Multi-peaked signatures of primordial gravitational waves from multi-step electroweak phase transition
António P. Morais Roman Pasechnik Thibault Vieu



 













         

   
   






















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    Multi-peak gravitational wave signaturesThe first-order electroweak phase transition in the early universe could occur in multiple steps leading to specific multi-peaked signatures 
    in the primordial gravitational wave (GW) spectrum. We argue that these signatures are generic phenomena in multi-scalar extensions of 
    the Standard Model. In a simple example of such an extension, we have studied the emergence of reoccurring and nested vacuum bubble 
    configurations and their role in the formation of multiple peaks in the GW spectrum. The conditions for potential detectability of these 
    features by the forthcoming generation of interferometers have been studied.
    
    European Physical Society Conference on High Energy Physics - EPS-HEP2019 -
    
    			10-17 July, 2019
    
    			Ghent, Belgium
    Introduction. Despite of the great success of measurements at the Large Hadron Collider (LHC), the persistent absence of new physics evidence 
    is driving an increasing discomfort among the particle physics community. 
    The current void of new phenomena either indicates that new physics can only be manifest at a larger energy scale than previously thought, or results from a lack of 
    sensitivity of the current experiments measuring rare events. In fact, the weaker the interaction strength between the SM and new physics, the greater the challenge 
    to probe it.
    
    On the other hand, the recent discovery of a binary neutron star merger, firstly observed by the gravitational waves (GW) interferometers of the LIGO-Virgo collaboration 
    <cit.>, a new era of multi-messenger astronomy has begun. Furthermore, the reach of GW observatories 
    is by no means exhausted and larger sensitivities are designed for future space-based interferometers such as those of the LISA <cit.>, 
    DECIGO <cit.> and BBO <cit.> collaborations. This opens up the door for a plethora of new studies including connections 
    with both cosmology and particle physics (see e.g. Refs. <cit.> and references therein). In particular, the potential observation 
    of a stochastic GW background produced by violent processes in the early universe, e.g. by expanding vacuum bubbles associated with strong cosmological 
    phase transitions <cit.>, may well become a gravitational probe for beyond-the-SM (BSM) physics and a complement for collider measurements.
    When extending the SM scalar sector, the underlying vacuum structure exhibits a growing complexity such that a possibility for transition patterns with several successive 
    first-order steps arises <cit.>. In this contribution, we discuss the key 
    implications of such successive transitions for GW signals.
    
    Multi-step electroweak phase transition. In cosmology, thermal evolution of the EW-breaking vacuum as the universe cools down is determined 
    by the temperature-dependent part of the one-loop effective potential <cit.>. Given the field content and its quantum numbers of an underlying multi-scalar fundamental 
    theory, the shape of the potential can be determined at any temperature T. In a configuration of two minima of the effective potential coexisting at the critical temperature T_c, 
    by using CosmoTransitions <cit.> one computes numerically the Euclidean action Ŝ_3 describing transitions 
    between the corresponding phases. The temperature T_n, at which the nucleation of vacuum bubbles effectively occurs, is estimated by setting the probability to nucleate 
    one bubble per horizon volume to unity, which translates into Ŝ_3 / T_n ∼ 140<cit.>. The sphaleron suppression criterion 
    v_c/T_c ≳ 1, with v_c=v(T_c) the Higgs vacuum expectation value (VEV), defines a strong first-order phase transition. This typically implies 
    that the transition produces strong GW signals detectable by the next generation of interferometers and may be used for constraining BSM scenarios 
    <cit.>. Here we study successive strong first-order EWPTs which we refer to as multi-step transitions. 
    As a result one has more than a single transition pattern for a particular point in the parameter space, which results in sequential nucleation of bubbles of different vacua. 
    
    A generic BSM scenario typically contains a large number of scalar degrees of freedom which can be advantageous e.g. for EW baryogenesis. Even reducing the scalar 
    sector to a few fields, new unexplored possibilities of transition patterns arise, in particular, transitions in several successive first-order steps. Therefore, a non-trivial EWPT is expected 
    and multi-step transitions may have occurred in the early universe <cit.>. The basic characteristics of such multi-peak spectra may also be 
    affected by dynamics of other sectors, in particular, by the neutrino sector suggesting the use of primordial GW data for probing the neutrino mass generation
    mechanisms <cit.>.
    
    In order to illustrate the generic features of multi-step first-order EW transitions, consider for instance a minimal extension of the SM scalar sector inspired by 
    the high-scale Grand-unified trinification theory <cit.>. Besides the SM Higgs field 
    ℋ_1, it contains an additional EW doublet ℋ_2 and singlet φ fields which are charged under a  family symmetry. 
    The resulting potential possesses an approximate descrete ℤ_2 symmetry acting as ℋ_j → - ℋ_j (j=1,2) and 
    φ→ - φ which significantly simplifies the vacuum structure of the model. An expansion of the scalar fields in terms of real components
    
        ℋ_j    = 1/√(2)[      χ_j + i χ_j'; ϕ_j + h_j + i η_j ] ,
          φ = 1√(2)ϕ_s + S_R + i S_I ,
    
    defines the quantum fluctuations h_1, h_2 and S_R about the classical configurations ϕ_α={ϕ_1,ϕ_2,ϕ_s}, respectively. With this expansion, 
    the classical potential reads
    
        V_ cl(ϕ_α) =   12 m_α^2 |ϕ_α|^2
        + 18λ_α|ϕ_α|^4 
        + 14λ_αβ|ϕ_α|^2 |ϕ_β|^2 .
    
    A comprehensive analysis of the tree-level vacuum structure was performed recently in Ref. <cit.>. It was shown that the basic characteristics of 
    EWPTs in this model, in particular, sequential first-order transitions, are generic for multi-Higgs extensions of the SM. Thus, this model, due to 
    the simplicity of its potential (<ref>), could serve as a good benchmark scenario for further in-depth explorations of cosmological implications of 
    multi-scalar BSM theories.
    
    
    Coexisting, nested and reoccurring bubbles. For simplicity, let us now consider a representative configuration of the parameter space <cit.> 
    where the only existing phases given in terms of the VEVs of the scalar fields v_α≡⟨ϕ_α⟩_ vac={v_1,v_2,v_s} are (0,0,0), (v_1,0,0), (0,v_2,0) 
    and (0,0,v_s), which we recast as [0], H_1, H_2 and Φ, respectively. The possible first-order transitions were found to be H_1 ↔ H_2, H_1 ↔Φ, 
    H_2 ↔Φ, which take place readily in the leading (m/T)^2 order of the thermal expansion and are expected to be strong. Here, without loss 
    of generality we identify the stable phase at T=0 with H_1 which is simple but represents the basic features of a generic 
    EW-breaking vacuum {v_1,v_2,0}.
    
    In a particular sequence of transitions to the true vacuum H_1 with the following two patterns
    
        Φ→ H_1  ,     Φ→ H_2 → H_1 ,
    
    one could expect several nucleation processes occurring in the same range of temperatures, e.g. Φ→ H_1 and Φ→ H_2. In this case, different sequences 
    could be realized during the same cosmological evolution time leading to a universe where  coexisting bubbles of different broken phases expand simultaneously 
    (left panel in Fig. <ref>). In addition to the coexisting bubbles, more exotic cosmological objects may emerge from multi-step phase transitions. In particular, 
    consider the second and third steps in the pattern [0] →Φ→ H_2 → H_1, occurring at typical nucleation temperatures T_n(Φ→ H_2) ≳ T_n(H_2 → H_1). 
    Between T_n(Φ→ H_2) and T_n(H_2 → H_1), the H_2-bubbles nucleate and expand in a universe filled with the Φ-phase. Then at T_n(H_2 → H_1), while 
    they are still expanding, the H_1-bubbles emerge and nucleate inside the H_2-bubbles. As such, the Φ-phase becomes populated with the H_2-bubbles 
    containing the H_1-bubbles inside. We denote such objects as  nested bubbles shown in Fig. <ref> (middle panel).
    
    
    
    Since the scalar potential keeps evolving as the universe cools down below T_n(H_2 → H_1), the initial phase Φ becomes unstable also along the H_1 direction, 
    so both transitions towards H_1 and H_2 can occur (coexisting bubbles scenario). In particular, if the potential barrier between the phases Φ and H_1 disappears, 
    the new H_1-bubbles would nucleate in the parts of the universe that still remain in the Φ-phase i.e. the direct Φ→ H_1 transition quickly eliminates 
    the Φ-phase outside of the H_2-bubbles formed at an earlier time. Such a mixed situation with the coexistence of the ordinary H_1 and nested H_2 → H_1 
    bubbles is depicted in Fig. <ref> (middle panel). In the end of this process, one ends up with the H_1-bubbles inside the H_2-bubbles which exist in a universe 
    filled with the H_1-phase. We denote these exotic cosmological objects as  reoccurring bubbles. Since the H_2-bubbles cannot expand in a universe filled with 
    the stable H_1-phase, they are pushed inwards and collapse while the H_1-bubbles nucleate inside them as illustrated in Fig. <ref> (right panel).
    
    In Fig. <ref> we show three realistic cosmological scenarios where the objects discussed above are expected to occur. In the left column, 
    we plot the evolution of the action Ŝ_3/T as a function of temperature for all possible transitions. Whenever a curve corresponding to a transition i → j 
    crosses the horizontal line Ŝ_3/T = 140, a bubble of phase j is nucleated inside the phase i, which is denoted as i(j) in what follows. In the right column, 
    we show a diagrammatic representation displaying all types of bubbles (co)existing at a given temperature and corresponding to the plots in the left 
    column. For example, the top panel describes a first-order phase transition Φ→ H_1 when H_1-bubbles nucleate in a universe filled with 
    the Φ-phase (i.e. Φ(H_1)). In particular, a universe in the symmetric phase [0] first collapses to the Φ-phase through a second-order 
    phase transition without generating any bubbles. Then, the H_1-bubbles are nucleated at T∼ 105 GeV and expand until the Φ-phase becomes 
    unstable at around T∼ 60 GeV leaving a universe entirely filled by the true vacuum H_1.
    
    
    
    While the graphs on the second line of Fig. <ref> represent a scenario where three successive steps lead to the nucleation of a nested 
    bubbles Φ(H_2(H_1)), those on the last line describe reoccurring bubbles that emerges from a nested ones. For instance, after nucleation of nested 
    bubbles Φ(H_2(H_1)), the potential barrier between the phases Φ and H_1 disappears (around T∼ 60 GeV), such that the parts of 
    the universe in the Φ-phase collapse to the H_1-phase transforming the nested bubbles Φ(H_2(H_1)) into the reoccurring ones H_1(H_2(H_1)). 
    This type of cosmological objects is only possible if the nucleation temperatures of the corresponding steps are not too different (percolation typically occurs 
    in the range of Δ T < 10 GeV), and further likely to occur when symmetries in the potential enforces them to be identical as in e.g. Ref. <cit.>. 
    
    Multi-peaked gravitational-wave spectrum. We analyse the multi-peaked signatures in the power spectrum of GWs using the well known 
    formalism of Ref. <cit.> 
    which describes the energy density per logarithmic frequency of the GW radiation, h^2 Ω_ GW. The net GW signal is typically considered to be produced by three different sources due to bubble wall collisions <cit.>, sound waves (SW) generated by the phase transitions <cit.>, 
    as well as magnetohydrodynamics (MHD) turbulences in the plasma <cit.>. The bubble wall collisions typically do not contribute to the GWs production 
    processes in the considering class of multi-scalar extensions of the SM <cit.> (see also Ref. <cit.>) so we account
    for the dominant SW and MHD contributions only.
    
    
    
    The results of the inclusive parameter scan searching for FOPTs in the considering 2HDSM scenario are illustrated in Fig. <ref> (left panel).
    Here, each point corresponds to a particular FOPT found in the scan, with calculated values of the induced GW peak amplitude and frequency.
    Scenarios corresponding to double-peak GW spectra are highlighted by red color. Here, dashed grey lines indicate sensitivities 
    of the LISA <cit.> and LIGO <cit.> interferometers, 
    as well as proposed DECIGO <cit.>, BBO <cit.> 
    and SKA <cit.> missions (see also Ref. <cit.>). The “ultimate-DECIGO”, “ultimate-DECIGO-corr” 
    and “DECIGO-corr” sensitivity curves are taken from Ref. <cit.>, while the sensitivities for other measurements
    can be found in Ref. <cit.>. We have selected three example scenarios whose largest peaks fall into the LISA sensitivity
    domain and whose GW spectra are shown in Fig. <ref> (right panel). The basic characteristics of the corresponding transitions
    and GW signals for each green, red and blue line are summarised in Table <ref>. Such scenarios can be further 
    considered as benchmarks for further explorations at GW interferometers. For a more detailed description of these and other benchmark
    scenarios, see Ref. <cit.>.
    
    We would like to point out that a complete knowledge of the bubble dynamics is needed in order to precisely describe the phase transitions, 
    from nucleation to percolation. A few immediate questions result from our analysis: do we expect new sources of GWs when nested 
    bubbles collide or when a nested bubble expands faster than its “mother bubble” reaching the wall of the latter? Given that in general such 
    objects have no reason to be spherically symmetric, what is their impact on the profile of the GW spectrum? What is the impact of nested 
    vacuum bubbles for baryogenesis? These are important questions for a further deeper analysis.
    
    
    
    A.P.M. is supported by Fundação 
    para a Ciência e a Tecnologia (FCT), within project UID/MAT/04106/2019 (CIDMA) and by national funds (OE), 
    through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, 
    of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. A.P.M. is also supported by 
    the Enabling Green E-science for the Square Kilometer Array Research Infrastructure (ENGAGESKA), 
    POCI-01-0145-FEDER-022217, and by the project From Higgs Phenomenology to the Unification 
    of Fundamental Interactions, PTDC/FIS-PAR/31000/2017. R.P. is supported in part by the Swedish 
    Research Council grants, contract numbers 621-2013-4287 and 2016-05996, by the Ministry of Education, 
    Youth and  Sports of the Czech Republic, project LT17018, as well as by the European Research Council (ERC) 
    under the European Union's Horizon 2020 research and innovation programme (grant agreement No 668679). 
    T. V. received a financial support Erasmus+ for international mobility from Université Paris-Sud.
    
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