From inflation to the CMB to today s universe I - How it all begins Raul Abramo Physics Institute - University of São Paulo abramo@fma.if.usp.br
redshift Very brief cosmic history 10 9 200 s BBN 1 MeV 10 3 380.000 yrs 1 ev Decoupling (surf. last scattering) 0 15Gy time energy
Some crucial observations to understand our Universe: Cosmic microwave background (COBE, WMAP, Boomerang, Dasi, QUAD,... PLANCK...) Matter distribution over large scales Matter Spectrum
History of cosmic domination: today radiation matter: z~10 4 Dark Energy z 1089: decoupling (recombination)
Some numbers I will often use... 1 pc = 3,26 l.yr. 1 Mpc = 3,1 x 10 24 cm Age of the Universe today: T 0 = (14 ± 0.5) Gy Density: ρ 0 = (1.9 ± 0.15) h 2 x 10-29 g cm -3 Hubble parameter: H 0 = 100 h Km s -1 Mpc -1 h = 0.72 ± 0.05 Baryon fraction: Ω b h 2 = (ρ b / ρ tot ) h 2 = 0.024 ± 0.003 Radiation (photons and massless neutrinos): Ω r = 2.5 x 10-5 h -2 Matter (dark matter + baryons) Ω m = 0.2-0.3... use to compute, e.g., matter-radiation equality:
The physics of the CMB involves propagation and scattering of photons The CMB is also the most distant direct observation we have of the universe in its infancy, hence it is a key observable to test correlations and causality over the largest observable scales of the Universe So, let s review some of the basic facts about the propagation of light and, therefore, of causality in a Friedmann-Robertson-Walker spacetime Light propagates over null geodesics. For a radial light ray:
In an FRW spacetime, proper distances for light-speed signals can be finite even when the travel time extends arbitrarily into the past or into the future. For instance, let s take a decelerating FRW: a t This spacetime can be continued to the past only down to t=0 (when a=0). Then: d Hp is the maximum physical distance a light ray can cover if it was emmitted at some arbitrary time in the past. This means that the past light cone in this scenario is bounded, and cannot be extended beyond that limit. t d
This maximal distance is called a horizon. Since in this case (p<1) the horizon refers to a truncation of the PLC, it is a past-like horizon, a.k.a. a particle horizon. This horizon is usually approximately equal to the curvature radius of the FRW, r ~ 1/ R -1/2 ~ H -1 - i.e., the Hubble radius! The particle horizon separates observers which never had causal contact prior to the time t. Therefore, when there is a particle horizon, the Universe can be separated into regions which are (up to that time) causally disconnected Since the Universe has been, for most of its history, dominated by either radiation (p=1/2) or matter (p~2/3), if that were true down to t=0 then our particle horizon today would be:??? How can the CMB be so homogeneous over the whole sky??? Ex: compute the particle horizon at the time of decoupling (t~380.000 y, z~1100), assuming that p=1/2. A: ~200 Kpc.
Now take an accelerating scale factor: a t We still have some initial time t=0, however: is an arbitrarily large distance as we take the lower limit ti 0, and hence there is no particle horizon in this case. But consider, instead, what happens if the upper limit is take to be tf, and take the lower limit to be t. This distance would then correspond to the maximal length that separates two objects such that they could exchange a light-speed signal emmitted at time t. If that maximal distance is not infinity, then there an event horizon:
The physical significance of a horizon is profound, as it clearly marks causality boundaries: A particle horizon sets a limit to the past light-cone of observers at time t: pairs of observers separated by a distance larger than dph at time t have never been in causal contact before t. An event horizon sets a limit to the future light-cone of observers at time t: pairs of observers separated by a distance larger than deh at time t will never again be in causal contact after t. t0 t1 dph(t1) tdec t2 deh(t2) t1 0 comoving distances comoving distances
The physical meaning of an event horizon is that it marks the boundary beyond which observers lose the possibility of causal connection in the future. v = c v = c v = c What happens when the accelerated expansion doesn t last forever? Mathematically, there isn t an event horizon anymore - but still we can define the notion of an effective horizon: Like the notion of thermal equilibrium, what really matters is the time interval during which there is no causal contact, compared to the typical times for other (causal) physical processes v = c
Can do that with scalar fields! small Eq. motion (Klein-Gordon) Background, φ(t): V(φ) Hubble drag Potential If the kinetic energy is << than the potential energy => slow roll : φ With slowroll, V(φ) works like a time-varying Λ
Explicit example: the power-law model characterized by the mass scales M and s: Using these into Friedman s and Klein-Gordon s equations: 0 = φ + 3H φ + V,φ ( ) 1 3H 2 = 8πG 2 φ 2 + V The solution (up to a transient) is: with: Where: a(t) = ( t t 0 ) p, H = p t φ(t) = φ 0 log t t 0 Just choose p (i.e., s) sufficiently large and inflation will ensue
How it all started: inflationary particle creation - the ultimate free lunch! QUANTUM MECHANICS: " ΔE Δt > h/2π " Vacuum is filled with virtual pairs of particles, which survive during brief moments before being annihilated back to vacuum Guth 1980 Starobinsky 1979 Linde 1982-85 Mukhanov & Chibisov 1979-1980 Hawking Guth & Pi...
Virtual pairs quantum fluctuations Right here, right now:
Virtual pairs in an expanding background (accelerated spacetime) accelerated expansion separates the pairs Horizon H -1 A contraction followed by expansion (bounce) would have a similar effect Inflation (accelerated expansion) converts virtuais pairs into real ones
Quantum fluctuations of a scalar field + metric perturbations during inflation Good reviews: Bassett, Tsujikawa & Wands astro-ph/0507632 L. Sriramkumar, arxiv:0904.4854 3H 2 = 8πG V (ϕ) ϕ + 3H ϕ + V = 0 After diagonalizing and integrating by parts, quadratic action: Solutions to modes @ U.V. (k 2 >>µ 2 ) and I.R. (k 2 <<µ 2 ):
Quantization: Vacuum: a k 0>=0 But: => Mixing of positive- and negative-energy modes => Amplification of zero-point energy by the external field (acc. expansion) => ~ Particle creation Inflation generates inhomogeneities in the primeval soup
v k + [k 2 + µ 2 (η)]v k = 0 t UV Today H -1 End of inflation Curvature perturbations: UV IR λ phys R k = v k z = ψ k + Ḣ ϕ δ ϕ k Constant on large scales ( IR ) convenient f/ normalization!
The primordial spectrum of curvature (-> density) perturbations: R k R k 2π 2 k 3 2 R(k) δ( k k ) ξ R ( x x ) = R( x)r( x ) = d 3 k d 3 k (2π) 3/2 (2π) 3/2 e i k x e i k x R k R k ξ R (r) = dk sin kr k 2 R(k) kr, r = x x It will also be useful to know the inverse of this relation ( Hankel transform ): 2 R(k) = 2 π k3 2 sin kr dr r kr ξ R(r) Many times we also define a dimensionful power spectrum: P R (k) = 2π 2 k 3 2 R(k) R k 2
Convergence arguments limit the form of the spectrum (Zel dovich, Harrison,...), so this function must be nearly scale-invariant: ( k 2 R(k) = 2 R(k 0 ) k 0 ) ns 1+... Rk arise from quantized harmonic oscillators their statistic is that of harmonic oscillators in their ground state GAUSSIAN! R k = e iφ kar ( k) Considering higher-order interactions lead to deviations from gaussianity! 0 φ k < 2π, P (φ) = 1 2π P (φ) P Gauss (A R ) e P G (A R ) A 2 R 4π 2 k 3 2 R 2π φ A R Nearly scale-invariant spectrum of Gaussian perturbations (Zel dovich, 70s)
The metric also has matter-independent fluctuations - gravity waves: 2 h(k) = 2 k3 2π 2 h k 2 Both density ( scalar ) and gravity wave ( tensor ) perturbations impact directly the CMB, and it is useful to set a pivot scale to determine the spectra: ( k 2 R(k) = 2 R(k 0 ) k 0 ) ns 1+... 2 R(k 0 ) 2.2 10 9, k 0 = 0.002Mpc 1 Amplitude nailed by WMAP ( k 2 h(k) = 2 h(k 0 ) k 0 ) nt +... Their relative power is also a key observable: r = 2 h (k 0) 2 R (k 0)
Density perturbations ( scalar ) and gravity waves ( tensor ) are both generated during inflation (or some other early-universe free restaurant). However, scalar and tensor perturbations test very different physical scales: V (ϕ)/m pl 2 V (ϕ)/h V (ϕ)/mpl 2 M pl 1 V 3/2 M pl V 1 V M 2 pl M 2 pl = G 1 Although density perturbations have been digested by processes at low energies, gravity waves come straight from the GUT era!
Summarizing: from quantum fluctuations to CMB, galaxies etc. Andromeda quantum fluctuation classical fluctuation perturbation in density & temperature We live here 10-35 s 10-33 s 3. 10 5 y 1.5 10 10 y (Adapted from Lineweaver 1997)
And how does inflation subvert the causality bounds of usual radiation/matter cosmology? t0 t1 dph(t1) tdec Inflation: the ultimate past-light-cone democracy
Tomorrow: Cosmic Microwave Background Radiation Z 1200 Tγ 10 ev Universe starts to become neutral Photons start to free stream While this happens, density and temperature fluctuations are imprinted on the CMB photons We detect these photons today - with their initial energies redshifted by the same factor 1/(1+z)