The lightcone gauge is a set of what are called the observational coordinates adapted to our past lightcone. We develop this gauge by producing a perturbed spacetime metric that describes the geometry of our past lightcone where observations are usually obtained. We then connect the produced observational metric to the perturbed Friedmann-Lemaître-Robertson-Walker metric in the standard general gauge or what is the so-called 1+3 gauge. We derive the relations between these perturbations of spacetime in the observational coordinates and those perturbations in the standard metric approach, as well as the dynamical equations for the perturbations in observational coordinates. We also calculate the observables in the lightcone gauge and rederive them in terms of Bardeen potentials to first order. A verification is made of the observables in the perturbed lightcone gauge with those in the standard gauge. The advantage of the method developed is that the observable relations are simpler than in the standard formalism. We use the perturbed lightcone gauge in galaxy surveys and galaxy number density contrast. The significance of the new gauge is that by considering the null-like light propagations, the calculations are much simpler since angular deviations are not considered. Standard cosmology based on General Relativity is generally believed to have serious shortcomings, such as the unexplained issues of dark matter and dark energy. As a remedy, many alternative theories of gravitation have been proposed over the years, one of which is ƒ(R) gravity. We explore classes of irrotational-fluid cosmological models in the context of ƒ(R) gravity in an attempt to put some theoretical and mathematical restrictions on the form of the ƒ(R) gravitational Lagrangian. In particular, we investigate the consistency of the linearised dust models for shear-free cases as well as in the limiting cases when either the gravito-magnetic or gravito-electric components of the Weyl tensor vanish. We also discuss the existence and consistency of classes of non-expanding irrotational spacetimes in ƒ(R)-gravity. Furthermore, we explore exact ƒ(R) gravity solutions that mimic Chaplygin-gas inspired ΛCDM cosmology. Starting with the original, generalized and modified Chaplygin gas equations of state, we reconstruct the forms of ƒ(R) Lagrangians. The resulting solutions are generally quadratic in the Ricci scalar, but have appropriate ΛCDM solutions in limiting cases. These solutions, given appropriate initial conditions, can be potential candidates for scalar field-driven early universe expansion (in ation) and dark energy-driven late-time cosmic acceleration
