Bridging the gap: spectral distortions meet gravitational waves
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 2 October 2020 (MN L A TEX style file v2.2)
Bridging the gap: spectral distortions meet gravitational waves
Thomas Kite (cid:63) , Andrea Ravenni † , Subodh P. Patil ‡ and Jens Chluba § Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, U.K. Instituut-Lorentz for Theoretical Physics, Leiden University, 2333 CA Leiden, The Netherlands
Accepted 2020 –. Received 2020 –
ABSTRACT
Gravitational waves (GWs) have the potential to probe the entirety of cosmological historydue to their nearly perfect decoupling from the thermal bath and any intervening matter afteremission. In recent years, GW cosmology has evolved from merely being an exciting prospectto an actively pursued avenue for discovery, and the early results are very promising. As wehighlight in this paper, spectral distortions (SDs) of the cosmic microwave background (CMB)uniquely probe GWs over six decades in frequency, bridging the gap between astrophysicalhigh- and cosmological low-frequency measurements. This means SDs will not only comple-ment other GW observations, but will be the sole probe of physical processes at certain scales.To illustrate this point, we explore the constraining power of various proposed SD missions ona number of phenomenological scenarios: early-universe phase transitions (PTs), GW produc-tion via the dynamics of SU(2) and ultra-light U(1) axions, and cosmic string (CS) networkcollapse. We highlight how some regions of parameter space were already excluded withdata from
COBE / FIRAS , taken over two decades ago. To facilitate the implementation of SDconstraints in arbitrary models we provide
GW2SD . This tool calculates the window function,which easily maps a GW spectrum to a SD amplitude, thus opening another portal for GWcosmology with SDs, with wide reaching implications for particle physics phenomenology.
Key words: cosmology: theory — gravitational waves — cosmic background radiation —spectral distortions.
Gravitational wave (GW) astronomy has become a reality. The nowroutine detection of compact object mergers by the LIGO / Virgocollaboration (Abbott et al. 2019) has made, for good reasons, thestudy of GWs one of the most active and current topics in cos-mology and astrophysics. Ongoing and planned observations of thetensor perturbation power spectrum currently span some 21 ordersof magnitudes of frequency: From cosmic microwave background(CMB) upper limits on primordial B-modes (Ade et al. 2018;Aghanim et al. 2020) measurements at the lowest frequencies, tointerferometry detections of GWs (e.g., Abbott et al. 2020b,a) andPulsar Timing Array (PTA) measurements (e.g., Perera et al. 2019;Alam et al. 2020) at higher frequencies. In the next few years, aplethora of experiments will test di ff erent scales between these ex-tremes (e.g., Campeti et al. 2020, for overview).Many physical processes can indeed lead to detectable tensorperturbations (see Caprini & Figueroa 2018, for review). These in-clude GWs from phase transitions (Caprini & Figueroa 2018; Nakaiet al. 2020), early universe gauge field production (Dimastrogio-vanni et al. 2017; Machado et al. 2019a,b), and cosmic string net- (cid:63) E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected] works (Buchmuller et al. 2019). Given these exciting theoreticaldevelopments, it is interesting to ask which cosmological and as-trophysical probes can help constrain these di ff erent scenarios. Inthis paper, we show that CMB spectral distortions (SDs) can pro-vide complementary information at frequencies f = − –10 − Hzunavailable to other probes. In this way, SDs o ff er a bridge betweenscales probed by next-generation CMB surveys (e.g., Ade et al.2019; Hazumi et al. 2019; Delabrouille et al. 2019), and astrophys-ical GW observatories such as current (e.g. Perera et al. 2019) andfuture (e.g. Weltman et al. 2020) PTA measurements.How do CMB SDs constrain tensor perturbations at the scalesthat they do? Spectral distortions are created by mechanisms thatlead to energy release into the photon-baryon fluid at redshifts z (cid:46) × , when thermalization processes cease to be e ffi -cient (Zeldovich & Sunyaev 1969; Sunyaev & Zeldovich 1970;Illarionov & Sunyaev 1975; Danese & de Zotti 1982; Buriganaet al. 1991; Hu & Silk 1993; Chluba & Sunyaev 2012). Manysources of distortions exist within standard Λ CDM cosmology aswell as scenarios invoking new physics (see Chluba et al. 2019b,for broad overview), and innovative experimental concepts (Kogutet al. 2016, 2019; Chluba et al. 2019a) have now reached criti-cal thresholds to significantly advance the long-standing distortionconstraints from
COBE / FIRAS (Mather et al. 1994; Fixsen et al.1996). A particular source of SDs is due to the dissipation of ten-sor modes while they travel almost unimpeded through the cosmicplasma (Ota et al. 2014; Chluba et al. 2015a). c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p Kite et al.
How do tensor perturbations distort the CMB spectrum? Ingeneral, perturbations in the photon fluid dissipate through electronscattering and free-streaming e ff ects. Dissipation of scalar pertur-bations provides one of the guaranteed sources of SDs in the earlyUniverse within the standard thermal history (e.g., Sunyaev & Zel-dovich 1970; Daly 1991; Hu et al. 1994; Chluba et al. 2012b,a).Similarly, tensor modes lose a small fraction of their energy by con-tinuously sourcing perturbations in the photon fluid which then alsodistort the CMB spectrum. In contrast to scalar modes, however, thedissipation is mainly mediated by free-streaming e ff ects. As shownin detail by Chluba et al. (2015a), this leads to dissipation of per-turbations over a vast range of scales, extending far beyond thoserelevant to scalar perturbations. Thus, although the tensor dissipa-tion rate is suppressed relative to scalar dissipation (tensor modesare not significantly damped by interactions with the photons), thisopens new avenues for model constraints from SDs.Building on Chluba et al. (2015a), we translate the relationsbetween µ -distortions and primordial tensor perturbation into quan-tities commonly used for GW searches. This makes it easier to com-pare SD limits to those from other probes. As examples we con-sider several inflationary models which source GWs beyond vac-uum fluctuations, early-universe phase transitions (PTs) and cos-mic string (CS) networks, all of which demonstrate how SD mea-surements are and will be important for excluding portions of theirrespective parameter spaces. Indeed we highlight that several ofthe widely discussed models could have already constrained someregions of their respective parameter spaces with SD limits from COBE / FIRAS , taken over a quarter-century ago. Future spectrom-eter concepts like
PIXIE (Kogut et al. 2016) and its enhanced ver-sions (e.g., PRISM Collaboration et al. 2014; Kogut et al. 2019)could, through their increased sensitivity, significantly increase therange of scales and parameter space covered, providing unique sci-entific opportunities for the next generation of cosmologists andparticle phenomenologists alike.
A GW can be represented as a transverse traceless tensor pertur-bation of the metric’s spatial component, h ij , and the energy den-sity it carries is ρ GW = (cid:104) h (cid:48) ij h (cid:48) ij (cid:105) / (32 π G ), where the prime denotesconformal time derivatives . If these GWs were produced primor-dially , we can define the GW fractional energy density per decadeof wavelengths as (e.g., Watanabe & Komatsu 2006) Ω GW ( k , η ) = ρ c ( η ) ∂ρ GW ( k , η ) ∂ ln k = P T ( k )12 a ( η ) H ( η ) (cid:2) T (cid:48) GW ( k , η ) (cid:3) , (1)where ρ c is the critical density, and in the second equality we fac-tored the primordial tensor power spectrum P T and the determinis-tic GW transfer function T GW .In Watanabe & Komatsu (2006), several analytical approxima-tions of the GW transfer function were developed. During radiationdomination (RD) we have (cid:2) T (cid:48) GW ( k , η ) (cid:3) ≈ k (cid:2) j ( k η ) (cid:3) . (2) We adopt the normalization conventions of Watanabe & Komatsu (2006). We consider the case of sub-horizon generation further on. whereas during matter domination (MD), one finds (cid:2) T (cid:48) GW ( k , η ) (cid:3) ≈ k η η (cid:2) A ( k ) j ( k η ) + B ( k ) y ( k η ) (cid:3) if k > k eq k (cid:104) j ( k η ) k η (cid:105) if k < k eq , A ( k ) = k η eq − cos(2 k η eq )2 k η eq + sin(2 k η eq )( k η eq ) , (3) B ( k ) = − + k η eq ) − cos(2 k η eq )( k η eq ) − sin(2 k η eq )2 k η eq . Here, k eq is the comoving wavenumber entering the horizon at thetime of matter-radiation equality η eq , and j (cid:96) and y (cid:96) are the sphericalBessel functions of first and second kind. For wavelengths muchsmaller than those entering the horizon today ( k η (cid:29)
1) we can ex-pand the GW transfer function derivatives at leading order in k . Ad-ditionally, since we always observe quantities that involve ( T (cid:48) GW ) integrated over some range of k , we can average over one period toobtain (e.g., Caprini & Figueroa 2018) (cid:104) (cid:2) T (cid:48) GW ( k , η ) (cid:3) (cid:105) k η (cid:29) ↓ ≈ η / η , (4)which is a smooth function of k valid during MD. Similarly, duringRD we can apply the same procedure to Eq. (2), and obtain (cid:104) (cid:2) T (cid:48) GW ( k , η ) (cid:3) (cid:105) ≈ / η . (5)For later use we point out that during RD, where Eq. (5) is valid, a ∝ η , while during MD relevant to Eq. (4) we have a ∝ η . To-gether with Eq. (1), this means that the GW energy density at agiven scale evolves as Ω GW ∝ a − H − ∝ const during RD and Ω GW ∝ a − H − ∝ (1 + z ) in the MD era.As pointed out in Watanabe & Komatsu (2006), the approxi-mations given above neglect some important details. One of theseis the process of neutrino damping, which has its greatest e ff ectson scales important to SD physics. The damping is e ff ective duringRD but only after neutrino decoupling ( T (cid:46) ff ect is a 35 .
6% decrease ofthe power available in GW (Weinberg 2004). To include this e ff ectthe transfer function given in Dicus & Repko (2005) is used: T (cid:48) GW = η (cid:88) n even a n (cid:2) n j n ( k η ) − k η j n + ( k η ) (cid:3) , (6)with the coe ffi cients a = a = . a = . × − and a = . × − . This is valid for the range of scales neededin the following section.Since gravitational wave upper limits are usually quoted asfunction of frequencies rather than wavelengths, we will use therelation k / Mpc − = . × f / Hz to change units. µ -DISTORTIONS FROM TENSOR PERTURBATIONS Much like scalar perturbations, tensor perturbations dissipate overtime, both transferring energy to neutrinos and (in smaller propor-tion) to photons. Primordial tensor perturbations entering the hori-zon during or slightly before the µ -era (5 × (cid:46) z (cid:46) × ),when dissipating, generate µ -distortions of the CMB that will beobservable today (Ota et al. 2014; Chluba et al. 2015a). c (cid:13) , 000–000 ridging the gap The average value of µ -distortions today is related to the pri-mordial tensor power spectrum via a window function W µ ( k ) (cid:104) µ GW (cid:105) ( η ) = (cid:90) d ln k W µ ( k ) P T ( k ) . (7)We calculate W µ ( k ) numerically according to Chluba et al. (2015a).The window function is shown in Fig. 1. In comparison to the cor-responding k -space window function of scalar perturbations (e.g.,Chluba et al. 2012a, 2015b), the dissipation e ffi ciency of tensors isabout five orders of magnitude smaller, highlighting how weaklytensor modes couple to the photon fluid.O ff setting this loss, we can see that tensor modes contributeto the generation of µ -distortions over a vast range of scales, with apower-law decay of contributions at k (cid:38) Mpc − (Fig. 1). This isin stark contrast to the dissipation of scalar perturbations, which arelimited to scales k (cid:39) − ,
000 Mpc − , with a strong exponentialdecay of contributions from k (cid:38) ,
000 Mpc − (e.g., Chluba et al.2012a). Scalar modes damp by photon di ff usion, which virtuallyerases all perturbations once the dissipation scale is crossed. Fortensors, the photon damping is minute and photon perturbationsare continuously sourced by the driving tensor force, explainingthis significant di ff erence (Chluba et al. 2015a). This makes SDs apotentially unique probe of GW backgrounds from early-universephysics. Equation (7) determines the SD signal from primordial perturba-tions that were created during inflation and only later enter the hori-zon to dissipate their energy. Another possibility is to have pertur-bations created on sub-horizon scales at later times. This requires ageneralisation of the window function formalism to account for thenew time dependence.An immediate di ff erence for sub-horizon injection is that neu-trino damping will not occur, as this only matters for GWs thatcross the horizon between neutrino decoupling and the start of MD.This means one can use the simpler versions of the transfer func-tion, valid in RD, given in Eqs. (2) and (5). The time dependence— which before was included in the physics underlying the win-dow function — has to be made more explicit. Using redshift z tobetter match Chluba et al. (2015a), Eq. (7) can be generalized to (cid:104) µ GW (cid:105) ( z = = (cid:90) ∞ d ln k (cid:90) ∞ d z W µ ( k , z ) P T ( k , z ) , (8)where we introduced the GW- µ -distortion window primitive W µ ( k , z ), which captures the physics behind the damping of GWs.Note that with P T ( k , z ) = P T ( k ) we recover Eq. (7) by defining (cid:82) ∞ W µ d z = W µ . The explicit form of the window primitive is W µ = . × H τ (cid:2) T (cid:48) GW ( k , z ) (cid:3) T Θ ( k , z ) e − Γ ∗ γ η J µ ( z ) , (9)and for convenience we summarize here the quantities which arerelevant to calculate the window function (for their derivation andfurther explanation we refer to Chluba et al. 2015a): ˙ τ is the timederivative of the Thomson optical depth. The terms T Θ e − Γ ∗ γ η containthe physics of how the GW transfer function T GW couples to thephoton fluid. These terms can be reliably approximated as T Θ ( k , z ) ≈ T Θ ( ξ ) ≈ + . ξ + ξ + . ξ + . ξ + ξ + ξ , (10a) e − Γ ∗ γ η ≈ , (10b)with ξ = k /τ (cid:48) . The final term J µ ( η ) is the energy branching ratio, − − k [Mpc − ]10 − − − − − − − − k − s p a ce w i nd o w f un c t i o n z max = 10 z max = 10 z max = 5 × z max = 9 × z max = 6 × Figure 1.
A series of curves demonstrating the form of the k -space windowfunction W z max µ for various upper limits in redshift. For practical purposes, z max = is equivalent to W ∞ µ ≡ W µ . The solid curve is the result forpower injection before the µ era, while the other curves su ff er from somereduced visibility at higher k . The faded line shows the results without neu-trino damping, leading to a (cid:39)
30% increase across the window function. which gives the fraction of total energy injected into the photonfluid that contributes to the µ distortion. We use the simple analyticapproximation of the branching ratio (‘method B’ in Chluba 2016): J µ ( z ) ≈ e − ( z / z th ) / for z > × , (11)with z th = . × denoting the redshift where thermalisationbecomes ine ffi cient (see also Hu & Silk 1993).We employ one further approximation in assuming the injec-tion happens instantaneously across all scales at a time η ∗ . There-fore, the tensor perturbations are uncorrelated ( P T ( k , η < η ∗ ) = η ∗ when their power spectrum abruptly jumps to some valuethat we will now determine. The spectrum is found at all times after η ∗ by redshifting Ω GW from the present-day value Ω GW ( k , η ) = a − ( η ) E ( η ) Ω GW ( k , η ) η > η ∗ η < η ∗ , E ( η ) ≡ H ( η ) H , (12)where we used a − / E ∝ ¯ ρ GW /ρ c . We will only consider powerinjection in the RD era, hence the tensor power spectrum is thenobtained using Eq. (1) together with Eq. (2), and reads P T ( k , η ) = H a k [ j ( k η )] Ω GW ( k , η ) η > η ∗ η < η ∗ (13)Notice that if not for the fact that the tensor perturbations appear at η ∗ , the power spectrum would always be time-independent: it is infact the equivalent of the primordial one in the “standard” scenariodescribed in Chluba et al. (2015a).Using that Eq. (13) is independent of time during RD and in-serting this into Eq. (8), we can remove the time dependence inthe integrand, leaving only changes in the upper limit of integra-tion. It is therefore su ffi cient to study a series of window functions W z max µ = (cid:82) z max W µ d z for di ff erent upper limits in time. Examples of W z max µ are shown in Fig. 1. We can observe that even modes origi-nating from z (cid:29) × contribute to the generation of distortions.The flat plateau of the window function at k (cid:39) . − Mpc − is nota ff ected until z max (cid:46) × , and will rapidly approach W z max µ (cid:39) z max approaches 5 × . c (cid:13)000
30% increase across the window function. which gives the fraction of total energy injected into the photonfluid that contributes to the µ distortion. We use the simple analyticapproximation of the branching ratio (‘method B’ in Chluba 2016): J µ ( z ) ≈ e − ( z / z th ) / for z > × , (11)with z th = . × denoting the redshift where thermalisationbecomes ine ffi cient (see also Hu & Silk 1993).We employ one further approximation in assuming the injec-tion happens instantaneously across all scales at a time η ∗ . There-fore, the tensor perturbations are uncorrelated ( P T ( k , η < η ∗ ) = η ∗ when their power spectrum abruptly jumps to some valuethat we will now determine. The spectrum is found at all times after η ∗ by redshifting Ω GW from the present-day value Ω GW ( k , η ) = a − ( η ) E ( η ) Ω GW ( k , η ) η > η ∗ η < η ∗ , E ( η ) ≡ H ( η ) H , (12)where we used a − / E ∝ ¯ ρ GW /ρ c . We will only consider powerinjection in the RD era, hence the tensor power spectrum is thenobtained using Eq. (1) together with Eq. (2), and reads P T ( k , η ) = H a k [ j ( k η )] Ω GW ( k , η ) η > η ∗ η < η ∗ (13)Notice that if not for the fact that the tensor perturbations appear at η ∗ , the power spectrum would always be time-independent: it is infact the equivalent of the primordial one in the “standard” scenariodescribed in Chluba et al. (2015a).Using that Eq. (13) is independent of time during RD and in-serting this into Eq. (8), we can remove the time dependence inthe integrand, leaving only changes in the upper limit of integra-tion. It is therefore su ffi cient to study a series of window functions W z max µ = (cid:82) z max W µ d z for di ff erent upper limits in time. Examples of W z max µ are shown in Fig. 1. We can observe that even modes origi-nating from z (cid:29) × contribute to the generation of distortions.The flat plateau of the window function at k (cid:39) . − Mpc − is nota ff ected until z max (cid:46) × , and will rapidly approach W z max µ (cid:39) z max approaches 5 × . c (cid:13)000 , 000–000 Kite et al.
For numerical applications, it is convenient to pre-tabulatethe tensor window function W z max µ across k and injection redshift z max . Since the background cosmology is fixed to high precision(Planck Collaboration et al. 2018), this procedure avoids additionalapproximations. However, a few comments are in place: we canfurther improve the treatment of the transition between the µ and y -distortion eras, which here we modelled as a step-function [seeEq. (11)]. Including the more gradual transition (e.g., see discus-sion in Chluba 2016), enhances the contributions from the largestscales ( k (cid:46) − Mpc − ), however, a more accurate treatment oftransfer e ff ects is also required and left to future work.With the procedure outlined in this section we can calcu-late the tensor dissipation contribution to the present day valueof µ -distortions. However, other processes, such as dissipation ofacoustic modes and Compton cooling also source µ -distortions,say a component µ other . Any non-detection of an enhanced level ofSD would straightforwardly constrain models that generate a large µ GW , comparable to or greater than µ other . However, things are moredelicate when µ GW becomes much smaller than the value of µ other expected in the standard cosmological model, µ other (cid:39) × − (e.g., Chluba 2016). In this regime, any actual analysis would re-quire a marginalization of other sources, that we do not take intoaccount here. However, assuming standard slow-roll inflation, wecan in principle accurately predict the expected standard contribu-tion given the power spectrum parameters measured at large angu-lar scales (Chluba et al. 2012b; Khatri & Sunyaev 2013; Chluba &Jeong 2014; Cabass et al. 2016; Chluba 2016). For simplicity, weshall thus assume perfect removal of other µ -contributions.Below we will consider the upper limit on µ -distortions set by COBE / FIRAS ( µ < × − PIXIE ( µ < × − )(Kogut et al. 2011), SuperPIXIE ( µ < . × − ) (Kogut et al.2019), Voyage 2050 ( µ < . × − ) and 10 × Voyage 2050 ( µ < . × − ) (Chluba et al. 2019b), all of which already account forthe presence of foregrounds following Abitbol et al. (2017). In this section, we calculate the constraining power of spectro-scopic CMB measurements in a minimally parametric fashion. Asin Campeti et al. (2020), we parametrize the primordial tensorpower spectrum using logarithmically spaced tophat functions cen-tered around some ln k i with ln k i + − ln k i = . ∀ i . This allow usan easy comparison with Fig. 8 of their paper: P T ( k ) = (cid:88) i A i W i ( k ) , (14a) W i ( k ) = k ∈ [ln k i − . , ln k i + . . (14b)Therefore, for each i , we insert Eq. (14a) into Eq. (7), and calcu-late the maximum value of A i that is compatible with the chosen (cid:104) µ GW (cid:105) ( η ) upper limit. With that information we then use Eq. (1) tocalculate the corresponding Ω GW constraint.In Fig. 2 we show the sensitivity curves for COBE / FIRAS , PIXIE , SuperPIXIE , Voyage 2050 and 10 × Voyage 2050 , whichall include estimated penalties from foregrounds. For comparison,we also report the sensitivity curves from Campeti et al. (2020), A simple interpolation routine to calculate W z max µ will be made availablehere: https://github.com/CMBSPEC/GW2SD.git which recently compiled the results of many planned experiments(Hazumi et al. 2019; Smith & Caldwell 2019; Sedda et al. 2019;Sesana et al. 2019; Kuroyanagi et al. 2015; Crowder & Cornish2005; El-Neaj et al. 2020; Reitze et al. 2019; Hild et al. 2011; Welt-man et al. 2020). Moreover, we show the NANOGrav 12.5 yearobservation (Arzoumanian et al. 2020), interpreted as GW stochas-tic background according to their 5 frequency power-law model.Since the extrapolation of a red spectra would be favourable for aSD detection, we conservatively assumed a flat spectrum.While the existing constraint derived from the COBE / FIRAS data is a few order of magnitude higher than other probes, nextgeneration satellites will start to bridge nicely the frequency gapexisting between CMB observation and direct GW detection. Inparticular, depending on the specific spectrometer configuration(e.g.,
PIXIE / SuperPIXIE ), future observations will be roughly onpar with those that SKA could provide using PTA (Weltman et al.2020), while extending to smaller scales. It is interesting to noticethat the upper bound from SDs will cover a very broad range of fre-quencies (more than 5 decades in f ). As such, any signal that is notsharply peaked in frequency will generate a comparatively higher µ -distortion, tightening the constraints on specific parametric mod-els, as we will see in the next section. In this section, we consider concrete models that generate GWsover a wide range of scales. For each of the following models itis enough to insert their corresponding tensor power spectrum intoEq. (7) or (8) to obtain the predicted µ -distortion, depending onwhether the injection is primordial or happens after reheating.Generally speaking, once accounting for the limits on r from Planck (Ade et al. 2018; Aghanim et al. 2020), we understandthat appreciable SDs can only be created by models with substan-tially enhanced tensor power at small scales. To also avoid fu-ture constraints at small scales, models with localized features at f = − − − Hz are most promising. In the context of PTs, forexample, this identifies low-scale dark or hidden sector transitionsat energies (cid:39)
10 MeV - 10 eV in the post-inflation era as a target.
As a benchmark we consider the tensor perturbations generated bysingle-field slow-roll inflation. This model predicts a very low, al-most scale invariant tensor spectrum, and as such we cannot expectSD constraints to be competitive with either CMB measurementsor future direct detections at small scales. We however include themodel for completeness, and as a point of comparison. The tensorspectrum from this model is given by P sf T = A T ( k / k ) n T , (15)where the amplitude of tensor and scalar perturbations A T and A S are related by the tensor to scalar ratio by r ≡ A T / A S , and n T = − r / Planck low- (cid:96) temperature and BICEP2 / Keck B -modesdata, (Ade et al. 2018; Aghanim et al. 2020) set the upper limit r . < .
06 (95%) at k = .
002 Mpc − . Upon noticing | n T | ≤ . ≈
0, one can approximate (cid:104) µ (cid:105) ( r ) ≈ . × − r whichgives the correct result to within ≤
5% for all values not ruled outby
Planck , while performing better for lower r . This shows that forany allowed value of r the SD signal will be out of reach for eventhe most sensitive SD mission concepts. c (cid:13) , 000–000 ridging the gap − − − − − − − − − − − − − − − − − − − f [Hz] − − − − − − − − − − − − − − − − − h Ω G W S u p e r P I X I E L i t e B I RD r = . i t e B I RD r = . SKA µ A r e s L I SA D E C I GO BB O AE D G E D O C on s e r v a t i v e D O O p t i m a l C E E T NANOGrav bin size F I R AS ( % C L ) P I X I EV o y age205010 × V o y age2050 − − − − k [Mpc − ] Figure 2.
Upper limits on the energy density of gravitational waves from measurements of µ -distortions for various experimental configurations ( COBE / FIRAS , PIXIE , SuperPIXIE , Voyage 2050 , 10 × Voyage 2050 ). For ease of comparison, we also report the upper limits for various other CMB, PTA and direct detectionexperiments (taken from Campeti et al. 2020), and the NANOGrav 12.5 years 95% confidence interval assuming a flat spectrum.
In principle this contribution is present as a component oftensor spectrum in the other models considered in the followingsections. However, since the amount of SD it generates is any-way negligible, we will omit it in the following. Note that the
Planck constraint on r will also be considered for other models.Strictly speaking, the aforementioned constraint only apply to apower-law tensor spectrum, a condition not necessarily met bythe models we will consider in the following. To provide somecontext to the SD constraints we will draw, we opt to employ anorder-of-magnitude estimate of the Planck constraint, simply re-quiring that any spectrum of tensor perturbations, P T ( k ), must sat-isfy P T ( k ) / P sf S ( k ) (cid:12)(cid:12)(cid:12) k = .
002 Mpc − < .
06. In principle, a proper analy-sis of the
Planck and BICEP2 / Keck data could be carried out to setconstraints on the models that will be discussed here. This, how-ever, goes beyond the scope of the paper.
LiteBIRD (Hazumi et al.2019), providing low multipole BB information at much higher pre-cision, will allow us to further improve the limits set by Planck onthe same range of scales in the near future.
Many inflationary models require the dynamics of additional spec-tator fields active during the inflationary period, itself driven bya separate scalar field. Generally speaking, the dynamics of thespectator field generate tensor perturbations in addition to thoseproduced by the vacuum fluctuations of the quasi de Sitter back-ground. In this section, following Campeti et al. (2020), we con-sider an axion-SU(2) spectator field based on the “chromo-natural”inflation model (Adshead & Wyman 2012; Dimastrogiovanni et al.2017). Here, the SU(2) gauge fields acquire an expectation value,the fluctuations around which include a tensor perturbation witha bilinear coupling to the graviton. The dynamics of the spectatorSU(2) axion are such that gravitons of a particular helicity are am-plified via a transient tachyonic instability, resulting in a (circularly polarized) contribution to the tensor power spectrum (see Thorneet al. 2018) P SU(2) T ( k ) = r ∗ P sf R ( k ) exp (cid:34) − σ ln (cid:32) kk p (cid:33)(cid:35) , (16)which relates to the spectrum of scalar perturbations P sf R = A S ( k / k ) n S − . (17)In order to constrain this model, we use the best-fit Planck param-eters (Planck Collaboration et al. 2018) for Eq. (17). However, r ∗ , k p and σ are related to the parameters of the gauge theory and areessentially free to vary here. We take as a reasonable set of values,those given in Campeti et al. (2020) (their model AX3) ( r ∗ , k p , σ ) = (50 , Mpc − , . µ = . × − . Entertain-ing the question as to which set of parameters would maximize the µ distortion signal while satisfying both observational and modelconstraints we find ( r ∗ , k p , σ ) = (265 , . × Mpc − , . µ = . × − . This result can be understood byconsidering the parabolic shape of the spectrum in log P -log k space,which due to model constraints cannot peak to sharply (e.g. seeEqs. (A8) and (A11) in Thorne et al. 2018). This means that a spec-trum which avoids the Planck constraints cannot simultaneouslypeak too high in the SD regime. In contrast, the models consideredin the following subsections next have spectra resembling brokenpower laws, and can be much more e ff ective in satisfying currentconstraints while simultaneously generating significant SDs. In this subsection we consider an axion model proposed inMachado et al. (2019a,b). In this scenario (generic in the contextof string compactifications), one has the presence of one or moreU(1) axions with mass m and decay constant f φ that couple to darksector photons. At early times during radiation domination, when c (cid:13)000
Many inflationary models require the dynamics of additional spec-tator fields active during the inflationary period, itself driven bya separate scalar field. Generally speaking, the dynamics of thespectator field generate tensor perturbations in addition to thoseproduced by the vacuum fluctuations of the quasi de Sitter back-ground. In this section, following Campeti et al. (2020), we con-sider an axion-SU(2) spectator field based on the “chromo-natural”inflation model (Adshead & Wyman 2012; Dimastrogiovanni et al.2017). Here, the SU(2) gauge fields acquire an expectation value,the fluctuations around which include a tensor perturbation witha bilinear coupling to the graviton. The dynamics of the spectatorSU(2) axion are such that gravitons of a particular helicity are am-plified via a transient tachyonic instability, resulting in a (circularly polarized) contribution to the tensor power spectrum (see Thorneet al. 2018) P SU(2) T ( k ) = r ∗ P sf R ( k ) exp (cid:34) − σ ln (cid:32) kk p (cid:33)(cid:35) , (16)which relates to the spectrum of scalar perturbations P sf R = A S ( k / k ) n S − . (17)In order to constrain this model, we use the best-fit Planck param-eters (Planck Collaboration et al. 2018) for Eq. (17). However, r ∗ , k p and σ are related to the parameters of the gauge theory and areessentially free to vary here. We take as a reasonable set of values,those given in Campeti et al. (2020) (their model AX3) ( r ∗ , k p , σ ) = (50 , Mpc − , . µ = . × − . Entertain-ing the question as to which set of parameters would maximize the µ distortion signal while satisfying both observational and modelconstraints we find ( r ∗ , k p , σ ) = (265 , . × Mpc − , . µ = . × − . This result can be understood byconsidering the parabolic shape of the spectrum in log P -log k space,which due to model constraints cannot peak to sharply (e.g. seeEqs. (A8) and (A11) in Thorne et al. 2018). This means that a spec-trum which avoids the Planck constraints cannot simultaneouslypeak too high in the SD regime. In contrast, the models consideredin the following subsections next have spectra resembling brokenpower laws, and can be much more e ff ective in satisfying currentconstraints while simultaneously generating significant SDs. In this subsection we consider an axion model proposed inMachado et al. (2019a,b). In this scenario (generic in the contextof string compactifications), one has the presence of one or moreU(1) axions with mass m and decay constant f φ that couple to darksector photons. At early times during radiation domination, when c (cid:13)000 , 000–000 Kite et al.
Figure 3.
A contour plot showing the expected SD signal arising from dif-ferent combinations of f φ and m in the U(1) model. Without loss of general-ity, fiducial values of α =
60 and θ = ff erent probes are most sensitive(from left to right: SD, PTA and Interferometry). Dotted lines continuing theSD mission contours show the estimates ignoring late time injection. the Hubble parameter H is greater than m , the axion field is over-damped and is frozen. Once H (cid:46) m , corresponding to the tempera-ture T ≈ (cid:112) mM pl , the axion starts to oscillate around the minimumof its potential, sourcing gauge field production of a particular he-licity that goes on to generate GWs. Since these GWs are only pro-duced on sub-horizon scales after the axion starts oscillating, theresults of Sect. 3.1 are essential in finding the µ signal accurately.This model is of particular interest to us as it produces a nar-rower spectrum of GWs. Thus, to constrain its parameter space itis important to have probes that can cover all phenomenologicallyrelevant frequencies. The GWs produced can be parametrized as aspectrum of the form Ω U(1)GW ( k ) = . Ω U(1)GW ( f AA ) (cid:16) k / ˜ k (cid:17) . + (cid:16) k / ˜ k (cid:17) . exp (cid:104) . (cid:16) k / ˜ k + (cid:17)(cid:105) , (18a)with ˜ k = . × (cid:2) f AA / Hz (cid:3) Mpc − . (18b)Here Ω GW ( f AA ) and f AA are a function of the free parametersof the model. These parameters, as introduced in Machado et al.(2019a,b), are f φ , m , α and θ , relating to the fit parameters in Eq.(18a) via f AA ≈ × − Hz (cid:20) αθ (cid:21) / (cid:20) m (cid:21) / , (19a) Ω U(1)GW ( f AA ) ≈ . × − g − / ρ, ∗ (cid:34) f φ M pl (cid:35) (cid:34) θ α (cid:35) / , (19b)which have both been redshifted to their present-day values. Thefirst two free parameters (i.e., f φ and m ) essentially dictate theheight and frequency of the peak in the power spectrum respec-tively. The second two parameters are limited to α ∼ −
100 and θ ∼ O (1), and do not significantly change the shape of the spectrumfor the range of allowed values. These parameters are therefore de-generate with the first two. We choose fiducial values of α = θ =
1, but the main results given here hold more generally. The direct dependence of f peak on m means that di ff erent typesof experiment will probe di ff erent mass scales. This is shown Fig. 3,where vertical dotted lines distinguish where di ff erent detectionmethods are dominant. From here it can be seen that SD are sen-sitive to the ultralight limit of the U (1) audible axion model, a re-sult which again holds for any valid combination of α and θ . Notethat Planck extends the limits from
COBE / FIRAS at low masses tosmaller values of f φ . Future SDs measurements could significantlyimprove the limits from Planck to higher masses, covering a widerrange of the parameter space of phenomenological interest.We note in particular how SDs can constrain masses in a rangenot accessible to other measurements (10 − − − eV). Such ultra-light axions may be ubiquitous in particular string compactifica-tions (Arvanitaki et al. 2010), and moreover, could be a viable darkmatter candidate were they to form a condensate at late times (Huiet al. 2017; Marsh 2016), further illustrating the utility of SDs forparticle phenomenology. The post-inflationary epoch may have seen a variety of first or-der phase transitions (PTs) in theories that go beyond the stan-dard model of particle physics. First order PTs are characterizedby the fact that latent energy is released, and phases of true vacuumnucleate within false vacuum domains, resulting in bubble colli-sions (BC) that generate a stochastic GW background. Moreover,magneto-hydrodynamic (MHD) turbulence and sound waves (SW)in the bulk plasma during and after the phase transition also sourcesub-horizon GWs at commensurate frequencies. If these processestake place during the µ -era or shortly before, they can potentiallyresult in measurable SDs. Here we once again use the results ofSect. 3.1 to calculate the associated SDs.Referring to the review of Caprini & Figueroa (2018), we seethat the spectra resulting from the three di ff erent mechanisms forGW production from PTs are given by h Ω BCGW ( f ) = . × − (cid:32) H ∗ β (cid:33) (cid:18) κ BC α + α (cid:19) (cid:32) g ∗ ( T ∗ ) (cid:33) (20a) × (cid:32) . (cid:51) w . + (cid:51) w (cid:33) . f / f BC ) . + . f / f BC ) . , h Ω SWGW ( f ) = . × − (cid:32) H ∗ β (cid:33) (cid:18) κ v α + α (cid:19) (cid:32) g ∗ ( T ∗ ) (cid:33) (20b) × (cid:51) w (cid:32) ff SW (cid:33) (cid:32) + f / f SW ) (cid:33) , h Ω MHDGW ( f ) = . × − (cid:32) H ∗ β (cid:33) (cid:18) κ MHD α + α (cid:19) (cid:32) g ∗ ( T ∗ ) (cid:33) (20c) × (cid:51) w ( f / f MHD ) (cid:2) + ( f / f MHD ) (cid:3) (1 + π f / h ∗ ) , with peak frequencies χ = (cid:34) β H ∗ (cid:35) (cid:20) T ∗ (cid:21) (cid:34) g ∗ ( T ∗ )100 (cid:35) , (20d) f BC = . × − Hz (cid:32) . . − . (cid:51) w + (cid:51) w (cid:33) χ , (20e) f SW = . × − Hz (cid:51) − w χ , (20f) f MHD = . × − Hz (cid:51) − w χ . (20g)Here, the three principal model parameters are α , β and (cid:51) w , which c (cid:13) , 000–000 ridging the gap Figure 4.
A series of contour plots showing the expected SD signal arising from low scale first order phase transitions. Dotted lines (visible on the left) give thesensitivity with standard window function, W µ ( k ), showing that late power injection leads to a decrease of less than an order of magnitude for 10 < z PT < .The limits from Planck are also shown for comparison. The temperature at the time of the PT can be found with T PT / MeV ≈ × − z PT . fix the relative energy content of the gauge field, inverse time dura-tion of the PT, and velocity of bubble walls respectively. Denotedwith ∗ are quantities at the time of the PT, making another keyparameter z PT . The first two parameters follow 0 ≤ α ≤ β/ H ∗ >
1. The velocity of sound waves has been set to unity, sincebubble walls usually propagate close to the speed of light. Param-eters labelled κ i ∈ [0 ,
1] give the weighted contribution from eachmechanism. For this work we have used κ BC = κ MHD = κ v and κ v ≈ α . + . √ α + α , (21)the last of which is valid for (cid:51) w (cid:39)
1. The expected SD limits on PTsgiven these considerations are shown in Fig. 4. Even for low-energyPTs ( α = .
1) a
PIXIE -like mission would explore some of theparameter space not already excluded by
Planck ; however, it wouldonly see rather long PT. In the more energetic cases ( α ≥ . Another tell tale sign of physics beyond the Standard Model is theexistence of topological defects. Excluding textures, the standardmodel does not allow for any defects. However, larger gauge sym-metries (ubiquitous in models that go beyond the Standard Model)could admit symmetry breaking patterns that generate topologicaldefects in the early Universe (see Kibble 1980, 1982) which couldhave persisted into cosmologically observable epochs. Althoughthe simplest models of monopoles and domain walls are tightlyconstrained (see Sects. 13.5.3 and 14.3.3 in Vilenkin & Shellard1994), cosmic strings networks remain a theoretical possibility andcan impart potentially observable GW signals (see Sect. 10.4 ofVilenkin & Shellard 1994).As an example, we consider a model proposed by Buchmulleret al. (2019) which attempts leptogenesis within an SO(10) grandunified theory via a U(1) B − L phase transition, where a local U(1)baryon minus lepton number symmetry is spontaneously broken. The result of the B − L transition will be a meta-stable CS networkgenerated at the time of the transition, which over the course of thecollapse generates a mostly flat spectrum of GWs due to the decayof string loops. An approximate form of their spectrum is given interms of the model parameters κ and G µ as h Ω CSG W = h Ω plateauGW min (cid:104) ( f / f ∗ ) / , (cid:105) , (22a) f ∗ = × Hz e − πκ/ (cid:20) G µ − (cid:21) − / , (22b) h Ω plateauGW = . Ω r h (cid:20) G µ Γ (cid:21) / . (22c)Buchmuller et al. (2019) give a value of Γ ≈
50 for this particularmodel, and we use a value of Ω r h = . × − .In reality string network collapse would be a function of time,but to match the formalism outlined in Sect. 3.1 we conservativelyassume the entire spectrum emerges at the final moment of collapsegiven by Buchmuller et al. (2019) z collapse = (cid:32) H (cid:33) / (cid:32) Γ ( G µ ) π G e − πκ (cid:33) / . (23)The spectrum grows ∝ f / up to f ∗ , and is flat for higher fre-quencies. Furthermore, f ∗ only depends weakly on G µ but variessignificantly with κ . This means that once κ is large enough that thespectrum is flat across the entire window of visibility for a givenexperiment, the probe will only be sensitive to G µ . With SD mis-sions probing lower frequencies than astrophysical probes they willbe complementary in limiting the lower bounds of the κ parameter.The potential of SD missions for constraining this model is shownin Fig. 5. It is again noteworthy that COBE / FIRAS placed early lim-its on the energy scale of this class of SO(10) GUT models thatproduced CS networks.Given that the GW spectra produced by CS network collapsehas a plateau at smaller scales, for any given sensitivity depicted inFig. 2, we see that any one of the probes depicted will be equallygood at detecting the GW background produced. However, we notethe decade window in which a
Voyage 2050 -type observation candetect this signal beyond the forecast sensitivities of PTAs. It is Not to be confused with the SD amplitude µ . The combination G µ willalways be in reference to the energy scale of the CS physics.c (cid:13)000
Voyage 2050 -type observation candetect this signal beyond the forecast sensitivities of PTAs. It is Not to be confused with the SD amplitude µ . The combination G µ willalways be in reference to the energy scale of the CS physics.c (cid:13)000 , 000–000 Kite et al.
Figure 5.
A contour plot showing the expected µ signal from a CS networkarising from a U(1) B − L phase transition at the GUT scale. The limit placedby COBE / FIRAS is shown in red, and similarly for
Planck in orange. Dashedcontours show the sensitivity of various proposed SD missions. Faint dottedlines show the contours without using the reduced window function. also worth noting that the type of spectrum considered here willhold more generally for a wide range of CS models (see Figueroaet al. 2020). We also note that CS networks can directly produceSDs from dissipation of induced scalar perturbations (e.g., Tashiroet al. 2013), however, with sensitivity in a di ff erent range of scales. Highly energetic events in the early Universe, either during in-flation or subsequently during radiation domination, can injectpower into the GW spectrum. This can include GWs from sourceswithin the standard Λ CDM cosmology, or from models that invokephysics beyond it. Detecting the GW spectrum is therefore key tofurther scrutinizing our current paradigm, as well as pushing ourknowledge of the early Universe to new and exciting areas. Futureexperiments will probe these stochastic backgrounds, each sensi-tive to a range of frequencies / wavelengths dictated by the natureof the experiment. As we have highlighted here, a wide range ofGW frequencies ( f = − –10 − Hz) can only be probed by SDobservations. This large span of wavelengths compensates for therelatively low e ffi ciency of generating SDs from GWs, thus makingthem a potentially powerful probe of physics beyond the StandardModels of both particle physics and cosmology.This work aims to introduce SDs as a complimentary probethrough which one can detect and constrain stochastic GW back-grounds. The fundamental element to link these two messengersis the k -space window function, which maps a given GW spec-trum into a terminal SD signal imprinted before last scattering [seeEqs. 7 and 8]. In order to study the injection of power on sub-horizon scales, the window function for primordial tensor pertur-bations has been generalised [see Eq. 8], leading to minor changesin some models (Fig. 3) but large changes in others (Fig. 5). Thisis essentially related to the fact that GWs have less cosmic historyto dissipate their energy to the photon-baryon plasma. A simplepython tool is provided at GW2SD and allows one to easily esti-mate SD limits on various models, given the tensor power spec-trum, P T ( k , z ), that comes into existence at a single redshift z . This https://github.com/CMBSPEC/GW2SD.git is certainly a good approximation for 1’st order phase transitions,and holds to a good approximation for scenarios that dynamicallygenerate GWs over a short duration. Refinements to account for theexact time-dependence of the process are left to future work.To illustrate the utility of SDs for GW cosmology, a series ofphenomenological models were discussed, and their resulting SDsignals studied: As expected, the tensor perturbations generated bysingle-field slow-roll inflation are too weak to be measured withSDs (Ota et al. 2014; Chluba et al. 2015a). Spectator axion-SU(2)fields too, even in more favourable cases that we considered, willrealistically be out of reach in the foreseeable future. The Audibleaxion model (Sect. 5.3) on the other hand, can have a large regionof its parameter space constrained by SDs, particularly for a widerange of masses in the ultra-light regime (Fig. 3). Similarly, theGWs from low scale (10 eV - 10 MeV) dark sector phase transitionsin the early Universe will be visible with future SD missions ifthe relative energy content of the participating field is su ffi cientlylarge, and the duration su ffi ciently long (see Sect. 5.4 and Fig. 4).The typically flat GW spectra produced by CS networks can beseen by many instruments, but SDs will be complementary to otherprobes in being sensitive to string collapse especially in the µ -era.It is noteworthy that all the aforementioned models were alreadyconstrained with COBE / FIRAS long before first limits from
Planck existed. Future CMB spectrometers like
SuperPIXIE (Kogut et al.2019) could establish a new frontier in this respect.Moving forward, while here we focused on the GW-inducedSDs, the limits could be made even more compelling by consider-ing the signals arising from both the tensor and scalar perturbationsthat energetic events could produce (e.g., Tashiro et al. 2013; Amin& Grin 2014, for SDs from scalar perturbations of CS and PTs,respectively). This potential for combining sources is another ad-vantage SD experiments have over GW-based experiments, sincethe latter are only sensitive to the direct tensor perturbations. Hereit is again important to highlight that SDs from tensor perturba-tions cover a wider range of physical scales than SDs from scalarsources, thus extending the reach of SDs to earlier epochs. In addi-tion, some scenarios do not produce any significant scalar perturba-tions [e.g., the axion-SU(2) model], making it crucial to account forSDs caused by tensor perturbations. Overall, SDs uniquely probethe presence of small-scale perturbations in regimes that are notdirectly accessible, thus highlighting the important role that futureCMB spectrometers could play in GW cosmology, and, by exten-sion, beyond the Standard Model phenomenology.
ACKNOWLEDGMENTS
This work was supported by the ERC Consolidator Grant
CMBSPEC (No. 725456) as part of the European Union’s Horizon 2020 researchand innovation program. TK was further supported by STFC grantST / T506291 /
1. JC was also supported by the Royal Society as a Royal So-ciety URF at the University of Manchester, UK. We also acknowledge useof
WebPlotDigitizer (Rohatgi 2020) for data compiled in Fig. 2.
REFERENCES
Abbott B., et al., 2019, Phys. Rev. X, 9, 031040Abbott B. P., et al., 2020a, The Astrophysical Journal, 892, L3Abbott R., et al., 2020b, The Astrophysical Journal, 896, L44Abitbol M. H., Chluba J., Hill J. C., Johnson B. R., 2017, Monthly Noticesof the Royal Astronomical Society, 471, 11261140Ade P., et al., 2018, Phys. Rev. Lett., 121, 221301c (cid:13) , 000–000 ridging the gap Ade P., et al., 2019, JCAP, 02, 056Adshead P., Wyman M., 2012, Phys. Rev. Lett., 108, 261302Aghanim N., et al., 2020, Astron. Astrophys., 641, A6Alam M. F. et al., 2020, arXiv e-prints, arXiv:2005.06490Amin M. A., Grin D., 2014, Physical Review D, 90, 083529Arvanitaki A., Dimopoulos S., Dubovsky S., Kaloper N., March-RussellJ., 2010, Phys. Rev. D, 81, 123530Arzoumanian Z., et al., 2020Buchmuller W., Domcke V., Murayama H., Schmitz K., 2019, arXiv e-prints, arXiv:1912.03695Burigana C., Danese L., de Zotti G., 1991, Astronomy & Astrophysics,246, 49Cabass G., Melchiorri A., Pajer E., 2016, Physical Review D, 93, 083515Campeti P., Komatsu E., Poletti D., Baccigalupi C., 2020Caprini C., Figueroa D. G., 2018, Class. Quant. Grav., 35, 163001Chluba J., 2016, Monthly Notices of the Royal Astronomical Society, 460,227Chluba J. et al., 2019a, arXiv e-prints, arXiv:1909.01593Chluba J., Dai L., Grin D., Amin M. A., Kamionkowski M., 2015a,Monthly Notices of the Royal Astronomical Society, 446, 2871Chluba J., Erickcek A. L., Ben-Dayan I., 2012a, The Astrophysical Jour-nal, 758, 76Chluba J., Hamann J., Patil S. P., 2015b, International Journal of ModernPhysics D, 24, 1530023Chluba J., Jeong D., 2014, Monthly Notices of the Royal AstronomicalSociety, 438, 2065Chluba J., Khatri R., Sunyaev R. A., 2012b, Monthly Notices of the RoyalAstronomical Society, 425, 1129Chluba J. et al., 2019b, BAAS, 51, 184Chluba J., Sunyaev R. A., 2012, Monthly Notices of the Royal Astronom-ical Society, 419, 1294Crowder J., Cornish N. J., 2005, Phys. Rev. D, 72, 083005Daly R. A., 1991, The Astrophysical Journal, 371, 14Danese L., de Zotti G., 1982, Astronomy & Astrophysics, 107, 39Delabrouille J., et al., 2019Dicus D. A., Repko W. W., 2005, Physical Review D, 72, 088302Dimastrogiovanni E., Fasiello M., Fujita T., 2017, JCAP, 01, 019El-Neaj Y. A., et al., 2020, EPJ Quant. Technol., 7, 6Figueroa D. G., Hindmarsh M., Lizarraga J., Urrestilla J., 2020, arXiv e-prints, arXiv:2007.03337Fixsen D. J., Cheng E. S., Gales J. M., Mather J. C., Shafer R. A., WrightE. L., 1996, The Astrophysical Journal, 473, 576Hazumi M., et al., 2019, J. Low Temp. Phys., 194, 443Hild S., et al., 2011, Class. Quant. Grav., 28, 094013Hu W., Scott D., Silk J., 1994, The Astrophysical Journal, 430, L5Hu W., Silk J., 1993, Physical Review D, 48, 485Hui L., Ostriker J. P., Tremaine S., Witten E., 2017, Phys. Rev. D, 95,043541Illarionov A. F., Sunyaev R. A., 1975, Soviet Astronomy, 18, 413Khatri R., Sunyaev R. A., 2013, Journal of Cosmology and AstroparticlePhysics, 6, 26Kibble T. W. B., 1980, Physics Reports, 67, 183Kibble T. W. B., 1982, Acta Physica Polonica B, 13, 723Kogut A., Abitbol M. H., Chluba J., Delabrouille J., Fixsen D., Hill J. C.,Patil S. P., Rotti A., 2019, in BAAS, Vol. 51, p. 113Kogut A., Chluba J., Fixsen D. J., Meyer S., Spergel D., 2016, inProc.SPIE, Vol. 9904, SPIE Conference Series, p. 99040WKogut A., Fixsen D., Chuss D., Dotson J., Dwek E., et al., 2011, JCAP,1107, 025Kuroyanagi S., Nakayama K., Yokoyama J., 2015, PTEP, 2015, 013E02Lyth D. H., Riotto A., 1999, Phys. Rept., 314, 1Machado C. S., Ratzinger W., Schwaller P., Stefanek B. A., 2019a, JHEP,01, 053Machado C. S., Ratzinger W., Schwaller P., Stefanek B. A., 2019bMarsh D. J. E., 2016, Phys. Rept., 643, 1Mather J. C. et al., 1994, The Astrophysical Journal, 420, 439Nakai Y., Suzuki M., Takahashi F., Yamada M., 2020, arXiv e-prints,arXiv:2009.09754 Ota A., Takahashi T., Tashiro H., Yamaguchi M., 2014, Journal of Cos-mology and Astroparticle Physics, 10, 29Perera B., et al., 2019, Mon. Not. Roy. Astron. Soc., 490, 4666Planck Collaboration et al., 2018, ArXiv:1807.06209PRISM Collaboration et al., 2014, Journal of Cosmology and Astroparti-cle Physics, 2, 6Reitze D., et al., 2019, Bull. Am. Astron. Soc., 51, 035Rohatgi A., 2020, WebPlotDigitizerSedda M. A., et al., 2019Sesana A., et al., 2019Smith T. L., Caldwell R., 2019, Phys. Rev. D, 100, 104055Sunyaev R. A., Zeldovich Y. B., 1970, Astrophysics and Space Science,9, 368Sunyaev R. A., Zeldovich Ya. B., 1970, Astrophys. Space Sci., 7, 20Tashiro H., Sabancilar E., Vachaspati T., 2013, Journal of Cosmology andAstroparticle Physics, 8, 35Thorne B., Fujita T., Hazumi M., Katayama N., Komatsu E., Shiraishi M.,2018, Phys. Rev. D, 97, 043506Vilenkin A., Shellard E. P. S., 1994, Cosmic strings and other topologicaldefectsWatanabe Y., Komatsu E., 2006, Phys. Rev. D, 73, 123515Weinberg S., 2004, Physical Review D, 69, 023503Weltman A., et al., 2020, Publ. Astron. Soc. Austral., 37, e002Zeldovich Ya. B., Sunyaev R. A., 1969, Astrophys. Space Sci., 4, 301c (cid:13)000