We develop a parameterized post-Friedmann (PPF) framework which describes three regimes of modified gravity models that accelerate the expansion without dark energy. On large scales, the evolution of scalar metric and density perturbations must be compatible with the expansion history defined by distance measures. On intermediate scales in the linear regime, they form a scalar-tensor theory with a modified Poisson equation. On small scales in dark matter halos such as our own galaxy, modifications must be suppressed in order to satisfy stringent local tests of general relativity. We describe these regimes with three free functions and two parameters: the relationship between the two metric fluctuations, the large and intermediate scale relationships to density fluctuations and the two scales of the transitions between the regimes. We also clarify the formal equivalence of modified gravity and generalized dark energy. The PPF description of linear fluctuation in f (R) modified action and the Dvali-Gabadadze-Porrati braneworld models show excellent agreement with explicit calculations. Lacking cosmological simulations of these models, our non-linear halo-model description remains an ansatz but one that enables well-motivated consistency tests of general relativity. The required suppression of modifications within dark matter halos suggests that the linear and weakly non-linear regimes are better suited for making complementary test of general relativity than the deeply non-linear regime.

We study a class of metric-variation f (R) models that accelerates the expansion without a cosmological constant and satisfies both cosmological and solar-system tests in the small-field limit of the parameter space. Solar-system tests alone place only weak bounds on these models, since the additional scalar degree of freedom is locked to the high-curvature general-relativistic prediction across more than 25 orders of magnitude in density, out through the solar corona. This agreement requires that the galactic halo be of sufficient extent to maintain the galaxy at high curvature in the presence of the low-curvature cosmological background. If the galactic halo and local environment in f (R) models do not have substantially deeper potentials than expected in ΛCDM, then cosmological field amplitudes |fR| 10 −6 will cause the galactic interior to evolve to low curvature during the acceleration epoch. Viability of large-deviation models therefore rests on the structure and evolution of the galactic halo, requiring cosmological simulations of f (R) models, and not directly on solar-system tests. Even small deviations that conservatively satisfy both galactic and solar-system constraints can still be tested by future, percent-level measurements of the linear power spectrum, while they remain undetectable to cosmological-distance measures. Although we illustrate these effects in a specific class of models, the requirements on f (R) are phrased in a nearly model-independent manner.

Euclid is a European Space Agency medium-class mission selected for launch in 2020 within the cosmic vision 2015–2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission.

We present a turnkey solution, ready for implementation in numerical codes, for the study of linear structure formation in general scalar-tensor models involving a single universally coupled scalar field. We show that the totality of cosmological information on the gravitational sector can be compressed -without any redundancy -into five independent and arbitrary functions of time only and one constant. These describe physical properties of the universe: the observable background expansion history, fractional matter density today, and four functions of time describing the properties of the dark energy. We show that two of those dark-energy property functions control the existence of anisotropic stress, the other two -dark-energy clustering, both of which are can be scale-dependent. All these properties can in principle be measured, but no information on the underlying theory of acceleration beyond this can be obtained. We present a translation between popular models of latetime acceleration (e.g. perfect fluids, f (R), kinetic gravity braiding, galileons), as well as the effective field theory framework, and our formulation. In this way, implementing this formulation numerically would give a single tool which could consistently test the majority of models of late-time acceleration heretofore proposed.

The detection of an electromagnetic counterpart (GRB 170817A) to the gravitational-wave signal (GW170817) from the merger of two neutron stars opens a completely new arena for testing theories of gravity. We show that this measurement allows us to place stringent constraints on general scalar-tensor and vector-tensor theories, while allowing us to place an independent bound on the graviton mass in bimetric theories of gravity. These constraints severely reduce the viable range of cosmological models that have been proposed as alternatives to general relativistic cosmology.

We study the evolution of linear cosmological perturbations in f (R) models of accelerated expansion in the physical frame where the gravitational dynamics are fourth order and the matter is minimally coupled. These models predict a rich and testable set of linear phenomena. For each expansion history, fixed empirically by cosmological distance measures, there exists two branches of f (R) solutions that are parameterized by B ∝ d 2 f /dR 2 . For B < 0, which include most of the models previously considered, there is a short-timescale instability at high curvature that spoils agreement with high redshift cosmological observables. For the stable B > 0 branch, f (R) models can reduce the large-angle CMB anisotropy, alter the shape of the linear matter power spectrum, and qualitatively change the correlations between the CMB and galaxy surveys. All of these phenomena are accessible with current and future data and provide stringent tests of general relativity on cosmological scales.PACS numbers:

We introduce a large class of scalar-tensor models with interactions containing the second derivatives of the scalar field but not leading to additional degrees of freedom. These models exhibit peculiar features, such as an essential mixing of scalar and tensor kinetic terms, which we have named kinetic braiding. This braiding causes the scalar stress tensor to deviate from the perfect-fluid form. Cosmology in these models possesses a rich phenomenology, even in the limit where the scalar is an exact Goldstone boson. Generically, there are attractor solutions where the scalar monitors the behaviour of external matter. Because of the kinetic braiding, the position of the attractor depends both on the form of the Lagrangian and on the external energy density. The late-time asymptotic of these cosmologies is a de Sitter state. The scalar can exhibit phantom behaviour and is able to cross the phantom divide with neither ghosts nor gradient instabilities. These features provide a new class of models for Dark Energy. As an example, we study in detail a simple one-parameter model. The possible observational signatures of this model include a sizeable Early Dark Energy and a specific equation of state evolving into the final de-Sitter state from a healthy phantom regime.1 Similar models were introduced before in another context [17,18]. Also, recently, such models have been extended to vectors and p-forms [19].2 Generic choices of K, G give equations of motion with explicit dependence on the first derivative ∇µφ.Hence, the Lagrangians (1.1) do not have the Galilean symmetry which characterizes the interactions of [7] and [8].We will therefore study the equation of state of the fluid representing the excess over the attractor, with energy density J = − * . Assuming that one can invert J(X, ρ ext ) to find X(J, ρ ext ), we can simply expand the total energy density (J, ρ ext ) =φ(J, ρ ext )J − K(X(J, ρ ext )) for small J to find, at lowest order in Notice that the factor in front of J is a function of ρ ext , implying that J does not dilute like dust as could have been naively expected and as is seen in the case of ghost condensates.

We introduce a novel class of field theories where energy always flows along timelike geodesics, mimicking in that respect dust, yet which possess non-zero pressure. This theory comprises two scalar fields, one of which is a Lagrange multiplier enforcing a constraint between the other's field value and derivative. We show that this system possesses no wave-like modes but retains a single dynamical degree of freedom. Thus, the sound speed is always identically zero on all backgrounds. In particular, cosmological perturbations reproduce the standard behaviour for hydrodynamics in the limit of vanishing sound speed. Using all these properties we propose a model unifying Dark Matter and Dark Energy in a single degree of freedom. In a certain limit this model exactly reproduces the evolution history of ΛCDM, while deviations away from the standard expansion history produce a potentially measurable difference in the evolution of structure. I. DUSTY FLUID WITH PRESSURE?How can one obtain dust from a scalar field? One can imagine a canonical scalar-field where the kinetic term is constrained to be equal to the potential. We can implement this property by introducing a Lagrange multiplier, λ, in the Lagrangian,We then find that the pressure is identically vanishing on all solutions and energy follows geodesics. This model describes the usual dust without vorticity. How can we obtain "dust with pressure"? We can generalise the above by adding some function of the scalar field and its derivatives to the Lagrangian,The constraint remains in effect and standard scalar-field dynamics are not restored. In fact, we will show that fluid elements in all such theories also always flow along geodesics, mimicking in that respect standard dust, yet the fluid has non-vanishing pressure. With this simple idea we have separated the notion that the pressure of the fluid is tied to the motion of a fluid element as is the situation in the usual case, e.g. radiation or cold dark matter. A parcel of such fluid will flow along geodesics, yet a manometer will record a pressure changing with time.In this paper, we introduce this new class of scalar-field models, which we will call λϕ-fluids. These theories are described by an action containing two scalar fields, ϕ and * Electronic address: eugene.a.lim@gmail.com † Electronic address: ignacy.sawicki@nyu.edu ‡ Electronic address: alexander.vikman@nyu.edu λ, where the latter plays the role of a Lagrange multiplier and enforces a constraint relating the value of the scalar field ϕ to the norm of its derivative. This constraint forces the dynamics of the λϕ-fluid to be driven by a system of two first-order ordinary differential equations, one for the field ϕ, the other for the Lagrange multiplier. As a consequence, there are no propagating wave-like degrees of freedom and the sound speed for perturbations is exactly zero irrespective of the background solution. However, the initial-value problem still requires the specification of two functions on the initial time slice. Thus, effectively, a single d...