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. Hence, it is necessary and constructive to present a brief of the dark energy. In combination with the second Fried- review on wormholes. mann equation, one then knows that ω<−1/3 is a nec- essary condition for the cosmic acceleration expansion the 123 Eur. Phys. J. C (2016) 76:484 Page 3 of 13 484
−1 <ω<−1/3 case is coined the quintessence region, [102] proposed an unknown vacuum energy model according ω =−1 is the well-known cosmological constant case (also to the holographic principle, in which the fine-tuning problem named phantom divide or cosmic barrier) and the ω<−1 at the cosmological scale as the dark energy and the coinci- case corresponds to the phantom region. At the same time, dence problem is also alleviated. But this model has an essen- we can easily find that in the phantom range, the NEC is natu- tial defect: the universe is decelerating and the effective equa- rally violated. Thus, whatever mysterious dark energy model tion of state parameter is zero. Subsequently, Fischler et al. one has, if one expects to explore the special wormhole solu- [103,104] proposed the particle horizon could be used as the tions, one must have a phantom or phantom-like equation of length scale. Nonetheless, as Hsu [105] and Li [33] pointed state as regards the real cosmic background evolution. out, the equation of state parameter is still greater than −1/3, In this paper, we intend to investigate the specific geomet- so this model could not explain the expansion mechanism of ric RDE wormholes corresponding to a non-ideal equation of the universe. For this reason, Li proposed the future event state. In particular, traversable wormholes (the second class), horizon could be used as the characteristic length. This holo- whose existence could provide an effective tool for the rapid graphic dark energy model could be expressed as follows: interstellar travel, may be of much more interest to causal physics development. So we would like to dedicate some ρ = 2 2 −2, H 3c MP L (1) efforts to the study of RDE traversable wormholes, since they ρ 2 may contain more constructive physical insights. Especially, where H is the holographic dark energy density, c a dimen- the most important result of this paper is that we explore the sionless constant, MP the Planck mass, and L the size of a traversable wormholes constrained by modern astronomical box containing the total energy. This model does not only observations, i.e., constrain the equation of state parameter give an accelerated universe but also is well compatible with by using various astrophysical data-sets (SNe Ia [89], baryon the current astronomical observations. Before long, Gao et al. acoustic oscillations (BAO) [90–93] and OHD [94–99]), in [40] had proposed a new HDE model, called RDE, in which order that one can determine when the exotic matter appears the future event horizon area is replaced by the inverse of with the evolution of the global universe background in dark the Ricci scalar curvature. They showed this model does not energy dominated cosmological models. As a result, one can only avoid the causality problem and is phenomenologically obtain a essentially clear picture in which the stage of the evo- viable, but also naturally solves the coincidence problem. lution of the universe the wormholes can exist (open) and/or In the following, we consider the spatially flat Friedmann– may disappear (close) in another different stage. This new Robertson–Walker (FRW) universe, and the Ricci scalar cur- connection, to our knowledge, which never had been clari- vature is given by fied in the previous literature about wormholes, between the ˙ 2 wormhole physics and exact cosmology modelings, can give R =−6(H + 2H ), (2) us a completely new perspective to investigate the evolution behavior of the wormhole spacetime configurations. where H =˙s/s is the Hubble parameter, the dot denotes a The present paper is organized in the following manner. derivative to the cosmic time t. The RDE model states the In the next section, we present a brief review on the RDE dark energy density is proportional to the Ricci curvature, model. In Sect. 3, we constrain the RDE model by the SNe 3α α Ia, BAO, and OHD data-sets. In Sect. 4, we present a general ρ = (H˙ + 2H 2) =− R, (3) solution of a traversable wormhole supported by the RDE 8π 16π cosmological fluid. In Sect. 5, we investigate several specific where α is a constant that can be determined by the current wormhole geometries and their physical properties and char- observations. The factor 3 is introduced for simplicity in acteristics, including three special choices for the redshift 8π the following calculations. Combining with the Friedmann function, a specific choice for the shape function, a constant equation, Gao et al. [40] obtain the result energy density and, finally, the isotropic pressure case. In Sect. 6, we present a discussion, point out the possible future α 2 −3x −(4− 2 x) direction of study, and conclude the present efforts. ρ = 3H m e + f e α , (4) 0 2 − α 0 0 We adopt the units 8πG = c = h¯ = 1.
where the subscript 0 denotes the present-day value, m0 ≡ 2 ρm /3H , x ≡ ln s (s denotes the scale factor) and f is an 2 Review on RDE 0 0 0 integration constant. Replacing Eq. (4) in the energy conser- vation equation, The holographic principle [100,101] is realized in QGT, which indicates that the entropy of a system increases not 1 dρ 2 p =−ρ − , (5) with its volume, but with its surface area L . Cohen et al. 3 dx 123 484 Page 4 of 13 Eur. Phys. J. C (2016) 76:484 one could easily obtain the dark energy pressure where 580 2 2 1 −( − 2 ) μ (z ) − μ (z ; δ; μ = 0) p =− H 2 − f e 4 α x . (δ) = obs i th i 0 , 3 0 α 0 (6) A (12) 3 3 σi i=1 For convenience of the calculations, the Friedmann equation can be rewritten as 580 μ (z ) − μ (z ; δ; μ = 0) B(δ) = obs i th i 0 , (13) σ 2 2( ) α i=1 i 2 H z 3 2(2− 1 ) E (z) = = m0(1+z) + f0(1+z) α (7) 2 2 − α 580 H0 1 C = . (14) σ 2 where E(z) denotes the dimensionless Hubble parameter, z i=1 i = 1 − the redshift, and z s 1. In the following, we will constrain χ 2 μ = B Therefore, S is minimized when 0 C by calculating the the parameters of the RDE model by the SNe Ia, OHD, and transformed χ 2 : BAO data-sets. SN [B(δ)]2 χ 2 = A(δ) − . (15) SN C 3 Astronomical observations constraints χ 2 One can constrain the RDE model by using SN, which is 3.1 Type Ia Supernovae μ χ 2 independent of 0, instead of S .
The observations of SNe Ia provide a forceful tool to probe 3.2 Observational Hubble parameter the expansion history of the universe. As is well known, the absolute magnitudes of all SNe Ia are considered to be the In the literature, there are two main methods of independent ≈ same, since all SNe Ia almost explode at the same mass (M observational H(z) measurement, which are the “radial BAO − . ± . 19 3 0 3). For this reason, SNe Ia can theoretically be method” and “differential age method”, respectively. More used as the standard candles. In the present paper, we adopt details can be found in [107]. The χ 2 for OHD is the Union 2.1 data-sets without systematic errors for data fitting, consisting of 580 points covering the range of the 29 H E(z ) − H (z ) 2 ∈ ( . , . ) χ 2 χ 2 = 0 i obs i . redshift z 0 015 1 4 . For performing the so-called O (16) σi fitting, the theoretical distance modulus is defined as i=1
μ (z ) = 5log D (z ) + μ , (8) Using the same trick as above, the minimization with respect th i 10 L i 0 χ 2 to H0 can be made trivially by Taylor-expanding O as μ = . − where 0 42 39 5log10 h, h is the dimensionless Hubble − − χ 2 (δ) = 2 − + , parameter today in units of 100 km 1 s 1 Mpc, O AH0 2BH0 C (17) z where = ( + ) dz DL 1 z (9) 0 E(z ; δ) 29 E2(z ) A = i , (18) is the Hubble luminosity distance in a spatially flat FRW σ 2 i=1 i universe, and δ denotes the model parameters. The corre- sponding χ 2 function to be minimized is S 29 E(z )H (z ) B = i obs i , (19) σ 2 580 μ ( ) − μ ( ; δ) 2 = i χ 2 = obs zi th zi , i 1 S σ (10) = i 29 2 i 1 H (zi ) C = obs . (20) σ 2 = where σi and μobs(zi ) are the corresponding 1σ error and i 1 i the observed value of the distance modulus for every super- Therefore, χ 2 is minimized when H = B by calculating μ O 0 A nova. The minimization with respect to 0 can be obtained the following transformed χ 2 : χ 2 OHD by Taylor-expanding S as [106] B2 χ 2 = − μ + μ2, χ 2 =− + C. (21) S A 2B 0 C 0 (11) OHD A 123 Eur. Phys. J. C (2016) 76:484 Page 5 of 13 484
χ 2 0.40 One can constrain the RDE model by using OHD, which is χ 2 independent of H0, instead of O .
3.3 Baryon acoustic oscillations 0.35
In addition to the SNe Ia and OHD data-sets, another con-
straint is from BAO traced by the Sloan Digital Sky Survey b 0.30 (SDSS). We use the distance parameter A to measure the BAO peak in the distribution of SDSS luminous red galax- A ies, and the distance parameter can be defined as 0.25 z 2 − 1 1 i dz 3 A = E(z ) 3 , m0 i ( ) (22) zi 0 E z 0.20 0.30 0.35 0.40 0.45 0.50 2 a where zi denote different redshifts. The χ for the BAO data- sets is Fig. 1 1σ,2σ,and3σ confidence ranges for parameter pair (a, b) of the RDE model, constrained by SNe Ia and BAO data-sets (here 7 A ( ) − A ( ; δ) 2 we fix the parameter c at the best fitting value c = 0.644957). For χ 2 = obs zi th zi . BAO (23) simplicity, we denote the model parameters α = a, = b,and σA m0 i=1 f0 = c, respectively
In the following context, for simplicity, we denote the model parameters α = a, m0 = b, and f0 = c, respectively. At 0.40 first, we compute the joint constraints from SNe Ia and BAO data-sets. The χ 2 can be defined as
0.35 χ 2 = χ 2 + χ 2 . 1 SN BAO (24)
In the second place, we calculate the combined constraints
b 0.30 from SNe Ia, OHD, and BAO data-sets. The χ 2 can be defined as
χ 2 = χ 2 + χ 2 + χ 2 . 0.25 2 SN BAO OHD (25)
The likelihoods of the parameters (a, b) in the two different constraints (χ 2 and χ 2) are depicted in Figs. 1 and 2, respec- 0.20 1 2 0.30 0.35 0.40 0.45 0.50 tively. The best fitting values of the parameters and the values a χ 2 χ 2 of the reduced 1 and 2 are listed in Table 1.Atthesame time, it is very constructive to show the relation between the Fig. 2 1σ,2σ,and3σ confidence ranges for parameter pair (a, b) distance modulus and redshift (Fig. 3). As a result, one can of the RDE model, constrained by SNe Ia, OHD and BAO data-sets (here we fix the parameter c at the best fitting value c = 0.650594). naturally get the evolution behavior of the universe when For simplicity, we denote the model parameters α = a, m0 = b,and taking the parameters as the best fitting values (Fig. 4). In f0 = c, respectively addition, the effective equation of state parameter ωX with respect to the redshift z from data fitting (see Table 1)are depicted in Fig. 5. Table 1 The best fitting values of the parameters for the RDE model by In Fig. 3, one can easily get the conclusion that the theo- using two different kinds of joint constraints: 580 SNe Ia gold samples, retical curve of the distance modulus with respect to red- 7 BAO data points and 29 OHD data points shift is well behaved by a comparison with the 580 SNe SNe Ia + BAO SNe Ia + OHD + BAO Ia samples. In Fig. 4, one can find that the cosmological χ 2 /d.o.f. 566.071/(587) 589.777/(616) background evolution of the RDE model is consistent with min . . the CDM model in the past. However, when z approaches a 0 4164098 0 423469 . . 0, corresponding to the present universe, the discrepancy b 0 280558 0 268549 . . occurs, the Hubble parameter of the RDE model will be a c 0 644957 0 650594 123 484 Page 6 of 13 Eur. Phys. J. C (2016) 76:484
46 –0.4 44
42 –0.6 (Z)
40 x (z)
μ ω –0.8 38
36 –1.0
34 –1.0 –0.5 0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z z Fig. 5 The relation between the effective equation of state parame- Fig. 3 The relation between distance modulus and redshift. The solid ter ωX (z) and the redshift z:thesolid (red) line and the dotted (blue) line represents the theoretical curve calculated from the model con- line correspond to the CDM model and the RDE model, respectively. cerned. The dots with errors bar correspond to the 580 data points from One can apparently discover that the RDE model represents a quintom- the supernova observations. One can find that the theoretical curve is like model crossing the phantom divide ωX (z) =−1(orz < 0.109), well consistent with the observations which mainly includes two types of exotic matter: phantom-like and quintessence-like. It is noteworthy that when ωX (z)<−1, the worm- holes will exist in the universe (open)andwhenz →−1, the wormholes will disappear (close) since the universe tends to be a super rip 180
160 physical observations, we can explore the related physics for 140 a concrete type of cosmic matter. In particular, we are much interested in the attractive and H(z) 120 elegant objects, wormholes. Hence, in this paper, we intro- 100 duce astrophysical observations into the field of wormhole physics, which seems to be the first attempt in the literature. 80 Through astronomical observations, we define a constraint on the parameters of a cosmological model, explore the type –1.0 –0.5 0.0 0.5 1.0 1.5 z of cosmic matter, limit the number of available models for wormhole research, reduce the number of the wormholes Fig. 4 The relation between the Hubble parameter and redshift. The corresponding to different parameters for a concrete cosmo- solid (red) line and the dotted (blue) line correspond to the CDM logical model, and provide a clearer picture for wormhole model and the RDE model, respectively. One can easily find that the RDE model is well consistent with the CDM model in the past. research from the point of view of observation cosmology. As Nonetheless, when the redshift z → 0 gradually, one may discover an illustration, in the following, we investigate the traversable that the discrepancy occurs: the Hubble parameter of the RDE model wormholes in the RDE model. will be a little higher than the standard cosmological model. In the remote future, the expansion velocity of the universe in RDE model will diverge, i.e., the universe tends to be a super rip 4 RDE traversable wormholes
4.1 Basic equations little higher than the standard cosmological model. In the future, the expansion velocity of the universe in RDE model In the present paper, we consider the spacetime geometry will diverge, namely, the universe tends to be a super rip. In representing a static and spherically symmetric wormhole Fig. 5, one cannot only find the evolution behavior of the RDE model, but also one apparently discovers the change of 2 ( ) dr the type of the cosmic matter (phantom-like or quintessence- ds2 =−e2 r dt2 + + r 2(dθ 2 + sin2 θdφ2), (26) − b(r) like) with the evolution of the universe by comparing with the 1 r CDM model. One can easily see that the RDE model corre- sponds to a quintom-like matter (virtually, still phantom-like where b(r) and (r) are arbitrary functions of the radial or quintessence-like). It is worth noting that this is the essen- coordinate r, denoting the shape function and redshift func- tial starting point of our work. Since we can describe quanti- tion, respectively [81]. The radial coordinate r runs in the tatively the evolution of the type of cosmic matter by astro- range r0 ≤ r < ∞ where r0 corresponds to the radius of 123 Eur. Phys. J. C (2016) 76:484 Page 7 of 13 484 the wormhole throat, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π are the 2 1 3αη pr =− − ρ − . (34) angular coordinates. One can also consider a cutoff of the 3α 3 2 − α stress-energy tensor at a junction radius a. As mentioned above, the most fundamental requirement Using Eqs. (30) and (31), we can obtain the following rela- to form a wormhole is violating the NEC. Besides, there tion: are also two fundamental requirements to form a traversable − 3( 2 − 1 )( b − 3αη ) b r α 2 −α wormhole. The first fundamental ingredient is satisfying the (r) = 3 3 r 2 . (35) 2( − b ) so-called flaring out conditions that can be expressed as fol- 2r 1 r lows: One can denote the solutions of Eq. (26) satisfying Eq. (35)as b(r0) = r0, (27) “RDE wormholes”. Furthermore, if the solutions also satisfy b (r0)<1, (28) the traversability condition, we denote it as “RDE traversable wormholes”. b(r)
where dl = (1 − b )−1/2dr is the proper radial distance, 5.1 Constant redshift function r and we assume that the space stations are located at a radius r = a −l l For a constant redshift function, namely, = 0, one can get ,at 1 and 2, respectively. the following shape function: It is obvious that inequalities (45) and (46) are well satis- fied in the case of a constant redshift function. Substituting 3α 3α Eqs. (43) and (44) into inequality (47) evaluated at the throat, r 2−α αη r 2−α b(r) = r + r 3 − r 3 . neglecting the essentially small term that contains the redshift 0 ( − α) 0 (43) r0 2 1 r0 z, considering a constant non-relativistic traversal velocity, namely, γ ≈ 1, we obtain the new conclusion for the veloc- It is easy to see that b(r) α (2 − α)g⊕ 3 2 v ≤ r . (50) b (r ) = (1 + ηr ). (44) 0 ( − α)λ2 0 2 − α 0 1 2 123 Eur. Phys. J. C (2016) 76:484 Page 9 of 13 484 In the following, through considering the equality case, IV = [pr (r) + ρ]dV , with a cutoff of the stress-energy assuming that the throat radius is given by r0 ≈ 100 m and tensor at a, one can obtain α = . taking into account the best fitting value 0 423469, we a 2 get the traversal velocity v ≈ 710.42 m/s. Furthermore, if one e IV = (r − b) ln takes into consideration that the matching radius is provided 1 − b r r0 by a = 10,000 m, then one can obtain τ ≈ t ≈ 28.15 a 2 e s from the traversal times τ ≈ t ≈ 2a/v (one can com- − (1 − b ) ln dr − b pare this case with that in [108] in order to get more useful r0 1 r information). a e2 = (r − b) ln dr, (54) 1 − b ( ) = ( r0 ) r0 r 5.2 r ln r where we have considered the asymptotical flat case, so the One can also make another choice of the redshift function, first boundary term vanishes. Then, the above integral can be which seems to be a little more complex than the first case, calculated, by using the above-mentioned redshift function ( ) = ( r0 ) i.e., r ln r . Substituting it into Eq. (35), one can get and shape function, as follows (for simplicity, we neglect the term containing redshift z): αη b − r 3 2 − 1 b − 3 1 3α 3 r2 2−α − = . (51) 3α b a α− r 2 − 2(2α − 1)[3aα − (α + 1)r0 − r0( ) 2 (2α − 1)] 2r 1 r r0 IV = . 3α(1 + α) Solving this equation, it follows that (55) 3α α−2 − α αηr 2 α αη 3 It is worth noting that if we adopt the best fitting parame- ( ) = r 1 2 − 0 + 3 + r . b r r0 r ter α = 0.423469 from the astrophysical observations, the r0 1 + α 2 1 + α 2 integral can be expressed as (52) I =−0.16928 V Furthermore, by differentiating both sides with respect to r, r 0.805824 × 1.27041a − 1.42347r + 0.153062r 0 . one can see that 0 0 a 2(2α−1) (56) 3α r 2−α 1 − 2α αηr 2 b(r) = − 0 α − 2 r0 1 + α 2 It is easy to see that when taking the limit a → r0 the inte- I → 3α 3αη2r 2 gral will be zero, namely, V 0. This indicates that we can + + . (53) theoretically construct a traversable wormhole with infinites- 1 + α 2 imal amounts of ANEC violating RDE matter. Furthermore, It is easy to check that the shape function satisfies the flaring one can find that this method may provide more information out conditions and b (r0) is still the same with the case of a for us as regards the “total amount” of ANEC violating mat- constant redshift function (Eq. 44). Unfortunately, this solu- ter in the whole spacetime (see [112] for more details). In tion is not asymptotically flat. However, one can glue it to an addition, actually, the quantity IV → 0 is satisfied for any exterior vacuum spacetime at a matching radius a. Moreover, possible value of α based on the observational constraints this solution is a traversable wormhole, since the correspond- when z < 0.109, and here we just choose the best fitting ing redshift function is finite in the range r0 ≤ r ≤ a. value of α as a concrete case. Ten years ago, Visser et al. discovered that one could the- oretically construct traversable wormholes with infinitesimal 5.3 (r) = ln( r ) amounts of average null energy condition (ANEC) violating r0 matter [112,113]. To be precise, taking into consideration the ( ) = ( r0 ) As a comparison with the case r ln r , we consider notion of “volume integral quantifier”, one could quantify the ( ) = ( r ) r ln r here. Similarly, one obtains total amounts of exotic matter by computing the definite inte- 0 μ ν μ ν grals TμνU U dV and Tμνk k dV , and the amount of 9α r 2−α 3α αηr 3 exotic mater is defined as how negative the values of these b(r) = r − − 0 0 1 α − ( − α) integrals become. It is of much interest for us to apply this r0 5 1 2 1 2 method to the RDE model, in order to study whether it is 3α αηr 3 + r + (57) the same case. Then, using this effective method, given by 5α − 1 2(1 − 2α) 123 484 Page 10 of 13 Eur. Phys. J. C (2016) 76:484 and (r) →∞when r →∞. Similarly, one can construct a 2(5α−1) traversable wormhole by matching the interior geometry to −α 3 9α r 2 3α αηr b (r) = 1 − − 0 an exterior vacuum geometry. 2 − α r0 5α − 1 2(1 − 2α) 3α 3αηr 2 5.5 Constant dark energy density + + . (58) 5α − 1 2(1 − 2α) Taking into account a constant dark energy density as in Evaluating at the throat, one gets the same expression as Eq. [114], namely, ρ = ρ0,fromEq.(30) one can obtain (44)again: ρ ( ) = 0 ( 3 − 3) + . 3α b r r r0 r0 (62) b(r ) = ( + ηr 2). 3 0 − α 1 0 (59) 2 ρ = 0 By a new definition A 3 ,Eq.(28) can be expressed as Obviously, this solution reflects a non-asymptotically flat 2 < = β 3Ar0 1. We should take into consideration A 2 , with 3r0 wormhole. Adopting the same step as mentioned above, we 0 <β<1, in order that Eq. (62) can be rewritten as can also construct a traversable wormhole in the finite range. It is more noteworthy that, as we think, the three choices of the β r 3 redshift function can give us a mathematical paradigm as Eq. b(r) = r0 − 1 + 1 . (63) 3 r0 (44). This means, at the throat r = r0, that the violation of the NEC can provide the same result for us (p (r )+ρ(r )<0). r 0 0 To form a wormhole geometry, Eq. (29) must be satisfied. Furthermore, these three wormholes have a high degeneracy Through some calculations, one can find that b(r) = r has at the same throat. More physically, when three different 12 −3−1 travelers cross the throats of three different wormholes at the = = β two positive roots: r1 r0 and r2 r0 2 , and r lies same moment, respectively, they may see the same bending of in the finite range light and feel the same radial pressure, gravitation accelera- α = . tion, etc. In addition, if we take the parameter 0 423469 12 − 3 − 1 ( ) ≈ . < β once again, then b r0 0 805824 1. Therefore, the r0 < r < r0 . (64) introduction of the astrophysical observations will provide 2 more useful information to investigate the behavior of the To be clearer, this constraint condition is shown graphically objects. in Fig. 6. One can get the conclusion that the dimensions of the wormholes decrease when the values of β increase. 5.4 b(r) = r ( r ) 0 r0 Substituting Eq. (63) into Eq. (35), one can find that the redshift function can be expressed as ( ) = ( r ) 1 β(2r + r ) Taking into account the case b r r0 , one can obtain (r) = C + √ √ arctanh √ 0 [3(αβ + ηr 2) r0 1 β( − β) 0 the redshift function as follows from Eq. (35): 2 3(β − 1) β(1 − β) r0 3 4 +(α − ) 2ρ ]− 1 ( ) αη 2 r0 0 ln r r ( r ) − r 3( 2 − 1 )[ ( r )−1 − 3 ] 2 0 r0 3α 3 r2 r0 2−α (r) = . (60) 2 2 2 2[ − ( r0 )1−] ln[(β − 3)r + βr0r + βr ]{r (2β − 3)[3η + (α − 2)ρ0]−3αβ} 2r 1 r + 0 0 12αβ(1 − β) Unfortunately, this equation cannot be solved analytically. ln(r − r )[3(α + r 2η) + (α − 2)r 2ρ ] + 0 0 0 0 , (65) Thus, one can solve it numerically for every given parameter 6(1 − β) . Moreover, by using inequality (28), one can discover that where C is an integration constant. One may find that < 1 1. In the following, we only consider the particular case when r = r there is an event horizon, so the solution is a = 1 0 2 . So the redshift function becomes α rη[6r + 3r + 4 r0 (3r + r)]−(7α − 2)[2arctanh r0 + ln(1 − r0 )]+6r 2η[ln( r ) − Beta[ r , 1 , 1]] 0 r 0 (1+α)r r 0 r0 r0 2 (r) = , (61) 12α [ r , 1 , ] non-traversable wormhole. Nonetheless, assuming the con- where Beta r 2 1 is the incomplete Beta function, which 0 3(α+ηr2) equals 2 evaluated at the throat r = r0. Through some sim- dition ρ = 0 ,Eq.(65) will reduce to 0 (2−α)r2 ple calculations, one can see that (r) is finite everywhere. 0 Nonetheless, this solution is not asymptotically flat, since 123 Eur. Phys. J. C (2016) 76:484 Page 11 of 13 484 7 3.5 6 3.0 5 0 r 0 2.5 r 4 b(r)/ b(r)/ 2.0 3 1.5 2 1.0 1.0 1.5 2.0 2.5 3.0 1 1 2 3 4 5 6 7 r/r0 r/r0 Fig. 6 To form a wormhole geometry, according to the restriction Fig. 7 To form a wormhole geometry, according to the restriction ( )< b r r, one can find that the range below the solid (purple) line b(r) 3α(1+ηr2) ρ( ) = 0 Noting that r0 2( −α) , we can obtain √ r0 2 3 β(2r + r ) (r) = C + √ arctanh √ 0 1 (β − ) β( − β) β( − β) 2 1 1 r0 3 4 2 ( α− ) ( ) 3α[2(2α − 1) + 3ηr ] 2 2 1 r0 0 −α C1 = e 2 . (69) ×[α(β + 1) + 2ηr 2] ( − α) 2 0 2 r0 ln[(β − 3)r 2 + βr r + βr 2]{r 2(2β − 3)[3η + (α − 2)ρ ]−3αβ} + 0 0 0 0 Then, replacing Eq. (68)inEq.(30) and solving it when 12αβ(1 − β) taking the redshift function as (r) = ln( r ), one can get r0 1 − ln(r). (66) the shape function in the following manner: 2 ⎧ ( α− ) ⎫ 2 2 1 6(1−α) It is not difficult to prove that (r) given by Eq. (66) is finite ⎨ 2−α ⎬ η αr r 2−α (4α + 3ηr 2 − 2) in the range (64). Thus, as the above-mentioned case, match- b(r)=3r + 0 0 . ⎩2(1−2α) 2(1 − 2α)(7α − 8) ⎭ ing the interior spacetime geometry into an exterior vacuum geometry, this solution represents a traversable wormhole (70) now. Comparing with the same case in [114], one can obtain a mathematical paradigm like Eq. (63) for the shape function. It is easy to check that this solution satisfies the flaring out This means for different cosmological models that one can conditions b (r0) ≈ 0.9 < 1 and b(r) It follows that Since the discovery, by elegant methods, that our universe is undergoing an accelerated expansion, cosmologists have 2(2α−1) (r) C e 2−α − 3η proposed many alternatives to explain the accelerated mecha- ρ(r) = 1 . (68) 2(2α − 1) nism, which mainly include two classes: physical dark energy 123 484 Page 12 of 13 Eur. Phys. J. C (2016) 76:484 models and extended theories of gravity. Actually, one can a traversable wormhole by matching the interior spacetime find that the two classes of models are physically equivalent to an exterior vacuum spacetime. The fifth case also reflects by rearranging the terms in the Einstein field equations. Up a non-asymptotically flat spacetime, where the exotic matter to now, we still cannot determine what the nature of dark from the RDE fluids is distributed in the vicinity of the throat. energy is. In this paper, we are very interested in exploring For the case of isotropic pressure, one may discover that the the wormhole physics of the RDE model, which is one of the dimension of the wormhole is essentially finite. popular dark energy models, by assuming the dark energy is After the general relativity’s centennial, we are still con- distributed homogeneously in the whole spacetime. fused as regards the attractive and mysterious nature of dark In the past few years, there have been a lot of papers inves- energy and dark matter in different scales, if we assume tigating the wormholes spacetime and constraining the model the dark sector is permeated globally in the whole universe. parameters for various kinds of dark energy models. But none Wormholes are theoretical objects in the universe which now study wormholes by using the astrophysical observations as appear to attract more observational astrophysics interest and the data support. In particular, we can find that, through the may provide a new window for new physics. In this situation, constraints of observational data-sets on some well-known we apply the astrophysical data-sets to wormhole physics dark energy model, the evolution behavior of the equation of and investigate six specific solutions quantitatively, which state parameter during the whole history of the universe could seems to be the first time in the wormhole research. Through be well studied quantitatively. To be more precisely, when the astronomical observations, one can make a constraint on the NEC is violated, namely, the equation of state parameter is parameters of a cosmological model, explore the type of cos- less than −1, wormholes may exist (open). Since we think the mic matter in different stages of the universe, limit the num- astrophysical observations contain the most realistic physics, ber of available models for wormhole research, reduce the so we introduce naturally the astrophysical observations into number of the wormholes corresponding to different param- the wormhole research, which seems to be the first try in the eters for a concrete cosmological model, and provide a clearer literature. For a concrete instance, we explore the traversable picture for wormhole research from the new perspective on wormholes in the RDE model. the observational cosmology background. In the present paper, we present a brief review on the RDE The future work could be to consider an obvious rela- model and constrain this model by astrophysical observa- tion between the energy density and the transverse pressure, tions. Subsequently, we obtain the best fitting values of the explore the profound connection between wormholes and parameters in this model by using the usual χ 2 statistics, in energy conditions, and investigate the evolution of the struc- order that we can know more about the evolution behavior ture of the wormhole with time. of the universe and determine the evolution of the effective Acknowledgments Useful and long-time communications with Saibal equation of state parameter ωX with cosmic time. Then one can investigate the RDE traversable wormholes better after Ray are highly appreciated. We thank Prof. Jing-Ling Chen for helpful discussions and comments, and Guang Yang and Sheng-Sen Lu for an accurate equation of state is obtained. We have analyzed programming. This work is supported in part by the National Science the effective equation of state of the RDE model and give Foundation of China. the constraint relation of the parameters by using the flaring Open Access This article is distributed under the terms of the Creative out conditions. Furthermore, we have investigated some spe- Commons Attribution 4.0 International License (http://creativecomm cific solutions and the related physical properties and char- ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, acteristics by considering three different redshift functions, and reproduction in any medium, provided you give appropriate credit one specific shape function, constant dark energy density, to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. and isotropic pressure. 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