Infinitesimal geometry of spatial mappings

1  OBSERVATIONAL EVIDENCE FOR DARK ENERGY
	1.2 Dynamics of expansion of the homogeneous isotropic multicomponent Universe
	1.3 The luminosity distance — redshift relation and SNe Ia evidence
	1.4 The angular diameter distance — redshift relation and acoustic peak tests
		1.4.1 CMB acoustic peaks
		1.4.2 Baryon acoustic oscillations
		1.4.3 X-ray gas fraction in clusters
	1.5 Evidence for dark energy from study of large scale structure
		1.5.1 Linear power spectrum of matterdensity perturbations
	1.6 Angular power spectrum of CMB temperature fluctuations
		1.6.1 Integrated Sachs—Wolfe effect
		1.6.2 Weak gravitational lensing of CMB
	1.7 Age of the Universe
	1.8 Constraints on dark energy parameters from combined data
	1.9 Summary
2  SCALAR FIELD MODELS OF DARK ENERGY
	2.2 Cosmological constant as vacuum energy
	2.3 Scalar fields as dark energy
	2.4 Scalar perturbations of the scalar field
	2.5 Specifying the scalar-field models of dark energy
		2.5.1 Lagrangian
		2.5.2 Potential
		2.5.3 EoS parameter
		2.5.4 The effective sound speed
	2.6 Quintessential scalar fields with barotropic EoS
		2.6.1 Classical scalar field
		2.6.2 Tachyonic scalar field
		2.6.3 Quintessential scalar fields in the phase plane
		2.6.4 Best-fit parameters of QSF
	2.7 Phantom scalar fields with barotropic EoS
		2.7.1 Gravitation instability of PSF and large scale structure formation
		2.7.2 Best-fit parameters of PSF
	2.8 Distinguishing of scalar field models of dark energy
	2.9 Summary
3 KALUZA—KLEIN MODELS
	3.2 Dimensional reduction
		3.2.1 General setup
		3.2.2 Stable compactification with minimal scalar fields
		3.2.3 Perfect fluid: no-go theorem
		3.2.4 Conventional cosmology from multidimensional models
	3.3 Abelian gauge fields in KK models, dimensional reduction
	3.4 Gravitational excitons
		3.4.1 Effective equation of motion for gravexcitons
		3.4.2 Light and ultra-light gravexcitons: m 10-2GeV
		3.4.3 Heavy gravexcitons: m 10-2GeV
		3.4.4 Variation of the fine-structure constant
		3.4.5 Lorentz invariance violation
	3.5 Dark energy in multidimensional models
		3.5.1 Internal space stabilization for pure geometrical f(R) models
		3.5.2 Dark energy in f(R) models with form fields
	3.6 Sp-branes
		3.6.1 Dark energy in pure geometrical Sp-brane model with hyperbolic internal space
	3.7 Problematic aspects of Kaluza—Klein models
		3.7.1 Equations of state in general case
		3.7.2 Latent solitons
		3.7.3 Experimental restrictions on the equations of state of a multidimensional perfect fluid
	3.8 Summary
4  BRANEWORLD MODELS
	4.2 General setup and notation
	4.3 Cosmological solutions
	4.4 Vacuum and static branes
	4.5 Properties of braneworld gravity
	4.6 Phantom property of braneworld dark energy
	4.7 Disappearing dark energy
	4.8 Cosmic mimicry
	4.9 Loitering
		4.9.1 Loitering Universe
		4.9.2 Loitering in braneworld models
		4.9.3 The parameter space in loitering models
		4.9.4 Inflation in braneworld models with loitering
	4.10 Quiescent singularities
		4.10.1 Homogeneous case
		4.10.2 Inhomogeneous case
	4.11 Asymmetric branes
		4.11.1 Induced cosmological constant on the brane
		4.11.2 Cosmic mimicry
		4.11.3 Phantom branes
		4.11.4 Disappearing dark energy
		4.11.5 Quiescent singularities
		4.11.6 Stability issues
	4.12 Gravitational instability on the brane
		4.12.1 Scalar cosmological perturbations on the brane
		4.12.2 Simplified boundary conditions for scalar perturbations
		4.12.3 Scalar perturbations in the DGP model
		4.12.4 Scalar perturbations in the mimicry model
	4.13 Perturbations of the bulk
		4.13.1 General system of equations
		4.13.2 Perturbations on the flat background bulk geometry
		4.13.3 Quasi-static approximation
	4.14 Summary
5 ENERGY IN GENERAL RELATIVITY IN VIEW OF SPINOR AND TENSOR METHODS
	5.2 Connection between spinor and tensor methods
		5.2.1 Sen—Witten equation in Petrov type N space-time
		5.2.2 Nodal surfaces of selfadjoint elliptic second order equations
		5.2.3 The solutions of Sen—Witten equation have no zeros
		5.2.4 Sen—Witten equations and SOF
		5.2.5 Conclusion
	5.3 Nodal points of elliptic equations
		5.3.1 Conditions for the absence of nodal points
		5.3.2 The conditions of nodal points absence for the solutions of Sen—Witten equation
		5.3.3 Towards Sen—Witten equation, special orthonormal frame and preferred time variables
	5.4 Sen—Witten orthonormal three-frame
		5.4.1 Direct link between the 4-covariant spinor 3-form and the Einstein Hamiltonian
		5.4.2 In which cases the conditions of Theorem 4 are fulfilled?
	5.5 Summary