Oblig6 FYS9130
Deadline: Thursday 21/10 at 14.15 (beginning of class)
1.
Double exponential potential and cosmological scaling solutions
Start with the standard Friedmann equations
H 2 =
2 ˙ =
1
3 M
1
2
−
M 2
( ρ tot
)
( ρ tot
+ p tot
)
(1)
(2)
Assume an universe filled with a non-relativistic perfect fluid (”matter”, p m
0), radiation ( p r
= 1
3
ρ r
) and a scalar field ( ρ
φ
= ρ k
+ ρ p 1
+ ρ p 2
, ρ k
= 1
2
φ
˙
=
2 ,
ρ p 1
= Ae −
λ
M
φ , ρ p 2
= Be −
γ
M
φ , p k
= ρ k
, and p px
=
−
ρ px
). That is, the scalar field potential is
V ( φ ) = Ae −
λ
M
φ
γ
+ Be −
M
φ = ρ p 1
+ ρ p 2
(3)
Assume no extra couplings between matter and radiation, between matter and
φ or between radiation and φ , so we have:
ρ ˙ m
=
−
3 H ( ρ m
+ p m
)
ρ ˙ r
=
ρ ˙
φ
=
−
3 H ( ρ r
+ p r
)
−
3 H ( ρ
φ
+ p
φ
)
(4)
(5)
(6) a) Show that we have
ρ ˙
ρ
ρ ˙
˙
ρ ˙ m r p 1 p 2
=
−
3 Hρ m
=
=
=
−
4 Hρ r
−
√
6 λHρ p 1
√
Ω k
−
√
6 γHρ p 2
√
Ω k
ρ ˙ k
=
−
6 Hρ k −
ρ ˙ p 1 −
ρ ˙ p 2 b) Find the derivative of the relative densities with respect to the natural logarithm of the scale factor (as a function of Ω’s, β and γ ). Show that
(7)
(8)
(9)
(10)
(11)
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these can be written
Ω ′ m
=
−
3Ω m
+ 3Ω m
µ
Ω m
+
4
3
Ω r
+ 2Ω k
¶
Ω ′ r
=
−
4Ω r
+ 3Ω r
µ
Ω m
+
4
3
Ω r
+ 2Ω k
¶
(12)
(13)
Ω ′ k
=
−
6Ω k
+
√
6 λ Ω p 1 p
Ω k
+
√
6 γ Ω p 2 p
Ω k
+ 3Ω k
µ
Ω m
+
4
3
Ω r
+ 2Ω k
¶
(14)
Ω ′ p 1
=
−
√
6 λ Ω p 1 p
Ω k
+ 3Ω p 1
µ
Ω m
+
4
3
Ω r
+ 2Ω k
¶
Ω ′ p 2
=
−
√
6 γ Ω p 2 p
Ω k
+ 3Ω p 2
µ
Ω m
+
4
3
Ω r
+ 2Ω k
¶
(15)
(16) c) Find the fixpoints of the above equation set.
If possible, deliver a paper copy, handwritten is ok. Otherwise, e-mail to ingunnkw@fys.uio.no
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