There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

Dynamics of dark energy

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10(-21). It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203,000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of 410(-180)(+160)  Mpc corresponding to a redshift z=0.09(-0.04)(+0.03). In the source frame, the initial black hole masses are 36(-4)(+5)M⊙ and 29(-4)(+4)M⊙, and the final black hole mass is 62(-4)(+4)M⊙, with 3.0(-0.5)(+0.5)M⊙c(2) radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

/pdf/the-observation-of-gravitational-waves-from-a-binary-black-58srpupgg9.pdf

The Observation of Gravitational Waves from a Binary Black Hole Merger

We discuss the phase-space of conservation laws in Lagrangian and Hamiltonian formalism from a very general point of view deriving all the geometrical properties of this reduced configuration space. Such a mathematical approach, based on an integration of the system dependent on the inequality between the number of dimensions in the configuration space and the number of conservation laws, is extremely useful in connection to the derivation of a General Conservation Principle (from which particular conservation laws can be derived). Properties and behaviours of general solutions are discussed in relation to the existence of first integrals of motion. Mathematics subject classification (2000): 70H33, 70H06.

/pdf/the-phase-space-view-of-conservation-laws-27j69zu07g.pdf

The phase-space view of conservation laws

Abstract We investigate how the cosmological equation of state can be used for scrutinizing extended theories of gravity, in particular, the Palatini $$f({{\mathcal {R}}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity. Specifically, the approach consists, at first, in investigating the effective equation of state produced by a given model. Then, the inverse problem can also be considered in view of determining which models are compatible with a given effective equation of state. We consider and solve some cases and show that, for example, power-law models are (the only models) capable of transforming barotropic Equations of State into effective barotropic ones. Moreover, the form of equation of state is preserved (only) for $$f({{\mathcal {R}}})={{\mathcal {R}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> , as expected. In this perspective, modified Equations of State are a feature capable of distinguishing extended gravity with respect to general relativity. We investigate also quadratic and non-homogeneous effective Equations of State showing, in particular, that they contain the Starobinsky model and other ones. 

/pdf/the-effective-equation-of-state-in-palatini-f-mathcal-r-f-r-1pyctlky.pdf

The effective equation of state in Palatini 
              
                
              
              $$f({{\mathcal {R}}})$$
              
                
                  f
                  (
                  R
                  )
                
              
             cosmology

The so-called f(R)-gravity could, in principle, explain the accelerated expansion of the Universe without adding unknown forms of dark energy/dark matter, but more simply extending the general relativity by generic functions of the Ricci scalar. However, as a part of several phenomenological models, there is no final f(R)-theory capable of fitting all the observations and addressing all the issues related to the presence of dark energy and dark matter. Astrophysical observations are pointing out huge amounts of “dark matter” and “dark energy” needed to explain the observed large-scale structures and cosmic accelerating expansion. Up to now, no experimental evidence has been found, at a fundamental level, to explain such mysterious components. The problem could be completely reversed considering dark matter and dark energy as “shortcomings” of general relativity.

Higher-order theories of gravity matched with large-scale structure and cosmological observations

One of the most crucial areas of gravity research, after the direct observation of gravitational waves, is the possible modification of General Relativity at ultraviolet and infrared scales. In particular, the possibility of parity violation should be considered in strong field regime. The Chern-Simons gravity takes into account parity violation in strong gravity regime. For all conformally flat spacetimes and spacetimes with a maximally symmetric two-dimensional subspace, Chern-Simons gravity is identical to General Relativity. Specifically, the Anti-de Sitter (A)dS-Kerr/Kerr black hole is not a solution for Chern-Simons gravity. The slow-rotating BH and the quadratic order in spin solutions are some of the known solutions to quadratic order in spin and they are rotating solutions in the frame of dynamical Chern-Simons gravity. In the present study, for the (A)dS slow-rotating situation (correct to the first order in spin), we derive the linear perturbation equations controlling the metric and the dynamical Chern-Simons field equation corrected to the linear order in spin and to the second order in the Chern-Simons coupling parameter. We show that the black hole of the (A)dS-Kerr solution is stronger (i.e. more compact and energetic) than the Kerr black hole solution and the reason for this feature comes form contributions at Planck scales. Moreover, we calculate the thermodynamical quantities related to this black hole. Finally, we calculate the geodesic equation and derive the effective potential of the black hole.

Salvatore Capozziello

Papers

The phase-space view of conservation laws

The effective equation of state in Palatini $$f({{\mathcal {R}}})$$ f ( R ) cosmology

Higher-order theories of gravity matched with large-scale structure and cosmological observations

Spinning (A)dS black holes with slow-rotation approximation in dynamical Chern-Simons modified gravity