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Register a new userHomogeneity and the causal boundary
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Enrique Gaztanaga
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A Universe with finite age also has a finite causal scale $chi_S$, so the metric can not be homogeneous for $chi>chi_S$, as it is usually assumed. To account for this, we propose a new causal boundary condition, that can be fulfil by fixing the cosmological constant $Lambda$ (a free parameter for gravity). The resulting Universe is inhomogeneous, with possible variation of cosmological parameters on scales $chi simeq chi_S$. The size of $chi_S$ depends on the details of inflation, but regardless of its size, the boundary condition forces $Lambda/8pi G $ to cancel the contribution of a constant vacuum energy $rho_{vac}$ to the measured $rho_Lambda equiv Lambda/8pi G + rho_{vac}$. To reproduce the observed $rho_{Lambda} simeq 2 rho_m$ today with $chi_S rightarrow infty$ we then need a universe filled with evolving dark energy (DE) with pressure $p_{DE}> - rho_{DE}$ and a fine tuned value of $rho_{DE} simeq 2 rho_m$ today. This seems very odd, but there is another solution to this puzzle. We can have a finite value of $chi_S simeq 3 c/H_0$ without the need of DE. This scale corresponds to half the sky at $z sim 1$ and 60deg at $z sim 1000$, which is consistent with the anomalous lack of correlations observed in the CMB.
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The cosmological constant $Lambda$ is usually interpreted as Dark Energy (DE) or modified gravity (MG). Here we propose instead that $Lambda$ corresponds to a boundary term in the action of classical General Relativity. The action is zero for a perfect fluid solution and this fixes $Lambda$ to the average density $rho$ and pressure $p$ inside a primordial causal boundary: $Lambda = 4pi G <rho+3p>$. This explains both why the observed value of $Lambda$ is related to the matter density today and also why other contributions to $Lambda$, such as DE or MG, do not produce cosmic expansion. Cosmic acceleration results from the repulsive boundary force that occurs when the expansion reaches the causal horizon. This universe is similar to the $Lambda$CDM universe, except on the largest observable scales, where we expect departures from homogeneity/isotropy, such as CMB anomalies and variations in cosmological parameters indicated by recent observations.
Quasinormal modes describe the return to equilibrium of a perturbed system, in particular the ringdown phase of a black hole merger. But as globally-defined quantities, the quasinormal spectrum can be highly sensitive to global structure, including distant small perturbations to the potential. In what sense are quasinormal modes a property of the resulting black hole? We explore this question for the linearized perturbation equation with two potentials having disjoint bounded support. We give a composition law for the Wronskian that determines the quasinormal frequencies of the combined system. We show that over short time scales the evolution is governed by the quasinormal frequencies of the individual potentials, while the sensitivity to global structure can be understood in terms of echoes. We introduce an echo expansion of the Greens function and show that, as expected on general grounds, at any finite time causality limits the number of echoes that can contribute. We illustrate our results with the soluble example of a pair of $delta$-function potentials. We explicate the causal structure of the Greens function, demonstrating under what conditions two very different quasinormal spectra give rise to very similar ringdown waveforms.
The purpose of this paper is two-fold. First, we would like to get rid of common assumption that causal set is bounded and attempt to model its scalar field action under the assumption that it isnt. Secondly, we would like to propose continuous measurement model in this context.
Teukolsky equations for $|s|=2$ provide efficient ways to solve for curvature perturbations around Kerr black holes. Imposing regularity conditions on these perturbations on the future (past) horizon corresponds to imposing an in-going (out-going) wave boundary condition. For exotic compact objects (ECOs) with external Kerr spacetime, however, it is not yet clear how to physically impose boundary conditions for curvature perturbations on their boundaries. We address this problem using the Membrane Paradigm, by considering a family of fiducial observers (FIDOs) that float right above the horizon of a linearly perturbed Kerr black hole. From the reference frame of these observers, the ECO will experience tidal perturbations due to in-going gravitational waves, respond to these waves, and generate out-going waves. As it also turns out, if both in-going and out-going waves exist near the horizon, the Newman Penrose (NP) quantity $psi_0$ will be numerically dominated by the in-going wave, while the NP quantity $psi_4$ will be dominated by the out-going wave. In this way, we obtain the ECO boundary condition in the form of a relation between $psi_0$ and the complex conjugate of $psi_4$, in a way that is determined by the ECOs tidal response in the FIDO frame. We explore several ways to modify gravitational-wave dispersion in the FIDO frame, and deduce the corresponding ECO boundary condition for Teukolsky functions. We subsequently obtain the boundary condition for $psi_4$ alone, as well as for the Sasaki-Nakamura and Detweilers functions. As it also turns out, reflection of spinning ECOs will generically mix between different $ell$ components of the perturbations fields, and be different for perturbations with different parities. We also apply our boundary condition to computing gravitational-wave echoes from spinning ECOs, and solve for the spinning ECOs quasi-normal modes.
We show that the Chern-Simons (CS) state when reduced to mini-superspace is the Fourier dual of the Hartle-Hawking (HH) and Vilenkin (V) wave-functions of the Universe. This is to be expected, given that the former and latter solve the same constraint equation, written in terms of conjugate variables (loosely the expansion factor and the Hubble parameter). A number of subtleties in the mapping, related to the contour of integration of the connection, shed light on the issue of boundary conditions in quantum cosmology. If we insist on a {it real} Hubble parameter, then only the HH wave-function can be represented by the CS state, with the Hubble parameter covering the whole real line. For the V (or tunnelling) wave-function the Hubble parameter is restricted to the positive real line (which makes sense, since the state only admits outgoing waves), but the contour also covers the whole negative imaginary axis. Hence the state is not admissible if reality conditions are imposed upon the connection. Modifications of the V state, requiring the addition of source terms to the Hamiltonian constraint, are examined and found to be more palatable. In the dual picture the HH state predicts a uniform distribution for the Hubble parameter over the whole real line; the modified V state a uniform distribution over the positive real line.
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