Inflation in the early Universe is one of the most promising probes of
gravity in the high-energy regime. However, observable scales give access to a
limited window in the inflationary dynamics. In this essay, we argue that
quantum corrections to the classical dynamics of cosmological fields allow us
to probe much earlier epochs of the inflationary phase and extend this window
by many orders of magnitude. We point out that both the statistics of
cosmological fluctuations at observable scales, and the field displacements
acquired by spectator fields that play an important role in many
post-inflationary processes, are sensitive to a much longer phase of the
inflationary epoch.Comment: 5 pages, 2 figures. Honourable Mention from the Gravity Research
  Foundation 2017 Awards for Essays on Gravitation. Version to appear in
  October 2017 Special Issue of the International Journal of Modern Physics D.
  Updated to match the published versio

Hardwick, Robert J.

Vennin, Vincent

Wands, David

English

arXiv

Portsmouth University Research Portal (Pure)

A Quantum Window Onto Early Inflation
Robert J. Hardwick, Vincent Vennin, David Wands
Institute of Cosmology & Gravitation,
University of Portsmouth, Dennis Sciama Building,
Burnaby Road, Portsmouth, PO1 3FX, United Kingdom
Corresponding author: robert.hardwick@port.ac.uk
Keywords: Inflation, Physics of the Early Universe, Quantum Gravity Phenomenology
PACS Number(s): 98.80.Cq 04.60.Bc 04.62.+v 05.10.Gg
October 6, 2017
Abstract
Inflation in the early Universe is one of the most promising probes of gravity in the high-energy regime.
However, observable scales give access to a limited window in the inflationary dynamics. In this essay, we
argue that quantum corrections to the classical dynamics of cosmological fields allow us to probe much
earlier epochs of the inflationary phase and extend this window by many orders of magnitude. We point
out that both the statistics of cosmological fluctuations at observable scales, and the field displacements
acquired by spectator fields that play an important role in many post-inflationary processes, are sensitive
to a much longer phase of the inflationary epoch.
Essay written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation.
Cosmological inflation1–7 is one of the only places in physics where an effect based on general relativity
(the accelerated expansion) and quantum mechanics (the amplification of vacuum quantum fluctuations of
the gravitational and matter fields to large-scale cosmological perturbations8–13) gives rise to predictions that
can be tested experimentally. For this reason, it is an ideal framework to discuss fundamental issues related
to the nature of gravity and even quantum gravity. Indeed, the cosmological perturbations produced in this
early epoch have been measured to very high accuracy14,15 in the temperature and polarisation anisotropies
of the cosmic microwave background (CMB), and can be probed more generally in the large-scale structure
of our Universe.
At a given physical length scale a/k, where a is the scale factor of the Universe and k is a fixed comoving
wavenumber, the statistical properties of cosmological fluctuations are mostly determined by the properties
of the inflationary classical dynamics around the time when a/k crosses the Hubble radius H−1 ≡ a/a˙, where
a dot denotes differentiation with respect to cosmic time t. For example, if inflation is driven by a single
scalar field φ with potential V (φ) ≡ 24pi2M4Plv(φ) (where MPl is the Planck mass), the power spectrum of
curvature perturbations ζ at scale k is given by
Pζ(k) = 2
M2Pl
v3[φ∗(k)]
v′2[φ∗(k)]
, (1)
where φ∗(k) is the value of φ when a/k exits the Hubble radius. The range of scales probed e.g. in the
CMB then translates into a time interval during inflation of length N ∼ 7, measured by the number of e-
folds N ≡ ln(a). If one includes the large-scale structure of our Universe, this window is extended but cannot
exceed the last ∼ 60 e-folds of inflation. But can we ever learn about larger scales, hence earlier times?
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During inflation, the coarse-grained fields (i.e. scales larger than the Hubble radius) are constantly sourced
by the small-wavelength quantum fluctuations as they cross the Hubble radius. This quantum backreaction
on the dynamics of the Universe can be modeled through the stochastic inflation formalism,16 where φ does
not only evolve classically under the gradient of its potential V (φ), but also receives quantum corrections
through an additional noise term
dφ
dN
= −V
′(φ)
3H2
+
H
2pi
ξ, (2)
where ξ is a normalised white Gaussian noise. The system then explores parts of the potential that would be
inaccessible under the classical dynamics. For example, the power spectrum (1) is now given by17–19
Pζ(k) =2
{∫ ∞
φ∗
dx
M
Pl
1
v (x)
exp
[
1
v (x)
− 1
v (φ∗)
]}−1
×∫ ∞
φ∗
dx
M
Pl
{∫ ∞
x
dy
M
Pl
1
v (y)
exp
[
1
v (y)
− 1
v (x)
]}2
.
(3)
Contrary to Eq. (1), this expression does not only depend on the potential evaluated at φ∗(k), but relies on
the properties of the potential in the entire inflationary domain. For this reason, even the limited range of
scales probed in the CMB may contain imprints from early features of the inflationary dynamics and in this
sense, quantum diffusion in an expanding background greatly extends the observational window. In practice,
when v  1, i.e. V  M4Pl, Eq. (3) is well approximated by Eq. (1) so the dependence on the potential
function outside the standard observational window is usually Planck suppressed. This is however not the
case when several fields drive inflation,18–20 or in very flat regions of the potential that can drive the dynamics
at smaller (but still accessible21) scales than the ones probed in the CMB.
Figure 1: Toy inflationary potential considered in this work, made of a plateau (i.e. asymptotically constant)
part between φend and φLF, and a monomial large-field part (where V ∝ φp) at φ > φLF. CMB observations
constrain the number of e-folds spent on the plateau to be Nplateau > 60, while the dynamics of spectator
fields is sensitive on a much wider part of the inflationary potential.
Another, less direct but more sensitive, cosmological probe sensitive to the early stages of inflation through
quantum diffusion is the field displacement acquired by light spectator fields.?, 22, 23 Such fields do not
participate in the accelerated expansion of inflation but can play an important role afterwards (e.g. Higgs
fields24 or dark matter candidates25) that depends on the field displacement they acquire during inflation.
Let us consider the toy model depicted in Fig. 1 where the inflaton potential V (φ) is made of a plateau
(i.e. asymptotically constant) part between φend and φLF and a monomial large-field (i.e. V ∝ φp) part at
φ > φLF. Observations of the CMB constrain the potential to be of the plateau type in the last few e-folds of
inflation26 so in the standard setup, the only constraint one has is that φLF should be located at least ∼ 60
e-folds before the end of inflation.
2
10−12 10−11 10−10 10−9 10−8 10−7
m
M
Pl
101
102
103
104
105
106
107
108
109
1010
1011
1012
1013
N
p
la
te
au
Hend = 10
−6M
Pl
∼ CMB
10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4
λ
101
102
103
104
105
106
107
N
p
la
te
au
Hend = 10
−6M
Pl
∼ CMB
Figure 2: Minimum number of e-folds spent on the plateau part of the inflationary potential so that
the spectator field displacement at the end of inflation is independent of the large-field correction to the
inflaton potential. The left panel corresponds to a quadratic spectator, V (σ) = m2σ2/2 and the right panel
corresponds to a quartic spectator, V (σ) = λσ4. Through CMB observations interpreted in the standard
way, one gets the constraint Nplateau > 60 (denoted with the dashed line), while spectator fields are sensitive
to a much wider part of the inflationary dynamics.
A spectator field σ on top of this inflationary background evolves under its potential V (σ) and according
to Eq. (2) (where φ is to be replaced by σ). If H is constant, the probability distribution Pr(σ) relaxes
towards the stationary attractor profile16
Pr(σ) ∝ exp
[
−8pi
2V (σ)
3H4
]
(4)
where any initial condition is erased. However, since H is not exactly constant during inflation, this does not
always happen on the large-field part of the inflationary potential, since the relaxation time towards Eq. (4)
can be larger than the variation time scale of H there. For example, if the spectator potential is quadratic,
V (σ) = m2σ2/2, Eq. (4) can never be attained in the early phase of large-field evolution where the typical
field displacement remains strongly dependent on initial conditions. By setting σ = 0 at the exit point of
eternal inflation (where the dynamics of φ is itself dominated by stochastic corrections), one can derive a
lower bound on the number of e-folds Nplateau spent on the plateau part of the inflaton potential so that the
details of the large-field phase are erased from the distribution of σ at the end of inflation,23
Nplateau ≥
3H2plateau
2m2
ln
[
8pipm2M2Pl
3H4plateau(p+ 2)
]
(5)
for p ≥ 2. It is displayed in the left panel of Fig. 2 for p = 2 (but the result depends only mildly on p).
Compared to the standard constraint Nplateau ≥ 60, one can see that the observational window on the inflaton
potential extends by orders of magnitude. For a quartic spectator V (σ) = λσ4, it turns out that Eq. (4) is
adiabatically tracked at early time in the large-field phase (more precisely, when H > λ−p/8Hplateau). In this
case, initial conditions on the spectator field displacement can be erased during this adiabatic epoch, and the
minimal number of e-folds spent on the plateau such that no imprint is left from the large-field epoch on the
distribution of σ at the end of inflation is given by23
Nplateau ≥
4piΓ
(
3
4
)
Γ
(
1
4
) √ 2
3λ
ln(2), (6)
3
for p ≥ 2. It is displayed in the right panel of Fig. 2 where one can see again that the observational window
on the inflaton potential extends by orders of magnitude.
Thus the quantum dynamics of cosmological fields in the early Universe gives access to a vast range of
scales that extend the classical window by orders of magnitude and allow us to explore high-energy gravity
beyond the observable horizon.
Acknowledgments
We are grateful to our collaborators Chris Byrnes, Alexei Starobinsky and Jesu´s Torrado. This work was
supported by STFC grants ST/N000668/1, ST/K502248/1 and ST/N504245/1.
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