Gravitational waves from first order electroweak phase transition in models with the $U(1)_X^{}$ gauge symmetry

We consider a standard model extension equipped with a dark sector where the $U(1)_X^{}$ Abelian gauge symmetry is spontaneously broken by the dark Higgs mechanism. In this framework, we investigate patterns of the electroweak phase transition as well as those of the dark phase transition, and examine detectability of gravitational waves (GWs) generated by such strongly first order phase transition. It is pointed out that the collider bounds on the properties of the discovered Higgs boson exclude a part of parameter space that could otherwise generate detectable GWs. After imposing various constraints on this model, it is shown that GWs produced by multi-step phase transitions are detectable at future space-based interferometers, such as LISA and DECIGO, if the dark photon is heavier than 25 GeV. Furthermore, we discuss the complementarity of dark photon searches or dark matter searches with the GW observations in these models with the dark gauge symmetry.


Introduction
Due to the discovery of the Higgs boson with a mass of m h ≃ 125 GeV [1,2], the standard model (SM) has been experimentally established as an effective theory that describes spontaneous breaking of the electroweak (EW) symmetry. However, the properties of the discovered Higgs boson as well as the dynamics behind the electroweak symmetry breaking (EWSB) are still unknown. In addition, there exist phenomena that require new physics beyond the SM (BSM), such as neutrino oscillations, baryon asymmetry of the Universe, the existence of dark matter (DM) and inflation. One of the most intriguing ideas is to relate new physics that accounts for these phenomena with physics in the Higgs sector.
One of the interesting phenomena that relate such new physics and the Higgs sector is electroweak baryogensis (EWBG) [3]. For generating baryon asymmetry, one must satisfy Sakharov's conditions: Violation of baryon number; simultaneous violation of C and CP symmetry; and departure from thermal equilibrium [4]. EWBG scenarios can be realized by extending the Higgs sector since the first condition is provided by the sphaleron process, the second by additional CP violating phases other than the Kobayashi-Maskawa phase and the third by strongly first order phase transition (1stOPT) (see e.g. Refs. [5][6][7]).
We emphasize that the nature of EWPT can be probed by exploring the Higgs sector at ongoing and future experiments. Models predicting significant deviations in various Higgs boson couplings can be tested at the LHC [57] as well as at future lepton colliders including, the International Linear Collider (ILC) [58], the Compact Linear Collider (CLIC) [59] and the Future Circular Collider of electrons and positrons (FCC-ee) [60]. As for the hhh coupling, the high-luminosity LHC will constrain the deviation up to a factor [61,62]. If the International Linear Collider (ILC) with √ s = 1 TeV is realized the hhh coupling will be determined with 10% accuracy [63][64][65] 1 . The capability of measuring the hhh coupling at future hadron colliders with √ s = 100 TeV has been discussed [68].
On the cosmological side, the strongly 1stOPT that occurs in the early Universe produces stochastic GWs detectable at future space-based interferometers [22,23,28,[69][70][71][72][73][74][75][76][77][78][79][80][81]. Until now, several GWs events generated by the mergers of binary black holes and binary neutron stars have been observed at the Advanced LIGO and Advanced VIRGO [82]. The worldwide network of GW detectors including Advanced LIGO [83], Advanced VIRGO [84] and KAGRA [85] will reveal astronomical problems. In future, planned space-based interferometers such as LISA [86], DECIGO [87] and BBO [88] will survey GWs in the millihertz to decihertz range, which is the typical frequency of GWs from the first order EWPT. Even the above-mentioned nightmare scenarios can be investigated by measuring GWs at these future interferometers [31,55]. Therefore, using the synergy between the measurement of the Higgs boson couplings at colliders and the observation of GWs at interferometers, one can scrutinize the nature of EWPT and distinguish EWBG scenarios.
Among various extensions of the Higgs sector, models with Higgs portal DM can account for the relic abundance of DM, and thus have been extensively studied in recent years.
In this paper, we shall focus on a model with gauged dark U (1) X symmetry as one of the viable Higgs portal DM models. First, we address a more general case where the dark photon decays into SM particles through the U (1) gauge kinetic mixing term and investigate the nature of EWPT and dark PT. We revisit the complementarity of dark photon searches and GW observations in this model with strongly 1stOPT [103], and find the lower bound of m X in the light of current collider bounds. We then explore the Higgs portal DM model with VDM, which is stabilized by introducing a discrete Z 2 , and investigate the complementarity of the detection of GWs from the strongly 1stOPT, collider bounds and DM searches.
This paper is organized as follows. In Sec. 2, we briefly review the model with the U (1) X dark gauge symmetry and show formulae about the properties of the Higgs bosons and the finite temperature effective potential. We discuss PT patterns at finite temperature and introduce quantities which characterize the spectrum of GWs produced by bubble collisions in Sec. 3. In Sec. 4, our numerical results about the prospect of the detection of GWs are shown for various benchmark points after imposing theoretical and experimental constraints. Sec. 5 is devoted to discussion and conclusions. The tree-level unitarity of the Higgs self-couplings is discussed using analytic formulae in Appendix A. The oneloop renormalization group equations for the model parameters are given in Appendix B. Constraints on the DM properties are discussed in Appendix C.
2 Model with dark U(1) X gauge symmetry We consider a model with a dark sector where the U (1) X Abelian gauge symmetry is spontaneously broken by the so-called dark Higgs mechanism. We introduce a complex scalar (called dark Higgs boson) S with U (1) X -charge Q X and the U (1) X gauge field (dark photon) X 0 µ . In generic, there appears the gauge kinetic mixing term between the U (1) X gauge boson X 0 µ and the hypercharge U (1) Y gauge boson B µ [104], and the Lagrangian for the newly introduced fields is (e.g. Ref. [103]) Here, the Higgs potential is given by We normalize the U (1) X charge of S, Q S ≡ Q X (S), as Q S = 1. Since the viable parameter range for ǫ is too small to affect PT [103], we focus on the rest six parameters, i.e. µ 2 Φ , µ 2 S , λ Φ , λ S , λ ΦS and g X . We refer to the model with the kinetic mixing as "Model A" in this paper. In this paper, we focus on the cases where µ Φ and µ S are of the same order since we address the complementarity between collider experiments and cosmological observations in exploring new physics at around the EW scale. In the large µ S limit, the singlet field decouples from the SM. GW production from such high-scale phase transition is discussed in e.g. Refs. [78,[105][106][107][108].
If we introduce a discrete Z 2 symmetry under which the vector boson X 0 µ is odd, the kinetic mixing term X µν B µν is prohibited, stabilizing X 0 µ . In this case, the vector boson X 0 µ can be an excellent candidate for DM [56,[99][100][101][102] 2 . The case without the kinetic mixing is referred to as "Model B". As far as PT is concerned, Model A and Model B can be discussed on the same footing.
After the EWSB, the two Higgs multiplets can be expanded as where v Φ (= 246 GeV) and v S are the corresponding vacuum expectation values (VEV), φ Φ and φ S are physical degrees of freedom which mix with each other through the λ ΦS term in the Higgs potential (Eq. (2)). The Nambu-Goldstone modes w ± , z 0 and x 0 are absorbed by the gauge bosons W ± µ , Z 0 µ and X 0 µ . The mass of X 0 µ is m X = g X |Q S |v S (see also Ref. [100]).
The phase structure of our model is analyzed in the classical field space spanned by Φ = (0, ϕ Φ / √ 2) and S = ϕ S / √ 2. The Higgs potential is modified from its tree-level form due to radiative corrections. At zero temperature, the effective potential at the oneloop level is given by [110] where Q is the renormalization scale, which is set at v Φ in our analysis 3 . Here, n i and M i (ϕ Φ , ϕ S ) stand for the degrees of the freedom and the field-dependent masses for particles i = h, H, w ± , z 0 , x 0 , W ± µ (T,L) , Z 0 µ (T,L) , X 0 µ (T,L) , γ 0 µ (T,L) , t and b, respectively. We adopt the mass-independent MS scheme, where the numerical constants c i are set at 3/2 (5/6) for scalars and fermions (gauge bosons). We impose the conditions that the tadpole terms at the one-loop level vanish as for α = Φ and S. The angle bracket · · · denotes the corresponding field-dependent value evaluated at our true vacuum The interaction basis states φ Φ and φ S are relations with their mass eigenstates h and H through with c θ ≡ cos θ, s θ ≡ sin θ. The one-loop improved mass squared matrix of the real scalar bosons in the (φ Φ , φ S ) basis is then diagonalized as We denote h and H as the discovered Higgs boson with the mass m h = 125 GeV and the additional neutral Higgs boson with mass eigenvalue m H , so that the absolute value of the mixing angle |θ • | is less than 45 • . In our analysis, we regard v Φ , m h , m H , θ, g X and m X as input parameters, and µ 2 Φ , µ 2 S , λ Φ , λ S , λ ΦS and v S as derived parameters from them. The tree-level interactions of h and H with the SM gauge bosons V (=W ± µ , Z 0 µ ) and with the SM fermions F are given by respectively. These Higgs boson couplings in our model normalized by the corresponding SM ones are universally given by Using the effective potential approach, the hhh coupling is computed as Its SM prediction is approximately given by [111,112] The deviation in the hhh coupling is defined as At finite temperatures, the effective potential receives additional contributions from thermal loop diagrams, and is modified to [113] for bosons (−) and fermions (+), respectively. In order to take ring-diagram contributions into account, we replace the field-dependent masses in the effective potential by [114] where Π i (T ) denote the finite temperature contributions to the self energies of the fields i. The thermally corrected field-dependent masses of the Higgs bosons are Here, g, g ′ and g X (y t and y b ) are the gauge couplings of SU (2) L , U (1) Y and U (1) X (the top and bottom Yukawa couplings). In the (W + µ , W − µ , W 3 µ , B 0 µ ) basis, the field-dependent masses of the EW gauge bosons are thermally corrected as with a L g = 11/6, a T g = 0 and that of the U (1) X gauge boson as with a L X = 1/3, a T X = 0 [114][115][116] 4 . On the other hand, fermion counterparts do not receive such thermal corrections.
The one-loop effective potential at finite temperature has the notorious problem of gauge dependence, which has been known for a long time, but no complete treatment has been invented. A gauge invariant treatment for evaluating the critical temperature T c has been discussed in Ref. [45]. However, the computation of the transition temperature T t , which is relevant to the GW production, requires the high temperature approximation. The uncertainties in the prediction of the GW spectrum under specific gauge choices are discussed in Ref. [116]. In this paper, we take the Landau gauge, where the gauge-fixing parameter vanishes ξ = 0, as a reference although we are aware of the problem pf gauge dependence.

The first order electroweak phase transition
As discussed in Refs. [42,43,45], there are typically four different types of PT path as shown in Fig. 1. In our numerical analysis, we impose the condition that the EW phase with massive dark photon (ϕ Φ , ϕ S )=(v Φ , v S ) becomes the global minimum at T = 0 [32,117]: In order to discuss GWs originating from the first order EWPT in an analytic manner, we introduce several important quantities that parametrize the dynamics of vacuum bubbles following Ref. [74]. The transition temperature T t is defined such that the bubble nucleation probability per Hubble volume per Hubble time reaches the unity: The produced GWs are enhanced as the released energy density ǫ is increased. A dimensionless parameter α is defined as the ratio of ǫ to the radiation energy density ρ rad = (π 2 /30)g * T 4 at the transition temperature T t : For simplicity, the relativistic degrees of freedom is set at g * = 110.75, and the temperature dependence of g * is neglected. The bubble nucleation rate can be parametrized as Γ(t) = Γ 0 exp(βt) at around the transition temperature T t . We introduce another dimensionless parameter β as the ration of the inverse of the time variation scale of the bubble nucleation rate β to the Hubble parameter at T = T t : where S 3 (T ) is the three-dimensional Euclidean action of the bounce solution of the classical fields that is stretched between the true and false vacua at finite temperature T . The predicted GW spectrum is expressed in terms of T t , α and β. According to Ref. [118], the contribution from sound waves is the main source for stochastic GWs from 1stOPT while those from the bubble wall collision and the turbulence are not significant [118]. We employ the approximate analytic formula provided in Ref. [76] for computing the spectrum of the GWs.

Numerical results
For our numerical analysis of PT, we implement the model introduced in Sec. 2 into the public code CosmoTransitions 2.0a2 [119], which computes quantities related to cosmological PT in the multi-field space. Our analysis is focused on the six benchmark points blobbed in Fig. 2. In the following, we detail theoretical and experimental constraints taken into account in our numerical analyses. The conditions for vacuum stability for the Higgs potential are given by By the requirement of perturbative unitarity [120], the magnitudes of the eigenvalues of S-wave scattering amplitudes for the longitudinal weak gauge bosons and the scalars must be smaller than 1/2, leading to [121,122], Further discussions on perturbative unitarity are given in Appendix A. Electroweak precision measurements constrain parameters in the Higgs sector of our model. Since the mass of the discovered Higgs boson is m h ≃ 125 GeV, the mixing angle  of the Higgs bosons is bounded as θ 23 • when the mass of the additional Higgs boson is m H 400 GeV [94,95]. The measurements of the Higgs boson decay into weak gauge bosons give constraints on the hV V couplings as κ Z = 1.03 +0.11 −0.11 and κ W = 0.91 +0.10 −0.10 from the ATLAS and CMS combination of the LHC Run-I data (68% CL) [123]. In our numerical analysis, we take the 68% CL bound κ Z > 0.92 as the lower bound on the mixing angle, namely The exclusion limits from the direct searches for the H boson at the LEP and LHC Run-II are examined in Ref. [124]. We will show that a large portion of the model parameter space where strongly 1stOPT and detectable GW signals are possible is excluded by the collider bounds on the Higgs bosons discussed above 5 .
Our numerical results about the EWPT and GW signals for the six benchmark points defined in Fig. 2    ∆κ Z to 0.38% [66]. If the ILC with √ s = 1 TeV is realized, the hhh coupling can be measured with an accuracy of 16% (10%) for L = 2000 fb −1 (5000 fb −1 ) [65]. The limit obtained from direct searches for the H-boson is discussed in Ref. [127] for the small mass region and in Ref. [128] for the large mass region. The high-luminosity LHC will extend the discovery reach. Therefore, the measurements of the properties of the Higgs bosons play an important role in pinning down viable model parameters.
The detectability of GWs from strongly 1stOPT is shown in Fig. 4 and the right frames of Figs. 5-9. All the points of multi-step PT with the first order EWPT (closed blue square, closed green star and closed green triangle in Fig. 3 and the left two frames of Figs. 5-9) are displayed in these right frames. Among these parameter sets, points surviving the collider bounds (the black solid line in the corresponding left and middle frames) are marked with colored symbols in these right frames while the gray points are not compatible with the collider bounds.
In these figures, the areas with light colors are within the expected reach of the future space-based interferometers, LISA [76,129,130] and DECIGO [87]. The expected sensitivities of different LISA (DECIGO) designs are labeled by "C1" and "C2" ("Correlation", "1 cluster" and "Pre") following Ref. [76] (Ref. [87]). Notice that the transition temperature T t depends on the model parameters, we take T t = 100 GeV for the purpose of illustra- tion. The amplitude of produced GWs is enhanced as the velocity of the bubble wall v b is increased. Since the uncertainty of the evaluation of v b is large, we here consider an optimistic case of v b = 0.95. For successful EWBG scenarios, on the other hand, subsonic wall velocities are preferable. In such cases, detecting GWs require smaller β and larger α. PT one-step PT (1st order) one-step PT (2nd order) ⋆ two-step PT (1st order → 1st order) two-step PT (2nd order → 1st order) △ two-step PT (1st order → 2nd order) ( , ⋆, ) in (m H , θ) insensitive at GW observation ( , ⋆, ) in (α, β) excluded by the collider constraints ( ) DM contours of log 10 (Ω X /Ω obs ) [ ] excluded by XENON1T [125] in terms of log 10 (σ X × Ω X /Ω obs ) Table 2. Summary of our numerical results. For each benchmark points, the types of the multistep PT are listed and the different solutions in terms of the detectability of the GW observations and the DM constraints are labeled. In the GW column, (△) denotes that there are (no) regions which reach to the planned sensitivity of LISA and DECIGO. In the DM column for Model B, (△) denotes that the VDM and nucleon elastic scattering cross section rescaled with relic density satisfy the bound from XENON1T [125] with the predicted relic density which accommodates the observed one (a subdominant DM component). We summarize the results of our benchmark point study in Table 2. Several parameter sets in each benchmark point are classified in light of the detectability of the GWs at the future interferometers, the DM direct detection constraints by XENON1T. The patterns of PT are detailed as follows: • One-step phase transition: In the U (1) X gauge model, the limit of g X → 0 corresponds to the Higgs singlet model (HSM) with the spontaneously broken Z 2 symmetry, where a real isospin scalar singlet S is introduced in addition to the Higgs doublet Φ. In this case, the 1stOPT is realized as the one-step PT with type-D in Fig. 1. As mentioned in Ref. [36], however, such a case is excluded by the collider bounds. In contrast, we can find some points satisfying the collider bounds in the U (1) X model by the existence of the dark photon (X 0 -boson) contribution. However, the strength of the PT is not so strong and it is not enough to detect by the future GW observations as shown by the blue point in Fig. 4 and Figs. 5-9.
• Two-step phase transition: There are two cases for two-step PT with the first order EWPT with type-C in Fig. 1: "2nd order → 1st order" or "1st order → 1st order". In the former case, EWPT with 1stOPT is shown by the triangle point in Figs. 3-9. In the latter case, two 1stOPTs can be calculated as shown by star and triangle point connected by the dashed line in Fig. 4 and Figs. 5-9. In most of the parameter region, the 1stOPT is strong and it can be detected by the future GW observations.
As we can see from the results, large values of m X ( 25 GeV) and large values of g X ( 0.5) are preferred for detectable GW signals. We can understand analytically that m X is correlated with the detectability of GWs as follows. The difference of the vacuum energies of the I-and the EW phases as well as that of the II-and the EW phases are given by [101] (4.4) and (4.5) respectively, withv Using Eq. (4.1), the EW vacuum becomes always the global minimum at the tree level. On the other hand, the latent heat is approximately given by the difference of the potential minima between the false vacuum and the true vacuum. As we know in our numerical results, since the detectable GWs are realized by the transitions of type-C in this model, the typical strength of the GWs is parametrized by This correlation shows that α is controlled by m X . In general, the strength of the GWs as the functions of α and β, which is given in Ref. [76], is enhanced by increasing α.
In addition, we find that g X also contributes to the GW detectability as shown in our numerical results, e.g. by comparing Fig. 7 and Fig. 8. Notice that large values of g X (small values of v S ) also give the upper bound of m H by the perturbativity condition (see Appendix A). As the result, the lower bound is at least m X > 100 GeV, but m X > 25 GeV can be possible depending on g X contribution at narrow parameter space shown in the left frame of Fig. 7. DECIGO is capable of detecting stochastic GWs from the sound wave source in a part of the model parameter region with the strongly 1stOPT. It might be challenging to detect by LISA because all points are in the β > 10 3 region.

Discussion and conclusions
In the following, we summarize our results and discuss some relevant points one by one.
In this paper, we define the Landau pole Λ LP as the scale where any of the Higgs couplings is as strong as [23] |λ Φ,S,ΦS (Λ LP )| = 4π. (5.1) The one-loop level β functions for these couplings are provided in Appendix B. In the U (1) X gauge model, the Landau pole can be above the Planck scale as discussed in Ref. [102]. Since our model has only one Φ-S mixing term, namely |Φ| 2 |S| 2 , we need large Higgs couplings for strongly first order EWPT to occur. Then, the Landau pole Λ LP appears at around O(10 4 ) GeV 7 . In contrast, if there are two Φ-S mixing terms, e.g. |Φ| 2 |S| 2 and |Φ| 2 S in the Higgs singlet model, the Landau pole can be as large as Λ LP ∼ O(10 14 ) GeV [36]. In our U (1) X gauge model, two-step PT along the path of the type C is realized in most of the parameter points predicting first order EWPT and detectable GWs, As discussed in Ref. [42], even if EWPT is strong enough to suppress the sphaleron process after the transition, the type C transition cannot produce a sufficient amount of baryon asymmetry. In view of this, we have not imposed the condition for strongly 1stOPT, with ζ sph being typically close to the unity. The wall velocity v b is a key parameter describing the dynamics of the bubble wall. In generic, there is a tension between strong GWs and baryon asymmetry in EWBG scenarios. The amplitude of GWs is suppressed by v p b with p 3 for small wall velocity. Large wall velocity v b ∼ 1 is preferred for detecting GWs. On the other hand, successful EWBG scenarios favor lower wall velocity v b 0.15 − 0.3 (for the calculation of v b in the singlet models, see Ref. [132]), which allows the effective diffuse of particle asymmetries near the bubble wall front [133]. In Ref. [134], however, it is pointed out that EWBG is not necessarily impossible even in the case with large v b . Further discussion is beyond the scope of this paper.
In Ref. [103], the complementarity of dark photon searches and GW observations is discussed for the mass region m X ≃ O(10 −2 − 1) GeV in the cases with strongly 1stOPT in Model A. However, the collider constraints are not properly included. Taking these collider bounds on the Higgs boson properties into consideration, we have found that GW signals are detectable only for larger dark photon mass, say m X O(25 − 100) GeV. As shown in Ref. [135], the recent data from LHCb [136] and LHC Run-II [137] give constraints on ǫ, which is roughly smaller than 10 −2 at least, for the mass regions 10.6 GeV < m X < 70 GeV and 150 GeV < m X < 350 GeV, respectively. It is also shown that the mass region 20 GeV < m X < 330 GeV can be constrained at future lepton colliders in Ref. [135]. We expect that 1stOPT with such a heavy dark photon will be tested by synergy between future GW observations and dark photon searches.
As for Model B with VDM, constraints on DM properties are discussed in Appendix C and shown in Fig. 3 and Figs. 5-9. If the thermal relic density of the VDM fulfills the total amount of the observed DM, GWs signals cannot reach the future sensitivities of GW observations (solution 2 in Table 2). On the other hand, in the case where only a fraction of the total DM is composed of VDM, there is an allowed region than can be probed by GW observations (solution 1 in Table 2).
In conclusion, we have comprehensively explored models with a dark photon whose mass stems from spontaneous U (1) X gauge symmetry breaking by the nonzero VEV of the dark Higgs boson S in light of the patterns of PT and the detectability of GWs from strongly 1stOPT as well as various collider and theoretical bounds. After imposing these constraints on the model parameter space, we have found that GWs produced from multistep PT can be detected at future observations such as LISA and DECIGO if the dark photon mass is m X 25 GeV with the U (1) X gauge coupling being g X 0.5. Some of the parameter regions predicting detectable GWs are covered by the measurements of ∆κ V and ∆λ hhh at future colliders including the HL-LHC and ILC. The model where the dark photon becomes a candidate for DM has been also investigated in view of the thermal relic density and the current constraint by DM direct detection. In order for the future interferometers to observe GW signals, the VDM component should be at most about 10% of the total DM abundance. Our results have been summarized in Table 2

A Perturbative unitarity
Here, we discuss constrains from perturbative unitarity using analytic formulae at the tree level for illustrating the behavior of the model parameters. The Higgs couplings in Eqs. (2.7) are expressed in terms of the masses of the Higgs bosons and the mixing angle as Then, the constraints obtained from Eq. (4.2) can be projected on the (m H , θ) plane as shown in Fig. 10. One can see that the excluded regions (indigo) in Fig. 3 (v S = 100 GeV) and Figs. 5, 7 and 9 (v S = 50 GeV) are consistent with the corresponding contours in Fig. 10.

C Dark matter relic abundance and direct detection
The parameter space of Model B is constrained in light of the thermal DM relic abundance and direct detection. The observed DM abundance reported by the Planck collaboration is [138] Ω obs h 2 = 0.1199 ± 0.0022. where µ X = m X m p /(m X + m p ) is the reduced mass of the DM and the proton, and f p = q=u,d,s f p q + (2/9)(1 − q=u,d,s f p q ) ≈ 0.468 [141]. The public code micrOMEGAs 4.3.2 [142] is used to calculate the thermal VDM relic density Ω X h 2 and the cross section σ X in this paper. Fig. 11 shows the contours of g X (black) and σ X = 10 −46 cm 2 (red) on the (m X , m H ) plane. Here, we assume that the VDM accounts for all the DM energy density i.e. Ω X = Ω obs , and take θ = 5 • as a reference point. The observed thermal relic density can be explained at the Higgs poles, m X = m h /2 and m X = m H /2 9 as well as for larger VDM masses m X > m h (see Fig. 3). In the latter case, the channels XX → hh, HH, hH contribute to the DM annihilation cross section, which is enhanced as g X increases (see the figures of Ω X h 2 plotted as a function of m X in Refs. [101,102]). Figure 11. Contours of g X (black) and σ X = 10 −46 cm 2 (red) on the (m X , m H ) for Ω X = Ω obs and θ = 5 • .