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De Sitter metric as a self-consistent solution of the back reaction problem
Volume 132B, number 4,5,6
PHYSICS LETTERS
1 December 1983
DE SITTER METRIC AS A SELF-CONSISTENT SOLUTION OF THE BACK REACTION PROBLEM Sumio WADA
Institute of Physics, University of Tokyo Komaba, Komaba 3-8-1, Tokyo 153, Japan and Takahiro AZUMA
Dokkyo University, Sakaemachi 600, Soka, Saitama 340, Japan Received 18 August 1983
The de Sitter metric is a solution of the Einstein equation even when quantum effects of matter fields are included. However between the curvature and the cosmological constant A changes drastically. For some values of A, there are two or three solutions, while there is none in some other cases. There can be even a positive curvature solution for negative A. These features depend on mass and spin of matter fields and the matter-curvature coupling.
In this letter, we calculate the relation between the cosmological constant (A) and the curvature of the de Sitter space (R), taking into account quantum effects of free matter fields in the background curved space. In general, quantum effects of matter fields make it difficult to solve the Einstein equation. The curvature of space time gives rise to a non-zero vacuum expectation value of the e n e r g y - m o m e n t u m tensor (Tuv) , which in turn acts as the source of curvature. This is the so-called back reaction problem. A special feature of the de Sitter space is that (Tuv) is proportional to g~u in the de Sitter invariant vacuum [1,2], which is just the tensor structure of the cosmological term. Therefore, the de Sitter metric is still a solution of the Einstein equation (with A 4= 0 in general) with back reaction included. Of course, the naive relation R = 4 A should be modified and its numerical study is the purpose of the present letter. We will find that the effects of(Tuv) depend on various parameters, such as the number of fields, their mass and spin, and the strength of the coupling to scalar curvature (~). In some cases there are two or three de Sitter solutions for a single cosmological constant, while there is none in some other cases. The statement that the de Sitter metric is a consis0 . 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
tent solution of the back reaction problem is not new. The massless conformal (~ = 1/6) case was mentioned in ref. [1] and a cosmological model based on it was presented in ref. [3]. The massive conformal case was discussed in ref. [4], along with their "cosmogenesis" picture (though their renormalization of (Tuv) is not standard). However, all these arguments assumed A = 0, which may not be true in the early universe from the point of view of grand unification. As far as we know, a detailed study of the relation between R and A is new. We first explain the basic formula. By using natural units (G = 1), we write the Einstein equation as l R~zv ~guv R +g~vA = 8n(Tuv)
(1)
(Tuv) is the renormalized value of the matrix element for some state. In general, we should add the higher derivative terms ( t ) H v and (2)H (which are variations of f x/--'gR 2 an~ f x/~Ru.ut~v in g~v, respectively), to the left-hand side ofeq. (1). These terms are necessary to absorb the logarithmic divergence of the unrenormalized (T~v). In the de Sitter space, however, both terms are identically zero and we need not worry about them. Eq. (1) is further simplified in the de Sitter space. Because it is maximally symmetric, R , v = 313
Volume 132B, number 4,5,6
¼Rguv. Then the left-hand side o f e q . (1) is proportional to guy" So is the right-hand side if we choose the de Sitter invariant vacuum. In this vacuum we know [1,2] that, for free scalar fields with mass m and the coupling to scalar curvature ~,
(Tuv) = (N/64rr2)g~v(m2[m 2 + (~ - ~)R] X [qJ(~ + v)+ q¢(32 - v)+ l o g R / 1 2 m 2]
- (N/64n2)g~vf(R),
(2)
where
~= (~
-
and N is the number of scalar fields. Finally, eq. (1) reduces to (3)
or
A = ¼[R - (N/2rr)f(R)] .
(3')
The relation eq. (3) is shown in figs. l a - l d . The results are presented for the massless case and for m = 1 (= Planck mass).The results for 0 < m < 1 lie in between, and when m -+ "~, the relation eq. (3) becomes the naive one R = 4A [because f ( R ) -+ 0 when R/rn 2 0 as will be explained below]. As for ~, we take four values: (a) ~ = 0 (minimal coupling) * 1 (b) ~ = 1/40. (c) ~ = 1/6 (conformal coupling). (d) ~ = 1. These values represent four typical behaviours for > 0. We do not consider the case ~ < 0. When m 2 + ~fl becomes negative, the theory is not well-defined. From these figures, we can see that the results depend much on ~. The general features for ~ = 0 and = 1/6 are rather common. First of all, when the cosmological constant A is large, there is no de Sitter solution. When A is small, on the other hand, there are two solutions. One is essentially a perturbation around the naive solution R = 4A, while the other is an abnor,1 The ~ = 0 massless case is defined as the m ~ 0 limit of the ii = 0 theory. This differs from the ~ --*0 limit of the massless theory [see eq. (4) below]. 314
mal solution which continues to exist even when A becomes negative! For ~ = 1/40 and ~ = 1, the curves lie below R = 4A, except the m = 1 case o f ~ =1/40. In the last case, there are three solutions for moderate values of the cosmological constant. Such a ~ and mass dependence, of course, reflects that o f (T~v) [or f ( R ) ] in eq. (2). Some typical behaviours o f f ( R ) are shown in fig. 2. The curves A, A',B and C correspond to (~ = 0, ~ = 1/6 (m = 0)}, ~ = 1/6 (m = 1), (~ = 1/40 (m = 0), ~ = 1} and ~ = 1/40 (m = 1), respectively. Then, the qualitative features of fig. 1 can easily be understood by noting that the relation between R and A is given by the intercepts of the two lines
y = (N/2n)f(R),
12~- 12m2/R) i n ,
- ~1R + A = - ( N / 8 7 r ) f ( R ) ,
1 December 1983
PHYSICS LETTERS
y = 1 (R - 4 A ) ,
in the R ~ y plane. To see the qualitative behaviour o f f ( R ) for general values of ~ and m, we calculate the asymptotic forms o f f ( R ) for R ~ 0 and R -~ oo. For the massive case we find that
f(R)
_~
(R--.0)
[-~
1 + ~(~ - 6) - 288(~ _~)3]
(1R)3
m
+ O(R4/m4). Therefore, f ( R ) < 0 for small positive R when > 0.1023. When R becomes large, on the other hand,
f(R)
"-'
[ 1(~ _ 6 ) 2 + 2_~6o1R2
(R'-*~)
+ ( ± - ¼ D [~ +(¼ - ~ ) 1 / 2 ] R 2 / ( 1 + ~R/m2) . 24
(4)
[The second term is non-leading but not negligible if
~R/rn 2 <~ O(1).] Except for the narrow region around = 1 / 6 , f ( R ) -+ - ~ f o r ~ > 0. When ~ "~ l, however, there is a positive peak due to the second term before
f ( R ) begins to decrease, and f ( R ) l o o k s like the curve C in fig. 2. Next we turn to higher spin fields. Expressions like eq. (2) are not available for general cases. For conformally invariant theories (such as massless fermion fields and gauge fields), however, (Tuv), for a state which is conformally related to the Minkowski vacuum, can be obtained from general arugments [5]. (Tuv) for such cases is a linear combination of (1)Hu u (~ (2)Huv) and (3)Huv (see ref. [5] for its definition), and their coefficients are determined from the trace
2
10
(a')
2 R R
5
0 -0.I
i l l l s
0.0
0.1
0.2
0
0.5
1.0
^
A
10 40 f
,,,s ,,, ,'''/''°
(b')
R 5
I
*'°i 0
4
1
~
^
3
^
(c)
o
100 R
100 R
50
50
,
0
I
2
5
10
0
I
i
I
10
i
I
20
A
30
A
1.0
/
/
2.0
(d)
s
(d')
°'°
R
R 0.5
1.0
I
0.1
0.2 A
i
I
0.3
I
0
I
,
I
0.4
0.2
,
I
0.6
A
Fig. 1. The relation between the curvature R of de Sitter space and the cosmological constant A for four values of ~, (a), (b), (c) and (d) correspond to ~ = 0, 1/40, 1/6 and 1, respectively. (a-d) are for massless cases, and (a'-d') are for rn = 1 (= Planck mass). In each diagram the bold (broken) curve represents the relation when the number of fields is 100 (10). The naive relation R = 4A is shown by the thin straight line when it does not overlap other lines.
PHYSICS LETTERS
Volume 132B, number 4,5,6
f(R)
A
C
Fig. 2. The four typical behaviours off(R) [eq. (2)1.
1 December 1983
back reactions are included. However, the relation between its curvature and the cosmological constant changes drastically in some regions due to the quantum effects of matter. Recently, there is growing interest in the de Sitter space. The grand unification theory suggests the existence of a de Sitter stage in the early universe. Even the origin of the universe may be a de Sitter space [3,4,6]. We do not know the relevance of the present calculation to such problems. To proceed our work, we should include interactions between matter and quantum gravity effects. The stability of our de Sitter solution is also an important question.
R efegences [I] ].S. Dowker and R. Critchley, Phys. Rev. 1313 (1976)
anomaly. In the de Sitter space, (1)Huu = 0 and (3)Huu = ~ guy R 2. Therefore, we get
(T.v) = (N/647r2) 21-5~6--ag, uR 2 , where N is the number o f fields and a = 11/2 (62) for fermions (gauge fields). Remember that (Tuu) for massless scalar fields has the same form but with a = 61 (1) for = 0 (1/6). Therefore, the self-consistent solutions for R can be deduced from figs. l a and lc and we do not present numerical results for higher spins. In conclusion, we find that the de Sitter metric is a consistent solution of the Einstein equation even when
316
3224. [2] P.C.W. Davies and T.S. Bunch, Proc. R. Soc. London A360 (1978) 117. [3] A.A. Starobinsky, Phys. Lett. 91B (1980) 99. [4] R. Brout, F. Englert and P. Spindel, Phys. Rev. Lett. 43 (1979) 417; R. Brout et al., Nucl. Phys. B170 (1980) 228; Ph. Spindel, Phys. Lett. 107B (1981) 361; [5] L.S. Brown and J.P. Cassidy, Phys. Rev. D15 (1977) 2810; T.S. Bunch and P.C.W. Davies, Proc. R. Soc. London, A356 (1977) 569. [6] A. Vilenkin, Phys. Rev. D27 (1983) 2848.
PHYSICS LETTERS
1 December 1983
DE SITTER METRIC AS A SELF-CONSISTENT SOLUTION OF THE BACK REACTION PROBLEM Sumio WADA
Institute of Physics, University of Tokyo Komaba, Komaba 3-8-1, Tokyo 153, Japan and Takahiro AZUMA
Dokkyo University, Sakaemachi 600, Soka, Saitama 340, Japan Received 18 August 1983
The de Sitter metric is a solution of the Einstein equation even when quantum effects of matter fields are included. However between the curvature and the cosmological constant A changes drastically. For some values of A, there are two or three solutions, while there is none in some other cases. There can be even a positive curvature solution for negative A. These features depend on mass and spin of matter fields and the matter-curvature coupling.
In this letter, we calculate the relation between the cosmological constant (A) and the curvature of the de Sitter space (R), taking into account quantum effects of free matter fields in the background curved space. In general, quantum effects of matter fields make it difficult to solve the Einstein equation. The curvature of space time gives rise to a non-zero vacuum expectation value of the e n e r g y - m o m e n t u m tensor (Tuv) , which in turn acts as the source of curvature. This is the so-called back reaction problem. A special feature of the de Sitter space is that (Tuv) is proportional to g~u in the de Sitter invariant vacuum [1,2], which is just the tensor structure of the cosmological term. Therefore, the de Sitter metric is still a solution of the Einstein equation (with A 4= 0 in general) with back reaction included. Of course, the naive relation R = 4 A should be modified and its numerical study is the purpose of the present letter. We will find that the effects of(Tuv) depend on various parameters, such as the number of fields, their mass and spin, and the strength of the coupling to scalar curvature (~). In some cases there are two or three de Sitter solutions for a single cosmological constant, while there is none in some other cases. The statement that the de Sitter metric is a consis0 . 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
tent solution of the back reaction problem is not new. The massless conformal (~ = 1/6) case was mentioned in ref. [1] and a cosmological model based on it was presented in ref. [3]. The massive conformal case was discussed in ref. [4], along with their "cosmogenesis" picture (though their renormalization of (Tuv) is not standard). However, all these arguments assumed A = 0, which may not be true in the early universe from the point of view of grand unification. As far as we know, a detailed study of the relation between R and A is new. We first explain the basic formula. By using natural units (G = 1), we write the Einstein equation as l R~zv ~guv R +g~vA = 8n(Tuv)
(1)
(Tuv) is the renormalized value of the matrix element for some state. In general, we should add the higher derivative terms ( t ) H v and (2)H (which are variations of f x/--'gR 2 an~ f x/~Ru.ut~v in g~v, respectively), to the left-hand side ofeq. (1). These terms are necessary to absorb the logarithmic divergence of the unrenormalized (T~v). In the de Sitter space, however, both terms are identically zero and we need not worry about them. Eq. (1) is further simplified in the de Sitter space. Because it is maximally symmetric, R , v = 313
Volume 132B, number 4,5,6
¼Rguv. Then the left-hand side o f e q . (1) is proportional to guy" So is the right-hand side if we choose the de Sitter invariant vacuum. In this vacuum we know [1,2] that, for free scalar fields with mass m and the coupling to scalar curvature ~,
(Tuv) = (N/64rr2)g~v(m2[m 2 + (~ - ~)R] X [qJ(~ + v)+ q¢(32 - v)+ l o g R / 1 2 m 2]
- (N/64n2)g~vf(R),
(2)
where
~= (~
-
and N is the number of scalar fields. Finally, eq. (1) reduces to (3)
or
A = ¼[R - (N/2rr)f(R)] .
(3')
The relation eq. (3) is shown in figs. l a - l d . The results are presented for the massless case and for m = 1 (= Planck mass).The results for 0 < m < 1 lie in between, and when m -+ "~, the relation eq. (3) becomes the naive one R = 4A [because f ( R ) -+ 0 when R/rn 2 0 as will be explained below]. As for ~, we take four values: (a) ~ = 0 (minimal coupling) * 1 (b) ~ = 1/40. (c) ~ = 1/6 (conformal coupling). (d) ~ = 1. These values represent four typical behaviours for > 0. We do not consider the case ~ < 0. When m 2 + ~fl becomes negative, the theory is not well-defined. From these figures, we can see that the results depend much on ~. The general features for ~ = 0 and = 1/6 are rather common. First of all, when the cosmological constant A is large, there is no de Sitter solution. When A is small, on the other hand, there are two solutions. One is essentially a perturbation around the naive solution R = 4A, while the other is an abnor,1 The ~ = 0 massless case is defined as the m ~ 0 limit of the ii = 0 theory. This differs from the ~ --*0 limit of the massless theory [see eq. (4) below]. 314
mal solution which continues to exist even when A becomes negative! For ~ = 1/40 and ~ = 1, the curves lie below R = 4A, except the m = 1 case o f ~ =1/40. In the last case, there are three solutions for moderate values of the cosmological constant. Such a ~ and mass dependence, of course, reflects that o f (T~v) [or f ( R ) ] in eq. (2). Some typical behaviours o f f ( R ) are shown in fig. 2. The curves A, A',B and C correspond to (~ = 0, ~ = 1/6 (m = 0)}, ~ = 1/6 (m = 1), (~ = 1/40 (m = 0), ~ = 1} and ~ = 1/40 (m = 1), respectively. Then, the qualitative features of fig. 1 can easily be understood by noting that the relation between R and A is given by the intercepts of the two lines
y = (N/2n)f(R),
12~- 12m2/R) i n ,
- ~1R + A = - ( N / 8 7 r ) f ( R ) ,
1 December 1983
PHYSICS LETTERS
y = 1 (R - 4 A ) ,
in the R ~ y plane. To see the qualitative behaviour o f f ( R ) for general values of ~ and m, we calculate the asymptotic forms o f f ( R ) for R ~ 0 and R -~ oo. For the massive case we find that
f(R)
_~
(R--.0)
[-~
1 + ~(~ - 6) - 288(~ _~)3]
(1R)3
m
+ O(R4/m4). Therefore, f ( R ) < 0 for small positive R when > 0.1023. When R becomes large, on the other hand,
f(R)
"-'
[ 1(~ _ 6 ) 2 + 2_~6o1R2
(R'-*~)
+ ( ± - ¼ D [~ +(¼ - ~ ) 1 / 2 ] R 2 / ( 1 + ~R/m2) . 24
(4)
[The second term is non-leading but not negligible if
~R/rn 2 <~ O(1).] Except for the narrow region around = 1 / 6 , f ( R ) -+ - ~ f o r ~ > 0. When ~ "~ l, however, there is a positive peak due to the second term before
f ( R ) begins to decrease, and f ( R ) l o o k s like the curve C in fig. 2. Next we turn to higher spin fields. Expressions like eq. (2) are not available for general cases. For conformally invariant theories (such as massless fermion fields and gauge fields), however, (Tuv), for a state which is conformally related to the Minkowski vacuum, can be obtained from general arugments [5]. (Tuv) for such cases is a linear combination of (1)Hu u (~ (2)Huv) and (3)Huv (see ref. [5] for its definition), and their coefficients are determined from the trace
2
10
(a')
2 R R
5
0 -0.I
i l l l s
0.0
0.1
0.2
0
0.5
1.0
^
A
10 40 f
,,,s ,,, ,'''/''°
(b')
R 5
I
*'°i 0
4
1
~
^
3
^
(c)
o
100 R
100 R
50
50
,
0
I
2
5
10
0
I
i
I
10
i
I
20
A
30
A
1.0
/
/
2.0
(d)
s
(d')
°'°
R
R 0.5
1.0
I
0.1
0.2 A
i
I
0.3
I
0
I
,
I
0.4
0.2
,
I
0.6
A
Fig. 1. The relation between the curvature R of de Sitter space and the cosmological constant A for four values of ~, (a), (b), (c) and (d) correspond to ~ = 0, 1/40, 1/6 and 1, respectively. (a-d) are for massless cases, and (a'-d') are for rn = 1 (= Planck mass). In each diagram the bold (broken) curve represents the relation when the number of fields is 100 (10). The naive relation R = 4A is shown by the thin straight line when it does not overlap other lines.
PHYSICS LETTERS
Volume 132B, number 4,5,6
f(R)
A
C
Fig. 2. The four typical behaviours off(R) [eq. (2)1.
1 December 1983
back reactions are included. However, the relation between its curvature and the cosmological constant changes drastically in some regions due to the quantum effects of matter. Recently, there is growing interest in the de Sitter space. The grand unification theory suggests the existence of a de Sitter stage in the early universe. Even the origin of the universe may be a de Sitter space [3,4,6]. We do not know the relevance of the present calculation to such problems. To proceed our work, we should include interactions between matter and quantum gravity effects. The stability of our de Sitter solution is also an important question.
R efegences [I] ].S. Dowker and R. Critchley, Phys. Rev. 1313 (1976)
anomaly. In the de Sitter space, (1)Huu = 0 and (3)Huu = ~ guy R 2. Therefore, we get
(T.v) = (N/647r2) 21-5~6--ag, uR 2 , where N is the number o f fields and a = 11/2 (62) for fermions (gauge fields). Remember that (Tuu) for massless scalar fields has the same form but with a = 61 (1) for = 0 (1/6). Therefore, the self-consistent solutions for R can be deduced from figs. l a and lc and we do not present numerical results for higher spins. In conclusion, we find that the de Sitter metric is a consistent solution of the Einstein equation even when
316
3224. [2] P.C.W. Davies and T.S. Bunch, Proc. R. Soc. London A360 (1978) 117. [3] A.A. Starobinsky, Phys. Lett. 91B (1980) 99. [4] R. Brout, F. Englert and P. Spindel, Phys. Rev. Lett. 43 (1979) 417; R. Brout et al., Nucl. Phys. B170 (1980) 228; Ph. Spindel, Phys. Lett. 107B (1981) 361; [5] L.S. Brown and J.P. Cassidy, Phys. Rev. D15 (1977) 2810; T.S. Bunch and P.C.W. Davies, Proc. R. Soc. London, A356 (1977) 569. [6] A. Vilenkin, Phys. Rev. D27 (1983) 2848.