CORE Metadata, citation and similar papers at core.ac.uk Provided by IACS Institutional Repository Indian J. Phys. 76B (6), 747-751 (2002) U P B — an international journal Higher dimensional string cosmology in Lyra geometry F Rahaman*^, S Chakraborty^, M Ho|sain» N Begum and J Bera Khodar Bazar. Baruipur-743 302, 24 I'argaiiis (South), West Bengal, India ^Department or Mathematics. Jadavpur tJnlrcrsily. Kolkata-700 032, India b-mail . Tarook [email protected] Received 16 May 2002, accepted S September 2002 Abstract . Some cosmological solutions Tor string model are derived in higher dimensional spherically symmetric space time based on Lyra’s geometry. The physical behaviour ot the models is also discussed Keywords ' String cosmology. Lyra geometry, higher dimension PACS Nos. : 98.80.Cq. 04.20.Jb, 04,50.+h 1. Introduction [4 {.These strings have stress energy and they couple to the In last few years, there are attempts to unify gravity with gravitational field so that it may be interesting to study the other fundamental forces in nature. Latest studies of super gravitational effects which arise from strings. Cosmic strings string and super gravity theories and the unification of as source of gravitational field in general relativity (OR) fundamental forces with gravity, reveal that the space time was discussed by many authors [5,6]. dimension should be different from four [1]. As a result, Since the discovery of general relativity theory by higher dimensional theory is receiving great attention both Einstein, there have been numerous modification of it. Lyra in Cosmology and in Particle Physics. It is argued that the [7] proposed a modification of Riemannian geometry by extra dimensions are observable at the present time owing introducing a gauge function into the structureless manifold to their size being assumed to be of the order of the Planck that bears a close resemblance to Weyl’s geometry. length, but they may perhaps be relevant for the very early Subsequent investigations were done by several authors [8] Universe [2]. The detection of time variation of fundamental in scalar tensor theoiy and cosmology within the frame constants may be a strong evidence for the existence of work of Lyra geometry. extra dimension [1,2]. But as far as our knowledge goes, there has not been The concept of string theory was developed to describe any work in literature where Lyra’s geometry has been events at the early stages of the evolution of the Universe. considered for study of string cosmology in higher dimensional space time. Therefore, it is interesting to study It is believed that strings may be one of the sources of string theory in higher dimension as both concepts are density perturbations that are required for formation of large important at the early stages of the Universe. scale structure in the Universe [3]. The existence of the large scale network of strings are well accustomed to the In the present paper, we shall study string cosmology in present day Universe. higher dimensional space time based on Lyra’s geometry in normal gauge i.e. displacement vector Moreover, the vacuum strings can, generate density fluctuations which explain the formation of the galaxy ,^,=(A /),o,o,o,o) (1) ’Conxsponding Author © 2002 lACS 748 F Rahaman, S Chakraborty, M Hossain, N Begum and J Bera 2. The basic equations + 2 ^ - - 0 \ 0 1 ) We consider a five dimensional space time with topology 4 of 4-space S*XS^ as The different equations of state for string mode! be [6] ds^ =r ^dt^ +a^dr^ ^b^dD \ , (2) (a) p = p{X) (barotropic equation of state), (b) - yi (geometric string), where a = a(t), b " h(t) and (c) (I + w)A (Takabayasi string Le. /7-string). dQ \ ^ d&\ -f sin^ 0\d0\ -f sin^ 0\ sin^ O^dOX In the following section, we shall determine the exact is the metric on unit 3-sphere. solutions of the field equations using above equations of The coordinate r is periodic and varies in the range of state for string model in Lyra geometry. [0, 2;r]. 3. Solutions The field equations in normal gauge for Lyra’s manifold as Case I. Barotropic equation o f state : obtained by Sen [8] are In this case, wc take displacement vector to be constant Le. 1 xTab^ (3) P ^ constant. h To solve the field equations, one notes that there are Here, (p^ is the displacement field vector defined in (1) and three field equations connecting 4-unknowns. So one more other symbols have their usual meanings as in Riemannian relations connecting these variables is needed. geometry. The energy momentum tensor for the string dust Here, we assume system is [6] a - pF ' (12) T'ab = (4) (/^ n are arbitrary constants) between the scale factors for where p is the rest energy density of the cloud of strings unique solutions of the field equations. with particles attached to them (/7-strings) and A is the string’s Using this relation, we get from eq. (8) tension density. They are related by the relation b‘^ C (13) p = pp+ A (5) b b with Pp being particle energy density. Va is the five velocity 1 for the cloud of particles and is the direction of anisotropy where ^ = " i ± ^ , fl = 1 - ^ ' , C n+2 4 «+2 n+2 /.e. the string’s direction and they satisfy [6] This equations has a first integral of the form = .^1 = -x^x^^ and V„x" = 0 in (-, -H, +, +) signature. p = ._ .^ _ £ + £ ) f t- 2 y (14) A + \ A If we use comoving coordinate systems (where D is an integration constant). Le. = (1, 0, 0, 0, 0) This differential equation can be written in the integral and x^ parallel to t9 i.c jc, (0, a \ 0, 0, 0), then the form non vanishing components of field eq. (3) for the metric (2) db are I = ± ( / - t o ) (15) 2b^ ^2ab ^ 3 3 = p (6) 'b^ ah />2 L A + \ a \ (to is another integration constant). (7) h b‘ The above integral can be solved only when A 1 t.e d 2 b « = 1. — H----- k L -L + 1 ^ 2 =0. (8) a b ab *2 4 ^ Hence, we obtain, The proper volume F*, expansion scalar 0 and shear scalar »2 = 1 [V c 2T2SD sin V2fl( / - to) - Cj. (16) c? are respectively given by V*=ab^, (9) The other parameters have the following expressions }b 0 = -+ - (10) a = ^ [ V c ^ 3 bD sin V2B(t-to)-C]2, (17) a b ' Higher dimensional string cosmology in Lyra geometry 749 -y/b^Af-b Al( . , V* =;/-^[Vc2+2BZ)sin>/2B{r-/o)-c]\ (18) („,)------3L------- + _^l s,n-i ^ 2y/2BC^ +4BD^ cosy/2B(t-to) (19) 2V2 [ V c ^ 3 i B sin V2B (ro - ro ) - c] ’ it-fo ) (27) \2 n ^ + 5n + 2 o* == 0, (20) (for >4 = -), ^ 3 i2BC^+4DB^)cos^y/2B(l-to) (2w^S« + 2)£) 2 [VC2+2B£» sin V lfi(/ - /q) - c]^ -42 = (28) T^e other parameters are : + 3J?[VC^ +25Z)sinV2B(r-fo)]”'-|> S 2 , (21) Fbr A ^--1 \ ;l = 2 b [ VC^ + 2 i9£) sin V ^ ( r - /o) - c ]‘‘, (22) a = fiB** cosh" ^fD(t - /o)» (29) (30) 3 i2BC^+4DB^)cos^y/2B(t-to) 2 [VC’2 + 2Z)fi sin y/lBQo -to )- c]^ 0-2 =^(«-l)Z>tanh2 Vo(/~/„), (31) . 4 iW4 3 + fi[Vc2l3BDsinV2fl(/o-/o)-C]"'---/?2. K'* =/i5"^3^cosh-/D(/-ro)]"‘\ (32) (22a) 3«</tanh^ -/Dil -to)-3D, (33) Case II. Geometric string (p - X) : 4 Here, we also assume the same relations between the metric coefficients Le. a = but the displacement vector is not p = —£)+3Dtanh^ ■/D (l-lo) constant. Using the above relation and after some calculations, we get + -^scch^y/D(t - to). (34) (23) For A - 1, the solution shows a contracting model, and is b b> (2«+l)/>2 ’ not of much physical interest. 2n^ +5n + 2 where >4 For A - we can not get explicit form of 6 in terms 2« + 1 of t and consequently ail physical parameters can not be Hence, b takes the following integral form determined in terms of t. Therefore, no physical conclusion db can be drawn from the solution. -y- = ± (/-/o ), (24) } Case III Takabayasi string (Le. p-String) : ---- ------------- 1^ L 2n^ Here, the equation of state p = A(l w) where w > 0, a where D and to are integration constants. constant and it is small for string dominant era and large for particle dominant era. From the above integral equation, b can be obtained in closed form only for Further, using the polynomial relation a ^ pb^ between metric coefficients, from the field equations, we get >4 = ~L 1 and — and we obtain 2 i A b ^ jB (35) (i) 6 = BcoshVO(r-/o). = (25) b b^ b^ ’ 2n^ ’^Sn + 2 (for y4A = -1)- i.e. n - -6, where A ~ (2 -f w)- n(2 + w) + 7 n -2 fy + 2nw+2 2n+2wn + l-w (ii) b^ = B^~ ^ (t-/o )^. 2«2 +5rt + 2 2 -2 w B = - £>(2n2 +Sn+2) 2n+2wn + l~ w (26) 2 For 0 < w < 1, ihe nature of the solutions is same as geometric (for A = 1), string. 750 F Rahaman, S Chakraborty, M Hossain, N Begum and J Bera For w = 1, eq. (35) transforms to n+3 (49) b 3w + 6 V» *0/ B where A (36) .(50) f t - I ] Hence, we get ' 1 4 b = b(^{t - ^ (37) 3B 3 B H t-to y where /o» ^oo are integration constants and j *0 =(*00(>1 + 1)) (5 1 ) The other physical parameters are [««-..).-I]' 4« a = (t -to) 3n+l6, (38) For A = + j .