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 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 subsequent to horizon 2 a˙(t ) ρφ (t ) + ρ (t ) reentry. For any expansion history after inflation [specified 2 = = R , H (t ) 2 (6) by the function a(u)], the transfer function can be found by a(t ) 3 MPl solving Eq. (2)forχ as a function of u with boundary con- k . × 18 ditions χ = 1 and ∂ χ = 0atu = 0. The same factorization where MPl 2 44 10 GeV is the reduced Planck mass k u k ρ = / φ˙2 + / 2 φ2 also carries over to  ( f ), i.e., the GW energy density per and φ 1 2 1 2 mφ denotes the energy density GW φ ρ logarithmic frequency interval d(ln f ) in units of the critical stored in the field . The energy density of radiation R is energy density of a flat universe. For modes deep inside the described by an evolution equation that follows from covariant energy conservation in an expanding universe, horizon, one obtains the following GW spectrum at the present   time: ρ d g∗,s( R(t )) 2 + 4 H(t ) ρ (t ) = φ φ˙ (t ). (7) 1 k2 k ρ R 0 P |χ |2, = dt g∗,ρ ( R(t )) GW( f ) 2 2 t (k) k f (3) 12 a H 2π a0 0 0 Here, g∗,s and g∗,ρ are the effective numbers of relativistic where a0 and H0 = 100 h km/s/Mpc represent the present degrees of freedom that contribute to the entropy and energy values of the scale factor and the Hubble rate, respectively. densities of radiation, respectively. We will evaluate g∗,s and The primordial tensor spectrum Pt depends on the initial g∗,ρ using the numerical data tabulated in [26]. conditions set by the inflationary period and is conventionally Let us now discuss the system of Eqs. (5)–(7). First, we parametrized by a simple power law around the CMB pivot observe that φ only begins to roll, and eventually oscillate −1 scale kCMB = 0.05 Mpc : around the origin in field space, once the Hubble rate has    φ 3 nt dropped below the scalar mass, H m . At earlier times, the k + 2 × 2 k φ φ P = (|h¯ | +|h¯ | ) = A . (4) Hubble friction term in (5) keeps more or less fixed at ini. t π 2 k k t kCMB For a sufficiently high reheating temperature after inflation,

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comoving entropy density S = a3 s:     SD 1/3 RD 1/4 g∗, g∗,ρ D = ec/3 d s η1/6. (10) RD SD g∗,s g∗,ρ The factor d 0.80, which needs to be determined numeri- cally, accounts for entropy production at T < TRD. The factor η is defined as the ratio of the two values of the Hubble rate at T = TRD and TSD, respectively,   3 4 H 18 α α φ M η = RD = SD RD Pl . (11) 4 HSD b c mφ φini

The two temperatures TSD and TRD also single out two frequencies in the GW spectrum, fSD and fRD, which corre- spond to the tensor modes that reenter the horizon at these temperatures, respectively. For fRD, we find       ω = / − 0 1/3 RD 1/2 0 1/2 FIG. 1. Pseudo EOS parameter 2 (3 tH) 1 as a function f g∗, g∗,ρ  T ω RD = d s R RD , (12) of temperature T for three parameter examples (see Fig. 3). is RD 0 α f g g 1/2 T defined such that a ∝ t ,whereα = 2/3/(1 + ω). It equals the EOS 0 ∗,s ∗,ρ 0 parameter w = p/ρ as long as w = const. −13 where T0 2.725 48 K 2.3 × 10 GeV is the current 20 . × −5 temperature of the CMB [66], h R 4 2 10 denotes the energy density parameter of radiation at the present time, and the expansion is initially driven by radiation. ρR is, however, 4 f0 = H0/(2π ) corresponds to the frequency of the GW mode rapidly redshifted, ρR ∝ 1/a , such that, at some temperature that currently stretches across one Hubble radius, f0/h TSD, the energy density ρφ begins to dominate the expansion: 5.2 × 10−19 Hz. The frequency f can be computed in terms   SD 1/4  2 φ 2 of the quantities in (11) and (12): b 5 ini 1/2   T (mφ M ) . (8) 1/3 SD α3/2 36π 2 gSD M Pl ec SD ∗,ρ Pl f = f . SD η RD (13) = / / / . Here, b (5 4) (3 2) 1 02 stems from the analytic η solution of (5) in a radiation-dominated (RD) background, This shows that the parameter determines the duration of the scalar era in frequency space. In our numerical analysis, while αSD ≈ 2/3 characterizes the behavior of the scale factor α η ∝ SD we will scan over different values of by varying the ratio at the onset of scalar-field domination a(t ) t .Theresult / φ = φ  mφ φ, while keeping the initial field value fixed at ini in (8) is only valid for ini MPl. Otherwise, the scalar field 18 has not yet begun to oscillate at the onset of the scalar era, 10 GeV. This benchmark represents a typical value that which leads to a short period of vacuum domination. Such can be realized in many models as an initial condition after a second phase of inflation results in a nontrivial oscillatory inflation [67–70]. Aside from that, it is important to note that, feature in the transfer function [25]. On the other hand, it also to first approximation, it is straightforward to generalize our analysis to different values of φini by simply rescaling all causes a stronger dilution of the primordial SGWB, which / weakens the expected GW signal. For this reason, we will not values of mφ φ. consider the case φini  MPl in this work. After many oscillations around the potential minimum t IV. TRANSFER FUNCTION 1/mφ, the energy density of the scalar field behaves like the 3 For a mode with wave number k deep inside the horizon, energy density of pressureless dust ρφ ∝ 1/a . The scalar era the transfer function roughly scales like χk ∼ ak/a0, where ak therefore effectively mimics, at least for some intermediate is the scale factor at the time of horizon re-entry, implicitly period, a phase of matter-dominated expansion (see Fig. 1). = / = /ρ = φ / defined by ak k H(ak )[20,71,72]. For an EOS w p It lasts until decays into SM radiation around tRD c φ, const (with pressure p and energy density ρ), this results in a which marks the onset of the final radiation-dominated era at simple power-law scaling of the final GW spectrum, temperature TRD, −   0 n 2(3 w 1)   1/4  ( f ) ∝ f , n = n + , (14) α 1/2 GW t 3 w + 1 RD 90 1/2 T = ( φ M ) , (9) RD 2 RD Pl c 2π g∗,ρ which illustrates the main impact of the scalar era on the GW spectrum: For frequencies fRD  f  fSD, the spectral index where αRD ≈ 2/3. Assuming a constant EOS throughout the n changes by n −2 compared to the spectrum at smaller scalar era, one is able to analytically derive c 1.07. and larger frequencies. This results in a steplike feature in the The nonstandard EOS and the late-time entropy injection GW spectrum with two kinks located at f fRD and f fSD, during the scalar era result in the dilution of subhorizon respectively (see Fig. 2). The ratio of the spectral amplitudes tensor modes. This effect is captured by the dilution fac- at these two frequencies is controlled by the dilution factor 1/3 tor D = [S(T TSD)/S(T TRD)] , where S denotes the D. Neglecting any changes in g∗,ρ and g∗,s, we approximately

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FIG. 2. Final GW spectrum for three parameter examples (see Fig. 3) alongside the power-law integrated sensitivity curves [95]of existing and upcoming GW experiments.

find FIG. 3. Experimental prospects to observe a signal or a even a   0 nt feature in the GW spectrum (see text for details) in dependence of  ( fSD) fSD GW D4. (15) the model parameters mφ and φ . The three points 1 , 2 , 3 mark the 0 GW( fRD) fRD parameter examples used in Figs. 1 and 2. The meshes drawn in the feature regions indicate where the frequencies f (meshes extending To obtain a precise and accurate result for the transfer RD up to arbitrarily large mφ )and fSD (meshes extending up to a finite function, we numerically solve the coupled set of equations mφ ) fall into the frequency ranges of the individual experiments (see

(2), (5), (6), and (7) for any choice of mφ and φ. Together Fig. 2). with (3) and (4), this allows us to compute the final GW spectrum across our entire parameter space. In principle, it is also necessary to account for confusion noise from astrophysical sources (see, e.g., [93]). The sig- V. SIGNAL-TO-NOISE RATIOS nificance of confusion noise, however, decreases over time, A GW experiment with strain noise power spectrum as larger observing times allow one to resolve and subtract = 2/ 3 a larger number of individual foreground sources. In what Snoise 6 f0 f noise is able to detect a stochastic signal signal at the following SNR [73]: follows, we will therefore consider an idealized scenario    without any foreground contamination. In this sense, our SNR fmax  ( f ) 2 forecasts may also be regarded as upper limits on what will be 2 = signal , SNR Ntobs df (16) realistically achievable. f noise( f ) min For each experiment, we compute two kinds of SNRs (see where tobs denotes the experiment’s observing time and N = Fig. 3): First, we compute the total SNR based on the full =  → 0 1(N 2) for experiments that perform an autocorrelation GW signal signal GW. If this SNR is larger than 1, we (crosscorrelation) measurement of the SGWB. With the full conclude that the respective experiment will have “at least GW spectrum at hand, we are now able to forecast the a slight chance” of detecting a signal. Then, we compute SNRs for future ground-based interferometers [(1) the four- a second, reduced SNR based on a subtracted GW signal  → 0 − fit fit detector network (HLVK) consisting of advanced LIGO in signal GW GW, where GW represents a power-law Hanford and Livingston [2,3], advanced Virgo [4], and KA- fit of the GW spectrum around the frequency at which a given GRA [74,75], (2) Cosmic Explorer (CE) [76], and (3) Einstein experiment is most sensitive. This construction is in accord Telescope (ET) [77–79]], future spaced-based interferometers with the matched-filter approaches in [37,94]. If the reduced [(1) LISA [80], (2) DECIGO [81,82], and (3) BBO [83–85]], SNR is larger than 100, we conclude that the respective and future pulsar timing arrays [(1) IPTA [86–88] and (2) experiment is “very likely” to detect a feature in the SGWB, SKA [89–91]]. We assume an observing time tobs = 4yr i.e., a deviation from a pure power law, related to the scalar for all interferometer experiments and the equivalent [92]of era. In practice, this feature will correspond to one or both of tobs = 20 yr for IPTA and SKA. In future work, one might the two kinks at f fRD and f fSD. Our choice of SNR consider refining our analysis by means of a Fisher informa- threshold values allows us to assign the labels “at least a tion analysis. We, however, expect that such an analysis would slight chance” and “very likely” to the different regions in only marginally improve over our SNR-based approach (see Fig. 3 in a conservative manner. Meanwhile, we stress that [37], which finds no difference between both approaches in our quantitative conclusions are nearly insensitive to O(1) the specific case of LISA). changes of these threshold values. Our qualitative conclusions

013010-4 IMPRINT OF A SCALAR ERA ON THE PRIMORDIAL … PHYSICAL REVIEW RESEARCH 1, 013010 (2019) are not affected at all by the precise values of the SNR understanding of BSM physics and early-Universe cosmology thresholds. alike. The analysis presented in this article should only be re- VI. DISCUSSION garded as a first step. Next, it will be necessary to relax our assumptions regarding the primordial tensor spectrum, the Our results in Fig. 3 are a powerful testament to the initial field value, and the shape of the scalar potential. Aside complementarity of future GW experiments. Consider, e.g., from that, it would be interesting to identify a self-consistent = = φ = a scalar field with mφ 10 GeV, φ 10 eV, and ini embedding of our scenario in a concrete inflationary model 18 10 GeV (point 2 in Fig. 3). In this case, the scalar era oc- that is capable of generating a strongly blue-tilted tensor . × 4  /  . × 8 curs at temperatures 5 4 10 T GeV 1 7 10 , which spectrum. We leave these steps for future work, concluding . × −2 leads to a dilution factor D 7 9 10 and a modification with the observation that a scalar era in the early Universe . × −3  /  of the GW spectrum at frequencies 1 6 10 f Hz represents a fascinating possibility that awaits further explo- . × −1 4 9 10 . This scenario can be directly probed by LISA, ration. 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 ACKNOWLEDGMENTS will fail to observe a primordial signal. On top of that, future CMB observations may confirm the blue tilt of the tensor We thank S. Kuroyanagi for sharing with us her numer- spectrum. Along these lines, it is possible to translate every ical results on the DECIGO overlap reduction function; K. point in Fig. 3 into a list of predictions, i.e., a characteristic Saikawa and S. Shirai for sharing with us their numerical experimental fingerprint, for the GW experiments that we results on the GW transfer function at frequencies below have considered in this article. 1 nHz, which takes into account damping effects caused by The results of our analysis are applicable to a wide range free-streaming photons and neutrinos at low temperatures; and of particle physics models that lead to a scalar era in the H. Yuzurihara for useful comments on the KAGRA sensitivity early Universe. In addition, it is important to note that the curve. We are grateful to all of these authors as well as modified expansion history and late-time entropy production to K. Kamada, M. Peloso, and A. Ricciardone for helpful during the scalar era not only affect the spectrum of primordial discussions. This project has received funding/support from GWs, but also the evolution of other cosmological relics, such the European Union’s Horizon 2020 research and innovation as dark matter [96–103] and the baryon asymmetry of the programme under the Marie Skłodowska-Curie Grant Agree- Universe [104]. Finding evidence for a scalar era in future GW ment No. 690575. This work was supported by INFN through experiments would therefore have a profound impact on our the “Theoretical Astroparticle Physics” (TAsP) project.

[1] B. P. Abbott et al. (LIGO Scientific and Virgo Collab- [9] V. Kalogera et al., Deeper, wider, sharper: next-generation oration), Observation of Gravitational Waves from a Bi- ground-based gravitational-wave observations of binary black nary Black Hole Merger, Phys. Rev. Lett. 116, 061102 holes, arXiv:1903.09220. (2016). [10] N. J. Cornish, E. Berti, K. Holley-Bockelmann, S. Larson, [2] G. M. Harry (LIGO Scientific Collaboration), Advanced S. McWilliams, G. Mueller, P. Natarajan, and M. Vallisneri, LIGO: The next generation of detectors, The discovery potential of space-based gravitational wave Class. Quantum Gravity 27, 084006 (2010). astronomy, arXiv:1904.01438. [3] J. Aasi et al. (LIGO Scientific Collaboration), Advanced [11] D. Shoemaker et al. (LIGO Scientific Collaboration), Gravita- LIGO, Class. Quantum Gravity 32, 115012 (2015). tional wave astronomy with LIGO and similar detectors in the [4] F. Acernese et al. (VIRGO Collaboration), Advanced Virgo: a next decade, arXiv:1904.03187. second-generation interferometric gravitational wave detector, [12] C. Caprini and D. G. Figueroa, Cosmological backgrounds Class. Quantum Gravity 32, 024001 (2015). of gravitational waves, Class. Quantum Gravity 35, 163001 [5] B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration), (2018). GWTC-1: A gravitational-wave transient catalog of compact [13] N. Christensen, Stochastic gravitational wave backgrounds, binary mergers observed by LIGO and Virgo during the first Rep. Prog. Phys. 82, 016903 (2019). and second observing runs, arXiv:1811.12907. [14] M. Geller, A. Hook, R. Sundrum, and Y. Tsai, Primordial [6] M. Maggiore, Gravitational wave experiments and early uni- Anisotropies in the Gravitational Wave Background from Cos- verse cosmology, Phys. Rep. 331, 283 (2000). mological Phase Transitions, Phys. Rev. Lett. 121, 201303 [7] S. T. McWilliams, R. Caldwell, K. Holley-Bockelmann, S. L. (2018). Larson, and M. Vallisneri, Astro2020 Decadal Science White [15] L. P. Grishchuk, Amplification of gravitational waves in an Paper: The state of gravitational-wave astrophysics in 2020, istropic universe, Zh. Eksp. Teor. Fiz. 67, 825 (1975) [Sov. arXiv:1903.04592. Phys.–JETP 40, 409 (1975)]. [8] R. Caldwell et al., Astro2020 science white paper: cos- [16] A. A. Starobinsky, Spectrum of relict gravitational radiation mology with a space-based gravitational wave observatory, and the early state of the universe, Pis’ma Zh. Eksp. Teor. Fiz. arXiv:1903.04657. 30, 719 (1979) [JETP Lett. 30, 682 (1979)].

013010-5 FRANCESCO D’ERAMO AND KAI SCHMITZ PHYSICAL REVIEW RESEARCH 1, 013010 (2019)

[17] V. A. Rubakov, M. V. Sazhin, and A. V. Veryaskin, [36] R. Jinno, T. Moroi, and K. Nakayama, Probing dark radiation creation in the inflationary universe and the grand unification with inflationary gravitational waves, Phys.Rev.D86, 123502 scale, Phys. Lett. B 115, 189 (1982). (2012). [18] M. C. Guzzetti, N. Bartolo, M. Liguori, and S. Matarrese, [37] R. R. Caldwell, T. L. Smith, and D. G. E. Walker, Using a Gravitational waves from inflation, Riv. Nuovo Cimento 39, primordial gravitational wave background to illuminate new 399 (2016). physics, arXiv:1812.07577. [19] N. Seto and J. Yokoyama, Probing the equation of state of the [38] Y. Cui, M. Lewicki, D. E. Morrissey, and J. D. Wells, Cosmic early universe with a space laser interferometer, J. Phys. Soc. archaeology with gravitational waves from cosmic strings, Jpn. 72, 3082 (2003). Phys. Rev. D 97, 123505 (2018). [20] L. A. Boyle and P. J. Steinhardt, Probing the early universe [39] Y. Cui, M. Lewicki, D. E. Morrissey, and J. D. Wells, Probing with inflationary gravitational waves, Phys.Rev.D77, 063504 the pre-BBN universe with gravitational waves from cosmic (2008). strings, J. High Energy Phys. 01 (2019) 081. [21] L. A. Boyle and A. Buonanno, Relating gravitational wave [40] Y. Matsui and S. Kuroyanagi, Gravitational wave back- constraints from primordial nucleosynthesis, pulsar timing, ground from kink-kink collisions on infinite cosmic strings, laser interferometers, and the CMB: Implications for the early arXiv:1902.09120. Universe, Phys. Rev. D 78, 043531 (2008). [41] G. D. Coughlan, W. Fischler, E. W. Kolb, S. Raby, and G. G. [22] S. Kuroyanagi, T. Chiba, and N. Sugiyama, Precision calcu- Ross, Cosmological problems for the polonyi potential, Phys. lations of the gravitational wave background spectrum from Lett. B 131, 59 (1983). inflation, Phys. Rev. D 79, 103501 (2009). [42] J. R. Ellis, D. V. Nanopoulos, and M. Quiros, On the axion, [23] K. Nakayama and J. Yokoyama, Gravitational wave back- dilaton, polonyi, gravitino and shadow matter problems in ground and non-Gaussianity as a probe of the curvaton sce- supergravity and superstring models, Phys. Lett. B 174, 176 nario, J. Cosmol. Astropart. Phys. 01 (2010) 010. (1986). [24] S. Kuroyanagi, C. Ringeval, and T. Takahashi, Early universe [43] B. de Carlos, J. A. Casas, F. Quevedo, and E. Roulet, Model tomography with CMB and gravitational waves, Phys. Rev. D independent properties and cosmological implications of the 87, 083502 (2013). dilaton and moduli sectors of 4D strings, Phys. Lett. B 318, [25] R. Jinno, T. Moroi, and K. Nakayama, Inflationary grav- 447 (1993). itational waves and the evolution of the early universe, [44] T. Moroi and L. Randall, Wino cold dark matter from J. Cosmol. Astropart. Phys. 01 (2014) 040. anomaly mediated SUSY breaking, Nucl. Phys. B 570, 455 [26] K. Saikawa and S. Shirai, Primordial gravitational waves, (2000). precisely: The role of thermodynamics in the Standard Model, [45] R. Durrer and J. Hasenkamp, Testing superstring theo- J. Cosmol. Astropart. Phys. 05 (2018) 035. ries with gravitational waves, Phys.Rev.D84, 064027 [27] K. Nakayama, S. Saito, Y. Suwa, and J. Yokoyama, Space laser (2011). interferometers can determine the thermal history of the early [46] G. Kane, K. Sinha, and S. Watson, Cosmological moduli and Universe, Phys. Rev. D 77, 124001 (2008). the post-inflationary universe: A critical review, Int. J. Mod. [28] K. Nakayama, S. Saito, Y. Suwa, and J. Yokoyama, Probing Phys. D 24, 1530022 (2015). reheating temperature of the universe with gravitational wave [47] C. D. Froggatt and H. B. Nielsen, Hierarchy of quark masses, background, J. Cosmol. Astropart. Phys. 06 (2008) 020. cabibbo angles and CP violation, Nucl. Phys. B 147, 277 [29] S. Kuroyanagi, K. Nakayama, and S. Saito, Prospects for (1979). determination of thermal history after inflation with fu- [48] T. Kugo, I. Ojima, and T. Yanagida, Superpotential symmetries ture gravitational wave detectors, Phys.Rev.D84, 123513 and pseudo Nambu-Goldstone supermultiplets, Phys. Lett. B (2011). 135, 402 (1984). [30] W. Buchmüller, V. Domcke, K. Kamada, and K. Schmitz, [49] T. Banks, M. Dine, and M. Graesser, Supersymmetry, axions The gravitational wave spectrum from cosmological B − L and cosmology, Phys.Rev.D68, 075011 (2003). breaking, J. Cosmol. Astropart. Phys. 10 (2013) 003. [50] M. Kawasaki and K. Nakayama, Solving cosmological prob- [31] W. Buchmuller, V. Domcke, K. Kamada, and K. Schmitz, A lems of supersymmetric axion models in inflationary universe, minimal supersymmetric model of particle physics and the Phys. Rev. D 77, 123524 (2008). early universe, arXiv:1309.7788. [51] K. Harigaya, M. Ibe, K. Schmitz, and T. T. Yanagida, Peccei- [32] R. Jinno, T. Moroi, and T. Takahashi, Studying inflation with Quinn symmetry from a gauged discrete R symmetry, Phys. future space-based gravitational wave detectors, J. Cosmol. Rev. D 88, 075022 (2013). Astropart. Phys. 12 (2014) 006. [52] K. Harigaya, M. Ibe, K. Schmitz, and T. T. Yanagida, Peccei- [33] S. Kuroyanagi, K. Nakayama, and J. Yokoyama, Prospects Quinn symmetry from dynamical supersymmetry breaking, of determination of reheating temperature after inflation by Phys. Rev. D 92, 075003 (2015). DECIGO, Prog. Theor. Exp. Phys. 2015, 013E02 (2015). [53] R. T. Co, F. D’Eramo, and L. J. Hall, Supersymmetric axion [34] S. Schettler, T. Boeckel, and J. Schaffner-Bielich, Imprints grand unified theories and their predictions, Phys.Rev.D94, of the qcd phase transition on the spectrum of gravitational 075001 (2016). waves, Phys.Rev.D83, 064030 (2011). [54] R. T. Co, F. D’Eramo, L. J. Hall, and K. Harigaya, Saxion [35] F. Hajkarim, J. Schaffner-Bielich, S. Wystub, and M. M. cosmology for thermalized gravitino dark matter, J. High Wygas, Effects of the QCD equation of state and lepton Energy Phys. 07 (2017) 125. asymmetry on primordial gravitational waves, Phys.Rev.D [55] Y. Akrami et al. (Planck Collaboration), Planck 2018 results. 99, 103527 (2019). X. constraints on inflation, arXiv:1807.06211.

013010-6 IMPRINT OF A SCALAR ERA ON THE PRIMORDIAL … PHYSICAL REVIEW RESEARCH 1, 013010 (2019)

[56] J. L. Cook and L. Sorbo, Particle production during inflation [75] Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. and gravitational waves detectable by ground-based interfer- Sekiguchi, D. Tatsumi, and H. Yamamoto (KAGRA Collab- ometers, Phys.Rev.D85, 023534 (2012); 86, 069901(E) oration), Interferometer design of the KAGRA gravitational (2012) wave detector, Phys.Rev.D88, 043007 (2013). [57] N. Barnaby, E. Pajer, and M. Peloso, Gauge field produc- [76] B. P. Abbott et al. (LIGO Scientific Collaboration), Exploring tion in axion inflation: Consequences for monodromy, non- the sensitivity of next generation gravitational wave detectors, gaussianity in the CMB, and gravitational waves at interfer- Class. Quantum Gravity 34, 044001 (2017). ometers, Phys.Rev.D85, 023525 (2012). [77] M. Punturo et al., The : A third-generation [58] M. M. Anber and L. Sorbo, Non-Gaussianities and chiral gravitational wave observatory, Class. Quantum Gravity 27, gravitational waves in natural steep inflation, Phys. Rev. D 85, 194002 (2010). 123537 (2012). [78] S. Hild et al., Sensitivity studies for third-generation gravita- [59] V. Domcke, M. Pieroni, and P. Binétruy, Primordial gravita- tional wave observatories, Class. Quantum Gravity 28, 094013 tional waves for universality classes of pseudoscalar inflation, (2011). J. Cosmol. Astropart. Phys. 06 (2016) 031. [79] B. Sathyaprakash et al., Scientific objectives of Einstein [60] D. Jiménez, K. Kamada, K. Schmitz, and X.-J. Xu, Baryon Telescope, Class. Quantum Gravity 29, 124013 (2012); 30, asymmetry and gravitational waves from pseudoscalar infla- 079501(E) (2013). tion, J. Cosmol. Astropart. Phys. 12 (2017) 011. [80] H. Audley et al. (LISA Collaboration), Laser interferometer [61] A. Papageorgiou, M. Peloso, and C. Unal, Nonlinear pertur- space antenna, arXiv:1702.00786. bations from axion-gauge fields dynamics during inflation, [81] N. Seto, S. Kawamura, and T. Nakamura, Possibility of Direct J. Cosmol. Astropart. Phys. 07 (2019) 004. Measurement of the Acceleration of the Universe Using 0.1- [62] B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration), Hz Band Laser Interferometer Gravitational Wave Antenna in Upper Limits on the Stochastic Gravitational-Wave Back- Space, Phys.Rev.Lett.87, 221103 (2001). ground from Advanced LIGO’s First Observing Run, Phys. [82] S. Kawamura et al., The Japanese space gravitational Rev. Lett. 118, 121101 (2017); 119, 029901(E) (2017). wave antenna DECIGO, Class. Quantum Gravity 23, S125 [63] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo (2006). Collaboration), A search for the isotropic stochastic back- [83] J. Crowder and N. J. Cornish, Beyond LISA: Exploring fu- ground using data from Advanced LIGO’s second observing ture gravitational wave missions, Phys. Rev. D 72, 083005 run, arXiv:1903.02886. (2005). [64] S. Kuroyanagi, T. Takahashi, and S. Yokoyama, Blue- [84] V. Corbin and N. J. Cornish, Detecting the cosmic gravita- tilted tensor spectrum and thermal history of the universe, tional wave background with the big bang observer, Class. J. Cosmol. Astropart. Phys. 02 (2015) 003. Quantum Gravity 23, 2435 (2006). [65] P. D. Lasky et al., Gravitational-Wave Cosmology Across 29 [85] G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner, Decades in Frequency, Phys.Rev.X6, 011035 (2016). and E. S. Phinney, Laser interferometry for the big [66] D. J. Fixsen, The temperature of the cosmic microwave back- bang observer, Class. Quantum Gravity 23, 4887 ground, Astrophys. J. 707, 916 (2009). (2006); 23, 7361(E) (2006). [67] T. S. Bunch and P. C. W. Davies, Quantum field theory in de [86] G. Hobbs et al., The international project: sitter space: Renormalization by point splitting, Proc. R. Soc. Using pulsars as a gravitational wave detector, Class. Quantum London A 360, 117 (1978). Gravity 27, 084013 (2010). [68] A. D. Linde, Scalar field fluctuations in expanding universe [87] R. N. Manchester, The international pulsar timing array, Class. and the new inflationary universe scenario, Phys. Lett. B 116, Quantum Gravity 30, 224010 (2013). 335 (1982). [88] J. P. W. Verbiest et al., The international pulsar timing ar- [69] I. Affleck and M. Dine, A new mechanism for baryogenesis, ray: First data release, Mon. Not. R. Astron. Soc. 458, 1267 Nucl. Phys. B 249, 361 (1985). (2016). [70] A. A. Starobinsky and J. Yokoyama, Equilibrium state of a [89] C. L. Carilli and S. Rawlings, Motivation, key science selfinteracting scalar field in the de Sitter background, Phys. projects, standards and assumptions, New Astron. Rev. 48, 979 Rev. D 50, 6357 (1994). (2004). [71] M. S. Turner, M. J. White, and J. E. Lidsey, Tensor pertur- [90] G. Janssen et al., Gravitational wave astronomy with the SKA, bations in inflationary models as a probe of cosmology, Phys. PoS (AASKA14) 037 (2015). Rev. D 48, 4613 (1993). [91] P. Bull et al., Fundamental physics with the square kilometer [72] Y. Watanabe and E. Komatsu, Improved calculation of the array, arXiv:1810.02680. primordial gravitational wave spectrum in the standard model, [92] X. Siemens, J. Ellis, F. Jenet, and J. D. Romano, The stochastic Phys.Rev.D73, 123515 (2006). background: Scaling laws and time to detection for pulsar [73] B. Allen and J. D. Romano, Detecting a stochastic timing arrays, Class. Quantum Gravity 30, 224015 (2013). background of gravitational radiation: Signal processing [93] T. Robson, N. Cornish, and C. Liu, The construction and use strategies and sensitivities, Phys.Rev.D59, 102001 of LISA sensitivity curves, Class. Quantum Grav. 36, 105011 (1999). (2019). [74] K. Somiya (KAGRA Collaboration), Detector configuration of [94] S. Kuroyanagi, T. Chiba, and T. Takahashi, Probing the Uni- KAGRA: The Japanese cryogenic gravitational-wave detector, verse through the stochastic gravitational wave background, Class. Quantum Gravity 29, 124007 (2012). J. Cosmol. Astropart. Phys. 11 (2018) 038.

013010-7 FRANCESCO D’ERAMO AND KAI SCHMITZ PHYSICAL REVIEW RESEARCH 1, 013010 (2019)

[95] E. Thrane and J. D. Romano, Sensitivity curves for searches [100] R. T. Co, F. D’Eramo, L. J. Hall, and D. Pappadopulo, for gravitational-wave backgrounds, Phys. Rev. D 88, 124032 Freeze-In dark matter with displaced signatures at colliders, (2013). J. Cosmol. Astropart. Phys. 12 (2015) 024. [96] M. Hashimoto, K. I. Izawa, M. Yamaguchi, and T. Yanagida, [101] L. Roszkowski, S. Trojanowski, and K. Turzynski, Axino dark Axion cosmology with its scalar superpartner, Phys. Lett. B matter with low reheating temperature, J. High Energy Phys. 437, 44 (1998). 11 (2015) 139. [97] G. F. Giudice, E. W. Kolb, and A. Riotto, Largest temperature [102] S. Hamdan and J. Unwin, Dark matter freeze-out dur- of the radiation era and its cosmological implications, Phys. ing matter domination, Mod. Phys. Lett. A 33, 1850181 Rev. D 64, 023508 (2001). (2018). [98] B. S. Acharya, G. Kane, S. Watson, and P. Kumar, A non- [103] A. E. Nelson and H. Xiao, Axion cosmology with thermal WIMP miracle, Phys.Rev.D80, 083529 (2009). early matter domination, Phys.Rev.D98, 063516 [99] A. Monteux and C. S. Shin, Thermal Goldstino production (2018). with low reheating temperatures, Phys. Rev. D 92, 035002 [104] M.-C. Chen, S. Ipek, and M. Ratz, Baryogenesis from flavon (2015). decays, arXiv:1903.06211.

013010-8