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Cosmic string solution as the dislocations
0083-6656/93 $24.00 @ 1993 Pergamon Press Lid
Vistas in Astronomy, Vol. 37, pp. 519-522, 1993 Printed in Great Britain. All rights tmetved.
COSMIC STRING SOLUTION AS THE DISLOCATIONS Takchiko T. Fujishiro,* Mitsuo J. Hayashit and Shoji Takeshita~: *Tokai University, Hiratsulm,Japan tTokai University, Shimizu, Japan :~Shizuoka Prefectural Institute of Public Health and Environmental Science, Shizuoka, Japan
The cosmic strings which u e the topological defects caused by the symmetry breaking on the manifold with nontrivial ~rt, have been extensively investigated as a candidate of the seeds for galaxy and large scale structure formation (Kibble[1976], Villenkin[1985]). In our previous papers we have proposed that the cosmic strings can be described naturally by torsion formalism which has s direct analogy with dislocations in three-dimensional crystalline solid (Fnjishiro, Hayashi sad Takeshita[1991, 1992a, 19921>]). Here the four-dlmensional effective action of the heterotic sigma model where the target space had an absolute parallelism (Hayashi and Naksno[1967], Hayashi and Shlrafuji[1979]), is investigated to search for an exact cosmic string solution which seems consistent with the present observations (Caldwell and Allen[1992]). The action is given by
f
(1)
~-----
where ~ is the translational gauge field strength. Assuming that the four-dimensional metric is block diagonal sad the line element of this manifold can be written as
~., = ~'~(~(.))dy=~ ' + / ~ ( . ) ~ ' d . ~,
(2)
where A, B u d ce,~ are the uncompactified and compactified coordinate indices respectively. Here we have chosen the internal space to be a two-dimensional torus whose metric is given by
1(-1 ~--'(~ --("I"(Z))= ~=" #ii
iT1
i~,~ M/'
(3)
T.T. Fuflshiro et aL
520
where l"(z) = ~'x + ir2,detg-(2) = - 1 . The metric on the effectively two-dimensional space is d~ 2 = g ( ~ ( z ) d z ~ d z n = dp 2 + eD(P)dB2. (4) Then the following form for the two-dimensional effective action is obtained after dimensional reduction,
s~ )
=
i -ia
/ d2zh[TCV~v
O~r_._OA~.l ,.; ~,
(~)
where f~ is the volume of the compactified space. Thus the equstions with respect to ~, r and D are
02~- + 2(Or)~ -- o,
o2~ - 2(0~)~ - o,
402D + (0D) 2 + ~
= 0,
(6)
(7)
where 0 = d/dp. The eqs.(6) axe exactly solved in this case as
- ~ = ia sech b(p+c), 1"i+ ~ = - a tanh b(p + c) + d,
(8)
where a, b, c and d are constant. Thus this solution is solitonic though it is not self evident to interpret it as a cosmic string. The last term of eq.(7) is now ,~ constant, i.e. 01"0~ ~; = b2"
(9)
Then the solution of eq.(7) is now given by exp 1 D = A p * [ c o s ( ~ - ) + 1],
(10)
where A and a are constant and p* = v~/b. Since the energy-momentum tensor is sero outside the string core, the exterior solution is given by m a l ~ g the last term of eq.(7) to zero. Thus we obtain 1 exp ] D = ( ~ ' -t- 7) 2, (ll) where P0 = ~'0 _~ ~', Po is the boundaxy of the interior region and the exterior region of the string and ~ and 7 axe constants. We assume that the interior and the exterior metrics have to be joined together along the surface of the string at p = P 0 , ~' = ~'o and the first derivatives of both metrics coincide there, which may also satisfy the Israel's junction condition (Hiscock[1985], Gott IH[1985], Raychandhuri[1990]), i.e.
~2 =
,
:
A 4711 - cos(~--~-, a)], (Ap.)½ [ [ c o s ( ~ . a ) ÷ 1]½ _
1[ 1 -
,po + a u ~ P F0j]
(12)
521
Cosmic String Solution
The string is characterized by a mass per unit length p. If p is expressed unit of Planck mass per Pl~nck length (we use the system of units G = c = 1 ), then the mass per unit length is a dimensionless quantity; p = 1 corresponds to one Planck mass per Planck length ~ 1.347 × 102s g/cm. The mass per unit length is
/~ ---- ,tO
JO
~exp
----
yLsmt~j s,n(~:)+ 7],
(13)
where 8~re = l/p* 2 is the nonzero components of the energy-momentum tensor of the string at p < Po. The deficit angle is 6qb =
/
dOij =
/
S~j~,dz s' = 3 A
f0" sin(~-Y-.~)de= e~A~n(-~-Y-~), (14)
where Sij~, is the contorsion tensor. From eq.(13) and (14) the relation between the mass per unit length and the deficit angle is obtained
6@ = 2 4 f p -
6~Apo, p*
(15)
where we put a = 0. Now we could predict the existence of a solltonic solution starting from the four-dimensional action which may mean that the torsion formalism could predict the existence of the cosmic string solutions. Next we can aAopt new coordinates that p = p*(~o - ~r) st the interior metric and (j3~' + 7) z = ~" st the exterior one, and we set t, z = constant. Thus the two surfaces defined by the interior and exterior metrics are given by ds 2 = p*2d~o3 + (p* - p* cos ~o)2d02,
ds'
1 ' dr,= + 1"'dO'. = 4/~2,.
(16)
(17)
With the new coordinates we can rewrite the mass per unit length and the deficit angle, i.e. A j. = ¥ ( ~ , o - s i n ~ o - . ) , (18) 6~b = - 6 x A sin ~o,
(19)
where ~o = Po/P* + 7r. Since the deficit angle is 0 < 6~b < 20r as physical observational condition and the mass per unit length is 0 < p _~ 1, from eq.(18) and (19) the extent of ~o for A = 1 is lr < ~o < lr + sin -1 } and 2a- - sin -x xa < ~o < 2a'. So the permit sphere of ~oo is divided into the two part.
522
T. T.
Fujishiroet al.
The mass per unit length for A = 1 correspond to the above ~oo is 0 < p < ~1 ( i1 + sin - 1
and
+.
- siu
i) <
<
that is to say 0 <
< 2.267 × 10
g/ m and
1.056 × 10 =s g / c m < p < 1.058 × 10 =s g/cm. These results are consistent with those of grand unified vacuum strings that predict p ,,~ 1021 g / c m which large enough to promote galaxy and large scale structure formation (Gott III[1985]). However recent observations on the upper limit of p are favored with the value 10 - e (Caldwell and Allen[1992]), for which we will show that the fact corresponds to the constant A ~ 1 case. From similar discussion with the case of A = 1 the extent of ~o0 is ~" < ~oo < x + sin -1 a-~ and 2~" - sin -1 8-~ < ~Oo < 2~r f o r A > - i1, ~r < ~o0 < 2~ for A < ~. 1 When the parameter A >> 1, p follows to an equality. If sln-1 sAx-!"_-- 1__8.t,then 0 < p < ~. The constant parameter A is not fixed in this model but on the other hand the small enough value can be chosen as 10 -e. Then the maximum value of the mass per unit length is p ,~ 10-s(,~ 10== g/cm). This result is consistent with the limits on the gravitational radiation b ~ k g r o u n d that comes from millisecond pulsar timing measurement. REFERENCES Caldwe]] R.R. and Allen B. (1992), Cosmological constraints on cosmlc-strlng gravitational radiation. Phya.Re~. D45, 3447-3468. Fujishiro T.T., Hayashi M.J. and Takeshita S. (1991), Are the cosmic strings seen as the dislocations ? Mod.Phys.Leff. A6, 2237-2242. Fujiskiro T.T., Hayashi M.J. and Takeshlta S. (1992a), What is the role of torsion in the universe ? Progr. Theor.Phys.Supplement, to be published. Fujishiro T.T., Hayashi M.J. and Takeshita S. (1992b), The cosmic strings generated from the torsion. Mod.Phys.Lett. A, to be published. Gott Ill J.R. (1985), Gravitational lensing effects of vacuum strings: exact solutions. AaLrophys.J. 288, 422-427. Hayashi K. and Nakemo T. (1967), Extended translation invariance and associated gauge fields. Progr. Theor.Phys. 38, 491-507. Haynshi K. and Shirafuji T. (1979), New general relativity. Phys.Ree. D19, 3524-3553. Hiscock W.A. (1985), Exact gravitational field of a string. Phys.Ree. D31, 3288-3290. Kibble T.W. (1976), Topology of cosmic domains and strings. 3.Phys. Ag, 1387-1398. KrOner E. (1981), Continuum theory of defects. In Les Houehes, Session XXXV, lg80-Physque des D~fauts, Ed. Balian R. eted., pp. 215-315. North Holland. Raychaudhuri A.K. (1990), Cosmic strings in general relativity. Phys.Ree. D41, 3041-3046. Vilenkin A. (1985), Cosmic strings and domain walls. Phys.Rep. 121, 263-315.
Vistas in Astronomy, Vol. 37, pp. 519-522, 1993 Printed in Great Britain. All rights tmetved.
COSMIC STRING SOLUTION AS THE DISLOCATIONS Takchiko T. Fujishiro,* Mitsuo J. Hayashit and Shoji Takeshita~: *Tokai University, Hiratsulm,Japan tTokai University, Shimizu, Japan :~Shizuoka Prefectural Institute of Public Health and Environmental Science, Shizuoka, Japan
The cosmic strings which u e the topological defects caused by the symmetry breaking on the manifold with nontrivial ~rt, have been extensively investigated as a candidate of the seeds for galaxy and large scale structure formation (Kibble[1976], Villenkin[1985]). In our previous papers we have proposed that the cosmic strings can be described naturally by torsion formalism which has s direct analogy with dislocations in three-dimensional crystalline solid (Fnjishiro, Hayashi sad Takeshita[1991, 1992a, 19921>]). Here the four-dlmensional effective action of the heterotic sigma model where the target space had an absolute parallelism (Hayashi and Naksno[1967], Hayashi and Shlrafuji[1979]), is investigated to search for an exact cosmic string solution which seems consistent with the present observations (Caldwell and Allen[1992]). The action is given by
f
(1)
~-----
where ~ is the translational gauge field strength. Assuming that the four-dimensional metric is block diagonal sad the line element of this manifold can be written as
~., = ~'~(~(.))dy=~ ' + / ~ ( . ) ~ ' d . ~,
(2)
where A, B u d ce,~ are the uncompactified and compactified coordinate indices respectively. Here we have chosen the internal space to be a two-dimensional torus whose metric is given by
1(-1 ~--'(~ --("I"(Z))= ~=" #ii
iT1
i~,~ M/'
(3)
T.T. Fuflshiro et aL
520
where l"(z) = ~'x + ir2,detg-(2) = - 1 . The metric on the effectively two-dimensional space is d~ 2 = g ( ~ ( z ) d z ~ d z n = dp 2 + eD(P)dB2. (4) Then the following form for the two-dimensional effective action is obtained after dimensional reduction,
s~ )
=
i -ia
/ d2zh[TCV~v
O~r_._OA~.l ,.; ~,
(~)
where f~ is the volume of the compactified space. Thus the equstions with respect to ~, r and D are
02~- + 2(Or)~ -- o,
o2~ - 2(0~)~ - o,
402D + (0D) 2 + ~
= 0,
(6)
(7)
where 0 = d/dp. The eqs.(6) axe exactly solved in this case as
- ~ = ia sech b(p+c), 1"i+ ~ = - a tanh b(p + c) + d,
(8)
where a, b, c and d are constant. Thus this solution is solitonic though it is not self evident to interpret it as a cosmic string. The last term of eq.(7) is now ,~ constant, i.e. 01"0~ ~; = b2"
(9)
Then the solution of eq.(7) is now given by exp 1 D = A p * [ c o s ( ~ - ) + 1],
(10)
where A and a are constant and p* = v~/b. Since the energy-momentum tensor is sero outside the string core, the exterior solution is given by m a l ~ g the last term of eq.(7) to zero. Thus we obtain 1 exp ] D = ( ~ ' -t- 7) 2, (ll) where P0 = ~'0 _~ ~', Po is the boundaxy of the interior region and the exterior region of the string and ~ and 7 axe constants. We assume that the interior and the exterior metrics have to be joined together along the surface of the string at p = P 0 , ~' = ~'o and the first derivatives of both metrics coincide there, which may also satisfy the Israel's junction condition (Hiscock[1985], Gott IH[1985], Raychandhuri[1990]), i.e.
~2 =
,
:
A 4711 - cos(~--~-, a)], (Ap.)½ [ [ c o s ( ~ . a ) ÷ 1]½ _
1[ 1 -
,po + a u ~ P F0j]
(12)
521
Cosmic String Solution
The string is characterized by a mass per unit length p. If p is expressed unit of Planck mass per Pl~nck length (we use the system of units G = c = 1 ), then the mass per unit length is a dimensionless quantity; p = 1 corresponds to one Planck mass per Planck length ~ 1.347 × 102s g/cm. The mass per unit length is
/~ ---- ,tO
JO
~exp
----
yLsmt~j s,n(~:)+ 7],
(13)
where 8~re = l/p* 2 is the nonzero components of the energy-momentum tensor of the string at p < Po. The deficit angle is 6qb =
/
dOij =
/
S~j~,dz s' = 3 A
f0" sin(~-Y-.~)de= e~A~n(-~-Y-~), (14)
where Sij~, is the contorsion tensor. From eq.(13) and (14) the relation between the mass per unit length and the deficit angle is obtained
6@ = 2 4 f p -
6~Apo, p*
(15)
where we put a = 0. Now we could predict the existence of a solltonic solution starting from the four-dimensional action which may mean that the torsion formalism could predict the existence of the cosmic string solutions. Next we can aAopt new coordinates that p = p*(~o - ~r) st the interior metric and (j3~' + 7) z = ~" st the exterior one, and we set t, z = constant. Thus the two surfaces defined by the interior and exterior metrics are given by ds 2 = p*2d~o3 + (p* - p* cos ~o)2d02,
ds'
1 ' dr,= + 1"'dO'. = 4/~2,.
(16)
(17)
With the new coordinates we can rewrite the mass per unit length and the deficit angle, i.e. A j. = ¥ ( ~ , o - s i n ~ o - . ) , (18) 6~b = - 6 x A sin ~o,
(19)
where ~o = Po/P* + 7r. Since the deficit angle is 0 < 6~b < 20r as physical observational condition and the mass per unit length is 0 < p _~ 1, from eq.(18) and (19) the extent of ~o for A = 1 is lr < ~o < lr + sin -1 } and 2a- - sin -x xa < ~o < 2a'. So the permit sphere of ~oo is divided into the two part.
522
T. T.
Fujishiroet al.
The mass per unit length for A = 1 correspond to the above ~oo is 0 < p < ~1 ( i1 + sin - 1
and
+.
- siu
i) <
<
that is to say 0 <
< 2.267 × 10
g/ m and
1.056 × 10 =s g / c m < p < 1.058 × 10 =s g/cm. These results are consistent with those of grand unified vacuum strings that predict p ,,~ 1021 g / c m which large enough to promote galaxy and large scale structure formation (Gott III[1985]). However recent observations on the upper limit of p are favored with the value 10 - e (Caldwell and Allen[1992]), for which we will show that the fact corresponds to the constant A ~ 1 case. From similar discussion with the case of A = 1 the extent of ~o0 is ~" < ~oo < x + sin -1 a-~ and 2~" - sin -1 8-~ < ~Oo < 2~r f o r A > - i1, ~r < ~o0 < 2~ for A < ~. 1 When the parameter A >> 1, p follows to an equality. If sln-1 sAx-!"_-- 1__8.t,then 0 < p < ~. The constant parameter A is not fixed in this model but on the other hand the small enough value can be chosen as 10 -e. Then the maximum value of the mass per unit length is p ,~ 10-s(,~ 10== g/cm). This result is consistent with the limits on the gravitational radiation b ~ k g r o u n d that comes from millisecond pulsar timing measurement. REFERENCES Caldwe]] R.R. and Allen B. (1992), Cosmological constraints on cosmlc-strlng gravitational radiation. Phya.Re~. D45, 3447-3468. Fujishiro T.T., Hayashi M.J. and Takeshita S. (1991), Are the cosmic strings seen as the dislocations ? Mod.Phys.Leff. A6, 2237-2242. Fujiskiro T.T., Hayashi M.J. and Takeshlta S. (1992a), What is the role of torsion in the universe ? Progr. Theor.Phys.Supplement, to be published. Fujishiro T.T., Hayashi M.J. and Takeshita S. (1992b), The cosmic strings generated from the torsion. Mod.Phys.Lett. A, to be published. Gott Ill J.R. (1985), Gravitational lensing effects of vacuum strings: exact solutions. AaLrophys.J. 288, 422-427. Hayashi K. and Nakemo T. (1967), Extended translation invariance and associated gauge fields. Progr. Theor.Phys. 38, 491-507. Haynshi K. and Shirafuji T. (1979), New general relativity. Phys.Ree. D19, 3524-3553. Hiscock W.A. (1985), Exact gravitational field of a string. Phys.Ree. D31, 3288-3290. Kibble T.W. (1976), Topology of cosmic domains and strings. 3.Phys. Ag, 1387-1398. KrOner E. (1981), Continuum theory of defects. In Les Houehes, Session XXXV, lg80-Physque des D~fauts, Ed. Balian R. eted., pp. 215-315. North Holland. Raychaudhuri A.K. (1990), Cosmic strings in general relativity. Phys.Ree. D41, 3041-3046. Vilenkin A. (1985), Cosmic strings and domain walls. Phys.Rep. 121, 263-315.