Horizon feedback inflation
Eur. Phys. J. C
Horizon feedback inflation
Malcolm Fairbairn 1
Tommi Markkanen 0
David Rodriguez Roman 1
0 Blackett Laboratory, Department of Physics, Imperial College London , London SW7 2AZ , UK
1 Department of Physics, King's College London Strand , London WC2R 2LS , UK
We consider the effect of the GibbonsHawking radiation on the inflaton in the situation where it is coupled to a large number of spectator fields. We argue that this will lead to two important effects  a thermal contribution to the potential and a gradual change in parameters in the Lagrangian which results from thermodynamic and energy conservation arguments. We present a scenario of hilltop inflation where the field starts trapped at the origin before slowly experiencing a phase transition during which the field extremely slowly moves towards its zero temperature expectation value. We show that it is possible to obtain enough efolds of expansion as well as the correct spectrum of perturbations without hugely finetuned parameters in the potential (albeit with many spectator fields). We also comment on how initial conditions for inflation can arise naturally in this situation. 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Phase transition with decaying vacuum energy . . . 2.1 m2 0: gradual decay of ρ . . . . . . . . . . 2.2 m2 ∼ 0: large quantum fluctuations . . . . . . 2.3 m2 0: classical rolling . . . . . . . . . . . . 3 Inflationary predictions . . . . . . . . . . . . . . . . 3.1 Numerical solution . . . . . . . . . . . . . . . 4 Discussion/Conclusion . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

Contents
1 Introduction
Cosmological inflation is the leading paradigm which
explains the horizon, flatness and defect problem of the
H
TH = 2π ,
extremely successful FLRW hot big bang model as well
as explaining the source of the initial density perturbations
observed to exist in the CMB [
1
]. Furthermore the
exponential expansion seems to have an elegant explanation in field
theory as being sourced by the potential energy of a field
which rolls only slowly down to its minimum, its kinetic
energy being redshifted by this rapid expansion [
2–4
]. It is
also very difficult to think up alternatives to inflation which
are natural, even with significant modifications of general
relativity and those which do exist often create the wrong
spectrum of perturbations [
5
].
Unfortunately, there are several problems with the
standard inflationary scenario. The most difficult problem is
(arguably) the fact that in order for inflation to start in the first
place, one needs to find a Hubble patch in the early Universe
across the entirety of which the potential energy dominates
the kinetic energy of the field [
6
]. Another way of putting
this is that in order to solve the horizon problem, one creates
another one at earlier times. Another problem is that in order
for the kinetic energy to be redshifted by the expansion for
long enough to explain the horizon and flatness problems,
either the inflaton field has to be transplanckian during
inflation or the shape of the potential below the Planck scale has
to be extremely flat, in other words the extent in the scalar
field direction has to be much larger in GeV than its height
in the energy direction.
Cosmological inflation usually requires a period of
deSitter or at least quasi deSitter expansion in order to obtain
the many efolds required to solve the horizon problem. The
Gibbons–Hawking temperature associated with the
cosmological horizon in de Sitter space [
7
]
plays a central role in inflation as it acts as the source of
quantum fluctuations in the inflaton field which source
density perturbations. This thermal radiation is analogous to the
thermal population observed by an accelerating observer i.e.
(1.1)
Unruh radiation [
8
] and closely resembles the Hawking
radiation which surrounds a black hole [
9
]. In the case of a black
hole, the thermal Hawking radiation escapes to infinity and
energy is conserved only by postulating that the mass of the
black hole is correspondingly released – a hypothesis which
cannot be proved in semiclassical quantum gravity without
including backreaction but which does fit coherently into
the theory of black hole thermodynamics [
10
]. Black hole
evaporation has an interpretation of resulting from a flux of
particles with negative mass going into the black hole. A
similar physical interpretation for de Sitter radiation is possible,
where the Cosmological Constant is reduced from the
addition of negative vacuum (zeropoint) energy to the overall
energy density of de Sitter [
11
].
The author of [
8
] has famously said ’you could cook your
steak by accelerating it’ in regards to the physicalness of
Unruh radiation [
12
]. In light of the analogy between the
Unruh and de Sitter radiation, assigning similarly physical
characteristics to the radiation associated with the de
Sitter horizon motivates its inclusion in the Einstein equations
[
13,14
]. In a first principle approach this can be achieved
in a prescription where the energymomentum tensor
sourcing gravity is coarse grained to include only the observable
degrees of freedom, i.e. those inside the cosmological
horizon [
11,15,16
].
Including a source of continuous particle production in de
Sitter space is wellknown to bring about a qualitative change
in the system’s behaviour: when given enough time
exponential expansion will cease implying that de Sitter space with
particle production is not stable. In addition to thermal
radiation from the horizon a de Sitter instability has been
speculated to result from a variety other underlying mechanisms
ranging from quantum gravity and infrared divergences in
propagators to environment induced decoherence and the
second law of thermodynamics [
11,15–36,36–39
].
A de Sitter instability unavoidably leads to a dynamical
cosmological constant and multiple studies have looked at the
possibility that particle production could gradually reduce the
value of the cosmological constant from a phenomenological
point of view [
40–46
], see [
47,48
] for observational
investigations. It has been argued that since particle production
from gravitational fields seems to be an irreversible process,
this reduction is inevitable from a thermodynamical
perspective [
33,49–52
]. Normally this effect is extremely small and
would not really be a practical way of ensuring a small
cosmological constant. In particular the rate of decrease would
not get rid of the cosmological constant fast enough in the
late Universe to explain today’s cosmic acceleration.
It is interesting to consider the situation where the
cosmological constant is not identified as a geometric term in
the Einstein field equation, but is rather the energy density
of a field which is located at a stable point in its potential
where d V /dϕ = 0. In this situation the same logic would
imply that terms in the Lagrangian that set the scale of the
potential would also decay over time due to the gravitational
production of particles. For example for the potential
(1.2)
λ
V (ϕ) = 4
,
and for a field resting at the metastable origin ϕ = 0, the
particle production associated with the spacetime curvature
arising from the nonzero potential energy at the origin should
have the effect of making ϕ0 decrease with time.
Another implication of the Gibbons–Hawking
temperature would be the possibility that it might create
nonnegligible finite temperature contributions to the potential
energy of scalar fields which are evolving in the quasi de
Sitter background.
In thermal inflation, a thermal sector with the equation of
state of radiation and coupled to the inflaton keeps the field
trapped at the origin for a few efolds until the thermal
radiation has been redshifted away, decreasing its temperature
[
53
]. If however the origin of the thermal radiation is the
Gibbons–Hawking temperature associated with the horizon
then its temperature will be constant and will not decrease
over time if there is no corresponding change in the vacuum
energy. It is clear however that in this situation only a
radiation bath with a large number of degrees of freedom, all of
which are coupled to the scalar field, would be able to keep
the field trapped at the origin. Also once located and
stablised there due to this horizon temperature, the field would
essentially be stuck at the origin for all time, leading to a
graceful exit problem. A constant TH would also violate the
continuity equation [
14
].
If however ϕ0 changes over time due to the production of
radiation, one can imagine a situation where ϕ0 and
consequently the energy density at the origin, the rate of expansion
of the Universe and the temperature of the thermal radiation
all decrease with time. The stable point at the origin then
eventually becomes tachyonic, and field becomes free to roll
away from ϕ = 0, setting the initial conditions for inflation
(see also [
54
] for a similar idea in a different context). We
argue that this could happen and that thermal effects would
subsequently slow the phase transition sufficiently such that
successful inflation can take place.
In Sect. 2 we will go through the equations of this
scenario in more detail and study the dynamics of the field and
how it might produce enough efolds of expansion. Then in
Sect. 3 we will study the inflationary predictions, namely the
perturbations and the spectral tilt, as well as comparing our
analytic estimates to a numerical treatment.
2 Phase transition with decaying vacuum energy
Let us examine the situation with the potential (1.2) in the
presence of a thermal sector characterized by the Gibbons–
Haking temperature (1.1) in more detail. We start by writing
down the Friedmann equations
3H 2 M P2
−(3H 2 + 2H˙ )M P2
.
Our model will consist of a scalar field ϕ and importantly N
conformal fields that couple to ϕ and are in thermal
equilibrium with the horizon as described by (1.1). This will lead
to and additional temperature component for the energy and
pressure densities
ρ = ϕ˙22 + V (ϕ, TH ) + N π302 TH4 ; p = ϕ˙22 − V (ϕ, TH ) + N3 π302 TH4 ,
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
and in addition to (1.2) the potential contains a contribution
from the thermalised conformal fields
λ
V (ϕ, TH ) = 4
We will throughout work in the approximation where the
energy density of the thermal component is subdominant to
that of the potential and in particular to the vacuum energy
piece
λ4 ϕ04 ≡ ρ
≈ 3H 2 MP2
π 2
N 30 TH4 .
This condition will turn out to be easily satisfied for a large
parameter range i.e. 1 N (H/MP)2(1440π 2)−1.
From (2.1) we get the dynamical Friedman equation of
motion
− 2H˙ M P2 = ϕ˙2 + 43 N π302 TH4 ,
from which it is quite apparent that a thermal sector with the
GibbonsHawking temperature is inconsistent with strict de
Sitter space, but will lead to H˙ < 0. Furthermore, it is easy
to see that the continuity equation
ρ˙ + 3H (ρ + p) = 0 ,
can only be satisfied if ϕ0 is not strictly constant but still
provides the source for the continuous particle creation required
for maintaining TH in the conformal fields despite the
dilution from the expansion of space. This one may
understand from the situation when ϕ = 0 giving ρ ∼ ρ , but
ρ + p ∼ T 4 . However as we will see we will not be relying
H
on this feature of the current scenario in the current work
other than to set initial conditions, the dynamics of ϕ0 will
be irrelevant by the time we come to calculate observables.
Also we have deemed more natural to keep the usual
interpretation of a strict coupling constant for the other constants
(2.7)
(2.8)
(2.9)
of our theory, λ and g, and only allow ϕ0 change with time.
Since the continuity Eq. (2.6) provides only one constraint in
principle one could consistently allow also the other
parameters to vary, at least from a purely phenomenological point
of view.
From (2.5) we may conclude that the change in H due to
the thermal sector is very gradual: the first Hubble slow roll
parameter for ϕ = 0 reads
H˙ N H 2
= − H 2 = 720π 2 M P2
and if initially N (720π 2 MP /Hinit )2 then 1 is
clearly satisfied. For the parameters in the example situation
that we will present later 10−7. The evolution of the
Hubble expansion rate as a function of the number of efolds
is then given by
H SB
H (Ne)
2
Ne · N H S2B
= 1 + 360π 2 M P2
where H SB is the value of the Hubble expansion at symmetry
breaking. (The number of efolds Ne are obtained simply by
integrating d Ne = H dt .)
However, the Friedman equations also show for ϕ = 0
that since ρ = 3H 2 M P2 ∼ (λ/4)ϕ04, ϕ0 will decrease slower
than TH . Therefore, the effective mass squared for ϕ
m2 ≡ N g2TH2 − λϕ02 ,
will eventually cross over to negative values, even if initially
N g2TH2 λϕ02. Simply put, at some point the system will
undergo a phase transition. This phase transition in contrast
to [
53
] is extremely gradual. As we will show, it can take
several thousands of efolds to complete. This is due to the
special nature of the thermal radiation as given by TH : the
thermal bath is continuously replenished by the decay of ϕ0
and hence does not dilute in the usual fashion.
After the phase transition ϕ will start rolling to the new
minimum and plays the role of the inflaton in the usual sense.
The evolution of ϕ can be characterized with (2.9) to consist
of three regions, m2 0, m2 ∼ 0 and m2 0.
2.1 m2
0: gradual decay of ρ
When the effective mass of the field (2.9) is very large and
positive the minimum of the potential is at ϕ = 0 and
supposing the field is in the minimum (2.6) leads (in the
approximation that neglects H˙ ) to
H =
Hinit
24N0πH2i3nMit P2 t + 1
t→∞
1/3 −→
240π 2 1/3
N
(MP2 /t)1/3 , (2.10)
V (ϕ, TH )
TH5 ,
as given by the continuity equation (2.6). As discussed after
(2.5) for most cases the change in the Hubble is very
gradual which justifies neglecting the derivatives of H from the
right side of the Einstein equation. Also interestingly the
late time limit of (2.10) is independent of the initial
Hubble rate indicating that the case m2 0 exhibits late time
attractor evolution which is independent of initial conditions:
after a sufficiently long time with m2 0 the system will
always relax to a configuration with H ∼ (M P2 /t )1/3 and
ϕ = ϕ˙ = 0. For the remaining analysis we will choose the
attractor configuration as our initial condition.
What is apparent from this section is that when m 0
the system has fairly unremarkable behaviour: the field sits
put in its vacuum state and the vacuum energy ρ gradually
decays. The quantum fluctuations around the mean we will
denote as φˆ . When the field is heavy m 0 they are similarly
suppressed
(2.11)
(2.12)
φˆ2
∼ 0 .
The case m 0 is illustrated in Fig. 1.
As discussed after (2.9) eventually the effective mass will
vanish and the system will undergo a phase transition leading
to interesting dynamics for ϕ.
2.2 m2 ∼ 0: large quantum fluctuations
The phase transition happens when the effective mass of the
field (2.9) vanishes at the origin. The value of the Hubble
parameter at and soon after the symmetry breaking transition
is approximately given by the first Friedmann equation from
(2.1) as
V (ϕ, TH )
ˆ2
φ
ϕ˙
∼ H2
H2
where (ϕ0)2SB corresponds to the time of symmetry breaking.
We can neglect both the thermal and kinetic contributions to
the first Friedman equation and the kinetic term in the
dynamical Friedman equation (2.5) so long as ϕ˙ 2 N TH4 V .
N λ/g4 may be obtained via (2.4) and (2.9) and it implies
tThhaet NseTcHo4nd cVonidsiftuiolfinllϕ˙e2d at theNtTimH4ehooflsdysmcmlaestsriycablrlyeaskiinncge.
at symmetry breaking ϕ˙ = 0; but as it is showed later due
to the quantum fluctuations ϕ = 0. As will be discussed
more in Sect. 2.3 (see Eq. (2.18)) when the quantum
fluctuations dominate over the classical motion one may write
ϕ˙  ∼ H 2/(2π ) which immediately implies ϕ˙ 2 N TH4 for
large N .
However, importantly, TH and hence m change very
gradually and therefore immediately after symmetry breaking we
expect a long epoch of very large quantum fluctuations before
the rolling becomes dominant as illustrated in Fig. 2. The
behaviour and magnitude of these fluctuations may be
analytically solved via the stochastic formalism [
55, 56
]. This
epoch of large quantum fluctuations is quite important in our
model since as we will further discuss in Sect. 2.3, when the
TH has dropped enough the classical rolling will take over
for which the initial condition will be dynamically set by the
period of quantum jumping. We note that during this epoch
where m2 ∼ 0 we would expect the large density
perturbations to lead to the formation of domain walls, but since
we expect to obtain many more than 60 efolds of inflation
after this period, we expect these evils to be swept outside
the horizon.
In order to obtain a qualitative picture of the system’s
behavior as the representative value for the classical
dynamics one may take the square root of the variance ϕ2 ≡ φˆ 2 .
Here we settle for solving this expectation value by using the
Langevin equation in the Hartree approximation relegating a
more complete discussion to Sect. 3.1. The relevant equation
ddt φˆ 2
= 4Hπ32 − 23mH2 φˆ 2
During this epoch there is little dynamics in φˆ 2 and the two
relevant limiting cases from the above are
φˆ 2 m=→0 π √H28λ
and
φˆ 2 m→=−∞ −m2 ,
3λ
which are separated by the threshold
m2
t ≡
3H 2√λ
√2π
,
i.e. for −m2 mt2 (−m2 mt2) the results coincide with
those on the left (right) side of (2.15).
The solutions (2.15) are approached only at the saturated
limit when the system has been given enough time to
equilibrate. The exact time this takes depends on the parameters
of the potential. It is known that for a quartic theory the
equilibration time scale in terms of efolds is given by 1/√λ
[
57,58
]. If the time scale for the change in the Hubble rate in
terms of efolds or −1 is much longer than the equilibration
time our approximation of using the results at the saturated
limit is valid. With (2.4), (2.9) at m ≈ 0 and (2.7) we can
write the condition −1 λ−1/2 as
15g4 N
4π 2√λ
1 ,
which again is easy to satisfy for large N . Also, in our model
the scale of symmetry breaking (2.13) can be tuned by
choosing λ,N and g in our potential (2.3) with smaller scales
corresponding to a slower dynamics as is evident from (2.7). Hence
the condition in (2.17) can also be understood to imply the
freedom to choose H SB low enough such that the saturated
expressions in (2.15) are a good approximation.
In the absence of the horizon entropy thermalising a large
number of fields coupled to the scalar field, this kind of
symmetry breaking would not lead to good inflationary initial
conditions since the Kibble mechanism would lead to large
fluctuations from horizon to horizon with large field gradients
that would prevent inflation from starting in the first place. In
this scenario, the thermal corrections to the potential prevent
the kinetic energy dominated regions from running straight
down to ϕ = ϕ0. Eventually therefore as the thermal
dissipation continues to gently facilitate the slow phase transition
and the field’s classical position and motion starts to
dominate over its quantum fluctuations, a region should emerge
somewhere where the field is coherent enough across
several horizons such that the well known difficulties obtaining
inflationary initial conditions are overcome.
V (ϕ, TH)

φˆ2
ϕ
Fig. 3 The onset of classical rolling occurs when the steepness of
the potential has increased enough. The initial value for the field is
determined by the preceding randomwalking epoch when m ∼ 0
Strictly speaking one can only talk about a classical “rolling”
once the minimum is further away from the origin than the
stochastic vacuum expectation value that the field ϕ2 ≡ φˆ 2
would obtain from the quantum fluctuations. As a first
approximation one can say that the field starts rolling when
in one efold the size of a single quantum jump (H/2π ) is
smaller that the distance traveled due to the classical rolling
ϕ˙H −1
H
2π
,
which is the opposite to the usual condition for eternal
inflation [
59
]. More accurately we can use (2.14) for the dynamics
of φˆ 2 .
As shown in the previous section, at the time of symmetry
breaking (m = 0), the variance of the field is approximately
2 Hλ φˆ 2 2 = 4Hπ32 . At times soon after symmetry breaking the
mass term remains negligible m2 φˆ 2 3λ φˆ 2 2 and the
field will remain constant at the value acquired at symmetry
breaking until the mass is relevant, i.e. −m2 − 3λ φˆ 2 = 0.
At this point, the potential will be steep enough for classical
slow roll to start and this classical motion will dominate over
the quantum fluctuations (see Fig. 3). Note that this condition
agrees with the estimate (2.18) since ddt φˆ 2 = 4Hπ32 . Once
the classical rolling starts the mass is given by
(2.18)
−m2
3λ φˆ 2 = 3λ π √H28λ
and the value of the field and the speed are
d
d Ne
φˆ 2
=
H 2
2π
φˆ 2 = π √H28λ =
−m2
3λ
Once classical rolling has been triggered, our model gives
rise to the usual slowly rolling inflation. We emphasize that
(2.19)
(2.20)
the initial conditions for inflation are set dynamically by the
large quantum fluctuations prior to classical rolling and hence
are not free parameters. Similarly, the start of slow roll is
triggered dynamically once the potential has acquired sufficient
steepness and occurs for a wide range of values for ϕ0, in
particular also for ϕ0/H 1. Finally, we remind the reader that
the neat attractor behaviour of the solutions prior to
symmetry breaking (2.10) also exhibit an independence from initial
conditions. For these reasons we can conclude that the model
presented here successfully evades all the usual finetuning
issues of inflationary models.
3 Inflationary predictions
To have a successful inflationary scenario, we need enough
efolds (around 50–60) and we need to obtain the right
perturbations PR and spectral tilt ns 50–60 efolds before the
end of inflation. As shown in the previous section, more than
60 efolds of expansion can be obtained very easily in this
scenario – the scale of inflation H can be set to be much
smaller than the Planck mass so 1 continues for many
efolds before H differs significantly from the value at
symmetry breaking H SB, meaning that we get enough inflation.
Here we show how the perturbations are generated once the
field is classically rolling to the minimum. The perturbations
in the spatially flat gauge ( = 0) are defined as [
60
]
H ϕ˙ δϕ
R = ρ + p ,
which we note will in our model lead to the usual expression
encountered in models of warm inflation [
61
]. The power
spectrum takes the form
PR =
H ϕ˙ H
−2H˙ M P2 2π
2
,
where from now on we set ϕ2 ≡ φˆ2 . In contrast to single
field slow roll inflation, in the current setting the perturbations
and the tilt are given by
PR =
60π
m2ϕ + λϕ3 2
N H 3
,
ns − 1 = 6 − 2η , (3.3)
where we used the slow roll condition ϕ˙ = −V /3H , and
dropped the subdominant kinetic piece ϕ˙2 in (2.5).
Furthermore, we note that is negligible in the calculation of the
tilt.
The difference with the usual single field slow roll
inflation comes from the contribution of the thermal sector to the
dynamical Friedman equation (2.5), which both changes the
vacuum energy decay and modifies the usual calculation for
the spectrum because H˙ ∝ H 4. Because of this, the
perturbations follow the inverse of the usual PR ∝ H 4/ϕ˙2 behaviour.
This means that the usual expression for the tilt ns − 1 ∝ η
is not true and we get ns − 1 ∝ −η, where η = M P2 VV is the
usual slow roll parameter. So in order to obtain a red
spectrum the inflaton needs to be evolving in the regime where
the potential is convex (V > 0) i.e. beyond the inflection
point at V = 0, contrary to the usual small field case.
There are a variety of different combinations of
parameters that can give rise to the correct inflationary perturbations,
however in what follows we will present a situation where
the 50–60 efolds of inflation we are interested in starts very
soon after the symmetry breaking occurs. In this situation,
the value of η is naturally small enough to give us the right
tilt because soon after symmetry breaking V = 0 by
definition. There is therefore no need to fine tune our treelevel
parameters to give us a flat potential near the origin
effectively bypassing the issue that the natural treelevel values
are argued to give rise to η 1 [
62
].
Our system has 4 free parameters (N , λ, ϕ0 and g).
However for the sake of clarity we choose here the final efolds
of inflation to occur soon after symmetry breaking and hence
we can approximate the value of ϕ0 during inflation by
ϕ0
MP ≈
(ϕ0) SB
MP
4π √3
= √N g
,
(3.4)
which can be derived by making use of m SB = 0 in (2.9) and
(2.13). With the above we effectively reduce the degrees of
freedom from four to three. Also, if the observable part of the
inflationary spectrum is going to take place soon after
symmetry breaking then as soon as the classical rolling starts we
want the field to be in the red tilt regime. For this to occur we
need to ensure that ddt φˆ 2 > ddt −λm2 at the inflection point,
otherwise the minimum would move faster away from the
origin than the field. If this were to occur, the spectral tilt
would be blue (after symmetry breaking). By making use of
very similar steps that led to (3.4) and the Hartree
approximation for φˆ 2 from (2.14) one may show this condition to
lead to the order of magnitude constraint g12 < 41π52 .
So from now on, we will set for simplicity g = 1. However
we also emphasize that a red tilt can be obtained a long time
after symmetry breaking, so g = 1 is only a choice for the
forthcoming calculation for the estimates of the perturbations
and the tilt, and none of the above derivations relies on a
specific value of g.
After this choice, we effectively have 2 degrees of
freedom left (N , λ) that will determine the perturbations and
the spectral tilt soon after the classical roll to the
minimum starts. It proves convenient to define a new
parameter, α = ϕ/ −m2/3λ, where α = 1 means that the field is
at the inflection point and α = √3 means that the field is
at the minimum of the potential. We can then approximate
the perturbations and the tilt from (3.3) and with the help of
equations from Sect. 2.3 by
2025
PR = √2π
α3 2 √λ
− α + 3 N 2
ns − 1 =
−1 + α2 √
√2π
λ
(3.5)
It is interesting to note that the order of magnitude of the
spectral tilt does not depend on N while the magnitude of the
perturbations depends on both (λ and N ). A smaller value of
λ will make our spectrum become more scale invariant and
will allow us to reduce the number of conformal fields that
need to be present to give us the right spectrum. Hence we
can shift the fine tuning between a small value of λ and a
large number of spectator fields.
Since in this model inflation takes place between the
inflection point and the new minimum emerging due to
broken symmetry, the field excursion of ϕ scales as ϕ ∼
−m2/λ. Furthermore, since for our choice g = 1 this takes
place soon after symmetry breaking where the mass
parameter can be estimated as −m2 ∼ √λH 2, which with the
Hubble rate from (2.13) and (3.4) indicates that for a large number
of spectator fields the field excursions are subPlanckian. For
the parameters agreeing with observations this turns out to
be true. This indicates that despite begin a small field model
of inflation, the initial conditions are not finetuned but arise
naturally as the favoured attractor solutions unlike what is
usually encountered [
63
].
The approximations and estimates set out here agree with
the numerical solution that we will turn to now.
3.1 Numerical solution
In this section, we look at the numerical solution
corresponding to the parameter choices we have made above.
Before symmetry breaking the field lies at the origin ϕ = 0
until ϕ0 drops sufficiently for the potential at the origin to
become tachyonic. Classically of course, the field would then
not move anywhere because d V /dϕ = 0 at the origin, and
if we were to introduce a small perturbation away from ϕ =
0 as an initial condition, then classically our final solution
would depend strongly on this perturbation. To get around
this problem we evolve the field using the Langevin equation
which takes into account the stochastic quantum fluctuations
the field receives
dϕ
d Ne = −
m2ϕ + λϕ3
3H 2
H
+ 2π ξ
(3.6)
where ξ is a Gaussian white noise with zero mean and unit
variance.
For each combination of the remaining two free
parameters (N , λ) we perform 104 numerical realisations of the
Langevin equation and find the mean values of the
magnitude of the perturbations PR for each value of the tilt ns as the
field evolves. These simulations are also used to verify that
the dynamics of the field agrees with our analytical estimates
from Sect. 2.
We know that to obtain a red spectrum (ns < 1) the field
must lie between the inflection point and the minimum.
Figure 4 shows how the variance of the field φˆ 2 evolves
with respect to the number of efolds of expansion for the
parameters λ = 10−2, N = 105.5, and g = 1 (making
ϕ0 = 10−1.41 MP ). Note that the initial epoch where we set
the horizontal axis equal to zero is chosen arbitrarily. We also
plot in the same figure the variance of the field that one would
expect using the Starobinsky–Yokoyama prescription for the
variance in a perfect de Sitter spacetime corresponding to a
vacuum energy equal to our evolving vacuum energy [
55
],
assuming instant equilibration for the probability
distribution. The field expectation value lags behind this estimator,
showing the importance of solving the Langevin equation.
The field approximately remains constant at this value until
the minimum has dropped enough to make the potential steep
and for it to possess an inflection point. We also plot the
position of the minimum as it moves out towards its zero
temperature value and the position of the inflection point, showing
that we remain on the good side to obtain a red spectrum.
Note that in a period of time during inflation corresponding
to 60 efolds of expansion, the variation of ϕ0 is negligible
relative to the variation in ϕ so we assume that it is constant
for calculational simplicity.
Figure 5 shows the average evolution of the magnitude of
perturbations for the same parameters as a function of the
spectral tilt. We can see that these parameters can give rise
to the correct combination of amplitude and spectrum for
the perturbations to match what is observed in the Cosmic
Microwave Background, i.e. PR = 2.2 × 10−9 and ns =
0.96 [
64,65
].
The power spectrum of tensor fluctuations is given by the
energy scale of inflation ( for example see equation 216 in
[
60
]), therefore for the parameters chosen, the tensor to scalar
ratio that we get is r = 0.17. This value is still within the 95%
CL from the Planck data [
1
] although lower values can be
obtained by choosing a different set of parameters, since the
scale of symmetry breaking can be easily tuned, Eq. (2.13).
In order to end inflation, we assume the temperature
corresponding to the expansion rate during inflation falls below
the mass of the particles which form the thermal bath
affecting the potential for the scalar field. We set this by hand to
give us 60 efolds after the epoch corresponding to the good
values of PR and ns .
4 Discussion/Conclusion
In this work we have presented a possible new mechanism
for inflation the early Universe. We have argued that if one
takes the Gibbons–Hawking temperature associated with the
horizon of de Sitter space seriously, one is lead to a couple of
conclusions that can affect the evolution of quantum fields in
the early Universe significantly. In particular we have argued
that if there are enough fields coupled to the scalar field which
takes the role of the inflaton, their Horizon induced
temperature leads to thermal corrections to the potential which can
affect its expectation value. Since these thermal fluctuations
do not redshift as rapidly as normal radiation, this effect can
last for a significant number of efolds of expansion.
To summarise the mechanism – We consider a real scalar
field with a Z2 symmetric potential with minima at ±ϕ0. If
the field starts at the origin, the nonzero energy density leads
to de Sitter expansion and a cosmological horizon with an
entropy. The field is trapped at the origin due to its coupling
to a large number of conformal spectator fields which have
a nonzero temperature associated with this horizon entropy.
The height of the potential at the origin decays slowly as
a result of the backreaction of the thermal radiation on the
parameter ϕ0 in the Lagrangian. The expansion rate therefore
decreases as does the temperature TH of the thermal radiation
until the mass at the origin becomes tachyonic, at which point
a phase transition occurs and the field rolls away from the
origin towards its zero temperature minimum at ϕ = ϕ0.
This phase transition however occurs extremely slowly due
to the same finite temperature corrections to the potential.
During this period of rolling we are able to obtain not only the
correct number of efolds but also the correct perturbations
and spectral tilt as measured in the CMB. This setup has two
attractive
features:• Normally, for a field to act as an inflaton, it must have
a super Planckian expectation value and/or very finely
tuned parameters. The mechanism we outline in this
paper enables a potential which would otherwise not be
flat enough to give rise to enough efolds of inflation to do
so due to the thermal effects of multiple spectator fields
• The initial conditions for inflation may arise naturally as
attractor solutions due to a slowly occurring phase
transition. This happens despite the fact that inflation occurs
with subPlanckian values for the inflaton. This is the
result of the parameters in the Lagrangian changing due
to the backreaction of the thermal radiation.
It is clear that this scenario is not necessarily a panacea for
all of the problems of inflation. We need to assume that
parameters in the Lagrangian decay over time due to the
back reaction of Hawking radiation at the horizon. While
there are many respected physicists who believe that such
behaviour is probable and perhaps necessary, it is clear that
further theoretical investigation and debate is required to put
such speculation on a stronger footing. We also require quite
a large number of fields which are coupled to the inflaton
in order to obtain the thermal braking required to obtain
enough efolds of inflation, in the example we have put
forward here about 105.5. The masses of the fields need to
be chosen such that inflation ends 50–60 efolds after the
epoch where the good perturbations and spectral index are
set. In return for this cost, we obtain a theory of inflation
which doesn’t require transplackian field excursions, does
not require extremely small parameters in the Lagrangian (we
use a value of λ ∼ 0.01) and which may naturally explain the
initial conditions for inflation. Finally, it seems quite
challenging to understand how the difference between this
scenario and normal inflation could be distinguished
experimentally.
Acknowledgements This project was funded by the European
Research Council through the project DARKHORIZONS under the
European Union’s Horizon 2020 program (ERC Grant Agreement
no.648680). TM is supported by the STFC grant ST/P000762/1. The
work of MF was also supported by the STFC.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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