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Cosmological tests of models for the accelerating universe in terms of inhomogeneities

  • Astronomy
  • Mathematics


Chapter 4 Previous attempts to test alternative models to ΛCDM model 4.1 Previous tests of modified gravity theories 4.1.1 Parameterized post-Newtonian framework In this section, we will introduce the “Parameterised Post-Newtonian” (PPN) frame- work [107] that contains a wide class of different gravitational theories, and that contains parameters which can be constrained by observations. Observers can apply observational results to constrain a wide class of theories without considering the details of the indi- vidual theories themselves. Theorists can straightforwardly constrain new theories by comparing to the already established bounds on the PPN parameters without recalculat- ing individual gravitational phenomena. Then, this approach has been highly successful. In the weak gravitational system, such as the solar system, where gravitation is weak enough for Newton’s theory of gravity, there are the relation between the velocity v˜ and the Newtonian gravitational potential U˜ v2 ∼ U = GM c2R ∼ 2× 10−6 ( M M� )( R R� )−1 , (4.1.1) where v ≡ v˜/c and U ≡ U˜/c2 are dimensionless in geometrized units (c = G = 1), and 50 G = 6.7×10−8 cm3g−1s−2,M� = 2.0×1033 g and R� = 7.0×1010 cm are the gravitational constant, the solar mass and the solar radius, respectively. Then, we can find that U� ∼ 10−5, Uearth ∼ 10−11, UNS ∼ 10−1, UBH ∼ 100, (4.1.2) in the solar system, the system of earth, neutron star and black hole, respectively. When we begin to demand accuracies greater than a part in 105, the Newtonian limit no longer suffices. For example, it cannot account for Mercury’s additional perihelion shift of ∼ 5× 10−7 radians per orbit. Then, we need a more accurate approximation to the metric that goes “post”-Newtonian theory. The PPN framework is a perturbative treatment of weak-field gravity, and therefore requires a small parameter. For this purpose, we define an order of smallness: U ∼ v2 ∼ P ρ ∼ Π ∼ O (�2) , (4.1.3) where P is the pressu

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