From c62098b299ae1befb393bc80cb5f65f6ccf8cc1a Mon Sep 17 00:00:00 2001 From: Liwei Ji Date: Mon, 1 Jul 2024 14:08:52 -0400 Subject: [PATCH] Z4co: add Z4c_rhs.wl --- Z4co/wolfram/Z4c_rhs.wl | 126 +++++++++++++++++++++++++++++++++++++++- 1 file changed, 124 insertions(+), 2 deletions(-) diff --git a/Z4co/wolfram/Z4c_rhs.wl b/Z4co/wolfram/Z4c_rhs.wl index c9e04ec3..0ec884c5 100644 --- a/Z4co/wolfram/Z4c_rhs.wl +++ b/Z4co/wolfram/Z4c_rhs.wl @@ -2,6 +2,128 @@ (* Z4c_rhs.wl *) -(* (c) Liwei Ji, 06/2024 *) +(* (c) Liwei Ji, 07/2024 *) -SetEQN[chi[], 2/3 chi[](alpha[](hK[]+2 theta[]) - dbeta[])]; +(****************) + +(* Intermediate *) + +(****************) + +SetEQN[invgam[i_, j_], chi[] invgamt[i, j]]; + +SetEQN[gam[i_, j_], chi[] ^ -1 gamt[i, j]]; + +SetEQN[GtDDD[k_, i_, j_], 1/2 (dgamt[i, j, k] + dgamt[j, k, i] - dgamt[ + k, i, j])]; + +SetEQN[GtDDU[i_, j_, k_], gamt[k, l] GtDDD[i, j, -l]]; + +SetEQN[Gt[k_, i_, j_], gamt[k, l] GtDDD[-l, i, j]]; + +SetEQN[trGtd[i_], invgamt[k, l] Gt[i, -k, -l]]; + +SetEQN[exAtUU[i_, j_], invgamt[i, k] invgamt[j, l] exAt[-k, -l]]; + +SetEQN[tDtDchi[i_, j_], ddchi[i, j] - Gt[k, i, j] dchi[-k]]; + +SetEQN[DDalpha[i_, j_], ddalpha[i, j] - Gt[k, i, j] dalpha[-k] + 1/2 chi[ + ] ^ -1 (dchi[i] dalpha[j] + dchi[j] dalpha[i]) - 1/2 gamt[i, j] invgamt[ + k, l] chi[] ^ -1 dchi[-l] dalpha[-k]]; + +(* (8) *) + +SetEQN[Rtchi[i_, j_], 1 / (2 chi[]) tDtDchi[i, j] + 1 / (2 chi[]) gamt[ + i, j] invgamt[k, l] tDtDchi[-k, -l] - 1 / (4 chi[] ^ 2) dchi[i] dchi[j] + - 3 / (4 chi[] ^ 2) gamt[i, j] invgamt[k, l] dchi[-k] dchi[-l]]; + +(* (9) *) + +SetEQN[Rt[i_, j_], -(1/2) invgamt[l, m] ddgamt[-l, -m, i, j] + 1/2 (gamt[ + -k, i] dtrGt[j, k] + gamt[-k, j] dtrGt[i, k]) + 1/2 trGtd[k] (GtDDD[i, + j, -k] + GtDDD[j, i, -k]) + ((Gt[k, -l, i] GtDDU[j, -k, l] + Gt[k, -l, + j] GtDDU[i, -k, l]) + Gt[k, i, -m] GtDDU[-k, j, m])]; + +(* (10) *) + +SetEQN[R[i_, j_], Rtchi[i, j] + Rt[i, j]]; + +SetEQN[trR[], invgam[k, l] R[-k, -l]]; + +SetEQN[rho[], alpha[] ^ -2 (eTtt[] - 2 beta[j] eTt[-j] + beta[i] beta[j + ] eT[-i, -j])]; + +SetEQN[Sm[i_], -alpha[] ^ -1 (eTt[i] - beta[k] eT[-k, i])]; + +SetEQN[Ss[i_, j_], eT[i, j]]; + +SetEQN[trSs[], invgam[k, l] Ss[-k, -l]]; + +SetEQN[dexAtUU[k_, i_, j_], -2 invgamt[i, l] exAtUU[j, m] dgamt[k, -l, + -m] + invgamt[i, l] invgamt[j, m] dexAt[k, -l, -m]]; + +(* (13) *) + +SetEQN[ZtC[i_], trGt[i] - trGtd[i]]; + +(* (14) *) + +SetEQN[HC[], trR[] + exAt[-k, -l] exAtUU[k, l] - 2/3 (exKh[] + 2 Theta[ + ]) ^ 2 - 16 cpi rho[]]; + +(* (15) *) + +SetEQN[MtC[i_], dexAtUU[-j, i, j] + Gt[i, -j, -k] exAtUU[j, k] - 2/3 invgamt[ + i, j] (dexKh[-j] + 2 dTheta[-j]) - 2/3 exAtUU[i, j] chi[] ^ -1 dchi[-j] + - 8 cpi invgamt[i, j] Sm[-j]]; + +(*******) + +(* EOM *) + +(*******) + +(* (1) *) + +SetEQN[chi[], 2/3 chi[] (alpha[] (exKh[] + 2 Theta[]) - dbeta[-i, i])]; + +(* (2) *) + +SetEQN[gamt[i_, j_], -2 alpha[] exAt[i, j] + (gamt[-k, i] dbeta[j, k] + + gamt[-k, j] dbeta[i, k]) - 2/3 gamt[i, j] dbeta[-k, k]]; + +(* (3) *) + +SetEQN[exKh[], -invgam[k, l] DDalpha[-k, -l] + alpha[] (exAt[-k, -l] exAtUU[ + k, l] + 1/3 (exKh[] + 2 Theta[]) ^ 2) + 4 cpi alpha[] (trSs[] + rho[]) + + alpha[] ckappa1 (1 - ckappa2) Theta[]]; + +(* (4) *) + +SetEQN[exAt[i_, j_], chi[] ((-DDalpha[i, j] + alpha[] (R[i, j] - 8 cpi + Ss[i, j])) - 1/3 gam[i, j] invgam[k, l] (-DDalpha[-k, -l] + alpha[] (R[ + -k, -l] - 8 cpi Ss[-k, -l]))) + alpha[] ((exKh[] + 2 Theta[]) exAt[i, j + ] - 2 invgamt[k, l] exAt[-k, i] exAt[-l, j]) + (exAt[-k, i] dbeta[j, k] + + exAt[-k, j] dbeta[i, k]) - 2/3 exAt[i, j] dbeta[-k, k]]; + +(* (5) *) + +SetEQN[Gt[i_], -2 exAtUU[i, j] dalpha[-j] + 2 alpha[] (Gt[i, -j, -k] exAtUU[ + j, k] - 3/2 exAtUU[i, j] dchi[-j] / chi[] - 1/3 invgamt[i, j] (2 dexKh[ + -j] + dTheta[-j]) - 8 cpi invgam[i, j] Sm[-j]) + invgamt[j, k] ddbeta[- + j, -k, i] + 1/3 invgamt[i, j] ddbeta[-j, -k, k] - 2/3 trGtd[i] dbeta[-j, + j] - 2 alpha[] ckappa1 (trGt[i] - trGtd[i])]; + +(* (6) *) + +SetEQN[Theta[], 1/2 alpha[] (trR[] - exAt[-k, -l] exAtUU[k, l] + 2/3 (exKh[ + ] + 2 Theta[]) ^ 2) - alpha[] (8 cpi rho[] + ckappa1 (2 + ckappa2) Theta[ + ])]; + +(* (11) *) + +SetEQN[alpha[], -alpha[] ^ 2 cmuL exKh[]]; + +(* (12) *) + +SetEQN[beta[i_], alpha[] ^ 2 cmuS trGt[i] - ceta beta[i]];