Ratbay Myrzakulov

Let us rewrite this formula as
where
H 37 = D 2 F RR + D 1 F R + JF RT + E 1 F T + KF + 2a 3 ρ, (4.31)
D 2 = −6ɛ 1 Ṙa 2 ȧ, (4.32)
D 1 = 6ɛ 1 aä + a 3 uȧȧ, (4.33)
J = −6ɛ 1 a 2 ȧT, ˙
(4.34)
E 1 = 12ɛ 2 aȧ 2 + a 3 vȧȧ, (4.35)
K = −a 3 . (4.36)
As usual we assume that the total energy H 37 = 0 (Hamiltonian constraint). So finally we have
the following equations of the M 37 - model [10]-[11]:
D 2 F RR + D 1 F R + JF RT + E 1 F T + KF = −2a 3 ρ,
U + B 2 F TT + B 1 F T + C 2 F RRT + C 1 F RTT + C 0 F RT + MF = 6a 2 p, (4.37)
˙ρ + 3H(ρ + p) = 0.
It deserves to note that the M 37 - model (4.1) admits some interesting particular and physically
important cases. Some particular cases are now presented.
i) The M 44 - model. Let the function F(R,T) be independent from the torsion scalar T:
F = F(R,T) = F(R). Then the action (4.1) acquires the form
∫
S 44 = d 4 xe[F(R) + L m ], (4.38)
where
R = u + R s = u + ɛ 1 g µν R µν , (4.39)
is the curvature scalar. It is the M 44 - model. We work with the FRW metric. In this case R takes
the form
R = u + 6ɛ 1 (Ḣ + 2H2 ). (4.40)
The action can be rewritten as
where the Lagrangian is given by
∫
S 44 = dtL 44 , (4.41)
L 44 = a 3 [F − (R − u)F R + L m ] − 6ɛ 1 F R aȧ 2 − 6ɛ 1 F RR Ṙa 2 ȧ. (4.42)
The corresponding field equations of the M 44 - model read as
Here
and
D 2 F RR + D 1 F R + KF = −2a 3 ρ,
A 3 F RRR + A 2 F RR + A 1 F R + MF = 6a 2 p, (4.43)
˙ρ + 3H(ρ + p) = 0.
D 2 = −6ɛ 1 Ṙa 2 ȧ, (4.44)
D 1 = 6ɛ 1 a 2 ä + a 3 uȧȧ, (4.45)
K = −a 3 (4.46)
A 3 = −6ɛ 1 Ṙ 2 a 2 , (4.47)
A 2 = −12ɛ 1 Ṙaȧ − 6ɛ 1 ¨Ra 2 + a 3 Ṙuȧ, (4.48)
A 1 = 12ɛ 1 ȧ 2 + 6ɛ 1 aä + 3a 2 ȧuȧ + a 3 ˙uȧ − a 3 u a , (4.49)
M = −3a 2 . (4.50)
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