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Role of scalar fields in cosmological models
Available online at www.sciencedirect.com
Advances in Space Research 44 (2009) 1359–1365 www.elsevier.com/locate/asr
Role of scalar fields in cosmological models E.V. Chubaryan *, R.M. Avakyan, G.G. Harutyunyan, A.S. Piloyan Department of Physics, Yerevan State University, 1 Alex Manoogian Street, 0025 Yerevan, Armenia Received 9 December 2008; received in revised form 2 April 2009; accepted 23 July 2009
Abstract Recent astronomical observations of supernovae and cosmic microwave background indicate that the universe is accelerating. Scalar– tensor theories of gravity give rise to suitable cosmological models where a late-time accelerated expansion is naturally realized. In an alternative proposal the cosmic acceleration is generated by means of a scalar field (quintessence), in a way similar to the early-time inflation. In this paper, we consider two classes of cosmological models with scalar fields. The first one corresponds to the Jordan–Brans– Dicke tensor–scalar theory with a cosmological scalar and the second one contains a conformally coupled scalar field with quartic potential. In both type of models the cosmological dynamics is described and the deceleration parameter is evaluated. The values of the parameters are specified for which a late-time accelerated expansion is realized. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Cosmology; Scalar field; Dark energy
1. Introduction Recent discovery of accelerated expansion of the Universe (Garnavich et al., 1998; Perlmutter et al., 1999; Copeland et al., 2006) stimulated the revision of the established concept of its structure and evolution character. The cosmological constant is the most relevant candidate to interpret the cosmic acceleration (for the problem of cosmological constant see, Weinberg, 1989). This is equivalent to the consideration of the non-zero vacuum energy and the related negative pressure (Guth, 1981; Linde, 1982; Starobinski, 1982). An alternative explanation of the observed acceleration is the hypothesis for the existence of a special substance in the Universe, referred as quintessence (Caldwell and Steinhardt, 1998), for which the equation of state is Pq = aeq, 1 < a < 1/3. In the relativistic theory of gravitation (RTG) (Logunov, 2001), proposed by Logunov and collaborators, the presence of quintessence leads to interesting consequences: the acceleration of the Universe is replaced by deceleration with the subse*
Corresponding author. Tel.: +374 010 570 370. E-mail address: [email protected] (E.V. Chubaryan).
quent stop, after which the contraction to a minimum value of the scale factor begins with subsequent new expansion cycle (Gershtein et al., 2003; Chernin, 2008). Note that the idea of oscillating character of the Universe evolution was suggested earlier, but mainly from philosophical reasons (Aman and Markov, 1984), because in the Friedmann closed model the oscillation mode is interfered by the entropy growth from one cycle to another and the transition through the cosmological singularity (Tolmen, 1949). Reasoned enough the introduction of the cosmological constant in general relativity (GR) has led to the idea of the introduction of a similar quantity in the tensor–scalar theory, assuming that the new field is scalar (not dynamical but controllable by the gravitational scalar of the Jordan– Brans–Dicke (JBD) theory Avagyan et al., 2005; Avagyan and Harutunyan, 2008, 2005). In curved spaces, the scalar fields complying with the general covariance requirements are described by the equation @w ð1:1Þ gab ra wb m2 c2 =h2 þ cR w ¼ 0; wb b ; @x where m is the mass of the field, R is the scalar curvature, c is a dimensionless constant in the absence of which the usual generalisation of the Klein–Gordon equation is
0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.07.018
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obtained in the case of Riemannian space (c = 1/6 was proposed by Penrose for the case of a conformally invariant massless scalar field). It turned out that the JBD theory (Bekenshtein, 1974) can be translated by conformal transformation from the proper representation with a self-coordinated scalar field w, subject to the condition gab$awb = 0, to another representation described by GR equations with a source in the form of non-gravitational fields and conformally coupled massless scalar field. In the present paper, we investigate two classes of cosmological models with scalar fields. The organization of the paper is as follows. In the next section, the cosmological dynamics is considered within the framework of the Jordan–Brans–Dicke tensor–scalar theory with cosmological scalar. It is shown that if the scalar field dominates and the cosmological scalar is absent the expansion of the Universe is always decelerated. The presence of the cosmological scalar leads to the possibility of the late-time accelerated phase. In Section 3, we consider the cosmological model with a conformally coupled scalar field in the presence of the Higgs potential and non-gravitational matter with the barotropic equation of state. The models with accelerated expansion of the Universe are specified. Section 4 summarizes the main results of the paper. 2. Cosmological scalar in the Jordan–Brans–Dicke theory In this section, the cosmological problem in the proper representation of the JBD theory is considered in presence of nonminimally coupled scalar field. As it will be shown the introduction of the cosmological scalar provides a possibility for the transition from the decelerated expansion of the Universe to the accelerated one. It was noted in Avagyan et al. (2005), Garriga and Mukhanov (1999), Panagia (2005), Fujii and Maeda (2003), Weinberg (1972), Scherrer (2004) that the modern conceptions of the Universe give raise to the introduction of the cosmological constant in the GR, therefore it is worth to introduce a similar quantity in the JBD theory. Having assumed that the field corresponding to this quantity should be scalar but cannot be dynamical (its changes should be controlled by the gravitational scalar y = y(xl)), we introduce the cosmological scalar u = u(y) in the JBD theory action similar to the introduction of the cosmological constant in the GR action (Avagyan et al., 2005): Z ylyl 1 pffiffiffiffi c4 y R þ 2uðyÞ 1 2 þ Lm d 4 x: W¼ ð2:1Þ g 16p y c Here, f is a dimensionless coupling constant of the JBD theory. The presence of the cosmological scalar means that in addition to the kinetic term the potential one is also considered for the scalar field. Equating to zero the independent variations of the action (2.1) with respect of gab and y, we obtain the JBD theory equations with the cosmological scalar
ra y a ¼ Glm
m kT 2y @u þ uy ; 3 þ 2f 3 þ 2f @y
k¼
8p ; c4
ð2:2Þ
! m l 1 l k ml l T rm y l yy 1 y ya þ þ f m 2 dlm a 2 ¼ dm R ¼ T m dm 3 þ 2f y y y 2 y 2 0m 1 c dlm @u Tl ð1 þ 2fÞu þ 2y ð2:3Þ ¼ k @ m þ T lm A; þ 3 þ 2f y @y Rlm
where m
T lm ¼ P m dlm þ ðqm þ P m ÞU m U l ; c 1 1 ðrm y l dlm ra y a Þ T lm ¼ ð2:4Þ k y f 1 þ 2 y m y l dlm y a y a þ dlm yuðyÞ : y 2 pffiffiffiffiffi m ffi dð dgglmLm Þ is the energy–momentum tensor of Here T lm ¼ p2ffiffiffiffi g c
the matter, and T lm is caused by the presence of the scalar field y, Pm and qm are the pressure and the density of the matter (dust and radiation), considered as a perfect fluid, U a ¼ dxa =ds is four-dimensional velocity satisfying the condition U a U a ¼ 1. Conceivably, long range massless scalar field with potential y ¼ yðxl Þ (gravitational scalar) replaces the Newtonian gravitational constant G at each point, and according to Mach ideas (Brans and Dicke, 1961) it is a background created by the whole energy in the Universe. An essential feature of the JBD theory scalar field, which distinguishes it from the fields of other tensor– scalar theories (Kaluza–Klein theory, string theory), is that it does not interact directly with matter. This circumstance gives rise to consider the JBD scalar field in itself without matter and radiation, which could probably allow revealing its role in the formation of dark matter and dark energy. In connection with the aforesaid it is of interest to remind how the JBD theory was constructed. The idea of the existence of scalar field has very naturally arisen in the attempt to unify the gravitation and electromagnetism. It was found out that the five-dimensional formulation of the Einstein gravitation and Maxwell theory should be much easier than the four-dimensional one. The conclusion was made after the extended group-theoretical analysis according to which the property of the invariance of the Einstein–Maxwell theory in 5D is much more symmetric. It was noticed that the unified group of arbitrary fourdimensional coordinate transformations and gauge transformations of the electromagnetic field potential is isomorphic to the coordinate transformations group of five-dimensional Riemannian space, with respect to which the field equation of the projective theory (Jordan, 1951, 1948; Ludwig, 1951) as well as modifications of the unified theory proposed by Pauli (1933) are invariant. In this isomorphic group Xl are transformed as 5-vectors, which allows to construct an additional invariant y ¼ glm X l X m . To ensure that the GR equations reduced to 4D are
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equivalent to the Einstein–Maxwell field equations system, the requirement of constancy for the scalar y = 1 is necessary. So, directly and without speculations and additional hypotheses, as a result of physical analysis, an idea of the generalisation of the gravitation theory arose, in which in addition to the tensor and vector fields there is a scalar field with the potential y. One of physically substantial and fully developed versions is the JBD theory, in which the gravitational scalar does not interact directly with the matter. Its existence is only manifested by the impact on the particles motion. Now we turn to the investigation of the cosmological model in JBD theory caused by the presence of nonminimally coupled scalar field. It makes sense to omit the possible contributions from all kinds of the matter and determine the role of the scalar field, nonminimally coupled with the gravitation. We use spatially-flat Friedmann–Robertson–Walker metric, assuming the space-time homogeneity and isotropy: 2 dr2 2 2 2 2 2 2 þ r dh þ sin hdu ; ds ¼ dt R ðtÞ ð2:5Þ 1 kr2 y ¼ yðtÞ; where k = 0 stands for the spatially-flat case, R(t) is the scale factor. In the case of the flat model the cosmological equations in the Jordan proper frame take the following form: €y 3y_ R_ 2 @u ¼ uy ; ð2:6Þ þ yR 3 þ 2f @y y € R_ 2 _ y f y_ 2 €y 2R_ 2R þ u; þ 2¼ R R Ry 2 y2 y R_ 2 1 y_ 2 R_ y_ 3 2 ¼ f 2 3 þ u; 2 y Ry R
ð2:7Þ ð2:8Þ
where the dot denotes the time derivative. If we determine u from (2.8) and substitute it in (2.6) and (2.7), then (2.6) will exactly coincide with (2.7), that is for any choice of u we have two independent equations. The choice of u = Ky/y0, where y0 is the modern value of y and K is the Einstein theory cosmological constant, is caused by the fact that in this case under the conformal transformation g0ik ¼ ygik we obtain the Einstein theory with cosmological constant (Einstein frame of the JBD theory). With this form of the function u(y), the integration of the substraction of Eqs. (2.7) and (2.8), 2 € €y y_ 2 y_ 2 R_ y_ R R_ þ þ ðf þ 1Þ 2 ¼ 0; ð2:9Þ 2 2 y y y R R Ry by taking into account the Eq. (2.6), €y R_ y_ ¼ 3 ; Ry y allows one to obtain the following result: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y_ ¼ c y ðrþ1Þ=2 þ cþ y ðr1Þ=2 ; r ¼ 3ð3 þ 2fÞ: y
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After the substitution of (2.11) into (2.8) and taking into account (2.10), from the equation 2 2 €y 1 €y f y_ Ky=y 0 ¼ 0 3 y 2 y y we obtain the following relation between the constant c cþ ¼
K ; ð3 þ 2fÞy 0
ð2:12Þ
which allows to conclude that for K = 0 in (2.11) either c or c+ is zero, and for K – 0 the constants c and c+ have opposite signs if K > 0 and f > 3/2. We consider two separate cases. (a) First we assume that K = 0. From Eq. (2.8) we _ obtain the following relation between the functions R=R and y_ =y: ð3 rÞ y_ R_ : ¼ 6 y R
ð2:13Þ
From (2.10) it follows that the plus sign corresponds to y_ ¼ jc jy ðrþ3Þ=2 and the minus sign corresponds to the case y_ ¼ jcþ jy ð3rÞ=2 As a result, for the scale factor we have ðr3Þ=6 y ; R ¼ R0 0 y 1 2=ðr1Þ y0 r 1 ðr1Þ=2 y0 ¼ 1 þ jc j ðt t0 Þ ; ð2:14Þ y 2 where t t0, R = R0, y = y0 correspond to the present moment. It is convenient to present the deceleration parameter € R_ 2 in the following form: q ¼ RR= q¼
€ R00 R R R ¼ 3 2 R_ 2 R0
ð2:15Þ
where the prime means the derivative with respect to y. Then, from (2.14) we have 2r : r3
q ¼
ð2:16Þ
Based on (2.4), it is possible to introduce by analogy the concepts of the pressure and the energy density of the scalar field, T lym ¼ P y dlm þ ðqy þ P y Þum ul , in accordance with the relations 1 ðf þ r 3Þc2 y ð3rÞ ; 2k 1 r 3 2 ð3rÞ Py ¼ ; fþ c y 2k 3
q y ¼
ð2:17Þ ð2:18Þ
ð2:10Þ
þ þ from where we find 1=3 < P y =qy < 1; 1 < P y =qy < 1 at 0 < f < 1 and, accordingly, the effective velocity of the elementary excitations (sound) arising in such medium can be defined as (Garriga and Mukhanov, 1999)
ð2:11Þ
v2 ¼
dP 2ðr 3Þ 4 y : ¼1 dq 3f 2ðr 3Þ y
ð2:19Þ
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It follows from (2.16) that q = 0 is only possible in the case f = 3/2, however this value of the JBD theory dimensionless parameter is initially excluded from the consideration at formulation of the theory (Ludwig, 1951), because in this special case as a result of the variation of the action ten equations are obtained for 11 unknown quantities. It is interesting to note that for f = 3/2 one has P y ¼ qy =3 and c2 < 0. Negative pressure can be reasonably interpreted, however the imaginary sound speed in liquid is rather doubtful. Thus, under the condition f > 3/2 and in the presence of only scalar field in JBD theory, we conclude that q+ can only accept positive values, which corresponds to the deceleration of the expansion of the Universe, and q is positive for f < 0 and can be negative for 3/2 < f < 0; however lim , which confirms once f!0 again the advisability of the assumption that f is positive. (b) Now we consider the case K – 0. As it was already _ from (2.11) in the case of noted, in the expression for y=y K > 0, the constants c and c+ are distinct from zero and they have different signs. Their signs coincide in the case K<0 From (2.10) and (2.11) we have R0 1 c ðr þ 3Þy r=2 þ cþ ð3 rÞy r=2 ¼ 6y R c y r=2 þ cþ y r=2
ð2:20Þ
Taking into account (2.15), we find q¼2
6 ½c ðr þ
3Þy r=2
2
þ cþ ð3 rÞy r=2
½c2 ðr þ 3Þy r þ c2þ ð3 rÞy r þ c cþ ð6 2r2 Þ ð2:21Þ If one of the constants of integration (c or c+) is zero, the value of (2.21) coincides with that for (2.16). In the presence of K, in addition of positive values for q, it is possible to obtain q = 0 and q < 0. The zero acceleration of the expanding Universe corresponds to a certain value of y1 satisfying the equation c2 ðr
þ
3Þy 2r 1
þ
4rc cþ y r1
þ
c2þ ðr
3Þ ¼ 0;
from where one has pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cþ 2r 3r2 þ 9 r y1 ¼ c rþ3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Kð 1 þ 2f=3 1 þ f=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ y 0 c2 ð 1 þ 2f=3 þ 1Þð3 þ 2fÞ
1=r
ð2:22Þ
It is easy to see from the expression €y ¼ c
3 þ r ð1þrÞ=2 r 3 ð1rÞ=2 cþ ; y y 2 2
at the moment of the zero acceleration can be either larger or smaller than the limiting value yn. The system of Eqs. (2.10) and (2.11) can be explicitly solved. The corresponding solution is presented in the permanent form rþ1 rþ1 ðrþ1Þ=2r ;1 þ ;u ; t=t1 ¼ u 2 F 1 1; 2r 2r ð2:24Þ ðrþ3Þ=2r 1=3 u 1=r ; y=y n ¼ u R=R1 ¼ 1u parameter 0 u 1. In this formulae, is the hypergeometric function and we have ðr1Þ=2r =ðcþ ðr þ 1ÞÞ. In the used the notation t1 ¼ 2jcþ =c j limit u ? 0 one has t=t1 uðrþ1Þ=2r and we have the following asymptotic behavior with
the
2 F 1 ða; b; c; xÞ
y / t2=ðrþ1Þ ;
R / tðrþ3Þ=3ðrþ1Þ ;
t2 ¼ jcþ =c j
q 2r=ðr þ 3Þ; t ! 0;
ð2:23Þ
ðr1Þ=2r
=ðcþ rÞ;
and the asymptotics for the scalar field and scale factor have the form y y n ½1 þ ð1=rÞet=t2 ;
y_ ! 0 ) €y < 0;
:
from where it follows that in the limit y_ ! 0, one has € ! 1, which corresponds to the accelerated expansion Rþ of the Universe. Thus, the scalar field value pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=r 2r 3r2 þ 9 y1 ¼ yn rþ3
t t2 lnð1 uÞ;
and from €y ¼ 0, taking into account that the signs of c and c+ are different, we obtain the limiting value for y: y n ¼ jcþ =c j
€ 3_y ½jc jðr þ 3Þðr þ 1Þy r=2 jcþ jðr 3Þðr 1Þy r=2 RR ¼ 1 þ ; R_ 2 y 3=2 ½jc jðr þ 3Þy r=2 þ jcþ jðr 3Þy r=2 2
for the scalar field, scale factor and the deceleration parameter. This limit corresponds to early stages of the cosmological expansion. In the limit u?1 one has
It follows from the expansion of the Universe that R ¼ cy_ 1=3 ! 1;
that €y < 0 for c < 0 and cþ > 0, in this case r y_ ¼ jc jy ð3þrÞ=2 ðjy n =yj 1Þ For y(t0) > yn one has y_ < 0, and y decreases asymptotically tending to yn. However if y(t0) < yn, then y_ > 0 and y t ? 1 ffione asymptotically increases to yn. In both cases at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi has R ? 1, y ¼ y n þ Aeðtt0 Þ=s , where s ¼ 2=ðr jc cþ jy 0 cÞ is the characteristic time, c is the light velocity, and the sign _ of the constant of integration A depends on the sign of y. As regards the deceleration parameter q, from (2.22) it can be presented in the following form:
R R1 et=3t2
q 1; t ! 1
This limit corresponds to late stage of the cosmological expansion. Hence, at late stages the scalar field tends to the constant value yn and we have an exponential expansion 1 with the Hubble constant H ¼ ð3t2 Þ . In Fig. 1, we have plotted the time dependence of the scalar field, the scale factor and the deceleration parameter described by formulae (2.24). From the graphs, the transition from the decelerated expansion to the accelerated phase is clearly seen. Summarizing the analysis given above we conclude that if the nonminimally coupled scalar field dominates, then
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In this case, the contraction of the Eq. (3.2), by taking into account (3.3), gives R ¼ kT ¼ kðe 3P Þ
ð3:5Þ
It is commonly assumed that the geometry with the Friedmann–Robertson–Walker metric is adequate for the model of homogeneous and isotropic Universe. Accordingly, for the flat model with the matter having the equation of state P = ae the above equations become " # _ w_ 2 k R_ 2 R ww_ þ e ; ¼ ð3:6Þ R R2 3ð1 kw2 =6Þ 2 " # € R_ 2 w_ 2 2R k R_ _ € þ 3P ; ¼ 2 ww ww þ R R2 R 3ð1 kw2 =6Þ 2 Fig. 1. Scalar field, scale factor and the deceleration parameter as functions of the time for the value of the JBD theory parameter f = 2.
the decelerated expansion of the Universe is only possible. Introducing the cosmological scalar in the case of nonminimally coupled scalar field, the Universe with transition from the phase with the decelerated expansion to the accelerated expansion phase is realized. (c) In the case of minimally coupled scalar field (Avagyan et al., 2005) the situation is basically the same. The presence of scalar field promotes the decelerated expansion of the Universe, and only the introduction of a positive cosmological constant can lead to the transition from the decelerated phase to the accelerated one, which – in the absence of K – can only take place at negative pressures. 3. Cosmological problem with a conformally coupled scalar field In the case of a conformally coupled scalar field with the quartic potential, the action has the following form (Bronshtein and Semendjaev, 1980; Stanukovich and Melnikov, 1983; Zajtsev and Melnikov, 1979): Z pffiffiffiffiffiffiffi 4 1 w2 1 k 4 2 R þ gd x: ðrwÞ w þ L W ¼ m 2k 12 2 12 ð3:1Þ As a result of the variation over gik and w we obtain the following field equations: 1 h i 1 kw2 slm þ T mat ð3:2Þ Rlm glm R ¼ k 1 lm ; 6 2 r2 w
R k w þ w3 ¼ 0; 6 3
ð3:3Þ
ð3:7Þ _2
€ R k R þ 2 ¼ ðe 3P Þ; 6 R R _ € _2 R € þ 3 w_ þ w R þ R þ k w3 ¼ 0; w 3 R R R2 3R_ e_ ¼ ðe þ P Þ; e ¼ e0 =Rn ; n ¼ 3ð1 þ aÞ; R
ð3:8Þ ð3:9Þ ð3:10Þ
where R(t) is the scale factor. In the system (3.6)–(3.10) one of the equations is a consequence of the others. From (3.8) for the deceleration parameter we obtain the following expression k ðe 3P Þ; ð3:11Þ 6H 2 _ where H ¼ R=R is the Hubble parameter, which allows to estimate the possibility of uniform or accelerated expansion of the Universe, because it follows from the condition q 0 that q¼1
e 2
3H 2 þ 3P : 8pG
ð3:12Þ
(Recall that within the framework of GR one has ecrit ¼ 3H 2 =ð8pGÞ.) For the values of a, typical in cosmology, the factor q is expressed as follows: 2ke ; vacuum model ða ¼ 1Þ; 3H 2 ke ; matter dominated ða ¼ 0Þ; q¼1 6H 2 q ¼ 1; radiation dominated ða ¼ 1=3Þ; q¼1
ð3:13Þ
q ¼ 1; scalar field dominated ðe ¼ 0; P ¼ 0Þ; ke ; stiff fluid ða ¼ 1Þ: q¼1þ 3H 2 Using (3.8), the Hubble parameter can be presented in the following form:
where 2 1 w w k slm ¼ rl wrm w glm ðrwÞ2 rl rm w þ glm r2 w þ w4 glm ; 3 6 3 3 12 T mat lm ¼ ðe þ P ÞU l U m Pg lm ;
ð3:4Þ
H2 ¼
ke0 b þ ; 3R3ð1þaÞ R4
with an integration constant b. Let us consider special cases separately.
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(a) If the scalar field dominates (e = 0, P = 0) or in the case of the equation of state P = e/3, the scalar pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi curvature goes to zero, and as a result R ¼ 2ct þ b; H ¼ c=R2 ; w ¼ c2 c1 =ðcRÞ with c2 ð3 c22 k=2Þ ¼ 0 ) c22 ¼ 6=k; pffiffiffiffiffiffiffiffi w ¼ 6=k c1 =cR;
ð3:14Þ
pffiffiffiffiffiffiffiffi i.e. for sufficiently large R we have w ! 6=k . (b) If the radiation dominates (a = 1/3), in the presence of the scalar field, the analytic forms for the functions R, w, H remain the same, and condition (3.14) becomes ð3:15Þ c2 ð3 c22 k=2Þ ¼ ke0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi so now c2 ¼ 6=k 2e0 =c2 . In both cases (a) and (b) the expansion of the Universe is always decelerated and at late stages the scalar field tends to constant value. (c) The matter dominated era corresponds to P = 0, e = e0/R3, therefore for the Hubble parameter we obtain ke b ke0 b H2 ¼ þ 4 ¼ 3 þ 4 3 R R 3R
ð3:16Þ
and, correspondingly,
where v is related to the scale factor by formulae (3.18) and (3.19). For both cases of the sign for b one has w ? 0 at late stages of the expansion, t ? 1. At early stages of the expansion the dynamics of the field is completely different for positive and negative values of the parameter b: for positive values we have w ? 1, t ? 0, whereas for negative values the field tends to zero, w ? 0, t ? 0. (d) For the equation of state P = e we have the set of equations € R_ 2 2ke R ; þ ¼ 3 R R2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ke0 b 2ke0 € _ w þ 3w þ 4þ w ¼ 0; 3 3 R
ð3:17Þ
At late stages of the cosmological expansion (R ? 1) the first term on the right hand side of (3.16) dominates and we have the standard matter dominated evolution with the decelerated expansion. In this limit one has q ? 1/2 (as in GR). The Eq. (3.16) is explicitly integrated. In the case b > o the solution is written in the from pffiffiffiffiffiffiffiffiffiffiffi 3b v; t=t2 ¼ ðv 2Þ v þ 1 þ 2; R¼ ke0 0 v 1; ð3:18Þ 3=2
2
where we have introduced the notation t2 ¼ 6jbj =ðke0 Þ . At early and late stages of the cosmological expansion one has RðtÞ / t1=2 , t?0, and RðtÞ / t2=3 ; t ! 1, respectively. In this case the expansion is always decelerated. For b < 0, we have the following solution pffiffiffiffiffiffiffiffiffiffiffi 3jbj v; t=t2 ¼ ðv þ 2Þ v 11 v 1: ð3:19Þ R¼ ke0 Now the late-time evolution is similar to that for the previous case, whereas the dynamics at early stages is completely different. The expansion is started at finite value of the scale factor Rmin ¼ 3jbj=ðke0 Þ with the asymptotic behavior 3 R Rmin þ ðke0 Þ t2 =ð4b2 Þ, t ? 0. At early stages the expansion is accelerated and the requirement q 6 0 is reduced to the inequality 1 6 m 6 2. The time evolution of the scalar field w is simply found in the case k = 0 when the corresponding equation is written in the form ð3:20Þ
ð3:22Þ ð3:23Þ
By the substitution m = R2, (3.23) is reduced to v_ 2
ke0 R=6 : q¼1 ke0 R=3 þ b
€ þ U_ R=R _ U ¼ 0;
with the notation U = Rw. By taking into account (3.16), from here for the scalar field we find ( pffiffiffiffiffiffiffiffiffiffiffi v þ 1=v; 0 v < 1; for b > 0; w ¼ const pffiffiffiffiffiffiffiffiffiffiffi ð3:21Þ v 1=v; 1 v < 1; for b < 0;
4ke 2 b v ¼ : 3 4
ð3:24Þ
The solution of this equation is presented in the form: pffiffiffiffiffiffiffiffiffi ð3:25Þ RðtÞ ¼ R0 sinh1=2 s; s ¼ 2t ke=3; for b > 0 and RðtÞ ¼ R0 cosh1=2 s;
ð3:26Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for b < 0, where 4R20 ¼ 3jbj=ðkeÞ. Accordingly, (3.23) is reduced to the equation 3 1 w00 ðsÞ þ f ðsÞw0 ðsÞ þ w ¼ 0; 2 2
ð3:27Þ
where f = tanh s for b < 0 and f = cosh s for b > 0. The solution of Eq. (3.27) is expressed in terms of the hypergeometric function. For large values of the scale factor corresponding late stages of the cosmological expansion, t ? 1, one has R / es=2 , the Eq. (3.27) is reduced to rffiffiffiffiffi € þ 3 kew_ þ 2ke w ¼ 0; w 3 3 and as a result for w one finds w c1 es þ c2 es=2 ; with constants c1, c2. In this limit the scalar field vanishes one has q 1. 4. Conclusion The scalar fields play an important role in theoretical models describing the expansion of the Universe. In the present paper, we have investigated the cosmological dynamics for two classes of models with scalar fields. In the first model the gravity is described by the JBD theory with the cosmological scalar. The corresponding action
E.V. Chubaryan et al. / Advances in Space Research 44 (2009) 1359–1365
has the form (1.1) and we have considered a special case of the function /ðyÞKy=y 0 . With this choice the gravitational part of the JBD action in the Einstein frame coincides with the GR action with the cosmological constant. We have shown that when the kinetic term of the scalar field dominates the evolution, the cosmological expansion in the Jordan frame is always decelerated. When the cosmological scalar is present in the case of nonminimally coupled scalar field, the Universe with transition from the phase with the decelerated expansion to the accelerated expansion phase is realized. At late stages we have an exponential expansion and the scalar field tends to constant value. This behavior is clearly seen on the example presented in Fig. 1. In the case of minimally coupled scalar field the situation is basically the same. The action for the second model is given by expression (2.1) which describes a conformally coupled scalar field with quartic potential. As an additional gravitational source we have considered matter with the barotropic equation of state (see (2.10)). The conditions are specified under which the cosmological expansion contains an accelerated phase. In particular, we have shown that if the scalar field dominates or in the case of the radiation type matter the expansion is always decelerated. In the case of the dust matter two classes of models are realized. In the first one, corresponding to positive values of the integration constant b in (3.16), the expansion is always decelerated with standard GR solution as late-time attractor. The second class of models is realized for negative values of b. In this case the expansion of the Universe starts from a finite value of the scale factor and at early stages we have an accelerated expansion with the subsequent transition to the decelerated phase with scalar field testing to zero at late stages. When the additional source in the cosmological equations is described by the cosmological constant, the early-time evolution essentially depends on the sign of the integration constant b in the right hand side of (3.24). The corresponding scale factor is given by formulae (3.25) and (3.26) and has minimum non-zero value for negative b. At late stages of the cosmological expansion the scalar field vanishes exponentially and we have an exponential expansion driven by the cosmological constant. References Aman, E.G., Markov, M.A. Oscillating Universe in the case p – 0. TMF 58 (2), 163–168, 1984. Avagyan, R.M., Harutunyan, G.G. The cosmological scalar in Jordan– Brans–Dicke theory. II. Astrophysica 48, 633–639, 2005. Avagyan, R.M., Harutunyan, G.G. Role of a scalar field in the radiation dominates epoch of the Universe evolution. Astrophysica 51, 151–159, 2008.
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Avagyan, R.M., Harutunyan, G.G., Papoyan, V.V. The cosmological scalar in Jordan–Brans–Dicke theory. I. Astrophysica 48, 455–462, 2005. Avagyan, R.M., Harutyunyan, G.G., Chubaryan, E.V. Cosmological models in the presence of conform-connected scalar field. Astrophysica 51, 653–661, 2008. Bekenshtein, J.D. Exact solutions of Einstein-conformal scalar equations. Ann. Phys. 82, 535–547 (NY), 1974. Brans, C., Dicke, C.H. Mach’s principle and a relativistic theory of gravitation. Phys. Rev. 124, 925–935, 1961. Bronshtein, I.N., Semendjaev, K.A. Handbook of Mathematics, Moscow, 1980, p. 220. Caldwell, R.R., Steinhardt, P.J. Imprint of gravitational waves in models dominated by a dynamical cosmic scalar field. Phys. Rev. D 57, 6057– 6064, 1998. Chernin, A.D. Dark energy and universal antigravitation. Phys. Usp. 51, 253, 2008. Copeland, E.J., Sami, M., Tsujikawa, S. Dynamics of dark energy. Int. J. Mod. Phys. D15, 1753–1936, 2006. Fujii, Y., Maeda, K. The Scalar Tensor Theory of Gravitation. Cambridge University Press, 2003. Garnavich, R.M., Jha, S., Challis, P., Clocchiatti, A., et al. Supernova limits on the cosmic equation of state. Astrophys. J. 509, 74–79, 1998. Garriga, J., Mukhanov, V.F. Perturbations in k-inflation. Phys. Lett. B458, 219–225, 1999. Gershtein, S.S., Logunov, A.A., Mestvirishvili, M.A., Tkachenko, N.P. Graviton mass, quintessence and oscilliatory character of the Universe evolution, Preprint IHEP 2003-13, 2003. Guth, A.H. The inflationary Universe: a possible solution to the horizon and flatness problems. Phys. Rev. D23, 347–356, 1981. Jordan, P. Fu¨nfdimensionale Kosmologie. Astron. Nachr. 276, 193, 1948. Jordan, P. Schwerkraft und Weltall, Braunschweig, 1951. Linde, A.D. A new inflationary Universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Phys. Lett. B108, 389–393, 1982. Logunov, A.A. Theory of Gravitational Field. Nauka, Moscow, 2001. Ludwig, G. Fortschritte der projektivan Relativitats teorie“, Braunschweig, 1951. Panagia, N. High redshift supernovae: cosmological implications. Nuovo Cim. B120, 667–676, 2005. Pauli, W. The formalisation of natural law with five homogeneous coordinations. Ann. Phys. 18, 305–336, 1933. Perlmutter, S., Aldering, G., Goldhaber, G., Knopet, R.A., et al. Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565–586, 1999. Scherrer, R.J. Purely kinetic k-essence as unified dark matter. Phys. Rev. Lett. 93, 011301, 2004. Stanukovich, K.P., Melnikov, V.N. Field Hydrodynamics and Constants in Gravitation Theory. Moscow, Energoatomizdat, 1983. Starobinski, A.A. Dynamics of phase transition in the new inflationary Universe scenario and generation of perturbations. Phys. Lett. B117, 175–178, 1982. Tolmen, R.. Relativity, Thermodynamics and Cosmology. Clarendon Press, Oxford, 1949. Weinberg, S. Gravitation and Cosmology. John Wiley and Sons, New York, 1972. Weinberg, S. The cosmological constant problem. Rev. Mod. Phys. 61, 1– 23, 1989. Zajtsev, N.A., Melnikov, V.N. Problems of Gravitation an Elementary Particle Theory, Moscow, Atomizdat No. 10, 1979, p. 131.
Advances in Space Research 44 (2009) 1359–1365 www.elsevier.com/locate/asr
Role of scalar fields in cosmological models E.V. Chubaryan *, R.M. Avakyan, G.G. Harutyunyan, A.S. Piloyan Department of Physics, Yerevan State University, 1 Alex Manoogian Street, 0025 Yerevan, Armenia Received 9 December 2008; received in revised form 2 April 2009; accepted 23 July 2009
Abstract Recent astronomical observations of supernovae and cosmic microwave background indicate that the universe is accelerating. Scalar– tensor theories of gravity give rise to suitable cosmological models where a late-time accelerated expansion is naturally realized. In an alternative proposal the cosmic acceleration is generated by means of a scalar field (quintessence), in a way similar to the early-time inflation. In this paper, we consider two classes of cosmological models with scalar fields. The first one corresponds to the Jordan–Brans– Dicke tensor–scalar theory with a cosmological scalar and the second one contains a conformally coupled scalar field with quartic potential. In both type of models the cosmological dynamics is described and the deceleration parameter is evaluated. The values of the parameters are specified for which a late-time accelerated expansion is realized. Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Cosmology; Scalar field; Dark energy
1. Introduction Recent discovery of accelerated expansion of the Universe (Garnavich et al., 1998; Perlmutter et al., 1999; Copeland et al., 2006) stimulated the revision of the established concept of its structure and evolution character. The cosmological constant is the most relevant candidate to interpret the cosmic acceleration (for the problem of cosmological constant see, Weinberg, 1989). This is equivalent to the consideration of the non-zero vacuum energy and the related negative pressure (Guth, 1981; Linde, 1982; Starobinski, 1982). An alternative explanation of the observed acceleration is the hypothesis for the existence of a special substance in the Universe, referred as quintessence (Caldwell and Steinhardt, 1998), for which the equation of state is Pq = aeq, 1 < a < 1/3. In the relativistic theory of gravitation (RTG) (Logunov, 2001), proposed by Logunov and collaborators, the presence of quintessence leads to interesting consequences: the acceleration of the Universe is replaced by deceleration with the subse*
Corresponding author. Tel.: +374 010 570 370. E-mail address: [email protected] (E.V. Chubaryan).
quent stop, after which the contraction to a minimum value of the scale factor begins with subsequent new expansion cycle (Gershtein et al., 2003; Chernin, 2008). Note that the idea of oscillating character of the Universe evolution was suggested earlier, but mainly from philosophical reasons (Aman and Markov, 1984), because in the Friedmann closed model the oscillation mode is interfered by the entropy growth from one cycle to another and the transition through the cosmological singularity (Tolmen, 1949). Reasoned enough the introduction of the cosmological constant in general relativity (GR) has led to the idea of the introduction of a similar quantity in the tensor–scalar theory, assuming that the new field is scalar (not dynamical but controllable by the gravitational scalar of the Jordan– Brans–Dicke (JBD) theory Avagyan et al., 2005; Avagyan and Harutunyan, 2008, 2005). In curved spaces, the scalar fields complying with the general covariance requirements are described by the equation @w ð1:1Þ gab ra wb m2 c2 =h2 þ cR w ¼ 0; wb b ; @x where m is the mass of the field, R is the scalar curvature, c is a dimensionless constant in the absence of which the usual generalisation of the Klein–Gordon equation is
0273-1177/$36.00 Ó 2009 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2009.07.018
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obtained in the case of Riemannian space (c = 1/6 was proposed by Penrose for the case of a conformally invariant massless scalar field). It turned out that the JBD theory (Bekenshtein, 1974) can be translated by conformal transformation from the proper representation with a self-coordinated scalar field w, subject to the condition gab$awb = 0, to another representation described by GR equations with a source in the form of non-gravitational fields and conformally coupled massless scalar field. In the present paper, we investigate two classes of cosmological models with scalar fields. The organization of the paper is as follows. In the next section, the cosmological dynamics is considered within the framework of the Jordan–Brans–Dicke tensor–scalar theory with cosmological scalar. It is shown that if the scalar field dominates and the cosmological scalar is absent the expansion of the Universe is always decelerated. The presence of the cosmological scalar leads to the possibility of the late-time accelerated phase. In Section 3, we consider the cosmological model with a conformally coupled scalar field in the presence of the Higgs potential and non-gravitational matter with the barotropic equation of state. The models with accelerated expansion of the Universe are specified. Section 4 summarizes the main results of the paper. 2. Cosmological scalar in the Jordan–Brans–Dicke theory In this section, the cosmological problem in the proper representation of the JBD theory is considered in presence of nonminimally coupled scalar field. As it will be shown the introduction of the cosmological scalar provides a possibility for the transition from the decelerated expansion of the Universe to the accelerated one. It was noted in Avagyan et al. (2005), Garriga and Mukhanov (1999), Panagia (2005), Fujii and Maeda (2003), Weinberg (1972), Scherrer (2004) that the modern conceptions of the Universe give raise to the introduction of the cosmological constant in the GR, therefore it is worth to introduce a similar quantity in the JBD theory. Having assumed that the field corresponding to this quantity should be scalar but cannot be dynamical (its changes should be controlled by the gravitational scalar y = y(xl)), we introduce the cosmological scalar u = u(y) in the JBD theory action similar to the introduction of the cosmological constant in the GR action (Avagyan et al., 2005): Z ylyl 1 pffiffiffiffi c4 y R þ 2uðyÞ 1 2 þ Lm d 4 x: W¼ ð2:1Þ g 16p y c Here, f is a dimensionless coupling constant of the JBD theory. The presence of the cosmological scalar means that in addition to the kinetic term the potential one is also considered for the scalar field. Equating to zero the independent variations of the action (2.1) with respect of gab and y, we obtain the JBD theory equations with the cosmological scalar
ra y a ¼ Glm
m kT 2y @u þ uy ; 3 þ 2f 3 þ 2f @y
k¼
8p ; c4
ð2:2Þ
! m l 1 l k ml l T rm y l yy 1 y ya þ þ f m 2 dlm a 2 ¼ dm R ¼ T m dm 3 þ 2f y y y 2 y 2 0m 1 c dlm @u Tl ð1 þ 2fÞu þ 2y ð2:3Þ ¼ k @ m þ T lm A; þ 3 þ 2f y @y Rlm
where m
T lm ¼ P m dlm þ ðqm þ P m ÞU m U l ; c 1 1 ðrm y l dlm ra y a Þ T lm ¼ ð2:4Þ k y f 1 þ 2 y m y l dlm y a y a þ dlm yuðyÞ : y 2 pffiffiffiffiffi m ffi dð dgglmLm Þ is the energy–momentum tensor of Here T lm ¼ p2ffiffiffiffi g c
the matter, and T lm is caused by the presence of the scalar field y, Pm and qm are the pressure and the density of the matter (dust and radiation), considered as a perfect fluid, U a ¼ dxa =ds is four-dimensional velocity satisfying the condition U a U a ¼ 1. Conceivably, long range massless scalar field with potential y ¼ yðxl Þ (gravitational scalar) replaces the Newtonian gravitational constant G at each point, and according to Mach ideas (Brans and Dicke, 1961) it is a background created by the whole energy in the Universe. An essential feature of the JBD theory scalar field, which distinguishes it from the fields of other tensor– scalar theories (Kaluza–Klein theory, string theory), is that it does not interact directly with matter. This circumstance gives rise to consider the JBD scalar field in itself without matter and radiation, which could probably allow revealing its role in the formation of dark matter and dark energy. In connection with the aforesaid it is of interest to remind how the JBD theory was constructed. The idea of the existence of scalar field has very naturally arisen in the attempt to unify the gravitation and electromagnetism. It was found out that the five-dimensional formulation of the Einstein gravitation and Maxwell theory should be much easier than the four-dimensional one. The conclusion was made after the extended group-theoretical analysis according to which the property of the invariance of the Einstein–Maxwell theory in 5D is much more symmetric. It was noticed that the unified group of arbitrary fourdimensional coordinate transformations and gauge transformations of the electromagnetic field potential is isomorphic to the coordinate transformations group of five-dimensional Riemannian space, with respect to which the field equation of the projective theory (Jordan, 1951, 1948; Ludwig, 1951) as well as modifications of the unified theory proposed by Pauli (1933) are invariant. In this isomorphic group Xl are transformed as 5-vectors, which allows to construct an additional invariant y ¼ glm X l X m . To ensure that the GR equations reduced to 4D are
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equivalent to the Einstein–Maxwell field equations system, the requirement of constancy for the scalar y = 1 is necessary. So, directly and without speculations and additional hypotheses, as a result of physical analysis, an idea of the generalisation of the gravitation theory arose, in which in addition to the tensor and vector fields there is a scalar field with the potential y. One of physically substantial and fully developed versions is the JBD theory, in which the gravitational scalar does not interact directly with the matter. Its existence is only manifested by the impact on the particles motion. Now we turn to the investigation of the cosmological model in JBD theory caused by the presence of nonminimally coupled scalar field. It makes sense to omit the possible contributions from all kinds of the matter and determine the role of the scalar field, nonminimally coupled with the gravitation. We use spatially-flat Friedmann–Robertson–Walker metric, assuming the space-time homogeneity and isotropy: 2 dr2 2 2 2 2 2 2 þ r dh þ sin hdu ; ds ¼ dt R ðtÞ ð2:5Þ 1 kr2 y ¼ yðtÞ; where k = 0 stands for the spatially-flat case, R(t) is the scale factor. In the case of the flat model the cosmological equations in the Jordan proper frame take the following form: €y 3y_ R_ 2 @u ¼ uy ; ð2:6Þ þ yR 3 þ 2f @y y € R_ 2 _ y f y_ 2 €y 2R_ 2R þ u; þ 2¼ R R Ry 2 y2 y R_ 2 1 y_ 2 R_ y_ 3 2 ¼ f 2 3 þ u; 2 y Ry R
ð2:7Þ ð2:8Þ
where the dot denotes the time derivative. If we determine u from (2.8) and substitute it in (2.6) and (2.7), then (2.6) will exactly coincide with (2.7), that is for any choice of u we have two independent equations. The choice of u = Ky/y0, where y0 is the modern value of y and K is the Einstein theory cosmological constant, is caused by the fact that in this case under the conformal transformation g0ik ¼ ygik we obtain the Einstein theory with cosmological constant (Einstein frame of the JBD theory). With this form of the function u(y), the integration of the substraction of Eqs. (2.7) and (2.8), 2 € €y y_ 2 y_ 2 R_ y_ R R_ þ þ ðf þ 1Þ 2 ¼ 0; ð2:9Þ 2 2 y y y R R Ry by taking into account the Eq. (2.6), €y R_ y_ ¼ 3 ; Ry y allows one to obtain the following result: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y_ ¼ c y ðrþ1Þ=2 þ cþ y ðr1Þ=2 ; r ¼ 3ð3 þ 2fÞ: y
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After the substitution of (2.11) into (2.8) and taking into account (2.10), from the equation 2 2 €y 1 €y f y_ Ky=y 0 ¼ 0 3 y 2 y y we obtain the following relation between the constant c cþ ¼
K ; ð3 þ 2fÞy 0
ð2:12Þ
which allows to conclude that for K = 0 in (2.11) either c or c+ is zero, and for K – 0 the constants c and c+ have opposite signs if K > 0 and f > 3/2. We consider two separate cases. (a) First we assume that K = 0. From Eq. (2.8) we _ obtain the following relation between the functions R=R and y_ =y: ð3 rÞ y_ R_ : ¼ 6 y R
ð2:13Þ
From (2.10) it follows that the plus sign corresponds to y_ ¼ jc jy ðrþ3Þ=2 and the minus sign corresponds to the case y_ ¼ jcþ jy ð3rÞ=2 As a result, for the scale factor we have ðr3Þ=6 y ; R ¼ R0 0 y 1 2=ðr1Þ y0 r 1 ðr1Þ=2 y0 ¼ 1 þ jc j ðt t0 Þ ; ð2:14Þ y 2 where t t0, R = R0, y = y0 correspond to the present moment. It is convenient to present the deceleration parameter € R_ 2 in the following form: q ¼ RR= q¼
€ R00 R R R ¼ 3 2 R_ 2 R0
ð2:15Þ
where the prime means the derivative with respect to y. Then, from (2.14) we have 2r : r3
q ¼
ð2:16Þ
Based on (2.4), it is possible to introduce by analogy the concepts of the pressure and the energy density of the scalar field, T lym ¼ P y dlm þ ðqy þ P y Þum ul , in accordance with the relations 1 ðf þ r 3Þc2 y ð3rÞ ; 2k 1 r 3 2 ð3rÞ Py ¼ ; fþ c y 2k 3
q y ¼
ð2:17Þ ð2:18Þ
ð2:10Þ
þ þ from where we find 1=3 < P y =qy < 1; 1 < P y =qy < 1 at 0 < f < 1 and, accordingly, the effective velocity of the elementary excitations (sound) arising in such medium can be defined as (Garriga and Mukhanov, 1999)
ð2:11Þ
v2 ¼
dP 2ðr 3Þ 4 y : ¼1 dq 3f 2ðr 3Þ y
ð2:19Þ
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It follows from (2.16) that q = 0 is only possible in the case f = 3/2, however this value of the JBD theory dimensionless parameter is initially excluded from the consideration at formulation of the theory (Ludwig, 1951), because in this special case as a result of the variation of the action ten equations are obtained for 11 unknown quantities. It is interesting to note that for f = 3/2 one has P y ¼ qy =3 and c2 < 0. Negative pressure can be reasonably interpreted, however the imaginary sound speed in liquid is rather doubtful. Thus, under the condition f > 3/2 and in the presence of only scalar field in JBD theory, we conclude that q+ can only accept positive values, which corresponds to the deceleration of the expansion of the Universe, and q is positive for f < 0 and can be negative for 3/2 < f < 0; however lim , which confirms once f!0 again the advisability of the assumption that f is positive. (b) Now we consider the case K – 0. As it was already _ from (2.11) in the case of noted, in the expression for y=y K > 0, the constants c and c+ are distinct from zero and they have different signs. Their signs coincide in the case K<0 From (2.10) and (2.11) we have R0 1 c ðr þ 3Þy r=2 þ cþ ð3 rÞy r=2 ¼ 6y R c y r=2 þ cþ y r=2
ð2:20Þ
Taking into account (2.15), we find q¼2
6 ½c ðr þ
3Þy r=2
2
þ cþ ð3 rÞy r=2
½c2 ðr þ 3Þy r þ c2þ ð3 rÞy r þ c cþ ð6 2r2 Þ ð2:21Þ If one of the constants of integration (c or c+) is zero, the value of (2.21) coincides with that for (2.16). In the presence of K, in addition of positive values for q, it is possible to obtain q = 0 and q < 0. The zero acceleration of the expanding Universe corresponds to a certain value of y1 satisfying the equation c2 ðr
þ
3Þy 2r 1
þ
4rc cþ y r1
þ
c2þ ðr
3Þ ¼ 0;
from where one has pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cþ 2r 3r2 þ 9 r y1 ¼ c rþ3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Kð 1 þ 2f=3 1 þ f=2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ y 0 c2 ð 1 þ 2f=3 þ 1Þð3 þ 2fÞ
1=r
ð2:22Þ
It is easy to see from the expression €y ¼ c
3 þ r ð1þrÞ=2 r 3 ð1rÞ=2 cþ ; y y 2 2
at the moment of the zero acceleration can be either larger or smaller than the limiting value yn. The system of Eqs. (2.10) and (2.11) can be explicitly solved. The corresponding solution is presented in the permanent form rþ1 rþ1 ðrþ1Þ=2r ;1 þ ;u ; t=t1 ¼ u 2 F 1 1; 2r 2r ð2:24Þ ðrþ3Þ=2r 1=3 u 1=r ; y=y n ¼ u R=R1 ¼ 1u parameter 0 u 1. In this formulae, is the hypergeometric function and we have ðr1Þ=2r =ðcþ ðr þ 1ÞÞ. In the used the notation t1 ¼ 2jcþ =c j limit u ? 0 one has t=t1 uðrþ1Þ=2r and we have the following asymptotic behavior with
the
2 F 1 ða; b; c; xÞ
y / t2=ðrþ1Þ ;
R / tðrþ3Þ=3ðrþ1Þ ;
t2 ¼ jcþ =c j
q 2r=ðr þ 3Þ; t ! 0;
ð2:23Þ
ðr1Þ=2r
=ðcþ rÞ;
and the asymptotics for the scalar field and scale factor have the form y y n ½1 þ ð1=rÞet=t2 ;
y_ ! 0 ) €y < 0;
:
from where it follows that in the limit y_ ! 0, one has € ! 1, which corresponds to the accelerated expansion Rþ of the Universe. Thus, the scalar field value pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=r 2r 3r2 þ 9 y1 ¼ yn rþ3
t t2 lnð1 uÞ;
and from €y ¼ 0, taking into account that the signs of c and c+ are different, we obtain the limiting value for y: y n ¼ jcþ =c j
€ 3_y ½jc jðr þ 3Þðr þ 1Þy r=2 jcþ jðr 3Þðr 1Þy r=2 RR ¼ 1 þ ; R_ 2 y 3=2 ½jc jðr þ 3Þy r=2 þ jcþ jðr 3Þy r=2 2
for the scalar field, scale factor and the deceleration parameter. This limit corresponds to early stages of the cosmological expansion. In the limit u?1 one has
It follows from the expansion of the Universe that R ¼ cy_ 1=3 ! 1;
that €y < 0 for c < 0 and cþ > 0, in this case r y_ ¼ jc jy ð3þrÞ=2 ðjy n =yj 1Þ For y(t0) > yn one has y_ < 0, and y decreases asymptotically tending to yn. However if y(t0) < yn, then y_ > 0 and y t ? 1 ffione asymptotically increases to yn. In both cases at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi has R ? 1, y ¼ y n þ Aeðtt0 Þ=s , where s ¼ 2=ðr jc cþ jy 0 cÞ is the characteristic time, c is the light velocity, and the sign _ of the constant of integration A depends on the sign of y. As regards the deceleration parameter q, from (2.22) it can be presented in the following form:
R R1 et=3t2
q 1; t ! 1
This limit corresponds to late stage of the cosmological expansion. Hence, at late stages the scalar field tends to the constant value yn and we have an exponential expansion 1 with the Hubble constant H ¼ ð3t2 Þ . In Fig. 1, we have plotted the time dependence of the scalar field, the scale factor and the deceleration parameter described by formulae (2.24). From the graphs, the transition from the decelerated expansion to the accelerated phase is clearly seen. Summarizing the analysis given above we conclude that if the nonminimally coupled scalar field dominates, then
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In this case, the contraction of the Eq. (3.2), by taking into account (3.3), gives R ¼ kT ¼ kðe 3P Þ
ð3:5Þ
It is commonly assumed that the geometry with the Friedmann–Robertson–Walker metric is adequate for the model of homogeneous and isotropic Universe. Accordingly, for the flat model with the matter having the equation of state P = ae the above equations become " # _ w_ 2 k R_ 2 R ww_ þ e ; ¼ ð3:6Þ R R2 3ð1 kw2 =6Þ 2 " # € R_ 2 w_ 2 2R k R_ _ € þ 3P ; ¼ 2 ww ww þ R R2 R 3ð1 kw2 =6Þ 2 Fig. 1. Scalar field, scale factor and the deceleration parameter as functions of the time for the value of the JBD theory parameter f = 2.
the decelerated expansion of the Universe is only possible. Introducing the cosmological scalar in the case of nonminimally coupled scalar field, the Universe with transition from the phase with the decelerated expansion to the accelerated expansion phase is realized. (c) In the case of minimally coupled scalar field (Avagyan et al., 2005) the situation is basically the same. The presence of scalar field promotes the decelerated expansion of the Universe, and only the introduction of a positive cosmological constant can lead to the transition from the decelerated phase to the accelerated one, which – in the absence of K – can only take place at negative pressures. 3. Cosmological problem with a conformally coupled scalar field In the case of a conformally coupled scalar field with the quartic potential, the action has the following form (Bronshtein and Semendjaev, 1980; Stanukovich and Melnikov, 1983; Zajtsev and Melnikov, 1979): Z pffiffiffiffiffiffiffi 4 1 w2 1 k 4 2 R þ gd x: ðrwÞ w þ L W ¼ m 2k 12 2 12 ð3:1Þ As a result of the variation over gik and w we obtain the following field equations: 1 h i 1 kw2 slm þ T mat ð3:2Þ Rlm glm R ¼ k 1 lm ; 6 2 r2 w
R k w þ w3 ¼ 0; 6 3
ð3:3Þ
ð3:7Þ _2
€ R k R þ 2 ¼ ðe 3P Þ; 6 R R _ € _2 R € þ 3 w_ þ w R þ R þ k w3 ¼ 0; w 3 R R R2 3R_ e_ ¼ ðe þ P Þ; e ¼ e0 =Rn ; n ¼ 3ð1 þ aÞ; R
ð3:8Þ ð3:9Þ ð3:10Þ
where R(t) is the scale factor. In the system (3.6)–(3.10) one of the equations is a consequence of the others. From (3.8) for the deceleration parameter we obtain the following expression k ðe 3P Þ; ð3:11Þ 6H 2 _ where H ¼ R=R is the Hubble parameter, which allows to estimate the possibility of uniform or accelerated expansion of the Universe, because it follows from the condition q 0 that q¼1
e 2
3H 2 þ 3P : 8pG
ð3:12Þ
(Recall that within the framework of GR one has ecrit ¼ 3H 2 =ð8pGÞ.) For the values of a, typical in cosmology, the factor q is expressed as follows: 2ke ; vacuum model ða ¼ 1Þ; 3H 2 ke ; matter dominated ða ¼ 0Þ; q¼1 6H 2 q ¼ 1; radiation dominated ða ¼ 1=3Þ; q¼1
ð3:13Þ
q ¼ 1; scalar field dominated ðe ¼ 0; P ¼ 0Þ; ke ; stiff fluid ða ¼ 1Þ: q¼1þ 3H 2 Using (3.8), the Hubble parameter can be presented in the following form:
where 2 1 w w k slm ¼ rl wrm w glm ðrwÞ2 rl rm w þ glm r2 w þ w4 glm ; 3 6 3 3 12 T mat lm ¼ ðe þ P ÞU l U m Pg lm ;
ð3:4Þ
H2 ¼
ke0 b þ ; 3R3ð1þaÞ R4
with an integration constant b. Let us consider special cases separately.
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(a) If the scalar field dominates (e = 0, P = 0) or in the case of the equation of state P = e/3, the scalar pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi curvature goes to zero, and as a result R ¼ 2ct þ b; H ¼ c=R2 ; w ¼ c2 c1 =ðcRÞ with c2 ð3 c22 k=2Þ ¼ 0 ) c22 ¼ 6=k; pffiffiffiffiffiffiffiffi w ¼ 6=k c1 =cR;
ð3:14Þ
pffiffiffiffiffiffiffiffi i.e. for sufficiently large R we have w ! 6=k . (b) If the radiation dominates (a = 1/3), in the presence of the scalar field, the analytic forms for the functions R, w, H remain the same, and condition (3.14) becomes ð3:15Þ c2 ð3 c22 k=2Þ ¼ ke0 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi so now c2 ¼ 6=k 2e0 =c2 . In both cases (a) and (b) the expansion of the Universe is always decelerated and at late stages the scalar field tends to constant value. (c) The matter dominated era corresponds to P = 0, e = e0/R3, therefore for the Hubble parameter we obtain ke b ke0 b H2 ¼ þ 4 ¼ 3 þ 4 3 R R 3R
ð3:16Þ
and, correspondingly,
where v is related to the scale factor by formulae (3.18) and (3.19). For both cases of the sign for b one has w ? 0 at late stages of the expansion, t ? 1. At early stages of the expansion the dynamics of the field is completely different for positive and negative values of the parameter b: for positive values we have w ? 1, t ? 0, whereas for negative values the field tends to zero, w ? 0, t ? 0. (d) For the equation of state P = e we have the set of equations € R_ 2 2ke R ; þ ¼ 3 R R2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ke0 b 2ke0 € _ w þ 3w þ 4þ w ¼ 0; 3 3 R
ð3:17Þ
At late stages of the cosmological expansion (R ? 1) the first term on the right hand side of (3.16) dominates and we have the standard matter dominated evolution with the decelerated expansion. In this limit one has q ? 1/2 (as in GR). The Eq. (3.16) is explicitly integrated. In the case b > o the solution is written in the from pffiffiffiffiffiffiffiffiffiffiffi 3b v; t=t2 ¼ ðv 2Þ v þ 1 þ 2; R¼ ke0 0 v 1; ð3:18Þ 3=2
2
where we have introduced the notation t2 ¼ 6jbj =ðke0 Þ . At early and late stages of the cosmological expansion one has RðtÞ / t1=2 , t?0, and RðtÞ / t2=3 ; t ! 1, respectively. In this case the expansion is always decelerated. For b < 0, we have the following solution pffiffiffiffiffiffiffiffiffiffiffi 3jbj v; t=t2 ¼ ðv þ 2Þ v 11 v 1: ð3:19Þ R¼ ke0 Now the late-time evolution is similar to that for the previous case, whereas the dynamics at early stages is completely different. The expansion is started at finite value of the scale factor Rmin ¼ 3jbj=ðke0 Þ with the asymptotic behavior 3 R Rmin þ ðke0 Þ t2 =ð4b2 Þ, t ? 0. At early stages the expansion is accelerated and the requirement q 6 0 is reduced to the inequality 1 6 m 6 2. The time evolution of the scalar field w is simply found in the case k = 0 when the corresponding equation is written in the form ð3:20Þ
ð3:22Þ ð3:23Þ
By the substitution m = R2, (3.23) is reduced to v_ 2
ke0 R=6 : q¼1 ke0 R=3 þ b
€ þ U_ R=R _ U ¼ 0;
with the notation U = Rw. By taking into account (3.16), from here for the scalar field we find ( pffiffiffiffiffiffiffiffiffiffiffi v þ 1=v; 0 v < 1; for b > 0; w ¼ const pffiffiffiffiffiffiffiffiffiffiffi ð3:21Þ v 1=v; 1 v < 1; for b < 0;
4ke 2 b v ¼ : 3 4
ð3:24Þ
The solution of this equation is presented in the form: pffiffiffiffiffiffiffiffiffi ð3:25Þ RðtÞ ¼ R0 sinh1=2 s; s ¼ 2t ke=3; for b > 0 and RðtÞ ¼ R0 cosh1=2 s;
ð3:26Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for b < 0, where 4R20 ¼ 3jbj=ðkeÞ. Accordingly, (3.23) is reduced to the equation 3 1 w00 ðsÞ þ f ðsÞw0 ðsÞ þ w ¼ 0; 2 2
ð3:27Þ
where f = tanh s for b < 0 and f = cosh s for b > 0. The solution of Eq. (3.27) is expressed in terms of the hypergeometric function. For large values of the scale factor corresponding late stages of the cosmological expansion, t ? 1, one has R / es=2 , the Eq. (3.27) is reduced to rffiffiffiffiffi € þ 3 kew_ þ 2ke w ¼ 0; w 3 3 and as a result for w one finds w c1 es þ c2 es=2 ; with constants c1, c2. In this limit the scalar field vanishes one has q 1. 4. Conclusion The scalar fields play an important role in theoretical models describing the expansion of the Universe. In the present paper, we have investigated the cosmological dynamics for two classes of models with scalar fields. In the first model the gravity is described by the JBD theory with the cosmological scalar. The corresponding action
E.V. Chubaryan et al. / Advances in Space Research 44 (2009) 1359–1365
has the form (1.1) and we have considered a special case of the function /ðyÞKy=y 0 . With this choice the gravitational part of the JBD action in the Einstein frame coincides with the GR action with the cosmological constant. We have shown that when the kinetic term of the scalar field dominates the evolution, the cosmological expansion in the Jordan frame is always decelerated. When the cosmological scalar is present in the case of nonminimally coupled scalar field, the Universe with transition from the phase with the decelerated expansion to the accelerated expansion phase is realized. At late stages we have an exponential expansion and the scalar field tends to constant value. This behavior is clearly seen on the example presented in Fig. 1. In the case of minimally coupled scalar field the situation is basically the same. The action for the second model is given by expression (2.1) which describes a conformally coupled scalar field with quartic potential. As an additional gravitational source we have considered matter with the barotropic equation of state (see (2.10)). The conditions are specified under which the cosmological expansion contains an accelerated phase. In particular, we have shown that if the scalar field dominates or in the case of the radiation type matter the expansion is always decelerated. In the case of the dust matter two classes of models are realized. In the first one, corresponding to positive values of the integration constant b in (3.16), the expansion is always decelerated with standard GR solution as late-time attractor. The second class of models is realized for negative values of b. In this case the expansion of the Universe starts from a finite value of the scale factor and at early stages we have an accelerated expansion with the subsequent transition to the decelerated phase with scalar field testing to zero at late stages. When the additional source in the cosmological equations is described by the cosmological constant, the early-time evolution essentially depends on the sign of the integration constant b in the right hand side of (3.24). The corresponding scale factor is given by formulae (3.25) and (3.26) and has minimum non-zero value for negative b. At late stages of the cosmological expansion the scalar field vanishes exponentially and we have an exponential expansion driven by the cosmological constant. References Aman, E.G., Markov, M.A. Oscillating Universe in the case p – 0. TMF 58 (2), 163–168, 1984. Avagyan, R.M., Harutunyan, G.G. The cosmological scalar in Jordan– Brans–Dicke theory. II. Astrophysica 48, 633–639, 2005. Avagyan, R.M., Harutunyan, G.G. Role of a scalar field in the radiation dominates epoch of the Universe evolution. Astrophysica 51, 151–159, 2008.
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