Selecting models of first-order phase transitions using the synergy between collider and gravitational-wave experiments

We investigate the sensitivity of future space-based interferometers such as LISA and DECIGO to the parameters of new particle physics models which drive a first-order phase transition in the early Universe. We first perform a Fisher matrix analysis on the quantities characterizing the gravitational-wave spectrum resulting from the phase transition, such as the peak frequency and amplitude. We next perform a Fisher analysis for the quantities which determine the properties of the phase transition, such as the latent heat and the time dependence of the bubble nucleation rate. Since these quantities are determined by the model parameters of the new physics, we can estimate the expected sensitivities to such parameters. We illustrate this point by taking three new physics models for example: (i) models with additional isospin singlet scalars, (ii) a model with an extra real Higgs singlet, and (iii) a classically conformal $B-L$ model. We find that future gravitational-wave observations play complementary roles to future collider experiments in pinning down the parameters of new physics models driving a first-order phase transition.

Figure 1

Sensitivity curves for LISA (green solid), DECIGO (green dashed), and BBO (green dotted). Blue curves correspond to the GW spectra for the sample points 1–4 in the main text. Red lines show the contribution from compact white dwarf binaries $S_{\mathrm{WD}}$.

Figure 2

$1\sigma$ contours for points 1 (left column) and 2 (right column) for LISA (top), DECIGO (middle), and BBO (bottom). Three contours in each panel correspond to $T_{\mathrm{obs}}=1$, 3, and 10 yr, respectively. The spectral slopes are taken to be $(n_L,n_R)=(3,-4)$.

Figure 3

$1\sigma$ contours for points 3 (left column) and 4 (right column). Otherwise, the same as Fig. 2.

Figure 4

$1\sigma$ fractional error $\mathrm{\Delta }{f}_{\mathrm{peak}}/{\widehat{f}}_{\mathrm{peak}}$ (left) and $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{GW},\mathrm{peak}}/{\widehat{\mathrm{\Omega }}}_{\mathrm{GW},\mathrm{peak}}$ (right) for the fiducial values ${\widehat{f}}_{\mathrm{peak}}$ and ${\widehat{\mathrm{\Omega }}}_{\mathrm{GW},\mathrm{peak}}$ for LISA (top), DECIGO (middle), and BBO (bottom). The spectral slopes and observation period are taken to be $(n_L,n_R)=(3,-4)$ and $T_{\mathrm{obs}}=1\mathrm{yr}$.

Figure 5

The same as Fig. 4, except that the foreground is taken to be $S_{\mathrm{WD}}+S_{\mathrm{NSBH}}$.

Figure 6

Sensitivity curves for LISA (green solid), DECIGO (green dashed), and BBO (green dotted). The blue lines are signals from sound waves (2.36) (blue dashed) and turbulence (2.39) (blue dotted). The red lines show the foreground from compact white dwarf binaries $S_{\mathrm{WD}}$. Each panel corresponds to points A–C in the main text from left to right.

Figure 7

$1\sigma$ contours for point A for LISA (top), DECIGO (middle), and BBO (bottom). The left and right columns correspond to two- and three-parameter analysis, respectively. The three contours in each panel correspond to $T_{\mathrm{obs}}=1$, 3, and 10 yr, respectively.

Figure 8

$1\sigma$ contours for point B. Otherwise, the same as Fig. 7.

Figure 9

$1\sigma$ contours for point C. Otherwise, the same as Fig. 7.

Figure 10

Comparison between different values for the turbulence fraction $\epsilon =0.1$ (blue) and 0.5 (red). The left and right columns are the results for two-parameter analysis for point A and B, respectively. The blue contours are the same as the left columns in Figs. 7 and 8.

Figure 11

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for the fiducial values $\widehat{\alpha }$ and $\widehat{\beta }$ for two-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). The wall velocity is taken to be $v_w=1$.

Figure 12

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for three-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). Otherwise, the same as Fig. 11.

Figure 13

Gravitational-wave spectra from sound waves (blue dashed line) and turbulence (blue dotted line) for the parameter point in Sec. 5a. The panels correspond to the model with CCI (left, $N=2$) and without CCI (center, $N=8$; right, $N=12$), respectively. The upper panels correspond to $v_w=0.95$, while $v_w=0.1$ for the lower panels.

Figure 14

(Left) $1\sigma$ contours for $O(N)$ singlet extensions of the SM with CCI for LISA. (Center, right) $1\sigma$ contours for $O(N)$ singlet extensions of the SM without CCI ($N=8$, 12) for LISA.

Figure 15

(Left) $1\sigma$ contours for $O(N)$ singlet extensions of the SM with CCI for DECIGO (top) and BBO (bottom). (Center, right) $1\sigma$ contours for $O(N)$ singlet extensions of the SM without CCI ($N=8$, 12) for DECIGO (top) and BBO (bottom).

Figure 16

LISA $1\sigma$ contours for the $O(N)$ singlet models with and without CCI in the $\alpha \text{−}\beta /H_{*}$ plane. The red points correspond to $N=1$, 2, 4, 12, and 60 for the $O(N)$ models with CCI from left to right, while the gray points correspond to $N=1$, 2, 4, 12, and 60 with ${\mu }_S^2=0{\mathrm{GeV}}^2$ for the $O(N)$ models without CCI. The fiducial values for the red contours correspond to the $O(N)$ models with CCI with $N=2$, while the ones for the blue contours correspond to the $O(N)$ models without CCI with $N=8$ ($N=12$) and ${\mu }_S^2=0{\mathrm{GeV}}^2$ in the left (right) panel. Also, the three contours correspond to $T_{\mathrm{obs}}=1$, 3 and 10 yr, respectively.

Figure 17

(Left) GW spectrum from sound waves (blue dashed line) and turbulence (blue dotted line) for the parameter point in Sec. 5b with $v_w=0.95$. (Right) $1\sigma$ contours in the $m_H\text{−}\kappa$ plane for LISA. The narrow contours correspond to fixed ${\mu }_{\mathrm{\Phi }S}$, while the wide contours correspond to marginalized ${\mu }_{\mathrm{\Phi }S}$. In drawing both contours, $v_S$ and ${\mu }_S^{\prime }$ are fixed to be the fiducial values.

Figure 18

(Top) GW spectra from sound waves (blue dashed line) and turbulence (blue dotted line) for the parameter point in Sec. 5b with $v_w=0.1$. (Bottom) $1\sigma$ contours in the $m_H\text{−}\kappa$ plane for DECIGO (left) and BBO (right). Otherwise, the same as the right panel in Fig. 17.

Figure 19

(Blue) LISA $1\sigma$ contours for the real Higgs singlet model with the fiducial values $m_H=166.4\mathrm{GeV}$ and $\kappa =0.96$. Narrow and wide contours correspond to fixed and marginalized ${\mu }_{\mathrm{\Phi }S}$, respectively. The same as Fig. 17. (Green) The region where the condition for a strongly first-order phase transition is satisfied. (Yellow) ILC $1\sigma$ sensitivity region with $\sqrt{s}=250\mathrm{GeV}$ and $\mathcal{L}=2{\mathrm{ab}}^{-1}$. (Gray) The region excluded by the direct search for a heavy Higgs [160].

Figure 20

Contours of the temperature just before the transition $T_{*}$ (left) and the temperature just after the transition $T_R$ (right) in units of GeV. The regions shaded in red, green, and yellow show the region which develops a Landau pole below the Planck scale, the region excluded by $Z^{\prime }$ search, and the region where the transition does not complete, respectively.

Figure 21

Contours of the latent heat fraction $\alpha$ (left) and bubble nucleation speed $\beta /H_{*}$ (right). The regions shaded in red, green, and yellow are the same as in Fig. 20.

Figure 22

(Left) GW spectrum from sound waves (blue dashed line) and turbulence (blue dotted line) for point 1 in Sec. 5c. (Right) GW spectrum for point 2. Otherwise, the same as the left panel.

Figure 23

$1\sigma$ contours for point 1 for LISA (top), DECIGO (middle), and BBO (bottom). The left and right columns correspond to point 1 and 2, respectively. The three contours in each panel correspond to $T_{\mathrm{obs}}=1$, 3, and 10 yr, respectively.

Figure 24

$1\sigma$ fractional error for $\mathrm{\Delta }M/\widehat{M}$ (left) and $\mathrm{\Delta }{\alpha }_{B-L}/{\widehat{\alpha }}_{B-L}$ (right) for the fiducial values $\widehat{M}$ and ${\widehat{\alpha }}_{B-L}$. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). Regions shaded in red, green, and yellow are the same as Figs. 20 and 21.

Figure 25

Sensitivity curves for LISA (green solid), DECIGO (green dashed), and BBO (green dotted). Blue curves correspond to the sample points 1–4 in Sec. 3 in the main text. Red lines show the contribution from compact white dwarf binaries $S_{\mathrm{WD}}$.

Figure 26

$1\sigma$ fractional error $\mathrm{\Delta }{f}_{\mathrm{peak}}/{\widehat{f}}_{\mathrm{peak}}$ (left) and $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{GW},\mathrm{peak}}/{\widehat{\mathrm{\Omega }}}_{\mathrm{GW},\mathrm{peak}}$ (right) for the fiducial values ${\widehat{f}}_{\mathrm{peak}}$ and ${\widehat{\mathrm{\Omega }}}_{\mathrm{GW},\mathrm{peak}}$ for LISA (top), DECIGO (middle), and BBO (bottom). The spectral slopes and foreground are taken to be $(n_L,n_R)=(1,-3)$ and $S_{\mathrm{WD}}$.

Figure 27

$1\sigma$ fractional error $\mathrm{\Delta }{f}_{\mathrm{peak}}/{\widehat{f}}_{\mathrm{peak}}$ (left) and $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{GW},\mathrm{peak}}/{\widehat{\mathrm{\Omega }}}_{\mathrm{GW},\mathrm{peak}}$ (right) for the fiducial values ${\widehat{f}}_{\mathrm{peak}}$ and ${\widehat{\mathrm{\Omega }}}_{\mathrm{GW},\mathrm{peak}}$ for LISA (top), DECIGO (middle), and BBO (bottom). The spectral slopes and foreground are taken to be $(n_L,n_R)=(1,-3)$ and $S_{\mathrm{WD}}+S_{\mathrm{NSBH}}$.

Figure 28

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for the fiducial values $\widehat{\alpha }$ and $\widehat{\beta }$ for two-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). The wall velocity and foreground are taken to be $v_w=1$ and $S_{\mathrm{WD}}+S_{\mathrm{NSBH}}$.

Figure 29

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for three-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). Otherwise, the same as Fig. 28.

Figure 30

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for the fiducial values $\widehat{\alpha }$ and $\widehat{\beta }$ for two-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). The wall velocity and foreground are taken to be $v_w=0.3$ and $S_{\mathrm{WD}}$.

Figure 31

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for three-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). Otherwise, the same as Fig. 30.

Figure 32

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for the fiducial values $\widehat{\alpha }$ and $\widehat{\beta }$ for two-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). The wall velocity and foreground are taken to be $v_w=0.3$ and $S_{\mathrm{WD}}+S_{\mathrm{NSBH}}$.

Figure 33

$1\sigma$ fractional error $\mathrm{\Delta }\alpha /\widehat{\alpha }$ (left) and $\mathrm{\Delta }\beta /\widehat{\beta }$ (right) for three-parameter analysis. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). Otherwise, the same as Fig. 32.

Figure 34

LISA $1\sigma$ contours for the $O(N)$ singlet models with and without CCI in the $\alpha \text{−}\beta /H_{*}$ plane. The same as the left panel in Fig. 16 ($N=2$ and $N=8$ with and without CCI, respectively) except that the result of the three-parameter analysis is also shown.

Figure 35

LISA $1\sigma$ contours for the $O(N)$ singlet models with and without CCI in the $\alpha \text{−}\beta /H_{*}$ plane. The same as the right panel in Fig. 16 ($N=2$ and $N=12$ with and without CCI, respectively) except that the result of the three-parameter analysis is also shown.

Figure 36

$1\sigma$ fractional error for $\mathrm{\Delta }M/\widehat{M}$ (left) and $\mathrm{\Delta }{\alpha }_{B-L}/{\widehat{\alpha }}_{B-L}$ (right) for the fiducial values $\widehat{M}$ and ${\widehat{\alpha }}_{B-L}$. Each row corresponds to LISA (top), DECIGO (middle), and BBO (bottom). Regions shaded in red, green, and yellow are the same as in Figs. 20 and 21. The foreground is taken to be $S_{\mathrm{WD}}+S_{\mathrm{NSBH}}$.

Figure 37

LISA sensitivity curve (green solid), white dwarf foreground (red solid), and signals from sound waves (blue dashed) and from turbulence (blue, red, and green dotted). The left and right panels are for point A and B in Sec. 4, respectively. For the turbulence signal, we change $\epsilon =0.1$, 1, and 10 from bottom to top.

Figure 38

Results for the three-parameter analysis for the setup given in Fig. 37. The left and right columns are for points A and B, respectively, and each row corresponds to $\epsilon =0.1$, 1, and 10 from top to bottom.