We critically examine the magnitude of theoretical uncertainties in perturbative calculations of fist-order phase transitions, using the Standard Model effective field theory as our guide. In the usual daisy-resummed approach, we find large uncertainties due to renormalisation scale dependence, which amount to two to three orders-of-magnitude uncertainty in the peak gravitational wave amplitude, relevant to experiments such as LISA. Alternatively, utilising dimensional reduction in a more sophisticated perturbative approach drastically reduces this scale dependence, pushing it to higher orders. Further, this approach resolves other thorny problems with daisy resummation: it is gauge invariant which is explicitly demonstrated for the Standard Model, and avoids an uncontrolled derivative expansion in the bubble nucleation rate.

The study of cosmic phase transitions are of central interest in modern cosmology. In the standard model of cosmology the Universe begins in a very hot state, right after at the end of inflation via the process of reheating/preheating, and cools to its present temperature as the Universe expands. Both new and existing physics at any scale can be responsible for catalyzing either first, second or cross over phase transition, which could be either thermal or non-thermal with a potential observable imprints. Thus this field prompts a rich dialogue between gravity, particle physics and cosmology. It is all but certain that at least two cosmic phase transitions have occurred -the electroweak and the QCD phase transitions. The focus of this review will be primarily on phase transitions above such scales, We review different types of phase transitions that can appear in our cosmic history, and their applications and experimental signatures in particular in the context of exciting gravitational waves, which could be potentially be constrained by LIGO/VIRGO, Kagra, and eLISA.Let us briefly summarize the early Universe cosmology in chronological order. How the Universe began remains a profound question, for which we do not have direct experimental evidence yet. Nonetheless, we can speculate based on sound physical arguments and the observations confirmed by the detection of cosmic microwave background (CMB) radiation [46,47]. How did the Universe begin?Einstein's theory of gravity (GR) is extremely successful in the infrared (IR) matching of all possible observables [48], including the recent discovery of gravitational waves from mergers of two blackholes [49], and binary neutron star mergers [50]. However at short distances and small time scales, i.e. in the ultraviolet (UV), GR has pathologies, besides being a non-renormalizable theory, GR introduces cosmological and blackhole singularities, see [51], and in some cases naked singularities, see [52]. In GR, our Universe has a distinct starting point, a singular spacetime -as long as all the standard energy conditions are always satisfied, i.e. strong, weak, and null energy conditions, see [51]. It is possible to address the cosmological singularity problem without violating the matter energy conditions by weakening the gravitational interaction in the UV. This can happen in ghost free infinite derivative gravity inspired from string field theory [53,54]. There could be two consequences for such study; one could be a realization of a non-singular bounce [55,56], and the other scenario would be that Universe could be frozen in time in the UV, such that the Universe becomes conformal as t → 0 [57]. Bouncing cosmologies and cosmological density perturbations have been reviewed in this nice review [58,59]. There is a strong indication that this non-singular initial phase of the Universe has a key role to play towards understanding the subsequent phases of the Universe such as cosmic inflation, horizon, homogeneity and isotropy of the Universe , to create appropriate initial co...

We undertake a careful analysis of stochastic gravitational wave production from cosmological phase transitions in an expanding universe, studying both a standard radiation as well as a matter dominated history. We analyze in detail the dynamics of the phase transition, including the false vacuum fraction, bubble lifetime distribution, bubble number density, mean bubble separation, etc., for an expanding universe. We also study the full set of differential equations governing the evolution of plasma and the scalar field during the phase transition and generalize results obtained in Minkowski spacetime. In particular, we generalize the sound shell model to the expanding universe and determine the velocity field power spectrum. This ultimately provides an accurate calculation of the gravitational wave spectrum seen today for the dominant source of sound waves. For the amplitude of the gravitational wave spectrum visible today, we find a suppression factor arising from the finite lifetime of the sound waves and compare with the commonly used result in the literature, which corresponds to the asymptotic value of our suppression factor. We point out that the asymptotic value is only applicable for a very long lifetime of the sound waves, which is highly unlikely due to the onset of shocks, turbulence and other damping processes. We also point out that features of the gravitational wave spectral form may hold the tantalizing possibility of distinguishing between different expansion histories using phase transitions.

We present the possibility that the seesaw mechanism with thermal leptogenesis can be tested using the stochastic gravitational background. Achieving neutrino masses consistent with atmospheric and solar neutrino data, while avoiding non-perturbative couplings, requires right-neutrinos lighter than the typical scale of grand unification. This scale separation suggests a symmetry protecting the right handed neutrinos from getting a mass. Thermal leptogenesis would then require that such a symmetry be broken below the reheating temperature. We enumerate all such possible symmetries consistent with these minimal assumptions and their corresponding defects, finding that in many cases, gravitational waves from the network of cosmic strings should be detectable. Estimating the predicted gravitational wave background we find that future space-borne missions could probe the entire range relevant for thermal leptogenesis.