Lecture 6: EDS, Milne and other Universes

1.  Omega Notation 2.  Time in simple Universes

–  The Einstein de Sitter Universe (flat, matter only) –  The Milne Universe (empty) –  A Radiation Universe (flat, radiation only) –  Dark Energy Universe (flat, dark energy only)

3.  The Anthropic Principle 4.  Distances:

–  Einstein de Sitter –  Matter and curvature

5.  The LOT Universe: -  Time -  Distances

Course Text: Chapter 4 Wikipedia: Einstein de Sitter, Milne, Dark Matter, Large Scale

Structure

Omega Notation !M =

!M ,o

!crit= !M ,o

8"G3Ho

2

!R =!R,o

!crit= !R,o

8"G3Ho

2

!" =!"!crit

= !"8"G3Ho

2 ="

3Ho2

!K = #kc2

Ro2Ho

2

! =!M +!R +!"

!K =1#!1=!M +!R +!" +!K = FRIEDMANN EQUATION TODAY

!crit =3H0

2

8"G

! >1, implies a high-density Universe!=1, implies a flat Universe!<1, implies a low-density Universe

qo = !!!RR!R2

"

#$

%

&'o

= !!!R!R

"

#$

%

&'o

1Ho

Deceleration parameter, defn: Not used much these days.

Reminder:

Einstein de Sitter Universe •  ΩM=1, ΩR=0, ΩΛ=0, i.e., flat (k=0), matter only Universe

•  Universe will expand at a decreasing rate forever.

!RR!

"#

$

%&

2

=8!G"

3, and " = "o

RoR

'

()*

+,

3

- !R2 =8!G"oRo

3

31R

Lets try: R. tq - t2q/2 . t/q

Above only valid if: 2q-2=-q, or q=23

i.e., RRo

=tto

'

()

*

+,

23

differentiate to get: !R = 2Ro3to

23t

13

set to today's values and rearrange to get:

to =2

3Ho

0 9Gyrs

•  ΩM=0, ΩR=0, ΩΛ=0, i.e., empty, open (k=-1) Universe or Special Relativity case (no mass)

•  Universe will expand at a constant rate forever.

–  i.e., Infinite size after infinite time.

Milne Universe

!RR!

"#

$

%&

2

=c2

R2

' !R2 = c2

Integrate w.r.t t' R = ±ct i.e., scale factor proprtional to time.

to =1Ho

(13Gyrs

•  ΩM>1, ΩR=0, ΩΛ=0, i.e., closed (k=+1), matter only Universe

•  Universe will expand to a constant size and then re-collapse

Over-dense Universe

!RR!

"#

$

%&

2

=8!G"

3+c2

R2 , and " = "oRoR

'

()*

+,

3

- !R2 =8!G"oRo

3

31R. c2

i.e., !R = 0 when Rc =8!G"oRo

3

3c2

Radiation only Universe •  ΩM=0, ΩR=1, ΩΛ=0, i.e., flat (k=0), radiation only Universe

•  Universe will expand at a decreasing rate forever. [Note this was the relation we adopted in Lec 2.]

!RR!

"#

$

%&

2

=8!G"

3, and " = "o

RoR

'

()*

+,

4

- !R2 =8!G"oRo

4

31R2

Lets try: R. tq - t2q/2 . t/2q

Above only valid if: 2q-2=-2q, or q= 12

i.e., RRo

=tto

'

()

*

+,

12

differentiate to get: !R = 1Ro2to

12t

12

set to today's values and rearrange to get:

to =1

2Ho

0 6Gyrs

•  ΩM=0, ΩR=0, ΩΛ=1, i.e., flat (k=0), dark energy only Universe

•  Universe will expand at an exponential rate forever!

Dark Energy only Universe

!RR!

"#

$

%&

2

=8!G"

3, and " =Constant

' !R2 (R2

i.e., R( et

With Λ

Anthropic Principle •  So what is the value of Ω?

‒  ΩΜ+ΩR∼0.01 - 1.0 (e.g., counting mass of galaxies in a volume) –  Why so close to 1?

•  Anthropic Principle –  Weak

Only in a Universe with Ω~1 can life come about to ask the question why Ω~1, hence its no coincidence. ‒  Ω<<1 Universe too diffuse and expanding too fast for

stars/galaxies to form. ‒  Ω>>1 Universe re-collapses before life can evolve.

–  Strong Only Universe with Ω~1 are allowed by some as yet undiscovered physics.

–  E.g., Inflation (see Lec 8).

Weak Anthropic principle Lots of Universes possible each with very different physics with the laws universal within each bubble. Only those Universe which can support life will puzzle over their good fortune…

Image: James Schombert

Strong Anthropic principle Only one Universe possible, physics to be determined:

Image: James Schombert

Distance in an EdS Universe •  ΩM=1, ΩR=0, ΩΛ=0, i.e., flat (k=0), matter only Universe

•  What we want is an expression for, dp the proper distance

•  To get this we now need to combine R(t) with the metric

!!RR= !

4!G"3

, !RR"

#$

%

&'

2

=8!G"

3, " = "o

RoR

"

#$

%

&'

3

, RRo

=tto

"

#$

%

&'

23

Set t = to, get:

qoHo2 =

4!G"o3

,Ho2 =

8!G"o3

, qo =(M

2= 0.5

Distance in an EdS Universe

0 = c2dt2 ! R2dr2

R2dr2 = c2dt2

dr0

r1

" = c dtRt1

to

"

Consider a photon travelling from some point towards us . RWM reduced to:

Photon starts at r=0 and gets to r1 today

Photon leaves at time t1 and arrives at time to

(dp = Ror1)

R = Rotto

!

"#

$

%&

23

(to =23Ho

)

! r1 =cRoto23 1

t23t1

to

" dt

Distance in an EdS Universe Ror1 = cto

23 1

t2

3t1

to

! dt = cto2

3 3t1

3"#$

%&'t1

to

Ror1 = 3cto2

3 to1

3 ( t11

3"#$

%&'

Ror1 = 3cto 1( t1to

)

*+

,

-.

13"

#

$$

%

&

''

But: to =2

3Ho& t1

to

)

*+

,

-.

2 3

=R1

Ro=

11+ z

/ dp = Ror1 =2cHo

1( 1(1+ z)

12

"

#$$

%

&''

This is the standard EdS solution which most astronomers used to calculate distances for most of the last century (pre-Λ) Remember also:

dl = (1+ z)dp, da =dp

(1+ z)

Distance in an open matter dominated Universe

•  From 1990s onwards it was clear ΩM<1 this implied the possibility of an open Universe and we needed, in brief:

!RR!

"#

$

%&

2

=8!G"

3'kc2

R2

At to

Ho2 =

8!G"o3

'kc2

Ro2

1=(M 'kc2

Ho2Ro

2

kc2 = Ho2Ro

2[(M '1]

Subs for kc2into above replace "="oRoR

!

"#

$

%&

3

& divide by Ho2 (Ro R)2

) !R = RoHo[(MRoR'(M +1]

12

dr

[1' kr2 ]1

20

r1

* =cdtRt1

to

* =cRr1

ro

* dR!R

as !R = dR dt

Sub x = RRo

=1

(1+ z)

!dr

[1" kr2 ]1

20

r1

# =cHo1

(1+z)

1

# [$M

x"$M +1]"

12 x"1dx

“It is straightforward to show that…” Cosmology, Weinberg

Translation:

After 20 years of sporadic effort it was shown that:

valid for all k

dp = Ror1 =2cHo

[!Mz+ (!M " 2)( (1+!Mz) "1)!M2 (1+ z)

]

Mattig’s Formulae (1958)

The Lot Universe •  In the late 1990s the data favoured a non-zero Λ •  The full-monty has no analytic solution but we can

derive the integral following the same route:

1.  Start with Friedmann Equation 2.  Set variables to todays values 3.  Substitute for qo, Ho, ΩM, ΩR, ΩΛ

4.  Rearrange to get kc2 5.  Sub back into F.E. 6.  Express variables in terms of Ro/R 7.  Substitute for Ωs 8.  Rearrange and tidy to get dR/dt 9.  Put into RWM

•  Step1: Write down Friedmann equation:

•  Step 2: Set to todays values:

•  Step 3: Substitute for Ho & Ωs

•  Step 4: Rearrange to get kc2

•  Step 5: Substitute back into FE

!RR!

"#

$

%&

2

=8!G"3

+'3(kc2

R2

!RR!

"#

$

%&o

2

=8!G"o3

+'3(kc2

Ro2

1=)M +)' (kc2

Ro2Ho

2

kc2 = Ro2Ho

2[)M +)R +)' (1]

!RR!

"#

$

%&

2

=8!G"3

+'3(Ro2Ho

2[)M +)R +)' (1]R2

•  Step 6: Express all variables as Ro/R

•  Step 7: x by R2/(Ro2Ho

2 ) & sub for Ωs

•  Step 8: Rearrange to get dR/dt

•  Age of Universe:

•  No analytic solution known, good approx =

!RR!

"#

$

%&

2

=8!G"M ,o

3(RoR)3 + 8!G"R,o

3(RoR)4 + '

3(Ro2Ho

2[)M +)R +)' (1]R2

!R2

Ro2Ho

2 =)M ,oRoR+)R,o(

RoR)2 +)' (

RRo)2 ()M ,o ()R,o ()' +1

!R = RoHo[)M ,oRoR+)R,o(

RoR)2 +)' (

RRo)2 ()M ,o ()R,o ()' +1]

12

tage = dt0

t0

! =dR!R0

R0

!

tage =1Ho0

1

! ["M (1+ z)+"R (1+ z)2 +"# (1+ z)

$2 +"k ]$12d 1

(1+ z)

tage =23Ho

[0.7!M + 0.3" 0.3!# ]"0.3 =13Gyrs

Distance R2dr2

1! kr2 = c2dt2

dr

[1! kr2 ]1

2o

r1

" = c dtRt1

to

" = c 1RdR!RR

Ro

"

dr

[1! kr2 ]1

2o

r1

" =c

RoHo

1R

[#M ,oRoR+#R,o(

RoR

)2 +#$ ( RRo

)2 !#M ,o !#R,o !#$ +1]!1

2 dRR1

Ro

"

Rodr

[1! kr2 ]1

2o

r1

" =cHo

(1+ z)!1

1(1+z)

1

" [#M ,o(1+ z)+#R,o(1+ z)2 +#$ (1+ z)!2 +#k ]

!12d 1

(1+ z)

Rodr

[1! kr2 ]1

2o

r1

" =cHo

1[#M ,o(1+ z)

3 +#R,o(1+ z)4 +#$ +#k (1+ z)

2 ]1(1+z)

1

" d 1(1+ z)

Where: dp = Ror1

Messy, but easy to evaluate on a computer. This is the expression we all use today, made a little easier because k=0

Distances in cosmology

Luminosity distance Angular diameter distance