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Z4co: add Z4c_rhs.wl
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lwJi committed Jul 1, 2024
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126 changes: 124 additions & 2 deletions Z4co/wolfram/Z4c_rhs.wl
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(* 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]];

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