Reconstruction of the dark energy model 2006 9

.......................................................................... 2 Abstract....................................................................... 3..................................................... 4 1.1............................................................... 4............................... 8 2.1..................................................... 8 2.1.1 Ia.............................................. 8 2.1.2......................................... 11 2.1.3........................... 12 2.2....................................................... 12 2.2.1 I Λ.................................... 13 2.2.2 II................................... 14 2.2.3 ω........................................................... 20................................................... 22 3.1 Hubble............................. 22 3.1.1 Hubble.............................................. 25 3.2 ω............................................ 26 3.2.1 SN Ia ω.................................... 26 3.2.2 CMB ω.................................... 26 3.2.3 ω................................ 29 3.2.4 ω..................................... 30 3.3 ω....................... 31 3.3.1 Quintessence........................................... 31 3.3.2 K-essence.............................................. 33 3.3.3 Tachyon............................................... 35.......................................................... 39..................................................................... 41 1

Hubble 3.3.3 tachyon Λ CDM Λ CDM ω = 1 (fine-tuning problem) (coincidence problem) ω ω > 1 ω < 1 1 ω 1 [27, 28, 29] quintessence k-essence tachyon phantom quintom Hubble quintessence k-essence tachyon tachyon Guo [52] tachyon tachyon ω 1 Lamost Plank SNAP tachyon 2

Abstract The reconstruction of the dark energy model has been studied widely. The dark energy model can be reconstructed from the parametrized Hubble parameter. The parametrized equation of state of the dark energy can be got by fitting the data sets from observations, so the dark energy model can also be reconstructed from the parametrized equation of state. In chapter 3, we introduce the reconstruction of the dark energy model detailedly. This thesis consists of four chapters. General relativity and the Cosmological Principle are the two theoretical pillars of the big bang cosmology. In chapter 1, we introduce the basic equations for the studies of the dark energy - the differential equations for the scale factor and the continuity equation, which are all derived from the two pillars. In chapter 2, we introduce the observational evidences for the dark energy and some dark energy models, i.e. Λ CDM model, scalar field models, which are introduced to try to explain the nature of the dark energy. The Λ CDM model is a simple one, which can give the equation of state of the dark energy ω = 1. Although this model can fit almost all data sets from observations, it has two serous problems - finetuning problem and coincidence problem. The ω for the scalar field model is not a constant but evolves with the redshift z. Some scalar field models can give ω > 1, some can give ω < 1, and some even can give ω crossing 1. Quintessence, k- essence, tachyon, phantom, quintom models are mentioned in this chapter. In chapter 3, firstly we introduce the details of the method of reconstructing the dark energy model from Hubble parameter. Then we introduce the parametrization of the equation of state of the dark energy and the reconstructions of three dark energy models - quintessence field, k-essence field, tachyon field. The reconstruction of the tachyon model is our work. Using the method developed by Guo et al [52], we reconstruct the potential of the tachyon from the equation of state of the dark energy ω φ (z). The shapes of the potentials of the tachyon field are numerically reproduced for four typical parametrized models. We find that for these four models, the potentials are all monotonous and possess the same asymptotic behavior at the low redshift. In the last chapter, we make some conclusions, and then introduce the work that 3

we will do. The future Lamost program, SNAP program and Plank program will be able to determine the parameters in the dark energy parametrization to high precision, so the method using the parametrized equation of state to reconstruct the dark energy model is a potential one to learn more about the essence of the dark energy. Key Words: parametrization dark energy, tachyon field, the equation of state, reconstruction, 4

20 Edwin Hubble Hubble Hubble 20 10k 10k 1964 5 Arno Penzias Robert Wilson 1978 Nobel 2.7k Hubble 1964 Hoyle Taylor 23 25% Wagoner Fowler Hoyler 3 He D 7 Li 3 He 7 Li 9 1.1 Robertson-Walker 4

[1, 2, 3] ds 2 = dt 2 + a 2 dr 2 (t)[ 1 Kr + 2 r2 (dθ 2 + sin 2 θ dφ 2 )], (1.1) a(t) r θ φ (1.1) K K +1 0-1 (1.1) ds 2 = dt 2 + a 2 (t)[dχ 2 + f 2 K(χ)(dθ 2 + sin 2 θ dφ 2 )], (1.2) sinχ f K (χ) = χ sinhχ K=+1, K=0, K=-1. (1.3) (1.1)Einstein a(t) Einstein [1] G µ ν R µ ν 1 2 δµ ν R = 8π GT µ ν, (1.4) G µ ν Einstein R µ ν Ricci (Ricci ) R Ricci T µ ν Robertson walker ( (1.1)) [2] ( ) t R 0 0 = 3ä a, (1.5) (ä Rj i = a + 2ȧ2 a + 2K ) δ 2 a j, i 2 (1.6) (ä R = 6 a + ȧ2 a + K ), 2 a 2 (1.7), 5

T µ ν T µ ν = Diag( ρ, p, p, p), (1.8) ρ p (1.5) (1.8) (1.4) a(t) (ȧ ) 2 H 2 = a 8π Gρ 3 K a 2, (1.9) Ḣ = 4π G(p + ρ) + K a, (1.10) 2 H(= ȧ/a) Hubble ρ p (1.9) Friedmann (1.9) (1.10) ρ ρ + 3H(ρ + p) = 0, (1.11) (1.9) (1.11) (1.9) (1.10) K/a 2 ä a = 4π G (ρ + 3p), (1.12) 3 ρ + 3p < 0 (1.9) Ω(t) 1 = K (ah) 2, (1.13) Ω(t) ρ(t)/ρ c (t) ρ c (t) = 3H 2 (t)/8π G 6

(1.13) Ω > 1 or ρ > ρ c K = +1, Ω = 1 or ρ = ρ c K = 0, Ω < 1 or ρ < ρ c K = 1. (1.14) (Ω 1) [4] [3] 7

5% 30% 70% 2.1 2.1.1 Ia Ia Chandrasekhar Ia M Ia Minkowski L s d F :F = L s /(4π d 2 ) d L d 2 L L s 4π F, (2.1) χ s t 1 t 1 E 1 χ = 0 t 0 ( ) t 0 E 0 L s L 0 L s = E 1 t 1, L 0 = E 0 t 0, (2.2) 8

c = ν 1 λ 1 = ν 0 λ 0 λ 0 λ 1 = ν 1 ν 0 = t 0 t 1 = E 1 E 0, (2.3) (2.2) (2.3) L s L 0 = ( λ0 λ 1 ) 2, (2.4) z 1 + z = λ 0 λ = a 0 a, (2.5) χ s z (2.4) (2.5) L s = L 0 (1 + z) 2, (2.6) χ ds 2 = dt 2 + a(t)dχ 2 = 0 χ s = χs 0 dχ = t0 t 1 dt a(t) = 1 z a 0 H 0 0 dz h(z ) (2.7) h(z) = H(z)/H 0 (1.3) χ s χ = 0 t = t 0 S = 4π(a 0 f K (χ s )) 2 Robertson-Walker f K (χ s ) = χ s F = L 0 4π(a 0 f K (χ s )) 2 = L 0 4π(a 0 χ s ) 2 (2.8) (2.1 2.6 2.8) d L = a 0 χ s (1 + z), (2.9) (2.7) d L = 1 + z H 0 z 0 dz h(z ), (2.10) 9

ρ ( ) ρ = ρ i ω = p/ρ, (2.11) (1.11) ρ = ρ (0) i (a/a 0 ) 3(1+ω i), (2.12) (2.5) ρ = ρ (0) i (1 + z) 3(1+ω i), (2.13) (1.9)( K = 0) Hubble H 2 = H 2 0 Ω (0) i (1 + z) 3(1+ω i), (2.14) Ω (0) i 8π Gρ (0) i /(3H0) 2 = ρ (0) i /ρ (0) c (2.10) d L = (1 + z) H 0 z 0 dz, (2.15) (0) Ω i (1 + z) 3(1+ω i) m F M L s (2.1) ( ) dl m M = 5log 10 + 25, (2.16) Mpc M 1 + z 1 (2.15) d L z/h 0, (2.17) 10

2.1 H 0 d L z [5] Riess et al. Gold HST m z (2.16) (2.17) M [5] Tonry et al. Riess et al. (2.1) Ω 0 m = 0.31 Ω 0 Λ = 0.69(ω Λ = 1 ) 2.1.2 t 0 t 0 = t0 0 dt = 0 dz H(1 + z), (2.18) 11

Hubble (2.14)( Λ) H 0 t 0 = = dz 0 (1 + z) Ω 0 m(1 + z) 3 + Ω 0 Λ ( 2 3 1 + ) Ω 0 Λ ln, (2.19) Ω 0 Ω m 0 m Ω 0 m Ω 0 m = 0.3 Ω 0 Λ = 0.7 t 0 = 0.964H 1 0 13.1Gyr( h 0.72) t 0 > 11 12Gyr [6, 7] ω = 1 2.1.3 CMB [4] [8, 9] 1 Ω total 1 [10] CMB Ω 0 Λ (2.2) [11] SN Ia CMB Ω 0 Λ 0.7, Ω0 m 0.3 ω Λ = 1 Λ CDM Λ CDM 2.2 ω 1 Λ CDM Λ (fine-tuning problem) (coincidence problem) ω = 1 ( ) 12

3 No Big Bang 2 Supernovae vacuum energy density (cosmological constant) 1 0 Maxima SNAP Target Statistical Uncertainty CMB Boomerang expands forever recollapses eventually closed -1 Clusters open flat 0 1 2 3 mass density 2.2 Ia CMB [11] SNAP 2.2.1 I Λ Einstein Λ ω = 1 (fine-tuning problem) [12] (T µν ) vac = ρ vac g µν, (2.20) WMAP [13] Ω (0) Λ = 8π G ρ 3H0 2 Λ 0.73, H 0 71km sec 1 Mpc 1 1.5 10 42 GeV, (2.21) 13