1 Redshift and Conformal Time

1 Redshift and conformal time

Photon trajectories. Recall from GR that a photon’s trajectory is described by xµ(s) where s is the aﬃne parameter. The photon’s momentum is

dxµ P µ = . (1) ds The photon’s energy as seen by an observer with 4-velocity u is

E = u P µ. (2) − µ In vacuum the photon travels along a geodesic: dP µ = Γµ P αP β. (3) ds − αβ The photon travels on a null curve, so

µ ν gµν P P =0. (4)

Redshift. In the last lecture we considered the FRW metric,

ds2 = dt2 + a2(t)[dχ2 + f(χ)(dθ2 + sin2 θ dφ2)]. (5) − Let’s imagine that we’re at the origin χ = 0, looking in some direction (θ, φ), at time t0. A photon is emitted somewhere else in the universe at (ti,χi,θ,φ), initially has energy Ei, and starts traveling toward us. Our job is to determine the energy E0 that is observed. By spherical symmetry, a photon traveling radially will stay at constant θ, φ. (Literal statement: if P θ = P φ = 0 then dP θ/ds = dP φ/ds = 0.) The requirement of a null trajectory then means

E2 + a2(P χ)2 = 0; (6) − for inward-going photon, E P χ = . (7) − a So let’s get out the Christoﬀel symbols:

dE E E 2 = Γt E2 Γt E Γt . (8) ds − tt − tχ − a − χχ − a

But of these symbols the only nonzero one is

t Γχχ = aa,˙ (9) so dE a˙ = E2. (10) ds −a

1 We really want dE/dt so use dt = P t = E, (11) ds so dE dE/ds a˙ E˙ = = E. (12) ≡ dt dt/ds −a The solution to this, as expected, is E a−1, or in terms of wavelength λ E−1 a. ∝ ∝ ∝ If a photon was emitted at time ti then its wavelength observed today must be a(t0) λ0 = λi. (13) a(ti) It is conventional to deﬁne the redshift z according to λ λ a(t ) a(t ) z 0 − i = 0 − i . (14) ≡ λi a(ti)

So if we normalize the scale factor today, a(t0) = 1, then 1 a(t )= . (15) i 1+ z The redshift-time relation z(t) is simply another way to parameterize the expansion of the Universe. Conformal time. One of the most important quantities in cosmology is the relation between redshift and “distance” – if we see an object at z = 2, what is its radial coordinate χi where the photon was emitted? µ Since the photon’s trajectory is null, u uµ = 0: (ut)2 + a2(uχ)2 = 0 (16) − for a photon moving radially (uθ = uφ = 0). This means dχ uχ 1 = = . (17) dt ut ±a For a photon moving toward us we take the solution, so integrate: − t0 dt χ0 χi = . (18) − − Zti a

The radial coordinate today (observed) is χ0 =0 so t0 dt χi = . (19) Zti a Let’s deﬁne the conformal time η by dt η(t)= , (20) Z a

2 so χ = η η (21) i 0 − i where η0 is the conformal time today and ηi is the conformal time of emission. We haven’t defined the “zero point” of conformal time (constant of integration in Eq. 20). Usually take η = 0 at the Big Bang, except when studying inflation. The conformal time η, scale factor a, redshift z, and proper time t are all different possible time coordinates and we are free to choose among them. When we do perturbation theory it will be useful to write the metric with η instead of t. Since dt = a dη, (22) we have ds2 = a2(η)[ dη2 + dχ2 + f(χ)(dθ2 + sin2 θ dφ2)]. (23) − In the special case of a spatially flat universe where f(χ) = χ2 the quantity in brackets is the Minkowski metric (in polar coordinates). The factor a2 is a conformal transformation, hence the name “conformal time.” Horizon. The most distant objects that we can see are the ones whose light left them at the Big Bang. Their radial coordinate χ is

χ = η0. (24) The sphere centered on the observer at this distance is known as the horizon. Objects beyond the horizon cannot be seen. The horizon only exists if the integral dt/a is convergent at the Big Bang. (We’ll come back to this when we study inﬂation.)R

2 Distance measures

We’ve already described one measure of distance to a galaxy, χ. This is called the radial comoving distance. It is the distance that would be measured at the present epoch by laying down a sequence of rulers end-to-end from us to the galaxy that are at rest with respect to the comoving observers. That is,

galaxy χ = ds, (25) Zobserver where the path of integration is at constant t,θφ and is at the present, a = 1. The word “comoving” here refers to the fact that the distances are measured with the size of the Universe at the present epoch, not the fact that the rulers are at rest with respect to the comoving observer. Nobody can measure the radial comoving distance, so we have several other quantities. The angular diameter distance D , which is the distance that goes in • A the relation between angular diameter ∆θ and physical diameter Sphys: ∆θ = S/DA.

3 The comoving angular diameter distance r, which is the distance that goes • in the relation between angular diameter α and the comoving diameter Scom (i.e. the diameter of a stucture today if it expands with the Hubble ﬂow): ∆θ = Scom/r.

The luminosity distance DL, which is the distance that goes in the relation • between ﬂux F of a source (in W/m2) and luminosity L (in W): F = 2 L/(4πDL). So we need to calculate each of these. Angular diameter distance. From the metric we can see that the physical diameter subtended by an object is

2 2 2 Sphys = ds = √ds = a f(χ) dθ = a f(χ) ∆θ. (26) Z Z Z p p The angular diameter distance is the coeﬃcient of proportionality,

DA = a f(χ). (27) p A special case is the spatially ﬂat universe, where f(χ)= χ2. Then

DA = aχ. (28)

This distance can be measured if we have an object of known physical size (“standard ruler”). Comoving angular diameter distance. Now instad of using the physical diameter of an object, we will use the comoving diameter, that is the size if it keeps expanding with the Hubble ﬂow. This means that the object expands by a factor of a0/a =1/a. Thus S S = phys = f(χ) ∆θ. (29) com a p The comoving angular diameter distance is the coeﬃcient of proportionality,

χ K =0 − r = f(χ)=  K 1/2 sin(K1/2χ) K > 0 (30) p  ( K)−1/2 sinh[( K)1/2χ] K < 0 − −  Note that r = χ for the spatially ﬂat universe. This distance can be measured if we have an object of known comoving size. Example would be a statistical feature in the distribution of galaxies, which will grow as we go to later times because of Hubble expansion. For most objects it is easier to measure redshift than distance. If we know z and hence a =1/(1 + z) and one of r or DA then we can ﬁnd the other from χ D = aχ = . (31) A 1+ z

4 Luminosity distance. A third possible distance arises from the relation of ﬂux to luminosity, L F = 2 . (32) 4πDL Need to ﬁnd constant of proportionality between L and F . Consider a blackbody of diameter Sphys and temperature T . The phase space density of photons emerging from it is 1 f = , (33) eEi/kT 1 − where Ei is the energy of the emitted photon. When these photons get to the observer, the energy has declined to E E = i . (34) 0 1+ z By conservation of phase space density (Liouville’s theorem), the phase space density today is 1 f = , (35) eE0(1+z)/kT 1 − so the observer sees a blackbody at temperature T0 = T/(1 + z). Now the luminosity of the object is

S 2 L =4π phys σT 4. (36) 2

σ = Stefan-Boltzmann constant. The ﬂux for a blackbody of solid angle

∆θ 2 Ω= π . (37) 2 is Ω F = σT 4 (38) π 0 [Constant out front occurs because the net downward ﬂux in one direction, say 4 the e3 axis, from a blackbody that ﬁlls the sky is σT0 . Therefore

2π π/2 4 dF dF dF σT0 = cos θ dΩ= cos θ sin θ dθ dφ = π . (39) Zhemisphere dΩ Z0 Z0 dΩ dΩ

The cos θ comes from the fact that the ﬂux is actually a vector.] This simpliﬁes to ∆θ 2 T 4 F = σ . (40) 2 1+ z

5 Compare to L: L S 2 =4π phys (1 + z)4. (41) F ∆θ

We recognize the ratio Sphys/∆θ as the angular diameter distance:

L =4πD2 (1 + z)4. (42) F A

2 So the left hand side is 4πDL and thus

2 DL = DA(1 + z) = r(1 + z). (43)

3 Example: Einstein de Sitter Universe

Now that we’ve done the theory, let’s see how these equations play out in the simplest universe, the Einstein-de Sitter universe. This is a universe that is spatially ﬂat and consists only of matter (w = 0). It is NOT the real universe because it doesn’t have a Λ. Density today is given by the Friedmann equation in terms of the Hubble constant, 8 H2 = πGρ , (44) 0 3 0 so 3H2 ρ = 0 . (45) 0 8πG The scale factor as a function of time we solved in the previous lecture:

3(1 + w) 2/3(1+w) 8 1/3(1+w) a = πGρ t2/3(1+w). (46) 2 3 0

For w = 0 and above ρ0: 3 2/3 a = H t . (47) 2 0 The time today (age of the Universe!) is 2 t0 = . (48) 3H0 The Hubble constant as a function of time is

2 − H = = H a 3/2. (49) 3t 0 The time as a function of scale factor is 2 t = a3/2. (50) 3H0

6 The conformal time is

2/3 2/3 2/3 1/3 dt 2 dt 2 1/3 2 2 1/2 2 1/2 η = = 2/3 = 3t = 3 a = a . Z a 3H0 Z t 3H0 3H0 3H0 H0 (51) The conformal time today (horizon distance!) is 2 η0 = . (52) H0 Now let’s look at distance-redshift relations. The comoving radial distance is 2 1/2 2 1 χ = η0 η = (1 a )= 1 . (53) − H0 − H0 − √1+ z For a spatially ﬂat universe, r is the same as χ. The angular diameter and luminosity distances diﬀer by a factor of 1 + z:

2 1 DA = 1 . (54) H0(1 + z) − √1+ z and 2 D = 1+ z √1+ z . (55) L H − 0 In the limit where z 1 can show ≪ z χ,r,DA,DL . (56) → H0 This relation between the distance and redshift is known as Hubble’s law and we will show next time that it applies to all nonsingular FRW models.

4 Density parameters, general expressions for the distances

Density parameters. It is often helpful to define the dimensionless density parameters that describe the distribution of the Universe’s energy in different constituents. To do this let’s define the critical density:

3H2 ρ = . (57) crit 8πG From the Friedmann equation we know that this is the density that would make the Universe ﬂat. For each constituent in the Universe, we can deﬁne a density parameter: ρX ΩX = . (58) ρcrit

7 Here X could be baryons, dark matter, radiation, cosmological constant, etc. The Friedmann equation then says 8 K πGρ Ω = H2 + , (59) 3 crit X a2 XX or: K H2 Ω = H2 + . (60) X a2 XX It is common to deﬁne a curvature parameter, K Ω . (61) K ≡−a2H2 This way the Friedmann equation simply says:

ΩX +ΩK =1. (62) XX

Constraints on curvature are often written in terms of ΩK . Interpretation: For a spatially ﬂat universe, Ω = 0. • K For a closed universe, ΩK < 0 and the radius of curvature is aR = a/√K = • Ω −1/2H−1. | K | For an open universe, ΩK > 0. The imaginary radius of curvature is • 1/2 −1 iΩK H .

The curvature today (a = 1) can be expressed in terms of ΩK :

K = Ω H2. (63) − K 0

Warning: ΩK does not correspond to an actual density, it’s just a number introduced to make the Friedmann equation look simple. Warning 2: In the literature Ωm,ΩΛ, etc. usually refer to values today, even though physically they’re redshift-dependent. Expansion history. Given a Hubble constant H0 and a set of Ωs it’s possible to construct the Hubble constant and distances as a function of redshift. We’ll consider a universe that has Cosmological constant, Λ (density ρ =constant); • Λ Curvature, K; • (Nonrelativistic) matter, m (ρ a−3); • m ∝ Radiation, r (ρ a−4). • r ∝

8 So today the Friedmann equation says

ΩΛ +Ωm +Ωr +ΩK =1. (64) Let’s now go to some previous scale factor a. The total density of the universe back then was

−3 −4 ρ(a) = ρΛ0 + ρm0a + ρr0a −3 −4 = ρcrit,0(ΩΛ +Ωma +Ωra ). (65) The Hubble constant at a is: 8 K H2 = πGρ(a) 3 − a2 2 8 − − Ω H = πGρ (Ω +Ω a 3 +Ω a 4)+ K 0 3 crit,0 Λ m r a2 2 −3 −4 −2 = H0 (ΩΛ +Ωma +Ωra +ΩK a ). (66) Let’s deﬁne the energy function (a) by: E −3 −4 −2 (a)= ΩΛ +Ωma +Ωra +ΩK a . (67) E p Then the Hubble constant varies with redshift according to

H(z)= H0 (z). (68) pE The present age of the universe is:

1 1 1 1 da da −1 da t0 = dt = = = H0 . (69) Za=0 Z0 a˙ Z0 aH Z0 a (a) E p This integral can also be written in terms of redshift: ∞ −1 dz t0 = H0 . (70) Z0 (1 + z) (z) E p The age of the universe at some previous time can be written as: ∞ ′ −1 dz t(z)= H0 ′ ′ . (71) Zz (1 + z ) (z ) E p For observers these relations are extremely useful because z is usually observable whereas t, H, etc. are much harder. Distance measures. The conformal time in one of these models is

dt da da − da − dz η = = = = H 1 = H 1 . (72) Z a Z aa˙ Z a2H 0 Z a2 (a) − 0 Z (z) E E p p So at a particular redshift, and taking the initial condition η =0 at z = : ∞ ∞ ′ −1 dz η(z)= H0 ′ . (73) Zz (z ) E p 9 The radial comoving distance is:

z ′ −1 dz χ(z)= η0 η(z)= H0 ′ . (74) − Z0 (z ) E p We can then obtain the other distance measures from χ K =0 − r =  K 1/2 sin(K1/2χ) K > 0 (75)  ( K)−1/2 sinh[( K)1/2χ] K < 0 − −  and r D = r(1 + z); D = . (76) L A 1+ z