普朗克尺度的量子涨落
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Eur. Phys. J. C (2019) 79:455
https://doi.org/10.1140/epjc/s10052-019-6963-5
Letter
Quantum fluctuations at the Planck scale
Fulvio Melia
1,2,a
1
Department of Physics, the Applied Math Program, The University of Arizona, Tucson, AZ 85721, USA
2
Department of Astronomy, The University of Arizona, Tucson, AZ 85721, USA
Received: 18 February 2019 / Accepted: 17 May 2019 / Published online: 29 May 2019
© The Author(s) 2019
Abstract The recently measured cutoff, k
min
= 4.34 ±
0.50/r
cmb
(with r
cmb
the comoving distance to the last scat-
tering surface), in the fluctuation spectrum of the cosmic
microwave background, appears to disfavor slow-roll infla-
tion and the associated transition of modes across the hori-
zon. We show in this Letter that k
min
instead corresponds
to the first mode emerging out of the Planck domain into
the semi-classical universe. The required scalar-field poten-
tial is exponential, though not inflationary, and satisfies the
zero active mass condition, ρ
φ
+ 3p
φ
= 0. Quite reveal-
ingly, the observed amplitude of the temperature anisotropies
requires the quantum fluctuations in φ to have classicalized at
∼ 3.5 ×10
15
GeV, consistent with the energy scale in grand
unified theories. Such scalar-field potentials are often associ-
ated with Kaluza–Klein cosmologies, string theory and even
supergravity.
1 Introduction
Three major satellite missions have now confirmed the lack
of angular correlation at θ 60
◦
in the cosmic microwave
background (CMB) [1–3], contrasting with the predictions of
standard inflationary cosmology [4–7]. Though many pos-
sible instrumental and observational selection effects have
been considered as the cause of this deficiency, today we
appear to be left solely with cosmic variance as an explana-
tion for the missing correlations. But even this conclusion
disfavors the conventional picture at 3σ . A recent in-
depth analysis of the Planck measurements [8] reveals that
a far more likely reason for the missing angular correlations
is a non-zero minimum wavenumber, k
min
, in the fluctua-
tion power spectrum P(k). This evidence now shows quite
compellingly that a zero k
min
is ruled out at a level of con-
Communicated by John Woodruff Simpson Fellow.
a
e-mail: fmelia@email.arizona.edu
URL: http://www.physics.arizona.edu/~melia
fidence exceeding 8σ .ThePlanck data suggest, instead, a
cutoff k
min
= 4.34 ±0.50/r
cmb
, where r
cmb
is the comoving
distance to the surface of last scattering.
This measurement disfavors basic slow-roll inflation
because, in the standard picture, k
min
would have been the
first mode crossing the horizon during the near-exponential
expansion [9]. Thus, for an assumed inflaton potential V (φ),
k
min
would have signaled the precise time at which inflation
started. As it turns out, however, one could not simultane-
ously solve the temperature horizon problem and account
for the observed fluctuation spectrum in the CMB with this
minimum cutoff. If inflation is to work, the sequence of steps
preceding (or succeeding) the inflationary phase would nec-
essarily have to be more complicated than has been conjec-
tured thus far.
In this Letter, we take an alternative approach, and sim-
plify the concept of how a scalar field ought to have
behaved in order to comply with the observational con-
straints. Arguably, the most significant, relevant measure-
ments we have to date are (1) the scalar spectral index,
n
s
= 0.9649 ± 0.0042, in the power spectrum P(k) =
A
s
(k/k
0
)
n
s
−1
[3], (2) the amplitude A
s
= (2.1 ± 0.04) ×
10
−9
[3], and now (3) the hard cutoff k
min
= 4.34±0.50/r
cmb
[8]. To be clear, none of these measurements argues against
the influence of a scalar field, or anisotropies arising from its
quantum fluctuations (QFs), but there are now good reasons
to question whether its potential is truly inflationary.
Indeed, a significant body of evidence is accumulating in
favor of a zero active mass (ρ + 3 p = 0) equation-of-state
in the cosmic fluid, based on the analysis of over 25 different
kinds of observation at low and high redshifts. A summary of
these results may be found in Table 2 of Ref. [10]. A notable
feature of such a universe is that it lacks a horizon problem
[11,12], so our failure to build a fully successful inflationary
paradigm may simply reflect the fact that the universe does
not need it. This is the key assumption we shall make in
this Letter. Some additional support for such a cosmological
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