Observables

• So far we have talked about the Universe’s dynamic evolution with two Observables– a –the scale factor (tracked by redshift)– t – time

So a cosmological test would be to compare age with redshift.

But cosmic clocks are in short supply.

Proper Distance

• Defined as what a ruler would measure at an instantaneous time. Use R-W Metric.

For a fixed t, angle, going from r=0 to r=r.

€

ds2 = (cdt)2 − a2(t)dr2

1− kr2+ r2(dθ 2 + sin2θdφ2

⎛

⎝ ⎜

⎞

⎠ ⎟

dt = 0,dθ = 0,dφ = 0

dproper = ds∫ =a(t)dr

1− kr2

⎛

⎝ ⎜

⎞

⎠ ⎟

0

r

∫ = a(t)

arcsin(r)

r

arcsinh(r)

k = +1

k = 0

k = −1

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪= a(t)S(r)

dproper (t0,r) = a0S(r) =a0

adproper (t,r)

dcomoving ≡a

a0

dproper = aS(r)

0

First Go at Hubble Law

( ) ( )

small. is d if )(

)()(),(

0dHdtHv

da

aRSaraS

dt

drtd

dt

dv

proper

properproper

≈=

====&

&

Objects which have an apparent velocity proportional to their distance. The Hubble Law.

Somewhat tricky use of v here, which is not reallyA velocity, but rather a redshift.

Origin of Redshift

)(

1

0

1

)(arcsinh

)arcsin(

1

)sin(1

sin(1

)()(0

particles massless allfor 0

02

2/1

0

22222

2

22222

2222

2

0

0

rS

k

k

k

r

r

r

kr

dr

a

cdt

ddrkr

dr

a

cdt

ddrkr

drtacdtds

ds

rt

t

rt

t

≡⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−==+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−−==

=

∫∫

∫∫ φθθ

φθθ

Massless particles (e.g. photons) travel on geodesics in GR – e.g.

0 00,t

r,t0

t t+t

=ct=1/t

When light is emitted from a object, its wavelength is represents the time interval between the wavelength peaks.

frame)(rest light ofh wavelengtEmitted

frame) s(Observer'light ofh wavelengtObserved

Redshift theof Definition

0

0

e

e

ez

−

≡

S(r) represents a comoving coordinate – it is invariant as the Universe expands

1

11

/

/

)(

00

00

0000

00

0

0

00

0

−==−

−=−

===

=

=

== ∫∫++

aa

z

aa

aa

cc

tt

aa

tt

atc

atc

rSa

cdta

cdt

e

e

e

ee

tt

t

tt

t

In the limit that t (and hencet0) are small, integral becomes

As the Universe Expands,Light is redshifted!

The Expansion Stretches Light WavesThe Expansion Stretches Light Waves

The wavelength of light increases as it The wavelength of light increases as it travels through the expanding travels through the expanding Universe - “Redshift”Universe - “Redshift”

The longer the light has been The longer the light has been travelling, the more it’s Redshifted – travelling, the more it’s Redshifted – Not a Doppler Shift!Not a Doppler Shift!

Close, Far,Close, Far,Recent AncientRecent Ancient

Luminosity Distance• We observe the Universe via luminous objects.

How bright they appear as a function of redshift is a fundamental observable.

2/1

2

4

UniverseEuclidean in light for law square inverse 4

⎟⎟⎠

⎞⎜⎜⎝

⎛≡

=

fL

d

dL

f

L π

π

Imagine we have a monochromatic bit of lightEmitted in all direction. How bright will it appearAs a function of distance

• Observed flux is Luminosity spread out over the surface area of a sphere with radius of the proper distance.2

024 ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

aa

dL

fpπ

But! There is time dilation – photons arrive moreslowly (same idea of redshift) as scale factor changes

But! There is energy dimunition.Wavelength increases…So E per photon diminished.

Add two factors of a/a0

)1(

4 &

4

0

2/12

02

zda

add

f

Ld

a

a

d

Lf

ppL

Lp

+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛≡⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ππ

Angular Size Distance…

How big does a rod (of length l) look as a function of distance?

€

θ =l

d for Euclidean

dA ≡l

Δθ

ds2 = (cdt)2 − a2(t)dr2

1− kr2+ r2(dθ 2 + sin2θdφ2

⎛

⎝ ⎜

⎞

⎠ ⎟

l = ds∫ = a(t)rdθ∫ = arΔθ

dA = ar = dcomoving (k = 0) =a

a0

dproper =dp

(1+ z)=

dL(1+ z)2

True as derived here for flat UniverseBut always true!

dt,dr,d = 0

Move to observable space…

⎥⎦

⎤⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛ +−=−

+−⎟⎠

⎞⎜⎝

⎛ ++−=

⎥⎦

⎤⎢⎣

⎡ +−−−+=

≡

−≡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−

−+=

...2

11

)(

...)(2

1)(

...)(2

1)(1)(

Constant Hubble )(

Parameter.on Decelerati )(

)(

...)(

)(2

1)()(1)(

20

00

20

20

000

20

200000

0

0

0

20

002

2

0

20

20

02

2

0

000

zq

zH

tt

ttHq

ttHz

ttHqttHata

a

tdtda

H

tdtda

atdtad

q

a

ttt

dt

ad

a

ttt

dt

daata Taylor Expansion

Substitute in deceleration and Hubble Constant

Non-trivial algebra…

⎥⎦

⎤⎢⎣

⎡ ++−=

⎥⎦

⎤⎢⎣

⎡+

−+−=

=⎥⎦

⎤⎢⎣

⎡+−⎟

⎠

⎞⎜⎝

⎛ ++−+=+

==−

=

=

⎥⎦

⎤⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛ +−=−

+−⎟⎠

⎞⎜⎝

⎛ ++−=

∫∫

∫ ∫

...)1(2

1

...2

)()(

...)(2

1)(1)1(

caseflat for the )(1

0 and metricW -R using ray,light afor

...2

11

)(

...)(2

1)(

20

00

20

20

00

20

20

000

00

02

2

20

00

20

20

000

00

0

zqzHa

cr

ttHtt

a

cr

rdtttHq

ttHa

cdtz

a

c

rrSkr

dr

a

cdt

ds

zq

zH

tt

ttHq

ttHz

t

t

t

t

t

t

r

Substitute in For z

Integrate

Substitute in for t0-t

AproperL

proper

proper

r

proper

dzdzd

zqzzH

cd

zqzHa

cr

zq

zH

tt

ttHqttHata

ard

rSta

k

k

k

r

r

r

takr

drtadsd

2

20

0

20

00

20

00

20

200000

02

)1()1(

...)1(2

1

)1(

...)1(2

1

...2

11

)(

...)(2

1)(1)(

)()(

1

0

1

)(arcsinh

)arcsin(

)(1

)(

+=+=

⎥⎦

⎤⎢⎣

⎡ +−++

=

⎥⎦

⎤⎢⎣

⎡ ++−=

⎥⎦

⎤⎢⎣

⎡+⎟

⎠

⎞⎜⎝

⎛ +−=−

⎥⎦

⎤⎢⎣

⎡ +−−−+=

=

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−==+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−== ∫∫

Substitute in for t0-t in a(t)

Multiply a(t) and r expansions to get