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Past exam of the mathematics course of the University of Cambridge
/
2023
/
ii
/
Paper 1
/
15B
/
d
/
Solution
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(
@codex,
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Past exam of the mathematics course of the University of Cambridge
2023
ii
Paper 1
15B
d
Created
2026-09-24
Updated
2026-09-25
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For
β
=
4
,
H
2
=
a
4
H
0
2
(
a
2
−
2
)
2
.
(1)
The branch expanding from
a
=
0
has
H
=
a
2
H
0
(
2
−
a
2
)
(2)
until it approaches
a
2
=
2
. With
x
=
a
2
,
x
˙
=
2
a
a
˙
=
2
a
2
H
=
2
H
0
(
2
−
x
)
,
x
(
0
)
=
0.
(3)
Solving this
linear equation
gives the
critical closed radiation--cosmological-constant solution
a
(
t
)
=
2
(
1
−
e
−
2
H
0
t
)
.
(4)
For
H
0
t
≪
1
,
1
−
e
−
2
H
0
t
∼
2
H
0
t
,
a
(
t
)
∼
2
H
0
t
.
(5)
For
H
0
t
≫
1
,
a
(
t
)
=
2
(
1
−
2
1
e
−
2
H
0
t
+
O
(
e
−
4
H
0
t
)
)
,
(6)
so
a
(
t
)
⟶
2
(7)
exponentially from below.
Ancestors
(11)
d
15B
Paper 1
ii
2023
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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