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The general k-essence Lagrangian for the existence of cosmological scaling solutions is derived in the presence of multiple scalar fields coupled to a barotropic perfect fluid. In addition to the scaling fixed point associated with the dynamics during the radiation and matter eras, I also obtain a scalar-field dominated solution relevant to dark energy and discuss the stability of them in the two-field set-up. I apply this general results to a model of two canonical fields with coupled exponential potentials arising in string theory. Depending on model parameters and initial conditions, I show that the scaling matter-dominated epochs followed by an attractor with cosmic acceleration can be realized with/without the couplings to scalar fields. The different types of scaling solutions can be distinguished from each other by the evolution of the dark energy equation of state from high-redshifts to today.

I report the current progress of a new 2nd-order Einstein-Boltzmann solver for CMB anistoropy in C++, cmb2nd (tentative). The development of the 1st-order solver, being equivalent to the existing codes like CAMB, has been almost completed. cmb2nd solves the perturbed Boltzmann equations for the fluctuations of the photon temperature+polarisation and the mass-less neutrino temperature coupled to the CDM+baryon fluid and gravitational perturbations. The recombination and reionisation history can be customised. Currently, cmb2nd implemented the instanteneous recombination/reionisation and Peebles' model (modified by Weinberg), and capable of inputing the output from Recfast. I briefly mention our future plan towards implementing osbervables involving the 2nd-order quantities.

I will describe how the fifth force in modified gravity is suppressed in the vicinity of a source in the context of the Horndeski theory and its generalization. I will in particular focus on the recently proposed scalar-tensor theory beyond Horndeski, and discuss the impact of the new terms on small scales.

I will review the recent developments in computing the second order CMB anisotropies and the intrinsic bispectrum that arises due to non-linear physics at recombination and the late time propagation of photons through inhomogeneous spacetime. I will show the latest results from SONG, a numerical code to solve the second order Boltzmann-Einstein system developed at Portsmouth. I will also discuss a way to incorporate the late time gravitational effects such as lensing, redshift and time-delay using a new line of sight integration approach.

In this talk, we introduce a new approach to a treatment of the non-linear gravitational effects (redshift, time delay and lensing) on the observed cosmic microwave background (CMB) anisotropies based on the Boltzmann equation. From the Liouville’s theorem in curved spacetime, the intensity of photons is conserved along a photon trajectory when non-gravitational scatterings are absent. Motivated by this fact, we derive a second-order line-of-sight formula by integrating the Boltzmann equation along a perturbed geodesic (curve) instead of a background geodesic (line). In this approach, the separation of the gravitational and intrinsic effects are manifest. This approach can be considered as a generalization of the remapping approach of CMB lensing, where all the gravitational effects can be treated on the same footing.

We consider a test of the Copernican Principle through observations of the large-scale structures, and for this purpose we study the self-gravitating system in a relativistic huge void universe model which does not invoke the Copernican Principle. If we focus on the slowly moving and weakly self-gravitating system whose spatial extent is much smaller than the scale of the cosmological horizon in the homogeneous and isotropic background universe model, the cosmological Newtonian approximation is available. Also in the huge void universe model, the same kind of the approximation as the cosmological Newtonian approximation is available in the analysis of the perturbations contained in a region whose spatial size is much smaller than the scale of the huge void. By using this approximation, we derive the equations of motion for the weakly self-gravitating perturbations whose elements have relative velocities much smaller than the speed of light, and show the derived equations can be significantly different from those in the homogeneous and isotropic universe model, due to the anisotropic volume expansion in the huge void.

Galaxies are biased tracers of the underlying matter distribution in our Universe. The distribution of primordial density perturbations is a key discriminant between different models for the origin of structure, and deviations from a simple Gaussian distribution can give rise to scale-dependent bias on the largest observable scales. Studying large scales close to the Hubble-horizon requires a consistent relativistic interpretation of gravitational collapse on small-scales, but standard treatments (and numerical simulations) are based on Newtonian gravity. I will show how Einstein's general relativity gives rise to a characteristic signature in the distribution of galaxies on large-scales.

I would like to review the effect of the primordial non-Gaussianity in large scale structure. Scale-dependent bias in the halo/galaxy power spectrum has been known as a probe of the primordial non-Gaussianity. Following the brief introduction of the scale-dependent bias, I would like to discuss the higher order statistics of the halo/galaxy distribution, especially, halo/galaxy bispectrum. In the halo/galaxy bispectrum, in addition to the gravity-induced non-Gaussianity, the effect of the primordial non-Gaussianity would appear. We analytically investigate in detail the effect of such primordial non-Gaussianity on the large-scale halo/galaxy bispectrum. In our formalism, the effects of primordial non- Gaussianity are wholly encapsulated in the linear (primordial) polyspectra, and we systematically calculate the contributions to the large-scale behaviors arising from the three types of primordial bispectrum (local-, equilateral-, and orthogonal-types), and also primordial trispectrum of the local- type non-Gaussianity. We find that the equilateral- and orthogonal-type non-Gaussianities show distinct scale-dependent behaviors which can dominate the gravity-induced non-Gaussianity at very large scales. For the local-type non-Gaussianity, higher-order loop corrections are found to give a significantly large contribution to the halo/galaxy bispectrum of the squeezed shape, and eventually dominate over the other contributions on large scales. We also give a useful diagrammatic approach which helps us to systematically investigate an impact of such higher-order contributions to the large-scale halo/galaxy bispectrum.

We study the nature of constraints and the Hamiltonian structure in a scalar-tensor theory of gravity recently proposed by Gleyzes, Langlois, Piazza and Vernizzi (GLPV). For the simple case with A_5 = 0, namely when the canonical momenta conjugate to the spatial metric are linear in the extrinsic curvature, we prove that the number of physical degrees of freedom is three at fully nonlinear level, as claimed by GLPV. Therefore, while this theory extends Horndeski's scalar-tensor gravity theory, it is protected against additional degrees of freedom.

It was difficult to construct a theory which includes a massive graviton. However, such theory is recently found, which is called bimetric gravity. Bimetric gravity includes two metrics which are interacting with each other, produces a massive graviton and a massless graviton (corresponding to the total coordinate transformation invariance) and depends on five theoretical parameters. We calculate the tensor spectrum produced during inflation in minimal bimetric model and explain the feature of the spectrum. We see that the amplitude of the spectrum is suppressed through the mixing of the massless graviton and the massive graviton. We also see that the amplitude does not conserve on super-horizon scales under the slow-roll approximation.

When extracting cosmological information of power spectrum measurement of cosmological probes such as weak lensing and galaxy clustering, we need to take into account the impact of super-sample density fluctuations whose wavelengths are larger than the survey scale. In this talk, I discuss the physics of the super-sample effects and the observational implications.

I'll talk about the possiblly detectable modifications of gravity by the future direct observation of gravitational waves.

I will discuss a consistent theory for a self-interacting vector field, breakingan Abelian symmetry in such a way to obtain an interesting behavior for its longitudinal polarization. In an appropriate decoupling limit, the dynamics of the longitudinal mode is controlled by Galileon interactions. The full theory away from the decoupling limit does not propagate ghost modes, and can be investigated in regimes where non-linearities become important. When coupled to gravity, this theory provides a candidate for dark energy, since it admits de Sitter cosmological solutions characterized by a technically natural value for the Hubble parameter. Finally, I will describe how to spontaneously break the gauge symmetry so to generate the desired vector self-interactions via a Higgs mechanism.

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Recent CMB observations hint at there being a suppression of scalar power on large scales. Focusing on two toy models, we discuss how such a suppression may arise in the context of open inflation.

We discuss whether or not bigravity theory can be embedded into the braneworld setup. By considering Dvali-Gabadadze-Porrati 2-brane model with Goldberger-Wise radion stabilization, we show that we can construct a ghost-free model whose low energy spectrum is composed of a massless graviton and a massive graviton with a small mass, and that the behavior of this effective theory is shown to be identical to the ghost-free bigravity. However, this attempt faces two difficulties: 1) this correspondence breaks down at a relatively low energy due to the limitation of the adopted stabilization mechanism, and 2) Boulware-Deser ghost revives in the ghost-free bigravity with doubly coupled matter which can be naturally introduced in our braneworld model without ghost.

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We investigate a general framework for effective theories propagating two tensor and one scalar degrees of freedom. Geometrically, it describes dynamical hypersurfaces coupled to a general background, in which the scalar mode encodes the fluctuation of the hypersurfaces. Within this framework, various models in the literature --- including k-essence, Horndeski theory, the EFT of inflation, ghost condensate as well as the Ho\v{r}ava gravity --- get unified. Our framework generalizes the Horndeski theory in the sense that, it propagates the correct number of degrees of freedom, although the equations of motion are generally higher order.

We study the cosmology of an extended version of Horndeski theories with second-order equations of motion on the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background. In addition to a dark energy field associated with the gravitational sector, we take into account multiple scalar fields (I=1,2...,N-1) characterized by the k-essence Lagrangians. These additional scalar fields can model the perfect fluids of radiation and non-relativistic matter. We derive propagation speeds of scalar and tensor perturbations as well as conditions for the absence of ghosts. The theories beyond Horndeski induce non-trivial modifications to all the propagation speeds of N scalar fields, but the modifications to those for the matter fields are generally suppressed relative to that for the dark energy field. We apply our results to the covariantized Galileon with an Einstein-Hilbert term in which partial derivatives of the Minkowski Galileon are replaced by covariant derivatives. Unlike the covariant Galileon with second-order equations of motion in general space-time, the scalar propagation speed square associated with the dark energy field becomes negative during the matter era for late-time tracking solutions, so the two Galileon theories can be clearly distinguished at the level of linear cosmological perturbations.

Noether gauge symmetry (NGS) approach is a powerful tool to solve the governing equations in various modified theories of gravity. It also yields the symmetries and the conserved quantities of the underlying theory. Here we apply this scheme to f(T) and f(T) gravity theories and discuss the results.

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Vacuum energy remains the simplest model of dark energy which could drive the accelerated expansion of the Universe without necessarily introducing any new degrees of freedom. In general relativity, inhomogeneous vacuum energy is necessarily interacting, exchanging energy and/or momentum with other degrees of freedom. Although the four-velocity of vacuum energy is undefined, the vacuum energy transfer defines a particular foliation of spacetime with spatially homogeneous vacuum energy in cosmological solutions. It is possible to give a consistent description of vacuum dynamics and in particular the relativistic equations of motion for inhomogeneous perturbations given a covariant prescription for the vacuum energy, or equivalently the energy transfer four-vector, and we can construct gauge-invariant vacuum perturbations. I will review evidence from current CMB and LSS observations for a late-time interaction between vacuum and cold dark matter.