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Inflation with superstrings
Volume 166B, number 3
PHYSICS LETTERS
16 January 1986
INFLATION WITH SUPERSTRINGS " Phillial O H 1 Enrtco Ferret Instttute and Department of Phystcs, Umverstty of Chtcago, Chtcago, IL 60637, USA Received 15 October 1985
Cosmological solutions with three exponentially expanding space dimensions m the N = 1 supersymmetrm Yang-Mills supergravlty system are found under the a s s u m p t m n of glumo and "subgravitmo" condensation The potential has a long flat regmn proxadlng sufficmnt inflation.
Superstring theories [1] are leading candidates for a unified theory of fundamental interactions including gravity. Of the five known superstring theories, the heterotic string [2] with gauge group E 8 X E 8 [3] seems to be the most promising phenomenologically [4]. In this paper we will continue our earlier work [5] where we found cosmological solutions of the tendimensional N = 1 supersymmetric Yang-Mills supergravity system which is the low energy limit of the heterotic (or type I) superstring theory. In all those solutions three space dimensions expand faster than the extra dimensions, and the time dependence of the expansion followed a power law. It would be interesting to have cosmological solutions where the three space dimensions expand exponentially with the inflationary universe * t in mind. We will show that such solutions are possible if we take account of fermion condensation phenomena. It is well known that gluino condensation can be used to provide supersymmetry breaking and vanishing of cosmological constant in the superstring theory [7]. But it is plausible that condensation of the "subgravitino", the MajoranaWeyl member of the supermultiplet which contains the graviton and gravitino may also take place at the Work supported xn part by the NSF, Grant No PHY-8301221. i Submatted to the Department of Physms, the Umverslty of Ctucago, in partial fulfillment of the reqmrements for the Ph D degree
early stage of the universe. Therefore we assume that in addition to the gluino condensation, there is subgravitino condensation, and look for cosmological solutions in the Kaluza-Klein context [8]. The relevant terms in the N = 1 supersymmetric Yang-Mills supergravity [9] are
f-? = -~e{R +~3 e --0"[HMNe _ eal2(Tr-~FMNP×)]2 + ~ e-a/2(TrGMN GMN) + ~ aMoOMo + (Tr ~FMNPX)(~I'~INP~)}.
(1)
Here, R is the ten-dimensional scalar curvature, GMN are the Yang-Mills field strengths and e - ° is the dilation field. The three-form H = ]HMNpdxM dxN dxP is given in terms of the exterior derivative F = dB of the Kalb-Ramond potential two-form B = X BMNd.x'MdxN and of the Yang-Mills Cern-Simons three-form 6O3y by H = F - W3y. The Lorentz C e r n Simons three-form 6O3L which occurs in the effective string action [10] is neglected here because t.,33L = 0 for our solutions. Also ~ is the gluino field and )~ is the subgravitino field. The ~ and k fields are redefined so that (.V/2/24)Tr~FMNpX and (3/32vr2)~FMN1~ in ref. [9] become Tr~FMNpXand -~FMNPk.It was pointed out in ref. [7] that the term a--~s'4(Tr~PMNe×)2 was missing in ref. [9]. The Einstein equations become
*l For Kaluza-Klein inflationary universe, see ref. [6]. 292
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 166B, number 3
PHYSICS LETTERS
16 January 1986
-- ~ (Tr XrpQ R x)2gMN
S 2 g ~ . The field configuration for the K a l b - R a m o n d field FMNP corresponds to the magnetic monopole on each three-sphere [12] and we have explicitly shown its dependence on the radius of the three-spheres. The ansatze for Tr~PMNP× and ~rMNP~ are exactly of the same form as the ansatz for HMNP except H1, H 2 are to be replaced by X1, X 2 and A1, A 2 respectively. The solution is
+ 9 e-O/2 [HMPQ(Tr ~FNPQX )
H I = - H 2 = H 0,
o =o0,
A1 =+A 2=A0,
S1 = S 2 = S 0 ,
RMN = 9 e-a[HMpQHNPQ - ~ gMN(HpRQ) 2 ] -- e - ° / 2 [Tr GMpGNP -- ~gMN Tr(GpQ) 2 ]
- ~ (aMOaNO) - ~(Tr XPPQR ×)( ~FPQR ?OgMN
- ~-~4gMNHpQR(Tr ~F PQR X)].
(2)
The dilation field equation is
X 1 =+X 2 =e-°2/2H0,
R(t) = R(0) exp [(2/X/-6)$6-1 t],
(I/e) am(gnNaN o) + e-°(HMNe) 2
k 3 = O, --~ e-°/2 HMNp(Tr 2pgNe×) + ¼ e - ° / 2 ( T r GMN GMN) = 0.
(3)
Consider the line element of the form [8] ds20 = _ d t 2 + R2(t)'~mndxmdx n + ds 2 with rn, n, ... = 1, 2, 3; gmn is the metric of the maximally symmetric three-dimensional space and R(t) corresponds to the time dependent cosmological scale factors. The line element ds 2 corresponds to a sixmanifold which is Itself a direct product of maximally symmetric manifolds. We found a solution in which the three-dimensional space is flat, i.e.,gmn = 8mn and expands exponentiaUy while the six-manifold is a direct product of two three-spheres and static:
ds2 =Slg~Kdx2~
mdxn + S2"~.ffdxrndx n '
~00 = ¼e-°°/2HoAo
(HoA0 ~> 0).
(6)
So in this solution, the flat three-space expands exponentially while the six-manifold which is the direct product of two identical three-spheres remains static. This removes the problem noted by Chapline and Gibbons [13] in the case without condensations. Stability analysis shows that our solution is stable against small perturbation of the scale factors R(t), S 1 (t) and S2(t ). We check whether such a solution can give a sufficient inflation of the three-dimensional space. In inflation with extra dimensions, the scale factor S 1 (and $2) is regarded as a dynamical scalar field [14] replacing the Higgs scalar field. Let us introduce the field (I) defined by (S 1 = S 2 = S)
• = ln(S/S 0).
(7)
In terms of a new variable x = at with A = vt2S6 -1 , the ~ (or ~"ff) components of the Einstein equations (2) yield
with S~, S . equal to the radii of the two three-spheres 2 ~ a (m, ~ "~_ and S 2l ~'~ g ~ ~( ~--, -n- _- 5, 6, 7) and S2g~.n - 8, 9, 10) the metrics of the two three-spheres. We take the following Ansatz [ 1 1] :
The prime is differentiation with respect to x. The potential V is given by
HMNP = (H1/S3)e~-~ V~, for (M,N,P) = (~,ff,~),
V(qb) = --~ e-2 ¢ + ] e-6 ¢ + ] + V(0),
(H2/S32)e~.~.~X/ffg, for ( M , N , P ) = ( m , n ' , p ) , O, otherwise
(4)
and GMN = O. Here ~
(5)
is the determinant of $ 2 ~ ' ~ and V~: that of
e~" + [3(R'/R) + 6 ~ ' ] ~ ' + ~ V/a~b = 0.
(8)
(9)
where V(0) is a constant. The potential is shown in fig. 1. We see that ~b = 0 (or S = SO) is the minimum of the potential and the potential has a long flat region for large (I)- Such a long flat region allows sufficient inflation to occur. So our solution has the necessary ingredients for successful inflationary universe. A few remarks are in order. We assumed nonvanishing values for fermion bilinears. In principle, 293
Volume 166B, number 3
PHYSICS LETTERS
V
16 January 1986
It is a great pleasure to thank Professor P.G.O. Freund for very useful discussions, patience and encouragement. I am very grateful to Professor Y. Nambu for useful discussions. I also enjoyed helpful conversations with A. Kagan and Dr. Y. Kitazawa.
R eferen ces
0 Fig. I. The potential V(~) for V(0) = 0. such non-vanishing values could arise from classical field configuration as well [15]. However, it is easy to see from the symmetry properties o f the product of three Majorana 3' matrices in ten-dimensions that there does not exist a classical field configuration that satisfies the ansatz ~FMNP~ "" e~--fi-ff (or e~ff~,). Hence the nonvanishing value must arise trom the quantum condensation phenomena. Also in our solution H 1 = H2, so that the radii of the two three-spheres must be equal. In this case the r~h- (or ~h") component of the energy-momentum tensor Tt~ H (or Tr~H)
T~H = H~pQH~ PQ - ~ g~-a (HpQR )2, vanishes. This parallels the vanishing o f Tpq 09, q = 5, .... 10) for the complex version o f the F r e u n d - R u b i n field configuration on Calabi-Yau manifold [ 16]. In conclusion, we investigated a possibility o f inflationary universe in superstring theories. We found that the de-Sitter phase of inflation is possible under the assumption o f gluino and subgravitino condensation. The potential o f the dynamical scalar field in terms o f the radius o f the extra dimensions has a long flat region providing sufficient inflation.
294
[1 ] See, J.H. Schwarz, preprint CALT-68-1252 (1985); M.B. Green, preprint CALT-68-1257 (1985). [2] D. Gross, J. Harvey, E. Martinet and R. Rohm, Nucl. Phys. B256 (1985) 253. [3] P.G.O. Freund, Phys. Lett. 151B (1985) 387. [4] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nuel. Phys. B258 (1985) 46. [5] P.G.O. Freund and P. Oh, Nuel. Phys. B255 (1985) 688. [6] E.W. Kolb, preprint FERMILAB-Conf-85]17-A (1985) and references therein. [7] M. Dine, R. Rohm, N. Selberg and E. Witten, IAS preprint (1985). [8] A. Chodos and S. Detweiler, Phys. Rev. D21 (1980) 2167; P G.O. Freund, Nucl. Phys. B209 (1982) 146. [9] G.F. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105. [10] M.B. Green and J.H. Schwarz, Phys. Lett. 149B (1984) 117. [11] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233. [12] P.G.O. Freund, EFI Report No. 81/07, 1981 unpublished; R.I. Nepomeehi, Phys. Rev. D31 (1985) 1921. [13] G.F. Chapline and G. Gibbons, Phys. Lett. 135B (1984) 43. [141 Y. Okada, U. Tokyo preprint UT-429 (1984). [15] M.J. Duff and C.A. Orzalesi, Phys. Lett. 122B (1983) 37. [16 ] R.I. Nepomeehi, U.S. Wu and A. Zee, Phys. Lett 158B (1985) 311.
PHYSICS LETTERS
16 January 1986
INFLATION WITH SUPERSTRINGS " Phillial O H 1 Enrtco Ferret Instttute and Department of Phystcs, Umverstty of Chtcago, Chtcago, IL 60637, USA Received 15 October 1985
Cosmological solutions with three exponentially expanding space dimensions m the N = 1 supersymmetrm Yang-Mills supergravlty system are found under the a s s u m p t m n of glumo and "subgravitmo" condensation The potential has a long flat regmn proxadlng sufficmnt inflation.
Superstring theories [1] are leading candidates for a unified theory of fundamental interactions including gravity. Of the five known superstring theories, the heterotic string [2] with gauge group E 8 X E 8 [3] seems to be the most promising phenomenologically [4]. In this paper we will continue our earlier work [5] where we found cosmological solutions of the tendimensional N = 1 supersymmetric Yang-Mills supergravity system which is the low energy limit of the heterotic (or type I) superstring theory. In all those solutions three space dimensions expand faster than the extra dimensions, and the time dependence of the expansion followed a power law. It would be interesting to have cosmological solutions where the three space dimensions expand exponentially with the inflationary universe * t in mind. We will show that such solutions are possible if we take account of fermion condensation phenomena. It is well known that gluino condensation can be used to provide supersymmetry breaking and vanishing of cosmological constant in the superstring theory [7]. But it is plausible that condensation of the "subgravitino", the MajoranaWeyl member of the supermultiplet which contains the graviton and gravitino may also take place at the Work supported xn part by the NSF, Grant No PHY-8301221. i Submatted to the Department of Physms, the Umverslty of Ctucago, in partial fulfillment of the reqmrements for the Ph D degree
early stage of the universe. Therefore we assume that in addition to the gluino condensation, there is subgravitino condensation, and look for cosmological solutions in the Kaluza-Klein context [8]. The relevant terms in the N = 1 supersymmetric Yang-Mills supergravity [9] are
f-? = -~e{R +~3 e --0"[HMNe _ eal2(Tr-~FMNP×)]2 + ~ e-a/2(TrGMN GMN) + ~ aMoOMo + (Tr ~FMNPX)(~I'~INP~)}.
(1)
Here, R is the ten-dimensional scalar curvature, GMN are the Yang-Mills field strengths and e - ° is the dilation field. The three-form H = ]HMNpdxM dxN dxP is given in terms of the exterior derivative F = dB of the Kalb-Ramond potential two-form B = X BMNd.x'MdxN and of the Yang-Mills Cern-Simons three-form 6O3y by H = F - W3y. The Lorentz C e r n Simons three-form 6O3L which occurs in the effective string action [10] is neglected here because t.,33L = 0 for our solutions. Also ~ is the gluino field and )~ is the subgravitino field. The ~ and k fields are redefined so that (.V/2/24)Tr~FMNpX and (3/32vr2)~FMN1~ in ref. [9] become Tr~FMNpXand -~FMNPk.It was pointed out in ref. [7] that the term a--~s'4(Tr~PMNe×)2 was missing in ref. [9]. The Einstein equations become
*l For Kaluza-Klein inflationary universe, see ref. [6]. 292
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 166B, number 3
PHYSICS LETTERS
16 January 1986
-- ~ (Tr XrpQ R x)2gMN
S 2 g ~ . The field configuration for the K a l b - R a m o n d field FMNP corresponds to the magnetic monopole on each three-sphere [12] and we have explicitly shown its dependence on the radius of the three-spheres. The ansatze for Tr~PMNP× and ~rMNP~ are exactly of the same form as the ansatz for HMNP except H1, H 2 are to be replaced by X1, X 2 and A1, A 2 respectively. The solution is
+ 9 e-O/2 [HMPQ(Tr ~FNPQX )
H I = - H 2 = H 0,
o =o0,
A1 =+A 2=A0,
S1 = S 2 = S 0 ,
RMN = 9 e-a[HMpQHNPQ - ~ gMN(HpRQ) 2 ] -- e - ° / 2 [Tr GMpGNP -- ~gMN Tr(GpQ) 2 ]
- ~ (aMOaNO) - ~(Tr XPPQR ×)( ~FPQR ?OgMN
- ~-~4gMNHpQR(Tr ~F PQR X)].
(2)
The dilation field equation is
X 1 =+X 2 =e-°2/2H0,
R(t) = R(0) exp [(2/X/-6)$6-1 t],
(I/e) am(gnNaN o) + e-°(HMNe) 2
k 3 = O, --~ e-°/2 HMNp(Tr 2pgNe×) + ¼ e - ° / 2 ( T r GMN GMN) = 0.
(3)
Consider the line element of the form [8] ds20 = _ d t 2 + R2(t)'~mndxmdx n + ds 2 with rn, n, ... = 1, 2, 3; gmn is the metric of the maximally symmetric three-dimensional space and R(t) corresponds to the time dependent cosmological scale factors. The line element ds 2 corresponds to a sixmanifold which is Itself a direct product of maximally symmetric manifolds. We found a solution in which the three-dimensional space is flat, i.e.,gmn = 8mn and expands exponentiaUy while the six-manifold is a direct product of two three-spheres and static:
ds2 =Slg~Kdx2~
mdxn + S2"~.ffdxrndx n '
~00 = ¼e-°°/2HoAo
(HoA0 ~> 0).
(6)
So in this solution, the flat three-space expands exponentially while the six-manifold which is the direct product of two identical three-spheres remains static. This removes the problem noted by Chapline and Gibbons [13] in the case without condensations. Stability analysis shows that our solution is stable against small perturbation of the scale factors R(t), S 1 (t) and S2(t ). We check whether such a solution can give a sufficient inflation of the three-dimensional space. In inflation with extra dimensions, the scale factor S 1 (and $2) is regarded as a dynamical scalar field [14] replacing the Higgs scalar field. Let us introduce the field (I) defined by (S 1 = S 2 = S)
• = ln(S/S 0).
(7)
In terms of a new variable x = at with A = vt2S6 -1 , the ~ (or ~"ff) components of the Einstein equations (2) yield
with S~, S . equal to the radii of the two three-spheres 2 ~ a (m, ~ "~_ and S 2l ~'~ g ~ ~( ~--, -n- _- 5, 6, 7) and S2g~.n - 8, 9, 10) the metrics of the two three-spheres. We take the following Ansatz [ 1 1] :
The prime is differentiation with respect to x. The potential V is given by
HMNP = (H1/S3)e~-~ V~, for (M,N,P) = (~,ff,~),
V(qb) = --~ e-2 ¢ + ] e-6 ¢ + ] + V(0),
(H2/S32)e~.~.~X/ffg, for ( M , N , P ) = ( m , n ' , p ) , O, otherwise
(4)
and GMN = O. Here ~
(5)
is the determinant of $ 2 ~ ' ~ and V~: that of
e~" + [3(R'/R) + 6 ~ ' ] ~ ' + ~ V/a~b = 0.
(8)
(9)
where V(0) is a constant. The potential is shown in fig. 1. We see that ~b = 0 (or S = SO) is the minimum of the potential and the potential has a long flat region for large (I)- Such a long flat region allows sufficient inflation to occur. So our solution has the necessary ingredients for successful inflationary universe. A few remarks are in order. We assumed nonvanishing values for fermion bilinears. In principle, 293
Volume 166B, number 3
PHYSICS LETTERS
V
16 January 1986
It is a great pleasure to thank Professor P.G.O. Freund for very useful discussions, patience and encouragement. I am very grateful to Professor Y. Nambu for useful discussions. I also enjoyed helpful conversations with A. Kagan and Dr. Y. Kitazawa.
R eferen ces
0 Fig. I. The potential V(~) for V(0) = 0. such non-vanishing values could arise from classical field configuration as well [15]. However, it is easy to see from the symmetry properties o f the product of three Majorana 3' matrices in ten-dimensions that there does not exist a classical field configuration that satisfies the ansatz ~FMNP~ "" e~--fi-ff (or e~ff~,). Hence the nonvanishing value must arise trom the quantum condensation phenomena. Also in our solution H 1 = H2, so that the radii of the two three-spheres must be equal. In this case the r~h- (or ~h") component of the energy-momentum tensor Tt~ H (or Tr~H)
T~H = H~pQH~ PQ - ~ g~-a (HpQR )2, vanishes. This parallels the vanishing o f Tpq 09, q = 5, .... 10) for the complex version o f the F r e u n d - R u b i n field configuration on Calabi-Yau manifold [ 16]. In conclusion, we investigated a possibility o f inflationary universe in superstring theories. We found that the de-Sitter phase of inflation is possible under the assumption o f gluino and subgravitino condensation. The potential o f the dynamical scalar field in terms o f the radius o f the extra dimensions has a long flat region providing sufficient inflation.
294
[1 ] See, J.H. Schwarz, preprint CALT-68-1252 (1985); M.B. Green, preprint CALT-68-1257 (1985). [2] D. Gross, J. Harvey, E. Martinet and R. Rohm, Nucl. Phys. B256 (1985) 253. [3] P.G.O. Freund, Phys. Lett. 151B (1985) 387. [4] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nuel. Phys. B258 (1985) 46. [5] P.G.O. Freund and P. Oh, Nuel. Phys. B255 (1985) 688. [6] E.W. Kolb, preprint FERMILAB-Conf-85]17-A (1985) and references therein. [7] M. Dine, R. Rohm, N. Selberg and E. Witten, IAS preprint (1985). [8] A. Chodos and S. Detweiler, Phys. Rev. D21 (1980) 2167; P G.O. Freund, Nucl. Phys. B209 (1982) 146. [9] G.F. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105. [10] M.B. Green and J.H. Schwarz, Phys. Lett. 149B (1984) 117. [11] P.G.O. Freund and M.A. Rubin, Phys. Lett. 97B (1980) 233. [12] P.G.O. Freund, EFI Report No. 81/07, 1981 unpublished; R.I. Nepomeehi, Phys. Rev. D31 (1985) 1921. [13] G.F. Chapline and G. Gibbons, Phys. Lett. 135B (1984) 43. [141 Y. Okada, U. Tokyo preprint UT-429 (1984). [15] M.J. Duff and C.A. Orzalesi, Phys. Lett. 122B (1983) 37. [16 ] R.I. Nepomeehi, U.S. Wu and A. Zee, Phys. Lett 158B (1985) 311.