The paradigm of inflation –  a period of accelerated expansion in the very early Universe –  was introduced to give solutions to a number of problems encountered in Standard Big Bang cosmology. Additionally, due to its quantum nature, inflation is able to generate the necessary primordial inhomogeneity “seeds”, which eventually evolve into large-scale structures. The particular primordial inhomogeneities are imprinted on the Cosmic Microwave Background radiation (CMB) as very small deviations –  temperature fluctuations –  from a perfect blackbody spectrum. If the Standard Model (SM) Higgs is a light spectator field during inflation, it can acquire quantum fluctuations and seed additional, potentially observable, fluctuations. This takes place via an effective breaking of electroweak symmetry at very high energy scales, which results in the reheating process being different in different regions of the Universe. In the first part of the research work, we develop methods for calculating the amplitude, as well as the non-Gaussianity, of such Higgs-induced temperature fluctuations in the CMB. In the case of reheating via resonant inflaton decays to Abelian gauge bosons, we show that the amplitude of the Higgs-induced temperature fluctuations always exceeds the observed value and that, therefore, such decays cannot be the main reheating channel. In the case of reheating via perturbative inflaton decays to SM fermions, we place strong constraints on the relevant SM parameters, using the amplitude of the Higgs temperature fluctuations. By additionally using the associated non-Gaussianity, we are able to strengthen the particular constraints even further. Having made a connection between cosmological observations and SM parameters, such as the Higgs self-coupling, we suggest a way to probe the SM Higgs potential at very high energy scales and constrain New Physics. In the second part of the research work we study a specific inflationary model, known as chain inflation. In particular, we calculate the primordial gravitational wave (GW) signatures produced by chain inflation. We show that the latter can explain the GW stochastic background detected by the International Pulsar Timing Array (IPTA). Finally, we show that GW signatures of chain inflation are detectable both by current and/or by future GW instruments.