The Expanding Universe Distance Ladder & Hubble s Law Robertson-Walker metric Friedman equa7ons Einstein De SiKer solu7ons Cosmological distance Observed proper7es of the Universe 1
The distance ladder Measuring distances in astronomy is hard. The most direct way of measuring distance is parallax. d = 1AU tan p 1 p AU The parsec is the distance at which a parallax of 1 is observed. 1 pc = 2.06x10 5 AU = 3.26 light-year = 3.09x10 18 cm State of the art in measuring parallaxes is space based (Gaia satellite successor to Hipparchus). Up to 10 kpc distances can be measured with 10% precision 2
Cepheid variable stars Cepheid stars are variable stars. They are not main sequence stars. Instead they are red giants that undergo oscilla7ons in size, and hence in luminosity. The period of the oscilla7ons varies from a few days to a few months. Stefan-Boltzman L = AT 4 HenrieKa LeaviK no7ced a rela7onship between average luminosity and period this was done with Cepheid stars in the SMC and LMC. Cepheids were used by Edwin Hubble to measure the distance to nearby galaxies. 3
Hubble s law and redshi: Hubble observed that the redshib (receding speed) of galaxies was linearly correlated with distance. Redshib is now a distance indicator 0 = (1 + z) s 0 = 1+ 1 0! (1 + ) (nearby) The Universe is expanding. This rela7onship is Hubble s law: v = H 0 D. With Cepheids, and using the Hubble space telescope: H 0, Hubble s constant, is 73.00 ± 1.75 km.s -1.Mpc. (arxiv:1604.01424) Because the Hubble constant has (tradi7onally) been uncertain, it is common to see the defini7on H, with H = 100 km.s -1 0 = hh.mpc. 4
More on the distance ladder There are many more steps in the distance ladder Redshib SNe Ia Tully Fisher Grav Lens Quasars Main Sequence Fijng Proper Mo7on Parallax Cepheids & RR Lyrae 1 10 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 d(pc) Proxima Centauri Galac7c Center LMC M31 Virgo Cluster z = 1 (in some defini7on of distance) 5
Distances Phys 8803 Special Topics on Astropar7cle Physics Ignacio Taboada 6
From Special to General relabvity Special rela7vity Iner7al frames: ds 2 = c 2 d 2 = c 2 dt 2 dx 2 dy 2 dz 2 = c 2 dt 2 dr 2 r 2 (d +sin 2 d 2 ) Take an iner7al frame S and an accelerated frame S. S has small accelera7on aî with respect to S. S : x, y, z, t S 0 : x 0,y 0,z 0,t 0 Then Now So x 0 = x 1/2at 2 dx = @x @x 0 dx0 + @x @t dt = dx0 + at dt ds 2 =(c 2 a 2 t 2 )dt 2 2atdx 0 dt dx 02 dy 02 dz 02 For a clock in S : ds 2 = c 2 d 2 = c 2 dt 02 =(c 2 2ad)dt 2 Now all the way General rela7vity: ds 2 = g µ dx µ dx 7
The Robertson-Walker metric General Rela7vity provides a framework to describe the Universe expansion. The Robertson-Walker metric is apple ds 2 =(cdt) 2 a(t) 2 dr 2 1 Kr 2 + r2 (d 2 +sin 2 d 2 ) Here a(t) is dimensionless and known as the expansion parameter. The constant K is the Gaussian curvature and has units 1/length 2. K=0 is for an Euclidean universe. K>0 posi7ve curvature, K<0 nega7ve curvature. The above metric uses comoving coordinates, which are the coordinates on which the Cosmological Principle applies. Cosmological Principle: the Universe is homogeneous and isotropic, at least to first approxima7on. Comoving 7me is the 7me since the Big Bang as measured in a clock that follows the Hubble flow. 8
Friedman Cosmological EquaBons Using Einstein s equa7on and assuming the Universe is filled with a homogeneous and isotropic fluid of density ρc 2 and pressure p, results in: ä a = 3 ȧ a 2 = 8 3 G K a 2 + 3 In the equa7ons above, both p and ρ are func7ons of the comoving 7me too. In general, an equa7on of state rela7ng pressure and density is also needed. Adiaba7c expansion of the Universe is assumed so d( c 2 a 3 )/da 3 = 4 G 3 ( +3p c ) Note that the expansion of the Universe is measured by the Hubble parameter (not a constant anymore) H(t) =ȧ(t) a(t) p 9
CriBcal Density In the case of a flat Universe (K=0) and no cosmological constant ȧ The density takes the value: a 2 8 G 3 c (t) = 3H2 (t) 8 G =0 This is known as the cri7cal density. It s value is: c,0 = 3H2 0 8 G =1.88 10 29 h 2 g cm 3 The present es7mate of the baryonic density in the Universe is: B,0 =3 10 31 g cm 3 10
Density parameter ΛCDM cosmology The ra7o of the density to the cri7cal density is the density parameter (t) = (t) c (t) The current baryonic maker density parameter is B h 2 =0.02205 There is a (present) contribu7on from dark maker: cdm h 2 =0.1199 And a (present) contribu7on from dark energy: =0.6866 With h = 0.673 the Universe is flat. (Note that h had a different value in a previous slide) Planck Cosmological parameters: arxiv 1502.01589 11
Einstein De SiJer Model Equa7on ȧ 2 + Kc 2 = 8 G (no cosmological constant) 3 a2 can be re-wriken thus: ȧ a 2 2 8 G a 3 = a 0 Kc2 a 2 0 A generic equa7on of state takes the form p =! c 2 ω=0 represents pressure-less material (dust) this is also a good representa7on of a non-rela7vis7c ideal gas. ω=1/3 represents a radia7ve fluid (e.g. photons) ω=-1 represents dark energy the cosmological constant 12
Einstein De SiJer Model From d( c 2 a 3 )= pda 3 and p =! c 2 we get We can use this in equa7on (no cosmological constant) So that ȧ a 2 ȧ a c H 2 0 = 2 a 3(1+!) = 0 a 3(1+!) 0 ȧ a 8 G 3 a a 0 2 0 c H 2 0 2 = a0 a Kc2 a 2 0 3(1+!) = Kc 2 a 2 0 In the case of K=0, which implies ȧ a 2 H 2 0 a0 0 = 0 / c =1 3(1+!) =0 a 13
Einstein De SiJer Model The K=0 solu7on is: a(t) =a 0 t t 0 2/3(1+!) For Ω < 1 (open Universe, K=-1) and ω=0 the parametric solu7on is: a(x) =a 0 (cosh x 1) 2(1 ) t(x) = 1 (sinh x x) 2H 0 (1 ) 3/2 And for Ω>1 (closed Universe, K=1) and ω=0 the parametric solu7on is: a(x) =a 0 (1 cos x) 2(1 ) t(x) = 1 (x sin x) 2H 0 (1 ) 3/2 14
Dark MaJer dominated Universe For K=0 and ω=-1 p a(t) / e /3t ω=1 is not the only possibility and it doesn t even have to be constant with proper 7me t. The best measurement of w, assuming that it is constant is ω = 1.00 ± 0.06. 15
Einstein De SiJer Universe a(t) t The ω=1/3 general solu7on is: a(t) =a 0 (2H 0 1/2 0 t) 1/2 1+ 1 0 2 1/2 0 H 0 t! 1/2 16
Luminosity and proper distance Let s return to the Robertson-Walker metric apple ds 2 =(cdt) 2 a(t) 2 dr 2 1 Kr 2 + r2 (d 2 +sin 2 d 2 ) The proper distance assume that the two ends are measured simultaneously in proper 7me. If at t 0 two events have a proper distance r 0, then at 7me t they have a proper distance a(t)/a(t 0 ) r 0. But proper distance is not directly measurable in prac7ce. Let s look at a different defini7on of distance. The redshib of a photon is defined as: z = o e Where λ e is the emiked wavelength and λ 0 is the observed wavelength. e 17
Luminosity distance Photons travel along null geodesics, ds 2 = 0. This implies: Z t0 t e cdt a(t) = Light emiked the source at t 0 e = t e + is observed at t 0 0 = t 0 + t 0. Assuming small Δt e and Δt 0, f(r) does not change. Here Δt e can represent, e.g. the frequency of emiked light. Then, Z t0 t e Z 0 R p 1 cdt a = dr = f(r) Kr 2 Z t 0 0 t 0 e cdt a t e For small Δt e and Δt 0, you find t 0 a(t 0 ) = t e a(t e ) A photon frequency will thus be e a(t e )= 0 a(t 0 ) 18
Luminosity distance a(t From which it follows, e ) = a(t 0). e 0 And using the defini7on of redshib: 1+z = a(t 0 )/a(t e ) Let P be the power emiked by a source at 7me t and co-moving distance r. Let p 0 be the power per unit area (flux) observed at 7me t 0. We want Luminosity distance to be such that: p 0 = P 4 d 2 L The area of a sphere centered at the source, at 7me t 0 is 4 a 2 0r0 2. 19
Luminosity distance Photons emiked by the source are observed with a redshib a/a 0 And photons emiked within a small 7me Δt e are observed within a 7me Δt 0 = a(t)/a 0 Δt e. Thus: p 0 = P 4 a 2 0 r2 a a 0 2 = P 4 d 2 L Two factors: one for energy, another for 7me d L = a 2 0 There are other defini7ons of distance in cosmology I skip that. r a 20
Type Ia Supernovae: Luminosity distance Top plot shows the Hubble diagram obtained with SN type Ia This confirms that Hubble s law is good and that SN Ia are good standard candles This is the same plot, but shows. The data is best described for K=0 and Ω m =0.3. Because K=0 can only be achieved for Ω total =1, this implies that there is a cosmological constant. 21
Cosmic Microwave background. It s easy to see that the early Universe, was denser and hoker that it is today. The high density coupled photons to maker. At some past redshib (z~1100) the Universe became transparent to photons. This is era of last scakering. This background of photons con7nues to fill the Universe and it has redshibed, due to expansion. It s extremely well described by a Planck func7on with T = 2.7255 ± 0.0006 K Planck (satellite) Phys 8803 Special Topics on Astropar7cle Physics Ignacio Taboada 22
Density fluctuabons The universe is very homogeneous. But inhomogeneity is observable from individual galaxies to the 100 Mpc scale. The inhomogenei7es seen in the CMB grow into today s density inhomogenei7es. The density fluctua7ons are described as: (~x )= (~x ) < > < > It is conven7onal to describe the density fluctua7ons as a Fourier expansion (periodic boundary condi7ons are assumed over a very large volume V). The variance of fluctua7ons is: (~x )= X ~ k e i~ k.~x < (~x ) 2 >= X ~k 2 = X P (k) (There no preferred direc7on for inhomogenei7es) 23
Density fluctuabons Galaxy power spectrum measured by SDSS. The best fit of ΛCMD is shown. Baryon Acous7c Oscilla7ons are inset. 24
CMB power spectrum Power spectrum of the CMB as measured by Planck. 25
CMB / Sne Type Ia / acousbc baryon oscillabons Best fit region for Planck, SN type Ia data and acous7c baryon oscilla7ons. Contour lines are for 68.3% and 95.4% confidence levels (1,2 sigma) 26