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Applications of Elliptic and Theta Functions to Friedmann–Robertson–Lemaˆıtre–Walker Cosmology with Cosmological Constant

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Applications of Elliptic and Theta Functions to Friedmann–Robertson–Lemaˆıtre–Walker Cosmology with Cosmological Constant A Window Into Zeta and Modular Physics MSRI Publications Volume 57, 2010 Applications of elliptic and theta functions to Friedmann–Robertson–Lemaˆıtre–Walker cosmology with cosmological constant JENNIE D’AMBROISE 1. Introduction Elliptic functions are known to appear in many problems, applied and theo- retical. A less known application is in the study of exact solutions to Einstein’s gravitational field equations in a Friedmann–Robertson–Lemaˆıtre–Walker, or FRLW, cosmology [Abdalla and Correa-Borbonet 2002; Aurich and Steiner 2001; Aurich et al. 2004; Basarab-Horwath et al. 2004; Kharbediya 1976; Krani- otis and Whitehouse 2002]. We will show explicitly how Jacobi and Weierstrass elliptic functions arise in this context, and will additionally show connections with theta functions. In Section 2, we review the definitions of various elliptic functions. In Section 3, we record relations between elliptic functions and theta functions. In Section 4 we introduce the FRLW cosmological model and we then proceed to show how elliptic functions appear as solutions to Einstein’s gravitational equations in sections 5 and 6. The author thanks Floyd Williams for helpful discussions. 2. Elliptic functions An elliptic integral is one of the form R Rx; pP.x/ dx, where P.x/ is a polynomial in x of degree three or four and R is a rational function of its arguments. Such integrals are called elliptic since an integral of this kind arises in the computation of the arclength of an ellipse. Legendre showed that any el- liptic integral can be written in terms of the three fundamental or normal elliptic integrals x def: Z dt F.x; k/ p ; D 0 .1 t 2/.1 k2t 2/ 279 280 JENNIE D’AMBROISE s x 2 2 def: Z 1 k t E.x; k/ dt; (2.1) 2 D 0 1 t Z x 2 def: dt ˘.x; ˛ ; k/ p ; D 0 .1 ˛2t 2/ .1 t 2/.1 k2t 2/ which are referred to as normal elliptic integrals of the first, second and third kind, respectively. The parameter ˛ is any real number and k is referred to as the modulus. For many problems in which real quantities are desired, 0 < k; x < 1, although this is not required in the above definitions (for k 0 and k 1, D D the integral can be expressed in terms of elementary functions and is therefore pseudo-elliptic). Elliptic functions are inverse functions of elliptic integrals. They are known to be the simplest of nonelementary functions and have applications in the study of classical equations of motion of various systems in physics including the pendulum. One can easily show that if f .u/ denotes the inverse function of an elliptic integral y.x/ R Rx; pP.x/ dx, then since y.f .u// u, which D D implies y .f .u// 1=f .u/, we have 0 D 0 1 f .u/2 : (2.2) 0 2 D Rf .u/; pP.f .u// The Jacobi elliptic function sn.u; k/ is the inverse of F.x; k/ defined above, and eleven other Jacobi elliptic functions can be written in terms of sn.u; k/: cn.u; k/ and dn.u; k/ satisfy sn2 u cn2 u 1 and k2 sn2 u dn2 u 1, re- C D C D spectively, and def: 1 def: 1 def: 1 ns ; nc ; nd ; D sn D cn D dn def: sn def: sn def: cn sc ; sd ; cd ; (2.3) D cn D dn D dn def: 1 def: 1 def: 1 cs ; ds ; dc : D sc D sd D cd By (2.2) and (2.3), one can see that the Jacobi elliptic functions satisfy the differential equation f .u/2 af .u/2 bf .u/4 c (2.4) 0 C C D for a; b; c in terms of the modulus k according to the following table. APPLICATIONS OF ELLIPTIC AND THETA FUNCTIONS TO FRLW COSMOLOGY 281 f .u/ a b c sn.u; k/ 1 k2 k2 1 C cn.u; k/ 1 2k2 k2 1 k2 dn.u; k/ k2 2 1 k2 1 ns.u; k/ 1 k2 1 k2 C nc.u; k/ 1 2k2 k2 1 k2 nd.u; k/ k2 2 1 k2 1 sc.u; k/ k2 2 k2 1 1 sd.u; k/ 1 2k2 k2.1 k2/ 1 cd.u; k/ 1 k2 k2 1 C cs.u; k/ k2 2 1 1 k2 ds.u; k/ 1 2k2 1 k2.k2 1/ dc.u; k/ 1 k2 1 k2 C (The a-value for f .u/ dn.u; k/ seen above corrects an error in [Basarab- D Horwath et al. 2004].) The Weierstrass elliptic function 1 X 1 1 }.z ! ;! / 1 2 2 2 2 I D z C .z m!1 n!2/ .m!1 n!2/ .m;n/ Z Z .0;0/ C 2 f g (2.5) is a doubly periodic elliptic function of z C with periods !1;!2 C such that 2 2 Im.!1=!2/ > 0. The function } is the inverse of the elliptic integral Z 1 1 1 } .x g2; g3/ dt (2.6) p 3 I D x 4t g2t g3 where g2; g3 C are known as Weierstrass invariants. Given periods !1;!2, 2 the invariants are X 1 g 60 ; 2 4 D .m!1 n!2/ .m;n/ Z Z .0;0/ C 2 f g (2.7) X 1 g 140 : 3 6 D .m!1 n!2/ .m;n/ Z Z .0;0/ 2 f g C Alternately given invariants g2; g3, periods !1;!2 can be constructed if the def: 3 2 discriminant g2 27g3 is nonzero — that is, when the Weierstrass cubic 3 D 4t g2t g3 does not have repeated roots (see 21 73 of [Whittaker and Watson 1927]). Therefore we refer to }.z !1;!2/ as either }.z/ or }.z g2; g3/ and I I consider only cases where the invariants are such that g3 27g2. By (2.2) and 2 ¤ 3 282 JENNIE D’AMBROISE (2.6) one can see that the Weierstrass elliptic function satisfies 2 3 } .z/ 4}.z/ g2}.z/ g3: (2.8) 0 D Note that in Michael Tuite’s lecture in this volume, !m;n there is equal to m!1 C n!2 here with our !1;!2 specialized to 2i and 2i in his lecture. 3 In the special case that the discriminant > 0, the roots of 4t g2t g3 are real and distinct, and are conventionally notated by e1 >e2 >e3 for e1 e2 e3 3 C C D 0. In this case 4t g2t g3 4.t e1/.t e2/.t e3/, the Weierstrass invariants D are given in terms of the roots by g2 4.e2e3 e1e3 e1e2/; g3 4e1e2e3; (2.9) D C C D and } can be written in terms of the Jacobi elliptic functions by 2 2 }.z/ e3 ns . z; k/; D C 2 2 }.z/ e2 ds . z; k/; (2.10) D C 2 2 }.z/ e1 cs . z; k/ D C 2 def: 2 e2 e3 for e1 < z R, e1 e3 and modulus k such that k (similar 2 D D e1 e3 equations hold if z R and z is in a different range in relation to the real roots 2 e1; e2; e3, and alternate relations hold for nonreal roots when < 0; see chapter II of [Greenhill 1959]). Note that the Jacobi elliptic functions solve a differential equation which con- tains only even powers of f .u/, and } solves an equation with no squared or quartic powers of }. Weierstrass elliptic functions have the advantage of being 3 easily implemented in the case that the cubic 4t g2t g3 is not factored in terms of its roots. Elliptic integrals of the type R R.x/=pP.x/dx, where P.x/ is a cubic polynomial and R is a rational function of x, can be written in terms of three fundamental Weierstrassian normal elliptic integrals although we will not record the details here (see Appendix of [Byrd and Friedman 1954]). In Section 6 we will see a method which allows one to write the elliptic integral R x 1=pF.t/dt, for F.t/ a quartic polynomial, in terms of the Weierstrassian x0 normal elliptic integral of the first kind (2.6) by reducing the quartic to a cubic. 3. Jacobi theta functions Jacobi theta functions are functions of two arguments, z C a complex num- 2 ber and H in the upper-half plane. Every elliptic function can be written as 2 the ratio of two theta functions. Doing so elucidates the meromorphic nature of elliptic functions and is useful in the numerical evaluation of elliptic functions. One must be cautious with the notation of theta functions, since many different APPLICATIONS OF ELLIPTIC AND THETA FUNCTIONS TO FRLW COSMOLOGY 283 conventions are used. We will use the notation of [Whittaker and Watson 1927] to define def: 1=4 X1 n n.n 1/ Â1.z; / 2q . 1/ q sin..2n 1/z/; D C C n 0 D def: 1=4 X1 n.n 1/ Â2.z; / 2q q cos..2n 1/z/; D C C n 0 D (3.1) def: X1 n2 Â3.z; / 1 2 q cos.2nz/; D C n 1 D def: X1 n n2 Â4.z; / 1 2 . 1/ q cos.2nz/; D C n 1 D def: i def: where q e is called the nome. We also define the special values Âi D D Âi.0; /. In terms of theta functions, the Jacobi elliptic functions are 2 Â3 Â1.u=Â ; / sn.u; k/ 3 ; 2 D Â2 Â4.u=Â3 ; / 2 Â4 Â2.u=Â ; / cn.u; k/ 3 ; (3.2) 2 D Â2 Â4.u=Â3 ; / 2 Â4 Â3.u=Â ; / dn.u; k/ 3 ; 2 D Â3 Â4.u=Â3 ; / where is chosen such that k2 Â 4=Â 4.
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