Kaluza-Klein towers in the early universe: Phase transitions, relic abundances, and applications to axion cosmology

We study the early-universe cosmology of a Kaluza-Klein (KK) tower of scalar fields in the presence of a mass-generating phase transition, focusing on the time development of the total tower energy density (or relic abundance) as well as its distribution across the different KK modes. We find that both of these features are extremely sensitive to the details of the phase transition and can behave in a variety of ways significant for late-time cosmology. In particular, we find that the interplay between the temporal properties of the phase transition and the mixing it generates are responsible for both enhancements and suppressions in the late-time abundances, sometimes by many orders of magnitude. We map out the complete model parameter space and determine where traditional analytical approximations are valid and where they fail. In the latter cases we also provide new analytical approximations which successfully model our results. Finally, we apply this machinery to the example of an axion-like field in the bulk, mapping these phenomena over an enlarged axion parameter space that extends beyond that accessible to standard treatments. An important by-product of our analysis is the development of an alternate “UV-based” effective truncation of KK theories which has a number of interesting theoretical properties that distinguish it from the more traditional “IR-based” truncation typically used in the extra-dimension literature.

Figure 1

The mass spectrum ${\lambda }_k$ for $N=10$, plotted as a function of $m/M_c$. When $\overline{m}\ll M_c$, our mass eigenvalues follow the simple form given in Eq. (2.14). By contrast, when $\overline{m}\gg M_c$, all but the highest mass eigenvalue shift upwards by $M_c/2$, while the highest mass eigenvalue diverges according to Eq. (2.16) for $m^2/M_c^2\gtrsim \mathcal{O}(\sqrt{N})$. As discussed in the text, this anomalous behavior of the highest mass eigenvalue is an artifact of having truncated our KK tower to only a finite number of states and disappears as $N\to \infty$.

Figure 2

The late-time energy density ${\overline{\rho }}_{4\mathrm{D}}$, plotted within the $(\overline{m}{t}_G,{\delta }_G)$ plane and normalized to the value it would have had in the abrupt approximation (left panel) or adiabatic approximation (right panel). In each case the colors and black lines indicate the contours of the normalized ${\overline{\rho }}_{4\mathrm{D}}$ (i.e., the degrees to which the true late-time energy density is enhanced or suppressed relative to its approximated value), while the blue lines indicate the corresponding contours of $(t_{\zeta }-t_G)/{\mathrm{\Delta }}_G$. As additional relevant guidelines, the dashed line in the left panel indicates the contour along which ${\mathrm{\Delta }}_G=2\pi /\overline{m}$, while in the right panel it indicates the contour along which $\dot{m}(t_{\zeta })=m^2(t_{\zeta })$. In general, the energy densities in regions above or to the left of these guidelines in the left (right) panel tend to obey the abrupt (adiabatic) approximation more strongly than those below or to the right. We also find that neither approximation holds across the majority of the parameter space shown, with the true values of the late-time energy density ${\overline{\rho }}_{4\mathrm{D}}$ often experiencing significant suppressions (blue regions) or enhancements (red regions) as compared with the approximated expectations.

Figure 3

The absolute late-time energy density ${\overline{\rho }}_{4\mathrm{D}}$, plotted in units of $\frac{1}{2}⟨\varphi {⟩}^2{t}^{-2}$ within the $(\overline{m}{t}_G,{\delta }_G)$ plane. Note that our late-time energy density ${\overline{\rho }}_{4\mathrm{D}}$ is generally independent of ${\delta }_G$ for $\{\overline{m}{t}_G\lesssim 10^3,{\delta }_G\lesssim 0.5\}$ and scales roughly as $(\overline{m}{t}_G{)}^2$ within this region.

Figure 4

The abundance fractions ${\rho }_{\lambda }/\rho$ at time $t=t_G$, evaluated within the abrupt ${\delta }_G\to 0$ limit and plotted as functions of $\overline{m}/M_c$ for $N=10$ (left panel) and $N=100$ (right panel). In both panels we observe that the lightest mode carries the largest fractional abundance for small $m/M_c$, but that this abundance is more fully distributed amongst the modes as $m/M_c$ increases. For finite $N$, this abundance then ultimately collects in the heaviest mode as $m/M_c\to \infty$. This last feature is ultimately an artifact of the truncation to only finitely many modes and disappears in the $N\to \infty$ limit. In all cases, the sum of the fractional abundances shown is equal to 1 for all $m/M_c$.

Figure 5

The tower fractions $\eta (t_G)$ corresponding to the fractional abundances shown in Fig. 4, plotted as functions of $m/M_c$ for different values of $N$. We see that the tower fraction begins near zero for small $m/M_c$ but then begins to grow as $m/M_c$ increases. For finite $N$, the tower fraction ultimately hits a peak before returning to zero at large $m/M_c$, while in the $N\to \infty$ limit the tower fraction continues monotonically to 1.

Figure 6

The total late-time abundance $\overline{\rho }$ for $N$ modes, evaluated within the $(\overline{m}{t}_G,{\delta }_G)$ plane for $N=2$ (left panel) and $N=10$ (right panel) and expressed as a fraction of the corresponding value ${\overline{\rho }}_{4\mathrm{D}}$ for the 4D $N=1$ special case. We have chosen $t_G=10^2/M_c$ as a reference value for both plots above. We see that the introduction of additional modes relative to the 4D special case either preserves the 4D result nearly exactly for $\overline{m}{t}_G\lesssim 10$ or suppresses it for $\overline{m}{t}_G\gtrsim 10$, with this suppression becoming increasingly severe for larger $\overline{m}{t}_G$ (or equivalently, larger $\overline{m}/M_c$) and ${\delta }_G\lesssim 1$. Note that the gray regions in the upper right corners of these panels (and in similar subsequent figures throughout this paper) are excluded for the reasons discussed in the paragraph below Eq. (2.12).

Figure 7

The total late-time tower fraction $\overline{\eta }$ corresponding to the panels shown in Fig. 6. In each case we see that the $(\overline{m}{t}_G,{\delta }_G)$ plane is subdivided into two disjoint $\overline{\eta }\approx 0$ regions (white) by a narrow $\overline{\eta }>0$ ribbon (green): the lightest mode carries the bulk of the energy density in the region to the left of the ribbon, while the heaviest mode carries the bulk of the energy density in the region to the right. Interestingly, the division between these two regions is nonmonotonic as a function of not only $\overline{m}{t}_G$ but also ${\delta }_G$.

Figure 8

The total late-time abundance $\overline{\rho }$ and late-time tower fraction $\overline{\eta }$ of our $N$-mode system, plotted as functions of $N$ for different values of ${\delta }_G$. The left and center panels show the total late-time abundance expressed as a fraction of the value it would have in the abrupt ${\delta }_G\to 0$ and 4D limits, respectively, while the right panel shows the late-time tower fraction. For all three panels we have taken $t_G=10^2/M_c$ and $\overline{m}=10^2{M}_c$, thereby fixing $\overline{m}{t}_G=10^4$ as a benchmark value. The blue curves are calculated using the usual “IR-based” truncation which has been employed thus far in Secs. 2, 3, 4, while the red curves are calculated using the alternative “UV-based” truncation to be discussed in Sec. 5b; both truncations have the same $N\to \infty$ asymptotic behavior but the UV-based truncation reaches asymptotia far more rapidly and smoothly. The black dashed line that occurs in each of the energy-density panels indicates the results of the multicomponent adiabatic approximation of Eq. (5.1) which serves as a lower bound for the energy density of the ensemble.

Figure 9

Eigenvalues of the IR-based and UV-based mass matrices, plotted, respectively, in blue and red as functions of $N$ for $m/M_c=2$. For visual clarity, only even eigenvalues are shown. In each case the blue and red curves asymptote to the exact eigenvalues ${\lambda }_k$ as $N\to \infty$, but the UV-based red curves generally approach this limit more rapidly and more smoothly than do the IR-based blue curves. Note that each blue curve begins with an anomalously high eigenvalue; this is nothing but the truncation artifact already sketched in Fig. 1 for the highest mode ${\lambda }_{N-1}$ in each case. By contrast, such artifacts are entirely eliminated in the UV-based approach.

Figure 10

The late-time energy density $\overline{\rho }$ of the full KK tower, plotted in the $(\overline{m}{t}_G,{\delta }_G)$ plane for $t_G=10^2/M_c$. The left and right panels show contours of $\overline{\rho }/\overline{\rho }({\delta }_G=0)$ and $\overline{\rho }/{\overline{\rho }}_{4\mathrm{D}}$, respectively.

Figure 11

The absolute late-time energy density $\overline{\rho }$ in units of $\frac{1}{2}⟨\varphi {⟩}^2{t}^{-2}$, plotted within the $(\overline{m}{t}_G,{\delta }_G)$ plane for three different values of $M_c{t}_G$.

Figure 12

The late-time tower fraction $\overline{\eta }$ of the KK tower, plotted within the $(\overline{m}{t}_G,{\delta }_G)$ plane for $t_G=10^2/M_c$. Unlike the tower fractions illustrated in Fig. 7 for finite $N$, we see that the tower fraction for $N=\infty$ is now monotonic in both $\overline{m}{t}_G$ and ${\delta }_G$ and approaches unity for large $\overline{m}{t}_G$ and small ${\delta }_G$.

Figure 13

A graphical representation of the distribution of the energy densities ${\overline{\rho }}_{\lambda }$ across the entire KK tower, calculated for a square “lattice” of locations in the $(\overline{m}{t}_G,{\delta }_G)$ plane. Each pie chart indicates how the total $\overline{\rho }$ is distributed across the different KK modes, with the blue slices indicating the contribution associated with the lightest mode and the remaining slices (colored pink through dark red) indicating the contributions from successively heavier KK modes. The results for each pie correspond to the parameters $(\overline{m}{t}_G,{\delta }_G)$ associated with the location of the center of the pie, and the yellow contours indicate the corresponding values of $\overline{\eta }$, as taken from Fig. 12. We see that the total energy density is preferentially captured by the lightest mode for larger ${\delta }_G$ or smaller $\overline{m}{t}_G$, but that the total energy density is distributed more democratically across the KK tower for smaller ${\delta }_G$ and larger $\overline{m}{t}_G$. Thus a wide variety of energy-density distributions across the KK tower can be realized simply by adjusting the parameters associated with the mass-generating phase transition.

Figure 14

A grand summary of the results of this section, combining our information concerning the absolute magnitudes of the total late-time energy density $\overline{\rho }$ from Fig. 11 with our information concerning the distribution of that energy density from Fig. 13, all plotted within the $(\overline{m}{t}_G,{\delta }_G)$ plane. This figure is essentially the same as Fig. 13 except that the overall areas of our pie charts have been rescaled in proportion to the magnitudes of the total late-time energy densities $\overline{\rho }$ shown in Fig. 11. In the background we have also indicated the contours of $\overline{\rho }$ from Fig. 11 (dashed black lines) as well as the contours of $\overline{\eta }$ from Fig. 12 (solid blue lines). We see that there is a nontrivial correlation between the overall magnitude of $\overline{\rho }$ and the distribution of that energy density, with larger energy densities distributed more democratically across the KK tower and smaller energy densities captured more and more preferentially by the lightest KK mode. Both features are nevertheless extremely sensitive to the parameters governing the mass-generating phase transition, with the former behavior dominating for smaller ${\delta }_G$ and larger $\overline{m}{t}_G$ and the latter dominating for larger ${\delta }_G$ and smaller $\overline{m}{t}_G$.

Figure 15

The late-time energy density ${\overline{\rho }}_{4\mathrm{D}}$ of our generalized axion field in the 4D limit, plotted within the $({\mathrm{\Lambda }}_G,{\widehat{f}}_X)$ plane for ${\delta }_G=0.1$ (top row) and ${\delta }_G=1.0$ (bottom row). Note that the horizontal ${\mathrm{\Lambda }}_G$ axis (as shown along the bottom of each panel) is equivalently a $t_G$ axis (as shown along the top). The panels in the left and middle columns plot $\overline{\rho }$ as fractions of the abrupt and adiabatic approximations ${\overline{\rho }}_{4\mathrm{D}}({\delta }_G=0)$ and ${\overline{\rho }}_{4\mathrm{D}}{|}_{\mathrm{abs}}$, respectively, while the panels in the right column plot the absolute magnitude of ${\overline{\rho }}_{4\mathrm{D}}$ in units of $\frac{1}{2}⟨\varphi {⟩}^2{t}^{-2}$. Also shown in each panel is a purple dashed line indicating the contour along which ${\overline{m}}_X{t}_G=1$.

Figure 16

The total late-time energy density $\overline{\rho }$ of our generalized axion KK tower, plotted as fractions of $\overline{\rho }({\delta }_G=0)$ (left column) or ${\overline{\rho }}_{4\mathrm{D}}$ (right column) within the $({\mathrm{\Lambda }}_G,{\widehat{f}}_X)$ plane for ${\delta }_G=0.1$ (top row) and ${\delta }_G=0.3$ (bottom row). As in Fig. 15, the ${\mathrm{\Lambda }}_G$ axis is shown along the bottom of each panel while the equivalent $t_G$ axis is shown along the top. Also shown in each panel are three purple dashed lines indicating the contours along which ${\overline{m}}_X{t}_G=1$, ${\overline{m}}_X=M_c$, or $M_c{t}_G=1$. As with other figures in this paper, the gray regions are excluded for the reasons discussed in the paragraph below Eq. (2.12).

Figure 17

The absolute magnitude of the total late-time energy density $\overline{\rho }$ of our generalized axion KK tower, plotted in units of $\frac{1}{2}⟨\varphi {⟩}^2{t}^{-2}$ within the $({\mathrm{\Lambda }}_G,{\widehat{f}}_X)$ plane for ${\delta }_G=0$ (left panel), ${\delta }_G=0.1$ (middle panel), and ${\delta }_G=0.3$ (right panel).

Figure 18

The late-time tower fraction $\overline{\eta }$ of our generalized axion KK tower, plotted within the $({\mathrm{\Lambda }}_G,{\widehat{f}}_X)$ plane for ${\delta }_G=0$ (left panel), ${\delta }_G=0.1$ (middle panel), and ${\delta }_G=0.3$ (right panel).