Addendum to "Hyperbolic Space Cosmologies"

Abstract

In our earlier paper [JHEP 0310 (2003) 058], we considered higher dimensional cosmological models with hyperbolic spaces. In particular the eternal accelerating expansion was obtained by studying small perturbation around the critical non-accelerated solution for $D > 10$. In this addendum, we show that there is also such a solution in the critical case $D = 10$. Comment: 4 pages, v2: abstract added

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arXiv:hep-th/0609043 v2 27 Sep 2006
KU-TP 006
hep-th/0609043
Addendum to “Hyperbolic Space Cosmologies”
Chiang-Mei Chena1, Pei-Ming Hob2, Ishwaree P. Neupanec,d3,
Nobuyoshi Ohtae4and John E. Wangf5
aDepartment of Physics, National Central University, Chungli 320, Taiwan
bDepartment of Physics, National Taiwan University, Taipei 106, Taiwan
cDepartment of Physics and Astronomy, University of Canterbury, Private Bag 4800,
Christchurch 8020, New Zealand
dCentral Department of Physics, Tribhuvan University, Kathmandu, Nepal
eDepartment of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
fPhysics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan
Abstract
In our earlier paper [JHEP 0310 (2003) 058], we considered higher dimensional cosmolog-
ical models with hyperbolic spaces. In particular the eternal accelerating expansion was ob-
tained by studying small perturbation around the critical non-accelerated solution for D > 10.
In this addendum, we show that there is also such a solution in the critical case D = 10.
1cmchen@phy.ncu.edu.tw
2pmho@phys.ntu.edu.tw
3ishwaree.neupane@cern.ch
4ohtan@phys.kindai.ac.jp
5jwang@phys.cts.nthu.edu.tw

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In our earlier paper [1], we considered higher dimensional cosmological models with hy-
perbolic spaces. In particular the eternal accelerating expansion was analyzed by studying
small perturbations around the critical non-accelerated solution. We concluded that eternal
acceleration is possible for D > 10. In this note we want to show that there is also such a
solution in the case D = 10 as well, which was overlooked in [1]. The existence of such a
solution was recently argued in refs. [2, 3].
For this critical dimension, our metric is
ds2= e−6φ?−dt2+ a2ds2
H3
?+ e2φds2
H6, (1)
where ds2
Hndenotes the metric of an n-dimensional hyperbolic space. The critical solution is
a = 2t,φ =
1
2√6ψ +1
8ln30, (2)
where
ψ =1
cln
?t2
3
?
,c = 2
?2
3.(3)
Previously in our paper [1] we obtained accelerating solutions by perturbing the above
solution as
a = a0+ a1,ψ = ψ0+ ψ1,(4)
where a0and ψ0are the critical solutions in (2) and (3). The perturbative parts were written
as
a1= A,ψ1=B
t
(5)
where a1and ψ1were chosen to be polynomials which were then fixed using the equations
of motion. For perturbation theory to be valid, it was necessary to assume that a1 ≪ a0
and ψ1≪ ψ0. Our ansatz led to solutions with accelerated expansion behavior for generic
D > 10. For the case D = 10 the analysis resulted in constant A and B, and therefore there
was no eternal acceleration. However, we now re-examine this case and perform more detailed
analysis for the case D = 10 allowing the possibility that A and B depend on time in search
of accelerated expansion behavior.
Our approach is to re-examine the equations of motion for D = 10 and directly look for
solutions starting from the above ansatz with arbitrary functions A and B. The equations
of motion for the first order term (given as eqs. (5.11)-(5.13) in [1]) reduce in this particular
case to
˙A −3A
t+3B
4t−c
t2+3
4
?
˙B +3c
4
B
t
?
˙B
=0,(6)
¨B +
˙B
c
˙A
t−A
¨A +c
t2
=0, (7)
t
=0.(8)
1

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To obtain a solution we first eliminate A from eqs. (6) and (7) and get as a result
˙A − ct¨B − 2c˙B = 0, (9)
where the dot denotes the derivative with respect to time. If we differentiate this result once
and use eq. (8) to eliminate the variable A, we obtain the equation for B:
t2...
B +3t¨B +˙B = 0,(10)
whose solution is
B = c1(lnt)2+ c2lnt + c3. (11)
Finally we obtain the solution for the variable A by substituting the solution for B into eq. (9)
A = c?c1(lnt)2+ (2c1+ c2)lnt + c4
?. (12)
Consistency of the equations tells us that
c4=8
3c1+ c2+ c3. (13)
The expansion and acceleration of the cosmic evolution are
˙ a = 2 + ct−1(2c1lnt + 2c1+ c2),¨ a = −ct−2(2c1lnt + c2). (14)
The acceleration occurs for all
2c1lnt + c2< 0, or t > e−c2/2c1, (15)
if we choose c1< 0. The latter condition is necessary for the accelerated expansion to continue
in the future direction. Since the expanding condition, which is also the condition for the
perturbation theory to be valid,
t > −c
can always be satisfied for sufficiently large t, the accelerated expanding phase always occurs
after certain time of evolution
Although a1approaches infinity as t → ∞, the first order perturbative treatment above
is still reliable for large t because lnt/t ≪ 1 and we still have a1≪ a0and ψ1≪ ψ0for all
t. Thus hyperbolic compactification can give eternal acceleration for D ≥ 10 instead of just
D > 10.
As a final comment, our solution is related to eq. (3.19) in [2] via the time redefinition
2(2c1lnt + 2c1+ c2), (16)
t =
?5
8
?3/21
4
?τ4− 18c0(lnτ)2− 36d0lnτ?,(17)
where τ is the time coordinate and c0,d0are parameters of the result in [2]. Here the factor
5/8 comes from the different conventions for hyperbolic space by a constant scale factor.
Our seemingly extra parameter c3is a redundancy and can be absorbed by the coordinate
transformation of scaling t.
2

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Acknowledgements
IPN would like to thank the CERN Theory Division for their kind hospitality during the course
of this work. This work was carried out while NO was visiting the Institute of Theoretical
Physics, Chinese Academy of Sciences and Centre for Mathematical Sciences, Cambridge
University, and he would like to thank R.-G. Cai and P.K. Townsend for valuable discussions
and kind hospitality. The work of CMC, PMH and JEW is supported in part by the National
Science Council, the Center for Theoretical Physics at National Taiwan University, the Na-
tional Center for Theoretical Sciences, Taiwan, R.O.C. That of IPN is supported in part by
the Foundation for Research, Science and Technology (New Zealand) under Research Grant
No. E5229. That of NO is supported in part by Grant-in-Aid for Scientific Research Fund
of the JSPS No. 16540250. That of JEW is supported in part by the Academic Center for
Integrated Sciences at Niagara University and the New York State Academic Research and
Technology Gen“NY”sis Grant.
References
[1] C. M. Chen, P. M. Ho, I. P. Neupane, N. Ohta and J. E. Wang, “Hyperbolic space
cosmologies,” JHEP 0310 (2003) 058 [arXiv:hep-th/0306291]. See also references therein.
[2] L.Andersson
arXiv:hep-th/0602102.
andJ.M.Heinzle,“Eternal accelerationfromM-theory,”
[3] J. Sonner and P. K. Townsend, “Recurrent Acceleration in Dilaton-Axion Cosmology,”
arXiv:hep-th/0608068.
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