  There is an apparent power deficit relative to the Lambda-CDM prediction of
the CMB spectrum at large scales, which, though not yet statistically
significant, persists from WMAP to Planck data. We first present a simple toy
model corresponding to a network of frustrated topological defects of domain
walls or cosmic strings that exist previous to the standard slow-roll
inflationary era of the universe. Those features are phenomenologically modeled
by a Chaplygin gas that can interpolate between a network of frustrated
topological defects and a de Sitter-like or a power-law inflationary era. We
show that these scenarios can alleviate the quadrupole anomaly of the CMB
spectrum, based on the approximate initial conditions for the long-wavelength
perturbations. We then go further to show that the large-scale spectrum at the
end of inflation reflects the super-horizon spectrum of the initial state of
the inflaton field. By studying the curvature perturbations of a scalar field
in the FLRW universe parameterized by the equation of state parameter w, we
find that the large-scale spectrum is suppressed if the universe begins with
the adiabatic vacuum in a superinflation (w < -1) or positive-pressure (w > 0)
era. In the latter case, there is however no causal mechanism to establish the
initial adiabatic vacuum. To search for a more realistic initial condition of
the inflationary universe, we consider the Hartle-Hawking no-boundary wave
function, which is a solution to the Wheeler-DeWitt equation, as the initial
condition of the universe. We find that the power suppression can be the
consequence of a massive inflaton, whose initial vacuum is the Euclidean
instanton in a compact manifold. We calculate the primordial power spectrum of
the perturbations and show that, as long as the scalar field is moderately
massive, the power spectrum is suppressed at the long-wavelength scales.
