# Problem set 4 FYS9130 – Week 39 ### Problem set 4 FYS9130 – Week 39

1.

Journal club

Read one or more of the papers (ADD, RS1, RS2 or DGP) and summarize the main idea. Do you see any problems?

2.

Unnatural units (toy problem)

Assume new units defined by g = 1 = 9 .

81m /s 2

G = 1 = 6 .

67

·

10 − 11 N m 2 /kg 2 a) Find 1m, 1N, 1J and 1kg measured in seconds.

b) Calculate the Planck mass in seconds.

3.

Cosmolgy with a scalar field (quintessence)

The Friedmann equations are:

2 a

− a 2 a 2 a 2 a 2

=

1

3 M

P

2

µ 1

2

φ

˙ 2 + V ( φ )

=

1

M

P

2

µ 1

2

φ

˙ 2

V ( φ )

¶ which can also be combined into the scalar field equation of motion:

(1)

(2)

(3)

(4)

¨

+ 3 a

φ

˙

+ V ′ ( φ ) a) Solve these equations for V = 0 (Hint: φ = k ln t + φ

0

) b) Solve these equations for V ( φ ) = Ae −

λφ

M

(Hint: φ = k ln t + φ

0

, show that k = 2 M

P

λ and V ( φ

0

) =

6(2 − λ

2

λ

4

) M

2

P c) Find Ω k in and Ω p ot with the definitions

)

Ω k in

Ω p ot

ρ c r

=

ρ k in

ρ c r

=

1

2

ρ

φ

˙ 2 c r

=

ρ p ot

=

ρ c r

= 3 H 2 M

P

2

V ( φ )

ρ c r

H = a

(5)

(6)

(7)

(8)

(9)

1