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.

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The Observation of Gravitational Waves from a Binary Black Hole Merger

This paper gives a comprehensive review on thick brane solutions and related topics. Such models have attracted much attention from many aspects since the birth of the brane world scenario. In many works, it has been usually assumed that a brane is an infinitely thin object; however, in more general situations, one can no longer assume this. It is also widely considered that more fundamental theories such as string theory would have a minimal length scale. Many multidimensional field theories coupled to gravitation have exact solutions of gravitating topological defects, which can represent our brane world. The inclusion of brane thickness can realize a variety of possible brane world models. Given our understanding, the known solutions can be classified into topologically non-trivial solutions and trivial ones. The former class contains solutions of a single scalar (domain walls), multi-scalar, gauge-Higgs (vortices), Weyl gravity and so on. As an example of the latter class, we consider solutions of two interacting scalar fields. Approaches to obtain cosmological equations in the thick brane world are reviewed. Solutions with spatially extended branes (S-branes) and those with an extra time-like direction are also discussed.

Thick brane solutions

In C. A. S. Almeida, R. Casana, M. M. Ferreira, Jr., and A. R. Gomes, Phys. Rev. D 79, 125022 (2009), the simplest Yukawa coupling eta(Psi) over bar phi chi Psi was considered for a two-scalar-generated Bloch brane model. Fermionic resonances for both chiralities were obtained, and their appearance is related to branes with internal structure. Inspired on this result, we investigate the localization and resonance spectrum of fermions on a one-scalar-generated de Sitter thick brane with a class of scalar-fermion couplings eta(Psi) over bar phi(k)Psi with positive odd integer k. A set of massive fermionic resonances for both chiralities is obtained when provided large coupling constant eta. We find that the masses and lifetimes of left and right chiral resonances are almost the same, which demonstrates that it is possible to compose massive Dirac fermions from the left and right chiral resonances. The resonance with lower mass has longer lifetime. For a same set of parameters, the number of resonances increases with k and the lifetime of the lower level resonance for larger k is much longer than the one for smaller k.

Fermion localization and resonances on a de Sitter thick brane

A class of thick branes in the background of sine-Gordon kinks with a scalar potential $V(\ensuremath{\phi})=p(1+\mathrm{cos}﻿\frac{2\ensuremath{\phi}}{q})$ was constructed by R. Koley and S. Kar [Classical Quantum Gravity 22, 753 (2005)]. In this paper, in the background of the warped geometry, we investigate the issue of localization of spin-$1/2$ fermions on these branes in the presence of two types of scalar-fermion couplings: $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\Psi}$ and $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\mathrm{sin}﻿\ensuremath{\phi}\ensuremath{\Psi}$. By presenting the mass-independent potentials in the corresponding Schr\"odinger equations, we obtain the lowest Kaluza-Klein modes and a continuous gapless spectrum of Kaluza-Klein states with ${m}^{2}g0$ for both types of couplings. For the Yukawa coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\Psi}$, the effective potential of the right chiral fermions for positive $q$ and $\ensuremath{\eta}$ is always positive; hence only the effective potential of the left chiral fermions could trap the corresponding zero mode. This is a well-known conclusion which is discussed extensively in the literature. However, for the coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\mathrm{sin}﻿\ensuremath{\phi}\ensuremath{\Psi}$, the effective potential of the right chiral fermions for positive $q$ and $\ensuremath{\eta}$ is no longer always positive. Although the value of the potential at the location of the brane is still positive, it has a series of wells and barriers on each side, which ensures that the right chiral fermion zero mode could be trapped. Thus we may draw the following remarkable conclusion: for positive $\ensuremath{\eta}$ and $q$, the potentials of both the left and right chiral fermions could trap the corresponding zero modes under certain restrictions.

Fermions on Thick Branes in the Background of Sine-Gordon Kinks

In the literatures, several types of thick smooth brane configurations in a pure geometric Weyl integrable 5-dimensional space time have been presented. The Weyl geometry is a non-Riemannian modification of 5-dimensional Kaluza-Klein (KK) theory. All these thick brane solutions preserve 4-dimensional Poincare invariance, and some of them break Z2-symmetry along the extra dimension. In this paper, we study localization of various matter fields on these pure geometrical thick branes, which also localize the graviton. We present the shape of the potential of the corresponding Schrodinger problem and obtain the lowest KK mode. It is shown that, for both spin 0 scalars and spin 1 vectors, there exists a continuum gapless spectrum of KK states with m2 > 0. But only the massless mode of scalars is found to be normalizable on the brane. However, for the massless left or right chiral fermion localization, there must be some kind of Yukawa coupling. For a special coupling, there exist a series of discrete massive KK modes with m2 > 0. It is also showed that for a given coupling constant only one of the massless chiral modes is localized on the branes.

http://iopscience.iop.org/article/10.1088/1126-6708/2008/02/067/pdf

Localization of matters on pure geometrical thick branes

Scalar kinks propagating along the bulk in warped spacetimes provide a thick brane realisation of the braneworld. We consider here, a class of such exact solutions of the full Einstein-scalar system with a sine-Gordon potential and a negative cosmological constant. In the background of the kink and the corresponding warped geometry, we discuss the issue of localisation of spin half fermions (with emphasis on massive ones) on the brane in the presence of different types of kink-fermion Yukawa couplings. We analyse the possibility of quasi-bound states for large values of the Yukawa coupling parameter $\gamma_F$ (with $\nu$, the warp factor parameter kept fixed) using appropriate, recently developed, approximation methods. In particular, the spectrum of the low--lying states and their lifetimes are obtained, with the latter being exponentially enhanced for large $\nu \gamma_F$. Our results indicate quantitatively, within this model, that it is possible to tune the nature of warping and the strength and form of the Yukawa interaction to obtain trapped massive fermion states on the brane, which, however, do have a finite (but very small) probability of escaping into the bulk.

Scalar kinks and fermion localisation in warped spacetimes

Scalar kinks propagating along the bulk in warped spacetimes provide a thick brane realization of the braneworld. We consider here, a class of such exact solutions of the full Einstein-scalar system with a sine-Gordon potential and a negative cosmological constant. In the background of the kink and the corresponding warped geometry, we discuss the issue of localization of spin half fermions (with emphasis on massive ones) on the brane in the presence of different types of kink–fermion Yukawa couplings. We analyse the possibility of quasi-bound states for large values of the Yukawa coupling parameter ηF (with ν, the warp factor parameter, kept fixed) using appropriate, recently developed, approximation methods. In particular, the spectrum of the low-lying states and their lifetimes are obtained, with the latter being exponentially enhanced for large νηF. Our results indicate quantitatively, within this model, that it is possible to tune the nature of warping and the strength and form of the Yukawa interaction to obtain trapped massive fermion states on the brane, which, however, do have a finite (but very small) probability of escaping into the bulk.

https://iopscience.iop.org/article/10.1088/0264-9381/22/4/008/pdf

Scalar kinks and fermion localization in warped spacetimes

A novel braneworld model with a bulk scalar field

Six-dimensional bulk spacetimes with 3- and 4-branes are constructed using certain non-conventional bulk scalars as sources. In particular, we investigate the consequences of having the phantom (negative kinetic energy) and the Brans–Dicke scalar in the bulk while obtaining such solutions. We find geometries with 4-branes with a compact on-brane dimension (hybrid compactification) which may be assumed to be small in order to realize a 3-brane world. On the other hand, we also construct, with similar sources, bulk spacetimes where a 3-brane is located at a conical singularity. Furthermore, we investigate the issue of localization of matter fields (scalar, fermion, graviton, vector) on these 3- and 4-branes and conclude with comments on our six-dimensional models.

http://iopscience.iop.org/article/10.1088/0264-9381/24/1/004/pdf

Braneworlds in six dimensions: new models with bulk scalars

A generalized two-brane Randall-Sundrum warped braneworld model has solutions of the warp factor for both positive or negative cosmological constant on the visible 3-brane which can resolve the naturalness problem in connection with the fine-tuning of Higgs mass in the standard model of elementary particles. To explore the location of the standard model fermions in such a generalized warped model, we, in this work, determine the dependence of the localization profile of a bulk fermion on the brane cosmological constant brane tension and the bulk fermion mass. Our results reveal that for a positive and small value of the induced cosmological constant a bulk fermion is localized close to the brane. On the other hand for a visible brane with negative cosmological constant and positive tension, the fermions are localized inside the bulk leading to phenomenologically interesting possibilities.

Ratna Koley

Papers

Scalar kinks and fermion localisation in warped spacetimes

Scalar kinks and fermion localization in warped spacetimes

A novel braneworld model with a bulk scalar field

Braneworlds in six dimensions: new models with bulk scalars

Fermion localization in a generalized Randall-Sundrum model