Our understanding of cosmology, supported by different forms of observation have led to a standard model of cosmology called the$\Lambda$CDM model. This model combines the idea of isotropy and homogeneity, with the presence of a cosmological constant and cold dark matter along with usual components. Additionally, it requires a special set of initial conditions, such as those that can be generated by inflation. The $\Lambda$CDMmodel is consistent with all current data and most model parameters are known to high accuracy by combining information from CMB experiments, Galaxy surveys, supernova surveys and measurements of the Hubble Constant. However, current data cannot discriminate between a cosmologicalconstant and other dark energy or modified gravity models that also produce a late time acceleration. Similarly, it cannot distinguish between different models that could seed the primordial perturbations. Thus, the goal of future experiments is to distinguish between such models and thereby study the differences in the underlying physics. In this thesis, we discuss the observables for dark energy models witha time dependent equation of state, and methods of constraining cosmological parameters from current data as well forecasting constraints from future experiments.We apply these methods to forecast constraints on the dark energy with a time dependent equation of state for the LSST supernova survey, andto a combination of a CMB and supernovae survey similar to PLANCK and the Dark Energy Survey. We use examine parameter constraints in a cosmology with a time dependent equation of state by combining WMAP5, SDSS, SNe, HST data sets by comparing the power spectra. Wecarefully quantify the differences of these constraints to those obtained by using geometrical summaries for the same data sets. We find that(a) using summary parameters instead of the full data sets give parameterconstraints that are similar, but with discernible differences, (b) dueto degeneracies, the constraints on the standard parameters broadensignificantly for the same data sets. In particular, we find that in thecontext of CPL dark energy, (i) a Harrison-Zeldovich spectrumcannot be ruled out at $2\sigma$ levelswith our current data sets. and(ii) the SNe Ia, HST, and WMAP 5 data are not sufficient to constrainspatial curvature; we additionally require the SDSS DR4 data to achieve this.We then use large scale structure data that will be available in the future in a non-standard way to forecast theconstraints on the dark energy equation of state. We argue that the shapes of cosmic voids, as measured in spectroscopic galaxy redshift surveys, constitute a promising new probe of dark energy (DE). We do this by forecasting constraints on the DE equation of state and its variation from current and future surveys and find that the promise of void shape measurements compares favorably to that of standard methods such as supernovae and cluster counts even for currently available data. Owing to the complementary nature of the constraints, void shape measurements improve the Dark Energy Task Force Figure of Merit by two orders of magnitude for a future large scale experiment such as EUCLID when combined with other probes of dark energy available on a similar time scale. Modeling several observational and theoretical systematics has only moderate effects on these forecasts. We discuss additional systematics which will require further study using simulations.Finally, we study how the experiments in the future will be able to constrain a different kind of deviation from the expectations of the standard model: a position dependent rotation of the plane of the CMB polarization after recombination.Following Kamionkowski (2008), a quadratic estimator of the rotation of the plane of polarization of the CMB is constructed. This statisticcan estimate a spatially varying rotation angle $\alpha(n)$. We use this estimator to quantify the prospects of detecting such a rotation field with forthcomingexperiments. ForPLANCK and CMBPol we find that the estimator containing the product of the $E$ and $B$ components of the polarization fieldis the most sensitive. The variance of this EB estimator, $N(L)$ is roughly independent of the multipole $L$, and is only weakly dependent on the instrumental beam. For FWHM of the beam size $\Theta_{fwhm}\sim 5'-50'$, and instrument noise $\Delta_p \sim 5-50 \mu K$-arcmin, the scaling of variance $N(L)$ can be fitted by a power law $N(L)=3.3\times 10^{-7} \Delta^2_p \Theta^{1.3}_{fwhm}$ deg$^2$. For small instrumental noise $\Delta_p \leq 5 \mu K$-arcmin, the lensing B-modes become important, saturating the variance to $\sim10^{-6}$deg$^2$ even for an ideal experiment.Upcoming experiments like PLANCK will be able to detect a power spectrum of therotation angle, $C^{\alpha \alpha}(L)$, as small as $0.01$ deg$^2$, while futuristic experiment like CMBPol will be able to detect rotation angle power spectrum as small as $2.5 \times 10^{-5}$ deg$^2$.We discuss the implications of such constraints, both for the various physical effects that can rotate the polarization as photons travel from the last scattering surface as well as for constraints on instrumental systematics that can also lead to a spurious rotation signal. Rotation of the CMB polarization generates B-modes which will act as contamination for the primordial B-modes detection. We discuss an application of our estimator to de-rotate the CMB to increase the sensitivity for the primordial B-modes.
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