C.I. Lazaroiu, C.S. Shahbazi / Nuclear Physics B 936 (2018) 542–596 551
In this expression, we assumed that t
0
is chosen within a time interval on which the gradient flow
approximation holds and we set q(t
0
) = q
0
. Using (1.18), relation (1.8) reduces to the algebraic
equation:
H
4
−
3M
2
H
2
−
||d||
2
G
54M
2
=0 ,
which gives:
H(t(q))=
1
M
√
6
(ϕ
•
(q)) +
(ϕ
•
(q))
2
+
2
3
M
2
||(d)(ϕ
•
(q))||
2
G
1/2
, (1.20)
where we assumed H(t) > 0. Thus (1.19) reduces to:
t − t
0
=
1
M
3
2
q
q
0
dq
(ϕ
•
(q)) +
(ϕ
•
(q))
2
+
2
3
M
2
||(d)(ϕ
•
(q))||
2
G
1/2
. (1.21)
This relation can be used to determine t as a function of q given the approximating gradient flow
trajectory ϕ
•
(q).
Let us assume that (ϕ
(t)) > 0, as required by the first inflation condition in (1.12). Then the
second inflation condition amounts to the following inequality in the gradient flow approxima-
tion:
M
||(d)(ϕ
•
(q))||
G
(ϕ
•
(q))
<
3
√
2
.
Together with (1.21), the last inequality can be used to determine the inflationary time intervals
within the gradient flow approximation. Notice from (1.20) that H is determined by the point
ϕ
•
(q) ∈ in this approximation.
A gradient flo
w trajectory is determined uniquely by the gradient flow equation:
dϕ
•
(q)
dq
=−(grad
G
)(ϕ
•
(q)) (1.22)
(which is a first order differential equation), together with the initial condition:
ϕ
•
(q
0
) = ϕ
0
.
Validity of the gradient flow approximation at t =t
0
requires
dϕ
dq
(q
0
) =−(grad
G
)(ϕ
0
), i.e.:
v
0
=−
(grad
G
)(ϕ
0
)
3H
0
, (1.23)
where H
0
def.
= H(t
0
) is determined from (1.20)as:
H
0
=
1
M
√
6
(ϕ
0
) +
(ϕ
0
)
2
+
2
3
M
2
||(d)(ϕ
0
)||
2
G
1/2
. (1.24)
In particular, the gradient flow approximation constrains the initial velocity v
0
in terms of the
initial value ϕ
0
, as expected from the fact that, in this approximation, the second order equation
(1.9)is replaced by the first order equation (1.22).
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