Stasis, stasis, triple stasis: A theoretical study of cosmological stasis

Many theories of physics beyond the Standard Model predict the existence of large or infinite towers of decaying states. In a previous paper [K. R. Dienes et al., Phys. Rev. D 105, 023530 (2022)], we pointed out that this can give rise to a surprising cosmological phenomenon that we dubbed “stasis” during which the relative abundances of matter and radiation remain constant across extended cosmological eras even though the universe is expanding. Indeed, such stasis epochs are universal attractors, with the universe necessarily entering (and later exiting) such epochs for a wide variety of initial conditions. Matter/radiation stasis is therefore an important and potentially unavoidable feature of many BSM cosmologies. In this paper we extend our arguments to universes containing significant amounts of vacuum energy, and demonstrate that such universes also give rise to various forms of stasis between vacuum energy and either matter or radiation. We also demonstrate the existence of several forms of “triple stasis” during which the abundances of matter, radiation, and vacuum energy all simultaneously remain fixed despite cosmological expansion. We further describe several close variants of stasis which we call “quasi-stasis” and “oscillatory stasis,” and discuss the circumstances under which each of these can arise. Finally, we develop a general formalism for understanding the emergence of stasis within BSM cosmologies irrespective of the number or type of different energy components involved. Taken together, these results greatly expand the range of theoretical and phenomenological possibilities for the physics of the early universe, introducing new types of cosmological eras which may play an intrinsic and potentially inevitable role within numerous BSM cosmologies.

Figure 1

Matter/radiation stasis. Left panel: the individual matter abundances ${\mathrm{\Omega }}_{\ell }$ (shown with colors ranging from orange to blue) and the corresponding total matter abundance ${\mathrm{\Omega }}_M$ (red), plotted as functions of the number $\mathcal{N}$ of $e$-folds since the initial time of production. Even though the individual abundances ${\mathrm{\Omega }}_{\ell }$ exhibit complex behaviors which are affected by cosmological expansion as well as ${\varphi }_{\ell }$ decay, the system quickly evolves into a stasis state in which their sum ${\mathrm{\Omega }}_M$ becomes constant. These curves were generated through a direct numerical solution of the Boltzmann equations corresponding to our discrete tower of decaying states without invoking any approximations, and correspond to the parameter choices $(\alpha ,\gamma ,\delta )=(1,7,1)$—for which ${\overline{\mathrm{\Omega }}}_M=1/2$—with $\mathrm{\Delta }m=m_0$, $N=300$, and ${\mathrm{\Gamma }}_{N-1}/H^{(0)}=0.01$. Right panel: the total matter abundances ${\mathrm{\Omega }}_M$, plotted as functions of $\mathcal{N}$ for different values of $(\alpha ,\gamma )$ satisfying the constraints in Eq. (2.23). For each plot we have taken $\delta =1$, $\mathrm{\Delta }m=m_0$, $N={10}^5$, and $H^{(0)}/{\mathrm{\Gamma }}_{N-1}=0.1$. In each case we see that the system settles into a prolonged stasis epoch lasting many $e$-folds, with a corresponding stasis abundance ${\overline{\mathrm{\Omega }}}_M$ predicted by Eq. (2.22). Ultimately, in all cases, the stasis epoch ends as we approach the final decays of the lightest modes. Both figures are taken from Ref. [1].

Figure 2

Two different scenarios illustrating possible behaviors for the individual energy densities ${\rho }_{\ell }(t)$ (blue/orange), sketched as functions of time, as our corresponding tower of states ${\varphi }_{\ell }$ undergoes sequential transitions from an overdamped phase to an underdamped phase at times $t_{\ell }$ for which $3H(t_{\ell })=2m_{\ell }$. These energy densities ${\rho }_{\ell }(t)$ are interpreted as corresponding to vacuum energy (blue) or matter (orange) when the corresponding states ${\varphi }_{\ell }$ are overdamped or underdamped, respectively, with the dashed line (red) indicating the moments of transition between these two regimes. For the purpose of these sketches we have assumed that states with greater energy densities have greater masses (i.e., $\alpha >0$). We have also chosen to sketch the case in which $w\to -1$ for the vacuum energy, causing the blue lines to be almost exactly horizontal. The left and right panels show the behaviors that emerge for $\overline{\kappa }<\alpha$ and $\overline{\kappa }>\alpha$, respectively. In the latter case, the energy densities “reflect” off the red transition line, thereby giving rise to the sequential “interleaved” energy-density crossings that are the hallmark of the stasis phenomenon.

Figure 3

Vacuum-energy/matter stasis. Here the total vacuum-energy and matter abundances, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}(t)$ and ${\mathrm{\Omega }}_M(t)$ respectively, are plotted as functions of the number $\mathcal{N}$ of $e$-folds since the initial production time $t^{(0)}$, taking $\alpha =0.7$, $\delta =2$, and $w=-0.8$ as reference values. We have also taken $H^{(0)}/m_{N-1}=0.75$. As we see, the system eventually evolves into a stasis state with $\overline{\kappa }=4$, ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}=5/8$, and ${\overline{\mathrm{\Omega }}}_M=3/8$. This stasis state ends only when the last component of the tower transitions to matter. As with matter/radiation stasis, the total number of $e$-folds of stasis ${\mathcal{N}}_s$ is determined by the total number $N$ of states in the tower. In analogy with Eq. (2.24) for matter/radiation stasis, we expect ${\mathcal{N}}_s\sim \log N$. Similar stasis behavior emerges for all values of $(\alpha ,\delta ,w)$ within the ranges in Eq. (3.24), with corresponding stasis abundances ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$ determined by Eq. (3.25) and ${\overline{\mathrm{\Omega }}}_M=1-{\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$.

Figure 4

Vacuum-energy/radiation stasis. Here we plot the total vacuum-energy and radiation abundances, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ and ${\mathrm{\Omega }}_{\gamma }$ respectively, as functions of time, taking $\alpha =0.5$, $\delta =2$, $\gamma =1$, $w=-0.7$, and $H^{(0)}/{\mathrm{\Gamma }}_{N-1}=20$ as reference values and working with the exponential-decay formalism. As we see, these benchmark values lead to a stasis with $\overline{\kappa }=10/3$, implying ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}=22/31\approx 0.71$ and ${\overline{\mathrm{\Omega }}}_{\gamma }=9/31\approx 0.29$. This stasis state ends only when the last component of the tower decays to radiation. Similar stasis behavior emerges for all values of $(\alpha ,\delta ,\gamma ,w)$ within the range in Eq. (4.7), with corresponding stasis abundances ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$ determined by Eq. (4.8).

Figure 5

Left panel: the individual energy densities ${\rho }_{\ell }(t)$ for the states ${\varphi }_{\ell }$ with $m_{\ell }<m_X$, sketched as functions of time during a stasis epoch with $\gamma >1$ and $\overline{\kappa }>\alpha$. Each state undergoes a transition from an overdamped phase (blue) to an underdamped phase (yellow) at a time $t_{\ell }$ for which $3H(t_{\ell })=2m_{\ell }$ before subsequently decaying to radiation (green) at the time ${\tau }_{\ell }\equiv 1/{\mathrm{\Gamma }}_{\ell }$. The dashed lines (red) indicate the moments of transition between these different behaviors and have slopes given in Eq. (6.17). For the purposes of this sketch we have assumed that states with greater energy densities have greater masses (i.e., that $\alpha >0$). We have also assumed that $H(t)\sim 1/t$ with a fixed constant of proportionality, as appropriate during stasis, and we have taken $w\approx -1$ so that our vacuum-energy lines are essentially horizontal. We see by considering vertical time slices through this figure that the identity of the state with the greatest energy density is time dependent, with lighter and lighter states carrying increasing shares of the total energy density as time evolves. Right panel: same as left panel, but now sketched for $\overline{\kappa }$ within the range $\varphi \alpha <\overline{\kappa }<\alpha$ where $\varphi \equiv (1+\gamma /3{)}^{-1}$.

Figure 6

Left panel: same as Fig. 5, but now sketched for $\gamma <1$. Here it is the heavier portion of the tower with $m>m_X$ for which the states experience transitions from overdamped to underdamped behavior before decaying to radiation. Right panel: same as left panel, but for $\overline{\kappa }$ within the range $\varphi \alpha <\overline{\kappa }<\alpha$ where $\varphi \equiv (1+\gamma /3{)}^{-1}$.

Figure 7

Left panel: same as Figs. 5 and 6, but now sketched for $\gamma =1$. As long as $\xi >1$, each component across the entire tower experiences a transition from overdamped to underdamped behavior before decaying to radiation. Right panel: same as left panel, but now sketched for $\overline{\kappa }$ within the range $3\alpha /4<\overline{\kappa }<\alpha$.

Figure 8

Triple stasis in action: the structure of the tower ${\varphi }_{\ell }$ during a potential triple-stasis epoch. The individual states are shown in order of increasing $\ell$, just as they are at the initial time in Figs. 5 through 7, with colors determined by considering appropriate vertical time slices through these figures as time evolves. At any moment two separate transitions are occurring at different locations within the tower: an underdamping transition from vacuum energy (blue) to matter (yellow) occurring at a lower location within the tower, and a decay transition from matter (yellow) to radiation (green) simultaneously occurring at a higher location. As time evolves, both transitions work their way down the tower, with the color of a given state changing from blue to yellow and then from yellow to green as each of the two transitions sweeps past it. For $\gamma >1$ the decay transition proceeds down the tower more slowly than the underdamping transition, thereby allowing the matter region of the tower to continually increase in size. By contrast, for $\gamma <1$ the decay transition proceeds more quickly down the tower than the underdamping transition, ultimately catching up with it at $m=m_X$. For $\gamma =1$ both transitions move down the tower at the same rate. All of this occurs within an expanding universe, thereby potentially supporting a triple-stasis epoch.

Figure 9

Two branches of solutions to Eqs. (6.36). These branches are defined as in Eq. (6.37) with the convention that the point with $(\gamma ,\overline{\kappa })=(1,2)$ is considered to be part of Branch A but not Branch B. With this convention the two branches are mutually exclusive.

Figure 10

An illustration of triple stasis. The abundances ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ (solid dark blue curves), ${\mathrm{\Omega }}_M$ (solid cyan curves), and ${\mathrm{\Omega }}_{\gamma }$ (solid magenta curves) are plotted as functions of the number $\mathcal{N}$ of $e$-folds of cosmic expansion since the production time $t^{(0)}$. In each case these abundances approach and then remain fixed at the corresponding stasis values (dotted lines) predicted in Eqs. (6.58) and (6.59), with $X$ given in Eq. (6.61). These curves are calculated for reference values $\alpha =2/3$ (left panel), 1 (middle panel), and 1.1 (right panel), holding $\delta =3$, $\gamma =1$, and $w=-2/3$ fixed across all panels. The full exponential nature of the decay process was taken into account in modeling the transition of energy density from matter to radiation. These choices lead to stasis values $\overline{\kappa }=3$, 2, and 1.7, respectively, and we have adjusted ${\mathrm{\Gamma }}_0/m_0$ in each case in order to hold $\xi =20$ fixed for all panels. These figures confirm that we can achieve triple stasis for $\overline{\kappa }>2$, $\overline{\kappa }=2$, and $\overline{\kappa }<2$, respectively, and that it takes less time to achieve triple stasis as $\overline{\kappa }$ increases.

Figure 11

Top panels: the stasis abundances ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$ (blue), ${\overline{\mathrm{\Omega }}}_M$ (cyan), and ${\overline{\mathrm{\Omega }}}_{\gamma }$ (magenta), plotted as functions of $\eta$ for different values of $w$ with $\xi$ held fixed. The solid curves are calculated assuming full exponential decay, while the dashed curves are calculated within the instantaneous-decay approximation. The gray region on the right side of each panel is excluded by the constraint $0<\eta \le (1-3w)/2$, which follows from the constraint $\overline{\kappa }>3/2$ in Eq. (6.49). Bottom panels: the stasis abundances ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$ (blue), ${\overline{\mathrm{\Omega }}}_M$ (cyan), and ${\overline{\mathrm{\Omega }}}_{\gamma }$ (magenta), plotted as functions of $\xi$ for different values of $\eta$ with $w$ held fixed. The gray regions on the left sides of these panels correspond to $\xi <{\xi }_{\min }$ and are thus outside our regime of validity. In the instantaneous-decay approximation, we see that ${\overline{\mathrm{\Omega }}}_M\to 0$ as $\xi \to 0$ irrespective of the value of $\eta$, whereupon our triple stasis reduces to a pairwise stasis involving vacuum energy and radiation. By contrast, the behavior of the abundances as $\xi \to \infty$ limit depends on the relationship between $\eta$ and $w$. For $\eta <2w$, as illustrated in the left and middle panels, ${\overline{\mathrm{\Omega }}}_{\gamma }\to 0$ and the triple stasis reduces to a pairwise stasis involving vacuum energy and matter. By contrast, for $\eta >2w$, as illustrated in the right panel, ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}\to 0$ and the triple stasis reduces to a pairwise stasis involving matter and radiation.

Figure 12

Triple stasis “under the hood,” illustrating how the individual abundances ${\mathrm{\Omega }}_{\ell }(t)$ conspire to bring our system into—and then sustain—triple stasis. Working within the framework of the instantaneous-decay approximation and choosing $(\alpha ,\gamma ,\delta ,w)=(1,1,5,-0.7)$ in order to highlight critical features, we plot the three total abundances ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ (thick dark-blue solid line), ${\mathrm{\Omega }}_M$ (thick cyan solid line), and ${\mathrm{\Omega }}_{\gamma }$ (thick magenta solid line) as functions of the number $\mathcal{N}$ of $e$-folds since the initial production time. We also plot the individual abundances ${\mathrm{\Omega }}_{\ell }(t)$ (thin lines) as they evolve in time, starting as vacuum energy (dark blue) before transitioning to matter (cyan) and eventually to radiation (magenta). For the sake of graphical clarity we have only shown every 20th individual abundance ${\mathrm{\Omega }}_{\ell }(t)$, starting from the top of the tower, and adjusted ${\mathrm{\Gamma }}_0/m_0$ such that $\xi =300$. The underdamping transition between the vacuum-energy and matter phases occurs along the upper dashed black line, while the instantaneous-decay transition between the matter and radiation phases occurs along the lower dashed black line. This figure can be viewed as an explicit numerical realization of Fig. 7 except that it is plotted for individual abundances ${\mathrm{\Omega }}_{\ell }(t)$ rather than individual energy densities ${\rho }_{\ell }(t)$ and also includes the initial transient behavior of our system as it evolves from the production time $t^{(0)}$ into stasis. The red transition lines in Fig. 7 correspond to the dashed black lines here, asymptotically becoming parallel once stasis is reached. This figure can also be viewed as a triple-stasis analog of the left panel of Fig. 1.

Figure 13

A version of Fig. 12 illustrating that at any sufficiently late time $t$ during triple stasis it is possible to choose an earlier fiducial time $t_{*}$—also during triple stasis—such that all states ${\varphi }_{\ell }$ with significant abundances ${\mathrm{\Omega }}_{\ell }$ at time $t$ are still overdamped at $t_{*}$. Thus, when studying the properties of stasis at any sufficiently late time $t$ during triple stasis, it is a legitimate approximation to assume that all relevant states ${\varphi }_{\ell }$ have $t_{\ell }>t_{*}$, with all other states from Fig. 12 disregarded and hence shown in gray.

Figure 14

The “seesaw” structure of triple stasis with $\overline{\kappa }=2$ along Branch A in which the abundances of the different energy components along the $w$-axis are balanced around a “fulcrum” located at $⟨w⟩=w_{△}=0$. The location of the fulcrum corresponds to a $\overline{\kappa }=2$ universe which is effectively matter dominated. The pumps $P_{\mathrm{\Lambda }M}$ and $P_{M\gamma }$ (shown schematically in blue) transfer energy from vacuum energy to matter and from matter to radiation, respectively. These pumps exactly compensate for the effects of cosmological expansion wherein the effects due to gravitational redshifting (shown schematically in pink) either increase or decrease the relative abundances of these different components according to their distances from the fulcrum at $w_{△}=0$. Changing the value of $w$ for the vacuum energy component (shown schematically in green) corresponds to “sliding” the position of the corresponding abundance along the $w$-seesaw. This abundance is then rescaled according to Eq. (6.81) in order to maintain the balancing of the $w$-seesaw. However, this increased abundance experiences a weakened gravitational redshifting and thus the delicate balancing inherent in triple stasis is maintained.

Figure 15

Same as Fig. 14, except for stasis solutions with $\overline{\kappa }\ne 2$. Upper panel: in this case $\overline{\kappa }<2$, implying that $w_{△}>0$. This causes ${\mathrm{\Omega }}_M$ to tend to increase under the effects of cosmological expansion, which implies that stasis is achieved only for $P_{\mathrm{\Lambda }M}<P_{M\gamma }$. This inequality of the pumps is consistent with the fact that ${\mathrm{\Omega }}_{\mathrm{\Lambda }}/{\mathrm{\Omega }}_{\gamma }$ must be smaller than it was in Fig. 14 in order to achieve a balanced “seesaw.” Lower panel: the same situation, but for $\overline{\kappa }>2$. In this case the signs of all inequalities are reversed, with $P_{\mathrm{\Lambda }M}>P_{M\gamma }$ and ${\mathrm{\Omega }}_{\mathrm{\Lambda }}/{\mathrm{\Omega }}_{\gamma }$ now larger than it was in Fig. 14.

Figure 16

Attractor behavior for $M\gamma$ stasis (left panel), $\mathrm{\Lambda }M$ stasis (middle panel), and $\mathrm{\Lambda }\gamma$ stasis (right panel). Shown in each case are the trajectories (blue curves) within the (${\mathrm{\Omega }}_1$,$\mathcal{H}$)-plane, where ${\mathrm{\Omega }}_1$ in each case represents the abundance of the cosmological energy component with the smaller equation-of-state parameter. These trajectories correspond respectively to the dynamical system described in Eq. (7.9) with $w_1=0$ and $w_2=1/3$, the dynamical system described in Eq. (7.24) with $w_1=w=-0.8$ and $w_2=0$, and the dynamical system described in Eq. (7.9) with $w_1=w=-0.8$ and $w_2=1/3$. The red dot in each case indicates the corresponding attractor stasis solution. We have chosen $p$ in each case such that ${\overline{\mathrm{\Omega }}}_1=0.5$, and we have used the instantaneous-decay approximation in obtaining the results shown in the left and right panels. We emphasize that while we have taken $w=-0.8$ for each pairwise stasis involving vacuum energy, similar results are obtained for other values of $w$.

Figure 17

The abundances of vacuum energy (left panel), matter (center panel), and radiation (right panel) in a three-component system exhibiting a triple-stasis solution, plotted as functions of the number $\mathcal{N}$ of $e$-folds of cosmological expansion since the initial production time $t^{(0)}$. The different curves in each panel correspond to different values of the ratio $H(t_{N-1})/H^{(0)}$, where $t_{N-1}$ denotes the time at which the heaviest of the ${\varphi }_{\ell }$ fields becomes underdamped and begins oscillating. These results were calculated within the full exponential-decay framework and correspond to the parameter choices $\alpha =0.95$, $\delta =4$, $w=-0.7$, $\xi =20$, and $\mathrm{\Delta }m/m_0=1$. For all values of $H(t_{N-1})/H^{(0)}$, we observe that ${\mathrm{\Omega }}_{\mathrm{\Lambda }}(t)$, ${\mathrm{\Omega }}_M(t)$, and ${\mathrm{\Omega }}_{\gamma }(t)$ all evolve toward their stasis values. We also note that it takes longer for the universe to settle into its asymptotic stasis state for some values of $H(t_{N-1})/H^{(0)}$ than for others.

Figure 18

Same as Fig. 17, except that the different curves now correspond to different values for the initial vacuum-energy and radiation abundances ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{(0)}$ and ${\mathrm{\Omega }}_{\gamma }^{(0)}=1-{\mathrm{\Omega }}_{\mathrm{\Lambda }}^{(0)}$, with ${\mathrm{\Omega }}_M^{(0)}=0$ held fixed. These results were calculated within the full exponential-decay framework and correspond to the parameter choices $\alpha =0.95$, $\delta =4$, $w=-0.7$, $\xi =20$, $\mathrm{\Delta }m/m_0=1$, and $3H^{(0)}/(2m_{N-1})=1$. For all values of ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{(0)}$ and ${\mathrm{\Omega }}_{\gamma }^{(0)}$, we see that ${\mathrm{\Omega }}_{\mathrm{\Lambda }}(t)$, ${\mathrm{\Omega }}_M(t)$, and ${\mathrm{\Omega }}_{\gamma }(t)$ all ultimately evolve toward their stasis values, with the universe taking longer to settle into its asymptotic stasis state for some values of ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^{(0)}$ and ${\mathrm{\Omega }}_{\gamma }^{(0)}$ than for others.

Figure 19

“Phase diagrams” for triple stasis within the $(w_{△},\xi )$-plane for $w=-0.7$ (top panel) and for $w=-0.9$, $w=-0.5$, and $w=-0.25$ (corresponding to the three panels along the lower row). Working within the full exponential-decay framework, we have assigned the color of each point within the $(w_{△},\xi )$-plane according to the abundance-map palette shown on the top right, with the relative levels of blue, green, and red indicating the values of the stasis abundances ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$, ${\overline{\mathrm{\Omega }}}_M$, and ${\overline{\mathrm{\Omega }}}_{\gamma }$, respectively. The gray region on the left side of each panel is physically inaccessible, since self-consistency requires $w_{△}>w$. Likewise the shaded region with $\xi <{\xi }_{\min }$ lies outside our regime of validity for ${\epsilon }_{\mathrm{dec}}=0.05$. The dashed black vertical line with $w_{△}=0$ corresponds to universes with $\overline{\kappa }=2$, as shown in Fig. 14, while points to the left (respectively right) of this line correspond to universes with $\overline{\kappa }>2$ (respectively $\overline{\kappa }<2$), as shown in the upper (respectively lower) panel of Fig. 15. The points within Regions I through IV exhibit triple stases with ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}>{\overline{\mathrm{\Omega }}}_M>{\overline{\mathrm{\Omega }}}_{\gamma }$, ${\overline{\mathrm{\Omega }}}_M>{\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}>{\overline{\mathrm{\Omega }}}_{\gamma }$, ${\overline{\mathrm{\Omega }}}_M>{\overline{\mathrm{\Omega }}}_{\gamma }>{\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$, and ${\overline{\mathrm{\Omega }}}_{\gamma }>{\overline{\mathrm{\Omega }}}_M>{\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}$, respectively. By contrast, along the edges of this plane, our triple-stasis solutions reduce to simpler stasis solutions: either to a pairwise $\mathrm{\Lambda }M$ stasis (as discussed in Sec. 3 and shown along the upper edge with $w_{△}<0$), or to a pairwise $M\gamma$ stasis (as discussed in Sec. 2 and shown along the upper edge with $w_{△}>0$), or to universes with only vacuum energy (as shown along the left edge), radiation (as shown along the right edge), or matter (as indicated at the top point with $w_{△}=0$). All universes along the vertical $w_{△}=0$ line are effectively matter-dominated (as illustrated in Fig. 14), with counterbalancing abundances of vacuum energy and radiation; however, as we move up this line towards greater values of $\xi$, these other abundances maintain their ratio but shrink to zero, leaving behind a universe consisting only of matter as $\xi \to \infty$. The star represents the location of the “triple point” at which ${\overline{\mathrm{\Omega }}}_{\mathrm{\Lambda }}={\overline{\mathrm{\Omega }}}_M={\overline{\mathrm{\Omega }}}_{\gamma }$. This figure therefore encapsulates and illustrates the relationships between the different versions of stasis discussed in this paper.

Figure 20

Stasis versus quasi-stasis: the abundances ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ (blue), ${\mathrm{\Omega }}_M$ (cyan), and ${\mathrm{\Omega }}_{\gamma }$ (magenta), plotted as functions of $\mathcal{N}$ for a true stasis (solid lines) and two nearby quasi-stases, one with $\gamma =0.95$ (dashed lines) and the other with $\gamma =1.05$ (dotted lines). All curves are evaluated within the framework of an exponential decay for the matter-radiation transition, with $(\alpha ,\delta ,w)=(1,2,-0.8)$ taken as benchmark values. Corresponding curves for all other quasi-stases with $0.95<\gamma <1.05$ and the same $(\alpha ,\delta ,w)$ benchmark values lie within the shaded bands. In each case we see that variations in $\gamma$ induce deviations from true stasis, but in each case the abundances ${\mathrm{\Omega }}_i$ nevertheless remain relatively constrained to lie near these stasis values, with time evolutions that continue to be significantly suppressed. The closer $\gamma$ is to unity, the more closely the ${\mathrm{\Omega }}_i$ curves trace the true stasis values.

Figure 21

Pairwise oscillatory stasis involving matter and radiation. In the left panel, we plot ${\mathrm{\Omega }}_M$ (solid cyan curve) and ${\mathrm{\Omega }}_{\gamma }$ (solid magenta curve) as functions of time for the parameter choices $\alpha \approx 1.13$, $\delta =4$, $\gamma =9$, which imply ${\overline{\mathrm{\Omega }}}_M=3/4$ and ${\overline{\mathrm{\Omega }}}_{\gamma }=1/4$. The dotted horizontal cyan and magenta lines respectively indicate the corresponding constant abundances ${\overline{\mathrm{\Omega }}}_M$ and ${\overline{\mathrm{\Omega }}}_{\gamma }$ which would be obtained for a true stasis with the same values of $\alpha$, $\delta$, and $\gamma$. In the middle panel we show the abundance curves obtained for the parameter choices $\alpha \approx 2.32$, $\delta =4$, $\gamma =9$, which imply ${\overline{\mathrm{\Omega }}}_M={\overline{\mathrm{\Omega }}}_{\gamma }=1/2$. These parameter choices exemplify the special case in which ${\overline{\mathrm{\Omega }}}_M={\overline{\mathrm{\Omega }}}_{\gamma }$ and in which the abundances ${\mathrm{\Omega }}_M$ and ${\mathrm{\Omega }}_{\gamma }$ exhibit a “braiding” phenomenon, mutually oscillating around the same central value. In the right panel, we plot as a function of time the ratio of the scale factor $a(t)$ to the corresponding value $\overline{a}(t)\sim t^{\overline{\kappa }/3}$ that the scale factor would have in a true stasis for the same parameter choices as in the middle panel. We emphasize that similar oscillatory phenomena can also arise for the other realizations of stasis that we have considered in this paper.