PHYSICAL REVIEW RESEARCH 1, 013010 (2019) Imprint of a scalar era on the primordial spectrum of gravitational waves Francesco D’Eramo and Kai Schmitz Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Via Marzolo 8, 35131 Padova, Italy and Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy (Received 24 April 2019; published 19 August 2019) Upcoming searches for the stochastic background of inflationary gravitational waves (GWs) offer the exciting possibility to probe the evolution of our Universe prior to big bang nucleosynthesis. In this spirit, we explore the sensitivity of future GW observations to a broad class of beyond-the-standard-model scenarios that lead to a nonstandard expansion history. We consider a new scalar field whose coherent oscillations dominate the energy density of the Universe at very early times, resulting in a scalar era prior to the standard radiation-dominated era. The imprint of this scalar era on the primordial GW spectrum provides a means to probe well-motivated yet elusive models of particle physics. Our work highlights the complementarity of future GW observatories across the entire range of accessible frequencies. DOI: 10.1103/PhysRevResearch.1.013010 I. INTRODUCTION frequencies are redshifted to the present day. This property turns primordial GWs into a powerful tool that grants access to The detection of gravitational waves (GWs) [1]byLIGO the thermal history of our Universe prior to BBN. Primordial [2,3] and Virgo [4] has rung in the age of GW astronomy. GWs offer, e.g., an opportunity to measure the reheating While all signals observed thus far have been of astrophysical temperature after inflation [27–33]. Similarly, they may be origin, i.e., the coalescence of compact binaries [5], it is used to infer the equation of state (EOS) during the quark- expected that GWs will also open a window on cosmology hadron phase transition in quantum chromodynamics [34,35] in the foreseeable future [6]. Within the next decades, a or constrain properties of hidden sectors beyond the standard multitude of GW observatories will go online, ranging from model (BSM) [36,37]. The same may also be true for GWs third-generation ground-based interferometers over satellite- emitted by cosmic strings [38,39], whose detailed proper- borne interferometers in space to pulsar timing arrays moni- ties are currently under discussion (see [40] and references tored by new radio telescopes (see [7–11] for an outlook on therein). GW astronomy in the 2020’s). These experiments promise The main goal of this article is to demonstrate how future to be sensitive to a wealth of hypothetical cosmological GW observations of the primordial SGWB from inflation across a signals, such as the stochastic GW background (SGWB) from large range of frequencies will be instrumental in constraining inflation, GWs from phase transitions in the early Universe, BSM physics. To this end, we shall consider a broad class of or GWs emitted by topological defects (see [12,13] for recent new physics models in which the presence of a scalar field φ review articles), and even to anisotropies in the GW signal in the early Universe leads to a modified expansion history. [14]. The direct observation of any of these cosmological Such a situation can occur in a variety of BSM scenarios. signals would allow one to chart unexplored territory that Possible BSM scalar fields that φ may be identified as include, cannot be accessed by conventional cosmological probes such but are not limited to, the Polonyi field in models of gravity- as the cosmic microwave background (CMB) or the primor- mediated supersymmetry breaking [41,42], a modulus field in dial abundances of light elements generated during big bang four-dimensional compactifications of string theory [43–46], a nucleosynthesis (BBN). flavon field in extensions of the standard model (SM) that aim In this article, we will be concerned with the primordial at explaining the flavor structure of quark and leptons (such as, SGWB from inflation [15–17] (see [18]forareview).An e.g., the Froggatt-Nielsen flavor model [47]), and the saxion intriguing feature of inflationary GWs is that they act as a field in supersymmetric axion models [48–54]. logbook of the expansion history of our Universe throughout At some point prior to BBN, the energy density stored in its evolution [19–26]. Indeed, the detailed time evolution of the coherent oscillations of the scalar field begins to dominate the Hubble rate during the expansion determines the transfer over radiation. The ensuing era of scalar-field domination function that describes how gravitational waves at different (SD), or scalar era for short, leads to a characteristic feature in the transfer function for primordial GWs, which may be detected by future GW experiments. In a detailed numerical analysis, we calculate the transfer function for primordial Published by the American Physical Society under the terms of the GWs that captures the modified expansion during the scalar Creative Commons Attribution 4.0 International license. Further era. This allows us to determine the final GW spectrum across distribution of this work must maintain attribution to the author(s) the entire relevant parameter space of our scalar-field model. and the published article’s title, journal citation, and DOI. For each parameter point, we compute the signal-to-noise 2643-1564/2019/1(1)/013010(8) 013010-1 Published by the American Physical Society FRANCESCO D’ERAMO AND KAI SCHMITZ PHYSICAL REVIEW RESEARCH 1, 013010 (2019) ratios (SNRs) at which the GW spectrum as well as individual In this article, we are interested in assessing the maximal features in this spectrum are expected to be observed by reach of future GW experiments. For this reason, we decide to various future GW experiments. The outcome of our analysis fix the primordial tensor amplitude at the maximally allowed −10 is a global picture of the GW signature of the scalar era value At 1.5 × 10 , that is consistent with the most recent and the experimental prospects of detecting it. Our study upper bound on the tensor-to-scalar ratio r = At /As 0.07 −9 highlights the complementarity of future GW experiments [55], where As 2.1 × 10 is the measured amplitude of across the entire range of accessible frequencies and defines the primordial scalar spectrum. Similarly, we shall assume a an important benchmark scenario for the experimental GW blue-tilted tensor spectrum, fixing its index at an optimistic community in the coming decades. value of nt = 0.4. On the one hand, this value clearly vio- lates the consistency relation of standard single-field slow- =− / < II. PRIMORDIAL GRAVITATIONAL WAVES roll inflation, nt r 8 0. This means that we implicitly assume a nonminimal inflationary sector that is capable of GWs correspond to spatial tensor perturbations hij of the generating a strong SGWB at small scales. Such scenarios space-time metric exist in the literature, with one prominent example being 2 2 2 i j natural inflation coupled to gauge fields [56–61]. On the other ds =−dt + a (t )[δij + hij(t, x)]dx dx , (1) hand, our choice for the spectral index is phenomenologically where a denotes the cosmic scale factor and the symmetric still completely viable. n = 0.4 complies with (1) the CMB = t tensor hij is evaluated in transverse-traceless gauge hii constraints derived by the PLANCK Collaboration [55], (2) ∂ = i hij 0. The free propagation of GWs is governed by the the upper bound on the amplitude of an isotropic SGWB linearized Einstein equation with a vanishing source term. In at O(10) Hz frequencies by LIGO and Virgo [62,63], and Fourier space, one thus obtains the following equations of p (3) global bounds on primordial GWs (based, e.g., on BBN) motion for the tensor modes hk: across the full frequency spectrum [64,65]. d2 2 da(u) d + + 1 hp(u) = 0. (2) du2 a(u) du du k III. SCALAR ERA Here, k denotes the comoving 3-momentum with absolute We consider the dynamics of a real scalar field φ,employ- value k =k, p =+, × distinguishes the two possible po- ing an effective description in terms of three parameters: mass larization states of a GW, and the variable u = kτ (with τ mφ, decay rate φ, and initial field value φini at the end of denoting conformal time, dt = adτ) is a measure for the inflation. We assume a harmonic potential around the origin spatial extent of a GW. For u 1(u 1), a mode is located in field space and some (direct or indirect) coupling to the far outside (inside) the Hubble horizon. standard model that allows φ to decay to radiation at the end of An important property of inflationary GWs is that they its lifetime. The dynamical evolution of the field φ is governed are stretched to superhorizon size during inflation. Outside by the Klein-Gordon equation in an expanding background the horizon, GWs remain frozen, until they re-enter the hori- 2 d d 2 zon during the decelerating expansion after inflation. This + [3 H(t ) + φ] + m φ(t ) = 0, (5) 2 φ enables one to factorize the amplitude of a GW mode at the dt dt τ p τ = ¯ p χ τ,τ present time 0, hk( 0 ) hk k (¯ 0 ), into an initial value at where we included a friction term proportional to φ.The τ ¯ p = p τ ¯ determined by inflation, hk hk(¯), and a transfer function Hubble rate H follows from the Friedmann equation χ τ,τ k (¯ 0 ) that accounts for the redshift subsequent to horizon 2 a˙(t ) ρφ (t ) + ρ (t ) reentry.