Quantum Gravity Effects in Cosmology

EPJ Web of Conferences Quantum Gravity Effects in Cosmology Je-An Gu 2 Sang Pyo Kim 1 Che-Min Shen 0 0 Department of Physics and Leung Center for Cosmology and Particle Astrophysics, National Taiwan University , Taipei 10617 , Taiwan 1 Department of Physics, Kunsan National University , Kunsan 54150 , Korea 2 Leung Center for Cosmology and Particle Astrophysics, National Taiwan University , Taipei 10617 , Taiwan Within the geometrodynamic approach to quantum cosmology, we studied the quantum gravity effects in cosmology. The Gibbons-Hawking temperature is corrected by quantum gravity due to spacetime fluctuations and the power spectrum as well as any probe field will experience the effective temperature, a quantum gravity effect. 1 Introduction Recent remarkable observations have made cosmology a science of precision. The more precise and accurate cosmological observations are, the more likely the possibility of quantum gravity effects are to be measured. In this talk, we review a quantum gravity effect in cosmological observations. There have been proposed different approaches to quantum gravity (for a review, see [1]). In partciular, the geometrodynamical approach to quantum gravity and cosmology seems to be practically useful in that the transition from quantum gravity to semiclasical gravity and then to classical gravity including quantum corrections for the Einstein and the energy-stress tensors prescribes dynamical laws and correlations in terms of classical cosmology variables and parameters [2]. The Wheeler-DeWitt (WDW) equation is the quantum law for the geometrodynamical approach to quantum gravity, which becomes a relativistic wave equation in the superspace of the scale factor a and an inflaton φ for a Friedmann-Robertson-Walker universe. The scattering formulation of the two-component wave was advanced in Refs. [3, 4]. The Born-Oppenheimer interpretation of the WDW equation with respect to the Planck scale for gravity part and the energy scale for the inflaton separates the gravity part from the inflaton part and a further application of de Broglie-Bohm pilotwave theory results in the classical equation together with quantum corrections of gravitational and field fluctuations [5, 6] (for references, see [2]). The quantum cosmology for an inflationary model with inhomogeneous fluctuations predicted the power spectrum with quantum corrections, which is suppressed at large scales and provides a weaker upper bound on the Hubble constant H than the tensor-to-scalar ratio [7]. Other quantum gravity effects may be found in Ref. [8]. The main purpose of this proccedings’ article is to compliment the recent discovery that the quantum cosmology results in an effective Gibbons-Hawking temperature due to quantum fluctuations [9]. To do this, we provide two new methods to find approximately the solution to the WDW equation for the FRW universe with a cosmological constant: the WKB approximation method and the LiouvilleGreen function method. From the wave function we can calculate the quantum potential, which gives rise to the quantum corrections to the semiclassical gravity. We then solve the semiclassical equation for the Eucliean geometry to find a periodic solution, whose period is the inverse temperature for the dS space in the sense of Ref. [10]. This approach may be an alternative to the gravitational instanton method [11]. Finally, we show that a probe field experiences this periodicity, which justifies the physical meaning of the temperature for dS space with quantum corrections. 2 de Broglie-Bohm Pilot-Wave Theory for Quantum Cosmology The quantum cosmology based on the WDW equation leads to the semiclassical cosmology with quantum corrections as will be shown below. The WDW equation for a spatially closed FRW universe with a cosmological constant Λ, i.e, pure dS space, takes the form −l2p ∂∂a22 + l12p (a2 − HΛ2a4) Ψ(a) = 0, ( 1 ) where lp = √4πG/2 is the Planck length. Introducing a dimensionless variable x = a/lp and a constant H¯ Λ = HΛlp, the WDW equation describes a quantum problem with the zero energy for an inverted Mexican hat potential d2 − dx2 + x2 − H¯ Λ2 x4 Ψ(x) = 0. ( 2 ) We now apply the de Broglie-Bohm pilot theory to Eq. ( 2 ) by expressing the solution in the oscillatory region as Substituting the solution F = 1/|S |1/2 of Eq. ( 5 ) into Eqs. ( 3 ) and ( 4 ) is equivalent to the exact Wentzel-Kramers-Brillouin (WKB) solution or phase-integral formula. The essence of the de BroglieBohm pilot theory is to solve Eqs. ( 4 ) and ( 5 ) at the same time by regarding F as a functional of S , i.e., F(S ). In the nonoscillatory region corresponding to a Euclidean spacetime, Eqs. ( 3 )−( 5 ) may be understood as an analytical continuation from the oscillatory region. In fact, introducing the cosmological time as the directional derivative of x along the peak of the wave packet ∂ 1 ∂S ∂ ∂t = − x ∂x ∂x , ( 6 ) Eq. ( 4 ) becomes the semiclassical gravity equati (...truncated)