PACS: 04.50.Kd; 98.10.+z; 97.20.Vs.

1. Introduction

One of the most successful outcomes of string theory is that the space-time coordinates may become NC operators and lead us to a NC version of a gauge theory via the Seiberg and Witten map ^{1}. After this result was published, a large number of papers related with NC in gauge theories, quantum mechanics, classical mechanics, quantum field theory and gravity has been published. In addition it is possible to note that the NC parameter has been bounded by different observations or experiments, mainly leading to an extremely small value ^{2}.

A different NC formulation of quantum field theory, based on coherent state formulation, can be achieved using the Feynman path integral on the NC plane which is a used framework for quantum mechanics and field theory ^{3}. In a recent paper ^{4} the authors discuss the gravitational analogue of the NC modification of quantum field theory, pointing out that NC is an intrinsic property of the manifold itself and affects gravity in an indirect way. The energy-momentum density determines space-time curvature. Thus in General Relativity (GR), the effects of NC can be taken into account by keeping the standard form of Einstein curvature tensor and introducing a modified energy-momentum tensor. The NC eliminates the point-like structures and replace them by smeared objects. The effect of smearing is implemented by using a Gaussian distribution of minimal width

where *M*
_{
NC
} is the total mass parameter and ^{5}^{,}^{6}^{,}^{7}, only having important effects in quantum gravity theories.

Hence, in order to study large scales, we relate the following functions and we change the NC parameter which comes from a quantum theory for a new macroscopic parameter

The parameter are constrained by observations or under arguments of a behavior that fits with the the traditional literature. Indeed, if we assume that this fluid permeates the galaxies structures, it is natural to study the presence of this fluid, also in the stellar dynamics.

Therefore, to start the study we will call this new fluid as noncommutativity energy density (NED) for the inspiration that comes from NC; then we give ourselves to the task of study the previous statements, proposing two exercises to analyze the dynamics in a scenario where baryons and NED coexist (without mutual interaction), *i.e.*

In this sense, there has been extensive studies in stellar dynamics since the advent of GR ^{8}; in such a way that observing the imprints of NED in the dynamics must be clear, showing deviations to GR predictions. For instance it is possible to assume that hypothetical NED particles are so heavy, forming a dense core in the star center allowing the applied treatment in the following sections. In this vein, stars with uniform density and white dwarfs are studied due that are excellent laboratories to study possible extensions to the GR background. It is important to remark that extensive studies about NC, can be seen in ^{9}.

In addition, the functional form of Eq. (1) with the recipe (2), inspire to be used as a density profile to reproduce the velocity rotation curves of galaxies, due to its similarity with Einasto’s model ^{10} with *n*' = 0.5. In galactic dynamic the NED matter is concentrated mainly in the halo, with a Gaussian density profile and thus giving restrictions to the free parameter ^{11}^{,}^{12}^{,}^{13}^{,}^{14}^{,}^{15}^{,}^{16} or for an excellent review see Ref. ^{17}.

Recapitulating, we remark that the recipe shown in Eq. (2) as a macroscopic phenomena, will generate constraints of the ^{18}, contributing to strengthen these previous research. In this sense, we will provide a table of the constraints of this parameter, which will be subject to the theoretical model under study or the observations with which it is contrasted.

From here, it is possible to organize the paper as follows: In Sec. 2, we analyze a Newtonian star in two cases, where it is composed by an uniform density (incompressible fluid) coexisting with the NED matter and when it is composed by a polytropic matter and the core contains a NED fluid, showing a modified Lane-Emden equation, constraining the NED free parameter and fixing the NED mass as a subdominant component. In Sec. 3 we implement an analysis of galactic rotation curves, assuming that the DM halo can be modeled by NED density matter. In this case, we use a sample of eighteen LSB galaxies without photometry with the aim of constraint the NED parameters (^{19} and ^{20}, it is an ultralight scalar model motivated by large scale simulations ^{21}. Finally in Sec. 4 we give a discussion and conclusions about the results obtained through the paper.

In what follows, we work in units in which

2. Toy model stars with a NED component

In this section, we study the stellar dynamics with a component of NED, together with the traditional baryonic matter. We start using the approach of a Newtonian star, composed by this fluid and baryonic matter with uniform density (incompressible fluid). After that, we study a most generic star through the Lane-Emden (LE) equation, under the assumption that the star is composed by a polytrope and also contains a NED. Both models are considered under the premise that ordinary matter does not interact with the NED and fixes one of the free parameters, with the argument of a subdominant NED.

2.1. NED matter on stars with uniform density in Newtonian approach

These stars are of interest, because they are simple enough to allow an exact solution in Newtonian background. Then, stars with uniform density consist of an incompressible fluid with equation of state (EoS),

In a Newtonian approach, dynamic equation for the evolution of a star can be written as:

being *G* the Newtonian gravitational constant,

Eqs. (3) and (4) are considered as the equations of motion for stellar dynamics in Newtonian approach. In addition, we propose that the star density is composed by an uniform component density and a NED in the following form:

where

and integrating Eq. (4) we have:

where Erf(x) is the error function defined as:

The behavior can be seen in Fig. 1 (Top) for different values of

Also, we analyze an extreme Newtonian star fulfilling the compactness relation *GM*/*R* = 0.44. Indeed, it is possible to observe that the dimensionless NED parameter

In addition, Eq. (3) can be written in terms of dimensionless variables as:

and its numerical integration can be observed in Fig. 1 (Bottom) for different values of the NED parameter. It is notorious how for values below

Thus, small values of NED parameter shown the convergence to the traditional behavior without NED for pressure and mass in a Newtonian star with uniform density.

From both results we conclude that it is necessary that

where for a star in the limit of stellar stability as it is our case, we have

2.2. Dwarf stars with NED

A dwarf star is mainly composed by a polytropic matter with equation of state (EoS) *K* the polytropic constant and *n* the polytropic index. This kind of stars are the most studied in literature due that it is possible to model them by just specifying the values of *K* and *n*, having in mind the appropriate limits of both stellar structures^{i}.

However now we consider a NED component in the effective density of the star. Starting from Eq. (3) it is possible to write the modified LE equation as:

where it was proposed the following dimensionless variables:

Notice that *n* =3. The results are shown in Fig. 2, restricting a subdominant

where

being *m*
_{
N
} is the nucleon mass ^{8} and ^{22}^{,}^{23}^{,}^{24}. Inside of the region, the NED presence is predominant and therefore, not having a traditional behavior of the stellar dynamics.

Also, we report the values of the NED parameters for the fifteen white dwarfs shown in Table I, constraining the NED parameter with observables to obtain a traditional behavior shown in literature.

3. NED in galaxy rotation velocities

To complement our analysis we study rotation curves of galaxies at the weak gravitational field limit in order to study NED parameters. We start modeling the halo of the galaxy with the energy density of the NED shown in Eqs. (1) and (2).

Moreover, it is important to notice some important characteristics of NED profile. Implementing an appropriate series expansion of the NED of the order ^{11}. In this context, this density profile is valid only at the center of galaxies. However, PISO density profile is an empirical profile designed for modeling DM in spirals galaxies and it has been applied not only at the center of galaxies but also in the outer spatial regions. Also, it is possible to observe that NED is a particular case of Einasto’s density profile ^{10} (see Eq. (14)) when *n*´ = 0.5

where *r*
_{-2} is the radius where the density profile has a slope -2 and *n*' is known as Einasto index which describes the shape of the density profile.

In general, the NED distribution can provide with extra information than other models can not (see for example ^{11}^{,}^{12}^{,}^{13}), mainly due to the advantage that comes naturally from the geometric properties of space-time and it is not just chosen by observations or numerical simulations.

On the other hand, we have that the rotation velocity is obtained from the absolute value of the effective potential as:

where *M*
_{
DM
} (*r*), steaming for a dark matter (DM) distribution.

3.1. NED rotation velocity

The rotation velocity for the NED can be obtained through the NED distribution, giving the following relationship:

where again, Erf(*x*) is the error function. This expression can be rewritten as

where ^{25}. It has units of surface density. The rotation curve equations for PISO, NFW, Burkert and WaveDM models can also be written similarly as ^{17} (see Appendix).

Let as first make a comparison of rotation curves associated with PISO, NFW and Burkert (See Appendix A) versus NED. Here we do not considere WaveDM model. The comparison between these three models and the NED rotation curves is shown in Fig. 3. In Fig. 3(a) we have plotted all the DM models with the same parameters values just to see their behavior. Given that *r*
_{
i
} , even do *r*
_{
i
} values are the same. In Fig. 3(b) we tried to reproduce almost the same behavior of the PISO rotation curve which has been produced with ^{2} and *r*
_{
p
} = 2.5 kpc, by adjusting the parameters values of the other three models: NED: ^{2} and ^{2} and *r*
_{
n
} = 20 kpc; and Burkert: ^{2} and *r*
_{
b
} = 4.8 kpc. It has been found that ^{25}.

3.2. Constraints with LSB galaxies

The main goal of this section, is to compare the results of galaxy rotation curves, comparing the NED parameters fit with parameters fit given by the four most successful models studied in literature which are PISO, NFW, Burkert and WaveDM densities profiles (See Appendix for details of the formulae used to fit observations).

In this sense, it is necessary to obtain the best fit, by maximizing the likelihood

here *N* is the total number of data and *p* is the number of free parameters ^{26}. Errors in the estimated parameters were computed using the covariance matrix as is described in Ref. ^{26}.

For the DM models analyzed in this work we have two parameters: a scale length, *r*
_{
i
} , and the density at the center of the galaxy, *i* is for PISO, NFW, Burkert, WaveDM or NED models, Eqs. (A.4), (A.7), (A.10), (A.13) and (A.16). As we have already say, we defined in Eq. (A.3), the surface density, ^{25}.

We analyze a sample of eighteen high resolution rotation curves of LSB galaxies with no photometry: an optical rotation curve were available but no optical or H I photometry ^{27}. Accordingly we neglect the visible components, such as gas and stars. The sample of analyzed galaxies is given in Table II. See Ref. ^{27} for technical details. We remark that in this subsection we use units such that *G* = 1, velocities are in km/s, and distances are given in kpc.

Results are summarized in Tables III for PISO model, IV for NFW model, V for Burkert model, VI for NED model and VII for WaveDM model. In those tables we show for each of the studied galaxies the fitting parameters *r*
_{
i
} for each one of the five models, columns (2) and (3) and the corresponding *r*
_{max}, columns (4), (5) and (6) respectively. Results for parameters *r*
_{
i
} for PISO and NFW models are very similar to those found in Ref. ^{27}, except in the estimated errors. This could be due to the method used to fit the data and the method to estimate the fitting errors. As we mentioned before we use the covariance matrix to compute the errors following Ref. ^{26} were its diagonal elements gives the errors in the estimated parameters. However, some programs do not include the factor ^{26} does take into account. In this way the estimated errors in the parameters reported here will diminish a great deal.

In Fig. 4, it is shown, for each galaxy in the sample of the LSB galaxies, the theoretical fitted curve to a preferred NED value (blue solid line) that best fit to the corresponding observational data (black symbols).

Notice how galaxies UGC 11648 and UGC 11748 are the worst fitted cases. This was also the case in Ref. ^{27} for PISO and NFW models. For NED

In Ref. ^{28} was found for PISO model that the surface density ^{2}. This work was extended to include more galaxies and other cored DM models in ^{25}^{,}^{29}. Authors found that its value is almost constant for galaxies in a width range of magnitude values. This indicates that ^{30} that used the newly SPARC galaxy catalog, a theoretical study was done in Ref. ^{19}. We redo these theoretical estimations to find the theoretical values for the DM models we are interested in: PISO, NFW, Burkert, NED and WaveDM. There were differences among PISO and Burkert values we have obtained with the values reported in Ref. ^{19}. In Tables III for PISO model, IV for NFW model, V for Burkert model, VI for NED model and VII for WaveDM model, we reported this surface density values, column (4) in each table. In Fig. 5 we have plotted

For PISO model the mean value of ^{2} while the theoretical value we found, following ^{19}, is 193 ^{2}. For NFW: ^{2} versus the theoretical value of 89 ^{2}. For Burkert: ^{2} versus the theoretical value of 348.69 ^{2}. For NED DM model we have that the mean value of ^{2} versus its theoretical value of ^{2}. And for WaveDM model we obtained that the mean value of ^{2} while the theoretical value is 648.58 ^{2}. This work should be extended to consider more galaxies in order to test more throughly the theoretical predictions as it was done in Ref. ^{25}.

In column (5) of each one of the Tables III, IV, V and VI, we have reported values of the masses of the DM models up to 300 pc, *M*
_{
DM
} (300 pc), and they were plotted versus *M*
_{abs}(*B*) (lower filled circles) for all models in Fig. 6. To guide the eye we have plotted the mean values of *M*
_{
DM
} (300 pc). The mean values we found for each one of the models are: PISO: 4.85 × 10^{7}^{8}^{7}^{7}^{7}^{31} was reported this mass within the central 300 pc for 18 dSph Milky Way satellites. The value found is about 10^{7}

Also, in Tables III, IV, V, VI and VII, column (6), we reported values for the total mass of the DM models up to the last spatial point measured, *r*
_{max}. This mass, M_{
DM
} (*r*
_{max}), was plotted in Fig. 6 (upper filled circles). Values of this total mass fail to follow the constant mean value, almost of the order of 10^{10}^{9}. But we believe more work is needed in order to make a final conclusion.

Finally, for NED model, Fig. 7 shows in (a) the fitted values of parameters

4. Discussion and conclusions

In this paper we have assumed that NED is an important component in Universe dynamics affecting the stellar equations of motion and the dynamic of galaxies in special the rotation curves. From here, it is possible to discuss the research shown in this paper in two main cases:

In the first case, we use the dynamic equations for the evolution of stars with the aim of investigate the behavior dictated by the presence of an uniform density (incompressible fluid) and NED. We find a constraint for the parameter

In addition, with polytropic stars it is possible to model Dwarf Stars with index *n* = 3. Here we also add NED which does not interact with the polytropic fluid, presenting a modified Lane-Emden equation to describe the dynamics of the star, our results present important effects in the dynamics except for values that fulfill the conditions

The second case took into account the corresponding rotation velocity of galaxies, assuming that NED is related with the galaxy halo. From here, we compute the rotation velocity associated with this model and was compared with four of the most studied and accepted models in literature, which are: PISO, NFW, Burkert and WaveDM. The analysis was implemented through a

For each DM model and for each galaxy in the analyzed sample we computed three important quantities that should give us information about galaxy formation and evolution that are summarized in Table VIII: the surface density _{
DM
} (300 pc), and M_{
DM
} (*r*
_{max}) for the DM haloes. From other studies it was expected that these quantities were nearly constants, independent of the absolute magnitude of the galaxies ^{9}^{,}^{19}^{,}^{28}^{,}^{31}. The general tendency, found in this work, is that these quantities are roughly constant (except the behavior of M_{
DM
} (*r*
_{max})). We found like in ^{31}, for dwarf spheroidal galaxies, that also must be a common mass for spiral galaxies within 300 pc and with the same order of magnitud, 10^{7}^{-3}, independent of the DM model with the exception of NFW model whose value is an order of magnitud larger. If this result is confirmed NFW would have problems to explain the observed rotation curves of spiral galaxies. The theoretical and observational predictions give that the surface density, *r*
_{max} fail to have this behavior, this is in tension with a previous result found in Ref. ^{9}. However, we believe that it is needed a more systematic study with a better sample of galaxies to reach a final conclusion.

Our results about NED parameter are summarized in Table IX showing the difference between each constraint. It is possible to observe that stellar dynamics provides a lower bound for the free parameter meanwhile galaxy rotation curves provides a higher bounds: an interval of values for the NED parameter. Nevertheless, the rotation curves analysis is less restrictive unlike the toy model cases of stellar analysis; presenting the first one a better confidence in the value. We emphasize that it is necessary to perform a further analysis in both cases to uncover and narrow the free parameter for this *matter* inspired by NED models. Each model provide us with different results, which are subjected to the astrophysical system to which it is being addressed. However, it is possible to establish a range of values where the NED parameter must fit in.

The results provided by this paper, contributes to generate astrophysical bounds of the ^{18}, but leaving aside the contrast with real data. Of course, we remark that our results are not about NC, instead is just a model that was motivated by NC. NC it is a quantum-gravity theory which has not important effects at macroscopic scales.

As a final note, we remark that the density profile inspired by NC can be used in cosmological analysis like structure formation, the statistics of the distribution of galaxy clusters, the temperature anisotropies of the cosmic microwave background radiation (CMB) and with other astrophysical studies. However, this work is in progress and will be reported elsewhere.