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!! metric_p(:) = h(:)
m_p_k(:, i, j, k) = ( (/ &
( (1._dl - 2._dl * xi) * f_x(1, i, j, k) ** 2 - 2._dl * xi * fields_fk(1, i, j, k) * I44_D2F_XX_FLD(fields_fk, 1, i, j, k) + f_x(2, i, j, k) ** 2 ) , &
( (1._dl - 2._dl * xi) * f_y(1, i, j, k) ** 2 - 2._dl * xi * fields_fk(1, i, j, k) * I44_D2F_YY_FLD(fields_fk, 1, i, j, k) + f_y(2, i, j, k) ** 2 ) , &
( (1._dl - 2._dl * xi) * f_z(1, i, j, k) ** 2 - 2._dl * xi * fields_fk(1, i, j, k) * I44_D2F_ZZ_FLD(fields_fk, 1, i, j, k) + f_z(2, i, j, k) ** 2 ) , &
( (1._dl - 2._dl * xi) * f_y(1, i, j, k) * f_z(1, i, j, k) - 2._dl * xi * fields_fk(1, i, j, k) * I44_D2F_YZ_FLD(fields_fk, 1, i, j, k) + f_y(2, i, j, k) * f_z(2, i, j, k) ) , &
( (1._dl - 2._dl * xi) * f_x(1, i, j, k) * f_z(1, i, j, k) - 2._dl * xi * fields_fk(1, i, j, k) * I44_D2F_XZ_FLD(fields_fk, 1, i, j, k) + f_x(2, i, j, k) * f_z(2, i, j, k) ) , &
( (1._dl - 2._dl * xi) * f_x(1, i, j, k) * f_y(1, i, j, k) - 2._dl * xi * fields_fk(1, i, j, k) * I44_D2F_XY_FLD(fields_fk, 1, i, j, k) + f_x(2, i, j, k) * f_y(2, i, j, k) ) , &
/) &
/ (metric%dx) ** 2 / 144._dl * (2._dl / Mplsq) / (1 - xi * fields_fk(1, i, j, k) ** 2 / Mplsq) &
-(/ &
(2._dl * h12_xy(i, j, k) + 2._dl * h13_xz(i, j, k) - h11_yy(i, j, k) - h11_zz(i, j, k) - h22_xx(i, j, k) - h33_xx(i, j, k) ) , &
(2._dl * h12_xy(i, j, k) + 2._dl * h23_yz(i, j, k) - h22_xx(i, j, k) - h22_zz(i, j, k) - h11_yy(i, j, k) - h33_yy(i, j, k)) , &
(2._dl * h13_xz(i, j, k) + 2._dl * h23_yz(i, j, k) - h33_xx(i, j, k) - h33_yy(i, j, k) - h11_zz(i, j, k) - h22_zz(i, j, k)) , &
(h12_xz(i, j, k) + h13_xy(i, j, k) - h23_xx(i, j, k) - h11_yz(i, j, k)) , &
(h12_yz(i, j, k) + h23_xy(i, j, k) - h13_yy(i, j, k) - h22_xz(i, j, k)) , &
(h13_yz(i, j, k) + h23_xz(i, j, k) - h12_zz(i, j, k) - h33_xy(i, j, k)) , &
/) &
/ (metric%dx) ** 2 / 144._dl ) / ak ** 2 - 3._dl * Hk * metric_pk(:, i, j, k) + (R / 6._dl * 4._dl - 2._dl * Hk ** 2) * metric_hk(:, i, j, k)
