The Cosmological Model with a Wormhole and Hawking Temperature Near Apparent Horizon

Physics Letters B 780 (2018) 174–180

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Physics Letters B

www.elsevier.com/locate/physletb

The cosmological model with a wormhole and Hawking temperature near apparent horizon

Sung-Won Kim

Department of Science Education, Ewha Womans University, Seoul 03760, Republic of Korea a r t i c l e i n f o a b s t r a c t

Article history: In this paper, a cosmological model with an isotropic form of the Morris–Thorne type wormhole was Received 15 January 2018 derived in a similar way to the McVittie solution to the black hole in the expanding universe. By solving Received in revised form 1 March 2018 Einstein’s ﬁeld equation with plausible matter distribution, we found the exact solution of the wormhole Accepted 1 March 2018 embedded in Friedmann–Lemaître–Robertson–Walker universe. We also found the apparent cosmological Available online 6 March 2018 horizons from the redeﬁned metric and analyzed the geometric natures, including causal and dynamic Editor: M. Trodden structures. The Hawking temperature for thermal radiation was obtained by the WKB approximation using the Hamilton–Jacobi equation and Hamilton’s equation, near the apparent cosmological horizon. © 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction The research on wormhole is also an important issue in study of spacetime physics. The wormhole usually consists of exotic mat- The solution to the black hole embedded in expanding uni- ter which satisfies the flare-out condition and violates weak energy verse has been familiar to relativists and cosmologists for a long condition [7,8], even though there have been attempts to construct time since McVittie derived a model [1]. The cosmological black wormhole with non-exotic matter [9]. There were also solutions of hole solutions are more like a realistic model. Recently, the solu- cosmological wormhole model as well as the cosmological black tions also has attracted us because of the role of black hole in the hole solutions. There was the solution of a wormhole in inflation- expanding cosmological model. The evidences for the accelerat- ary expanding universe model [10]. In this solution, the wormhole ing universe and dark energy forced the study of the cosmological throat inflates at the same rate as that of the scale factor. Also black hole model. The interaction of black holes with dark energy there was a wormhole solution in FLRW cosmological model [11]. distributed over the universe can be one of the most important is- The solution also showed the expansion of the wormhole throat sues. Moreover, they can show a generalized theory of global and at the same rate as that of the scale factor. Hochberg and Kepart local physics, that is interested in the unification of interactions tried to extend the Visser type wormhole into a surgical connec- [2]. tion of two FLRW cosmological models [12]. Similarly there was After McVittie solution there were several models of black hole a solution for the connection of two copies of Schwarzschild–de in the universe. McVittie spacetime [1]was a spherically symmet- Sitter type wormhole as the cosmological wormhole model [13]. ric, shear free, perfect solution and was asymptotically Friedmann– There was a research on quantum cosmological approach by con- Lemaître–Robertson–Walker (FLRW) model. In this model, matters sidering wave function of the de Sitter cosmological model with a were isotropically distributed and there was no-accretion onto the wormhole [14]. Recently there was a cosmological wormhole so- black hole centered at the FLRW universe. As generalizations of lution [15]as a generalization of MT wormhole in FLRW universe, the McVittie solution, there were solutions of the charged black but there was a weak point that Einstein’s equation could not be hole in expanding universe [3,4]. Faraoni and Jacques obtained a guaranteed. generalized McVittie solution without no-accretion condition [2]. First of all, it is necessary to find the exact solution of the cos- Sultana–Dyer [5]got the extension of the geometry generated by mological wormhole model satisfying Einstein field equation. For conformally transforming the Schwarzschild metric with the scale reasons similar to black holes, the exact wormhole model embed- factor of flat FLRW universe. Kottler [6]derived the solution of the ded in the universe is very interesting to us. They will provide a Schwarzschild black hole in de Sitter background. lot of information to understand the relationship between wormhole matter and the background spacetime. Most of the previous cosmological wormhole solutions originated from the spacetime E-mail address: sungwon @ewha .ac .kr. assumed to be plausible models. If a wormhole were in the ex- https://doi.org/10.1016/j.physletb.2018.03.005 0370-2693/© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. S.-W. Kim / Physics Letters B 780 (2018) 174–180 175 r 1 panding universe, it would interact with dark energy in some way ˜ = = + 2 − 2 ˜ ∞ r (r r b0), (b0/2 < r < ). (4) that would be considered. The impacts on the spacetime are also B 2 very interesting. So we need to find the exact solution of worm- The old coordinate r and the radial factor B are given in terms of hole universe to see its influence to the evolution of spacetime. r˜ as In the spacetime structure with strong gravity, the Hawking 2 2 temperature is one of the issues focused on gravitation problem b r b r = r˜ + 0 , B = = 1 + 0 . (5) dealing with quantum phenomena. The Hawking temperature, de- 4r˜ r˜ 4r˜2 rived from the definition of surface gravity at event horizon, is a good example of the semi-classical handling of quantum gravity. Thus the isotropic form of the static wormhole is Usually the temperature is calculated from the surface gravity de- 2 fined by Killing vector in static case. In the dynamic case, such b2 ds2 =−dt˜2 + 1 + 0 (dr˜2 + r˜2d2), (6) as a black hole or a wormhole in expanding universe, the surface 4r˜2 gravity at the event horizon is not constant. In this case, we need to adopt the Kodama vector [16] instead of the Killing vector, and for the ultra-static case. In this isotropic wormhole model, the try to find the corresponding physical quantities for the spacetime metric does not diverge at r˜ = b0/2, while it diverges at r = b0 [17]. By using the WKB approximation to the tunneling method, in MT-wormhole. This is because this non-diversity is removed by the Hawking temperature is derived by comparing the thermal the coordinate transformation. We can also transform the matter distribution and probability amplitude from Hamilton–Jacobi equa- part of the old model [21]into new one by using the relationship tion [18]. There is also another way to find the probability ampli- between two coordinates (5). For wormhole solution, the matter tude from Hamilton’s equation, which was designed by Parikh and components in new coordinates are Wilczek [19]. There is an example of Hawking temperature at apparent horizon of the FLRW model in both methods [20]. ˜ 44b2r˜4 0 =− 0 = In this paper, we have found the exact solution of the wormhole T ˜ ρw , (7) 0 8π(b2 + 4r˜2)4 embedded in FLRW universe, the locations and the existence con- 0 ditions of apparent horizon. The influence of wormhole matter to ˜ 44b2r˜4 1 =− 0 =− the structure of the apparent horizons was studied. The Hawking T 1˜ τw , (8) 8π(b2 + 4r˜2)4 radiation was also discussed, and the temperature was also derived 0 4 2˜4 near the apparent horizon. 2˜ 4 b0r T ˜ =+ = P w , (9) 2 2 + ˜2 4 8π(b0 4r ) 2. Cosmological wormhole model 4 2˜4 3˜ 4 b0r T ˜ =+ = P w , (10) 2.1. Isotropic wormhole 3 2 + ˜2 4 8π(b0 4r ) where ρ , τ , and P are wormhole energy density, tension, and Now we derive the exact model of the wormhole embedded in w w w pressure, respectively. The negative density is still required for the FLRW cosmology. First, we need to find the isotropic form of the isotropic form of wormhole model. wormhole model in order to derive the wormhole solution embedded in a cosmological model because of the isotropy of the 2.2. Exact solution cosmological models in this paper. The Morris–Thorne type wormhole (MT-wormhole) is given by [7] From now on, we use un-tilde coordinate in isotropic form for 1 convenience if only there are no confusions in the rest of this pa- ds2 =−e2dt2 + dr2 + r2d2, (1) 1 − b(r)/r per. The FLRW spacetime in isotropic form is given by [1] where (r) is red-shift function and b(r) is the shape function. a2(t) ds2 =−dt2 + (dr2 + r2d2). (11) The geometric unit, that is, G = c = h¯ = 1is used here. The ra- (1 + kr2)2 dial coordinate r is in the range of b < r < ∞. Two functions (r) Here a(t) is the scale factor and k = 1/4R2, where R is the curva- and b(r) are restricted by the ‘flare-out condition’ to maintain the ture, and k goes zero in case of flat FLRW spacetime. We start from shape of the wormhole. Because the wormhole has the structure the general isotropic metric element to see the unified wormhole that prevents the existence of the event horizon, wormhole can be in FLRW cosmological model as used for two-way travel. Since MT-wormhole is spherically symmetric form, we introduce the new coordinates (t˜, r˜) to define the ds2 =−eζ(r,t)dt2 + eν(r,t)(dr2 + r2d2). (12) isotropic form of a wormhole as Similar to the McVittie solution, the following matter distribution 2 2 ˜2 2 2 2 2 ds =−A dt + B (dr˜ + r˜ d ). (2) in the universe is assumed: A spherical symmetric distribution of

b2 matter around the origin where there is a wormhole, no ﬂow of In this paper, we treat the example of b(r) = 0 and e2 = 1tosee r the matter as a whole either towards or away from the origin, the nature of wormhole geometry more simply. Then B becomes and isotropic pressure of matter in the universe. In addition to these assumptions, we add one more assumption that the local = 2 B , (3) and the global matters are separate. Since the wormhole is a local- b2 1 + 1 − 0 ized object, the matter of wormhole is separated from the cosmic r2 matter, which is distributed over the universe. When we adopt where the integration constant is determined by the asymptoti- the ansatz, the cosmological matter term is time-dependent and cally ﬂat condition. The new coordinate r˜ is given in terms of old isotropic, while the wormhole matter term depends only on space coordinates r as and not necessarily isotropic as, 176 S.-W. Kim / Physics Letters B 780 (2018) 174–180

2 2 a (t)ρ(r,t) = a (t)ρc(t) + ρw (r), (13) is the negative value of the tension. If we use ansatz to separate 2 2 temporal parts, as in the previous case (14)and (15), the equation a (t)p1(r,t) = a (t)p1c(t) + p1w (r), (14) contains only pure spatial parts as 2 2 a (t)p2(r,t) = a (t)p2c(t) + p2w (r), (15) 4b2 2 = 2 + 1 1 2 −β 0 a (t)p3(r,t) a (t)p3c(t) p3w (r), (16) [β − β − (β ) ]e = 2κ(p2w − p1w ) = . r 2 r4(1 + b2/4r2)4 which were already used in the previous wormhole cosmologi- 0 2 (26) cal model [11]. If we estimate the orders of magnitude, ρcc ∼ × −10 3 2 ∼ × 6 10 m 4 × 9 10 J/m over the universe and ρw c 6 10 ( r ) Here we use the wormhole matter components of (8) and (9). The b0 2 3 ( 10 m ) J/m near throat, where ρc is roughly the critical density general solution to this inhomogeneous differential equation is the for flat universe. The wormhole matter is concentrated at or near solution to the homogeneous equation plus the special solution. 4 throat and it decreases rapidly as 1/r , as wee see in (7)–(10). Thus For the black hole cosmological model, the right hand side is can- ρc ρw near throat at least. celed and the equation is homogeneous [1]. The solution for this Einstein’s equation is given by wormhole is β = βc + βw , where βc is the same as that of the black hole cosmology [1] and β is the special solution that sat- G = κ T , w αβ αβ isfies (26)and boundary condition (24). The physical meaning of μ where κ = 8π . The non-zero components of Einstein tensor G ν βc is the global cosmological background which dominates at far are region. The meaning of βw is the local wormhole solution which dominates near origin. The global cosmological solution is 0 1 2 −ν+ζ 2 −ζ G 0 = {[(8ν + 4ν r + ν r)e − 3ν˙ r]e }, (17) 4r 2 βc =−2ln(kr + 1) (27) 0 1 −ζ G 1 = (2ν˙ − ν˙ζ )e , (18) 2 with the boundary condition for FLRW model (24). The local 1 3 wormhole solution βw (special solution) that satisfies (26)with 1 = {[ − ¨ + − ˙ + ˙ ˙ −ζ+ν + + + G 1 r( 2ν ( ν ζ)ν)e 2ν 2ζ ζ ν r the boundary condition (24) becomes 2r 2 1 + 2] −ν} 2 rν e , (19) b0 2 βw = 2ln 1 + . (28) 4r2 1 1 −ν G 0 = (−2ν˙ + ν˙ζ )e , (20) 2 Thus the general solution to the equation (23)is 1 − 2 = 3 = {[ + + + + 2 ] ν 2 G 2 G 3 2ζ 2ν 2ν r 2ζ r ζ r e α(t) 2 4r + + e b eν(r,t) = eα(t) βc (r) βw (r) = 1 + 0 (29) 3 − + (kr2 + 1)2 4r2 + 2r(−2ν¨ − ν˙ 2 + ζ˙ν˙)e ζ ν}. (21) 2 and the cosmological wormhole solution finally becomes Here dot denotes the derivation with respect to t and prime de- 2 notes the derivative with respect to r. 2 2 1 2 2 a (t) b0 2 2 2 For the case of ultra-static observer (ζ = 0), G 0 = 0 becomes ds =−dt + 1 + (dr + r d ). (30) (kr2 + 1)2 4r2 ν˙ = 0. (22) When we inverse-transform this spacetime, at least in k = 0case, The general solution to (22)is into the spherically symmetric type, it returns to the MT-wormhole times the time-dependent scale factor a2(t). For flat FLRW uni- = + ν(t,r) = α(t) β(r) ν(r,t) α(t) β(r) or e e e . (23) verse, the spacetime is same form as the previous one [11]. The time-dependent part α(t) relates with the scale factor, eα(t) ≡ dr2 a2(t), while β(r) determines curvature of background and worm- a2(t) + r2d2 . (31) − b(r) hole spacetime, with the boundary condition as 1 r b2 b2 Now we found the exact solution of the wormhole in FLRW cos- eβ(r) = (1 + 0 )2 or β(r) = 2ln(1 + 0 ), for → b /2, 4r2 4r2 0 mological model. In this model, matters are separated into time- − eβ(r) = (1 + kr2) 2 or β(r) =−2ln(1 + kr2), for r →∞. dependent part for cosmological background ρc(t), and space- dependent part for wormhole ρw (r). Apparently the cosmic part (24) is coupled with wormhole part in spacetime metric of the cosmo- Since these two boundary conditions are dominant in the local and logical wormhole as in (29). However, the wormhole matter does global regions, we can find the unified form of eβ(r). The limit r → not affect cosmological background, since ρw does not affect the + 2 b0/2(or k = 0) represents the value of the wormhole throat, which term (1 kr ) determined by ρc(t). is a local solution to the wormhole. The limit r →∞ (or b0 = 0) is an asymptotic solution away from the origin, that is, the metric of 3. Apparent horizons and causal structures the FLRW cosmological model. 1 2 When we compare G 1 and G 2, we get To understand the causal structure of the spacetime, we should find and analyze the apparent horizons. If we redefine the new 1 1 2 −ν [ν − ν − (ν ) ]e = 2κ(p2 − p1). (25) coordinate R from the spacetime (30)as r 2 + 2 2 For the cosmological term, p = p due to the isotropy, while 1 b0/4r 1c 2c R ≡ a r = a(t)A(r), (32) 2 p1w = p2w for the wormhole case. In case of the wormhole, p1w 1 + kr S.-W. Kim / Physics Letters B 780 (2018) 174–180 177

Fig. 2. The apparent horizons (R±) and the Hubble horizon (R ) at a constant time Fig. 1. The function = (1 − R2 H2/r2 J 2) = (R2 − b2 − R4 H2)/(R2 − b2). Here we H t in terns of b are shown. Solid line is R+, dashed line is R−, dash-dot line is R set b = 1. The three cases for the number of solutions to = 0are shown. Two 1 H which is asymptote of R+, and the dotted line is the asymptote of R−. Here we set solutions are shown for b < R H /2(solid line), one solution is shown for b = R H /2 H = 0.05 at a constant time. (dashed line), and no solution for b > R H /2(dotted line). The dash-dot line is the case of b = 0. 1/(2H) = R H /2. Since we treat the isotropic coordinate in cosmo- 2 a b 2 2 the metric can be rewritten as ds = habdx dx + R d , where logical wormhole, the minimum is the half value of the spherically a = = − 2 + 2 2 + 2 symmetric case as in (4) and the comparison of Rmin to R H /2is x (t, R), hab diag( 1, a (1 b0/4r )/(1 kr )). The location of the apparent horizon is defined from the relation hab∂ R∂ R = 0. reasonable. We will consider the three cases for the solution to a b = When we see the metric, the apparent horizon is located at the 0. position which satisfies the relation as Case 1: b(t) < R H /2. In this case there are two horizons: the larger is R+ and the smaller is R−. The former is the cosmologi- H2 R2 cal horizon and the latter is the size of the wormhole throat in the ≡ 1 − = 0, (33) r(R)2 J(R)2 expanding universe. The fact that R+ < R H means that the wormhole reduces the Hubble horizon to a smaller size and b(t) < R− where indicates that the cosmological background enlarges the wormhole ˙ 2 3 throat size. Only in R− < R < R+, the coordinate t is timelike and a A 1 b0/2r 2kr = H, = J = − − . R is spacelike. In the region b < R < R− or R+ < R < R H , the role a A r + 2 2 1 + kr2 1 b0/4r of time and space coordinates is reversed. The coordinate R is lim- In the redefined coordinates system, appears as the coefficient ited to R > R− similar to a static wormhole, so we will do not of g and g for Schwarzschild-like metric. Simply we here see consider the region smaller than R−. tt RR = the k = 0case, which is the wormhole spacetime in flat FLRW uni- Case 2: When b(t) R H /2, two horizons√ coincide and there is only one horizon, such as R± = 1/( 2H). In this case the metric verse. In this case, r is given in terms of R by coefficient is negative except for that point. 2 2 Case 3: If b(t) > R H /2, there is no apparent horizon. In this R R b r = ± − 0 . (34) case, the coefficient is always negative (never zero). Thus the 2a 2a 4 coordinate t is spacelike and R is timelike. There is no regular region. This relationship shows that the size of the scaled wormhole There are two relations between r and R, but the plus sign is cho- throat can’t be larger than the half of the FLRW Hubble horizon, if sen only due to the positive nature of r. When R = a(t)b0, two horizons should exist. relations meet and r = R/2a. By putting this (34)into relation (33), Fig. 2 shows the apparent horizons in the case of b(t) ≤ R H /2at we get the location of the apparent horizons at a given time as a function of b(t). Three horizons, Hubble horizon without wormhole (R H ) and two horizons with wormhole (R±) at 1 1 2 1/2 R± = √ [1 ± 1 − (2b(t)H) ] < ≡ R H , (35) a constant time t1, are depicted in the figure. As mentioned above, 2 2H H when b(t1) = R H /2, two horizons meet. At the limit of b → 0, R+ approaches the FLRW horizon R H and R− goes to zero. At the limit where b(t) ≡ a(t)b0 is the size of the wormhole throat scaled by of H → 0, R− approaches b(t1). a(t) and is equal to the minimum of R at r = b0/2. R H = 1/H is the apparent cosmological horizon of FLRW model in absence The apparent horizons R± are produced from the wormhole of wormhole (b0 = 0) and is called ‘Hubble horizon’. When b0 = 0, matter ρw of b0 and its coupling to the cosmic matter ρc of a(t). the apparent horizons approach R+ = R H and R− = 0, respectively. The equation (7)shows that the b0 can be expressed by the peak As shown in Fig. 1, the equation = 0 provides two, one, of wormhole energy density. This value is equal to the energy den- or zero solutions, depending on the relative values of b(t) for sity at throat of the wormhole as 178 S.-W. Kim / Physics Letters B 780 (2018) 174–180

Hawking radiation, we use coordinates (t, R) to rewrite the metric as 2 2 2 R /(R+ + R−) 2HR ds2 =− 1 − dt2 − dtdR 1 − b2/R2 1 − b2/R2 1 + dR2 + R2d2, (40) 1 − b2/R2

where b = a(t)b0. The Kodama vector is

2 a a ab b ∂ K ≡−ε ∂b R = 1 − , (41) R2 ∂t

ab 1 where ε = (dt)a ∧ (dR)b. We should consider a particle 1−b2/R2 of mass m moving radially under the background of FLRW wormhole spacetime. The Hamilton–Jacobi equation is

Fig. 3. The apparent horizons (R±) and the Hubble horizon (R H ) in terms of μν 2 | p | g ∂ S∂ S + m = 0. (42) ρw /ρc0 are shown. Solid line is R+, dashed line is R−, and dash-dot line is R H . μ ν p The apparent horizons begin to appear at |ρ |/ρ = 4/3. Here we set H = 0.05. w c0 One can define the energy ω and momentum k with the Kodama R 1 1 1 1 vector similar to the case of FLRW universe [20]as 2 = ≡ b0 p , (36) 8π |ρw (r)| = 8π |ρ | a r b0/2 w b2 ∂ =−K a∂ S =− 1 − ∂ S, k = ∂ S = ∂ S. (43) p ω a 2 t R a R where ρw is the peak of ρw (r). The most part of energy is dis- R ∂ R tributed near the throat of the wormhole. The term a2 H2 is deter- Therefore, the action S can be written as mined by the distribution of the cosmic matter ρ , for example, c ω 2 1+3ω S =− dt + kRdR. (44) 8π + + 2 2 = 3(1 ω) 3(1 ω) 1 − b2/R2 a H ρ ρc (37) 3 c0 The Hamilton–Jacobi equation with the action is in case of Pc = ωρc , where ρc0 is the current mass density of the 2 2 2 2 2 cosmic matter. Thus the term in square root of (35)is ω 2HRω (R+ − R )(R − R−) − + k + k2 + m2 2 2 R 2 2 R 1 − b /R 1 − b2 R2 (R+ + R−)R2 4 ρc0 / 1 − 4b2a2 H2 1 − ≥ 0 (38) 0 | p | = 0, (45) 3 ρw in case of ρc ≈ ρc0. The apparent horizons are determined by the and the solution to kR is | p | peak value of the energy density of the wormhole. When ρw 2 2 m2 2 2 is large, b0 is smaller and the horizons are separated. The larger −HR± H R + λ[1 − (1 − b /R )] ω2 2 2 2 the peak value, the closer the horizon R+ approaches R and kR = R (R+ + R−)ω, H 2 2 | p | 1 − b2/R2(R+ − R2)(R2 − R−) R− approaches zero. As ρw becomes smaller, b0 becomes larger. Because of the positivity of (38), the horizons disappear when (46) | p | = 4 ρw ρc0 (see Fig. 3). The exotic matter shrinks the horizons of 2 2 2 2 3 = (R+−R )(R −R−) the universe to a smaller size in both microscopic and macroscopic where λ 2 2 . We choose minus sign for incoming (R++R−)R2 directions. In Fig. 2 and 3, the horizons appear and disappear in re- mode since the observer is inside the apparent horizon similarly | p | verse, because b0 and ρw have a relationship (36). to the FLRW case [20]. Through the contour integral, we get In Schwarzschild-like coordinate, the Misner–Sharp–Hermandez 2 mass is derived as usual, −HR− H2 R2 + λ[1 − m (1 − b2/R2)] ω2 Im S = Im 3 2 2 2 R 4π 3 2 −1 1 − b2/R2(R − R2)(R2 − R ) M = H = R [1 − (b/R) ] (39) + − MSH 2 2 ρc 2(1 − b /r ) 3 2 2 2 × R (R+ + R−)ω from the definition = π R+ω. (47) 2 2 2MMSH H R 1 − = 1 − . Here we integrate out over the region larger than R− such that 2 2 R r J there is only one pole on the contour. In the WKB approximation, − The value calculated here is modified by a factor of [1 − (b/R)2] 1 the emission rate is proportional to the square of the tunneling ∝ − compared to that of the FLRW cosmological model without any amplitude, exp( 2ImS). By comparing with the form of the wormhole, the value of which is the mass of sphere of radius R thermal spectrum ∼ exp(−ω/T ), we thus obtain the Hawking filled with cosmic matter ρc . temperature as 1 4. Hawking temperature T = . (48) 2π R+ We can see the quantum nature of the spacetime by calculat- We can also derive the Hawking temperature using the Hamil- ing the Hawking temperature near horizon. In order to discuss the ton’s equation along the way by Parikh and Wilczek [19]. As the S.-W. Kim / Physics Letters B 780 (2018) 174–180 179 case of de Sitter space [23] and FRW spacetime [20], we will take guaranteed. They simply showed additive generalization in their fi- the s-wave approximation of a massless particles tunneling across nal solution with the wormhole part and the spatial curvature part the horizon. The emission rate can be related to the imaginary part of cosmological model. of the action of a system. The radial null geodesic of the metric is Since this was not unique one, we had alternative generalization here as R˙ = HR± (HR)2 + (1 − b2/R2 − R2/(R2 + R2 )), (49) + − b2 q(t) 1 + 0 → w(t, r) 1 + where we take the minus (−) sign which corresponds to a incom- 4r2 4r2 ing null geodesic. The imaginary part can be 1 b2 → + 0 R R a(t) 1 , f f pR 1 + kr2 4r2 Im S = Im pRdR = Im dp dR, (50) R and we got the solutions satisfying Einstein’s field equation. In this Ri Ri 0 case, the multiplicative generalization was shown by the wormhole part and the curvature part of the cosmological model. There was where pR is the radial momentum, Ri is the initial position, the coupling term of k · b0, that is spacetime curvature-wormhole slightly outside the apparent horizon and R f is a classical turn- coupling term that did not appear in the previous generalization. ing point. By the Hamiltonian equation, The coupling refers to local and global physics, such as unified field ˆ ˆ theory, which links two interactions of extreme regions. ˙ ∂ H dH R = = | . Second, we also found that the number of apparent horizons ∂ p dp R R R of the model are depends on the value of b(t)H. The values also Here Hˆ is the Hamiltonian of the particle, the generator of the cos- shown the physically reasonable regions such as timelike t- and mic time t as we see in the action. We can calculate the imaginary spacelike R-coordinates. The apparent horizons were similar to the Vaidya–de Sitter spacetime with a cosmological horizon and part of the action as a black hole horizon [22]. Of course, in the black hole spacetime, the causal structure inside the event horizon was reversed com- Im S pared to the outer spacetime. However, we did not need the region R f with smaller radius than the wormhole horizon. As a result, the = ˆ 1 Im dR dH ˙ larger cosmological horizon was reduced by wormhole and the R smaller wormhole throat size was enlarged by the cosmological Ri background. =−ω Im Third, the Hawking temperature was calculated along the con- R f tour of the action from Hamilton–Jacobi equation. Even if we use dR × Hamilton’s equation, the result was the same. Along the contour, 2 2 2 2 2 2 2 2 1 − b /R ( (HR) + (1 − b /R − R /(R + + R − )) − HR) − Ri we adopted only one pole due to the limitation of R > R , while there was the effective temperature from both temperatures ac- = π R+ω. (51) cording to the previous two horizons in Vaidya–de Sitter spacetime From this value, we also get the Hawking temperature as [22]. ω 1 T = = . (52) Acknowledgements 2ImS 2π R+ This work was supported by National Research Foundation The result gives the same Hawking temperature as the previous of Korea (NRF) funded by the Ministry of Education (2017- derived one (48)by Hamilton–Jacobi equation. R1D1A1B03031081). 5. Conclusion Appendix A. Supplementary material In conclusion, we would like to remark the following three Supplementary material related to this article can be found on- points. line at https://doi .org /10 .1016 /j .physletb .2018 .03 .005. First, we derived the exact solution of a wormhole embedded in expanding universe in this paper. The background uni- References verse model was FLRW. There were several solutions based on the generalization of the isotropic wormhole solution. For exam- [1] M.L. Mc Vittie, Mon. Not. R. Astron. Soc. 93 (1933) 315. ple [15], they have generalized the static wormhole metric as a [2] V. Faraoni, A. Jacques, Phys. Rev. D 76 (2007) 063510. [3] V. Faraoni, A.F. Zambrano Moreno, A. Prain, Phys. Rev. D 89 (2013) 103514. time-dependent cosmological wormhole metric by introducing two [4] C.J. Gao, S.N. Zhang, Phys. Lett. B 595 (2004) 28. functions w(t, r) and q(t, r) similar to the solution of the charged [5] J. Sultana, C.C. Dyer, Gen. 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