Why is the Universe Expanding?

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Angela Patterson
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1 Why is the Universe Expanding? In general relativity, mass warps space. Warped space makes matter move, which changes the structure of space. Thus the universe should be dynamic! Gravity tries to collapse the universe (7 years before Hubble s Law): Alexander Friedmann used Einstein s field equations to derive the Friedmann equation: ȧ a 2 = 8 G 3c 2 applec 2 R a(t) 2

2 Classical Derivation of Friedmann Equation Consider a uniform expanding sphere with mass density ρ and radius R. Total energy of the outer shell of mass m is E = 1 2 mṙ2 GMm R R(t) M = mass os sphere m = mass of shell M = 4 3 (t)r3 (t) E m = 1 2Ṙ2 4 3 G R2 Rearranging:! 2 Ṙ = 8 2E G + R 3 mr 2

3 Indroduce a dimensionless scale factor: r s represents comoving radius. a(t) is the dimensionless scale factor. Friedmann Equation in Newtonian form: R(t) =a(t)r s ȧ a 2 = 8 3 G + 2E m 1 r 2 sa 2

4 Possible Fates: Fate of expanding sphere depends on energy E. If E>0, sphere can expand forever. Shell s velocity greater than the escape velocity. If E=0, sphere expands forever, but at infinity it finally stops. If E<0, sphere reaches a maximum radius, and then begins contracting inward.

5 Friedmann Equation Derived classical equation: ȧ 2 = 8 a 3 G + 2E 1 m rsa 2 2 Friedmann Equation (based on GR): ȧ a 2 = 8 3c 2 G applec 2 R 2 0 a2 Fundamental differences: Energy density (of all sorts) determines acceleration (ε=ρc 2 if mass dominates the energy density) Energy term exchanged for curvature constant κ

6 Clicker question If κ=0, the universe will a) Expand forever (but stop an infinite time from now) b) Expand forever c) Eventually stop expanding and begin collapsing

7 Clicker question If κ=-1, the universe will a) Expand forever (but stop an infinite time from now) b) Expand forever c) Eventually stop expanding and begin collapsing

8 Fate of universe and geometry are related! open flat closed

9 Curvature Only Consider an empty universe. The Friedmann equation becomes If κ=0, then the universe is static. The other option is κ=-1 (open universe). Then ȧ = ȧ 2 = applec2 R 2 0 c R 0 = constant ) a(t) =ȧt Age of universe: t 0 = a(t 0) ȧ t 0 = a 0 /ȧ =1/H 0 = H y H 1 70

10 Critical Density Recall Friedmann Equation: ȧ 2 = 8 a 3c 2 G applec 2 R0 2a2 This can be expressed in terms of the Hubble constant today: H 2 0 = 8 G 3c 2 0 If we know the Hubble constant and the energy density, we can determine fate/geometry. Critical density: If κ=0, c c 2 = c = 3H2 0 8 G applec 2 R 2 0 a2 0

11 Using H 0 =70 km/s/mpc, Critical Density c = kg/m 3 = 6 H atoms/m 3 = M /Mpc 3 Observationally we have found that M = c 0.3 Note: H 2 0 = 8 G 3c 2 0 applec 2 R 2 0 a2 0 ) 1= M applec 2 R 2 0 a2 0 H2 0 apple c2 R 2 0 H2 0 = M 1

12 Clicker Question How does the current age of the universe depend on the mass density? a) a higher mass density means an older universe b) a lower mass density means an older universe c) The age of universe doesn t depend on the mass density! d) Are you kidding me? How do I figure this out? I m still trying to figure out what an expanding universe means

14 Fluid Equation Friedmann equation has two unknowns: a(t) and ε(t) Use first law of thermodynamics: dq = de + PdV For a homogeneous universe (and a large enough comoving volume) dq =0 Consider sphere of comoving radius r s : Energy in sphere: Fluid Equation: ) Ė + P V =0 V (t) = 4 3 r3 sa(t) 3 ) V = 4 3 r3 s3a 2 ȧ = V E = V ) Ė = V + V = V ) +3ȧ a ( + P )=0 +3ȧ a 3ȧ a

15 ) +3ȧ a Fluid Equation In general, for a given type of component, the equation of state can be expressed as P i = w i ( + P )=0 For a non-relativistic gas, w 0 For radiation, w=1/3 For each component the fluid equation holds: d i i = 3(1 + w) da a ) i (a) / a 3(1+w)

16 An Example: Flat universe with only matter For this case the Friedmann equation is: 2 3 a3/2 = ȧ 2 = 8 G 3c 2 a2 = 8 G 3c 2 0 a 1 r a 1/2 8 G 0 da = 3c 2 dt r 8 G 0 3c 2 t a(t) 3/2 = a(t) = t t 0 2/3 r 6 G 0 c 2 t = t/t 0

17 a(t) = t t 0 2/3 The Hubble constant is given by H 0 = ȧ a = 2 t=t 0 3 t 0 1 Thus the age of the universe is t 0 = 2 3 H 1 0 How does redshift depend on lookback time (t 0 t e )? z = a 0 a(t e ) z( t) = t 0 t0 1= t0 t e 2/3 1 2/3 2/3 1 1= 1 t 1 t/t 0 z δt to

18 Proper Distance The proper distance to a galaxy today is r (using Ryden s co-moving coordinate). But what is r and how does it relate to look-back time, t 0 - t e, or redshift? Use the null geodesic of the Robertson-Walker metric for light moving radially towards us: ds 2 =0=c 2 dt 2 a(t) 2 dr 2 c dt a(t) = dr ) r = Z t0 t e c dt a(t) For our matter-only example: Z " t0 dt d p (t 0 )= c 2/3 =3ct 0 1 t e t t 0 te t 0 1/3 # = 2c H 0 apple 1 1 p 1+z

19 Models with the Cosmological Constant Einstein introduced (or more properly didn t set to zero) a term Λ in his field equations such that a static universe is possible. From a Newtonian point of view, basically he added a fudge factor to Poisson s equation: r 2 + =4 G (~a = r ) This cosmological constant causes the Friedmann equation to become: ȧ 2 = 8 a 3c 2 G applec 2 R0 2 + Dark energy density: a2 3 ȧ a 2 = 8 G 3c 2 ( + ) applec 2 R 2 0 a2 = c2 8 G

20 Energy Density and Pressure due to Cosmological Constant Notice this dark energy density is constant! Recall the fluid equation: + 3(ȧ/a)( + P )=0 In order for the energy density to be a constant, dark energy must produce a negative pressure! P = = Thus w = -1 c 2 8 G This negative pressure competes with gravity.

21 Evaluating the Friedmann equation today: H 2 0 = 8 G 3c This eqn. can be re-expressed as 1= 8 G 3H H 2 0 applec 2 R 2 0 a2 0 1= 0 c c 2 + c c 2 applec 2 H 2 0 R2 0 1= M + + apple applec 2 H 2 0 R2 0 a2 0

22 Model: Flat universe with just dark energy ȧ a 2 = 8 3c 2 G applec 2 R 2 0 a2 + 3 For this model, ε=0 and κ=0. Thus, ȧ = r 3 a a(t) =a 0 exp "r # 3 (t t 0) How old is the universe for this model?

23 Clicker Question Which model is the solid line? a) Ω m = 0.3, Ω Λ =0.7 b) Ω m = 0.0, Ω Λ =1.0 c) Ω m = 1, Ω Λ =0

24 The two parameters Ω m and Ω Λ (combined with eqn of state) determine rate of expansion, fate, and age of universe.

25 Standard WMAP Model WMAP model parameters: Ω M = 0.27, Ω Λ = 0.73, Ωκ =0 Notice that, for the WMAP model, early on the universe was de-accelerating and then more recently started to accelerate. As universe expands, gravity gets weaker but dark energy density remains constant

26 Acceleration Equation Now let s combine the Friedmann equation and fluid equation: ȧ 2 = 8 a 3c 2 G applec 2 R0 2a2 2ȧä = 8 G 3c 2 ä a = 4 G 3c 2 use fluid equation: Acceleration Equation: a 2 +2 aȧ ȧ a +2 +3ȧ a ( + P )=0 ) ȧ a = 3( + P ) Note: energy density changes for two reasons: increasing volume and Pdv work ä a = 4 G ( +3P ) 3c2

27 The acceleration equation with dark energy is ä a = 4 G 3c 2 ( +3P )+ 3

28 Observational Tests Observationally, we observe cosmological redshift and apparent brightness (flux). Cosmological redshift is related to how much the scale factor increased since the light was emitted: z = em = obs / em 1= 1 a(t em ) Apparent brightness depends on how far away the galaxy is from us, which also depends on t em. Relationship between brightness, lookback time, and proper distance is model dependent Convert a(t) and F(d p [t]) models into predicted apparent brightness vs redshift: F(z) vs z 1

29 Clicker Question Consider these two cosmological models: Suppose we observed a SN with a certain redshift (say z=1). For which model would the SN be dimmer (further away)? a) The WMAP model b) The model with no cosmological constant c) Both models predict the same brightness

30 Clicker Question For which model would the look-back time of a white-dwarf supernovae having a redshift of 1.0 be the longest? a) top curve b) middle curve c) bottom curve

31 Luminosity Distance We define a luminosity distance by the equation d L = L 4 F 1/2 This would be the proper distance IF space is not curved AND space were not expanding. Suppose a standard candle is located at a co-moving distance r away from us (d p (t 0 )= r). The light we are receiving today was emitted at time t e. Today, the light emitted at time t e spans a geometric area A p (t 0 )=4 S apple (r) 2

32 Luminosity Distance Due to cosmological redshift, each photon loses energy by a fraction Due to cosmological time dilation, the rate of photons reaching us decreases by a factor of (1+z) Thus the observed flux is 0 = 1 e 1+z F = The luminosity distance is give by d L = S apple (r)(1 + z) L 4 S 2 apple(r)(1 + z) 2 Assuming r<<r 0, d L d P (t 0 )(1 + z)

33 Brightness vs. Redshift For a given model of a(t), we can predict F (z) = L 4 S 2 apple(r[z])(1 + z) 2 r(t e )=c Z t0 t e z(t e )= 1 a(t e ) dt a(t) 1 t e = a z

34 Brightness vs. Redshift Let s return to our model of a matter-dominated flat universe. t t0 a(t) = t 0 2/3 z = a 0 a(t e ) 1= t e 2/3 1 3/2 1 t e = t 0 = 2 z +1 3 H 0 1 r(z) = Z t0 t e cdt 2/3 =3ct 0 t t 0 " 1 1 3/2 z +1 te t 0 1/3 # = 2c H 0 apple 1 1 p 1+z F (z) = L 4 (1 + z) 2 r(z) 2 = LH0 2 i 2 16c 2 (1 + z) h1 2 p 1 1+z

35 Flux vs redshift Flux vs redshift

36 Dim m = m 0 Apparent Brightness vs. Redshift 2.5 log 10 F F 0 Bright Redshift Which model predicts the dimmest brightness (for a given redshift)? a) Ω m = 0.3, Ω Λ =0.7 b) Ω m = 0.3, Ω Λ =0.0 c) Ω m = 1, Ω Λ =0

37 Dim Apparent Brightness vs. Redshift Bright Redshift Objects with long lookback times are far away and consequently dimmer than objects with short lookback times.

38 High z supernovae

45 Supernova Cosmology Project Far Near

46 Data with Gamma Ray Bursts as Standard Candles

The Friedmann Equation R = GM R 2. R(t) R R = GM R GM R. d dt. = d dt 1 2 R 2 = GM R + K. Kinetic + potential energy per unit mass = constant

The Friedmann Equation R = GM R R R = GM R R R(t) d dt 1 R = d dt GM R M 1 R = GM R + K Kinetic + potential energy per unit mass = constant The Friedmann Equation 1 R = GM R + K M = ρ 4 3 π R3 1 R = 4πGρR