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Физика
UDC 53, 531.123.6+521.12+523.11
On Cosmological Solutions with Sigma-Model Source
A. A. Golubtsova*, V. D. Ivashchuk
* Institute of Gravitation and Cosmology Peoples' Friendship University of Russia 6 Miklukho-Maklaya str., Moscow, Russia, 117198Laboratoire de Univers et Theories (LUTh),
Observatoire de Paris Place Jules Janssen 55, Meudon, France 92190 ^ Institute of Gravitation and Cosmology Peoples' Friendship University of Russia 6 Miklukho-Maklaya str., Moscow, Russia, 117198 Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Str., Moscow, Russia, 119361
A multidimensional model of gravity with a sigma-model action for scalar fields is considered. The gravitational model is defined on the manifold, which contains n Einstein factor spaces. General cosmological-type solutions to the field equations are obtained when n — 1 factor-spaces are Ricci-flat. The solutions are defined up to solutions of geodesic equations corresponding to a sigma-model target space. Several examples of sigma-models are considered. A subclass of non-singular solutions is singled-out for the case when all factor-spaces are Ricci-flat.
Key words and phrases: cosmological solutions, sigma-model, acceleration.
1. Introduction
Scalar-tensor theories are well-known and important as alternatives to Einstein's general relativity. They are widely used, in particular, for explaining present-day accelerated expansion of the Universe [1] and in many other applications.
Here we consider a gravitational model governed (in essence) by a Lagrangian
C = R[g] — hab(^)gMNdM<paBN(1)
where g is a metric and non-linear "scalar fields" come to the Lagrangian in a sigma-model form with a target space metric h assumed. For a review of sigma-models see [2] and refs. therein.
The Lagrangian (1) with hab = const describes the truncated NS — NS sector of various D = 10 and D = 11 supergravity theories in the Einstein frame [3]. Usually these theories contain form-fields (fluxes) in addition to massless scalar fields and Chern-Simons (CS) terms. In this sense, the Lagrangian (1) matches zero flux (and CS) limit. For D = 3 the Lagrangians of such types are generic ones when dimensional reductions of (bosonic sectors of) supergravity models are considered, see [4, 5] and refs therein.
Here we deal with cosmological-type solutions defined on the product of n Einstein spaces (e.g. Ricci-flat ones). The integrable cosmological configurations were studied in numerous papers, see [6,7] (without scalar fields), [8-11] (with one scalar field), [12] etc. The authors of these papers restricted their attention to a linear sigma-model for which components hab are constant. Here we study the solutions for hab('^) with arbitrary dependence on (e.g. for a non-linear sigma-model).
Received 22nd March, 2012. This work was supported in a part by Russian Foundation for Basic Research (Grant Nr. 09-02-00677-a) and by the FTsP "Nauchnie i nauchno-pedagogicheskie kadry innovatsionnoy Rossii" for the years 2009-2013.
2. The Model
We start by considering an action of the form
dDx^\i\{R[g] - hab(p)gMNdMpadNpb] + SYGH, (2)
5= 1 1 ^
2k2
m
where k2 is a ^-dimensional gravitational coupling, g = gMNdxM ® dxw is a metric defined on a manifold M, p : M ^ Mv is a smooth sigma-model map and Mv is a I-dimensional manifold (target space) equipped with the metric h = hab(p)dpa ® dp6 (pa are coordinates on Mv). Here sygh is the standard York-Gibbons-Hawking boundary term [13,14].
The field equations for the action (2) read as follows
rMN - ^ 9MNr = tMN , (3)
-jgdM{gMNVW\hab(p)dNpb) - 2dhpp1dKPcdLpbgKL = 0, (4)
where1
tMn = hab(p)dMpadNpb - ^hab(p)gMNdKpadL'PbgKL■ (5)
is the stress-energy tensor.
Here we consider a cosmological-type ansatz for the metric and "scalar fields"
g = we2^(u)du <8> du + ^ e2l3%(u)g\ (6)
i=1
pa = pa(u), (7)
where a = 1,..., 1.
The metric is defined on the manifold
M = R* x Mi x ... x Mn, (8)
where R* = (u-,u+) and any factor-space Mi is a ¿¿-dimensional Einstein manifold with the metric g% obeying
Rm,nt [g i] = iigU ini, (9)
where i = 1,... ,n.
To find solutions for the equations (3)-(4) seems to be complicated due to the nonlinear structure of the Einstein equations and intricacy having scalar fields. However it may be shown that the field equation for the model (2) with the metric and "scalar fields" from (6), (7) are equivalent to the Lagrange equations corresponding to the Lagrangian of the one-dimensional (n + I)-component a-model
L = 1 AT-1 [Gl3ff%ff° + hab(p)paph] -NVs. (10)
n
Here N = exp(7 — 70) > 0 is a modified lapse function, 70 = ^^ dip1,
i=1
Gij = di5ij - didj, (11)
where i,j = 1,... ,n, are components of the gravitational part of the minisuperspace metric and
n
Ъ = f £ №(12)
i=1
is the potential. For the constant Наъ(ф) = hab the reduction to the sigma-model was proved (for more general set up) in [15]. We note that Наь(ф) can be interpreted as a
dA
scalar part of the target space metric. Here and in what follows A = ——.
du
When all Mi have finite volumes the substitution of (6) and (7) into the action (2) gives us the following relation
5 = v J Ldt (13)
n
where L is defined in (10), v =--2 П Vi, and V = /м ddiy (л/det (gгт,n.)) is the
^ ■ -1 г
г= 1
volume of Mi, г = 1,... ,n.
The relation (13) can be derived using the following expression for the scalar curvature
n 2 n
R = -fe-^(27o - 2707 + 702 + ^ (>) ) + £e-2?R[g% (14)
=1 =1
where R[gl] = ^di is the scalar curvature corresponding to the Mi-manifold. To obtain (13) one should extract the total derivative term in (14) which is canceled by the York-Gibbons-Hawking boundary term.
We write the Lagrange equations for (10) and then put N = 1, or equivalently 7 = 70, i.e. when u is a harmonic variable. We get
G^' + i3d3 (-5j + di) e-2ß° = 0, (15)
d(h сЪ(ф)ф>b) 1 dhab (v).a,-b
3 = 1
where г = 1,... ,n,
d(h , (ф)фb) 1 дh , (ф
-ф афb = 0, (16)
du 2 дфс
с = 1,..., I, and
]^GiJßi ß! + 1 /йаб(ф)фаф6 + П = 0. (17)
In fact, equations (15) are nothing else but Lagrange equations corresponding to the Lagrangian
1
Lß = 2Gi,ßiß' - Vè (18)
with the energy integral of motion
1
Eß = 2Gi,ff ß> + ^. (19)
Likewise (15), the equations (16) are Lagrange equations corresponding to the Lagrangian
= 2 hab^ афъ (20)
118 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 115-128 with the energy integral of motion
Ev = 1 hab{p)p b. (21)
Equations (16) are equivalent to geodesic equations corresponding to the metric h. The relation (17) is the energy constraint
E = Ep + EV = 0, (22)
coming from dL/dM = 0 (for M =1).
Equations (15) may be rewritten in an equivalent form
- w^ie-2pi+2l° = 0, (23)
where i = 1,... ,n. These expressions may be obtained from (15) by using the inverse matrix (Gij) = (G^)-1:
5ij 1
G" = * + 2-D (24)
and the following relations for u(k)-vectors:
u(k) = - 6lk + di, u(k^ = Giju[k) = - ^. (25)
z 1 3 di
where i,j,k = 1,... ,n.
In what follows we will use the following relation
(u(k), u(k)) = G^u^uf = 1 - 1, (26)
dk
where k = 1, . . . , n.
Hence, the problem of finding the cosmological-type solutions for the model (2) (with u being harmonic variable) is reduced to solving the equations of motion for the Lagrangians Lp and Lv with the energy constraint (22) imposed. Geodesies for a flat metric h. For the constant hab('p) = hab eqs (16) read
Pa = 0, (27)
or, equivalently,
,na = v
where v^,pg are integration constants, a = 1,..., 1. The energy for scalar fields (21) takes the form
Ev = 2 hahv%v%. (29)
More examples of geodesic solutions will be given in Section 4.
= v%u + (28)
3. Cosmological-Type Solutions
In this section we deal with certain examples of cosmological-type solutions with the metric and "scalar fields" from (6) and (7), respectively.
3.1. Solutions with n Ricci-flat Spaces
In this subsection, we focus on the solutions for the case when all factor-spaces Mi are Ricci-flat:
Ricbi = 0, (30)
where = 1, . . . , n.
Due to (30) the potential V^ is equal to zero and the equations of motion (23) for f now become
f = 0, (31)
where = 1, . . . , n.
Integration of the equations (31) yields
n
p = Vu + P0, 70 = ^ d (Vi + P0) , (32)
=1
where the parameters V and f0 are integration constants and the energy (19) takes the form
Ep = hSijj v*, (33)
where the minisuperspace metric Gij is given by (11). The metric reads
g = w exp
2 ^d^u + A0 )
i= 1
du ® du + ^exp [2 [vlu + /30)]g\ (34)
=1
The "scalar fields" obey eqs. (16) with the energy constraint
Ev = 2 hab(p)p apb = - ^GijV* v*. (35)
In a special case of one (non-fantom) scalar field (h11 = 1) and w = —1 this solution was obtained in [8].
The scalar curvature for the metric (34) reads (see (14))
R[g] = — w (GijvW) e-2j0. (36)
In what follows we use a parameter
n
E = E(v) = ^dV (37)
=1
to classify the solutions.
Non-special Kasner-like solutions.
First we shall consider the non-special case when £(v) = 0. Let us define a "synchronous" variable
1 n
t = \ i exp + r3
|E(v)f exp [E(v'u + $)dj] (38)
obeying e2'yo(^ du2 = dr2.
We introduce new parameters:
ai = vi/E(v), (39)
where i = 1,... ,n, and
£v = Eb/(E(v))2. (40)
Then the metric reads
g = wdr ®t + ^c2ir2a^g\ (41)
c2Jla.i i= 1
where t > 0. "Scalar fields" are solutions to equations of motion (see (16))
&r
rh c b(v)~r~ d
^ _i dhab&) d^a V = 0 (42)
2 dr dr
where a = 1,..., I. The parameters (39) obey the Kasner-like conditions
=1
=1
where
^diai = 1, (43)
=1
YJdl(al)2 = 1 - 2£v, (44)
d d
is the integral of motion for eqs. (42). In (41) a are constants
2£b = r2hab(^ ^, (45)
n
a = |srl exp -a1 ^Pidj , (46)
=1
where i = 1,... ,n, obeying cAii = |S(-u)|.
, n,
i=l
Flat h. For the special case of the flat target space metric hab(<p) = hab we get
^ = a£ lnr + v^a, (47)
where <^a are constants, a = 1,..., I, and
1 2
£v = 2 habaavabv. (48)
The scalar curvature (36) reads
R[g] = 2w£vt-2. (49)
It diverges for t ^ +0 if £b = 0. Hence all solutions with £b = 0 are singular. For £b = 0 the solutions with non-Milne-type sets of the Kasner parameters are singular when all g% have Euclidean signatures since the Riemann tensor squared is divergent at t ^ +0 [16]. For Milne-type sets of parameters, i.e. when di = 1
and ai = 1 for some i (aJ = 0 for all j = i) the metric is regular, when either i) g(i) = —wdy1 <8> dy1, Mi = R (—to < yi < or ii) g(i) = wdy1 <8> dyi, Mi is a
circle of length Li (0 < yi < Li) and ciLi = (i.e. when the cone singularity is absent).
Special (steady state) solutions. Now we consider the special case when £(v) = 0. Due to (35) we obtain
1 n
= — ^W2 < 0. (50)
i=1
We get in this case 70 = ^^ di^00 = const and hence the scalar curvature
=1
R[g] = 2wEv e-2'70 (51)
and the volume scale factor v = e70 are constants.
The "synchronous" variable is proportional to u (t = e70u).
Hence, we obtained a restriction for the energy Ev < 0. For Ev = 0, all vi = 0, and we are led to a static Ricci-flat solution.
For Ev < 0 we get R[g] = 0. This possibility occurs if the target space metric h is not positive-definite (e.g. there are phantom scalar fields for flat h). For solutions with one (phantom) scalar field see [8].
Solutions with acceleration. Let d1 =3 and M1 = R3. The factor-space M1 may be considered as describing our space. In both cases there exist subclasses of solutions describing accelerated expansion of our space.
Indeed, for Kasner-like solutions with w = —1 one could make a replacement t 1—y t0 — t where t0 is a constant (corresponding to the so-called "big rip"). For such replacement the scale factor of M1 reads
01(t) = C1(to — T)ai, (52)
where c1 > 0.
If a1 < 0 we get accelerated expansion of 3-dimensional factor space M1. For the Hubble parameter we get
H = «,'1/01 = (—a{)/( to — t), (53)
while the variation of the effective gravitational constant reads
G/G = (1 — 3«1)/(ro — t). (54)
n
Here we used the relation G = constJJ(r0 — T)-diai = const(r0 — T)3a1-1 [17] (see
=2
(43)). This implies the relation
S = G/(GH) = (3a1 — 1)/a1. (55)
The condition a1 < 0 yields the huge value |5| > 3 which does not obey the observational limits [17]: |<5| < 0.1. Thus, accelerated expansion of M1 factor-space is incompatible with tests on G — dot.
Analogous consideration may be carried out for special (steady state) solutions. For w = — 1, d1 = 3 and v1 > 0 we get accelerated expansion of 3-dimensional factor-space M1. In this case due to £ = 0 one get 5 = 3 which also does not pass the G — dot test.
3.2. Solutions with One Curved Einstein Space and n — 1 Ricci Flat
Spaces
Here we put
Ricb 4= Î191, Î1 = 0, ШФЧ = 0, i> 1, (56)
i.e. the first space (Mi, g1) is an Einstein space of non-zero scalar curvature and other spaces (Mi,gi) are Ricci-flat.
The Lagrangian (18) reads in this case
1 • in
Lß = ß> — -iidi exp (—2ß1 + 27o), (57)
where — ß1 + = u(1 ßi and u^ = — 61 + di.
The Lagrange equations corresponding to the Lagrangian (57) are integrated in Appendix. The solution reads
P1 =ln |/|t-3! +vlu + /1, (58)
pi =vi t + P0, i> 1, (59)
where /30, vi are constants obeying
n n
v1 = ^rfdi, /1 = Y,Pidi- (60)
i=l i=1
The function is following
'Rsinh(VC(u -u0)), C> 0, > 0;
=
|£i(di — 1)|1/2(u — uo), C = 0, 1 > 0; Rsin(^=C(u — u0)), C< 0, w^ > 0; , Rcosh(VC(u — uo)), C> 0, < 0,
where u0 and C are constants and
(61)
|
R = . (62)
For 70 we get
70 =31 - ln |/|. (63)
The energy integral of motion Ep corresponding to Lp reads (see Appendix)
Eß = ЦТ—d1) + ^<64>
Using (58), (59) and (63) we are led to the relation for the metric
2d n
g = If 11^ exp [2(v1u + 30)] (wdu ® du + fg1) + ^ exp [2( viu + P0 )]gi. (65)
Golubtsova A. A., Ivashchuk V. D. On Cosmological Solutions with Sigma- . . . 123
The "scalar fields" obey eqs (16) with the energy constraint
Ev = 1 hab(f)<j>a<i>b = Cd\, - iGi.vV. (66)
Here the constraints (60) on 00, V should be kept in mind, and the function f is defined in (61).
In a special case of one (non-fantom) scalar field (h11 = 1) and w = —1 this solution was obtained earlier in [9,10], see also [11].
4. More Examples of Geodesic Solutions
In this section we consider three examples of solutions to geodesic equations corresponding to the metric h that may be used for the cosmological-type solutions above.
4.1. Metric on S2
Let h be a metric on a two-dimensional sphere S2
h = d$ ® d$ + sin2 -ddp ® dp. (67)
The simplest solution to geodesic equations (16) for the metric reads
p = ши, = n/2, (68)
where ш is constant. Here Ev = 2ш2. The general solution to geodesic equations may be obtained by a proper isometry SO(3)- transformation of the solution from (68).
4.2. Metric on dS2
Now we put h to be a metric on a two-dimensional de Sitter space dS2
h = —dx <g) dx + cosh2 xdp <8> dp. (69)
There are three basic solutions to geodesic equations (16) in this case
i = ши, X = 0 (70)
X = vu p = (71)
tan p = sinh x = mu, (72)
where ш, v and m are constants. For the energy we have Ev = 1ш2, — 1v2 and 0, for space-like, time-like,and null geodesics, respectively. The general solution to geodesic equations may be obtained by a proper isometry SO(1, 2) — transformation of the solutions from (70)-(72).
4.3. A Diagonal Metric h
Here we deal with a diagonal metric
-1
h = e0dp ® dp + ^ ekA2k(p)d^k ® d^k, (73)
k=1
where e0 = ±1, ek = ±1 (k > 0) and all Ak(p) > 0, are smooth functions.
The Lagrange function for the non-linear sigma-model is given by
i-i
L
v
eop2 + £ e-A- (v)(t- )2
k=1
Equations of motion for cyclic variables
d_ du
(e-A- (<^k ) = 0
yield the following of integrals of motion
£ kAl (p)^k = Mk,
where k = 1,... ,1 — 1.
Another integration constant is energy Ev
Ev=2
i-i
eov2 + £ ekAl (p)(^k )2
which due to (76) reads
Ev = 2
k=1
1-1
£0P2 + Y1 £1MIAI 2(V)
k=1
This relation implies the following quadrature
dip
Vo
y^oEv - £q Ek"=\ e-M-A- 2(v)
u — u0,
which implicitly defines the function p = p(u). Another quadratures just following from (76)
(74)
(75)
(76)
(77)
(78)
(79)
tl — t- = J due-M-A-2(u),
uo
(80)
complete the integration of the geodesic equations for the metric (73).
For Ak (p) = exp (Xp), X = 0, the metric (73) may describe either a part of de-Sitter space (if e o = —1, £ i = 1, k > 0) or a part of anti-de-Sitter space (if £i = — 1, £ r = 1, r = 1). The case I = 3 is of interest in connection with the so-called the AWE hypothesis [18].
v
u
5. Conclusions
Here we have considered a multidimensional model of gravity with a sigma-model source (for scalar fields). The model is defined on the manifold M, which contains n Einstein spaces.
We have obtained exact cosmological-type solutions to the field equations in two cases: i) when either all factor-spaces are Ricci-flat or ii) when only one factor-space space has nonzero scalar curvature.
In the first case i) the solutions have either Kasner-like form or describe steady-state solutions, generalizing those from [8]. The Kasner-like solutions are mostly singular with certain exceptions (of Milne-type).
For the case when all factor-spaces are Ricci-flat we have singled-out a subclass of solutions describing accelerated expansion of 3-dimensional manifold. We have shown that these solutions do not obey the tests on variation of G.
The second subclass of solutions ii) (e.g. for spherically symmetric configurations) will be considered in a separate publication (e.g. a possible fitting of acceleration with bounds on G-dot, see [19] and ref. therein).
6. Appendix: Solutions Governed by Liouville Equation
Here we consider a Toda-like system with the following Lagrangian
L =2{3,3)-Aexp (2(b,/)), (81)
where / G Rn, A = 0, b G Rn. The scalar product for vectors belonging to Rn is defined by
{/1,32) = GijP1fi2, (82)
where Gij is a non-degenerate symmetric matrix (e.g. given by (11)).
The Lagrange equations corresponding to the model (81) read (in a condensed vector form)
3 + 2Ab exp (2{b,/)) = 0. (83)
Let { , ) = 0.
Eqs (83) is exactly integrable and the solution has the following form
3 = ^TlA(i + vt + 30, (84)
{ , )
where {b, b) = 0 and v, 30 G Rn are constant vectors obeying
{v, b) = {30, b) =0. (85)
The function = ( ) obeys the Liouville equation
q + 2A{b, b)e2q = 0, (86)
The solution to Liouville equation reads
q= - ln If I, (87)
where
'R sinh(VC(t - t0)), C> 0, A < 0; l2Al1/2(t - t0), C = 0, A < 0; Rsin(V-C(t - to)), C< 0, A < 0; , Rcosh(VC(t - to)), C> 0, A > 0;
here we put A = A(b, b) and
f
(88)
R =J™. (89)
The energy corresponding to the model (81) reads
E =1 {3,3) + Ae 2< b'Pl (90) After substitution of (84) to (90) we obtain
E = ET + 1 {v, v), (91)
where
ET = 2{b)^ + Ae 2q■ (92)
Due to (88) we get
C
ET = 2(M). (93)
Proposition. For {b, b) = 0 all solutions to Lagrange equations (83) are covered by the relations (84), (85), (87) and (88).
Proof. It is obvious that the solutions (84), (85) with q from (87), (88) obey the equations of motion (83).
Let us show that the relations (84), (85), (87) and (88) follow from (83). Let q = {b, 3) and y = 3 — (bq)/{b, b). It is obvious that {b, y) = 0. It follows from (83) that that the equation (86) and
q = 0, ^ y = vt + 3o, (94)
where constant vectors and 3 obey (due to { , ) = 0)
{b, y) =0 b, v) = { b,3o) = 0. (95)
Hence
3 = T^A +y = ITiA +vt + 3o (96)
{ , ) { , ) 0
where q = q(t) obeys (86) and hence it is given by relations (87) and (88).
The Proposition is proved. □
Let us introduce a dual vector u = (ui): ui = GijV. Then we get u(3) = ui3i = {b ,3), (u,u) = GijUiUj = {b, b) ((Gij) = (Gij)-1) and the solution (84) reads
V
3% = --,-r ln lfl + VH + 31, (97)
(u, u)
where i = 1,... ,n, where (u, u) = 0,
u( ) = ui = 0, u(3 ) = ui 3 = 0, (98)
and function is defined in (88) with
f
R ^/2!A(UfU)i, A = A(u, u). (99)
For the energy (92) we obtain from (91), (93)
C 1
2(u,u) + 2
E + vj- (100)
Example. Let us consider the Lagrange system from Section 3 with parameters: A = f Ui = u\1) = -5} +di,d! > 1. Then due to (25) and (26) we get ui = -^ and (u, u) = } — 1 < 0. The solution reads
I = in l/l + vH + i3i, (101)
where i = 1,..., n, with constraints
n n
v1 = ^ydi, ii = ^m, (102)
i=1 i=1
imposed. In (88) we should put A = f £1 (1 — di) and fl = ^|gl||q-1)-. For the energy we get
Cd 1
£ =2(1—(103>
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УДК 53, 531.123.6+521.12+523.11
О космологических решениях с сигма-модельным
источником
А. А. Голубцова*, В. Д. Иващук^
* Институт гравитации и космологии Россиийский университет дружбы народов уд. Миклухо-Маклая, д. 6., Москва, Россия, 117198 Лаборатория теории Вселенной (ЬиТН), Обсерватория Парижа пл. Жуль Жансен, д.5, Медон, Франция, 92190 ^ Институт гравитации и космологии Россиийский университет дружбы народов уд. Миклухо-Маклая, д. 6., Москва, Россия, 117198 Центр гравитации и фундаментальной метрологии ВНИИМС, ул. Озерная, д. 46, Москва,, Россия, 119361
Рассматривается многомерная модель скалярно-тензорной гравитации с сигма-модельным действием для скалярного сектора. Гравитационная модель определена на многообразии, которое содержит п фактор-пространств Эйнштейна. Получены общие решения космологического типа для полевых уравнений, когда все фактор-пространства, за исключением одного, риччи-плоские. Решения определены с точностью до решения уравнений геодезических на пространстве мишеней. В случае, когда все фактор-пространства риччи-плоские, выделен подкласс несингулярных решений.
Ключевые слова: космологические решения, сигма-модель, ускорение.
