Particle creation in asymptotically Minkowskian spacetimes

Journal of Geometry and Physics 59 (2009) 173–184

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Particle creation in asymptotically Minkowskian spacetimes Shahpoor Moradi Department of Physics, Razi University, Kermanshah, Iran

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Article history: Received 8 January 2008 Received in revised form 31 August 2008 Accepted 10 October 2008 Available online 29 October 2008 MSC: 81T20 Keywords: Particle creation Bogoliubov transformations

a b s t r a c t Exact solutions of Klein–Gordon and Dirac equations are obtained for two classes of Robertson–Walker (RW) spacetimes with asymptotically Minkowskian regions. One class is Minkowskian in the remote past and future. In this class in and out vacua are well defined, because the scale factor reduces to a constant at the asymptotic regions. Another class is asymptotically flat only in the far past. Using the obtained exact solutions we calculate the density of scalar and Dirac particles created through the Bogolubov transformations technique. For Dirac field it is shown that the rates of creation of particles and antiparticles are equal. © 2008 Elsevier B.V. All rights reserved.

1. Introduction The possibility of particle production in curved spacetime was first discussed by Schrödinger [1]. A long time after that Parker discovered that the expansion of the universe can create pairs of particles [2–5]. He showed that fields obeying conformally invariant wave equations will not be produced, because the RW models are conformally flat. When the conformal symmetry of the RW models is broken by anisotropy, conformally invariant particles will be produced. The calculation of particle production rate in strong gravitational fields has had considerable results in cosmology and astrophysics [2–8]. It should be noted that at the present time, because of the very small rate of expansion, the particle production is almost zero but at the early stage of the universe was much larger than at present [2–5]. One of the interesting results of particle production in an expanding universe is that the particles are created in pairs with equal amount of matter and antimatter [2–5], which is in favour of a cosmological theory with equal amount of matter and antimatter such as the theory of Alfvén [9]. The creation of equal amounts of matter and antimatter occurs for non-interacting fields because the particles are created in pairs that conserve quantum numbers. However, it is possible to devise interactions that violate both baryon number conservation and CP symmetry, and these can cause asymmetric creation of matter and antimatter [10]. One of the important but difficult problems of the quantum field theory in curved spacetime is the problem of finding analytic solutions of Dirac equation on given backgrounds. Exact solutions of Dirac equation in RW spacetime have been discussed in various contexts [11–13]. Barut and Duru investigated the exact solutions of the Dirac equation for three typical models of the RW metrics with spatially flat hypersurfaces. In this paper, we introduce the spacetimes which are manifestly Minkowskian in the remote past and future. For these models of spacetimes we obtain the exact solutions of Klien–Gordon and Dirac equations. We also obtain the particle creation rates. The paper is organized as follows: In Section 2 we present a brief discussion on scalar field and it’s Bogolubov transformations in spatially flat RW spacetimes. We solve the Klein–Gordon equation for some models of RW spacetimes. Using the obtained solutions we compute the particle creation rates. In Section 3 we obtain the exact solutions of Dirac equation and rates of creation of particles and antiparticles. Finally we conclude with a discussion in Section 4.

E-mail address: [email protected]. 0393-0440/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2008.10.008

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S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

2. Scalar field The spatially flat RW line element has the form ds2 = dt 2 − a2 (t )dxi dxi = a2 (η)(dη2 − dxi dxi ).

(1)

It is conformally flat, as are all RW metrics. The coordinate η = dt /a(t ), is called the conformal time, to distinguish it from the proper time t of the isotropic observers. Klein–Gordon equation in curved spacetime is [6]

R

x + m2 + ξ R(x) ϕ(x) = 0,




(2)

where ξ is a constant and R is the scalar curvature. ξ R(x) is necessary to make the equation conformally invariant in the massless case. One must set ξ equal to 61 in order to have that property, in this case we say there is conformal coupling to gravitational field. If ξ = 0 we say there is minimal coupling. Because of the spatial homogeneity, we can sperate the mode solutions of the wave equation

ϕ(x) = (2π )−3/2 C −1/2 eik·x χk (η),

(3)

where k = |k| and C (η) = a (η). If we set b (η) = a (η), then the equation satisfied by χk (η) is 2

s

2

 2  ∂η + k2 + bs (m2 + (ξ − 1/6)R) χk (η) = 0,

(4)

where b˙ 2



R(η) = 3sb−s (s/2 − 1)

b2

+

b¨ b



.

(5)

Eq. (4) possesses the formal WKB-type solution [6]

 Z  1 χk = (2Wk )− 2 exp −i Wk (η0 )dη0 ,

(6)

η

where Wk satisfies the nonlinear equation Wk2 (η) = ωk2 (η) −

1

 ¨ Wk

2

Wk

−

˙ k2 3W



2 Wk2

,

(7)

with

ωk2 (η) = k2 + bs (m2 + (ξ − 1/6)R).

(8)

If the spacetime is slowly varying, then the derivative terms in (7) will be small compared to ω (η), so a zeroth order approximation is to substitute 2 k

(0)

Wk

= ωk (η),

(9)

into the integrand of (6). This solution obviously reduces to the standard Minkowskian spacetime modes as C (η) −→ constant. Assume that ϕ in and ϕ out are positive frequency solutions of Klein–Gordon equation in the remote past and far future. Each set is orthonormal so that

(ϕkin , ϕkin0 ) = (ϕkout , ϕkout 0 ) = δkk0 , (ϕkin , ϕk∗0in ) = (ϕkout , ϕk∗0out ) = 0.

(10)

Although these functions are defined by their asymptotic properties in different regions they are solutions of the wave equation every where in spacetime. As both sets are complete, we can expand in-modes in terms of out-modes

ϕjin =

X

αjk ϕkout + βjk ϕk∗out ,

(11)

k

where the Bogolubov coefficients αjk , βjk possess the following properties

X (αik αjk∗ − βik βjk∗ ) = δij ,

(12)

k

X (αik βjk − βik αjk ) = 0. k

(13)

S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

175

Bogolubov coefficients relate the in and out operators and give directly the number of particles created, i.e., the number of out particles contained in the in vacua. Two sets of creation and annihilation operators can be expanded in terms of one another by Bogolubov transformations aout = k

X ∗ inĎ (αjk ain j + βjk aj ),

(14)

j

ain j =

X ∗ out Ď (αjk∗ aout ). k − βjk ak

(15)

k

There are two different vacuum |0in i and |0out i associated with two Fock spaces F in and F out ain k |0in i = 0, aout k

∀k,

|0out i = 0,

∀k.

(16)

The expectation value of number operator for the out-modes in the state |0in i is Ď out hNk i = h0in |aout ak |0in i = k

X

|βjk |2 .

(17)

j

Therefore an initially vacuum state in far past acquires at remote future a background of massive scalar particles. In a spatially flat spacetime because of spatial translation symmetry, the spatial modes are orthogonal, so the Bogolubov coefficients will be diagonal in k

βkk0 = βk δ−kk0 ,

αkk0 = αk δkk0 ,

(18)

i.e. Bogolubov coefficients mix only modes of wave vectors k and −k, and they depend only the magnitude of the k. Now the Bogolubov transformation (11) reduces to

ϕkin = αk ϕkout + βk ϕ−∗out k .

(19)

Taking into account normalization condition for scalar field |αk |2 − |βk |2 = 1, the average number of particles created will be

h0in |Nkout |0in i = nk =

γk 1 − γk

,

(20)

where γk = |βk |2 /|αk |2 . Consider now the special case where there is no time dependence in the far past and future C (η) =



1+δ 2

+

1−δ 2

tanh(λη)

s

,

s, δ, λ = constants,

(21)

this spacetime is Minkowskian in the far past and future, i.e., C (η) −→ δ s in the in region and C (η) −→ 1 at the out region. Case s = 1 was first discussed by Bernard and Duncan [14] and with more details in [15,16]. Note that this spacetime has no big bang, instead the universe starts out as Minkowskian space, then expands smoothly and ends up as another Minkowskian space. It is interesting to analyze this universe in the limit δ → −1 and 0 < η  1/λ. For s = 2, a(t ) ∝ t 1/2 , so spacetime behaves like radiation-dominated Friedman cosmology. In similar situation for s = −2, the spacetime approaches to de Sitter model with a(t ) ∝ eλt . For massive conformally coupled case and scale factor (21), Eq. (4), to be



∂η2 + k2 +



1+δ 2

+

1−δ 2

tanh(λη)

s



m2 χk (η) = 0.

(22)

This equation can be solved in terms of hypergeometric functions for s = ±1, ±2. In asymptotic regions equation (22) reduces to


∂η2 + k2 + δ s m2 χk (η) = 0, η −→ −∞,
∂η2 + k2 + m2 χk (η) = 0, η −→ +∞.

(23) (24)

Then the solutions in the asymptotic regions are e±iΩin η and e±iΩout η where 1

Ωin = [k2 + m2 δ s ] 2 , 1

Ωout = [k2 + m2 ] 2 .

(25) (26)

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S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

We use the method of Bogolubov transformations to compute the density of particles created. To apply this method it is necessary to specify positive modes in the remote past and future. After some algebra, the solutions of (22) behaving as positive frequency modes as η → −∞(t → −∞), are found to be

   i Ω−  ln[2 cosh λη] ( 2Ωin )−1/2 exp −iΩ+ η −   λ      ¯s ¯s  1 iΩ i Ω− 1 iΩ i Ω− Ωin 1 + tanh(λη)   − + , + + ,1 − i , s = 1, 2,  ×F 2 2λ λ 2 2λ λ λ  2  χkin (η) =
i Ω−   ln 2b(η)δ −1 cosh λη (2Ωin )−1/2 exp −iΩ+ η −    λ      ¯ ¯ 1 + tanh(λη)   × F 1 − iΩs + iΩ− , 1 + iΩs + iΩ− , 1 − i Ωin , 2 2λ λ 2 2λ λ λ 1 + δ + (1 − δ) tanh(λη)

(27)

s = −1, −2,

where

Ωin = Ω+ − Ω− , Ωout = Ω+ + Ω− , Ω± =

1 2

(28) (29)

(Ωout ± Ωin ),

(30)

¯ ±1 = iλ, Ω

(31)

¯ ±2 = [m2 (1 − δ ±1 )2 − λ2 ]1/2 . Ω

(32)

In the in region when η −→ −∞, these modes reduce to Minkowskian exponential modes

χkin (η) −→ (2Ωin )−1/2 e−iΩin η .

(33)

However in the out region, where η → +∞, these modes are not simple exponentials, but are more complicated functions of time. Similarly, one may find a complete set of modes of the field that behaving as positive frequency modes as η → +∞(t → +∞)

  −1/2  ( 2 Ω ) exp −i Ω + η −  out      ¯s  1 iΩ iΩ− 1   − + ,  ×F 2 2λ λ 2 out χk (η) =  (2Ωout )−1/2 exp −iΩ+ η −        ¯   × F 1 − iΩs + iΩ− , 1 2 2λ λ 2

iΩ−

λ +

¯s iΩ

+

+

2λ

i Ω−

λ

ln[2 cosh λη] iΩ−

λ



,1 + i

Ωout 1 − tanh(λη) , λ 2



s = 1, 2, (34)

ln (2b(η) cosh λη)

¯s iΩ

+

2λ

iΩ−

λ

,1 + i

Ωout δ(1 − tanh(λη)) , λ 1 + δ + (1 − δ) tanh(λη)



s = −1, −2.

Now we are ready to calculate the particle creation rate by the expanding universe. Using the linear transformation properties of hypergeometric functions [17] F (a, b, c , z ) =

0 (c )0 (c − a − b) F (a, b, a + b − c + 1, 1 − z ) 0 (c − a)0 (c − b)

+ (1 − z )c −a−b

0 (c )0 (a + b − c ) F (c − a, c − b, c − a − b + 1, 1 − z ), 0 (a)0 (b)

(35)

and F (a, b, c ; z ) = (1 − z )c −a−b F (c − a, c − b, c ; z ),

(36)

we can evaluate Bogolubov transformation between χ (η) and χ in k

χ (η) = αk χ in k

out k

(η) + βk χ

out ∗ k

out k

(η),

(η) (37)

where the Bogolubov coefficients are given by

αk =

βk =





Ωout Ωin

1/2

Ωout Ωin

1/2



0 1 − i Ωλin 0 −i Ωλout   ¯ ¯ 0 12 + i2Ωλs − iΩλ+ 0 12 − i2Ωλs −  
0 1 − iΩλi n 0 i Ωλout    ¯ ¯ 0 12 − i2Ωλs + iΩλ− 0 12 + i2Ωλs + 

iΩ+

,

(38)

iΩ−

.

(39)

λ

λ

S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

177

Fig. 1. Plot of rates of created particles as a function of m for scale factor (21) with λ = 100, δ = 10, k = 10. s = 1 (solid line), s = 2 (dash line), s = −1 (dotted line), s = −2 (thickdotted line).

Using the following relations [17]

|0 (iy)|2 =

π

,

(40)

y sinh π y πy 2

|0 (1 + iy)| =

sinh π y

,

(41)

  2 π 0 1 + iy = , 2 cosh π y

(42)

we get

|αk |2 =

|βk |2 =

cosh πλ



¯s Ω 2

   ¯ − Ω+ cosh πλ Ω2s + Ω+

sinh πλ Ωin sinh πλ Ωout cosh πλ



¯s Ω 2

   ¯ − Ω− cosh πλ Ω2s + Ω−

sinh πλ Ωin sinh πλ Ωout

,

(43)

.

(44)

Using the normalization of the solutions |αk |2 − |βk |2 = 1, the density of created particles reads

  + cosh π 2Ωλ−   .  nk = |βk |2 = 2Ω+ − cosh π 2Ωλ− cosh π λ cosh π



¯s Ω λ



(45)

In Fig. 1 we plot the density of created particles versus mass m. In the massless case limit, nk −→ 0: no particle production occurs, i.e., particle creation occurs when the conformal symmetry is broken by mass, on the other hand particle production is a consequence of coupling of the spacetime expansion to the quantum field via the mass. It is obvious from (45), the rate of particles creation depends on the expansion rate. In the weak expansion limit, i.e., λ → 0, 2π λ Ωin s π ¯ ±2 ) − λ (2Ω+ −Ω e

( nk =

e−

= ±1 s = ±2.

(46)

For heavy particles the particle creation rate reduces to 2π m s/2 nk ≈ e − λ δ ,

0 < δ ≤ 1,

(47)

this is the planck spectrum for radiation at temperature T = 2π λδ s/2 . The creation of heavy particles for m  λ, approaches zero sharply, which is reasonable, because production of high energy particles needs much more changing of gravitational field.

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S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

Another exactly soluble model is C (η) = (1 + eλη )s ,

(48)

which has an asymptotically static in region C (η) −→ 1.

η → −∞.

(49)

We will calculate the exact solutions of scalar field for two special cases s = −1, −2. In these models, cosmological time t is related to conformal time η by t =η+ t =η+

2e

λη 2

,

λ eλη

λ

,

s = −1,

(50)

s = −2,

(51)

then asymptotic values η → ±∞ correspond to t → ±∞. The zeroth order adiabatic solution for minimally coupled field is complicated but for conformally coupled field is

 Z η  1 1 1 χk(0) = 2− 2 (k2 + m2 bs )− 4 exp −i (k2 + m2 bs ) 2 dη0

(52)

so for s = −1, −2 respectively we have (0)

χk

− 21



= 2

× ×

k +

− 41

m2

2

1 + eλη



m2 /2k + k(1 + eλη ) +



2k (1 + e 2

−λη

p

−ik/λ

k2 (1 + eλη )2 + m2 (1 + eλη )

) + m (1 + 2e

−λη

2

) + 2ω

p

k2

(1 +

e−λη )2

+

m2

(1 +

 iλω 

e−λη )

,

 − 41 m2 1 χk(0) = 2− 2 k2 + (1 + eλη )2  −ik/λ × k(1 + eλη ) + (k2 (1 + eλη )2 + m2 )1/2 h i iλω p × k2 (1 + e−λη ) + m2 e−λη + 2 k2 + m2 (k2 (1 + e−λη )2 + m2 e−λη )1/2
−i m  × m2 (1 + e−λη ) + m(k2 + m2 (1 + e−λη )−2 )1/2 λ .

(53)

(54)

We can check for both cases (0)

χk

( 1 (1/2k) 2 e−ikη , η → +∞ ≈ 1 (1/2ω) 2 e−iωη . η → −∞.

(55)

Here we obtain the asymptotic solutions using the asymptotic forms of equations. For scale factor (48) we have

 

∂η + 2

∂η2 +

m2 − (3ξ − 1/2)λ2 eλη 1 + eλη m2 + (12ξ − 2)λ2 e2λη

(1 + eλη )2

+ k + (3ξ − 1/2) 2

+ k2 − (6ξ − 1)

3λ2 e2λη 2(1 + eλη )2



χ (η) = 0,

 λ2 eλη χ (η) = 0, (1 + eλη )

s = −1,

s = −2.

(56)

(57)

In the in region as η → −∞ the equations for both cases reduce to


∂η2 + ω2 χ (η) = 0.

(58) 1

Then the solutions are e±iωη with ω = (k2 + m2 ) 2 . In the out region (η → +∞)


∂η2 + κ 2 χk (η) = 0, s = −1,
∂η2 + ωˆ 2 χk (η) = 0, s = −2,

(59) (60)

S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

179

where 1

κ = (k2 − λ2 /4 + 3λ2 ξ /2) 2 ,

(61)

1 2

ωˆ = (k2 − λ2 + 6λ2 ξ ) .

(62)

One may then write down a complete set of field modes that reduce to standard exponential modes in the in-region (i.e. when η → −∞)

     1 λη −1/2  − ϑ ln[1 + e ] (2ω) exp −iωη +    2      1 i 1 i 2iω  λη  × F − ϑ − (ω + κ), − ϑ − (ω − κ), 1 − , − e s = −1  2 2  λ λ  λ   χ in (η) = 1 iω   (2ω)−1/2 exp −iωη + − ln[1 + eλη ]    2 a      1 i 2iω   × F 1 − i (ω + ωˆ + ω), ¯ − (ω − ωˆ + ω), ¯ 1− , −eλη s = −2 2 λ 2 λ λ

(63)

where 1

ω¯ = (m2 − 9λ2 /4 + 12λ2 ξ ) 2 ,

(64)

1 2

ϑ = (1 − 9ξ /2) ,

(65)

Using the linear transformation properties of hypergeometric functions F (a, b, c , z ) =

  0 (c )0 (a − b) 1 (−z )−b F b, 1 − c + b, 1 − a + b; 0 (a)0 (c − b) z   0 (c )0 (b − a) 1 −a + (−z ) F a, 1 − c + a, 1 − b + a; , 0 (b)0 (c − a) z

(66)

when z → +∞ F (a, b, c , z ) ≈

0 (c )0 (a − b) 0 (c )0 (b − a) (−z )−b + (−z )−a , 0 (a)0 (c − b) 0 (b)0 (c − a)

(67)

the Bogoliubov transformations for s = −1 and s = −2 respectively are given by



0 1 − 2i ωλ 0 −2i κλ

e−iκη 0 12 − ϑ − λi (ω + κ) 0 21 + ϑ − λi (ω + κ)

0 1 − 2i ωλ 0 −2i κλ

eiκη , + 0 12 + ϑ − λi (ω − κ) 0 21 − ϑ − λi (ω − κ) 
 0 1 − 2i ωλ 0 −2i ωλˆ ˆ

e−iωη χin (η) ≈ 0 12 − λi (ω + ω ˆ + ω) ¯ 0 21 − λi (ω + ωˆ − ω) ¯
  0 1 − i ωλ 0 2i ωλˆ ˆ

eiωη + . 0 12 − λi (ω − ω ˆ + ω) ¯ 0 12 − λi (ω − ωˆ − ω) ¯ χin (η) ≈

(68)

(69)

Using properties of Gamma functions Bogoliubov coefficients are given by

 0      0 |βk |2 γk = = 2  0 |αk |     0

1 2 1 2 1 2 1 2


− ϑ − λi (ω + κ) 0
+ ϑ − λi (ω − κ) 0
− λi (ω + ωˆ + ω) ¯ 0
− λi (ω − ωˆ + ω) ¯ 0


2 + ϑ − λi (ω + κ)
2 s = −1 1 − ϑ − λi (ω − κ) 2
2 1 − λi (ω + ωˆ − ω) ¯ 2
2 s = −2. 1 − i (ω − ωˆ − ω) ¯ 1 2

2

(70)

λ

Taking into account normalization of the wave function |αk |2 − |βk |2 = 1 one obtains nk =

cosh cosh 2λπ

2π ω ¯

  + cosh 2λπ ω − ωˆ    , ω + ωˆ − cosh 2λπ ω − ωˆ λ




s = −2.

(71)

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S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

In the massless case we have 1

ω = k,

ω¯ = λ(12ξ − 9/4) 2 .

(72)

It’s obvious in conformally coupled case as ξ = have nk =

cosh 2λπ (k −

cosh 2λπ (k +

1 , 6

particle creation rate is zero. But in minimally coupled case as ξ = 0 we

√

√

k2 − λ2 ) − 1

k2 − λ2 ) − cosh 2λπ (k −

√

k2 − λ2 )

.

(73)

For s = −1

sin π γk = sin π


− ϑ − λi (ω − κ) sin π
− ϑ − λi (ω + κ) sin π

1 2 1 2

1 2 1 2


+ ϑ − λi (ω − κ)
+ ϑ − i (ω + κ)

(74)

λ

where we used the following identity

π | sin z |

|0 (z )0 (1 − z )| =

(75)

using 1

1

| sin z | = 2− 2 (cosh 2y − cos 2x) 2

(76)

(74) takes the form

γk =

cosh 2λπ (ω − κ) + cos 2π ϑ

. cosh 2λπ (ω + κ) − cos 2π ϑ

(77)

Finally taking into account normalization condition of the wave function the density of particles created is nk =

cosh 2λπ (ω − κ) + cos 2π ϑ

. cosh 2λπ (ω + κ) − cosh 2λπ (ω − κ)

(78)

Again for conformally coupled case particle creation rate is zero, but for minimally coupled case we have

nk =

cosh 2λπ



cosh 2λπ

k+

p



k−

p





k2 − λ2 /4 − 1

k2 − λ2 /4 − cosh 2λπ



k−

.

k2 − λ2 /4

p

(79)

3. Dirac field Let assume that {Uin (k, d, η), Vin (k, d, η)} and {Uout (k, d, η), Vout (k, d, η)} are two complete set of mode solutions of Dirac equation which define particles and antiparticles in asymptotic regions. The Dirac field operators can be written as

ψ(x) =

Z

d3 k

X [ain (k, d)Uin (k, d, η)eik·x + bĎin (−k, d)Vin (k, d, η)e−ik·x ],

(80)

d

and

ψ(x) =

Z

d3 k

X [aout (k, d)Uout (k, d, η)eik·x + bĎout (−k, d)Vout (k, d, η)e−ik·x ],

(81)

d

here ain , bin and aout , bout are respectively annihilation of particles and antiparticles. Since each set is complete we can write down one set in terms of another Uin (k, d, η) =

X

αdd0 Uout (k, d0 , η) + βdd0 Vout (k, d0 , η),

(82)

%dd0 Uout (k, d0 , η) + σdd0 Vout (k, d0 , η).

(83)

d0

Vin (k, d, η) =

X d0

Then the in and out creation operator of particles and antiparticles are related by aout (k, d) =

X

αdd0 ain (k, d0 ) + %dd0 bĎin (−k, d0 ),

(84)

∗ 0 ∗ Ď 0 σdd 0 bin (k, d ) + βdd0 ain (−k, d ).

(85)

d0

bout (k, d) =

X d0

S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

181

orthogonality of spinors yields

X

|αdd0 |2 + |βdd0 |2 = 1,

(86)

|%dd0 |2 + |σdd0 |2 = 1.

(87)

d0

X d0

The expectation value of the out particles in the in vacuum (i.e., created particles) is Ď

N p = h0in |aout (k, d)aout (k, d)|0in i =

X

|ρdd0 |2 .

(88)

d0

Similarly the expectation value of out antiparticles in the in vacuum (i.e., n created antiparticles) is Ď

N a = h0in |bout (k, d)bout (k, d)|0in i =

X

|βdd0 |2 .

(89)

d0

Then total number of particles created is N =

X

|ρdd0 |2 +

X

d0

|βdd0 |2 .

(90)

d0

The Dirac equation in the line element (1) becomes



iγ µ ∂µ + i

3 a˙ 2a

 γ 0 − ma Ψ = 0,

(91)

where dot refers to differentiation with respect to conformal time η and flat-Dirac matrices are defined as follows

γ0 =



I 0

0 −I



,

γ i = γ 0αi ,

αi =



0

σ

i

σi 0



.

(92)

By substituting

Ψ = (2π )−3/2 ei k·x a−3/2 Φ ,

(93)

into Eq. (91), the Dirac equation becomes iγ 0 ∂η + γ · k − ma Φ = 0,




(94)

where  = +, − respectively corresponds to positive and negative frequency mode solutions. Inserting


Φ = −iγ 0 ∂η − γ · k − ma φ,

(95)

into Eq. (94) we arrive at

 2  ∂η + m2 a2 ∓ ima˙ γ 0 + |k|2 φ = 0.

(96)

Since eigenvalues of γ 0 are ±1 the equation reduces to

 2  ∂η + m2 a2 ∓ ima˙ + |k|2 φ (∓) = 0.

(97)

where positive and negative frequency solutions respectively are given by

φ (−) = U (k, d, η) = f (−) u(m, 0),

(98)

φ (+) = V (k, d, η) = f (+) v(m, 0).

(99)

and

Here u(m, 0) and v(m, 0) are flat rest spinors satisfying

γ 0 u(m, 0) = u(m, 0),

(100)

γ v(m, 0) = −v(m, 0). 0

(101)

(−)

(+)

Note that f in the in region reduces to positive frequency mode and f reduces to negative frequency mode. We solve the Dirac equation and analyze the particle creation for two cases case I: C (η) = (1 + eλη )−2 ,

(102)

case II: C (η) = (A + B tanh(λη)) , 2

(103)

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S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

where A, B and λ are constants. For case I, positive and negative in-mode solution will be

 i i 2iω − (k + ω + m), (k − ω − m), 1 − , −eλη , λ λ λ   i i 2iω n (+) λη im iωη λη λ ϕin (η) = f (η) = (1 + e ) e F (k + ω + m), − (k − ω − m), 1 + , −e , λ λ λ ϕinp (η) = f (−) (η) = (1 + eλη )

−im −iωη λ e F



(104)

(105)

where we have introduced 1

ω = (k2 + m2 ) 2 .

(106)

By using the relation (67) f

(−)

(η) ≈

0 1−

2iω

0 − 2ik λ









e−ikη 0 − λi (k + ω + m) 0 1 − λi (k + ω − m)

0 1 − 2iλω 0 2ik λ

eikη , + 0 λi (k − ω − m) 0 1 + λi (k − ω + m) λ

(107)

and f

(+)



0 1 + 2iλω 0 2ik λ

eikη (η) ≈ 0 λi (k + ω + m) 0 1 + λi (k + ω − m)

0 1 + 2iλω 0 − 2ik λ

e−ikη , + 0 − λi (k − ω − m) 0 1 − λi (k − ω + m)

(108)

then the γk is given by



0 − i (m + ω + k) 0 1 + i (m − k − ω) 2 λ λ γk =

. 0 i (k − m − ω) 0 1 + i (k + m − ω) 2 λ

(109)

λ

after some algebra we obtain

γk = =

(m − k + ω)(m − k − ω) sinh πλ (m − k + ω) sinh πλ (m + k − ω) (m + k + ω)(m + k − ω) sinh πλ (m + k + ω) sinh πλ (m − k − ω) cosh 2λπ m − cosh 2λπ (k − ω)

. cosh 2λπ (k + ω) − cosh 2λπ m

(110)

Taking into account normalization condition for Dirac field |αk |2 + |βk |2 = 1 the density of total created particles is N = Na + Np =

cosh 2λπ m − cosh 2λπ (k − ω) sinh 2λπ k sinh 2λπ ω

.

(111)

For very light particles we arrive at the following distribution n≈

2π 2 m2



λ2

sinh

2π

λ

−1 k

+ O(m3 ),

(112)

then the total number of created particles is

Z N =

∞

nd3 k ≈

λm2 π 2

.

12

0

(113)

For massless neutrinos, as a result of conformal invariant, particle creation rate is zero. For case II the positive and negative modes respectively are p in

ϕ (η) = f

(−)

−1/2

(η) = (ωin ) 

×F

iω−

λ

+

imB

λ

   iω− exp −iω+ η − ln[2 cosh λη] λ

,1 +

iω −

λ

−

imB

λ

,1 −

iωin tanh(λη) + 1

λ

,

2



,

(114)

S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

183

and

   iω+ ϕinn (η) = f (+) (η) = (ωin )−1/2 exp −iω− η − ln[2 cosh λη] λ   iω+ imB iω+ imB iωin tanh(λη) + 1 ×F − ,1 + + ,1 + , , λ λ λ λ λ 2

(115)

where 1

ωin = [k2 + m2 (A − B)2 ] 2 ,

(116)

2 21

ωout = [k2 + m2 (A + B) ] , ωout = ω+ + ω− , ωin = ω+ − ω− , ω± =

1

(117) (118) (119)

(ωout ± ωin ).

(120)

2 Using the linear transformation properties of hypergeometric functions at η → +∞ we get p in

ϕ (η) ≈

 0 1− 

0 1−

iω+

λ

iωin




0 − iωλout λ   − imB 0 − iωλ+ + λ

imB

e

−iωout η

+

λ

 0 1−

iωin



iω−

imB

  0 iωλ− +



0 1+

λ

−

λ

λ

0

iωout

λ


 eiωout η ,

(121)

imB

 eiωout η .

(122)

imB

λ

and

ϕinn (η) ≈

 0 1+  0 1−

iω−

λ

iωin

 0 1+




0 iωλout λ   + imB 0 − iωλ− − λ

imB

λ

 e−iωout η +

 0 1+

iω+

λ

iωin




0 − iωλout   + imB 0 iωλ+ − λ λ

λ

Then the rates of created particles and antiparticles are the same. Using the orthogonality of the solutions we obtain the creation rates

 −1      0 1 − iω− + imB 0 − iω− − imB 2 λ λ λ λ       + 1 Np = Na =  iω+ iω+ imB imB 0 1+ λ + λ 0 λ − λ  = =

(123)

 −1 (ω− − mB)(ω+ − mB) sinh πλ (ω+ + mB) sinh πλ (ω+ − mB) +1 . (ω− + mB)(ω+ + mB) sinh πλ (ω− + mB) sinh πλ (ω− − mB) Ω+ sinh πλ (ω− + mB) sinh πλ (ω− − mB)

Ω− sinh πλ (ω+ + mB) sinh πλ (ω+ − mB) + Ω+ sinh πλ (ω− + mB) sinh πλ (ω− − mB)

(124)

,

(125)

where

Ω± = m(A + B) ± ωout .

(126)

In the massless case limit no particle production occurs. In the weak expansion limit, i.e., λ → 0 2π Np = Na ≈ e− λ ωin ,

(127)

which is a thermal distribution. 4. Conclusion


s

In this paper we introduce the spacetime with the scale factor 1+δ + 1−δ tanh(λη) , which is manifestly Minkowskian 2 2 in the remote past and future. In this model of spacetime in and out vacua are well defined because the scale factor reduces to a constant at asymptotic past and future times, therefore in these regions there is a timelike killing vector and one can define particle state in terms of positive frequency modes. We show that the massive conformally coupled scalar field is exactly soluble for s = ±1, ±2. With a straightforward calculation of the Bogolubov coefficients between the in and out modes, the rate of particle production during the expansion of the spacetime is obtained. We show that the density of particles created for high masses is thermal. For case s = 2 we obtain the exact solutions of Dirac equation. We found the positive and negative spinors and showed that particle and antiparticle creation rates are equal. Another soluble model is (1 + eλη )s , which has an asymptotically static in region. In this model, positive and negative frequency modes in the remote past are well defined, but in out region we used the WKB approximation method for

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S. Moradi / Journal of Geometry and Physics 59 (2009) 173–184

definition of positive frequency modes. Massive conformally coupled scalar field is soluble for s = ±1, ±2. We considered the cases s = −1, −2, in these cases we can solve exactly massive minimally coupled scalar field as well as massive conformally coupled scalar field. We show that in asymptotically regions, conformal time and cosmological time are equivalent. After the solution of the scalar field equation we obtain the particle creation rates. We also investigate creation of a massless minimally coupled scalar field. Also, we found the analytic solutions of Dirac equation which behave as positive frequency modes in the asymptotic regions for s = −2. The creation of equal amounts of particles and antiparticles is shown for this model. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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