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 elliptic 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/ ޚ ޚ .0;0/ C 2 f g (2.5) is a doubly periodic elliptic function of z ރ with periods !1;!2 ރ 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 ރ 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/ ޚ ޚ .0;0/ C 2 f g (2.7) X 1 g 140 : 3 6 D .m!1 n!2/ .m;n/ ޚ ޚ .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 ޒ, e1 e3 and modulus k such that k (similar 2 D D e1 e3 equations hold if z ޒ 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 ރ a complex num- 2 ber and ވ 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. By [Whittaker and Watson 1927, D 2 3 22 11], if 0 < k2 < 1 there exists a value of for which the quotient Â 4=Â 4 2 3 equals k2.

4. The FRLW cosmological model

The Friedmann–Robertson–Lemaˆıtre–Walker cosmological model assumes that our current expanding universe is on large scales homogeneous and iso- tropic. On a d-dimensional spacetime this assumption translates into a metric of the form Â dr 2 Ã ds2 dt 2 a.t/2 r 2d˝2 ; (4.1) D CQ 1 k r 2 C d 2 0 where a.t/ is the cosmic scale factor and k 1; 0; 1 is the curvature pa- Q 0 2 f g rameter. 284 JENNIE D’AMBROISE

The Einstein ﬁeld equations

Gij Ä Tij gij D d C then govern the evolution of the universe over time. In these equations, the Einstein tensor def: 1 Gij Rij Rgij D 2 is computed directly from the metric gij by calculating the Ricci tensor Rij and the scalar curvature R. Also Ä 8G , where G is a generalization of d D d d Newton’s constant to d-dimensional spacetime and > 0 is the cosmological constant. The form of the energy-momentum tensor Tij depends on what sort of matter content one is assuming, and in this lecture will be that of a perfect- ﬂuid — that is, Tij .p /gi0gj0 pgij , where .t/ and p.t/ are the density D C C and pressure of the ﬂuid, respectively. For the metric (4.1), Einstein’s equations are

.d 1/.d 2/ 2 k0 Á H Äd .t/ ; (i) 2 C a2 D C Q .d 1/.d 2/ 2 .d 2/.d 3/ k0 .d 2/H H Äd p.t/ ; (ii) P C 2 C 2 a2 D C def: Q for H.t/ a.t/=a.t/ and where dot denotes differentiation with respect to t. In D PQ Q this lecture, only equation (i) will be required to relate a.t/ to elliptic and theta Q functions. We rewrite equation (i) in terms of conformal time Á by deﬁning the def: conformal scale factor a.Á/ a.f .Á//, where f .Á/ is the inverse function of DQ Á.t/, which satisﬁes Á.t/ 1=a.t/. In terms of a.Á/, (i) becomes P D Q a .Á/2 a.Á/4 Ä .f .Á//a.Á/4 k a.Á/2; (4.2) 0 D z C zd 0 def: def: where we use notation 2=.d 1/.d 2/, Ä 2Ä =.d 1/.d 2/ and z D zd D d we take spacetime dimension d > 2.

5. FRLW and Jacobi elliptic and theta functions

In general, if f .u/ is a solution to f .u/2 af .u/2 bf .u/4 c, then 0 C C D g.u/ ˇf .˛u/ is a solution to D A2bc g .u/2 Ag.u/2 Bg.u/4 (5.1) 0 C C D a2B for ˛ pA=a and ˇ pAb=.aB/, where we may choose either the positive D D or negative square root for each of ˛ and ˇ. We will construct solutions to (4.2), given that Jacobi elliptic functions solve (2.4), and also proceed to write these solutions in terms of theta functions by the relations in (3.2). APPLICATIONS OF ELLIPTIC AND THETA FUNCTIONS TO FRLW COSMOLOGY 285

For the special case of density .t/ D=a.t/4 with D > 0, (4.2) becomes D Q a .Á/2 k a.Á/2 a.Á/4 Ä D: (5.2) 0 C 0 z D zd Therefore in (5.1) we take A k ; B , and a; b; c as in the table in Section D 0 D z 2. To restrict to real solutions we only consider entries in the table for which the ratios a=A and b=B are positive, and we also restrict D to be such that the right side of (5.2) agrees with the right side of (5.1). k2 For positive curvature k0 1, and for D with 0 < k < 1, D D Ä .1 k2/2 we solve zd z C 2 2 4 k2 a0.Á/ a.Á/ a.Á/ (5.3) C z D .1 k2/2 z C in conformal time in terms of Jacobi elliptic functions to obtain

q 4 4 Â1 Á= Â Â ; k Á Á Â2Â3 2 3 asn.Á/ sn ; k q q C ; 2 p 2 4 4 4 4 D p.1 k / 1 k D .1 Â Â / Â4 Á= Â Â ; z C C z C 2 3 2 C 3 q 4 4 Â4 Á= Â Â ; 1 Á Á Â2Â3 2 3 ans.Á/ ns ; k q q C ; 2 p 2 4 4 4 4 D p.1 k / 1 k D .1 Â Â / Â1 Á= Â Â ; z C C z C 2 3 2 C 3 (5.4) q 4 4 Â2 Á= Â Â ; k Á Á Â2Â3 2 3 acd.Á/ cd ; k q q C ; 2 p 2 4 4 4 4 D p.1 k / 1 k D .1 Â Â / Â3 Á= Â Â ; z C C z C 2 3 2 C 3 q 4 4 Â3 Á= Â Â ; 1 Á Á Â2Â3 2 3 adc.Á/ dc ; k q q C ; 2 p 2 4 4 4 4 D p.1 k / 1 k D .1 Â Â / Â2 Á= Â Â ; z C C z C 2 3 2 C 3 where is chosen such that k2 Â 4=Â 4. D 2 3 The ﬁrst two solutions, asn.Á/ and ans.Á/, reduce to hyperbolic trigonometric functions in the case of modulus k 1, since sn.u; 1/ tanh.u/ and ns.u; 1/ D D D coth.u/. That is, two additional solutions in terms of elementary functions are 1 1 p p a1.Á/ p tanh.Á= 2/; a2.Á/ p coth.Á= 2/: (5.5) D 2 D 2 z z For these two solutions, one may solve the differential equation Á.t/ 1=a.t/ P D Q D 1=a.Á.t// for Á.t/, and therefore obtain the cosmic scale factor a.t/ a.Á.t// Q D which solves the Einstein ﬁeld equation (i) in Section 4 for .t/ D=a.t/4 with D Q special value D k2=Ä .1 k2/2. Doing so, we obtain D zzd C Â q Ã 1 pt 2pt a1.t/ a1.Á.t// p tanh ln e z e z 1 ; Q D D 2 C z Â q Ã (5.6) 1 pt 2pt a2.t/ a2.Á.t//; p coth ln e z e z 1 ; Q D D 2 C C z for t > 0. 286 JENNIE D’AMBROISE

For negative curvature k 1, and for 0 D 1 k2 D D Ä .k2 2/2 z zd with 0 < k < 1, equation (5.2) becomes 2 2 2 4 1 k a0.Á/ a.Á/ a.Á/ : (5.7) z D .k2 2/2 z In terms of Jacobi elliptic functions and theta functions, we obtain the two solutions for the scale factor in conformal time, q q 2 4 4 2 4 4 p1 k Á Á Â3 Â2 Â3 Â1 Á=Â3 2Â3 Â2 ; a .Á/ sc ; k ; sc 2 q 2q 4 4 D p.2 k2/ p2 k D .2Â 4 Â 4/ Â Â Á=Â 2Â Â ; z z 3 2 4 2 3 3 2 (5.8) 2q 4 4 Â2 Á=Â 2Â Â ; 1 Á Á Â3 Â4 3 3 2 acs.Á/ cs ; k q q ; 2 p 2 4 4 2 4 4 D p.2 k / 2 k D .2Â Â / Â1 Á=Â 2Â Â ; z z 3 2 3 3 2 where is such that k2 Â 4=Â 4. D 2 3 Note that by the comments following equation (2.10), it is possible to express the solutions obtained in this section in terms of Weierstrass functions. E. Ab- dalla and L. Correa-Borbonet [2002] have also considered the Einstein equation (i) with .t/ D=a.t/4, and have found connections with Weierstrass functions D Q in cosmic time (as opposed to the conformal time argument given here). In Sec- tion 6 of this lecture we will ﬁnd more general solutions to the conformal time equation (4.2) in terms of }, for arbitrary curvature k0 and D-value. We will also 3 3 4 consider the density functions .t/ D=a.t/ and .t/ D1=a.t/ D2=a.t/ D Q D Q C Q in Section 6.

6. FRLW and Weierstrass elliptic functions

In general, if g.0/ x0 and g.u/ satisﬁes D g .u/2 F.g.u// (6.1) 0 D 4 3 2 for F.x/ A4x A3x A2x A1x A0 any quartic polynomial with no D C C C C repeated roots, then the inverse function y.x/ of g.u/ is the elliptic integral Z x dt y.x/ p : (6.2) D x0 F.t/

For the initial condition g0.0/ 0, x0 is a root of the polynomial F.x/ by D R ıp (6.1). In this case the integral (6.2) can be rewritten as 1 dz P.z/, where 1=.x x0/ and P.z/ is a cubic polynomial. To do this, first expand F.t/ D APPLICATIONS OF ELLIPTIC AND THETA FUNCTIONS TO FRLW COSMOLOGY 287 into its Taylor series about x0 and then perform the change of variables z R ıp D 1=.t x0/. Furthermore one may obtain the form 1 d Q./ for 1 1 3 D F .x0/ F .x0/ , where Q./ 4 g2 g3. This is done by setting 4 0 C 6 00 D z .4 B2=3/=B3 where B2 F .x0/=2 and B3 F .x0/ are the quadratic D D 00 D 0 and cubic coefficients of P.z/ respectively. Note that since F.x/ has no repeated roots, x0 is not a double root, F .x0/ 0 and the variable z is well-defined. 0 ¤ Therefore we have obtained Â Ã 6F 0.x0/ 1 y x0 } ./: 24 F .x0/ C D 00 Writing this in terms of x, setting x g.u/ and solving for g.u/, one obtains D the solution to (6.1): F .x / g.u/ x 0 0 ; (6.3) 0 1 D C 4}.u g2; g3/ F .x0/ I 6 00 where 1 1 2 g2 A0A4 A1A3 A and D 4 C 12 2 (6.4) 1 1 1 2 1 2 1 3 g3 A0A2A4 A1A2A3 A A4 A0A A D 6 C 48 16 1 16 3 216 2 are referred to as the invariants of the quartic F.x/. Since F.x/ has no repeated 3 2 roots, the discriminant g 27g 0. Here if x0 0, (6.3) becomes D 2 3 ¤ D A g.u/ 1 : (6.5) 1 D 4}.u g2; g3/ A2 I 3 If the initial condition on the first derivative is such that g .0/ 0 then x0 0 ¤ is not a root of F.x/ and a more general solution to (6.1) is due to Weierstrass. The proof (which we will not include here) was published by Biermann in 1865 (see [Biermann 1865; Reynolds 1989]). The solution is

p 1 1 1 F.x0/}0.u/ 2 F 0.x0/ }.u/ 24 F 00.x0/ 24 F.x0/F 000.x0/ g.u/ x0 C C D C 1 2 1 2 }.u/ 24 F 00.x0/ 48 F.x0/F 0000 .x0/ (6.6) where } is formed with the invariants of the quartic seen in (6.4) such that 0. Here if x0 0, (6.6) becomes ¤ D 1 1 1 pA0} .u/ A1 }.u/ A2 A0A3 g.u/ 0 C 2 12 C 4 : (6.7) D 1 2 1 2 }.u/ A2 A0A4 12 2 To generate a number of examples, we consider the conformal time Einstein 3 4 equation (4.2) for density .t/ D1=a.t/ D2=a.t/ with D1; D2 > 0. In D Q C Q 288 JENNIE D’AMBROISE this case (4.2) becomes

2 4 2 a .Á/ a.Á/ k a.Á/ Ä D1a.Á/ Ä D2 (6.8) 0 D z 0 C zd C zd and we take A4 ; A3 0; A2 k0; A1 Äd D1 and A0 Äd D2. The most D z D D D z D z def: general solution seen here is with initial conditions a .0/ 0 so that a.0/ a0 0 ¤ D is not a root of the polynomial

4 2 F.t/ t k t Ä D1t Ä D2: (6.9) D z 0 C zd C zd The solution is given by (6.6) as

p 1 1 1 F.a0/}0.Á/ 2 F 0.a0/ }.Á/ 24 F 00.a0/ 24 F.a0/F 000.a0/ a.Á/ a0 C C ; D C 1 2 1 2 }.Á/ 24 F 00.a0/ 48 F.a0/F 0000 .a0/ (6.10) with Weierstrass invariants 1 2 g2 Äd D2 .k0/ and D zz C 12 (6.11) 1 1 2 2 1 3 g3 Ä D2k Ä D .k / D 6 zzd 0 16 zzd 1 C 216 0 restricted to be such that g3 27g2 0 so that F.t/ does not have repeated D 2 3 ¤ roots. Since D2; Ä > 0, by (6.9) zero is not a root of F.t/ and therefore for initial zd condition a .0/ 0 and a0 0, the solution to (6.8) is given by (6.7) as 0 ¤ D p 1 k Äd D2} .Á/ Äd D1 }.Á/ 0 a.Á/ z 0 C 2 z C 12 (6.12) D k 2 1 2 }.Á/ 0 Ä D2 C 12 2 zzd for invariants g2; g3 as in (6.11) with 0. One can compare this with the ¤ results in papers by Aurich, Steiner and Then, where curvature is taken to be k 1 [Aurich and Steiner 2001; Aurich et al. 2004]. 0 D For a .0/ 0 and a0 a root of F.t/ in (6.9), the solution to (6.8) is given by 0 D (6.3), F .a / a.Á/ a 0 0 (6.13) 0 1 D C 4}.Á/ F .a0/ 6 00 again with invariants (6.11) such that 0. ¤ For a more concrete example, consider the density function .t/ D=a.t/3 D Q for D > 0 so that conformal time equation (4.2) becomes

a .Á/2 a.Á/4 k a.Á/2 Ä Da.Á/: (6.14) 0 D z 0 C zd Here zero is a root of the polynomial F.t/ with A4 ; A3 A0 0; A2 k D z D D D 0 and A1 Ä D. Therefore with initial conditions a .0/ a.0/ 0, the solution D zd 0 D D APPLICATIONS OF ELLIPTIC AND THETA FUNCTIONS TO FRLW COSMOLOGY 289 to (6.14) is given by (6.5) as 3Ä D a.Á/ zd (6.15) D 12}.Á/ k with invariants C 0

1 2 1 2 2 1 3 g2 .k / and g3 Ä D .k / (6.16) D 12 0 D 16 zzd C 216 0 restricted to be such that 0. As noted in the comments following equations ¤ (2.10), one can write this solution in terms of Jacobi elliptic functions (by using 3 equations (2.10) if the roots of the reduced cubic 4t g2t g3 are real). To demonstrate this, we choose s 1 2 D D Äd 27 z z 1 and k 1, so that g3 0 and g2 . For this positive curvature case, (6.15) 0 D D D 12 becomes p2 a.Á/ p ; (6.17) D 312}.Á/ 1 z C and the reduced cubic is 4t 3 .1=12/t 4t.t 1=4p3/.t 1=4p3/. Applying D C (2.10) with e3 1=4p3, e2 0, e1 1=4p3, (6.17) can be equivalently D D D written in terms of Jacobi elliptic functions as p p 2= 2= z z a.Á/ p p D p3 3 6 ns2 Á= 2p3; 1=p2 D p3 6 ds2Á= 2p3; 1=p2 C p C 2= z p (6.18) D p3 3 6 cs2Á= 2p3; 1=p2; C C 2 e2 e3 1 2 1 since k and e1 e3 . In terms of theta functions, D e1 e3 D 2 D D 2p3 this is p p 2= Â 2 Â 2Á= 2p3 Â 2; z 3 1 3 a.Á/ p p D p3 3 Â 2 Â 2Á= 2p3 Â 2; 6 Â 2 Â 2Á= 2p3 Â 2; 3 1 3 C 2 4 3 p p 2= Â 4 Â 2Á= 2p3 Â 2; z 3 1 3 p p D p3 Â 4 Â 2Á= 2p3 Â 2; 6 Â 2 Â 2 Â 2Á= 2p3 Â 2; 3 1 3 C 2 4 3 3 p p 2= Â 2 Â 2Á= 2p3 Â 2; z 3 1 3 p p (6.19) D p3 3 Â 2 Â 2Á= 2p3 Â 2; 6 Â 2 Â 2Á= 2p3 Â 2; C 3 1 3 C 4 2 3 where is taken such that 1 Â 4=Â 4. 2 D 2 3 290 JENNIE D’AMBROISE

As a ﬁnal example, we return to .t/ D=a.t/4 for D > 0 considered in D Q Section 5. That is, we will obtain alternate solutions to equation (5.2). Since D > 0, zero is not a root of the polynomial F.t/ with A4 ; A3 A1 D z D D 0; A2 k and A0 Ä D. Therefore for initial conditions a .0/ 0 and D 0 D zd 0 ¤ a.0/ 0, (6.7) gives the solution D pÄ D} .Á/ a.Á/ zd 0 (6.20) D 2.}.Á/ 1 k /2 1 Ä D C 12 0 2 zzd for invariants

1 2 1 1 3 g2 Ä D .k / and g3 Ä Dk .k / (6.21) D zzd C 12 0 D 6 zzd 0 C 216 0 restricted to be such that 0. (6.20) is more general than the solutions to (5.2) ¤ in Section 5, since here the curvature k0 and the constant D are unspeciﬁed. To see this solution expressed in terms of Jacobi elliptic functions, take curvature 1 k0 1 and D 1=.36Äd / for 0 < k < 1 so that g3 0 and g2 9 . Then the D D 3 z1z 1 1 D 1 D 1 reduced cubic is 4t t 4 t t so that e3 , e2 0, e1 9 D 6 C 6 D 6 D D 6 and (6.20) becomes p .12= / } .Á/ a.Á/ z 0 : (6.22) D 144.}.Á/ 1 /2 1 C 12 By (2.8) and (2.10), we write this solution in terms of Jacobi elliptic functions and obtain r Á Á 2 3 sn2 Á ; 1 sn4 Á ; 1 p3 p2 C p3 p2 a.Á/ p Á ÁÁ D 6 2 ns Á ; 1 sn Á ; 1 z p3 p2 p3 p2 v u 2 Á 1 Á u 2 ds ; 1 1 u p3 p2 p Át Á D 2 3 ds Á ; 1 2 ds2 Á ; 1 1 z p3 p2 p3 p2 C Á cs Á ; 1 p p 3 2 (6.23) D p r ÁÁ ÁÁ 6 1 cs2 Á ; 1 1 2 cs2 Á ; 1 z C p3 p2 C p3 p2 where each of the positive and negative square roots solve (5.2) for k0 1, 2 1 2 D D 1=.36Ä / and where e1 e3 and k .e2 e3/=.e1 e3/ D zzd D D 3 D 1 . Writing (6.23) in terms of theta functions, and deﬁning Á=p3Â 3, we D 2 D 3 APPLICATIONS OF ELLIPTIC AND THETA FUNCTIONS TO FRLW COSMOLOGY 291

ﬁnd that

q 4 4 2 2 2 2 4 4 Â3Â1./ 2Â2 Â4 ./ 3Â2 Â3 Â1 ./Â4 ./ Â3 Â1 ./ a.Á/ C D p 2 2 2 2 6 Â2Â4./ 2Â Â ./ Â Â ./ z 2 4 3 1 v 2 u 2 2 2 4 2 Â Â1./ u 2Â Â Â ./ Â Â ./ 3 2 4 3 3 1 p t 2 2 2 4 2 D 2 3 Â Â Â ./ 2Â Â Â ./ Â Â ./ z 2 4 3 2 4 3 C 3 1

Â4Â3Â2./Â1./ q (6.24) D 6 Â 2Â 2./ Â 2Â 2./ Â 2Â 2./ 2Â 2Â 2./ z 3 1 C 4 2 3 1 C 4 2 by the theta function representations for sn, ds and cs respectively. Here the that forms the theta functions is suppressed and is taken to satisfy 1 Â 4=Â 4. 2 D 2 3 7. Summary

There are a number of ways to see that elliptic and theta functions solve the d-dimensional Einstein gravitational field equations in a FRLW cosmology with a cosmological constant. Here we considered a scenario with no scalar 3 4 3 field and with density functions .t/ D1=a.t/ D2=a.t/ , .t/ D=a.t/ D Q C Q D Q and .t/ D=a.t/4 scaling in inverse proportion to the scale factor a.t/. In D Q Q these cases the first Einstein equation (i) takes the form a.t/2 an expression PQ D containing negative powers of the cosmic scale factor a.t/. At this point, one Q could have introduced the inverse function y.x/ of a.t/ to obtain an expression Q for y.x/ as the integral of a power of x divided by the square root of a polynomial in x. That is, y.x/ would be an elliptic integral that is not normal; other authors have taken this approach [Abdalla and Correa-Borbonet 2002; Kraniotis and Whitehouse 2002]. Here, we switched to conformal time by a change of def: variables a.Á/ a.f .Á//. This produced an equation of the form a .Á/2 an DQ 0 D expression containing nonnegative powers of the conformal scale factor a.Á/. After reviewing the definitions and properties of elliptic and theta functions in sections 2 and 3, we introduced the FRLW cosmological model in Section 4. In Section 5 for .t/ D=a.t/3, we obtained a differential equation for D Q a.Á/ containing only even powers of a.Á/ and constructed solutions in terms of Jacobi elliptic functions, restricted to particular values of the constant D, parameterized by modulus 0 < k < 1. The equivalent theta function representations for these solutions were recorded, and we noted the special cases for which the elliptic solutions reduce to elementary functions and the correspond- ing solution in cosmic time was also computed. In Section 6, we considered 3 4 3 4 each of .t/ D1=a.t/ D2=a.t/ , .t/ D=a.t/ and .t/ D=a.t/ D Q C Q D Q D Q 292 JENNIE D’AMBROISE with various initial conditions and obtained solutions in terms of Weierstrass functions for general curvature k0 and constants D1; D2; D > 0. By considering these solutions restricted to certain D-values (again, parameterized by modulus 0 < k < 1), we wrote a.Á/ equivalently in terms of Jacobi elliptic and theta functions. In current joint work with Floyd Williams [D’Ambroise and Williams 2010], we have seen that elliptic functions also appear in the presence of a scalar field .t/, for both the FRLW and Bianchi I d-dimensional cosmological models with a 0 and with a similar density function scaling in inverse pro- ¤ portion to a.t/. There we note that the equations of each of these cosmological Q models can be rewritten in terms of a generalized Ermakov–Milne–Pinney differential equation [Lidsey 2004; D’Ambroise and Williams 2007], a type which the square root of the second moment of the wave function of the Bose–Einstein condensate (BEC) also satisfies. On the cosmological side of the FRLW-BEC correspondence, imposing an equation of state .t/ wp .t/ (w constant) on D the density .t/ and pressure p.t/ of the scalar field .t/ allows one to obtain the differential equation for an elliptic function on the side of the BECs.

References

[Abdalla and Correa-Borbonet 2002] E. Abdalla and L. Correa-Borbonet, “The elliptic solutions to the Friedmann equation and the Verlinde’s maps”, preprint, 2002. arXiv:hep-th/0212205. [Aurich and Steiner 2001] R. Aurich and F. Steiner, “The cosmic microwave back- ground for a nearly ﬂat compact hyperbolic universe”, Monthly Not. Royal Astron. Soc. 323:4 (2001), 1016–1024. Also see arXiv:astro-ph/0007264. [Aurich et al. 2004] R. Aurich, F. Steiner, and H. Then, “Numerical computation of Maass waveforms and an application to cosmology”, preprint, 2004. arXiv:gr- qc/0404020. [Basarab-Horwath et al. 2004] P. Basarab-Horwath, W. I. Fushchych, and L. F. Baran- nyk, “Solutions of the relativistic nonlinear wave equation by solutions of the nonlinear Schrodinger¨ equation”, pp. 81–99 in Scientiﬁc works of W. I. Fushchych, vol. 6, 2004. [Biermann 1865] G. G. A. Biermann, Problemata quaedam mechanica functionum el- lipticarum ope soluta, Dissertatio Inauguralis, Friedrich-Wilhelm-Universitat,¨ 1865. Available at http://edoc.hu-berlin.de/ebind/hdiss/BIERPROB PPN313151385/XML. [Byrd and Friedman 1954] P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and physicists, Grundlehren der math. Wissenschaften 67, Springer, Berlin, 1954. APPLICATIONS OF ELLIPTIC AND THETA FUNCTIONS TO FRLW COSMOLOGY 293

[D’Ambroise and Williams 2007] J. D’Ambroise and F. L. Williams, “A non-linear Schrodinger¨ type formulation of FLRW scalar ﬁeld cosmology”, Int. J. Pure Appl. Math. 34:1 (2007), 117–127. [D’Ambroise and Williams 2010] J. D’Ambroise and F. Williams, “A dynamic correspondence between FRLW cosmology with cosmological constant and Bose– Einstein condensates”, to appear. [Greenhill 1959] A. G. Greenhill, The applications of elliptic functions, Dover, New York, 1959. [Kharbediya 1976] L. I. Kharbediya, “Some exact solutions of the Friedmann equations with the cosmological term”, Astronom. Z.ˇ 53 (1976), 1145–1152. [Kraniotis and Whitehouse 2002] G. V. Kraniotis and S. B. Whitehouse, “General rel- ativity, the cosmological constant and modular forms”, Classical Quantum Gravity 19:20 (2002), 5073–5100. arXiv:gr-qc/0105022. [Lidsey 2004] J. Lidsey, “Cosmic dynamics of Bose–Einstein condensates”, Classical and Quantum Gravity 21 (2004), 777–785. arXiv:gr-qc/0307037. [Reynolds 1989] M. J. Reynolds, “An exact solution in nonlinear oscillations”, J. Phys. A 22:15 (1989), L723–L726. Available at http://stacks.iop.org/0305-4470/22/L723. [Whittaker and Watson 1927] E. T. Whittaker and G. N. Watson, A course of mod- ern analysis, Cambridge Mathematical Library, Cambridge University Press, Cam- bridge, 1927.

JENNIE D’AMBROISE DEPARTMENT OF MATHEMATICS AND STATISTICS LEDERLE GRADUATE RESEARCH TOWER UNIVERSITYOF MASSACHUSETTS AMHERST, MA 01003-9305 UNITED STATES [email protected]