Loop Quantum Cosmology: Effective Theory and Related Applications

Abstract

The effort toward making quantum mechanics and general relativity compatible (quantum gravity) has lasted more than a century. The string theory and loop quantum gravity both stand out as strong candidates and currently they both make popular research subjects in quantum gravity. As we known, Loop quantum gravity (LQG)(Ashtekar & Lewandowski, 2004; Rovelli, 1998; 2004; Thiemann, 2007) is a background independent and non-perturbative canonical quantum gravity theory. LQG has made many breakthroughs in recent years: the establishment of the quantum Einstein equations Ashtekar & Tate (1994); Ashtekar et al. (1995a); Corichi & Zapata (1997); Lewandowshi & Thiemann (1999); Rovelli & Smolin (1994); Thiemann (1996; 1998a;b;c; 2001), the proof that the Riemannian operators have discrete eigenvalues Ashtekar et al. (1995b); Lewandowski (1997); Loll (1995a;b; 1997a;b); Rovilli & Smolin (1995); Thiemann (1998d;e),results concerning the entropy of the black hole horizon and cosmological horizon entropy with statistical mechanics Ashtekar et al. (1998; 1999; 2000; 2001; 2002; 2003a;b); Berreira et al. (1996); Rovelli (1996a;b); Smolin (1995), and so on. As an application of loop quantum gravity to cosmology, loop quantum cosmology (LQC) Bojowald (2005a; 2008); Date (2002) also presents itself as a possible path toward answers to the cosmological and astrophysical riddles. As a symmetry reduced model of LQG, LQC inherits the quantum schemes originated from LQG that dealt with the isotropic and homogeneous universe firstly and then extended to the inhomogeneous and anisotropic model Bojowald (2002a). It plays an important role in connecting the LQG theory and the measureable world. On one hand, it is used to test the full theory, which, in its own form, is extremely complex and difficult to directly apply.On the other hand, making connections to the real world sheds light on further improvement of the LQG theory. These reasons make LQC a promising and enlightening subject to study. In LQC, the collapsing and expanding phases are connected by the cyclic or oscillatorymodels Lidsey et al. (2004), and the universe is automatically born with a small scale factor at the fixed point near the Planck phase Bojowald (2005); Mulryne et al. (2005a). Unlike in the emergent universe model, this fixed point here is stable and allows the universe to start in an initial phase of oscillation. Then an inflationary phase is entered, which is the relevant regime for structure formation. In LQC, there aremany different inflationary scenarios Artymowski et al. (2009); Bojowald et al. (2004); Mulryne et al. (2005b); Zhang & Ling (2007), among which the one without inflation is the most attractive, mainly because it can explain the inflationary phase directly from LQG. But, unfortunately, it is difficult to study the structure formation. 16