Chapter 5: Cosmology Part 1: the large scale homogeneous Universe Some important things for Life about (homogeneous) cosmology
• The Universe as we know it is old, but had a finite beginning 13.8 Gyr ago,
• The (observable) Universe is large, but is finite in size,
• Baryonic material is a minority component of matter in the Universe, but is not zero,
• The Universe is expanding and is not (currently) in thermal equilibrium, with a characteristic temperature that is very low (2.7K),
• The Universe will continue to expand for a long time in the future,
• The average density of atoms in the Universe is very low (of order 1 m-3),
• The primordial composition after the Big Bang was only H and 4He (+ tiny amounts of 7Li) Hubble 1929 Expansion of the Universe Hubble (1929) redshift-distance relation
The “redshift” is the λ − λ shift of spectral features z = obs em λ in distant galaxies to em longer wavelengths v 1 + v Modern SN1a data Relativistic Doppler z c 1 z for z 1 = v − ⇒ = << effect (but see later) 1− c c H z = 0 d v = H d Hubble’s Law c 0 v µ d is a signature of an expanding “space”, but note: • This does not imply we are at a “center” • The scatter is due to the “peculiar motions” of individual galaxies due to local gravitational effects – typically 200 kms-1 3K Cosmic Microwave Background (CMB) dominates the radiation content of the Universe CMB has Planck Black Body spectrum at 2.728 ± 0.002 K, with exquisite precision.
But, matter and radiation are not in thermal equilibrium today (there is no matter at 2.73 K!). Photon to baryon ratio in the universe is 9 high: ng/nB ~ 2×10 . Also, the CMB radiation is isotropic at the level of 10-3 (dipole term due to our own 600 kms-1 motion w.r.t. CMB frame, otherwise isotropic at 10-5).
The extragalactic background light The last scattering surface of the CMB At very early times, matter and radiation were indeed in thermal equilibrium via creation/annihilation reactions of particle/anti-particle pairs. Prior to 300,000 years after Big Bang, all matter was ionized plasma (T > 4500 K) and the Universe was effectively “opaque” due to scattering of light off charged particles. After this epoch, matter was generally neutral (at least for a while) and the Universe was “transparent”, as it still is today. Since the CMB has been around since the beginning, the visual impression is of being located at the center of a clearing in a fog bank. We “see” the “Last Scattering Surface” as a section through the Universe at an immense distance, and as it was 3 x 105 yr after Big Bang. p + opaque
The Big Bang transparent
US Out in space and back in Appearance of time “inside of fog bank”
Isotropy of the CMB is really a statement about the homogeneity of the Universe 300,000 yr after Big Bang. Roberston-Walker metric for expanding isotropic homogeneous Universe The Minkowski metric in Special Relativity is: ds2 = c2dt 2 − dl 2 3 3 The most general metric will be of the ds2 = g dxidx j ≡ g dxidx j form: ij ∑∑ ij i=0 j=0
About our choice of co-ordinates: (a) It is convenient to define time as a cosmic time t: t is defined as the proper time of the “fundamental observers” who see an isotropic CMB and Hubble flow (and are thus “at rest” with respect to the Universe as a whole). (b) It is convenient to define a polar co-ordinate system centered on us with two angles (q,f = position on sky) and a radial coordinate. (c) Also, it is convenient to split the radial coordinate into a fixed (for a fundamental observer) “co-moving” position w and a time-varying cosmic scale factor R(t). q and f of any fundamental observer are anyway also constant with time. 3 3 2 i j i j ds = gijdx dx ≡ ∑∑gijdx dx i=0 j=0
Simple symmetry arguments then allow us to show that the metric for any homogeneous and isotropic, but possibly expanding, Universe must have the form of the Robertson-Walker metric. & dr 2 # ds2 c2d R2 ( ) $ r 2d 2 ! = τ − τ $ 2 2 + γ ! %1− kr / A " Conventional 2 2 2 2 2 2 angular ds = c dτ − R (τ)(dω + Sk (ω)dγ ) coordinates q,f “Cosmic time” Comoving radial Effects of the curvature of coordinate w a t = constant surface, Scale factor of with A = (comoving) expanding Universe radius of curvature
Again, simple symmetry arguments require that Sk(w) has one of only two forms, which both approach the straightward Sk(w) = w as the radius of curvature A tends to infinity: We measure the curvature and find ⎛ω ⎞ ⎛ω ⎞ Sk (ω) = Asin⎜ ⎟ or Asinh⎜ ⎟ that it is very small, i.e. A is very ⎝ A ⎠ ⎝ A ⎠ large. This 2-d analogue of our 3-d Universe illustrates • Isotropic expansion with v µ d • Patterns and geometry/topology remain constant: “comoving radius of curvature” doesn’t change. • Our “observable region” (horizon = light travel distance in the age of Universe) is likely to be finite. • We can measure the curvature in different ways. We know that we live in a Universe with A very much larger than the “horizon”.
Note: the volume of Universe (i.e. the number of galaxies) will also be finite if it is (uniformly) positively curved, but will be formally infinite if it is (uniformly) negatively curved. The volume will definitely be “very large” if it is time more or less flat in our observable Maps of galaxies (fixed in comoving region as it is. space) at different t Some straightforward consequences of the RW metric
The redshift of galaxies in terms of the change R(τ observed ) in scale factor R. (1+ z) = R(τ emitted )
Hubble’s “constant” (better: Hubble’s “parameter”) R H = is the normalized first derivative of R(t) R
The “deceleration parameter” is the normalized RR second derivative of R(t) q = − R 2
Behaviour of a black body radiation field RTbb = constant
The last equation gives us a natural explanation of the 2.7K microwave background: The Universe was in thermal equilibrium at very early times, in a -1 hot dense state. The black body Tbb will simply decline as R even after thermal equilibrium is lost. The CMB is the relic radiation of this hot dense phase. The Friedmann equation Solution of General Relativity field equations for homogenous isotropic Universe gives the Friedmann equation linking the dynamics of the expansion R(t), the r(R) “equation of state”, and the curvature c2/A2 2 2 8πG 2 kc & Λ 2 # R = ρR − 2 + R 3 A %$ 3 "! R - scale factor from the Robertson Walker metric R! -derivative w.r.t. cosmic time t ρ - density in whatever form (non-relativistic matter, radiation, others?) kc2 - curvature as in the Robertson Walker metric: A = comoving radius of curvature (fixed for all time) and k = ± 1 just to fix the sign. A2 Λ - Einstein’s “cosmological constant” (note that this is entirely equivalent to a r = constant term, also known as a “false vacuum”, i.e. a form of “Dark Energy”) To solve for R(t) we must know the (constant) curvature and r(R), which will depend on the form of the density. What are the possibilities for r(R)?
-3 rM µ R ”cold’ matter, either normal baryonic matter or “dark matter”
-4 rg µ R radiation, or relativistic matter rL = constant “false vacuum” … and possibly others Note: the different r(R) dependences mean that different components may dominate at different R. We can also can imagine a given component changing from one R dependence to another, e.g. relativistic particles lose energy à non-relativistic.
Dynamical effects on R(t)
It is easy to solve Friedmann equation if the density is dominated by one single component and the curvature term is negligible:
-3 2/3 rM µ R R µ t Decelerated expansion (as you’d expect)
-4 ½ rg µ R R µ t Decelerated expansion (as you’d expect)
Ht rL = constant R µ e (Exponentially) accelerated expansion under gravity!! ρi It is convenient to parameterize the density ri of different Ωi = components i in terms of a “critical density” that is given at any ρc epoch by the expansion rate, H, and gravitational constant, G, 3H 2 and call this W . We can then have a “total” W = SW . ρc = i i 8πG
The Friedmann Equation then can be re-arranged in different ways
2 € ⎛ ⎞ kc2 1 ⎛ c ⎞ 1 Ω −1 = Ω −1 = = k⎜ ⎟ ⎜∑ i ⎟ 2 ! 2 2 ⎝ i ⎠ A R ⎝ H ⎠ (RA)
• c/H is of order the size of the horizon, so |W-1| tells us the ratio of the horizon scale to the (physical) radius of curvature RA, i.e. it gives the “importance” of the curvature within our horizon. We know that in our Universe at the moment W ~ 1.
• Note that if the expansion decelerates, then R-dot decreases, |W-1| increases, and W diverges away from unity, with curvature becoming more important, and vice versa: if the expansion accelerates then W converges towards unity. For illustration, the following slides look at three cases that were, 25 years ago, thought to be all more or less consistent with observations.
Wm = 1: A flat “Einstein-de Sitter” universe, in which the density is dominated by a critical density of matter. This was preferred theoretically in the 1990’s.
Wm = 0.25: A spatially “open” (negatively curved) universe, with a low density of matter (dominated by dark matter). This was certainly preferred observationally in 1990’s.
Wm = 0.25 + WL = 0.75: The so-called “concordance cosmology”. It is spatially flat, with a low (matter) density, but with a substantial false-vacuum-like component that makes W = 1. “Concordance cosmology” is the one that is now strongly indicated by observations post-2000, even though it demands that the density is dominated by the not-at-all-understood “Dark Energy”. It reconciles: • Flat geometry W = 1 as indicated from CMB fluctuations
• Wm = 0.3 as the observers had always been saying • Evidence for accelerating expansion from supernovae SN1a. Main point: Despite these big differences in the models, the effect on the history of the Universe is small, because Dark Energy (with constant density) is insignificant at much earlier times compared with the matter R-3 and radiation R-4 components. Wm = 1 Einstein-de Sitter The scale factor R(t) Wm = 0.25; WL = 0.75 “Concordance” Wm = 0.25 Open Universe
All currently plausible models have R ~ 0 at some finite time in the past i.e. BIG BANG “creation event” The time since the Big -1 Bang is of order H0 (almost exactly so for the concordance model) Note: the late epoch acceleration of the concordance model is barely noticeable. Wm = 1 Einstein-de Sitter Distance that light has traveled since Wm = 0.25; WL = 0.75 “Concordance” Wm = 0.25 Open Universe redshift z (in units of c/H0)
The comoving distance w(z) converges at some finite value of w for very high z – this is the concept of “horizon” (limits to vision) t c ω (t) = dt H ∫ 0 R(t)
The region of the Universe that we can “see” (or physically interact with) is finite and has radius of order (c/H). This is what we mean by the “observable universe” Wm = 1 Einstein-de Sitter Changes in W Wm = 0.25; WL = 0.75 “Concordance” m Wm = 0.25 Open Universe
All models approach Wm = 1 in the past (at very early times, radiation and relativistic particles dominate with Wrel =1). This makes our treatment of early Universe simple….. e.g. universe was matter (or radiation) dominated with r = 3H2/8pG An example of divergence: The divergence of Wm if Wm ≠1
The Problem for Life: Three matter- dominated Universe turns Wm > 1 Universe quickly turns around and models with recollapses around and recollapses Wm = 1.1, 1.0 and 0.9 today Wm << 1 Universe cannot form structures like galaxies (it is “empty” and gravitational instability fails – see next week)
Universe rapidly Development of Life arguably becomes “empty” needs a Universe which has
Wm close to unity, i.e. had W very very close to W = 1 at early times, so as to produce log R/Rtoday a long-lived Universe able to form structures What about non-baryonic “dark matter”
• First evidence: “missing mass” in systems on scales of galaxies and larger. First seen in 1930’s, became compelling from 1980’s.
Van Albada et al 1985 NGC 3198 Galaxy rotation curves v(r) indicate mass has density going as r-2, whereas distribution of visible material goes as r-3.
Dynamical measurements of galaxies in clusters, hydrostatic equilibrium of hot gas, gravitational lensing, all consistently indicate mass ~ 7× mass of visible baryons Dark Matter explains so much (note the very varied evidence): • Analysis of primordial nucleosynthesis in early Universe constrains the photon-baryon ratio, and sets WB ~ 0.04, compared with total WM ~ 0.27 • Existence of non-baryonic dark matter is required to ensure survival of density fluctuations during phase when Universe was a fully ionized plasma. • Non-baryonic dark matter dominated Universe provides extremely good quantitative match to CMB fluctuations and structure in the Universe today. There is now little doubt that normal baryonic matter is a minority (~ 15%) of matter in the Universe. Dark matter (effectively) interacts only through gravity (and possibly through the Weak Force). Not good for Life!
Existence of Dark Matter not surprising in retrospect? We primarily observe the Universe with electromagnetic radiation ...
The challenge for Physics: • There are no candidate DM particles within the Standard Model of Particle Physics. • There are very many candidates outside the Standard Model, e.g. • axions (10-5 eV) • sterile neutrinos (wide range of masses) • super-symmetric WIMPs (100 GeV) • primordial black holes (very massive) Measuring the parameters that describe our Universe on the largest scales
H Expansion rate of the Universe (H ~ 70 kms-1Mpc-1) Need to measure distances to objects with known redshift • “distance ladder” approach based on parallax of nearest stars • well-understood astrophysical systems (e.g. expanding SN explosions) q Deceleration/acceleration of the expansion Need to measure deviations from Hubble’s Law at large distances • SN1a, assuming they are standard candles of fixed brightness
SN1a gave clear indication (in 2000) for accelerating expansion now with q ~ − 0.6,
which implies WL ~ 0.7 Measuring the parameters that describe our Universe on the largest scales (continued)
Wm matter density of the Universe today (Wm ~ 0.3) We get constraints from: • measure “mass-to-luminosity” ratio in e.g. rich clusters of galaxies and apply to the measured luminosity density of the Universe • growth rate of structures We get a separate constraint on the density of baryonic matter, Wb from “Big Bang Nucleosynthesis” (BBNS).
In the first three minutes of the Universe (t < 200 sec), various reactions p + n à 4He + traces of 2H, 3He, 7Li. The (equilibrium) n/p ratio evolves ith temperature but the ratio “freezes out” at t ~ 1 sec when the timescale for n-p conversion via weak interaction exceeds the expansion timescale, at which point the conversions effectively stop and equilibrium between n and p is lost.
The predicted abundance ratios (esp. minority 2H, 3He and 7Li) depend on the ratio of particles to photons, h ~ 10-9, which is constant from then till now. We know number density of photons today from 3K CMB. The observed ratios are beautifully consistent with theoretical predictions for matter density (today) of
Wb = 0.04
Aside: differences with stellar nucleosynthesis? • plenty of free neutrons at t ~ 1 sec -2 10 • all over in ~ 100 seconds: t(sec) ~ T10 (10 K) Measuring the parameters that describe our Universe on the largest scales (continued)
W Total W = SWi given by spatial curvature of universe, e.g. the angular size of something of known physical size at great distance Length scale of the length of the peak density fluctuations in universe on CMB last scattering surface is well understood theoretically and the angular scale was measured by the Boomerang circumpolar balloon experiment in 1999. Peak at 1 degree scale à W = 1
Conclusion: curvature is very small within our observable volume. Our universe is “flat”. Measuring the parameters that describe our Universe on the largest scales (continued)
The details of CMB fluctuations have become a “Rosetta Stone” of almost all of the cosmological parameters (Boomerang, then WMAP, Planck missions)
The “Concordance Cosmology”
H = 71 ± 3 kms-1Mpc-1 Curvature = - 0.02 ± 0.02
WM = 0.27 ± 0.03 of which WB = 0.044 ± 0.004 WL = 0.73 ± 0.03 n = 0.95 ± 0.02 s8 = 0.74 ± 0.05 We get an important consistency check on independent constraints on the age of the Universe t Current age of the Universe is constrained from ages of things in it using astrophysical models • Masses of (highest mass) surviving stars in Milky Way globular clusters (12.5±1.5 Gyr) • Temperatures of (coolest) white dwarfs in Milky Way disk (8.5±1.5 Gyr) • Radioactive dating of radioactive isotopes, using assumed production history in Milky Way (10-20 Gyr) • Age of the Earth itself (4.6 Gyr)!
All these different estimates involve very different physics, but are all consistent with the age ~ 13.8 Gyr from Concordance Cosmology, if the globular clusters formed about 1 Gyr after Big Bang, and the disk of our Galaxy after a few Gyr, and so on. All quite reasonable. Basic cosmology - some first implications for Life (a) The Universe is quite big: • 1011 galaxies within observable region, but the radius of curvature ≥ 100 × observable region, so the total volume is almost certainly at least ≥ 106 larger, possibly much larger. and therefore also quite old (13.8 Gyr since Big Bang) • time for multiple generations of stars and chemical enrichment of gas • temperature of the radiation is very low (2.7 K) so matter and radiation are far from equilibrium today.
and had Wm ~ 1 for an extended period, • allowing galaxies and other structure to form These three interrelated facts all follow from the fact that W was set to be very close to unity at early times. How was this done? INFLATION
(b) The present-day average density of baryonic matter is low (or order 1 atom per m3) but is non-zero. We might have expected exact symmetry between particles and anti-particles in the early hot Universe, producing no residual of matter today.
How did the hot dense Big Bang universe come to have non-zero Baryon number? BARYOGENESIS Inflation Why is the Universe large and long-lived? = what set W = 1 at the beginning The concept of inflation: “Inflation” is a period of very rapid, accelerating expansion at the beginning of the Big Bang - i.e. t ~ 10-35 sec. Proposed ca. 1980 to “explain” several shortcomings of standard Big Bang scenario, by invoking new Physics at very high energies (of order 1015 GeV) What is the relevance of Inflation for Life in the Universe?
• It naturally sets W very close to W=1 at early times, producing a long-lived large Universe; • It naturally explains why different parts of the Universe are the same (same physics etc); • Next lecture: the small (quantum) density fluctuations present during inflation will grow into the large scale structure seen in the Universe today (e.g. galaxies). The “spectrum” of fluctuations predicted from inflation matches what we see. Inflation: Setting W = 1 at early times From the Friedmann equation, any decelerating 2 Universe, W progressively diverges from W ~ 1. " kc2 % 1 (c / H) Ω =1+$ ' =1+ k But in an accelerating Universe, W progressively # A2 & R 2 (AR)2 converges towards W ~ 1. Idea of “inflation” was introduced in 1980. • Postulate that at early times (E ~ 1016 GeV), r was dominated by a non-
zero energy-level of the vacuum, rvac. Originally GUT symmetry- breaking field, but works better with more general “inflaton field”. 1/2 ⎛8πG ⎞ • This produces exponential expansion R µ eHt, with H = ⎜ ρ ⎟ ~ constant ⎝ 3 vac ⎠
• At the end of inflation, the energy in rvac was thermalized into hot matter and radiation rr, from then on a standard “decelerating” hot Big Bang universe. • Beyond this basic idea, there are a great variety of more speculative ideas
about how it got into this state, how it got out of it, how rvac may have evolved, and so on.
Idea of inflation pre-dated by 20 years the discovery of the “similar” Dark Energy, that is now emerging to (again?) dominate density of the Universe. Are they connected? But, note the energy scale is ×1027 different. t 2 c w = dt An attraction of Inflation: the “Horizon Paradox” e ò R(t ) t1
In any decelerating Universe, the horizon wH increases monotonically as time passes. For instance, the mass of matter within the horizon increases as t (during matter-dominated era) and as t3/2 (during radiation dominated era).
Inevitable consequence: all that we see today within our horizon was, at earlier times, in many disjoint regions that were apparently always outside of each other’s horizons
f(t) O
A B
0 t=0 w Changing R(t) at early times (and A and B have not, at the epoch at which we observe having accelerating universe) enables them to be, had a chance to exchange information, if the Universe has always been decelerating! all these areas to have been in causal contact at early times. Baryogenesis Why is there any matter in the Universe? Baryogenesis and the matter-antimatter asymmetry • Matter and anti-matter are equally respectable in Physics. • Yet, the Universe today contains almost no (naturally occurring) anti-matter at all. The dominance of matter over antimatter is established on at least galactic scales and is almost certainly “universal”. Why? • At high energies, matter and anti-matter pairs were in equilibrium with creation and annihilation reactions: e.g e + e ↔ 2γ • As energies fell during the expansion, pair-annihilation dominated over pair-creation. Therefore, the present day particle-to-photon n p ≈ 1+ η ratio (h ~ 10-9) reflects an initial small asymmetry in number of € np particles and anti-particles before the last annihilation occurred. • If this had not been present, all the particles and all the anti-particles would have annihilated and the Universe would have contained no particles (or anti-particles) at all Þ Life would presumably be impossible! • Inflation strongly predicts net baryon (and lepton) number was precisely zero at the end of inflation. So, where did the non-zero baryon (and lepton) number of the Universe come from? 2 Sakharov’s three conditions X !!r→ qq B = − L = 0 1 3 1 1 1. Baryon non-conservation X !1!−r→ ql B = + L = 1 2 3 2 e.g. postulate an X-boson with two decay paths that do not conserve B 2 X !!r → qq − B = + − L = 0 (OK in GUTs(?) but not so far 1 3 1 observed) 1 X !1!−r → ql − B = − − L = −1 2 3 2 2. CP-violation e.g. so that branching ratios of ε=(rB1 + (1− r)B2 )−(rB1 − (1− r)B2 ) X and anti-X are different. CP- violation is seen in Nature but =(r − r)(B1 − B2 ) not understood.
3. Loss of equilibrium When? m ~ 1014 GeV, so that back reactions do not destroy B X implying 10-34 sec, i.e. very gained in this way. Easy to achieve in soon after inflation expanding Universe when treaction >> tH. Are there unknown connections? Note: Dark Matter, Inflation and Baryogenesis all represent unknown (or at least untested, and possibly untestable) Physics, lying outside of the “Standard Model + GR”, for which the Universe itself may represent the only evidence.
Strange co-incidence: WB ~ WCDM And today at least: ~ WL
Is this coincidence telling us something? Would expect very different physics: WIMP etc. for Dark Matter, Inflaton field at 1016 GeV, GUTS and CPT violation. Or is it simply anthropic (i.e. required to allow our existence)? Arguably there would be no Life as we know it:
• if WB was too low to allow cooling to form galaxies or if WCDM too low to prevent Silk eradication of density fluctuations on galaxy scales (i.e. need to -3 have 10 > WB/ WCDM < 10 ?) 6 • if L dominated at z > 100 (i.e. WL > 10 WM) …. More on this next week Some important things for Life about (homogeneous) cosmology
• The Universe as we know it is old, but had a finite beginning 13.8 Gyr ago.
• The (observable) Universe is large, but is finite in size.
• Baryonic material is a minority component of matter in the Universe, but is not zero.
• The Universe is expanding and is not (currently) in thermal equilibrium, with a characteristic temperature that is very low (2.7K).
• The Universe will continue to expand for a long time in the future.
• The average density of atoms in the Universe is very low (of order 1 m-3).
• The primordial composition after the Big Bang was only H and 4He (+ tiny amounts of 7Li).