CMB quadrupole suppression: I. Initial conditions of inflationary perturbations

Abstract

We investigate the issue of initial conditions of curvature and tensor perturba- tions at the beginning of slow roll inflation and their effect on the power spectra. Renormalizability and small back reaction constrain the high k behavior of the Bogoliubov coefficients that define these initial conditions.We introduce a transfer function D(k) which encodes the effect of generic initial conditions on the power spectra. The constraint from renormalizability and small back reaction entails that D(k) < mu^2/k^2 for large k, implying that observable effects from initial conditions are more prominent in the low multipoles. This behavior affects the CMB quadrupole by the observed amount \~10-20% when mu is of the order of the energy scale of inflation. The effects on high l-multipoles are suppressed by a factor ~1/l^2 due to the fall off of D(k) for large wavevectors k. We show that the determination of generic initial conditions for the fluc- tuations is equivalent to the scattering problem by a potential V(eta) localized just prior to the slow roll stage. Such potential leads to a transfer function D(k) which automatically obeys the renormalizability and small backreaction constraints. We find that an attractive potential V(eta) yields a suppression of the lower CMB multipoles.Both for curvature and tensor modes, the quadrupole suppression depends only on the energy scale of V(eta) and on the time interval where V(eta) is nonzero. A suppression of the quadrupole for curvature pertur- bations consistent with the data is obtained when the scale of the potential is of the order of k^2_Q where k_Q is the wavevector whose physical wavelength is the Hubble radius today.

Figures (4)

• Figure 1: The quadrupole correction $(\Delta C_2/C_2)/|v_0 \eta_0\Delta|$ vs, $\kappa |\eta_0|$ in the Born approximation for an attractive potential of the form (\ref{['potential']}), for $|\Delta/\eta_0|=0.01, \; 0.05, \; 0.1$. Here, $\kappa = \frac{a_0 H_0}{3.3}$. It clearly reveals a suppression for $\kappa \eta_0\sim 1$, that is for the modes whose wavelengths correspond to the Hubble radius today
• Figure 2: The corrections $(\Delta C_l/C_l)/ |v_0 \; \eta_0\; \Delta|$ vs. $l$ in the Born approximation for an attractive potential of the form (\ref{['potential']}), for $\kappa \; |\eta_0|=1; \; |\Delta/\eta_0|=0.01$. The corrections for the higher multipoles are substantially smaller than the quadrupole correction, vanishing very rapidly for $l \geq 3$
• Figure 3: $(\Delta C_2/C_2)/|v_0 \; \eta_0 \; \Delta|$ vs $\kappa| \; \eta_0|$ for the square well potential, for $|v_0| \; \eta_0^2 = 1, \; 3, \; 5~;~|\Delta/\eta_0|=0.01$ (left panel) and $|\Delta/\eta_0|=0.1$ (right panel). There is a substantial suppression of the quadrupole when the potential is localized at a time scale $\eta_0 \sim 1/[a_0 \; H_0]$. This time scale is approximately $55$ e-folds before the end of inflation when the wavelengths corresponding to today's Hubble radius exited the horizon.
• Figure 4: $(\Delta C_l/C_l)/|v_0 \; \eta_0 \; \Delta|$ vs $l$ for the square well potential , for $|v_0| \; \eta_0^2 = 1;\kappa \; |\eta_0|=1$ and $|\Delta/\eta_0|=0.01, \; 0.1$. We see that the suppresion of higher multipoles are negligibly small (they fall off as $1/l^2$) and observationally irrelevant.