# A power-law coupled three-form dark energy model

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## Abstract

We consider a field theory model of coupled dark energy which treats dark energy as a three-form field and dark matter as a spinor field. By assuming the effective mass of dark matter as a power-law function of the three-form field and neglecting the potential term of dark energy, we obtain three solutions of the autonomous system of evolution equations, including a de Sitter attractor, a tracking solution and an approximate solution. To understand the strength of the coupling, we confront the model with the latest Type Ia Supernova, Baryon Acoustic Oscillations and Cosmic Microwave Background radiation observations, with the conclusion that the combination of these three databases marginalized over the present dark matter density parameter \(\Omega _{m0}\) and the present three-form field \(\kappa X_{0}\) gives stringent constraints on the coupling constant, \(-\,0.017< \lambda <0.047\) (\(2\sigma \) confidence level), by which we present the model’s applicable parameter range.

## 1 Introduction

According to cosmological observations, the universe has entered a stage of an accelerated expansion with a redshift smaller than 1 [1, 2]. Since all usual types of matter with positive pressure decelerate the expansion of the universe, a sector with negative pressure named dark energy was suggested to account for the invisible fuel that accelerates the expansion rate of the current universe [3, 4].

The simplest cosmological model of dark energy is the so-called Lambda cold dark matter (\(\Lambda \)CDM) model, in which vacuum energy plays the role of dark energy. Although \(\Lambda \)CDM model provides an excellent fit to a wide range of astronomical data so far, such a model in fact is theoretically problematic because of two cosmological constant problems, the fine tuning problem, concerning why is the observational vacuum density is so small compared to the theoretical one, and the coincidence problem which asks why the observational vacuum density is coincidentally comparable with the critical density at the present epoch in the long history of the universe. In order to alleviate the latter, various ideas as regards evolving and spatially homogeneous scalar fields, including quintessence [5], phantom [6], dilatonic [7], tachyon [8] and quintom [9] models, etc. were suggested to take the vacuum energy’s place. In these models, the resolution of the coincidence problem typically leads to a fine tuning of the model parameters.

Since experimental evidence of cosmology-specific scalar particles has not been discovered yet, there is no reason to exclude the possibility of some other high form field to be dark energy. Indeed, the three-form cosmology proposed in [10, 11] could be a good alternative to scalar cosmology, because such a high form field not only respects the Friedmann–Robertson–Walker (FRW) symmetry naturally but also can accelerate the expansion rate of the current universe without a slow-roll condition. Moreover, some interesting results, e.g. three-forms with simple potentials, lead to models of inflation with potentially large non-Gaussian signatures [12], etc., in three-form cosmology.

The coincidence problem mentioned above is just that the amount of dark matter is comparable to that of dark energy in the present universe, so it is natural to consider an interaction between these two components. As was pointed out in Ref. [13], in comparison to coupled scalar dark energy model [14], some new features appear in coupled three-form dark energy model, including one that the stress tensor is modified by the interaction between two dark sectors, hence it is problematic to consider a coupled three-form dark energy model in a phenomenological way [15], and one needs to construct it in a Lagrangian formalism. Different from modeling dark matter as point particles [13], we follow the thread of describing the interaction between dark energy and dark matter from a fundamental field theory point of view [16] and consider dark matter to be a Dirac spinor field.

The contents of this paper are as follows. In Sect. 2, we present a type of Lagrangians describing the interaction between a three-form field and a Dirac spinor field in curved space-time and then derive the field equations from such Lagrangians. In Sect. 3, we consider these field equations in a FRW space-time by assuming the effective mass of dark matter as a power-law function of the three-form field and setting the potential of dark energy to be zero. In Sect. 4, we carry out a simple likelihood analysis of the model with the use of 580 SN Ia data points from recently released Union2.1 compilation [17] and BAO data from the WiggleZ Survey [18], SDSS DR7 Galaxy sample [19] and 6dF Galaxy Survey datasets [20], together with CMB data from WMAP7 observations [21]. In the last section, we present a brief conclusion with this paper.

## 2 A type of field theories of three-form and Dirac spinor in curved space-time

This section involves some concepts that are used to include fermionic sources in the Einstein theory of gravitation; for a more detailed analysis the reader is referred to [22, 23, 24, 25].

*R*denotes the Ricci scalar and \(\kappa =\sqrt{8\pi G}\) is the inverse of the reduced Planck mass.

## 3 Cosmological evolution of the power-law coupled three-form dark energy model

*a*(

*t*) refers to the scale factor.

^{1}for simplicity, we set the potential zero in the following discussions, together with choosing the coupling function as the following power-law form:

^{2}:

*m*with mass dimension is, in fact, the mass of dark matter in a free field theory. Since typically it is very difficult for dark energy to couple dark matter with mass larger than milli-eV [26], we choose

*m*to be smaller than milli-eV. Although the phenomenological bounds on the dark matter mass coming from large scale structure require that most of the dark matter is considerably heavier than milli-eV [27], some dark matter’s mass can be smaller than milli-eV. Indeed, Rajagopal, Turner and Wilczek considered the axino in the keV range and they obtained the axino mass bound \(m_{a}<2\) keV for an axino that constitutes warm dark matter [28].

*x*(

*N*) and

*y*(

*N*) with a wide range of initial conditions and the assumption that \(\lambda =0.01\) (we will see that this is a good choice in the next section) can be visualized by Fig. 1.

As is showed in Fig. 1, the trajectories run toward the de Sitter attractor, coasting along the saddle point. Also, one should note that the present value of *x* must be equal to or larger than a certain value \(x_{*}\), which depends on both \(\lambda \) and *y*(0), to make sure in high redshift that the autonomous system does not encounter a singularity. After specifying \(\lambda \) and *y*(0), if \(\widetilde{x}\) is the present value of *x* leading to the condition that *x*(*N*) is positive-definite for arbitrary non-infinite *N*, i.e. \(x(N)>0(-\infty<N< +\infty )\), then \(x_{*}\) is the lower limit of \(\widetilde{x}\).

It can be inferred from (40) and (41) that the energy transfer between two dark sectors keeps the density of dark energy constant even when its EOS deviates from \(-1\).

*L*, we have such a formula

## 4 Confront the power-law coupled three-form dark energy model with observations

In this section, we perform a simple likelihood analysis on the free parameters of the model with the combination of data from Type Ia Supernova (SN Ia), Baryon Acoustic Oscillations (BAOs) and Cosmic Microwave Background (CMB) radiation observations.

*P*,

*Q*and

*R*are defined as

Finally, the total \(\chi ^{2}\) function for the combined observational datasets is given by \(\chi ^{2}=\chi _{\mathrm{SN}\, \mathrm{Ia}}^{2}+\chi _{\mathrm{BAO/CMA}}^{2}\), from which we can construct the likelihood function as \(L=L_{0} e^{-\frac{1}{2}\chi ^{2}}\); here \(L_{0}\) is a normalized constant which is independent of the free parameters.

One may note that the marginalized likelihood in the right panel of Fig. 2 appears to be a little non-Gaussian; this is mainly because of the non-Gaussian structure of the likelihood that has not been marginalized. One also can see from Fig. 2 that the observations favor a small positive coupling constant which, as we mentioned above, allows for the existence of a tracking solution that can be used to alleviate the coincidence problem with fine-tuning of the model parameters.

## 5 Conclusions

In this paper we have studied a power-law coupled dark energy model which considers dark energy as a three-form field and dark matter as a spinor field. By performing a dynamical analysis on the field equations with the introduction of three dimensionless variables, we obtained two fixed points of the autonomous system of evolution equations, among which one is a de Sitter attractor, and the other is a tracking solution, assuming \(\lambda >0\), which provides a possible solution of the coincidence problem.

By marginalizing over \(x_{0}\), we have also carried out a likelihood analysis on the free parameters \(\lambda \) and \(\Omega _{m0}\) with the combination of SN *I*a+BAO/CMB datasets, through which we have a best-fit value of the pair \((\lambda ,\Omega _{m0})\) as (0.013, 0.269). In addition, the likelihood function marginalized over \(x_{0}\) and \(\Omega _{m0}\) showed that \(\lambda \) is restricted by \(-\,0.017<\lambda <0.047\) (\(2\sigma \) confidence level, with a best-fit value 0.01), indicating that the measurements considered here are quite consistent as regards the \(\Lambda CDM\) and our three-form model. However, future measurements might allow us to tell them apart.

This notwithstanding, it can be told from the fitting result that \(\lambda \) and \(\Omega _{m0}\) are strictly restricted, \(x_{0}\) can be any value beyond \(x_{*}\). However, as mentioned above, future measurements might decrease the uncertainty on \(x_{0}\).

## Footnotes

## Notes

### Acknowledgements

The authors warmly thank Jia-Xin Wang and Deng Wang for beneficial discussions.

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