Lecture 3: From the CMBR to the first star Galaxies 626
Firstly, some very brief cosmology for background and notation: Summary: Foundations of Cosmology
1. Universe is homogenous and isotropic - need to consider large volumes - very good observational evidence from CMB
2. Universe is expanding - Hubble law compatible with homogenous / isotropic assumptions
3. Universe was once hot - existence of the microwave background with a thermal spectrum
4. Evolution described by General Relativity Expansion of the Universe Expansion of Universe cannot Homogeneity + isotropy alter the relative orientations of galaxies expanding with the Universe Means that if the present separation between two galaxies
is d0, then the separation at time t can be written as:
d = d0a(t) a(t) is the scale factor - it is dimensionless and depends upon time but not on position. Relative velocity of the two galaxies is: a˙ ! v = d˙ = d a˙ (t) = d 0 a
! Deﬁnition of the Hubble parameter is v = H x d, so: a˙ H = a
H is a function of time, present value is denoted H0
Sometime useful to deﬁne comoving coordinates. If we divide distances! by a(t), then two galaxies which simply recede from each other due to the Hubble expansion always have the same separation in comoving coordinates.
We will usually express densities as per comoving volume equivalent to the density they would have after expansion to the present time. Can derive the evolution of a(t) using mostly Newtonian mechanics, provided we accept two results from General Relativity: 1) Birkhoff’s theorem: this states (in part) that for a spherically symmetric system, the force due to gravity at radius r is determined only by the mass interior to that radius.
2) Energy contributes to the gravitating mass density, which equals: u energy density " + -3 m c 2 (ergs cm ) of radiation and relativistic particles density of matter ! Consider the evolution of a spherical volume of the Universe, radius L:
Sphere expands with the Universe, L so L = L0a(t)
Since expansion is described entirely by a(t), can consider any size sphere we want - if L is small `reasonable’ to assume that space is approximately Euclidean. Expansion of the sphere will slow due to the gravitational force of the matter (+energy) inside: d 2L GM = - dt 2 L2
! Note: no pressure forces because Universe is homogenous Contributions to the gravitating mass come from matter plus energy density from radiation: • Matter density ρm • Radiation with energy density u has pressure: 1 P = u 3 gravitating mass density is: 3P " = " + ! m c 2 Mass within sphere, radius L, is:
4 3 ! M = "V = #L " 3
! Substitute into acceleration equation: d 2L G 4 = - " #L3$ dt 2 L2 3
Since L = L0a(t), with L0 a constant, can write this as an equation for the evolution of the scale factor a(t): ! 4"G $ 3P' a˙˙ = - & # + ) a 3 % m c 2 ( (also substituting for ρ in the above expression)
• Matter density ρm > 0 !• Pressure of radiation is also positive RHS of the equation is always negative Impossible to have a static Universe Lack of static solutions is not a problem - Universe is expanding. But this was not known in 1917. Einstein therefore modiﬁed the equations of General Relativity so the equation becomes:
4"G $ 3P' * a˙˙ = - & # + ) a+ a 3 % m c 2 ( 3 Λ is the cosmological constant (the factor 3 is just convention). A positive cosmological constant tends to ! accelerate the expansion - i.e. as if the Universe is ﬁlled with material with a negative pressure.
Is a static solution stable? Properties of the cosmological constant Cosmological constant is assumed to be a smooth component… i.e. it does not cluster or clump together in the same way as ordinary matter. Original cosmological constant was… constant in time! This is just an assumption, however - models in which the vacuum energy varies with time are called quintessence.
For Λ to be important today, it must have a value comparable to the ﬁrst term in the equation: %36 -2 (for ρ ~ 10-30 g cm-3) " ~ 4#G$m ~10 s Gh t = =10"43 s `Fundamental’ unit of time is the Planck time: Planck c 5
#2 120 Might guess that " ~ tPlanck …bad guess by factor 10 . ! ! Cosmological constant problem… ! Which terms are most important?
4"G $ 3P' * a˙˙ = - & # + ) a+ a 3 % m c 2 ( 3 • Early times - energy density of radiation is large compared to the energy density of matter !• Later, matter dominates • Finally, if Λ is non-zero, eventually it dominates
Radiation dominated Each of these changes Matter dominated in different way as Universe expands - Cosmological constant distinct expansion dominated laws Galaxies 626
The cosmic microwave background Cosmic Microwave Background
Following recombination, photons that were coupled to the matter have had very little subsequent interaction with matter. Now observed as the cosmic microwave background. Arguably the most important cosmological probe, because it originates at a time when the universe was very nearly uniform: • Fluctuations were small - easy to calculate accurately (linear rather than non-linear) • Numerous complications associated with galaxy and star formation (cooling, magnetic ﬁelds, feedback) that inﬂuence other observables not yet important.
Basic properties: isotropy, thermal spectrum Anisotropies: pattern of ﬂuctuations Basic Properties of the CMB Excellent ﬁrst approximation: CMB has a thermal spectrum with a uniform temperature in all directions
Thermal spectrum: support for the hot big bang model Isotropy: evidence that the universe is homogenous on the largest observable scales The thermal radiation ﬁlling the universe maintains a thermal spectrum as the universe expands
Suppose that at recombination the radiation has a thermal spectrum with a temperature T ~ 3000 K. Spectrum is given by the Planck function: 2h" 3 1 B = " c 2 eh" / kT #1 At time t, number of photons in volume V(t) with frequencies between ν and ν + dν is: ! 8"# 2 1 dN(t) = V(t)d# c 3 eh# / kT $1
! Now consider some later time t’ > t. If there have been no interactions, the number of photons in the volume remains the same: dN(t) = dN(t' ) However, the volume has increased with the expansion of the universe and each photon has been redshifted: a3 (t' ) V(t' ) =V(t) ! a3(t) a(t) "' =" a(t' ) a(t) d"' = d" a(t' )
Substitute for V(t), ν and dν in formula for dN(t), and use fact that dN(t’) = dN(t) ! Obtain: '2 ' # 1 ' ' dN(t ) = 8" 3 ' V(t )d# c e(h# / kT )$(a(t')/ a(t )) %1 which is a thermal spectrum with a new temperature: a(t) T' = T ! a(t' ) Conclude: radiation preserves its blackbody spectrum as the universe expands, but the temperature of the blackbody decreases: ! T " a#1 "(1+z)
Recombination happened when T ~ 3000 K, at a redshift z = 1090. ! CMB Anisotropies
Universe at the time of recombination was not completely uniform - small over (under)-densities were present which eventually grew to form clusters (voids) etc.
In the microwave background sky, ﬂuctuations appear as: • A dipole pattern, with amplitude: "T #10$3 T Origin: Milky Way’s velocity relative the CMB frame. Reﬂects the presence of local mass concentrations - clusters, !superclusters etc. • Smaller angular scale anisotropies, with ΔT / T ~ 10-5 Experiments detect any cosmic source of microwave radiation - not just cosmic microwave background: • Low frequencies - free-free / synchrotron emission • High frequencies - dust
CMB dominates at around 60 GHz Also different spectra - can be separated given measurements at several different frequencies WMAP results K band - 22 GHz
W band - 94 GHz Directly `see’ the primordial CMB anisotropy at these frequencies Full sky map from WMAP
• Dipole subtracted (recall dipole is much larger than the smaller scale features) • Galactic foreground emission subtracted as far as possible Characterizing the Microwave Background Sky
First approximation - actual positions of hot and cold spots in the CMB is `random’ - does not contain useful information
Cosmological information is encoded in the statistical properties of the maps:
• What is the characteristic size of hot / cold spots? ~ one degree angular scale
• How much anisotropy is there on different spatial scales? CMB is a map of temperature ﬂuctuations on a sphere - conventionally described in terms of spherical harmonics Spherical harmonics Any quantity which varies with position on the surface of a sphere can be written as the sum of spherical harmonics:
"T (#,$) = almYlm (#,$) T % l,m spherical harmonic measured anisotropy function map as function! of weight - how much spherical polar angles of the signal is θ and φ accounted for by this particular mode The spherical harmonic functions themselves are just (increasingly complicated) trignometric functions, e.g.:
5 2 2i# Y22(",#) = 3sin "e 96$
! l = 6, m = 0 l = 6, m = 3 Having decomposed the observed map into spherical harmonics, result is a large set of coefﬁcients alm. Next compute the average magnitude of these coefﬁcients as a function of l: 2 Cl " |alm|
Plot of Cl as a function of l is described as the “angular power spectrum” of the microwave background. Each
Cl measures how much anisotropy there is on a particular angular scale,! given by: 180o " ~ l Angular power spectrum is basic measurement to compare with theory ! Observational determinations of CMB anisotropy
amount of anisotropy
large scales small scales Red curve is a theoretical model - evidence for a peak but curve is not signiﬁcantly constrained by the data at high l Compilation of all available data includes WMAP and some ground based / balloon experiments sensitive to smaller angular scales
Peak at degree scales Decline toward very small Plateau at scales large scales
Want to understand physical origin of each of these features The power spectrum reﬂects ﬂuctuations in the density at the time of recombination:
Photons escape from the overdense region
Recombination Consider a slight overdensity collapsing during the radiation dominated phase. Photons escaping at recombination: • Escape from a hotter, denser region • Are redshifted escaping from a deeper potential well • Have a Doppler shift due to relative velocity How this works in detail depends upon the scale of the ﬂuctuations: Largest scales (low l) On the largest scales, perturbations have not had time to collapse signiﬁcantly prior to recombination. At low l, directly see the ﬂuctuations generated at an earlier epoch.
Intermediate scales (~degree) Overdensities start to collapse, but increased pressure causes them to bounce - leading to oscillations. Maxima and minima of these oscillations lead to the strongest signals in the microwave background: • Doppler peaks • First peak (compression) occurs at degree scales
Recombination is not instantaneous Photons will `leak’ out of small over / under-densities during the process - damping very small scale ﬂuctuations Exponential suppression Start End of anisotropy at the smallest scales Dependence on Cosmology
1. Is the universe ﬂat, open, or closed? Doppler peaks deﬁne a physical scale at recombination Angular scale this corresponds to depends upon the geometry of the universe: Blue curve – effect of changing the geometry
Open universe – position of the peaks is shifted to smaller angular scales (i.e., larger multipole l) Observed position of the ﬁrst peak is at
l = 220 "total =1.02 ± 0.02
i.e., the universe is ﬂat (or very close to being ﬂat) What does this imply about the cosmological constant? ! Directly: almost nothing! - CMB anisotropy is mainly sensitive to the total energy density, not to the individual contributions from matter and cosmological constant Indirectly: estimate (by other means) that the total matter density is perhaps Ωm = 0.3 (mostly dark matter). Need `something else’ to make up the inferred value of Ωtotal = 1. A cosmological constant with ΩΛ ~ 0.7 as deduced from SN is consistent with this. 2. Baryon content of the universe
Increasing the fraction of baryons: • Increases the amplitude of the Doppler peaks • Changes the relative strength of the peaks - odd peaks (due to compressions) become stronger relative to even peaks (due to rarefactions) Full power spectrum from WMAP and other experiments is consistent with the predictions of ΛCDM (i.e., the family of cosmological models that includes dark matter plus a cosmological constant):
Simplest such models have 6 free parameters: • Being able to ﬁt the data is a genuine success! • Parameters are mostly well constrained by the data Adding in other cosmological information, e.g., from the supernovae measurements, further constrains the model:
Further information provided by: Lyman-α forest, galaxy clustering (2dF, Sloan Digital Sky Survey), weak gravitational lensing Today’s best guess universe Age: Best ﬁt CMB model - consistent with ages of oldest stars t0 =13.7 ± 0.2 Gyr Hubble constant: CMB + HST Key Project -1 -1 measure Cepheid distances H0 = 71 km s Mpc ! Density of ordinary matter: CMB
"baryon = 0.04 ! Density of all forms of matter: CMB + SNe
"matter = 0.27 ! Cosmological constant: CMB + SNe
"# = 0.73 !
! Theory of inﬂation links the power spectrum of ﬂuctuations to the nature of the vacuum energy driving inﬂation:
Power spectrum of inﬂationary ﬂuctuations is parameterized (k is the wave-number in the Fourier transform from spatial coordinates) P(k) = Ak n
n =1+2" # 6$
Original H!arrison-Zeldovich expectation was n=1 (a scale invariant spectrum). With inﬂation, generically one expects n slightly different! than 1 (either lower or higher). Inﬂation predicts very little variation of n with k. Observational evidence not yet deﬁnitive but WMAP3 data favor n~0.95. Summary: • The universe is ﬂat and will expand forever • Ordinary matter (stars, gas, dark baryons) is negligible • Cold dark matter and dark energy dominate the evolution of the universe, and currently make roughly equal contributions to the total energy density
Suggests that the universe at the time of recombination is well understood - we know the initial conditions that eventually gave rise to galaxies, stars, quasars, etc…
The big unknown: the nature of dark energy Is it a cosmological constant or something more interesting that varies with redshift? Joint Dark Energy Mission is being planned (SNe at higher z, larger red galaxy survey; read the task force report on the web)