LIGO as a probe of Dark Sectors
UUCI-HEP-TR-2021-06
LIGO as a probe of Dark Sectors
Fei Huang a,b , ∗ Veronica Sanz c,d , † Jing Shu a,e,f,g,h,i , ‡ and Xiao Xue a,e § a CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China b Department of Physics and Astronomy, University of California, Irvine, CA 92697 USA c Instituto de F´ısica Corpuscular (IFIC), Universidad de Valencia-CSIC, E-46980 Valencia, Spain d Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK e School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China f CAS Center for Excellence in Particle Physics, Beijing 100049, China g Center for High Energy Physics, Peking University, Beijing 100871, China h School of Fundamental Physics and Mathematical Sciences,Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China and i International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China
We show how current LIGO data is able to probe interesting theories beyond the StandardModel, particularly Dark Sectors where a Dark Higgs triggers symmetry breaking via a first-orderphase transition. We use publicly available LIGO O2 data to illustrate how these sectors, even ifdisconnected from the Standard Model, can be probed by Gravitational Wave detectors. We linkthe LIGO measurements with the model content and mass scale of the Dark Sector, finding thatcurrent O2 data is testing a broad set of scenarios where the breaking of SU ( N ) theories with N f fermions is triggered by a Dark Higgs at scales Λ (cid:39) − GeV with reasonable parameters forthe scalar potential.
I. INTRODUCTION
Much of the Universe is dark, and many theories havebeen built trying to explain it. Our hopes for prob-ing these theories often rely on their possible connec-tion to regular matter via some form of non-gravitationalinteraction For example, direct searches for Dark Mat-ter hinges on some sort of coupling to nucleons or elec-trons, and constraints on those couplings usually assumea mechanism of communication between the Dark Sectorand the rest of the Universe which establishes some formof tracking between these two sectors.The first observation of Gravitational Waves (GW) bythe LIGO and Virgo collaborations [1–3] in 2015 initiateda new way to see the Universe, and since then excitingnew observations have provided information about as-trophysical objects like Black Holes [4, 5]. However, thephysics reach for LIGO is not circumscribed to detectionof mergers [6–9]. The detection, or the lack of, a stochas-tic GW background allows us to explore interesting, non-standard sectors. We will explain how, with the currentpublic data from LIGO, one can probe plausible DarkSector scenarios, regardless of their non-gravitational in-teraction with visible matter.These Dark Sectors could resemble Standard-Modeldynamics, with new forces, Dark Higgses and statescharged under them. Influenced by thermal contribu-tions from the degrees of freedom in the Dark Sector, the ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] thermal history of the Dark Higgs could then lead to first-order phase transitions. Many studies have been devotedto the prospects that future interferometers could offerto explore Dark Sectors, e.g. , Ref [10]. In this paper weexplore the possibilities that LIGO and its current pub-lic dataset present, and bridge the gap between genericstudies of thermal parameters, e.g. , Ref [11], and specificparticle-physics models.The paper is structured as follows. In Sec. II we firstdescribe the analysis of GWs from first-order phase tran-sitions, then discuss in Sec. III the connection betweenthe phase transition thermal parameters with classes ofparticle-physics models, especially of SU ( N ) /SU ( N − II. GRAVITATIONAL WAVES FROM PHASETRANSITIONS
The stochastic gravitational wave background(SGWB) is often considered as an isotropic, unpolar-ized, stationary and Gaussian background generatedby a large number of unresolved gravitational-wavesources [6–9]. Its power spectrum is characterized bythe dimensionless quantityΩ GW ( f ) ≡ ρ crit dρ GW d ln f , (1)where ρ GW is the energy density of the stochastic gravi-tational wave background, f is the frequency of the GW a r X i v : . [ h e p - ph ] F e b f [Hz] − − − − − S s w ( f ) f pk = 5 Hz f pk = 33 Hz f pk = 224 Hz f pk = 1495 Hz f pk = 10000 Hz ∼ f − ∼ f FIG. 1. The coloured curves show the spectral shape S sw ( f )by varying the peak frequency f sw , whereas the dashed anddotted black lines indicate its asymptotic behaviour. Thetwo solid vertical lines at 20 and 1726 Hz show the minimumand maximum frequency considered when obtaining the GWupper limits using LIGO data. and ρ crit ≡ c H πG (2)is the critical energy density of the universe today.In principle, the total SGWB is a superposition of allpossible astrophysical and cosmological sources. How-ever, we can obtain a conservative upper limit for theSGWB from phase transitions that occur in the earlyuniverse by assuming that phase transition dynamics isthe main source of SGWB.The SGWB generated from phase transitions in theearly universe consists of three parts [12–17]:Ω GW = Ω col + Ω sw + Ω turb , (3)in which the three terms on the right hand side corre-spond to the contribution from bubble collisions, soundwaves in the fluid and the turbulence, respectively. Forsimplicity, we shall assume in this work that contribu-tions from sound waves are always dominant. We em-phasize that this is typically the case for models in whichgauge bosons acquire masses during the phase transition[18]. However, the analysis we present can be easily gen-eralized to cases in which other types of contribution be-come more important.The phase transition is in general characterized by justa few parameters: the velocity of the bubble wall v w , theratio of the free energy density difference between thetrue and false vacuum and the total energy density, ξ ,the speed of the phase transition β/H , and the nucleationtemperature T N . With these parameters, the GW powerspectrum can be expressed as [19]Ω GW h (cid:39) . × − (cid:16) g ∗ (cid:17) / Γ ¯ U f (cid:18) βH (cid:19) − v w S sw ( f ) , (4) f pk [Hz] − − − − − − h Ω up . li m G W ( f r e f ) f ref = 25Hz f ref = f pk BBN bound at f pk FIG. 2. The upper limit of Ω ref for each different f pk usingthe 2 σ criterion. The curve with triangles corresponds to theupper limit at 25 Hz, while the curve with circles uses theupper limit at f pk . The solid horizontal line corresponds tothe BBN bound at f pk . where g ∗ is the effective number of relativistic degreesof freedom at the time of the transition, Γ ∼ / U f ∼ (3 / κ f ξ is the root-mean-squarefluid velocity with the efficiency parameter given by theapproximate expressions [15] κ f ∼ ξ .
73 + 0 . √ ξ + ξ v w → ξ / .
017 + (0 .
997 + ξ ) / v w ≈ . , (5)The spectral shape S sw is given by S sw ( f ) = (cid:18) ff pk (cid:19) (cid:32)
74 + 3 ( f /f pk ) (cid:33) / (6)with the peak frequency f pk = 8 . × − Hz (cid:18) v w (cid:19) (cid:18) βH (cid:19) (cid:18) T N GeV (cid:19) (cid:16) g ∗ (cid:17) / . (7)The shape of S sw is shown in FIG. 1, noting that S sw isequal to Ω GW ( f ) / Ω GW ( f pk ). In this figure one observesthat varying the peak frequency f sw amounts to simplyshifting the spectrum horizontally. Also note that theasymptotic behavior of S sw goes as ∼ f for f (cid:28) f sw (dotted line), whereas one expects a behaviour ∼ f − for f (cid:29) f sw (dashed line).The behavior of S sw in the LIGO frequency range (in-dicated by the two vertical lines) can therefore transitionfrom a simple descending power law, to one with a peakin between, and eventually to a ascending power law aswe increase f pk .With this spectrum, we follow the procedure laid outin Refs [20, 21] to compute the upper limit of Ω GW fordifferent values of f pk using data from LIGO O2 [21].More details can be found in Appendix A. The only dif-ference respect to these references is related to the choiceof the optimized estimator, i.e. , the estimator ˆΩ f ref is op-timized by putting Ω GW ( f ) / Ω GW ( f ref ) instead of somepower of f /f ref . Note that this means the upper limitessentially depends only on the shape of S sw within theLIGO frequency band, since all the other factors drop outwhen taking the ratio. The 95% confidence level upperlimit is then obtained by settingΩ up . lim . GW ( f ref ) = 2 σ ( f ref ) . (8)In FIG. 2 we show the upper limit at f ref = 25 Hz bythe curve with triangles. For f pk (cid:46)
10 Hz or f pk (cid:38) Hz, the bound approaches a constant since the spectrumis essentially a power law with fixed exponent within theLIGO frequency band. This behaviour allows us to eas-ily extrapolate the constraint to even smaller or biggervalues of f pk . However, for intermediate f pk there is asmooth transition on the upper limit between the ascend-ing and descending asymptotic behavior. We see thatLIGO provides a stronger constraint on ascending spec-tra than on the descending spectra, and the differencecan be as large as an order of magnitude.In practice, LIGO could only provide reliable con-straints on SGWB in the band 20 − f ref = 25 Hz as it is the frequency where LIGO ismost sensitive [20, 21]. However, the thermal parametersof the phase transition are connected to the amplitude atthe f pk . In order to place constraint on the thermal pa-rameters, we notice that, for a given GW spectrum as inEq. (6), the constraint at a particular frequency in theLIGO band can be mapped to the peak frequency f pk using Ω up . lim . GW ( f pk ) = Ω up . limGW ( f ref ) S sw ( f pk ) S sw ( f ref ) . (9)The curve made by circles in FIG. 2 shows the result ofthis mapping. For f pk very small or very large, the upperbound from LIGO becomes substantially weaker, evenwhen comparing with the constraint from BBN, which isobtained by integrating out the spectrum and requiring h (cid:82) d ln f Ω GW ( f ) < × − [9, 22].In what follows, we shall discuss the LIGO constraintson the thermal parameters and how such constraint canbe utilized to constrain particular models of phase tran-sition. III. SCENARIOS FOR PHASE TRANSITIONSAND THEIR THERMAL PARAMETERS
Typically, one represents phase transitions as driven bythe dynamics of a scalar field which transitions from onevacuum to another under the influence of the evolvingthermal potential. In that context, the thermal parame-ters β/H and ξ are defined by βH = T ( S E /T ) dT (cid:12)(cid:12)(cid:12)(cid:12) T = T N (10) ξ = 1 ρ N (cid:18) ∆ V − T ∆ dVdT (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) T = T N (11) in which S E is the Euclidean action, V is the thermalpotential of the scalar field, and the bubble nucleationtemperature can be obtained by solving S E T N ≈ − (cid:18) T N GeV (cid:19) − g (cid:63) ( T N ) , (12)in which ρ N = g (cid:63) ( T N ) π T N /
30. On the other hand, thecalculation of the bubble-wall velocity v w for a particularmodel is highly non-trivial. Therefore, instead of directlycalculating it, we shall follow the customary convention ofconsidering a few reference values, v w = 0 . v w = 1.To connect the thermal parameters, ξ and β/H tospecific models we will follow the approach described inRef. [10]. We will consider classes of potentials which con-sists of competing terms with alternating signs. Specif-ically, we will look into two types of finite-temperaturepotentials V ( H, T ) = 12 m ( T ) h D − c ( T ) h D + 14 λ ( T ) h D (13) V ( H, T ) = 12 m ( T ) h D − λ ( T ) h D + c ( T ) h D (14)in which the coefficients of the scalar field h D are allpositive at the time of transition.Indeed, most dark phase transitions can be mappedonto these effective scenarios. For example, in a DarkSector where its particles acquire mass from a Dark Higgsas its gauge group SU ( N ) breaks into SU ( N − SU ( N ) /SU ( N −
1) models with renormalisable and non-renormalisable operators specifically.
A. Exploring Dark Sectors with LIGO: SU ( N ) /SU ( N − models
1. Models with renormalisable operators
For the type of potential in Eq. (13), we canparametrise zero-temperature parameters as m (0) = − Λ v D , λ (0) = Λ v D (15)in which v D is the zero temperature vacuum expectationvalue, and Λ is the scale of the potential. With thisparametrisation, the finite temperature potential can beexpressed as V ( H, T ) = Λ (cid:34) − (cid:18) h D v D (cid:19) + 14 (cid:18) h D v D (cid:19) (cid:35) + T π (cid:34) (cid:88) i ∈ bosons n i J B ( m i /T ) − (cid:88) i ∈ fermions n i J F ( m i /T ) (cid:35) = Λ (cid:40) (cid:20) −
12 + (cid:18)
18 + N G (cid:19) T v D + 324 N GB g T v D Λ + y N f T v D Λ (cid:21) (cid:18) h D v D (cid:19) − (cid:34) N GB (cid:18) g (cid:19) / π v D T Λ (cid:35) (cid:18) h D v D (cid:19) + 14 (cid:18) h D v D (cid:19) (cid:41) , (16)where N GB = 2 N − g , and which get a mass from the Dark Higgs interac-tions. N G = 2 N − N f = N × N FL is the number of self-adjoint fermionswith Yukawa coupling y where N FL is the number of fla-vors. Note that for Dirac fermions, one would need todouble the number of degrees of freedom. For simplic-ity, we assume the Yukawa coupling is universal for thosefermions. See Ref. [10] for more details.In the second equality, the following high temperatureexpansions are used J B ( m /T ) ∼ π (cid:18) m T − m πT (cid:19) , (17)and J F ( m /T ) ∼ − π (cid:18) m T (cid:19) , (18)in which the field dependent masses can be read as m H = ∂ h D V = Λ (cid:18) h D v − v (cid:19) , (19) m G = 1 h D ∂ h D V = Λ (cid:18) h D v D − v D (cid:19) , (20) m GB = gh D , (21) m f = yh D √ . (22)Note that in the second line of Eq. (16), only the mas-sive gauge bosons are taken into account, i.e. , the higherorder term ( ∼ m /T ) from the Goldstone boson is ne-glected.Besides, the part of the expansion which givesrise to terms independent of h D is neglected, since it onlyamounts to a constant shift in the potential V . Finally,the mapping to the temperature dependent parametersin Eq. (13) is straightforward by matching the terms withthe same powers of h D . Eq. (16) is also often written in the following form V ( H, T ) = Λ ( T ) (cid:34) (cid:18) − α ( T )2 (cid:19) (cid:18) h D v D ( T ) (cid:19) − (cid:18) h D v D ( T ) (cid:19) + α ( T ) (cid:18) h D v D ( T ) (cid:19) (cid:35) , (23)in which the minimum of the potential can be easily ob-tained by minimizing the potential v D ( T ) = 3 c ( T ) + (cid:112) c ( T ) − m ( T ) λ ( T )2 λ ( T ) . (24)By identifying c ( T ) = Λ ( T ) v D ( T ) , λ ( T )4 = α ( T ) Λ ( T ) v D ( T ) , (25)one finds α ( T ) = λ ( T ) v D ( T )4 c ( T ) , Λ( T ) = (cid:0) c ( T ) v D ( T ) (cid:1) / . (26)With these, for α ( T ) ∈ [0 . , . S E T = v D ( T ) T Λ ( T ) 10 a + b | α ( T ) − . | c | α ( T ) − . | d (27)with the fitting parameters a = − . , b = 71 . c = 0 . d = 0 .
2. Models with non-renormalisable operators
For the type of potential in Eq. (14), similar to theprevious case, one can perform the following parametri-sation m (0) = (2 − α ) Λ v D , λ (0) = 4 Λ v D , c (0) = α Λ v D . (28)The finite temperature potential then becomes V ( H, T ) = Λ (cid:34) (2 − α ) (cid:18) h D v D (cid:19) − (cid:18) h D v D (cid:19) + α (cid:18) h D v D (cid:19) (cid:35) + T π (cid:34) (cid:88) i ∈ bosons n i J B ( m i /T ) − (cid:88) i ∈ fermions n i J F ( m i /T ) (cid:35) = Λ (cid:40)(cid:20) − α − (cid:18)
12 + N G (cid:19) T v D + 324 N GB g T v D Λ + y N f T v D Λ (cid:21) (cid:18) h D v D (cid:19) − (cid:20) − (30 + 6 N G ) αT v D (cid:21) (cid:18) h D v D (cid:19) + α (cid:18) h D v D (cid:19) (cid:41) . (29)Note that, in the high-temperature expansion, the cubicterm is assumed to be subdominant, i.e. , we have onlykept the part proportional to m /T [10]. Moreover, thefield-dependent masses of the Goldstone bosons and theDark Higgs which goes into the thermal correction are m H = Λ (cid:20) − α ) v D − h D v D + 30 αh D v D (cid:21) , (30) m G = Λ (cid:20) − α ) v D − h D v D + 6 αh D v D (cid:21) . (31)The terms proportional to αh D /v D would give rise to thethermal correction of the quartic term.Just as we have done in the previous section, one canwrite Eq. (29) in terms of temperature-dependent param-eters: V ( H, T ) = Λ ( T ) (cid:34) (2 − α ( T )) (cid:18) hv D ( T ) (cid:19) − (cid:18) hv D ( T ) (cid:19) + α ( T ) (cid:18) hv D ( T ) (cid:19) (cid:35) . (32)The non-vanishing VEV v D ( T ) = (cid:32) λ ( T ) + (cid:112) Λ ( T ) − c ( T ) m ( T )12 c ( T ) (cid:33) / (33)is obtained by minimizing the potential. Suppose thenon-vanishing VEV does exists (Λ ( T ) − c ( T ) m ( T ) > α ( T ) can be obtained by solving λ ( T ) = 4Λ( T ) /v ( T ) c ( T ) = α ( T )Λ( T ) /v D ( T ) . (34)Therefore, α ( T ) = 4 c ( T ) v D ( T ) /λ ( T ) . (35)Following [10], the Euclidean action can be fitted by S E = v D ( T )Λ ( T ) 10 (cid:80) i =1 a i ( α ( T ) − / i (36)with a i = ( − . , − . , − . α ( T ) ∈ [0 . , . g (cid:63) ( T N ), at high temperatures, i.e. , T (cid:29)O (100) GeV (cid:29) m i , m h D , we shall assume that the par-ticles in the Dark Sector are the only degrees of freedomin addition to the SM. Therefore, the effective number ofrelativistic degrees of freedom will be given by g (cid:63) ( T N ) ≈ . N GB +2( N − − N GB )+1+ 78 × × N f , (37)where we have included all degrees of freedom from theSM, as well as the massless and massive gauge bosonsand the dark fermions charged under SU ( N ).
3. Results
In FIG. 3, we present the LIGO constraints on the ther-mal parameters at 95% confidence level together with theconstraint from BBN. In the left panels, the constraintsare evaluated with v w = 0 .
5, whereas the bounds in theright panels are evaluated at v w = 1.Note that, with v w fixed, the thermal parameters( β/H, T N , ξ ) constitutes a 3-dimensional space. In ouranalysis, we find the the excluded region can be conve-niently projected on the 2-dimensional plane of β/H and T N as the contours of ξ do not intersect each other. Thisthus enables us to use colour variation to represent thevalues of ξ on the exclusion contours. Indeed, within aparticular contour, any value of ξ larger than the valueassociated with the contour is excluded. As a result, theregions in the 2D plots where the LIGO ellipsoidal con-tours have a colour lighter than the BBN vertical con-tours suggest that LIGO can set a better bound thanBBN, and viceversa.On the same figure, we also plot the values of the ther-mal parameters obtained from different models definedby Λ , N and N FL . We show examples with N = 2 and5, and a fixed number of flavours N FL = 3, as repre-sentative of the behaviour expected from the choices onmatter content and gauge groups. We also fix g = 1 and y = 1, although similar behaviour is found for similar O (1) values.If a set of thermal parameters obtained from a particu-lar model lies within a contour, and has a larger ξ (darker β/H T N [ G e V ] Λ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeV v w = 0 . N = 5 , N FL = 3 N = 2 , N FL = 3 − − − ξ β/H T N [ G e V ] Λ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeV v w = 1 N = 5 , N FL = 3 N = 2 , N FL = 3 − − − ξ β/H T N [ G e V ] Λ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeV v w = 0 . N = 5 , N FL = 3 , g = 1 N = 2 , N FL = 3 , g = 1 − − − ξ β/H T N [ G e V ] Λ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeVΛ = 10 GeV v w = 1 N = 5 , N FL = 3 , g = 1 N = 2 , N FL = 3 , g = 1 − − − ξ FIG. 3. 95% confidence level exclusion region from LIGO (curved contours) and the constraint from BBN (vertical contours),projected on the T N − β/H plane. The values of ξ corresponding to this exclusion are represented by the colour variation.The constraints are calculated for two reference bubble-wall velocity v w = 0 . v w = 1. Markers with different shapes areobtained from different models of phase transitions. The top two panels correspond to the cases with renormalisable operators,whereas the bottom two panels show the cases with non-renormalisable operators. colour) than the corresponding contour, then this set ofthermal parameter is excluded at 95% CL.From the top figures 3, one can deduce that the con-straint from LIGO is still unable to probe the renormal-isable classes of models, as the markers representing thethermal parameters barely touch the excluded region.However, in the non-renormalisable case, we indeedsee how LIGO is able to probe these Dark Sectors. Inparticular we observe that the dark scales Λ ∼ - 10 GeV are best constrained by LIGO. Note that modelswith larger group rank N and number of flavours N FL are more constrained as they tend to produce a larger ξ .The regions excluded by LIGO O2 correspond to Λ inthe region around 10 -10 GeV. In this region we havevaried v w from 0.5 and 1 and explored different optionsfor N and N FL as illustrated by FIG. 3. The range ofparameters α and x ≡ v D / Λ where LIGO currently has sensitivity is given by α ∈ [0 . , . , x = v D Λ ∈ [1 . , . (38)These regions can be translated into parameters in thescalar potential Eq. (14) by inspecting Eq. (28): m Λ ∈ [0 . , . , λ ∈ [0 . − . , c Λ ∈ [0 . , . . (39)Clearly, these are reasonable choices of parameter spaceand indicate that LIGO is testing interesting Dark-Sectortheories. IV. CONCLUSIONS
When discussing Dark Sectors and their GW signa-tures, we usually think on future probes like LISA, manyyears from now. Here we have shown that LIGO is al-ready probing interesting scenarios for Dark Sectors.In the context of first-order phase transitions andLIGO data, the emphasis has been placed in performingeffective analyses, such as Ref. [11] where the authors ex-plore the bounds from LIGO O3 using parametrisationsof the power spectrum. In this paper, we take a comple-mentary step and focus on examining whether concreteparticle-physics models could be related to the tested re-gions.In this work we answer the question whether the typeof first-order phase transitions LIGO is currently prob-ing could be represented by concrete, reasonable particle-physics models. For that reason, we have focused onclasses of models which capture a broad set of features ofDark Sectors, and, at the same time, capable of produc-ing interesting GW signatures. The renormalisable andnon-renormalisable benchmarks we used had been iden-tified in Ref. [10] as more promising for strong first-orderphase transitions from Dark Sectors.We choose to set up those models with a breaking SU ( N ) /SU ( N −
1) which should be understood as anexample in which some bosonic and fermionic degrees offreedom influence the thermal history of the Dark Higgs.Such a choice also allows a simple parametrisation interms of the group rank N and the number of flavours N FL . We find that scales around 10 − GeV are bet-ter probed by LIGO O2. We also find that the sensitiveregions correspond to moderate values for N and N FL ,evidencing that LIGO is not testing extreme regions inthe UV parameter space. Of course, various other typesof models for phase transition could also generate GWwhose spectrum lies in the frequency range relevant forLIGO, e.g. , models motivated by grand unification the-ories [24, 25] and models for confinement-deconfinementphase transition [26]. It is straightforward to see whetherLIGO might be able to constrain those models once thethermal parameters are computed.Note that, when obtaining the GW peak amplitude inEq. (4), we have used the standard formula from Ref. [19].Recent discussions in Ref. [27] suggest the existence ofan additional suppression factor due to the finite lifetimeof the sound waves. Moreover, it has been shown re-cently that theoretical uncertainties in GW productioncould lead to changes in the GW spectrum as large asseveral orders of magnitude [28]. In addition, we haveassumed in our analysis that sound waves are the dom-inant source for GW production. In other scenarios, forexample, strongly supercooled phase transitions [29–32],or scenarios in which new heavy particles can providesufficiently large friction [33], contribution from turbu-lence or bubble collision could also be important and oneneeds to care about their effects in the GW power spec-trum. Although subject to those uncertainties, our anal-ysis nevertheless continues to provide a concrete methodto constrain Dark-Sector models with LIGO data.Our analysis of the SGWB is based on publicly avail-able O2 data. In Appendix A, we have provided ex- planations on how to reproduce our analysis. Recentpapers from authors in the LIGO/Virgo collaboration, e.g. , [11, 34], make use of the O3 data. GW constraintson particle-physics models considered in this paper areexpected to improve as more data becomes available.We believe our results motivate a more systematicstudy of the particle-physics scenarios that the LIGO ex-periment is able to test. We emphasize again that tra-ditional direct or indirect searches for dark particles as-sume that the Dark Sector interacts non-gravitationallywith the Standard Model. On the other hand, since grav-ity is universal, methods for probing the Dark-Sector viaits gravitational effects such as structure formation (seeRef. [35, 36] and references in it for recent progress) andgravitational waves do not rely on those assumptions.These gravitational effects therefore offer unique oppor-tunities to access Dark Sectors, which would otherwise behidden from us if they lack a connection to the StandardModel. ACKNOWLEDGEMENTS
We would like to thank Djuna Croon for conversa-tions at the beginning of this project. V.S. acknowledgessupport from the UK Science and Technology FacilitiesCouncil ST/L000504/1. J.S. and F.H. are supported bythe National Natural Science Foundation of China un-der Grants No. 12025507, No. 11690022, No.11947302;and is supported by the Strategic Priority Research Pro-gram and Key Research Program of Frontier Scienceof the Chinese Academy of Sciences under Grants No.XDB21010200, No. XDB23010000, and No. ZDBS-LY-7003. F.H. is also supported by the National ScienceFoundation of China under Grants No. 12022514 andNo. 11875003.
Appendix A: On the use of LIGO data
In this Appendix, we describe what we actually dowith data downloaded from LIGO. This consists of twoparts: 1) the data selection in which we select data thatsatisfy certain criterion, and 2) the analysis in which weestimate the GW upper limit using data selected in thefirst part.
1. Data Selection
To perform this analysis, one needs to downloadthe data of the LIGO detectors at both Hanfordand Livingston from O2 data release ( ) [21] with a 16 kHz sam-pling frequency. Each file covers a 4096 s period of mea-surement, and the file name contains the start time ofthe measurement, which is referred to as the “GPS starttime”. In each file, there are in general two types of data– the strain time series h ( t i ) and some auxiliary datasuch as the data quality (DQ) mask label associated witheach strain measurement. The DQ mask is a 7-bit binarynumber each of which indicates whether a certain type ofcheck is passed (value=1) or not (value=0). We convertthis binary number into a decimal digit. For example,there is no data at time t i if DQ( t i ) = (0000000) = 0.On the contrary, data is present if this value is nonzero.Following the LIGO stochastic gravitational wave anal-ysis [20, 21], we first downsample the 16 kHz strain datato 4 kHz. We then select out the timestamps at whichboth detectors are taking data properly, i.e. , the times t i at which both DQ H ( t i ) (cid:54) = 0 and DQ L ( t i ) (cid:54) = 0. Afterdoing that we get, for each data file, a list of GPS timesat which data in both detectors is available. We thencombine the lists of GPS times within a file and acrossneighbouring files into continuous segments. The seg-ments whose duration ≥
600 s are further picked out toperform a stationarity cut following Ref. [20, 37]. Whenwe perform the stationarity cut, we also notch out fre-quencies at which the data exhibits narrowband coherentlines that are known to be instrumental or environmen-tal artifacts [38–41]. The list of the notched frequencybands we used can be found on the public data releasepage: https://dcc.ligo.org/LIGO-T1900058/public .
2. Analysis
After selecting a clean list of strain data and GPS timewhich satisfies the requirements on data quality and sta-tionarity, we then use the cross-correlation method toestimate the SGWB signal. The spectrum of the GWbackground is estimated with the cross-correlation statis-tic ˆ C ( f ) [21], defined asˆ C ( f ) ≡ T Re[˜ s ∗ ( f )˜ s ( f )] γ T ( f ) S ( f ) , (A1) (cid:104) ˆ C ( f ) (cid:105) = Ω GW ( f ) , (A2)where S = 3 H / (10 π f ), H is the Hubble parameter,˜ s , ( f ) are the Fourier transfroms of the strain data ofboth detectors, T = 192 s is the segment duration ofthe Fourier transforms, and (cid:104) ... (cid:105) indicates the averageover all such 192s-segments. In the limit that the GWsignal is negligible comparing to the instrumental noise,the variance of ˆ C ( f ) is given by σ ( f ) = 12 T ∆ f ¯ P ( f ) ¯ P ( f ) γ T ( f ) S ( f ) , (A3) where ∆ f = 1 /
32 Hz, and ¯ P , ( f ) are the one-sidedpower spectrum of each detector, which are obtained asan average over two neighbouring segments ,¯ P i,I ≡ P i,I − + P i,I +1 , (A4)in which the subscript i and I labels the detector andthe 192s-segments, respectively. For each 192s-segment,we use the broadband estimator for any spectral shapeof the gravitational wave background,ˆΩ ref ≡ (cid:80) k ω ( f k ) − ˆ C ( f k ) σ − ( f k ) (cid:80) k ω ( f k ) − σ − ( f k ) (A5) (cid:104) ˆΩ ref (cid:105) = Ω GW ( f ref ) (A6)where the weight function ω ( f ) ≡ Ω GW ( f ref ) / Ω GW ( f ), f k are discrete frequencies between 20 and 1726 Hz withthe interval of 1 /
32 Hz. The uncertainty of the optimalestimator is, σ − = (cid:88) k ω ( f k ) − σ − ( f k ) . (A7)After calculating ˆΩ ref and σ for all 192s-segments, theensemble average over all the segments is obtained from σ , tot = 1 (cid:80) I σ − ,I , (A8) µ = (cid:80) I ˆΩ ref ,I σ − ,I (cid:80) I σ − ,I . (A9)The signal-to-noise ratio (SNR) can be calculated bySNR = µ/σ , where σ ≡ (cid:113) σ , tot . In the absence ofdetection signal, we set the 95% confidence level upperlimit byΩ up . lim . ref = Ω up . limGW ( f ref ) = 2 × σ Ω , tot ( f ref ) . (A10)Note that this upper limit depends on the choice of thereference frequency f ref . In the calculation of Fouriertransforms ˜ s i ( f ) and power spectral density P i ( f ), weuse the 50% overlapping Hann windows to avoid spectralleakage [20, 21, 34]. This means we need at least 3 continuous 192s-segments to obtaina power spectrum. Those 192s-segments without a neighbor onboth sides do not have an associated power spectrum. [1] B. P. 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