the limit of a static STT, and then we need to describe the theory in its Newtonian approximation, that is, where gravity and the SF are weak (and time independent) and velocities of dark matter particles are nonrelativistic. We expect to have small deviations of the SF around the background field, defined here as ⟨𝜙⟩ and can be understood as the scalar field beyond all matter. Accordingly we assume that the SF oscillates around the constant background field𝜙=𝜙+𝑔𝜙,_𝜇𝜈=𝜂_𝜇𝜈+ℎ_𝜇𝜈, (8)

where 𝜂_𝜇𝜈 is the Minkowski metric. Then, Newtonian approximation gives [30, 32–34] 𝑅₀₀=12∇²ℎ₀₀=𝐺_𝑁11+𝛼4𝜋𝜌−2∇²∇𝜙,₂𝜙−𝑚²_SF𝜙=−8𝜋𝛼𝜌. (9)(10) We have set ⟨𝜙⟩=(1+𝛼)/𝐺_𝑁 and 𝛼≡1/(3+2𝜔). In the above expansion we have set the cosmological constant term equal to zero, since on small galactic scales its influence should be negligible. However, at cosmological scales we do take into account the cosmological constant contribution (see below).

Note that (9) can be cast as a Poisson equation for 𝜓≡(1/2)(ℎ₀₀+𝜙/⟨𝜙⟩)∇²𝐺𝜓=4𝜋^𝑁𝜌1+𝛼, (11) and the New Newtonian potential is given by Φ_𝑁≡(1/2)ℎ₀₀=𝜓−(1/2)𝜙/⟨𝜙⟩. The above equation together with∇²𝜙−𝜆⁻²𝜙=−8𝜋𝛼𝜌, (12) forms a Poisson-Helmholtz equation and givesΦ_𝑁1=𝜓−2𝐺_𝑁1+𝛼𝜙 (13)

which represents the Newtonian limit of the STT with arbitrary potential 𝑉(𝜙) and function 𝜔(𝜙) that were Taylor expanded around ⟨𝜙⟩. The resulting equations are then distinguished by the constants 𝐺_𝑁, 𝛼, and 𝜆=ℎ_𝑃/𝑚_SF𝑐. Here ℎ_𝑃 is Planck’s constant.

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. General solutions to (11) and (12) can be found in terms of the corresponding Green functions, and the new Newtonian potential is [30, 35] Φ_𝑁𝐺=−^𝑁1+𝛼d𝐫_𝑠𝜌𝐫_𝑠𝐫−𝐫_𝑠𝐺−𝛼^𝑁1+𝛼d𝐫_𝑠𝜌𝐫_𝑠𝑒^{−|𝐫−𝐫𝑠|/𝜆}𝐫−𝐫_𝑠+B.C. (14) The First term of (14) is the contribution of the usual Newtonian gravitation (without SF), while information about the SF is contained in the second term, that is, arising from the influence function determined by the modified Helmholtz Green function, where the coupling 𝜔 (𝛼) enters as part of a source factor.

The potential of a single particle of mass 𝑚 can be easily obtained from (14) and is given byΦ_𝑁𝐺=−^𝑁𝑚1+𝛼𝑟1+𝛼𝑒^−𝑟/𝜆. (15) For local scales, 𝑟≪𝜆, deviations from the Newtonian theory are exponentially suppressed, and for 𝑟≫𝜆 the Newtonian constant diminishes (augments) to 𝐺_𝑁/(1+𝛼) for positive (negative) 𝛼. This means that (15) fulfills all local tests of the Newtonian dynamics, and it is only constrained by experiments or tests on scales larger than—or of the order of—𝜆, which in our case is of the order of galactic scales. In contrast, the potential in the form of (4) with the gravitational constant defined as usual does not fulfill the local tests of the Newtonian dynamics [36].

It is appropriate to give some additional details on the Newtonian limit for the Einstein equations without scalar fields (see [31]). We are considering a small region compared to the Hubble length 𝑐𝐻⁻¹ but large compared to the Schwarzschild radii of any collapsed object. In this small region the metric tensor was written as 𝑔_𝜇𝜈=𝜂_𝜇𝜈+ℎ_𝜇𝜈, where ℎ_𝜇𝜈 is small as compared to the Minkowski metric 𝜂_𝜇𝜈. In this region Einstein’s field equations are simple because the standard weak field linear approximation applies. One finds𝑅₀₀1=−2𝜂^𝜇𝜈ℎ_𝜇𝜈,00−ℎ_𝜇0,𝜈0−ℎ_𝜈0,𝜇0+ℎ_00,𝜇𝜈,=∇²_𝑟𝑔Φ,₀₀=𝑐²+2Φ. (16)(17) Then, the zero-zero component of the field equations for an ideal fluid with density 𝜌, pressure 𝑝, and velocity 𝑣≪𝑐 becomes∇²_𝑟Φ=4𝜋𝐺_𝑁𝑝𝜌+3𝑐²−Λ. (18) For completeness the cosmological constant has been added. The geodesic equations, in the limit 𝑣≪𝑐, ℎ≪1, are𝑑²𝑟^𝑖𝑑𝑡²=−Φ_,𝑖. (19) Equations (18) and (19) are the standard equations of Newtonian mechanics, except that if there is an appreciable radiation background, one must take into account the active gravitational mass associated with the pressure, and of course if Λ≠0, there is the cosmic force Λ𝐫/3 between particles at separation 𝐫.

Equations (18) and (19) apply to any observer outside a singularity, though depending on the situation, the region within which these equations apply need not contain much matter. The region can be extended by giving the observer an acceleration 𝑔_𝑖 to bring the observer to rest relative to distant matter, which adds the term 𝑔_𝑖𝑟^𝑖 to Φ, and then by patching together the results from neighboring observers. This works (the acceleration and potentials can be added) as long as relative velocities of observers and observed matter are ≪𝑐 and Φ≪𝑐² (see (17)). For a region of size 𝑅 containing mass 𝑀∼𝜌𝑅³ with density 𝜌 roughly uniform, this second condition is𝐺_𝑁𝜌𝑅²≪𝑐². (20) In the Friedmann-Lemaître models Hubble’s constant is𝐻∼𝐺_𝑁𝜌^1/2. (21) If one assumes Λ is negligible and the density parameter Ω∼1, so (20) indicates𝑅≪𝑐𝐻⁻¹∼3000Mpc∼10²⁸cm. (22) That is, the region must be small compared to the Hubble length. Since the expansion velocity is 𝑣≈𝐻𝑟, this condition also says 𝑣≪𝑐.

The Newtonian approximation can fail at much smaller 𝑅 if the region includes a compact object like a neutron star or black hole, but one can deal with this by noting that at distances large compared to the Schwarzschild radius the object acts like an ordinary Newtonian point mass. It is speculated that in nuclei of galaxies there might be black holes as massive as 10⁹𝑀_⊙, Schwarzschild radius ~10¹⁴ cm. If this is an upper limit, Newtonian mechanics is a good approximation over a substantial range of scales, 10¹⁴cm≪𝑟≪10²⁸cm.

2.2. Multipole Expansion of the Poisson-Helmholtz Equations

The Poisson’s Green function can be expanded in terms of the spherical harmonics, 𝑌_𝑙𝑛(𝜃,𝜑),1𝐫−𝐫_𝑠=4𝜋^∞𝑙_{𝑙=0𝑛=−𝑙}1𝑟2𝑙+1𝑟^𝑙<_>^𝑙+1𝑌^∗_𝑙𝑛𝜃^′,𝜑^′𝑌_𝑙𝑛𝜃,𝜑, (23)

where 𝑟_< is the smaller of |𝐫| and |𝐫_𝑠| and 𝑟_> is the larger of |𝐫| and |𝐫_𝑠| and it allows us that the standard gravitational potential due to a distribution of mass 𝜌(𝑟), without considering the boundary condition, can be written as [37]𝜓(𝐫)=𝜓^(𝑖)+𝜓^(𝑒), (24)

where 𝜓^(𝑖) (𝜓^(𝑒)) are the internal (external) multipole expansion of 𝜓:𝜓^(𝑖)=−^∞𝑙_{𝑙=0𝑛=−𝑙}√4𝜋𝑞2𝑙+1^{(𝑖)𝑙𝑛}𝑌_𝑙𝑛𝜃,𝜑𝑟^𝑙,𝜓^(𝑒)=−^∞𝑙_{𝑙=0𝑛=−𝑙}√4𝜋𝑞2𝑙+1^{(𝑒)𝑙𝑛}𝑌_𝑙𝑛𝜃,𝜑𝑟^𝑙+1. (25)

Here, the coefficients of the expansions 𝜓^(𝑖) and 𝜓^(𝑒), known as internal and external multipoles, respectively, are given by𝑞^(𝑖)_𝑙𝑛=√4𝜋_{𝑉(𝑟≤𝑟′)}1𝑑𝐫′𝑟′^𝑙+1𝑌^∗_𝑙𝑛𝜌,𝑞𝜃′,𝜑′𝐫′_{(𝑒)𝑙𝑛}=√4𝜋_{𝑉(𝑟>𝑟′)}𝑑𝐫′𝑌^∗_𝑙𝑛𝜃′,𝜑′𝑟′^𝑙𝜌𝐫′. (26)

The integrals are done in a region 𝑉 where 𝑟≤𝑟′ for the internal multipoles and in a region 𝑉 where 𝑟>𝑟′ for the external multipoles. They have the property𝑞^(𝑖)_{𝑙(−𝑛)}=(−1)^𝑛𝑞^(𝑖)_𝑙𝑛^∗,𝑞^(𝑒)_{𝑙(−𝑛)}=(−1)^𝑛𝑞^(𝑒)_𝑙𝑛^∗. (27)

We may write expansions above in cartesian coordinates up to quadrupoles. For the internal multipole expansion we have𝜓^(𝑖)=−𝑀^(𝑖)−𝐫⋅𝐩^(𝑖)−12𝐫⋅𝐐^(𝑖)⋅𝐫, (28) and its force is𝐅_𝜓^(𝑖)=𝐩^(𝑖)+𝐐^(𝑖)⋅𝐫, (29) where𝑀^(𝑖)≡∫_{𝑉(𝑟≤𝑟′)}𝑑𝐫^′1/𝑟′𝜌𝐫^′,𝑝_𝑖^(𝑖)≡_{𝑉(𝑟≤𝑟′)}𝑑𝐫^′𝑥^′_𝑖1𝑟^′3𝜌𝐫^′,𝑄^(𝑖)_𝑖𝑗≡_{𝑉(𝑟≤𝑟′)}𝑑𝐫^′3𝑥^′_𝑖𝑥^′_𝑗−𝑟^′2𝛿_𝑖𝑗1𝑟^′5𝜌𝐫^′. (30)

For the external multipoles we have𝜓^(𝑒)𝑀=−^(𝑒)𝑟−𝐫⋅𝐩^(𝑒)𝑟³−12𝐫⋅𝐐^(𝑒)⋅𝐫𝑟⁵, (31) and its force is𝐅_𝜓^(𝑒)𝑀=−^(𝑒)𝑟³𝐩𝐫+^(𝑒)𝑟³𝐩−3^(𝑒)⋅𝐫𝑟⁵𝐫+𝐐^(𝑒)⋅𝐫𝑟⁵−52𝐫⋅𝐐^(𝑒)⋅𝐫𝑟⁷𝐫, (32) where𝑀^(𝑒)≡_{𝑉(𝑟>𝑟′)}𝑑𝐫^′𝜌𝐫^′,𝑝_𝑖^(𝑒)≡_{𝑉(𝑟>𝑟′)}𝑑𝐫^′𝑥^′_𝑖𝜌𝐫^′,𝑄^(𝑒)_𝑖𝑗≡_{𝑉(𝑟>𝑟′)}𝑑𝐫^′3𝑥^′_𝑖𝑥^′_𝑗−𝑟^′2𝛿_𝑖𝑗𝜌𝐫^′. (33)

The external multipoles have the usual meaning, that is, 𝑀^(𝑒) is the mass, 𝐩^(𝑒) is the dipole moment, and 𝐐^(𝑒) is the traceless quadrupole tensor, of the volume 𝑉(𝑟>𝑟′). We may attach to the internal multipoles similar meaning, that is, 𝑀^(𝑖) is the internal “mass,’’ 𝐩^(𝑖) is the internal “dipole’’ moment, and 𝐐^(𝑖) is the traceless internal “quadrupole’’ tensor, of the volume 𝑉(𝑟≤𝑟′).

In the case of the scalar field, with the expansionexp−𝑚𝐫−𝐫_𝑠𝐫−𝐫_𝑠=4𝜋𝑚^∞𝑙_{𝑙=0𝑛=−𝑙}𝑖_𝑙𝑚𝑟_<𝑘_𝑙𝑚𝑟_>𝑌^∗_𝑙𝑛𝜃^′,𝜑^′𝑌_𝑙𝑛𝜃,𝜑, (34)

the contribution of the scalar field to the Newtonian gravitational potential can be written as𝜙(𝐫)=𝜙^(𝑖)+𝜙^(𝑒), (35)

where, for simplicity of notation, we are using 𝑚=𝑚_SF=ℎ_𝑃/(𝑐𝜆) and12𝛼𝜙^(𝑖)=√4𝜋^∞𝑙_{𝑙=0𝑛=−𝑙}𝑖_𝑙(𝑚𝑟)(𝑚𝑟)^𝑙𝑞^(𝑖)_𝑙𝑛𝑟^𝑙𝑌_𝑙𝑛,1𝜃,𝜑2𝛼𝜙^(𝑒)=√4𝜋^∞𝑙_{𝑙=0𝑛=−𝑙}(𝑚𝑟)^𝑙+1𝑘_𝑙(𝑚𝑟)𝑞^(𝑒)_𝑙𝑛𝑌_𝑙𝑛𝜃,𝜑𝑟^𝑙+1. (36)

𝑖_𝑙(𝑥) and 𝑘_𝑙(𝑥) are the modified spherical Bessel functions.

We have defined the multipoles for the scalar field as𝑞^(𝑖)_𝑙𝑛=√4𝜋_{𝑉(𝑟≤𝑟′)}𝑑𝐫^′𝑌^∗_𝑙𝑛𝜃^′,𝜑^′𝑟^′𝑙+1𝑚𝑟^′𝑙+1𝑘_𝑙𝑚𝑟^′𝜌𝐫^′,𝑞^(𝑒)_𝑙𝑛=√4𝜋_{𝑉(𝑟>𝑟′)}𝑑𝐫^′𝑌^∗_𝑙𝑛𝜃^′,𝜑^′𝑖_𝑙𝑚𝑟^′𝑚𝑟^′𝑙𝑟^′𝑙𝜌𝐫^′. (37)

They, also, have the property𝑞_𝑙()^(𝑖)_−𝑛=(−1)^𝑛𝑞^(𝑖)_𝑙𝑛^∗,𝑞_𝑙()^(𝑒)_−𝑛=(−1)^𝑛𝑞^(𝑒)_𝑙𝑛^∗. (38)

The above expansions of SF contribution to the Newtonian potential can be written in cartesian coordinates. The internal multipole expansion of the SF contribution, up to quadrupoles, is12𝛼𝜙^(𝑖)=𝑖₀(𝑚𝑟)𝑀^(𝑖)𝑖+3¹(𝑚𝑟)𝑚𝑟𝐫⋅𝐩^(𝑖)1+52𝑖₂(𝑚𝑟)(𝑚𝑟)²𝐫⋅𝐐^(𝑖)⋅𝐫, (39) and its force is1𝐅2𝛼^𝜙(𝑖)=−𝑚²𝑖₁(𝑚𝑟)𝑚𝑟𝑀^(𝑖)𝑖𝐫−3¹(𝑚𝑟)𝑚𝑟𝐩^(𝑖)−3𝑚²𝑖₂(𝑚𝑟)(𝑚𝑟)²𝐩^(𝑖)𝑖⋅𝐫𝐫−5²(𝑚𝑟)(𝑚𝑟)²𝐐^(𝑖)1⋅𝐫+52𝑟²5𝑖₂(𝑚𝑟)(𝑚𝑟)²−𝑖₁(𝑚𝑟)𝑚𝑟𝐫⋅𝐐^(𝑖)⋅𝐫𝐫, (40) where𝑀^(𝑖)≡_{𝑉(𝑟≤𝑟′)}𝑑𝐫^′𝑚𝑟^′𝑘₀𝑚𝑟^′1𝑟^′𝜌𝐫^′,𝑝_𝑖^(𝑖)≡_{𝑉(𝑟≤𝑟′)}𝑑𝐫^′𝑚𝑟^′2𝑘₁𝑚𝑟^′𝑥^′_𝑖1𝑟^′3𝜌𝐫^′,𝑄^(𝑖)_𝑖𝑗≡_{𝑉(𝑟≤𝑟′)}𝑑𝐫^′𝑚𝑟^′3𝑘₂𝑚𝑟^′3𝑥^′_𝑖𝑥^′_𝑗−𝑟^′2𝛿_𝑖𝑗1𝑟^′5𝜌𝐫^′. (41)

In the exterior region the SF multipole contribution to the potential is12𝛼𝜙^(𝑒)=𝑚𝑟𝑘₀(𝑚𝑟)𝑀^(𝑒)𝑟+3(𝑚𝑟)²𝑘₁(𝑚𝑟)𝐫⋅𝐩^(𝑒)𝑟³1+52(𝑚𝑟)³𝑘₂(𝑚𝑟)𝐫⋅𝐐^(𝑒)⋅𝐫𝑟⁵, (42) and its force is1𝐅2𝛼^𝜙(𝑒)=(𝑚𝑟)²𝑘₁(𝑚𝑟)𝑀^(𝑒)𝑟³𝐫−3(𝑚𝑟)²𝑘₁(𝑚𝑟)𝐩^(𝑒)𝑟³+3(𝑚𝑟)𝑘₀()+()𝑚𝑟2+𝑚𝑟)(𝑚𝑟²𝑘₁(𝑚𝑟)𝐩^(𝑒)⋅𝐫𝑟⁵𝐫−5(𝑚𝑟)³𝑘₂(𝑚𝑟)𝐐^(𝑒)⋅𝐫𝑟⁵+523(𝑚𝑟)𝑘₀(𝑚𝑟)+3(𝑚𝑟)²𝑘₁()+()𝑚𝑟3+𝑚𝑟)(𝑚𝑟³𝑘₂(𝑚𝑟)𝐫⋅𝐐^(𝑒)⋅𝐫𝑟⁷𝐫, (43) where𝑀^(𝑒)≡_{𝑉(𝑟>𝑟′)}𝑑𝐫^′𝑖₀𝑚𝑟^′𝜌𝐫^′,𝑝_𝑖^(𝑒)≡_{𝑉(𝑟>𝑟′)}𝑑𝐫^′𝑖₁𝑚𝑟^′𝑚𝑟^′𝑥^′_𝑖𝜌𝐫^′,𝑄^(𝑒)_𝑖𝑗≡_{𝑉(𝑟>𝑟′)}𝑑𝐫^′𝑖₂𝑚𝑟^′𝑚𝑟^′23𝑥^′_𝑖𝑥^′_𝑗−𝑟^′2𝛿_𝑖𝑗𝜌𝐫^′. (44)

In the limit when 𝑚→0 we recover the standard Newtonian potential and force expressions.

Up to here the formulation is general; that is, mass distribution may have any symmetry or none at all. In order to take advantage of the symmetry of the spherical harmonics, the mass distribution must be spherically symmetric.

3. Cosmological Evolution Equations Using a Static STT

To simulate cosmological systems, the expansion of the universe has to be taken into account. Also, to determine the nature of the cosmological model we need to determine the composition of the universe, that is, we need to give the values of Ω_𝑖≡𝜌_𝑖/𝜌_𝑐, with 𝜌_𝑐=3𝐻²/8𝜋𝐺_𝑁, for each component 𝑖, taking into account in this way all forms of energy densities that exist at present. If a particular kind of energy density is described by an equation of state of the form 𝑝=𝑤𝜌, where 𝑝 is the pressure and 𝑤 is a constant, then the equation for energy conservation in an expanding background, 𝑑(𝜌𝑎³)=−𝑝𝑑(𝑎³), can be integrated to give 𝜌∝𝑎^−3(1+𝑤).

Then, the Friedmann equation for the expansion factor 𝑎(𝑡) is written as𝑎²̇𝑎²=𝐻²_0𝑖Ω_𝑖𝑎₀𝑎^{3(1+𝑤𝑖)}−𝑘𝑎², (45) where 𝑤_𝑖 characterizes equation of state of species 𝑖.

The most familiar forms of energy densities are those due to pressureless matter with 𝑤_𝑖=0 (i.e., nonrelativistic matter with rest-mass-energy density 𝜌𝑐² dominating over the kinetic energy density 𝜌𝑣²/2) and radiation with 𝑤_𝑖=1/3. The density parameter contributed today by visible, nonrelativistic, baryonic matter in the universe is Ω_𝐵≈(0.04−0.05) and the density parameter that is due to radiation is Ω_𝑅≈2×10⁻⁵.

In this work we will consider a model with only two energy density contributions. One is a pressureless and nonbaryonic dark matter with Ω_DM≈0.3 that does not couple with radiation, although in the numerical simulations we may include in Ω_DM the baryonic matter. The other will be a cosmological constant contribution Ω_Λ≈0.7 with and equation of state 𝑝=−𝜌. The above equation for 𝑎(𝑡) becomes𝑎²̇𝑎²=𝐻²₀Ω_DM𝑎₀𝑎³+Ω_Λ−𝑘𝑎². (46)

The above discussion gives us the standard cosmological model with cosmological constant, that is, ΛCDM model.

In the framework of a scalar-tensor theory the cosmology is given as follows. If we use the Friedmann metric [29]𝑑𝑠²=−𝑑𝑡²+𝑎²(𝑡)𝑑𝑟²1−𝜅𝑟²+𝑟²𝑑𝜃²+sin²𝜃𝑑𝜑² (47) in the time-time component of the Einstein field equations (Hamiltonian constraint) it gives𝐻²=8𝜋𝜙̇3𝜙𝜌−𝐻𝜙+𝜔𝜙6𝜙̇𝜙²−𝜅𝑎²+𝑉6𝜙 (48) while the equation for the scalar field is𝜙̈+3𝐻+𝜔̇𝜙̇=12𝜔+32𝜔+38𝜋𝜌−3𝑃−𝜙𝑑𝑉𝑑𝜙+2𝑉 (49) and the equation of the fluid is𝜌̇+3𝐻𝜌+𝑃=0. (50)

The cosmological evolution of the initial perturbed fields should be computed using the above equations for the expansion factor. However, here, we will employ a cosmological model with a static SF which is consistent with the Newtonian limit given above. Thus, the scale factor, 𝑎(𝑡), is given by the following Friedman model:𝑎³𝐻²=𝐻²₀Ω_DM0+Ω_Λ0𝑎³+Ω1+𝛼1−^DM0+Ω_Λ0𝑎1+𝛼, (51) where 𝐻=𝑎/̇𝑎 and Ω_DM0 and Ω_Λ0 are the matter and energy density evaluated at present, respectively. The denominator (1+𝛼) appears in (51) due to the fact that we have defined Ω_𝑖≡𝜌/𝜌_𝑐 and 𝜌_𝑐=3𝐻²/8𝜋𝐺_𝑁. Then, whenever appears the gravitational constant 𝐺_𝑁, we replace it by 𝐺_𝑁/(1+𝛼).

We notice that the source of the cosmic evolution is deviated by the term 1+𝛼 when compared to the standard Friedmann-Lemaître model. Therefore, it is convenient to define a new density parameter by Ω_𝑖^(𝛼)≡Ω_𝑖/(1+𝛼). This new density parameter is such that Ω^(𝛼)_DM+Ω_Λ^(𝛼)=1, which implies a flat universe, and this shall be assumed in the following computations, where we consider (Ω_𝑚^(𝛼),Ω_Λ^(𝛼))=(0.3,0.7).

For positive values of 𝛼, a flat cosmological model demands to have a factor (1+𝛼) with more energy content (Ω_𝑚 and Ω_Λ) than in standard cosmology. On the other hand, for negative values of 𝛼 one needs a factor (1+𝛼)with less Ω_𝑚 and Ω_Λ to have a flat universe. To be consistent with the CMB spectrum and structure formation numerical experiments, cosmological constraints must be applied on 𝛼 in order for it to be within the range (−1,1) [38–41]. In Figure 1 we show the effect of different values of the gravitational constant 𝐺_𝑁 on the anisotropies of the CMB.

Figure 1 

Cosmic microwave background anisotropies. With small dash symbol are the WMAP 7 years experimental observations [53]. We have plotted several cases in which the Newton constant has been modified. The standard case is shown with a small dash line. With a long dash line is shown an effective gravitational constant, 𝐺eff=0.5𝐺𝑁. With a dotted line is shown the case with 𝐺eff=1.5𝐺𝑁, and with a dash-dotted line is shown the case with 𝐺eff=2𝐺𝑁.

4. Vlasov-Poisson-Helmholtz Equations and the 𝑁-Body Method

The Vlasov-Poisson equation in an expanding universe describes the evolution of the six-dimensional, one-particle distribution function, 𝑓(𝐱,𝐩). The Vlasov equation is given by𝜕𝑓+𝐩𝜕𝑡𝑚𝑎²⋅𝜕𝑓𝜕𝐱−𝑚∇Φ_𝑁(𝐱)⋅𝜕𝑓𝜕𝐩=0, (52) where 𝐱 is the comoving coordinate, 𝐩=𝑚𝑎²𝐱̇, 𝑚 is the particle mass, and Φ_𝑁 is the self-consistent gravitational potential given by the Poisson equation,∇²Φ_𝑁(𝐱)=4𝜋𝐺_𝑁𝑎²𝜌(𝐱)−𝜌_𝑏(𝑡), (53) where 𝜌_𝑏 is the background mass density. The Vlasov-Poisson system, (56) and (53) form the Vlasov-Poisson equation, constitutes a collisionless, mean-field approximation to the evolution of the full 𝑁-body distribution.

An 𝑁-body code attempts to solve the Vlasov-Poisson system of equations by representing the one-particle distribution function as𝑓(𝐱,𝐩)=^𝑁_𝑖=1𝛿𝐱−𝐱_𝑖𝛿𝐩−𝐩_𝑖. (54) Substitution of (54) in the Vlasov-Poisson system of equations yields the exact Newton’s equations for a system of 𝑁 gravitating particles (see [42] for details):𝐱̈_𝑖+2𝐻𝐱_𝑖1=−𝑎³𝐺_{𝑁𝑗≠𝑖}𝑚_𝑗𝐱_𝑖−𝐱_𝑗𝐱_𝑖−𝐱_𝑗³, (55) where the sum includes all periodic images of particle 𝑗 and numerically is done using the Ewald method (see [43]). It is important to keep in mind, however, that we are not really interested in solving the exact 𝑁-body problem for a finite number of particles, 𝑁. The actual problem of interest is the exact 𝑁-body problem in the fluid limit, that is, as 𝑁→∞. For this reason, one important aspect of the numerical fidelity of 𝑁-body codes lies in controlling errors due to the discrete nature of the particle representation of 𝑓(𝐫,𝐩).

In the Newtonian limit of STT of gravity to describe the evolution of the six-dimensional, one-particle distribution function, 𝑓(𝐱,𝐩,𝑡), we need to solve the Vlasov-Poisson-Helmholtz equation in an expanding universe [44–48]. The Vlasov equation is given by𝜕𝑓+𝐩𝜕𝑡𝑚𝑎²⋅𝜕𝑓𝜕𝐱−𝑚∇Φ_𝑁(𝐱)⋅𝜕𝑓𝜕𝐩=0, (56) where Φ_𝑁=(1/2)ℎ₀₀=𝜓−(1/2)(𝐺_𝑁/(1+𝛼))𝜙 with 𝜓 satisfying Poisson equation∇²𝜓(𝐱)=4𝜋𝐺_𝑁𝑎²𝜌(𝐱)−𝜌_𝑏(𝑡) (57) and 𝜙 satisfying the Helmholtz equation∇²𝜙−𝜆⁻²𝜙=−8𝜋𝛼𝑎²𝜌(𝐱)−𝜌_𝑏(𝑡). (58) The above equations form the Vlasov-Poisson-Helmholtz system of equations that constitutes a collisionless, mean-field approximation to the evolution of the full 𝑁-body distribution in the framework of the Newtonian limit of a scalar-tensor theory.

Using the representation of the one-particle distribution function (54) the Newtonian motion equation for a particle 𝑖 is written as [45]𝐱̈_𝑖+2𝐻𝐱_𝑖1=−𝑎³𝐺_𝑁1+𝛼_𝑗≠𝑖𝑚_𝑗𝐱_𝑖−𝐱_𝑗𝐱_𝑖−𝐱_𝑗³×𝐹_SF𝐱_𝑖−𝐱_𝑗,𝛼,𝜆, (59) where the sum includes all periodic images of particle 𝑗, and 𝐹_SF(𝑟,𝛼,𝜆) is𝐹_SF()𝑟𝑟,𝛼,𝜆=1+𝛼1+𝜆𝑒^−𝑟/𝜆, (60) which, for small distances compared to 𝜆, is 𝐹_SF(𝑟<𝜆,𝛼,𝜆)≈1+𝛼(1+𝑟/𝜆) and, for long distances, is 𝐹_SF(𝑟>𝜆,𝛼,𝜆)≈1, as in Newtonian physics.

The function 𝐹_SF(𝑟,𝛼,𝜆) is shown in Figure 2 for several values of parameter 𝛼. The horizontal line at 𝐹_SF=1 is for the standard Newtonian case (𝛼=0). The long dashed line above the horizontal standard line is for 𝛼=1, the dotted line is for 𝛼=1/2 and the dotted-dashed line is for 𝛼=1/4. For negative values of 𝛼 the lines are below the standard horizontal case. Small dashed line is for 𝛼=−1/2 and double-dotted-dashed line is for 𝛼=−1/4. We should notice that even though 𝜆=5 Mpc/ℎ, 𝐹_SF gives an important contribution to the force between particles for 𝑟>𝜆.

Figure 2 

Function 𝐹SF(𝑟,𝛼,𝜆) for 𝜆=5 Mpc/ℎ and several values of parameter 𝛼. 𝛼=0, That is, the standard Newtonian case is the horizontal line. The long dashed line above the horizontal standard line is for 𝛼=1. The dotted line is for 𝛼=1/2, and the dotted-dashed line is for 𝛼=1/4. For negative values of 𝛼 the lines are below the standard horizontal case. Small dashed line is for 𝛼=−1/2 and the double-dotted-dashed line is for 𝛼=−1/4.

5. Results

In this section, we present results of cosmological simulations of a ΛCDM universe with and without SF contribution in order to study the large-scale structure formation.

We have used the standard Zel’dovich approximation [5] to provide the initial 256³ particles displacement of a uniform grid and to assign their initial velocities in a 256ℎ⁻¹ Mpc box [49]. In this approximation the comoving and the Lagrangian coordinates are related by𝐱=𝐪−𝛼_𝑍𝐤𝑏_|𝐤|(𝑡)𝐒_|𝐤|(𝐪),𝐩=−𝛼_𝑍𝑎²_𝐤𝑏_|𝐤|(𝑡)𝑏̇_|𝐤|(𝑡)𝑏_|𝐤|(𝑡)𝐒_|𝐤|(𝐪), (61) where the displacement vector 𝐒 is related to the velocity potential Φ and the power spectrum of fluctuations 𝑃(|𝑘|):𝐒_|𝐤|(𝐪)=∇_𝑞Φ_|𝐤|(𝐪),Φ_|𝐤|=_𝐤𝑎_𝐤cos(𝐤⋅𝐪)+𝑏_𝐤sin(𝐤⋅𝐪), (62) where 𝑎 and 𝑏 are gaussian random numbers with the mean zero and dispersion 𝜎²=𝑃(𝑘)/𝑘⁴,𝑎_𝐤=𝑃(|𝑘|)|𝑘|²𝐺(0,1),𝑏_𝐤=𝑃(|𝑘|)|𝑘|²𝐺(0,1), (63) where 𝐺(0,1) is a gaussian number with mean zero and dispersion 1.

The parameter 𝛼_𝑍 together with the power spectrum 𝑃(𝑘) defines the normalization of the fluctuations. The initial power spectrum was generated using the fitting formula by [50] for the transfer function. This formula is a slight variation of the common BBKS fit [51].

Therefore, for the standard ΛCDM we have for the initial condition that the starting redshift is 𝑧_in=50 and we choose the following cosmology: Ω_DM=0.314 (where Ω_DM includes cold dark matter and baryons), Ω_𝐵=0.044, Ω_Λ=0.686, 𝐻₀=71km/s/Mpc, 𝜎₈=0.84, and 𝑛=0.99. Particle masses are in the order of 1.0×10¹⁰𝑀_⊙. The individual softening length was 50 kpc/ℎ. This choice of softening length is consistent with the mass resolution set by the number of particles. All these values are in concordance with measurements of cosmological parameters derived from the seven-year data of the WMAP [6]. The initial condition—called the big box case—is in the Cosmic Data Bank web page: http://t8web.lanl.gov/people/heitmann/test3.html. See [49] for more details.

Because the visible component is the smaller one and given our interest to test the consequences of including an SF contribution to the evolution equations, our model excludes gas particles, but all its mass has been added to the dark matter. We restrict the values of 𝛼 to the interval (−1,1) (see [38–41]) and use 𝜆=1,5,10,20 Mpc/ℎ, since these values sweep the scale lengths present in the simulations.

In all the simulations we have done, with or without the scalar field contribution, we demand that the cosmological model be flat. In this model with a scalar field a flat universe is obtained if Ω_𝑚^(𝛼)+Ω_Λ^(𝛼)=1, with Ω_𝑚^(𝛼)≡Ω_𝑚/(1+𝛼) and Ω_Λ^(𝛼)≡Ω_Λ/(1+𝛼). Then, for positive values of 𝛼 we need a factor (1+𝛼) with more energy content (Ω_𝑚 and Ω_Λ) than that in the standard cosmology whereas for negative values of 𝛼 we need a factor of (1+𝛼) with less Ω_𝑚 and Ω_Λ to have a flat universe. With this recipe and for a given value of 𝛼 we modified the above ΛCDM initial condition accordingly.

In Figure 3 we show how the above initial conditions evolve and give us the LSS formation process without SF and with SF: in (a) without SF: in (b) with SF: 𝛼=1 and 𝜆=1Mpc/ℎ. In (c) with SF: 𝛼=1 and 𝜆=5 Mpc/ℎ in (d) with SF: 𝛼=1 and 𝜆=10 Mpc/ℎ in (e) with SF: 𝛼=1 and 𝜆=20 Mpc/ℎ in (f) With SF: 𝛼=−1/2 and 𝜆=5Mpc/ℎ in (g) With SF: 𝛼=−1/4 and 𝜆=5 Mpc/ℎ.

(a) Without SF

(b) With SF: 𝛼=1,𝜆=1Mpc/ℎ

(c) With SF: 𝛼=1,𝜆=5Mpc/ℎ

(d) With SF: 𝛼=1,𝜆=10Mpc/ℎ

(e) With SF: 𝛼=1,𝜆=20Mpc/ℎ

(f) With SF: 𝛼=−1/2,𝜆=5Mpc/ℎ

(g) With SF: 𝛼=−1/4,𝜆=5Mpc/ℎ

Figure 3 

(a) LSS formation process without the SF contribution. (b) and (c) LSS formation process with the SF contribution. (b) 𝛼=1 and 𝜆=1 Mpc/ℎ. (c) 𝛼=1 and 𝜆=5 Mpc/ℎ. (d) and (e) LSS formation process with the SF contribution. (d) 𝛼=1 and 𝜆=10 Mpc/ℎ. (e) 𝛼=1 and 𝜆=20 Mpc/ℎ. (f) and (g) LSS formation process with the SF contribution. (f) 𝛼=−1/2 and 𝜆=5 Mpc/ℎ. (g) 𝛼=−1/4 and 𝜆=5 Mpc/ℎ.

To study the structure formation in the universe we follow the evolution of the overdensity [4],𝛿(𝐱)≡𝜌(𝐱)𝜌₀−1, (64)

where 𝜌₀ is the average density over a volume 𝑉 and 𝐱 is the comoving distance related to the physical density by 𝐫=𝑎(𝑡)𝐱. In the linear regime 𝛿≪1.

The correlation function tells us how 𝛿 is correlated in two nearby points 𝐱′ and 𝐱′+𝐱,𝜉(𝐱)≡𝛿𝐱′𝛿𝐱′+𝐱, (65)

and the power spectrum is the Fourier transform of the correlation function𝜉(𝑥)=1𝑉_𝐤𝑃(𝐤)𝑒^{𝑖𝐤⋅𝐱}. (66)

In Figure 4(a) we show the evolution of the power spectrum for the big box 𝐿=256 Mpc/ℎ without SF, for several values of the redshift 𝑧. We have used the POWMES code to compute the matter power spectrum [52]. In this figure we can appreciate how it is forming structures, solid line is the power spectrum for the present epoch 𝑧=0. Figure 4(b) shows the power spectrum for the same values as in Figure 3. Continuous line is without SF. Dashed line is with SF: 𝛼=1 and 𝜆=5 Mpc/ℎ. Dashed-dotted line is with SF: 𝛼=−1/2 and 𝜆=5 Mpc/ℎ. The curve that is just below the continuous line (dotted line) is with SF: 𝛼=−1/4 and 𝜆=5 Mpc/ℎ. More greater values of the power spectrum means more structure formation. Therefore, the inclusion of an SF modifies the structure formation process. Depending on the values of its parameters, 𝛼 and 𝜆, we can obtain more structure or less structure at the present epoch, 𝑧=0.

(a)

(b)

Figure 4 

(a) Evolution of the power spectrum for the case which there is no SF. (b) The power spectrum including SF for several values of parameter 𝛼 and with 𝜆=5Mp𝑐/ℎ. Experimental data are from galaxies in the Sloan Digital Sky Survey (SDSS) catalog and from an analysis of the Lyman-𝛼 forest [54].

In our results as shown in Figures 3(c), 3(f), and 3(g), and Figure 4 we have used a fix value of 𝜆=5Mpc/ℎ. For a given 𝜆 the role of SF parameter 𝛼 on the structure formation can be inferred by looking at (59) and (60) and Figure 2 where we show the behavior of 𝐹_SF as a function of distance for several values of 𝛼. The factor 𝐹_SF augments (diminishes) for positive (negative) values of 𝛼 for small distances compared to 𝜆, resulting in more (less) structure formation for positive (negative) values of 𝛼 compared to the ΛCDM model. In the case of the upper curve in Figure 4(b), for 𝑟≪𝜆, the effective gravitational pull has been augmented by a factor of 2, in contrast to the case shown with the lower line in Figure 4(b) where it has been diminished by a factor of 1/2. That is why we observe for 𝑟<𝜆 more structure formation in the case of the upper curve in Figure 4(b) and lesser in the case of the lower curve in the same figure. The effect is then, for an increasing positive 𝛼, to speed up the growth of perturbations, then the halos, and then the clusters, whereas negative values of 𝛼(𝛼→−1) tend to slow down the growth. We also observe that for the large-scale regimen of our simulations (𝑘<0.1ℎ/Mpc) they tend to predict almost the same structure formation. From comparison with the experimental results, we see that the ΛCDM agrees well with SDSS observations, but predicts more structure formation than observations show in the Lyman-𝛼 forest power spectrum. In general the more favored model is the model with SF with 𝛼=−1/2 and 𝜆=5 Mpc/ℎ.

6. Conclusions and Final Comments

We have used a general, static STT, that is compatible with local observations by the appropriate definition of the background field constant, that is, ⟨𝜙⟩=(1+𝛼)/𝐺_𝑁, to study the LSS formation process. The initial condition for the several cases (different values of parameter 𝛼) was built in such a way that the geometry of the model universe was flat. Quantitatively, this demands that our models have Ω/(1+𝛼)=1 and this changes the amount of dark matter and energy of the models in order to have a flat cosmology.

Using the resulting modified dynamical equations, we have studied the LSS formation process of a ΛCDM universe. We varied the amplitude and sign of the strength of the SF (parameter 𝛼) in the interval (−1,1) and performed several 3D simulations with the same initial conditions. From our simulations we have found that the inclusion of the SF changes the local dynamical properties of the most massive groups however, the overall structure is very similar, as can be seen in Figure 3.

The general gravitational effect is that the interaction between dark matter particles given by the potential Φ_𝑁 (see (15)) changes by a factor 𝐹_SF, (see (60)), in comparison with the purely Newtonian case. Thus, for 𝛼>0 the growth of structures speeds up in comparison with the Newtonian case (without SF). For the 𝛼<0 case the effect is to diminish the formation of structures.

It is important to note that particles in our model are gravitating particles and that the SF acts as a mechanism that modifies gravity. The effective mass of the SF (𝑚_SF=1/𝜆) only sets an interaction length scale for the Yukawa term.

In this work we only varied the amplitude of the SF—parameter 𝛼—leaving the scale length, 𝜆, of the SF unchanged. However, in other studies we have done [47] we have found that increasing 𝜆 enhances the structure formation process for 𝛼 positive and decreasing 𝜆 makes the structures grow at a slower rate.

We have computed the mass power spectrum in order to study the LSS formation process. The theoretical scheme we have used is compatible with local observations because we have defined the background field constant ⟨𝜙⟩=𝐺_𝑁⁻¹(1+𝛼) or equivalently that the local gravitational constant is given by (1+𝛼)⟨𝜙⟩⁻¹, instead of being given by ⟨𝜙⟩⁻¹. A direct consequence of the approach is that the amount of matter (energy) has to be increased for positive values of 𝛼 and diminished for negative values of 𝛼 with respect to the standard ΛCDM model in order to have a flat cosmological model. Quantitatively, our model demands to have Ω/(1+𝛼)=1 and this changes the amount of dark matter and energy of the model for a flat cosmological model, as assumed. The general gravitational effect is that the interaction including the SF changes by a factor 𝐹_SF(𝑟,𝛼,𝜆)≈1+𝛼(1+(𝑟/𝜆)) for 𝑟<𝜆 in comparison with the Newtonian case. Thus, for 𝛼>0 the growth of structures speeds up in comparison with the Newtonian case. For the 𝛼<0 case the effect is to diminish the formation of structures. For 𝑟>𝜆 the dynamics is essentially Newtonian.

Comparison of the power spectrums from galaxies in the SDSS catalog versus those inferred from Lyman-𝛼 forest observations tells us that ΛCDM predicts more structure formation in the regime of smaller scales (𝑘>0.4ℎ/Mpc) whereas the model with SF with 𝛼=−1/2 and 𝜆=5 Mpc/ℎ follows the general trend of the observed power spectrum. In this way we are able to construct a model that predicts the observed structure formation in the regime of small scales, with lesser number of halo satellites than the ΛCDM model.