Diego Saez-Gomez

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General Relativity: Big Bang model R0 0 = 3ä a , (5) (ä Rj i = a + 2ȧ2 a 2 + 2K ) a 2 δj i , (6) (ä R = 6 a + ȧ2 a 2 + K ) a 2 , (7) FLRW cosmologies where a dot denotes a derivative with respect to t. Let us consider an ideal perfect fluid as the source of the energy momentum tensor T µ ν .Inthiscasewehave T ν µ =Diag(−ρ,p,p,p) , (8) B. The evolution of the universe filled with a where ρB. andThe p are evolution theperfect energy of the density fluid universe and the filledpressure with a density of the fluid, respectively. perfect Then fluid Eq. (4) gives the FLRW equations in General two Relativity independent equations Let us consider the evolution of the universe filled with B. The Let usevolution consider (ȧ the ofevolution the universe of the universe filled filledawith abarotropicperfectfluidwithanequationofstate ion of homogeneity and isotropy of the pproximately true on large scales. The m homogeneity at early epochs played oleinthedynamicalhistoryofourunil density perturbations grew via gravy into the structure we see today in temperature anisotropies observed in wave Background (CMB) are believed from quantum fluctuations generated nary stage in the early universe. See 3, 74, 75, 76] for details on density pered by inflationary cosmology. In this eview the main features of the homopic cosmology necessary for the subseic is given by [70, 77, 78, 79] ) 2 [ dr 2 ] abarotropicperfectfluidwithanequationofstate H 2 ≡ perfect = fluid 8πGρ − K ) 1 − Kr 2 + r2 (dθ 2 +sin 2 θdφ 2 ) , a 3 a 2 , (9) w = p/ρ , (1) Ḣ = −4πG(p + ρ)+ K a 2 , (10) (17) Let us consider the evolution of the universe filled with abarotropicperfectfluidwithanequationofstate where w is assumed to be constant. Then by solving the where factorwithcosmictimet. Thecoordire known as comoving For a perfect coordinates. fluid with Einstein H is the equations Hubble given parameter, in Eqs. ρ(9) and and p denote (10) with the K to-=0tal energy density and pressure of all the species present A wean inobtain equation we obtainof state, w = p/ρ , (17) the universe at a given epoch. icle comes to rest in these coordinates. The energy momentum tensor is conserved 2 by virtue of a purely kinematic statement. In this where w is assumed toH be= constant. 2 Then by solving the the Bianchi identities, leading to the continuity equation ics is associated with the scale factor– H = 3(1 + w)(t − t 0 ) , (18) Einstein equations given in Eqs. (9) and 2 (10) with K =0, ations allow us to determine the scale we obtain a(t) ∝ (t − t 0 ) 3(1+w) , (19) 2 e matter content of the universe is spect K in the metric (1) describes the ge- 2 a(t) ˙ρ +3H(ρ ρ ∝(t a −3(1+w) −+ p) t 0 ) =0. , (11)(20) tial section of space time, with closed, Equation (11) can H be = where t derived from Eqs. (9) and (10), 3(1 + w)(t − t erses corresponding to K =+1, 0, −1, which means that two of Eqs. (9), (10) 0 ) , (18) 0 is constant. We note that the above solution is valid for w ≠ −1. The radiation and (11) are independent. Eliminating a(t) ∝ the (t −K/a t 2 2 dominated universe 0 ) 3(1+w) term from , Eqs. (9) and which contains an initial singularity corresponds (and toperhaps w = 1/3, a future whereas one..). the dust dominated (19) enient to write the metric (1) in the (10), universe we obtain to w ρ=0. ∝ aInthesecaseswehave −3(1+w) , (20) ä t) [ dχ 2 + fK 2 (χ)(dθ2 +sin 2 θdφ 2 ) ] universe toRadiation w =0. a = Inthesecaseswehave : − a(t) 4πG ∝ (ρ(t +3p) − t 0 ) . (12) where t 3 , (2) 0 is constant. We note that 1/2 , ρ ∝ a above −4 , (21) solution is valid Hencefor thewaccelerated ≠Dust −1. : The a(t) expansion radiation ∝ (t − t 0 ) 2/3 occurs dominated , ρ ∝ a −3 for +3p
FLRW cosmologies 12 3 No Big Bang (54) 2 Supernovae t 0 → ig. 3 verse (0) Λ = t 0 = s the tellar s the to go a flat rized n the vacuum energy density (cosmological constant) 1 0 -1 Clusters Maxima SNAP Target Statistical Uncertainty CMB Boomerang open expands forever recollapses eventually flat closed 0 1 2 3 mass density arget the MB AP f prithe G. Aldering [SNAP Collaboration], “Future Research FIG. 4: The Direction Ω (0) m -Ω (0) Λand confidence Visions for regions Astronomy”, constrained Alan M. from the observations Dressler, of SN editor, Ia, Proceedings CMB and of the galaxy SPIE, Volume clustering. 4835, We also show the pp. expected 146-157 [arXiv:astro-ph/0209550]. confidence region from a SNAP satellite for a flat universe with Ω (0) m =0.28. From Ref. [106]. We need something Clearly the flat else: universe Dark with- energy or modified gravity....but how is the Equation of State the “cosmic triangle”). out a cosmological constant is ruled out. The compilation of three different cosmological data sets strongly reinforces the need for a dark energy dominated universe
Page 1 and 2: Cosmological constraints on modifie
Page 3: Brief Article The Author April 6, 2
Page 7 and 8: FLRW cosmologies Future singulariti
Page 9 and 10: Viable f(r) gravity Viability condi
Page 11 and 12: 3 Cosmological evolution in viable
Page 13 and 14: Cosmological evolution in viable f(
Page 15 and 16: Cosmological evolution in viable f(
Page 17 and 18: 8 − 3H 2 − 2Ḣ = 1 κ 2 p m +
Page 19 and 20: (z = 0) = H 0 = 100 hkms −1 Mpc
Page 21 and 22: (z = 0) = H 0 = 100 hkms −1 Mpc
Page 23 and 24: effective repulsive force 12h R ∼
Page 25 and 26: Testing F(R) gravity with Sne Ia da
Page 27 and 28: Testing F(R) gravity with Sne Ia da
Page 29 and 30: ere right lethand us remind side (r
Page 31 and 32: e previous section, we study two pa
Page 33 and 34: en at redshift z = 1000 where δ is

Diego Saez-Gomez

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