Possible origin of CMB temperature fluctuations: Vacuum fluctuations of Kaluza-Klein and string states during the inflationary era

We point out that the temperature fluctuations of cosmic microwave background can be generated in a way that is different from the one usually assumed in slow-roll inflation. Our mechanism is based on vacuum fluctuations of fields which are at rest at the bottom of the potential, such as Kaluza-Klein modes or string excited states. When there are a large number (typically of order $N\sim {10}^{14}$) of fields with small mass in units of Hubble parameter during the inflationary era, this effect can give significant contributions to the cosmic microwave background temperature fluctuations. This number $N$ makes it possible to enhance scalar perturbation relative to tensor perturbation. Comparison with the observed amplitudes suggests that models with string scale of order ${10}^{-5}$ of four-dimensional Planck scale are favorable.

Figure 1

One-loop diagram: One-loop diagram in field theory corresponds to the torus diagram in string theory. String theory can be regarded as a field theory with an infinite number of fields, which are the Fourier modes on the world sheet spatial direction.

Figure 2

Fundamental region: Integration over the moduli of the torus should be restricted to the fundamental region, since other regions are physically equivalent to this region due to modular invariance. $\mathrm{Im}(\tau )$, which corresponds to the Schwinger parameter, is cut off at the distance of order string scale.

Figure 3

${log}_{10}(L{m}_{\mathrm{pl}})$ as a function of $m_s/H$, with ${\delta }_T=2.6\times {10}^{-5}$, $r_{\mathrm{t}/\mathrm{s}}=0.22$, $RH\sim {10}^{67}$.

Figure 4

$g_s$ as a function of $m_s/H$, with ${\delta }_T=2.6\times {10}^{-5}$, $r_{\mathrm{t}/\mathrm{s}}=0.22$, $RH\sim {10}^{67}$.

Figure 5

Spectral index as a function of $m_s/H$.

Figure 6

Non-Gaussianity $f_{\mathrm{NL}}$ as a function of $m_s/H$.

Figure 7

Diagrams for the three-point function of $\Phi$. Left panel: Ignoring the interaction among the fields $\varphi$, this is of order $N$ (the number of fields). Right panel: Higher loop diagram where ${\varphi }_1$ and ${\varphi }_2$ are independent, but ${\varphi }_3$ will be determined from the conservation law (such as KK momentum conservation). This is of order ${\lambda }^2{N}^2$.