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The Inflationary Universe

The Inflationary Universe. By Tom Elmer PHYS 514 2004.05.07. Big Bang Cosmology. Robertson-Walker Metric. d t 2 = dt 2 – R 2 (t) [dr 2 /(1-kr 2 ) + r 2 (d q 2 + sin 2 q d f 2 )].

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The Inflationary Universe

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  1. The Inflationary Universe By Tom Elmer PHYS 514 2004.05.07

  2. Big Bang Cosmology Robertson-Walker Metric dt2 = dt2 – R2(t) [dr2/(1-kr2) + r2(dq2 + sin2q df2)] Where k is a curvature constant set to -1, 0, +1 to define open, flat, closed universes. The evolution of R is governed by the Einstein Equations d2R/dt2 = -4p/3 G (r + 3p) R H2 + k/R2 = 8p/3 G r H ≡ dR/dt / R (Hubble “constant”) Expansion of the universe traces back to some T0 ...Problems Tom Elmer

  3. Adding Grand unified Theories • Quantum field theory in the GUT accounts for the creation of particles. • A GUT model begins with a gauge group, G, which is a valid symmetry at the highest energies. As energy is lowered, the theory undergoes spontaneous symmetry breaking. • G → Hn→ … → H0 • Where H1 = SU3 X SU2 X U1 [QCD X Weinberg-Salam] and H0 = SU3 X U1EM • Problem: monopoles (very heavy ~1016 GeV) and domain walls abound in the theory in low temperature phases, but not in nature. Tom Elmer

  4. Horizon problem The universe seems to be homogenous, yet regions are causally disconnected. (Not enough time to ‘communicate’.) One way to view this is to look at the ratio of light volume to volume of the region that will evolve into our observed region of the universe. T2 = MP / (2gt) g2 = (4p3 / 45)n RT = const → R ≈ t½ l(t) = R(t) ∫ dt’ R-1(t’) = 2t = MP / (T2g) s = 2p2/45 n(T) T3 L(t) = [sp/s(t)]1/3Lp l3/L3 = 11/43(45/(4p3))3/2n-½(MP/(LpTgT))3 = 10-83 (where T0 = 1017 GeV and n ≈ 102 typical of GUT models ) Tom Elmer

  5. Flatness Problem For a fixed initial temperature, the expansion must be extremely fine-tuned. • W ≡ r / rcr (curvature) • S ≡ R3s (total entropy) • = p2/30 n(T) T4 (energy density) e(T) = k / (R2T2) = k[2p2/45 n(T)/S]2/3 |(r-rcr)/r| = 45MP2|e|/(4p3nT2) < 10-55 (where T = 1017 GeV and n ≈102 typical of GUT models ) Tom Elmer

  6. The Assumptions • Strictly conserved quantities • i.e. fixed baryon number • Expansion of the universe is constant • Expansion of the universe is adiabatic Tom Elmer

  7. A Possible Solution What if the adiabatic assumption was incorrect? Sp = Z3S0 Horizon, 10-83 l3/L3 = 11/43(45/(4p3))3/2n-1/2(MP/(LpTgT))3 1 If Z > 5 X 1027 L = Lp(Sp/S)1/3 Flatness, 10-55 Initial could be O(1) and it wouldn’t matter if Z > 3 X 1027 |(r-rcr)/r| =45MP2|e|/(4p3nT2) e aS-2/3 Tom Elmer

  8. Toy Inflationary Universe • Suppose we have a phase transition at some Tc • If Tc is high enough in a GUT model, we can delay baryon production until after (during) the transition • Supercooling allows Ts many orders of magnitude below Tc • When the phase transition finally occurs, energy is released and S is increased by a factor of (Tr/Ts)3 by the universal reheating to Tr (comparable to Tc) • Z≈(Tr/Ts) Tom Elmer

  9. Scalar Inflation Field, f Lagrangian density £ = ½ ∂mf∂mf – V(f) Various proposed V(f)s (of many more) V(f) = l (f2 – M2)2Higgs Potential V(f) = ½M2f2 Massive Scalar Field V(f) = lf4 Self-interacting SF V(f) = 2Hi2 (3 – 1/s) e-f/√s Dilation SF (string thy) Guth, Steinhardt Tom Elmer

  10. Problems With This Model • Bubbles expand too quickly, leaving the universe devoid of structure. • Fine tuning is necessary to avoid ‘bubbles’ colliding (forming monopoles) • However, tuned to agree with cosmic background observations, bubbles would collide far too often. Tom Elmer

  11. Resolved: • Toy model as described: • Fixes the flatness problem • Fixes the horizon problem • Modify to use a slow-roll phase transition instead of tunneling: • Fixes Domains • Fixes Monopoles • Modify to use quantum fluctuations to perturb initially: • Reduces fine-tuning necessary in initial conditions Tom Elmer

  12. Evolution of the Inflation Field Linde Tom Elmer

  13. References • Guth, Alan H. Inflationary universe: A possible solution to the horizon and flatness problems. Physical Review D. vol 23, number 2. Pp 347-356. • Guth, Alan H. and Paul J Steinhardt. The Inflationary Universe. Scientific American. 1984 May. Pp 116-128. • Linde, Andrei. The Self-Reproducing Inflationary Universe. Scientific American. 1994 Nov. Pp 48-55. • Linde, Andrei. http://physics.stanford.edu/linde. • Bucher, Martin A. and David N Spergel. Inflation in a Low-Density Universe. Scientific American. 1999 Jan. Pp 62-69. • Watson, Gary Scott. An Exposition on Inflationary Cosmology. http://nedwww.ipac.caltech.edu/level5/Watson/Watson_contents.html • Gribbin, John. Inflation for Beginners. http://www.biols.susx.ac.uk/home/John_Gribbin/cosmo.htm. Tom Elmer

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