Imprint of a scalar era on the primordial spectrum of gravitational waves

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.

Introduction. The detection of gravitational waves (GWs) [1] by LIGO [2, 3] and Virgo [4] has rung in the age of GW astronomy. While all signals observed thus far have been of astrophysical origin, i.e., the coalescence of compact binaries [5], it is expected that GWs will also open a window on cosmology in the foreseeable future [6]. Within the next decades, a multitude of GW observatories will go online, ranging from third-generation groundbased interferometers over satellite-borne interferometers in space to pulsar timing arrays monitored by new radio telescopes (see [7][8][9][10][11] for an outlook on GW astronomy in the 2020's). These experiments promise to be sensitive to a wealth of hypothetical cosmological GW signals -such as the stochastic GW background (SGWB) from inflation, GWs from phase transitions in the early Universe, or GWs emitted by topological defects (see [12,13] for recent review articles) -and even to anisotropies in the GW signal [14]. The direct observation of any of these cosmological signals would allow one to chart unexplored territory that cannot be accessed by conventional cosmological probes such as the cosmic microwave background (CMB) or the primordial abundances of light elements generated during Big Bang nucleosynthesis (BBN).
In this paper, we will be concerned with the primordial SGWB from inflation [15][16][17] (see [18] for a review). An intriguing feature of inflationary GWs is that they act as a logbook of the expansion history of our Universe throughout its evolution [19][20][21][22][23][24][25][26]. Indeed, the detailed time evolution of the Hubble rate during the expansion determines the transfer function that describes how gravitational waves at different frequencies are redshifted to the present day. This property turns primordial GWs into a powerful tool that grants access to the thermal history of our Universe prior to BBN. Primordial GWs offer, e.g., an opportunity to measure the reheating temperature after inflation [27][28][29][30][31][32][33]. Similarly, they may be used to infer the equation of state (EOS) during the quark-hadron phase transition in quantum chromodynamics [34,35] or constrain properties of hidden sectors beyond the Standard Model (BSM) [36,37].
The same may also be true for GWs emitted by cosmic strings [38,39], whose detailed properties are currently under discussion (see [40] and references therein).
The main goal of this paper is to demonstrate how future observations of the primordial SGWB from inflation across a large range of frequencies will be instrumental in constraining BSM physics. To this end, we shall consider a broad class of new physics models in which the presence of a scalar field φ in the early Universe leads to a modified expansion history. Such a situation can occur in a variety of BSM scenarios. Possible BSM scalar fields that φ may be identified as include, but are not limited to: the Polonyi field in models of gravity-mediated supersymmetry breaking [41,42], a modulus field in four-dimensional compactifications of string theory [43][44][45][46], a flavon field in extensions of the Standard Model (SM) that aim at explaining the flavor structure of quark and leptons (such as, e.g., the Froggatt-Nielsen flavor model [47]), and the saxion field in supersymmetric axion models [48][49][50][51][52][53][54].
At some point prior to BBN, the energy density stored in the coherent oscillations of the scalar field begins to dominate over radiation. The ensuing era of scalar-field domination (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 GWs that captures the modified expansion during the scalar era. This allows us to determine the final GW spectrum across the entire relevant parameter space of our scalar-field model. For each parameter point, we compute the signal-to-noise ratios (SNRs) at which the GW spectrum as well as individual features in this spectrum are expected to be observed by various future GW experiments. The outcome of our analysis is a global picture of the GW signature of the scalar era and the experimental prospects of detecting it. Our study highlights the complementarity of future GW experiments across the entire range of accessible frequencies and defines an important benchmark scenario for the experimental GW community in the coming decades.
Primordial gravitational waves. GWs correspond to spatial tensor perturbations h ij of the spacetime metric, where a denotes the cosmic scale factor and the symmetric tensor h ij is evaluated in transverse-traceless gauge, h ii = ∂ i h ij = 0. The free propagation of GWs is governed by the linearized Einstein equation with a vanishing source term. In Fourier space, one thus obtains the following equations of motion for the tensor modes h p k , Here, k denotes the comoving 3-momentum with absolute value k = k , p = +, × distinguishes the two possible polarization states of a GW, and the variable u = kτ (with τ denoting conformal time, dt = a dτ ) is a measure for the spatial extent of a GW. For u 1 (u 1), a mode is located far outside (inside) the Hubble horizon.
An important property of inflationary GWs is that they are stretched to super-horizon size during inflation. Outside the horizon, GWs remain frozen, until they re-enter the horizon during the decelerating expansion after inflation. This enables one to factorize the amplitude of a GW mode at the present time τ 0 , h p k (τ 0 ) =h p k χ k (τ , τ 0 ), into an initial value atτ determined by inflation,h p k = h p k (τ ), and a transfer function χ k (τ , τ 0 ) that accounts for the redshift subsequent to horizon re-entry. For any expansion history after inflation [specified by the function a (u)], the transfer function can be found by solving Eq. (2) for χ k as a function of u with boundary conditions χ k = 1 and ∂ u χ k = 0 at u = 0. The same factorization also carries over to Ω GW (f ), i.e., the GW energy density per logarithmic frequency interval d (ln f ) in units of the critical energy density of a flat universe. For modes deep inside the horizon, one obtains the following GW spectrum at the present time, where a 0 and H 0 = 100 h km/s/Mpc represent the present values of the scale factor and the Hubble rate, respectively. The primordial tensor spectrum P t depends on the initial conditions set by the inflationary period and is conventionally parametrized by a simple power law around the CMB pivot scale, k CMB = 0.05 Mpc −1 , In this paper, we are interested in assessing the maximal reach of future GW experiments. For this reason, we decide to fix the primordial tensor amplitude at the maximally allowed value, A t 1.5 × 10 −10 , that is consistent with the most recent upper bound on the tensor-to-scalar ratio, r = A t /A s 0.07 [55], where A s 2.1×10 −9 is the measured amplitude of the primordial scalar spectrum. Similarly, we shall assume a bluetilted tensor spectrum, fixing its index at an optimistic value of n t = 0.4. On the one hand, this value clearly violates the consistency relation of standard single-field slow-roll inflation, n t = −r/8 < 0. This means that we implicitly assume a nonminimal inflationary sector that is capable of generating a strong SGWB at small scales. Such scenarios exist in the literature, with one prominent example being natural inflation coupled to gauge fields [56][57][58][59][60][61]. On the other hand, our choice for the spectral index is phenomenologically still completely viable. n t = 0.4 complies with (1) the CMB constraints derived by the PLANCK Collaboration [55], (2) the upper bound on the amplitude of an isotropic SGWB at O (10) Hz frequencies by LIGO and Virgo [62,63], and (3) global bounds on primordial GWs (based, e.g., on BBN) across the full frequency spectrum [64,65]. The scalar era. We consider the dynamics of a real scalar field φ, employing an effective description in terms of three parameters: mass m φ , decay rate Γ φ , and initial field value φ ini at the end of inflation. We assume a harmonic potential around the origin in field space and some (direct or indirect) coupling to the Standard Model that allows φ to decay to radiation at the end of its lifetime. The dynamical evolution of the field φ is governed by the Klein-Gordon equation in an expanding background, where we included a friction term proportional to Γ φ . The Hubble rate H follows from the Friedmann equation, where M Pl 2.44 × 10 18 GeV is the reduced Planck mass and ρ φ = 1/2φ 2 + 1/2 m 2 φ φ 2 denotes the energy density stored in the field φ. The energy density of radiation, ρ R , is described by an evolution equation that follows from covariant energy conservation in an expanding universe, Here, g * ,s and g * ,ρ are the effective numbers of relativistic degrees of freedom that contribute to the entropy and energy densities of radiation, respectively. We will evaluate g * ,s and g * ,ρ using the numerical data tabulated in [26].
Let us now discuss the system of equations (5) to (7). First, we observe that φ only begins to roll -and eventually oscillate around the origin in field space -once the Hubble rate has dropped below the scalar mass, H m φ . At earlier times, the Hubble friction term in (5) keeps φ more or less fixed at φ ini . For a sufficiently high reheating temperature after inflation, the expansion is initially ω is defined such that a ∝ t α , where α = 2/3/ (1 + ω). It equals the EOS parameter w = p/ρ as long as w = const. driven by radiation. ρ R is, however, rapidly redshifted, ρ R ∝ 1/a 4 , such that, at some temperature T SD , the energy density ρ φ begins to dominate the expansion, Here, b = Γ (5/4) /Γ (3/2) 1.02 stems from the analytic solution of (5) in a radiation-dominated (RD) background, while α SD ≈ 2/3 characterizes the behavior of the scale factor at the onset of scalar-field domination, a (t) ∝ t αSD . The result in (8) is only valid for φ ini M Pl . Otherwise, the scalar field has not yet begun to oscillate at the onset of the scalar era, which leads to a short period of vacuum domination. Such a second phase of inflation results in a nontrivial oscillatory feature in the transfer function [25]. On the other hand, it also causes a stronger dilution of the primordial SGWB, which weakens the expected GW signal. For this reason, we will not consider the case φ ini M Pl in this work.
After many oscillations around the potential minimum, t 1/m φ , the energy density of the scalar field behaves like the energy density of pressureless dust, ρ φ ∝ 1/a 3 . The scalar era therefore effectively mimics, at least for some intermediate period, a phase of matter-dominated expansion (see Fig. 1). It lasts until φ decays into SM radiation around t RD c/Γ φ , which marks the onset of the final radiation-dominated era at temperature T RD , where α RD ≈ 2/3. Assuming a constant EOS throughout the scalar era, one is able to analytically derive c 1.07.
The nonstandard EOS and the late-time entropy injection during the scalar era result in the dilution of subhorizon tensor modes. This effect is captured by the dilution factor D = [S (T T SD ) /S (T T RD )] 1/3 , where S denotes the comoving entropy density, S = a 3 s, D = e c/3 d g SD * ,s g RD * ,s 1/3 g RD * ,ρ g SD * ,ρ The factor d 0.80, which needs to be determined numerically, accounts for entropy production at T < T RD . The factor η is defined as the ratio of the two values of the Hubble rate at T = T RD and T = T SD , respectively, The two temperatures T SD and T RD also single out two frequencies in the GW spectrum, f SD and f RD , which correspond to the tensor modes that re-enter the horizon at these temperatures, respectively. For f RD , we find where T 0 2.72548 K 2.3 × 10 −13 GeV is the current temperature of the CMB [66], h 2 Ω 0 R 4.2×10 −5 denotes the energy density parameter of radiation at the present time, and f 0 = H 0 / (2π) corresponds to the frequency of the GW mode that currently stretches across one Hubble radius, f 0 /h 5.2 × 10 −19 Hz. The frequency f SD can be computed in terms of the quantities in (11) and (12), This shows that the parameter η determines the duration of the scalar era in frequency space. In our numerical analysis, we will scan over different values of η by varying the ratio m φ /Γ φ , while keeping the initial field value fixed at φ ini = 10 18 GeV. This benchmark represents a typical value that can be realized in many models as an initial condition after inflation [67][68][69][70]. Aside from that, it is important to note that, to first approximation, it is straightforward to generalize our analysis to different values of φ ini by simply rescaling all values of m φ /Γ φ .
Transfer function. For a mode with wave number k deep inside the horizon, the transfer function roughly scales like χ k ∼ a k /a 0 , where a k is the scale factor at the time of horizon re-entry, implicitly defined by a k = k/H (a k ) [20,71,72]. For an EOS w = p/ρ = const (with pressure p and energy density ρ), this results in a simple power-law scaling of the final GW spectrum, which illustrates the main impact of the scalar era on the GW spectrum: For frequencies f RD f f SD , the spectral index n changes by ∆n −2 compared to the spectrum at smaller and larger frequencies. This results in a step-like feature in the GW spectrum with two kinks located at f f RD and f f SD , respectively (see Fig. 2). The ratio of the spectral amplitudes at these two frequencies is controlled by the dilution factor D. Neglecting any changes in g * ,ρ and g * ,s , we approximately find To obtain a precise and accurate result for the transfer function, we numerically solve the coupled set of equations (2), (5), (6), and (7) for any choice of m φ and Γ φ . Together with (3) and (4), this allows us to compute the final GW spectrum across our entire parameter space.
Signal-to-noise ratios. A GW experiment with strain noise power spectrum S noise = 6f 2 0 /f 3 Ω noise is able to detect a stochastic signal Ω signal at the following SNR [73],  -91]]. We assume an observing time t obs = 4 yr for all interferometer experiments and the equivalent [92] of t obs = 20 yr for IPTA and SKA. In future work, one might consider refining our analysis by means of a Fisher information analysis. We, however, expect that such an analysis would only marginally improve over our SNRbased approach (see [37], which finds no difference between both approaches in the specific case of LISA). In principle, it is also necessary to account for confusion noise from astrophysical sources (see, e.g., [93]). The significance of confusion noise, however, decreases over time, as larger observing times allow one to resolve and subtract a larger number of individual foreground sources. In what follows, we will therefore consider an idealized scenario without any foreground contamination. In this sense, our SNR forecasts may also be regarded as upper limits on what will be realistically achievable.
For each experiment, we compute two kinds of SNRs (see Fig. 3): First, we compute the total SNR based on the full GW signal, Ω signal → Ω 0 GW . If this SNR is larger than 1, we conclude that the respective experiment will have "at least a slight chance" of detecting a signal. Then, we compute a second, reduced SNR based on a subtracted GW signal, Ω signal → Ω 0 GW −Ω fit GW , where Ω fit GW represents a power-law fit of the GW spectrum around the frequency at which a given experiment is most sensitive. This construction is in accord with the matched-filter approaches in [37,94]. If the reduced SNR is larger than 100, we conclude that the respective experiment is "very likely" to detect a feature in the SGWB, i.e., a deviation from a pure power law, related to the scalar era. In practice, this feature will correspond to one or both of the two kinks at f f RD and f f SD . Our choice of SNR threshold values allows us to assign the labels "at least a slight chance" and "very likely" to the different regions in Fig. 3 in a conservative manner. Meanwhile, we stress that our quantitative conclusions are nearly insensitive to O (1) changes of these threshold values. Our qualitative conclusions are not affected at all by the precise values of the SNR thresholds.
Discussion. Our results in Fig. 3 are a powerful testament to the complementarity of future GW experiments. Consider, e.g., a scalar field with m φ = 10 GeV, Γ φ = 10 eV, and φ ini = 10 18 GeV (point in Fig. 3). In this case, the scalar era occurs at temperatures 5.4 × 10 4 T /GeV 1.7 × 10 8 , which leads to a dilution factor D 7.9 × 10 −2 and a modification of the GW spectrum at frequencies 1.6 × 10 −3 f /Hz 4.9 × 10 −1 . This scenario can be directly probed by LISA, DECIGO, and BBO, all of which will observe a departure from a pure power law. At the same time, CE, IPTA, and SKA will detect the SGWB from inflation, while HLVK and ET will fail to observe a primordial signal. On top of that, future CMB observations may confirm the blue tilt of the tensor spectrum. Along these lines, it is possible to translate every point in Fig. 3 into a list of predictions, i.e., a characteristic experimental fingerprint, for the GW experiments that we have considered in this paper.
The results of our analysis are applicable to a wide range of particle physics models that lead to a scalar era in the early Universe. In addition, it is important to note that the modified expansion history and late-time entropy production during the scalar era not only affect the spectrum of primordial GWs, but also the evolution of other cosmological relics, such as dark matter [96][97][98][99][100][101][102][103] and the baryon asymmetry of the Universe [104]. Finding evidence for a scalar era in future GW experiments would therefore have a profound impact on our understanding of BSM physics and early-Universe cosmology alike.
The analysis presented in this paper should only be regarded as a first step. Next, it will be necessary to relax our assumptions regarding the primordial tensor spectrum, the initial field value, and the shape of the scalar potential. Aside from that, it would be interesting to identify a self-consistent embedding of our scenario in a concrete inflationary model that is capable of generating a strongly blue-tilted tensor spectrum. We leave these steps for future work, concluding with the observation that a scalar era in the early Universe represents a fascinating possibility that awaits further exploration.
Acknowledgements. We thank Sachiko Kuroyanagi for sharing with us her numerical results on the DECIGO overlap reduction function; Ken'ichi Saikawa and Satoshi Shirai for sharing with us their numerical results on the GW transfer function at frequencies below 1 nHz, which takes into account damping effects caused by freestreaming photons and neutrinos at low temperatures; and Hirotaka Yuzurihara for useful comments on the KA-GRA sensitivity curve. We are grateful to all of these authors as well as to Kohei Kamada, Marco Peloso, and Angelo Ricciardone for helpful discussions. This project has received funding/support from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 690575. This work was supported by INFN through the "Theoretical Astroparticle Physics" (TAsP) project.