Can the quintessence be a complex scalar field?

In light of the recent observations of type Ia supernovae suggesting an accelerating expansion of the Universe, we wish in this paper to point out the possibility of using a complex scalar field as the quintessence to account for the acceleration. In particular, we extend the idea of Huterer and Turner in deriving the reconstruction equations for the complex quintessence, showing the feasibility of making use of a complex scalar field (instead of a real scalar field) while maintaining the uniqueness feature of the reconstruction for two possible situations, respectively. We discuss very briefly how future observations may help to distinguish the different quintessence scenarios, including the scenario with a positive cosmological constant.


Introduction
The Supernovae Cosmology Project 1 and the High-Z Supernova Search 2 reported on their observations of type Ia supernovae (SNe Ia), suggesting that the expansion of the Universe is still accelerating [1,2]. In addition, recent measurements of the power spectrum of the cosmic microwave background (CMB) from BOOMERANG [3] and MAXIMA-1 [4] detected a sharp peak around l ≃ 200, indicating that the Universe is flat. Combining these two classes of observations, we may conclude that the Universe has the critical density (to make it flat) and that it consists of 1/3 of ordinary matter and 2/3 of dark energy with negative pressure (such that p total < −ρ total /3 at present time) [5,6]. At the moment, the most often considered candidates for the dark energy include (1) the existence of a positive cosmological constant [7] and (2) the presence of a slowly-evolving real scalar field called "quintessence" [8]. In the case of the quintessence, Zlatev, Wang and Steinhardt [9] considered a real scalar field as the "tracker field" which, regardless of a wide range of possible initial conditions, will join a path more or less common to the evolving radiation, the dominant energy density in the early universe, before the matter-dominated era. In addition, Huterer and Turner [10] considered basically the inverse problem of reconstructing the quintessence potential from the SNe Ia observational data (see also Refs. [11] and [12]).
In this paper, we wish to point out the possibility of using a complex scalar field as the quintessence to account for the accelerating expansion of the Universe. In particular, we extend the idea of Huterer and Turner in deriving the reconstruction equations for the complex quintessence, showing the feasibility of making use of a complex scalar field (instead of a real scalar field) while maintaining the uniqueness of the solution of the inverse problem. We also discuss briefly how future observations may help to distinguish the different quintessence scenarios (including the scenario with a positive cosmological constant).

The Basics
We consider a spatially flat Universe which is dominated by the non-relativistic matter and a spatially homogeneous complex scalar field Φ and which is described by the flat Robertson-Walker metric: The action for the Universe is where g is the absolute value of the determinant of the metric tensor g µν , G is the Newton's constant, R is the Ricci scalar, ρ M is the matter density, and L Φ is the Lagrangian density for the complex scalar field Φ: In Eq. (3), we have assumed that the potential V depends only on the absolute value (or the amplitude) of the complex scalar field: |Φ|. Instead of Φ and Φ * , we would like to use the alternative field variables: the amplitude φ(x) and the phase θ(x) (of the complex scalar field Φ(x)), which are defined by (More precisely, Φ(t) = φ(t)e iθ(t) .) The usage of the field variables-φ(x) and θ(x)-will benefit the derivation of the reconstruction equations which relate the quintessence potential V (φ) to SNe Ia data. Using Eq. (4), the Lagrangian density for Φ (Eq. (3)) becomes The variation of the action (Eq. (2)) with the above Lagrangian density yields the Einstein equations and the field equations of the complex scalar field. Using the metric tensor in Eq.
(1), these equations can be rearranged and become where H is the Hubble parameter, dot and prime denote derivatives with respect to t and φ respectively, ρ is the energy density, and p is the pressure. We note that the non-relativistic matter contributes the energy density ρ M and pressure p M = 0, while the evolving complex scalar field contributes the energy density ρ Φ and pressure p Φ as follows: Eqs. (6)-(9) are the fundamental equations which govern the evolution of the Universe. We first note that Eq. (9) can be solved and the solution for the "angular velocity"θ is given byθ where ω is an integration constant determined by the initial condition ofθ (or the value oḟ θ at some specific time). Using Eq. (12), the fundamental equations become Eq. (15) can be rearranged and becomë The term ω 2 /(2a 6 φ 2 ) in the bracket, coming from the "angular motion" of the complex scalar field Φ, can be treated as an effective potential (to be called "centrifugal potential"). It produces a "centrifugal force" and tends to drive φ away from zero if ω = 0 (i.e. the "angular velocity"θ is nonzero). In Eqs. (13) and (14), the contributions from the "angular motion" of Φ to the energy density ρ and pressure p (or the right hand sides of these two equations) are both proportional to a −6 φ −2 . The factor a −6 may make these contributions decreasing very fast, faster than the matter density ρ M , provided that φ does not decrease as fast as a −3/2 (i.e. along with the expansion of the Universe). Moreover, the contributions from the quintessence should have been insignificant just a short time ago, otherwise their gravitational influence would have prevented the ordinary matter from forming structures, and also back to the epoch of primordial nucleosynthesis for preserving the concordance between theories (Big Bang Cosmology + Inflation + Cold Dark Matter) and observations (for instance, measurements of CMB and light element abundances). Therefore, it may be reasonable to assume that the contributions from the "angular motion" of the complex scalar field Φ to the energy density and pressure are negligible.
On the other hand, under the situation that φ decrease no slower than a −3/2 , the angular-motion contributions to the energy density and pressure are no longer necessary to be negligible, while the contributions from evolving φ (proportional toφ 2 ) in Eqs. (13) and (14) would decrease no slower than a −3 H 2 . Likewise, these evolving-φ contributions should have been insignificant just a short time ago (as explained in the last paragraph), and hence can be neglected in Eqs. (13) and (14).
To sum up, we have two possible situations, depending on how fast a −6 φ −2 falls off as the Universe expands. One situation is that the contributions from the "angular motion" of the complex scalar field Φ to the energy density and pressure are negligible. The other is that the contributions from evolving φ are negligible. We note that, when both the contributions from evolving φ and the "angular motion" of the complex scalar field Φ are negligible such that the quintessence potential dictates, it is equivalent to the case of having a positive cosmological constant.

Reconstruction Equations
The quantities introduced in Sec.
where ρ c is the critical density, and we have set the present scale factor a 0 to be one. In addition, for the quantityä/a in Eq. (14), which is related to the acceleration of the expansion, we haveä Using Eqs. (19)-(23), we can obtain the reconstruction equations from the fundamental equations (13) and (14). These reconstruction equations relate V (φ),φ, andθ to the observationally relevant quantities z, r(z), H 0 , and Ω M as follows: where the term (ω 2 /φ 2 )(1 + z) 4 (dr/dz) 2 in Eq. (25) is the contribution from the "angular motion" of the complex scalar field Φ. Through the reconstruction equations, given the data r(z) from SNe Ia experiments and the values of the parameters Ω M and H 0 obtained from other experiments, it seems that we can reconstruct the quintessence potential V (φ) after inputting the values of ω and φ 0 , which correspond to some specific initial conditions. But, unlike the case of a real scalar field discussed in [10], the values of ω and φ 0 are not the parameters we can input arbitrarily. They will determine the proportion of the contributions from the "angular motion" of the complex scalar field Φ to the energy density and pressure. As discussed in Sec. 2, we have two possible situations, the one in which the contributions from the "angular motion" of the complex scalar field Φ are negligible and the other one in which the contributions from evolving φ are negligible. In the following, however, we will show that the reconstruction of V (φ) is still possible for both situations.
For the situation in which the angular-motion contributions are negligible, the reconstruction equations become the same as those in [10] as follows: We note that in this case only the shape of V (φ) and the corresponding region of V (φ) (which corresponds to the observational region of redshift z) can influence the evolution of the Universe, while the value of φ 0 has no influence. Thus, in this case, the initial value φ 0 can be put in by hand, and ω is no longer the parameter we need to input. With the initial value φ 0 , the parameters: Ω M and H 0 , and the data r(z), we can obtain the information on V (z) and φ(z) through the reconstruction equations (26) and (27), respectively, and then reconstruct the quintessence potential V (φ) for some specific region of V (φ) corresponding to the observational region of redshift z. We note that the reconstruction of V (φ) in this case is in the same way as the case of a real scalar field discussed in [10]. On the other hand, for the situation in which the evolving-φ contributions are negligible, the reconstruction equations become With the initial value φ 0 , the parameters: Ω M and H 0 , and the data r(z), Eq. (29) may be used to determine the value of ω and yield the information on φ(z), and Eq. (28) gives the information on V (z) in the observational region of redshift z. Then the reconstruction of the quintessence potential V (φ) can be achieved for some region of V (φ) corresponding to the observational region of redshift z. Unlike the previous case, the reconstruction of V (φ) in this case is different from the case of a real scalar field discussed in [10], because the possible existence of a significant angular-motion contribution here is the very feature that the real quintessence does not possess.

Discussion and Summary
In this work, we have investigated the scenario of using a complex scalar field as the quintessence for accelerating the expansion of the Universe. In the present scenario, there are two kinds of "kinetic-energy" type contributions to the energy density and pressure, one coming from the evolving amplitude φ and the other from the "angular motion" of the complex scalar field Φ. In many cases, the contribution from the "angular motion" may decrease very fast along with the expansion of the Universe, and is negligible in the process of reconstructing the quintessence potential V (φ) (which is responsible for the possible accelerating expansion of the Universe). Nevertheless, there is also a situation in which the contribution from the evolving amplitude φ is negligible while the part from the angular motion need to be treated with care. Making use of the reconstruction equations (24) and (25) (as derived from the fundamental equations (6)-(9)), we may reconstruct the quintessence potential V (φ) from the observational data r(z) (the coordinate distance as a function of the redshift z, as may be deduced from SNe Ia experiments), respectively for the two situations mentioned above. Accordingly, the complex scalar fields may be used as the candidate for the quintessence and our analysis indicates that, depending on how fast a −6 φ −2 falls off as the Universe expands, the quintessence potential may be reconstructed in a fairly unique manner.
It is useful to note that the observation data on r(z) may be converted uniquely into the information on the effective equation of state of the dark energy: w(z) ≡ p X (z)/ρ X (z) (where 'X' denotes the dark energy) [12,13]. Such information may in turn be used to distinguish different quintessence scenarios: The scenario with a positive cosmological constant corresponds to w = −1 so that a significant variation of w, especially differing from the value of −1, would help to rule out such scenario. The distinction between the real and complex scalar field scenarios is obviously more subtle: When the "angular motion" part is negligible, the complex quintessence behaves like the real one. However, the situation when the "angular motion" part is more important by comparison needs to be further studied since it poses new possibilities for the Universe.
In any event, the complex scalar fields as the quintessence should be seriously considered since such fields, unlike the real scalar field, have been invoked in many different sectors of elementary particle physics, such as the possibility for gauging (i.e., interacting with the various gauge fields), responsible for mass generations (Higgs mechanisms), as well as arising from condensates into Goldstone "pions" (from techni-color quark-antiquark pairs). In our opinion, the rich physics associated with the complex fields should be taken seriously in constructing a workable model, or theory, for the early universe.