Eur. Phys. J. C (2016) 76:484 DOI 10.1140/epjc/s10052-016-4321-4 Regular Article - Theoretical Physics Traversable geometric dark energy wormholes constrained by astrophysical observations Deng Wang1,a, Xin-he Meng2,3,b 1 Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China 2 Department of Physics, Nankai University, Tianjin 300071, China 3 State Key Lab of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing 100080, China Received: 2 February 2016 / Accepted: 18 August 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In this paper, we introduce the astrophysical 1 Introduction observations into the wormhole research. We investigate the evolution behavior of the dark energy equation of state Modern astronomical observations with increasing evidence, parameter ω by constraining the dark energy model, so that such as high redshift Type Ia supernovae (SNe Ia), matter we can determine in which stage of the universe wormholes power spectra, observational Hubble parameter data (OHD), can exist by using the condition ω<−1. As a concrete cosmic microwave background radiation (CMBR), etc, have instance, we study the Ricci dark energy (RDE) traversable strongly suggested that the universe is undergoing an acceler- wormholes constrained by astrophysical observations. Par- ated phase at present [1–5]. To explain the accelerated mech- ticularly, we find from Fig. 5 of this work, when the effective anism, cosmologists have proposed a new negative pressure equation of state parameter ωX < −1(orz < 0.109), i.e., the fluid named dark energy. The simplest candidate of dark null energy condition (NEC) is violated clearly, the worm- energy is the so-called cosmological constant, i.e., the - holes will exist (open). Subsequently, six specific solutions cold-dark-matter (CDM) model, which has proved to be of statically and spherically symmetric traversable wormhole very successful in describing many aspects of the observed supported by the RDE fluids are obtained. Except for the universe. For instance, the spectrum of anisotropies of the case of a constant redshift function, where the solution is not CMBR, the large scale structure (LSS) of matter distribution only asymptotically flat but also traversable, the five remain- at linear level, and the expansion phenomena are very well ing solutions are all non-asymptotically flat, therefore, the described by the standard cosmological model. However, exotic matter from the RDE fluids is spatially distributed in this model has faced two fatal problems, namely, the “fine- the vicinity of the throat. Furthermore, we analyze the phys- tuning” problem and the “coincidence” problem [6]. The for- ical characteristics and properties of the RDE traversable mer indicates that theoretical estimates for the vacuum den- wormholes. It is worth noting that, using the astrophysical sity are many orders of magnitude larger than its observed observations, we obtain the constraints on the parameters of value, i.e., the famous 120-orders-of-magnitude discrepancy the RDE model, explore the types of exotic RDE fluids in that makes the vacuum explanation suspicious, while the lat- different stages of the universe, limit the number of available ter implies that why the dark energy and dark matter are at the models for wormhole research, reduce theoretically the num- same order today since their energy densities are so different ber of the wormholes corresponding to different parameters during the evolution of the universe. In addition, a positive for the RDE model, and provide a clearer picture for worm- cosmological constant is inconsistent with perturbed string hole investigations from the new perspective of observational theory [7]. Therefore, a realistic interpretation of dark energy cosmology. should not be simply in terms of the cosmological constant (interpreting it as quantum vacuum). In recent years, to alle- viate or even solve these two problems, cosmologists have proposed a variety of dark energy models, partly as follows: • Exotic equation of state: a linear equation of state [8], van der Waals equation of state [9,10], Chaplygin gas [11–13], a e-mail: [email protected] generalized Chaplygin gas [14,15], modified Chaplygin gas b e-mail: [email protected] [16,17], superfluid Chaplygin gas [18–20], inhomogeneous 123 484 Page 2 of 13 Eur. Phys. J. C (2016) 76:484 equation of state [21], barotropic fluid model [22], Cardassian Wormholes could be defined as handles or tunnels in the model [23–26]. spacetime topology linking widely separated regions of our • Viscosity: bulk viscosity in the isotropic space, bulk and universe or of different universes altogether [81]. The most shear viscosity in the anisotropic space [27–32]. It is worth fundamental ingredient to form a wormhole is violating the μ ν noting that the perfect fluid that occurs in many papers is just Null Energy Condition (NEC), i.e., Tμνk k > 0, and con- an approximation of the medium of the universe. Nowadays, sequently all of other energy conditions, where Tμν is the all the observations indicate that the medium of the universe stress-energy tensor and kμ any future directed null vector. is not an idealized fluid and the viscosity is investigated in In general, the wormholes in the literature can be divided into the evolution of the universe. three classes: • The holographic principle: holographic dark energy [33– • Ordinary wormholes – this class just satisfies the viola- 36], Ricci dark energy [37–40], agegraphic dark energy [41], tion of the NEC and is usually non-asymptotically flat, sin- tachyon model [42,43]. gular and consequently non-traversable. • Dynamical scalar fields: quintessence (or cosmon) [44– • Traversable wormholes – in light of ordinary wormholes, 52], ghost condensates [53,54], phantom [55] and quintom one can obtain the traversable wormholes by an appropriate [56], the model potential ranging from power laws to expo- choice of redshift function or shape function. Subsequently, nentials and, to some extent, the quintom is an interesting one can analyze conveniently the traversability conditions of combination of quintessence and phantom. the wormholes and the stabilities. • Modified gravity: f(R) gravity [57–59], braneworld • Thin shell wormholes – one can theoretically construct models [60–63], Gauss–Bonnet models [64–67], Chern– a geodesically complete traversable wormhole with a shell Simons gravity [68], Einstein–Aether gravity [69], cosmo- placed in the junction surface by using the so-called cut- logical models from scalar–tensor theories of gravity [70– and-paste technique. This class has attracted much attention 76]. since the exotic matter required for the existence of spacetime In the above, a part of various models on this topic are configuration is only located at the shell, and it avoids very mentioned, since there are too many. Hereafter, we plan to naturally the occurrence of any horizon. focus our attention on the geometric contributions, namely, Wormholes like other extreme astrophysical objects also the so-called Ricci dark energy model based on the holo- have a long theoretical formation history. The earliest graphic principle. Although a complete quantum gravity the- remarkable contribution we are aware of is the 1935 introduc- ory (QGT) has not been developed, we could still explore tion of the object now referred to as an Einstein–Rosen bridge partly the nature of the dark energy by using the holographic [82]. Twenty years later, Wheeler first introduced the famous principle which acts as an important result of present QGT (or idea of “spacetime foam” and coined the term “wormhole” string theory) for gravity phenomena. Thus, the holographic [83]. The actual revival of this field is based on the 1988 paper dark energy model (HDE) constructed in light of the holo- by Morris and Thorne [81], in which they analyzed in detail graphic principle can bring us a new perspective on the under- the construction of the wormhole, energy conditions, time lying theory of dark energy. Recently, Gao et al. [40]pro- machines, stability problem, and traversabilities of the worm- posed a new HDE model called the Ricci dark energy model holes. In succession, Visser and Possion introduce the famous (RDE), in which the future event horizon area is replaced by “thin shell wormholes” by conjecturing that all “exotic mat- the inverse of the Ricci scalar curvature. They have shown ter” is confined to a thin shell between universes [84–88]. this model does not only avoid the causality problem and After that, there were a lot of papers to investigate the above is phenomenologically viable, but also naturally solves the three classes of wormholes and related properties. coincidence problem. In the past few years, in light of the important discovery In this study, we intend to investigate the astrophysical that our universe is undergoing a phase of accelerated expan- scale properties of the RDE during the evolution of the uni- sion, an increasing and significant interest in the subject of verse through assuming the dark energy fluid is permeated wormholes has arisen in connection with the globally cos- everywhere. In particular, as in our previous work [77–79], mological scale discovery. Due to the violation of NEC in the existence of wormholes is always an important problem both cases (astrophysics wormholes and cosmic dark energy in physics both at micro and macro scales. There is no doubt for simple terms), an unexpected and subtle overlap between that wormholes together with black holes, pulsars (physi- the two seemingly separated subjects occurs. To be precise, cal neutron stars), and white dwarfs [80], etc., constitute the one can usually parameterize the dark energy behaviors by most attractive, extreme, strange, and puzzling astrophysical an equation of state of the form ω = p/ρ, where p is the objects that may provide a new window for physical discov- spatially homogeneous pressure and ρ the energy density ery.