# Wave correlations and quantum noise in cosmology

Ulf Leonhardt
Department of Physics of Complex Systems,
Weizmann Institute of Science,
Rehovot 7610001, Israel

###### Abstract

Wave noise is correlated. While it may look random in space, correlations
appear in space–time, because the noise is carried by wave propagation. These
correlations of wave noise give rise to fluctuation forces such as the Casimir
force, they are responsible for the particle creation in the dynamical Casimir
effect and in the expanding universe. This paper considers the noise
correlations for light waves in non-exponentially expanding flat space. The
paper determines the high-frequency asymptotics of the correlation spectrum in
the conformal vacuum. These noise correlations give rise to a nontrivial
vacuum energy that may appear as the cosmological constant.

##  1 Introduction

Explorers have mapped every corner of the Earth, but the time of exploration
has only just began: $95\%$ of the current content of the universe is
completely unknown. The uncharted $95\%$ are called the “dark sector” with
$25\%$ belonging to dark matter and $70\%$ to dark energy [1]. While there are
many ideas from particle physics on the nature of dark matter, and several
experimental programmes for detecting dark–matter particles [2] dark energy
has been an enigma [3, 4]. However, it might actually be the other way round:
dark energy could be the easier problem to solve, but not as a problem of
high–energy physics. Rather, it might belong to an area of low–energy physics,
extrapolated to cosmological scales. In this paper I will follow up on the
hypothesis [5, 6, 7] that dark energy, this arcane force that drives the
universe apart, is a form of much more mundane forces, the van der Waals and
Casimir forces, that cause ordinary things to stick. These are forces of the
quantum vacuum [8, 9].

This is not a new idea. In 1968 Zel’dovich [10] suggested that vacuum
fluctuations create Einstein’s cosmological constant $\Lambda$ [11].
Einstein’s $\Lambda$ is what was later called dark energy [12]. However,
Zel’dovich’s and similar suggestions [13] disagree with the measured value of
$\Lambda$ by some 120 orders of magnitude. The idea that $\Lambda$ comes from
the quantum vacuum is not new — and seem to have failed spectacularly. What is
new is a better theory of the quantum vacuum, inspired by precision
measurements and manipulations of Casimir forces [14, 15, 16], by the analogy
between dielectric media and space–time geometries [17, 18, 19, 20, 21, 22,
23, 24] tried and tested in transformation optics [23, 24, 25, 26] and in
optical analogues of black holes [27, 28, 29, 30, 31, 32, 33], and inspired by
the person to whom this volume is dedicated: Michael Berry. Not only did he
encourage me to pursue unconventional ideas, these ideas resonate with his
work on the infinite intricacies of light [34].

The theory [5, 6, 7] is still mostly a hypothesis, but it appears to agree
with astronomical data [7] and seems to resolve [7] a major inconsistency in
the conventional interpretation of that data [35]: the $5\sigma$ tension
between the directly measured Hubble constant [36] and the Hubble constant
inferred from the Cosmic Microwave Background [1]. There are some $10^{2}$
theories to explain the Hubble tension [37]. All of them require modifications
of known physics — changes to the standard model of particle physics, general
relativity or the cosmological principle; all make some experimentally
untested modifications, with one exception. The theory advocated here is the
only one in the field rooted on experiments and relying on “new things in old
things” — to quote a phrase of Michael Berry.

These results are encouraging, but much more work needs to be done to prove or
disprove the theory on astronomical data [7], to test its physical mechanism
in laboratory analogues [38] and also to improve the theory itself. Let me
explain. The renormalized vacuum expectation value
$\varepsilon_{\mathrm{vac}}$ of the electromagnetic energy density can be
expressed such that [5]

$\frac{4\pi G}{3c^{2}}\,\varepsilon_{\mathrm{vac}}=-\alpha_{\Lambda}\Delta$
(1)

in terms of the gravitational constant $G$, the speed of light in vacuum $c$
and the dimensionless coupling parameter $\alpha_{\Lambda}$. The parameter
$\alpha_{\Lambda}$ depends on the inverse squared of the cutoff length
$\ell_{\Lambda}$ with [5] $\alpha_{\Lambda}=(9\pi)^{-1}$ if $\ell_{\Lambda}$
is the Planck length $\ell_{\mathrm{p}}=\sqrt{\hbar G/c^{3}}$ ($\hbar$ being
the reduced Planck constant). The energy density $\varepsilon_{\mathrm{vac}}$
does two things: it gravitates and it generates a trace anomaly [5, 38, 39]
with energy density $\varepsilon_{\Lambda}$ that appears as the cosmological
term $\Lambda$, but is no longer constant. The total vacuum energy
$\varepsilon_{\Lambda}+\varepsilon_{\mathrm{vac}}$ grows with
$-4\varepsilon_{\mathrm{vac}}$ times the Hubble parameter [5]. The
cosmological term $\varepsilon_{\Lambda}$ thus accumulates
$\varepsilon_{\mathrm{vac}}$ during the cosmic evolution, it grows with
negative $\varepsilon_{\mathrm{vac}}$ and falls with positive
$\varepsilon_{\mathrm{vac}}$. The cosmological constant still appears in the
theory, yet not as a fundamental constant of nature but only as an integration
constant [7] that depends on the initial conditions and presumably was zero at
the beginning of time.

The quantity $\Delta$ in the vacuum energy density (1) carries the physical
units of a frequency squared and depends on the nature of the quantum vacuum.
In the first version [5] of the theory $\Delta$ was found to be

$\Delta=\partial_{t}^{3}\frac{1}{H}+H\partial_{t}^{2}\frac{1}{H}$ (2)

where $H$ denotes the Hubble parameter [40]. One sees from a scale analysis
that $\varepsilon_{\mathrm{vac}}$ carries the correct order of magnitude of
the cosmological constant111The argument [5] goes as follows. According to the
Friedman equation [40, 41] expression (1) gives $\frac{1}{2}H^{2}$ for the
realistic case of zero spatial curvature [1]. As $H$ varies on the scale of
$H$ the energy density $\varepsilon_{\mathrm{vac}}$ goes like $H^{2}$ and thus
plays a role in the cosmic dynamics.. In the second incarnation [7] of the
theory222Actually, this was the result of my first, unpublished version of the
theory. the expression

$\Delta=\partial_{t}^{3}\frac{1}{H}$ (3)

was published and used to compare theory with data [7] assuming
$\varepsilon_{\mathrm{vac}}$ as a perturbation of the cosmic dynamics [7].
While Eqs. (2) and (3) agree on the leading term, they differ in the
subdominant term. The data ruled out Eq. (2) whereas Eq. (3) agrees with the
astronomical data with the precision of that data for exactly the Planck–scale
value $\alpha_{\Lambda}=(9\pi)^{-1}$. However, this is only true within
first–order perturbation theory; the full solution of the cosmic dynamics
contains oscillatory modulations, suggesting that some vital ingredient was
missing that dampens these oscillations. In this paper I hope to have
identified the missing component and to have finally deduced the correct
vacuum energy. The paper also clarifies the role the quantum vacuum plays in
cosmology and it offers an explanation why quantum electromagnetism, and
quantum electromagnetism alone, is responsible for what appears as dark energy
in the current era. The heart of the problem of explaining dark energy from
vacuum fluctuations is the physics of wave noise.

Figure 1:  Wave noise. Space–time diagram of waves with Gaussian noise.
Although the wave field looks random in space $\\{x\\}$ features appear in
space–time $\\{ct,x\\}$ following the causal cones of wave propagation (with
speed $c$). For this picture 128 normalized left–moving and 128 right–moving
plane waves [Eqs. (5) and (6)] with periodic boundary conditions and of random
Gaussian complex coefficients were summed up. Increasing the number of waves
produces finer and finer structures, but ultimately the noise field diverges,
illustrating the divergence of the bare vacuum noise.

Wave noise is organized. In space, it may look completely random, but in
space–time patterns of correlations are clearly visible (Fig. 1). There we see
the characteristic diagonal features of wave propagation. Waves are traveling
to the left or the right with the wave velocity $c/n$, and the noise they
carry travels with them. If $n$ varies the noise pattern varies as well. The
most dramatic of such modifications are reflections, for example at obstacles
where $n$ is discontinuous. Reflected wave noise gives rise to fluctuation
forces [8, 9] such as the Casimir forces [42]. If $n$ varies in time, waves
may be reflected in time as well [43, 44]. A reflection in space is the change
of sign in the wave number, in time it is a sign change in frequency. In the
dynamical Casimir effect [45, 46, 47, 48, 49] these negative–frequency
components correspond to newly–created particles, simply because if part of a
wave of positive frequency $\omega$ is converted to $-\omega$ the energy
$\hbar\omega$ of the remaining positive–frequency component must grow,
particles are created. Here we focus less on the particle aspects, but rather
on the amplitude correlations of wave noise. We begin with a brief review on a
familiar example, the noise seen by accelerated observers [50, 51, 52]. Then
we show how this is related to the noise perceived by an observer at rest in
an exponentially expanding universe [53] before turning to the discussion of
vacuum modes in a universe of arbitrary expansion [54]. We confirm the
extension [54] of Gibbons’ and Hawking’s formula for the radiation temperature
[53] and find a new feature not present in exponential expansion: the Hawking
partners appear as red–shifted thermal radiation. The multiple interference of
all Hawking processes in the expanding universe gives the effective vacuum
energy; to calculate it we use the Wigner function of wave noise.

##  2 Uniform acceleration

Wave noise is organized, because waves can be organized in terms of modes, and
the noise appears solely in the amplitudes and phases of the mode
coefficients. Consider a simple 1+1 dimensional example: a scalar wave field
$\widehat{A}$ in empty Minkowski space given by the mode decomposition

$\widehat{A}=\int_{-\infty}^{+\infty}\left(\widehat{a}_{k}A_{k}+\widehat{a}_{k}^{\dagger}A_{k}^{*}\right)\mathrm{d}k$
(4)

where the $A_{k}$ are the mode functions $A_{k}(x,t)$ describing how the modes
propagate in space $x$ and time $t$. The $\widehat{a}_{k}$ are the mode
coefficients, and only they are subject to statistical or quantum
fluctuations. The mode functions should be normalized such that each mode
accounts for the field of exactly one particle. This is conveniently done with
the help of the scalar product [55]

$\left(A_{1},A_{2}\right)=\frac{\mathrm{i}}{\hbar}\int_{-\infty}^{+\infty}\left(A_{1}^{*}\,\partial_{t}A_{2}-A_{2}\,\partial_{t}A_{1}^{*}\right)\mathrm{d}x$
(5)

requiring

$\left(A_{1},A_{2}\right)=\delta(k_{1}-k_{2})\,,\quad\left(A_{1}^{*},A_{2}\right)=0\,.$
(6)

For example, if the modes are plane waves $A_{k}={\cal
A}\exp(\mathrm{i}kx-\mathrm{i}\omega t)$ with $\omega=c|k|$ we must require
${\cal A}^{2}=\hbar/(4\pi\omega)$. From the canonical commutation relations
between field and momentum density then follow [55] — for Bosonic fields like
the electromagnetic field — the standard Bose commutation relations:

$[\widehat{a}_{k_{1}},\widehat{a}_{k_{2}}^{\dagger}]=\delta(k_{1}-k_{2})\,,\quad[\widehat{a}_{k_{1}},\widehat{a}_{k_{2}}]=0\,.$
(7)

The Minkowski vacuum $|0\rangle$ is the quantum state annihilated by all the
plane–wave operators:

$\widehat{a}_{k}|0\rangle=0\,.$ (8)

The Minkowski vacuum is the vacuum with respect to an observer at rest in
Minkowski space. It also appears as the vacuum to observers in uniform motion,
because they perceive the modes $A_{k}$ as plane waves as well,
Doppler–shifted of course. But this is no longer true for accelerated
observers [50, 51, 52].

Figure 2:  Accelerated observers. Space–time diagram of accelerated observers
(black curves) in Minkowski space with Cartesian coordinates $x$ and $t$. The
observers follow the Rindler trajectories of Eq. (9) with fixed $\xi$ and
variable parameter $\eta$. The acceleration is given by $c^{2}/\xi$ while
$(\xi/c)\eta$ gives the proper time of each observer. For negative $\xi$ the
parameter $\eta$ needs to run backwards (reversed arrow) as proper time always
runs forwards. The observer on the right (R) is separated from the observer on
the left (L) by horizons (red). Neither left– nor right–moving light from R
can reach the shaded region in L.

Uniform acceleration is described by the transformation to Rindler coordinates
[56] as follows. Suppose we write the Cartesian space–time coordinates in
terms of hyperbolic polar coordinates:

$x=\xi\cosh\eta\,,\quad ct=\xi\sinh\eta\,.$ (9)

The Rindler coordinates $\\{\xi,\eta\\}$ cover the two wedges with $x\geq|ct|$
for $\xi\geq 0$ on the right and $-x\geq|ct|$ for $\xi\leq 0$ on the left of
the space–time diagram (Fig. 2). In analogy to the regular polar coordinates
$\\{r,\phi\\}$ with spatial metric $\mathrm{d}r^{2}+r^{2}\mathrm{d}\phi^{2}$
we get for the hyperbolic space–time metric

$\mathrm{d}s^{2}=c^{2}\mathrm{d}t^{2}-\mathrm{d}x^{2}=\xi^{2}\mathrm{d}\eta^{2}-\mathrm{d}\xi^{2}\,.$
(10)

A space–time metric measures the proper time $\tau$ with increment
$\mathrm{d}\tau=\mathrm{d}s/c$. In particular, as
$\mathrm{d}s=\xi\mathrm{d}\eta$ for $\mathrm{d}\xi=0$, the proper time along a
trajectory with fixed $\xi$ is $(\xi/c)\eta$. We can draw another conclusion
from the analogy of the Rindler coordinates with polar coordinates. In space a
rotation corresponds to a shift in the angle. In Minkowski space–time, a
hyperbolic rotation corresponds to a Lorentz transformation to a frame moving
with velocity $u$. An infinitesimal Lorentz boost shifts the hyperbolic angle
by $\mathrm{d}u/c$. A sequence of infinitesimal boosts thus draws an entire
Rindler coordinate line along varying $\eta$ for $\xi=\mathrm{const}$. Now,
uniform acceleration is just such a sequence of infinitesimal Lorentz
transformations. We thus conclude that the Rindler line is the world line of a
uniformly accelerated observer with acceleration
$\mathrm{d}u/\mathrm{d}\tau=c^{2}/\xi$.

Consider such a uniformly accelerated observer. Suppose the observer is
equipped with a spectrometer. A spectrometer consists of a spectral element to
decompose the field $\widehat{A}$ into frequencies, and a detector to measure
the spectral components. It is not important what the detector is. It may be a
particle detector [52] or an amplitude detector [57], the physically important
feature of the spectrometer is the ability to perform a frequency analysis,
and there the important aspect is the fact that the spectrometer responds to
its proper time $\tau$ and not to the coordinate time $t$. As
$\tau=(\xi/c)\eta$ we may describe the effect of the spectrometer as a Fourier
transformation with respect to $\eta$. Note, however, that for $\xi<0$ (on the
left side L of the Rindler diagram of Fig. 2) $\eta$ needs to run backwards,
since proper time always runs forwards.

Imagine now a pair of accelerated observers — one with positive $\xi$ on R and
one with the exact opposite $-\xi$ on L. Figure 2 reveals that the two
observers are separated by horizons. The entire world line of observer L lies
in the shadow of left– or right–moving waves that touch observer R. But it
turns out the two observers can and must communicate by sharing the same noise
field. To work this out, consider the spectral components they measure:

$\widetilde{A}_{\mathrm{R}}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\left.\widehat{A}\right|_{\mathrm{R}}\mathrm{e}^{\mathrm{i}\nu\eta}\,\mathrm{d}\eta\,,\quad\widetilde{A}_{\mathrm{L}}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\left.\widehat{A}\right|_{\mathrm{L}}\mathrm{e}^{-\mathrm{i}\nu\eta}\,\mathrm{d}\eta$
(11)

in terms of the dimensionless Fourier components $\nu$. Here the R and L
indicate the space–time trajectories of the two observers. They sample the
plane–wave Minkowski modes (Fig. 3) as oscillations with phases

$\varphi_{\mathrm{R}}=\left.k(x\mp
ct)\right|_{\mathrm{R}}=k\xi\,\mathrm{e}^{\mp\eta}\,,\quad\varphi_{\mathrm{L}}=\left.k(x\mp
ct)\right|_{\mathrm{L}}=-k\xi\,\mathrm{e}^{\mp\eta}\,.$ (12)

Now, components with positive Rindler frequencies $\nu$ may also sample
negative Minkowski frequencies, i.e. the complex–conjugated modes $A_{k}^{*}$.
In fact, moving the contour of the Fourier integral by $+\mathrm{i}\pi$ on R
and by $-\mathrm{i}\pi$ on L changes the sign in the phases (12) while
preserving the convergence of the Fourier integrals (11). We thus see that the
Fourier transform of the conjugate $A_{k}^{*}$ is exactly
$\mathrm{e}^{-\pi\nu}$ times the Fourier transform of $A_{k}$, on both sides
of the Rindler wedge.

Figure 3:  Plane wave. The accelerated observer (Fig. 2) samples noise made of
plane waves with random amplitudes and phases. Each plane wave is sampled
along the Rindler trajectory of Eq. (9) with proper time $(\xi/c)\eta$. The
panel shows the real and imaginary part of the wave sampled along the path
with parameter $\eta$. Fourier analysis reveals that the positive–frequency
components for $\eta$ contain negative–frequency components for $t$ enhancing
the quantum noise perceived by one observer at $+\xi$ by correlations with its
partner at $-\xi$ (Fig. 2).

Accelerated observers sample negative Minkowski frequencies. To see how this
affects the wave noise perceived by the accelerated observers, we introduce a
set of modes that are monochromatic with respect to those observers (Fig. 4).
Any mode in Minkowski space must be a superposition of left– or right–moving
waves. The left–moving waves are functions of $x_{-}=x+ct$ while the
right–moving modes depend on $x_{+}=x-ct$. From
$x_{\pm}=\xi\mathrm{e}^{\mp\eta}$ follows that the phases of monochromatic
Rindler modes must be logarithmic in $x_{\pm}$, which means that the Rindler
modes are purely imaginary powers of $x_{\pm}$. There we have two
possibilities: $x_{\pm}$ or $-x_{\pm}$ to an imaginary power. In the first
case the wave is predominately localized on the right side of the space–time
diagram (Fig. 2), in the second case on the left side. On R we should give the
Rindler wave a positive $\eta$–frequency $\nu$, i.e. the power $\pm\nu$ of
$x_{\pm}$, while on L it should oscillate with $-\nu$ as $\eta$ runs backwards
for forward–running proper time, which also corresponds to the power $\pm\nu$
but this time of $-x_{\pm}$. We thus define

Figure 4:  Rindler modes. The figure shows examples of modes that are
monochromatic for the two accelerated observers (white hyperbolas, see also
Fig. 2). For a monochromatic mode the phase increases linearly with time, but
for the observers this is proper time, not coordinate time. Each accelerated
observer comes in with asymptotically the speed of light and leaves
asymptotically with the speed of light. For such velocities proper time ticks
exponentially slowly, and so the phase grows only logarithmically. Near the
horizon (Fig. 2) the phase diverges logarithmically [Eq. (13)]. An
exponentially small part of the wave crosses to the other side if this wave is
made of a superposition of positive–norm plane waves, describing the quantum
vacuum.

$A_{\nu}={\cal A}\begin{cases}(x_{\pm})^{\pm\mathrm{i}\nu}&:\nu>0\\\
(-x_{\pm})^{\pm\mathrm{i}\nu}&:\nu<0\end{cases}\quad\mbox{with}\quad
x_{\pm}=x\mp ct$ (13)

and represent the field as

$\widehat{A}=\sum_{\pm}\int_{-\infty}^{+\infty}\left(\widehat{a}_{\nu}A_{\nu}+\widehat{a}_{\nu}^{\dagger}A_{\nu}^{*}\right)\mathrm{d}\nu\,.$
(14)

It only remains to determine the normalization factor ${\cal A}$ from Eqs.
(6). We substitute the modes (13) into the scalar product (5) with the
understanding that $(A_{1},A_{2})$ differs from zero only when
$\nu_{1}\sim\nu_{2}$. We define $\delta=\pm(\nu_{2}-\nu_{1})$ and obtain for
$\nu>0$:

$\left(A_{1},A_{2}\right)=\frac{2c\nu}{\hbar}{\cal
A}^{2}\int_{-\infty}^{+\infty}(x\mp
ct)^{\mathrm{i}\delta-1}\,\mathrm{d}x=\frac{2c\nu}{\hbar}{\cal
A}^{2}\left(1-\mathrm{e}^{-2\pi\nu}\right)\int_{0}^{\infty}\xi^{\mathrm{i}\delta}\,\frac{\mathrm{d}\xi}{\xi}\,.$
(15)

Writing $\xi$ as an exponential gives $2\pi$ times the standard Fourier
representation of the delta function. Defining the parameter $\zeta$ by

$\tanh\zeta=\mathrm{e}^{-\pi\nu}$ (16)

with $\cosh\zeta=(1-\mathrm{e}^{-2\pi\nu})^{-1/2}$ we thus get

${\cal A}={\cal B}\cosh\zeta\,,\quad{\cal B}^{2}=\frac{\hbar}{4\pi c\nu}\,.$
(17)

This concludes the normalization of the Rindler modes and hence the Rindler
representation of the field. Only one important, subtle point remains to be
discussed.

The Rindler modes (13) are understood to be analytic on the upper half complex
plane for $x_{+}$ and on the lower half plane for $x_{-}$ such that the left
side is suppressed for $\nu>0$ and the right side for $\nu<0$. In either case,
the $A_{\nu}$ are then analytic on the lower complex plane for the time $t$.
From this follows that we can always close the contour of a Fourier
transformation with respect to Minkowski time $t$ for negative frequencies
$\omega$ and get zero. In other words, the Rindler modes (13) have only
positive Minkowski frequencies. Therefore, they are superpositions of
positive–norm Minkowski waves, and so their associated annihilation operators
$\widehat{a}_{\nu}$ are also just superpositions of the Minkowski
$\widehat{a}_{k}$, which implies that both share the same vacuum state
$|0\rangle$.

Having established the vacuum in the Rindler representation, it is elementary
to work out the spectral components seen by the two accelerated observers. We
obtain from Eqs. (11) and (14) for the modes (13) with norm (17) and
$x_{\pm}=\xi\mathrm{e}^{\mp\eta}$ the expressions

$\widetilde{A}_{\mathrm{R}}={\cal
B}\left(\widehat{a}_{\nu}\cosh\zeta+\widehat{a}_{-\nu}^{\dagger}\sinh\zeta\right)\,,\quad\widetilde{A}_{\mathrm{L}}={\cal
B}\left(\widehat{a}_{-\nu}\cosh\zeta+\widehat{a}_{\nu}^{\dagger}\sinh\zeta\right)\,.$
(18)

We see here again that the observers sample negative–frequency components
$\widehat{a}^{\dagger}$ with relative weight
$\tanh\zeta=\mathrm{e}^{-\pi\nu}$. Representing the mode operators in terms of
their real and imaginary parts (Hermitian and anti–Hermitian parts) we see
that the sampled field amplitudes are connected — the real parts are
correlated and the imaginary parts anti–correlated. This means that the wave
noise perceived by the observer on R is correlated with the noise perceived by
observer L. Observer R is influenced by some extra randomness that comes from
this connection to observer L and vice versa. That excess noise appears in the
intensity as an additional contribution to the standard vacuum noise:

$\langle\widetilde{A}_{\mathrm{R}}^{\dagger}\widetilde{A}_{\mathrm{R}}\rangle=\langle\widetilde{A}_{\mathrm{L}}^{\dagger}\widetilde{A}_{\mathrm{L}}\rangle={\cal
B}^{2}\left(\frac{1}{2}+\frac{1}{\mathrm{e}^{2\pi\nu}-1}\right).$ (19)

As the dimensionless $\eta$ is related to the proper time by the factor
$c/\xi$, the frequencies measured in the spectrometers of the accelerated
observers are related to the dimensionless $\nu$ by the same factor. We may
read the $(\mathrm{e}^{2\pi\nu}-1)^{-1}$ in Eq. (19) as the Planck
distribution $(\mathrm{e}^{\hbar\omega/k_{\mathrm{B}}T}-1)^{-1}$ with Unruh
temperature [52]

$k_{\mathrm{B}}T=\frac{\hbar c}{2\pi\xi}$ (20)

where $k_{\mathrm{B}}$ denotes Boltzmann’s constant. Each one of the two
observers perceives the vacuum as thermal radiation with temperature (20).
Each one receives this extra noise, because the noise is correlated. These
correlations do appear when the field amplitudes are Fourier–transformed: they
are spectral correlations. In terms of particles, they appear as entangled
Einstein–Podolski–Rosen pairs [55]. When the spectrometer of observer R
detects a particle at frequency $\omega$ so does the spectrometer of observer
L (provided they are perfectly efficient). But here we are primarily concerned
with amplitude noise and its cosmological implications.

##  3 Exponential expansion

Turn now from accelerated observers in static Minkowski space to an observer
at rest in the expanding universe. Consider first the conceptually simplest
case: pure exponential expansion (de Sitter space [58]). This is the phase of
the cosmic evolution we are entering at the present time and, presumably, it
was the phase of inflation [59] just after the Big Bang (although with a much
higher expansion rate then in the current era). Assume in agreement with
astronomical observations [60] that the universe is homogeneous and isotropic,
and spatially flat [1]. In this case, the space–time geometry is given by the
flat–space Friedmann–Lemaitre–Robertson–Walker metric [40]:

$\mathrm{d}s^{2}=c^{2}\mathrm{d}t^{2}-a^{2}\mathrm{d}\bm{r}^{2}$ (21)

with time–dependent scale factor $a(t)$. The scale factor describes how
spatial distances expand, as the physical distance between two points at the
same time $t$ is given by $a$ times the coordinate difference $r$. The spatial
coordinates $\bm{r}$ are called comoving coordinates, because they do not move
relative to the universe. The coordinate time $t$ is called cosmological time
and, physically, it is the proper time of an observer at rest with the
universe ($\mathrm{d}\bm{r}=\bm{0}$). We may introduce a new time $\tau$
called conformal time, defined as

$\tau=\int\frac{\mathrm{d}t}{a}$ (22)

such that the metric becomes conformally flat:

$\mathrm{d}s^{2}=a^{2}\left(c^{2}\mathrm{d}\tau^{2}-\mathrm{d}\bm{r}^{2}\right).$
(23)

For light rays ($\mathrm{d}s=0$) the conformal factor $a^{2}$ is irrelevant,
and so light rays travel in conformal time and comoving space like in empty
Minkowski space. As Maxwell’s equations are conformally invariant [24] this
remains true for full electromagnetic fields and their quantum fluctuations.
We assume that the quantum vacuum is carried by plane waves in conformal time.
The notation is the exact opposite as in the case of uniform acceleration:
there $t$ is the time the vacuum propagates with and $\tau$ denotes the proper
time of the accelerated observer, whereas in the expanding universe the vacuum
waves propagate with $\tau$ while $t$ is the proper time of the observer at
rest with the universe.

Note that the gravitational field of the universe (the space–time geometry)
does distinguish a global frame — only in this frame the metric is homogeneous
and isotropic. We can of course move this frame to any point (as the universe
is homogeneous) and rotate it (as it is isotropic) but the metric is different
for an observer in uniform motion. Note also that although the universe is
spatially flat, it is curved in space–time. One obtains for the curvature
scalar [41]

$R=-\frac{6}{c^{2}}\left(\partial_{t}H+2H^{2}\right)$ (24)

in terms of the Hubble parameter

$H=\frac{\partial_{t}a}{a}\,.$ (25)

In the case of exponential expansion the Hubble parameter is a constant
$H_{0}$ such that

$a=a_{0}\,\mathrm{e}^{H_{0}t}\,.$ (26)

In this case, the space–time curvature is negative and constant333The
space–time of exponential expansion (de Sitter space) is a maximally symmetric
space with constant Riemann tensor
$R^{\alpha\beta}_{\mu\nu}=-(H_{0}/c)^{2}\,(\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}-\delta^{\alpha}_{\nu}\delta^{\beta}_{\mu})$.
The negative prefactor indicates the negative curvature. as we also see from
$R=-12H_{0}^{2}/c^{2}$.

Figure 5:  Exponential expansion. An observer at rest samples a plane wave in
the exponentially expanding universe. The wave oscillates with conformal time
[Eq. (27)] that differs exponentially from the proper time of the observer
(the cosmological time $t$) in perfect analogy to the Minkowski wave sampled
by the accelerated observer (Fig. 3).

Suppose the observer at rest with the universe samples the plane waves of the
quantum vacuum (Fig. 5). They oscillate with frequencies $\Omega$ in the
conformal time $\tau$ of Eq. (22). We obtain for the case of exponential
expansion:

$\tau=-\frac{1}{aH_{0}}\,.$ (27)

Note that conformal time is negative and ends at $\tau=0$ in the infinite
future ($t=+\infty$). The observer samples the phase

$\varphi=\Omega\tau=\frac{\Omega}{a_{0}H_{0}}\,\mathrm{e}^{-H_{0}t}\,.$ (28)

This is the same phase as the one of a right–moving wave sampled by Rindler
observer R (Fig. 4). We see from Eq. (12) that $\Omega/(a_{0}H_{0})$
corresponds to $k\xi$ and $H_{0}t$ to the dimensionless Rindler time $\eta$.

Figure 6:  Extended de Sitter space. Radial space–time diagram $\\{c\tau,r\\}$
in conformal time $\tau$ and comoving radius $r=|\bm{r}|$. Cosmological time
$t$ runs according to the arrows indicated and ends ($t=+\infty$) at the
horizontal line ($\tau=0$).. Light travels along diagonal lines in the
conformal diagram and may cross over to the next world, the extension, for
$\tau>0$. Light beyond the horizon (red line) cannot reach the observer (black
vertical line up until $t=+\infty$) before this world ends ($\tau=0$). Light
coming in within the white area — within the horizon — leaves in the shaded
area, but cannot reach the double–shaded region in the extended world, in
perfect analogy to the Rindler horizon of uniform acceleration (Fig. 2).

The observer at rest with the exponentially expanding universe thus perceives
waves in the same way as the uniformly accelerated observer in Minkowski
space, including the waves of the quantum vacuum. Like in uniform
acceleration, the observer is surrounded by a horizon (Fig. 6). Seen in
conformal time and comoving space, incoming rays outside of the radius
$r_{H}=-c\tau$ will never arrive at the observer before the world ends in
conformal time ($\tau=0$). From Eq. (27) we get

$r_{H}=\frac{c}{aH}\,.$ (29)

Unlike the accelerated observer, there is no partner L to the observer R, at
least in this universe. We may construct an artificial partner by extending de
Sitter space to $\tau>0$ (similar to the Kruskal extension of the black hole
[56]). For this we imagine another universe with infinite cosmological time
related to positive conformal time by $\tau=H_{0}^{-1}\mathrm{e}^{-H_{0}t}$.
In this netherworld time runs backwards from $+\infty$ to $-\infty$ such that
conformal time and light smoothly passes from one world into the other (Fig.
6). The partner observer in the netherworld is then shrouded behind a horizon
(Fig. 6) from the observer in this world, in perfect analogy to uniform
acceleration. In particular, we may conclude that the de Sitter observer
perceives the vacuum as thermal radiation as well [53]. From the
correspondence to the case of the accelerated observer with Unruh temperature
(20) we obtain the Gibbons–Hawking temperature [53]

$k_{\mathrm{B}}T=\frac{\hbar H_{0}}{2\pi}\,.$ (30)

Exponential expansion is a clear, simple, perfectly understood case of quantum
noise in cosmology, but it is largely an academic case. In reality, the
universe does not expand exponentially yet nor did it in the past. Very few
papers have tackled the problem beyond the case of de Sitter space [54, 61,
62], because it is a difficult problem of — apparently — hardly any relevance,
as the Gibbons–Hawking temperature of the real universe is astronomically
small ($T$ lies in the order of $10^{-29}\mathrm{K}$ for $1/H_{0}$ of
$10\mathrm{Gy}$). But if the quantum noise of general cosmological horizons is
indeed the key to understanding the cosmological constant [5], understand it
we must.

##  4 Expanding flat space

Apart from exponential expansion, there is no other case when an expanding
flat space establishes a genuine event horizon [54, 63] (Fig. 7a). One sees
this as follows. The cosmological horizon [64] is the spherical surface around
a given point where the expansion velocity reaches the speed of light. The
expansion velocity $u$ is the derivative of the proper length $\ell=ar$ with
respect to cosmological time $t$. Differentiating $\ell$ gives Hubble’s law,
$u=H\ell$, in terms of the Hubble parameter $H$ defined in Eq. (25). We see
that $u$ reaches $c$ at $r_{H}$ of Eq. (29). For the cosmological horizon to
be an event horizon it needs to be light–like, parallel to light rays in the
$\\{c\tau,r\\}$ space–time diagram, because otherwise light may cross it.
Since

$\tau=\int\frac{\mathrm{d}a}{a^{2}H}$ (31)

the conformal time $\tau$ does only agree with $-1/(aH)$ for
$H=\mathrm{const}$, i.e. exponential expansion, which proves that cosmological
horizons are not event horizons, except in the exponential case. In fact, the
light of distant galaxies and the Cosmic Microwave Background reaches us from
beyond our horizon [40, 63]. Therefore, it is not clear from the outset how to
generalize the Gibbons–Hawking formula (30) to the case of expanding flat
space in general.444This section closely follows Ref. [54] but corrects an
error in the conformal factor. Despite this error, the ideas and results of
the paper [54] are correct, as we show here and in Sec. 5.

Figure 7:  Cosmological horizon. Space–time diagrams of the horizon (red
curve) based on actual cosmological data [1, 40] (plotted in units $c/H_{0}$
with Hubble constant $H_{0}$). a: in co–moving spatial coordinates $\bm{r}$
and conformal time $\tau$ light (black and white lines) propagates like in
Minkowski space. The region outside the horizon is shaded in grey. Light may
cross the horizon, except when, in the final stage of cosmic evolution, the
horizon becomes light–like and hence a genuine event horizon. b: vacuum modes
in analogy to the Rindler modes (Fig. 4). The modes are defined with respect
to a specific time, here $\tau=0$ (the present time). The figure shows the
phase pattern of the incident light only, not the outgoing light; Eq. (36)
describes both.

Consider light in a universe with metric (21). Space shall be expanding,
$H>0$. For conceptual simplicity we do not start from Maxwell’s equations, but
rather describe each polarization component by a conformally–coupled scalar
field with modes satisfying the wave equation [24, 65]:

$\frac{1}{\sqrt{-g}}\,\partial_{\alpha}\sqrt{-g}\,g^{\alpha\beta}\partial_{\beta}A-\frac{R}{6}\,A=0$
(32)

in terms of the metric tensor $g_{\alpha\beta}$, its determinant $g$ and
matrix–inverse $g^{\alpha\beta}$, and the curvature scalar $R$ of Eq. (24).
Einstein’s summation convention over repeated indices is adopted. The modes
shall be normalized according to Eq. (6) with the scalar product [65]:

$\left(A_{1},A_{2}\right)=\frac{\mathrm{i}c}{\hbar}\int\left(A_{1}^{*}\,\partial^{0}A_{2}-A_{2}\,\partial^{0}A_{1}^{*}\right)\sqrt{-g}\,\mathrm{d}^{3}x\,,\quad\partial^{0}=g^{0\alpha}\partial_{\alpha}\,.$
(33)

One sees from the wave equation that the scalar product (33) is a conserved
quantity for arbitrary wave packets satisfying Eq. (32). Writing $A$ as
$A_{0}/a$ reduces the wave equation (32) to the free wave equation for $A_{0}$
with respect to the conformal time $\tau$ of Eq. (22), which shows that light
waves propagate in the expanding universe like in free Minkowski space
$\\{c\tau,\bm{r}\\}$ (not just light rays). We may use the plane waves

$A=({\cal
A}/a)\,\mathrm{e}^{\mathrm{i}\bm{k}\cdot\bm{r}-\mathrm{i}\omega\tau}\quad\mbox{with}\quad\omega=c|\bm{k}|\,,\quad{\cal
A}^{2}=\frac{\hbar}{16\pi^{3}\omega}$ (34)

as normalized modes. We assume that the cosmological quantum vacuum is in the
vacuum state (8) with respect to these conformal plane waves. This
cosmological vacuum is called the conformal vacuum [65]. However, as we know
from the case of exponential expansion, an observer at rest may not perceive
the conformal vacuum as vacuum fluctuations.

Imagine a point–like observer at rest with the expanding universe. We use
spherical coordinates with the origin attached to the point of the observer.
Only radial waves will matter, because all waves with higher orbital angular
momentum vanish at the origin. Write the radial modes as

$A=\frac{1}{\sqrt{4\pi}\,ar}A_{\nu}(r,\tau)\,.$ (35)

From the wave equation (32) follows that the $A_{\nu}$ satisfy one–dimensional
wave propagation, which means that $A_{\nu}$ consists of a superposition of
incoming and outgoing waves $f(c\tau\pm r)$. As $A$ must not diverge for
$r\rightarrow 0$ we need to require $A_{\nu}=f(c\tau+r)-f(c\tau-r)$, the
outgoing wave is the ingoing wave reflected at the focus. Inspired by the
cases of uniform acceleration and exponential expansion, we wish to define
modes in close analogy to the Rindler modes of Eq. (13). These modes can only
capture the cosmological horizon at a given moment in time, i.e. for a given
scale factor $a_{0}$ and corresponding Hubble parameter $H_{0}$. We define
[54] (Fig. 7b) in analogy to the Rindler modes [Eq. (13), Fig. 4]:

$A_{\nu}={\cal
A}\begin{cases}(\eta-\rho)^{\mathrm{i}\nu}-(\eta+\rho)^{\mathrm{i}\nu}&:\nu>0\\\
(\rho-\eta)^{-\mathrm{i}\nu}-(-\eta-\rho)^{-\mathrm{i}\nu}&:\nu<0\end{cases}$
(36)

where $\eta$ (not to be confused with the Rindler $\eta$) and $\rho$ are
defined as (Fig. 8)

$\eta=1+a_{0}H_{0}(\tau_{0}-\tau)\,,\quad\rho=\frac{a_{0}H_{0}}{c}\,r\,.$ (37)

Like in the case of the Rindler modes, the modes (36) are analytic on the
lower half $\tau$ plane. Consequently, they consist entirely of
positive–frequency plane–wave modes (34) and share the conformal vacuum. Let
us call them vacuum modes. The phase of each of the vacuum–mode components,
incoming or outgoing, is logarithmic:

$\varphi=\nu\ln\left[1+a_{0}H_{0}(\tau_{0}-\tau\mp r/c)\right]\,.$ (38)

Like the Rindler modes (Fig. 4) the vacuum modes (36) are not monochromatic
(Fig. 7b); the frequency $\omega=-\partial_{t}\varphi$ varies in space and
time. At the defining time of the modes $t_{0}$ we have

$\left.\omega\right|_{t=t_{0}}=\frac{\omega_{0}}{1\mp u/c}\,,\quad
u=H_{0}\ell\,,\quad\ell=a_{0}r$ (39)

where $\omega_{0}$ denotes the frequency at the origin and at $t=t_{0}$. This
frequency is related to the dimensionless parameter $\nu$ by

$\omega_{0}=\nu H_{0}\,.$ (40)

Equation (39) shows that the vacuum modes are Doppler–shifted in the expanding
universe. Incoming waves propagate against the Hubble flow $u$ and are
blue–shifted, outgoing waves are red–shifted. Note that the Doppler profile
(39) was originally used to define the modes (36). Here we have derived them
from the analogy to the case of uniform acceleration.

Figure 8:  Characteristic events. Space–time diagram showing a part of the
actual cosmological horizon (Fig. 7a). Vacuum modes (Fig. 7b) are established
in analogy to the Rindler modes (Fig. 4). The vacuum modes are characterized
by the time parameter $\eta$ and the space parameter $\rho$ defined in Eq.
(37). The $\eta$ parameter runs backwards from $\eta=1$ when the vacuum mode
is defined ($t=t_{0}$) to $\eta=0$ when the Hawking partners arrive at the
origin. At the time $t_{0}$ ($\eta=1$) the spatial parameter reaches unity at
the horizon.

It remains to normalize the radial vacuum modes. For this we express the
scalar product (33) in conformal time $\tau$ and spherical coordinates
$\\{r,\theta,\phi\\}$ with metric tensor
$g_{\alpha\beta}=a^{2}\,\mathrm{diag}(1,-1,-r^{2},-r^{2}\sin^{2}\theta)$. We
obtain for the radial waves (35):

$\left(A_{1},A_{2}\right)=\frac{\mathrm{i}}{\hbar}\int_{0}^{\infty}\left(A_{\nu_{1}}^{*}\,\partial_{\tau}A_{\nu_{2}}-A_{\nu_{2}}\,\partial_{\tau}A_{\nu_{1}}^{*}\right)\mathrm{d}r\,.$
(41)

For the vacuum modes (36) with definitions (37) we have
$\partial_{\tau}=-a_{0}H_{0}\,\partial_{\eta}$ and
$a_{0}H_{0}\,\mathrm{d}r=c\,\mathrm{d}\rho$ and get

$\left(A_{1},A_{2}\right)=-\frac{\mathrm{i}c}{\hbar}\int_{0}^{\infty}\left(A_{\nu_{1}}^{*}\,\partial_{\eta}A_{\nu_{2}}-A_{\nu_{2}}\,\partial_{\eta}A_{\nu_{1}}^{*}\right)\mathrm{d}\rho\,.$
(42)

We may normalize the vacuum modes at a convenient moment ($\eta=0$) as the
scalar product remains constant at any time. We find exactly the same norm as
for the Rindler waves, Eqs. (16) and (17).

Finally, consider the mode overlap between the vacuum modes defined at
different times. The most relevant case is the overlap between the vacuum
modes at one horizon, say at $t_{2}$, with the modes at the previous horizon
at $t_{1}$. By this we mean that $t_{2}$ is the time when the Hawking partners
generated at $t_{1}$ arrive. The overlap tells how the modes at one instant of
creating Gibbons–Hawking radiation are related to the modes at the next stage
of creation. In particular, the phases between the modes are important, as the
acts of creation will interfere with each other. This is because particle
creation works like parametric amplification [55] where the phase of the
incident light determines whether particles are created or annihilated. We
calculate the scalar product $(A_{1},A_{2})$ at time $t_{2}$ where
$\eta_{1}=0$ (arrival of the partners) and $\eta_{2}=1$ (primary Hawking
radiation). We denote the scale factors and Hubble parameters as $a_{1},H_{1}$
and $a_{2},H_{2}$, and use $\rho=\rho_{2}$ as integration variable with
$\rho_{1}=\rho_{2}(a_{1}H_{1})/(a_{2}H_{2})$ from Eq. (37). In this way we get

$\left(A_{1},A_{2}\right)=\frac{c}{\hbar}\,(\nu_{1}+\nu_{2})\,{\cal
A}_{1}{\cal
A}_{2}\left(\frac{a_{2}H_{2}}{a_{1}H_{1}}\right)^{\mathrm{i}\nu_{1}}\cosh^{2}\zeta\,I_{12}$
(43)

with definition (16) and the remaining overlap integral

$\displaystyle I_{12}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\rho^{-\mathrm{i}\nu_{1}}\,(1+\rho)^{\mathrm{i}\nu_{2}}\,\frac{\mathrm{d}\rho}{\rho}-\int_{0}^{1}\rho^{-\mathrm{i}\nu_{1}}\,(1-\rho)^{\mathrm{i}\nu_{2}}\,\frac{\mathrm{d}\rho}{\rho}$
(44) $\displaystyle=$
$\displaystyle\frac{\Gamma(-\mathrm{i}\nu_{1})}{\Gamma(-\mathrm{i}\nu_{2})}\,\Gamma(\mathrm{i}\nu_{2}-\mathrm{i}\nu_{1})-\frac{\Gamma(1+\mathrm{i}\nu_{2})}{\Gamma(1-\mathrm{i}\nu_{1}+\mathrm{i}\nu_{2})}\,\Gamma(-\mathrm{i}\nu_{1})$
(45)

in terms of Gamma functions. Note that we gave the $\nu$ an appropriate small
imaginary part such that the integrals (44) converge. The dominant
contribution to the mode overlap appears for $\nu_{1}\rightarrow\nu_{2}$ where
$\Gamma(\mathrm{i}\nu_{2}-\mathrm{i}\nu_{1})\sim
1/(\mathrm{i}\nu_{1}-\mathrm{i}\nu_{2})$. In the mode expansion
$\int(\widehat{a}_{\nu}A_{\nu}+\widehat{a}_{\nu}^{\dagger}A_{\nu}^{*})\mathrm{d}\nu$
the overlap $(A_{1},\widehat{A})$ picks out a single mode with
$\nu_{1}=\nu_{2}=\nu$ by Cauchy’s theorem. Taking into account the
normalization (17) we arrive at the simple result:

$\widehat{a}_{2}\sim\left(\frac{a_{2}H_{2}}{a_{1}H_{1}}\right)^{\mathrm{i}\nu}\widehat{a}_{1}\,.$
(46)

Therefore, to a good approximation, the coefficients of the vacuum modes at
time $t_{2}$ are given by the mode coefficients at time $t_{1}$ multiplied by
the characteristic logarithmic phase factor $\nu(\ln a_{2}H_{2}-\ln
a_{1}H_{1})$ of the cosmological horizons. This concludes our discussion of
the vacuum modes in the expanding universe.

##  5 Radiating horizons

Consider now the noise the observer perceives. The observer, at rest with the
expanding universe at $r=0$, samples the field with respect to cosmological
time $t$, but the field oscillates with conformal time $\tau$. In the radial
vacuum modes we have organized all the superpositions of conformal plane waves
the observer perceives, such that

$\left.\widehat{A}\right|_{r=0}=\int_{-\infty}^{+\infty}\left(\widehat{a}_{\nu}A_{0,\nu}+\widehat{a}_{\nu}^{\dagger}A_{0,\nu}^{*}\right)\mathrm{d}\nu$
(47)

where according to Eq. (35) the $A_{0,\nu}$ are given by

$A_{0,\nu}=\left.\frac{1}{\sqrt{4\pi}\,ar}\,A_{\nu}\right|_{r=0}\,.$ (48)

We obtain from expressions (36) and (37) for the modes:

$\lim_{r\rightarrow 0}\frac{A_{\nu}}{r}=\mp 2\mathrm{i}\nu{\cal
A}\,(\pm\eta)^{\pm\mathrm{i}\nu-1}\,\frac{a_{0}H_{0}}{c}\,.$ (49)

Consider the radiation field around two times in the cosmic evolution: near
the time $t_{0}$ when particles are produced in the Gibbons–Hawking effect for
the Hubble parameter $H_{0}$ and then around the time when the corresponding
Hawking partners arrive, given by the condition $\eta=0$ (Fig. 8). The time
$t_{0}$ is arbitrary, but for each $t_{0}$ a new system of modes needs to be
constructed according to Eqs. (36) and (37). Since any such system is a
superposition of positive–frequency plane waves, Eq. (34), the vacuum state
with respect to the mode operators $\widehat{a}_{\nu}$ is the cosmic vacuum,
regardless of $t_{0}$.

As in the cases of uniform acceleration and exponential expansion, imagine the
observer as equipped with a spectrometer measuring the Fourier transformation
of the field with respect to the proper time of the observer, cosmological
time. Consider the Fourier transform near the time $t_{0}$. We write for each
vacuum mode

$\widetilde{A}_{0,\nu}=\int_{-\infty}^{+\infty}A_{0,\nu}\,\mathrm{e}^{\mathrm{i}\omega
t}\,\mathrm{d}t$ (50)

with the understanding that the integration is performed near $t_{0}$. There
we get from Eqs. (37) and (22):

$\mathrm{d}\eta=-\frac{a_{0}H_{0}}{a}\,\mathrm{d}t\,,\quad
t\sim-\frac{1}{H_{0}}\,\ln\eta\,,$ (51)

and hence from Eqs. (48) and (49):

$\widetilde{A}_{0,\nu}=\sqrt{4\pi}\,\mathrm{i}\nu c{\cal
A}\,\delta(\nu-\nu_{0})\,,\quad\nu_{0}=\frac{\omega}{H_{0}}\,.$ (52)

For positive frequencies $\omega$ the Fourier transform
$\widetilde{A}_{0,-\nu}$ of the negative–index modes vanishes. However, like
in the case of the accelerated observer, the Fourier transform of the complex
conjugate negative–index modes $A^{*}_{0,-\nu}$ does not disappear:

$\widetilde{A^{*}}_{0,-\nu}=\mathrm{e}^{-\pi\nu}\widetilde{A}_{0,\nu}\,.$ (53)

From relation (16) and the normalization (17) of the vacuum modes we obtain
the compact result:

$\left.\int_{-\infty}^{+\infty}\widehat{A}\,\mathrm{e}^{\mathrm{i}\omega
t}\,\mathrm{d}t\right|_{r=0}=\frac{\sqrt{\hbar\nu}}{c}\,\mathrm{i}\left(\widehat{a}_{\nu}\cosh\zeta+\widehat{a}_{-\nu}^{\dagger}\sinh\zeta\right)\,,\quad\nu=\frac{\omega}{H_{0}}\,.$
(54)

The result shows that the observer, sampling the vacuum noise with respect to
cosmological time around $t_{0}$, experiences the creation of Hawking
particles [54], even in the case of non–exponential expansion when the
cosmological horizon is not an event horizon [63].

Now turn to the time $t_{1}$ when the Hawking partners are expected to arrive,
i.e. when $\eta\sim 0$ (Fig. 8). It follows from Eqs. (22) and (37):

$\eta\sim-\frac{a_{0}H_{0}}{a_{1}}\,t$ (55)

where $a_{1}$ denotes the scale factor at $t_{1}$. Defining now
$\nu_{1}=(a_{1}/a_{0})(\omega/H_{0})$ we thus obtain

$\widetilde{A}_{0,-\nu}=\frac{\mathrm{i}\nu}{\sqrt{\pi}}\,\frac{{\cal
A}}{c}\int_{-\infty}^{+\infty}(-\eta)^{-\mathrm{i}\nu-1}\,\mathrm{e}^{-\mathrm{i}\nu_{1}\eta}\,\mathrm{d}\eta\sim\mathrm{i}\sqrt{2\mathrm{i}\nu}\,\frac{{\cal
A}}{c}\,(\nu_{1}/\nu)^{\mathrm{i}\nu}\,\mathrm{e}^{\mathrm{i}\nu}$ (56)

in the saddle–point approximation for $\nu\gg 1$. Similarly, for the
negative–frequency Fourier–transform of the complex conjugate modes with
positive index $\nu$ we get

$\widetilde{A^{*}}_{0,+\nu}=\mathrm{e}^{-\pi\nu}\widetilde{A}_{0,-\nu}\,.$
(57)

Substituting these results in the mode expansion (47) we calculate the
integral over the mode index in the saddle–point approximation as well. The
phase of the integrand, $\varphi=\nu\ln(\nu_{1}/\nu)+\nu$, is stationary
($\partial_{\nu}\varphi=0$) for $\nu=\nu_{1}$. We obtain in perfect analogy to
Eq. (54):

$\left.\int_{-\infty}^{+\infty}\widehat{A}\,\mathrm{e}^{\mathrm{i}\omega
t}\,\mathrm{d}t\right|_{r=0}=\frac{\sqrt{\hbar\nu}}{c}\,\mathrm{i}\left(\widehat{a}_{-\nu}\cosh\zeta+\widehat{a}_{\nu}^{\dagger}\sinh\zeta\right)\,,\quad\nu=\frac{a_{1}}{a_{0}H_{0}}\,\omega\,.$
(58)

Like the accelerated observer [Eq. (18)] the observer at rest with the
expanding universe measures spectral correlations expressed in the Bogoliubov
transformations (54) and (58). These correlations appear as extra noise with
Planck spectrum (19).

##  6 Cosmic cascade

We have thus derived the thermal radiation of cosmological horizons in
expanding flat space from the physical picture of wave noise (Fig. 1). This
picture reproduces the generalization [54] of Gibbons’ and Hawking’s [53]
result, Eq. (30), to arbitrary expansion. In this general case $H_{0}$ in Eq.
(30) refers to the Hubble parameter (25) at any given time $t_{0}$, not
required to be constant as in Gibbons’ and Hawking’s case [53] of exponential
expansion, de Sitter space (Fig. 6). In addition, we also derived a new aspect
of Gibbons–Hawking radiation not seen in de Sitter space. There the Hawking
partners never arrive before the world ends in conformal time (Fig. 6) whereas
in reality they do (Fig. 8). The light of distant galaxies and the Cosmic
Microwave Background easily cross the cosmological horizon [40, 63] and so do
the Hawking partners. We have found that the partners are correlated with the
primary particles, Eqs. (54) and (58), for the same dimensionless frequency
$\nu$. For the primary particles, $\nu$ is given by the frequency $\omega$
divided by the Hubble parameter $H_{0}$, which gives in the Planck spectrum
(19) the Gibbons–Hawking temperature (30). For the Hawking partners, $\nu$ is
given by $\omega$ divided by $(a_{0}/a_{1})H_{0}$ where $a_{1}$ denotes the
scale factor at their time of arrival. This means that the Hawking partners
also arrive as thermal radiation, but with the red–shifted temperature

$k_{\mathrm{B}}T_{1}=\frac{a_{0}}{a_{1}}\,\frac{\hbar H_{0}}{2\pi}\,.$ (59)

These results are simple and intuitive, but they are still incomplete. If the
present Hawking partners arrive in the future as thermal radiation, so should
the Hawking partners of the past arrive in the present. Call the scale factor
and Hubble parameter of the past cosmological horizon $a_{-1}$ and $H_{-1}$.
The present radiation of Hawking partners should then have the temperature

$k_{\mathrm{B}}T_{-1}=\frac{a_{-1}}{a_{0}}\,\frac{\hbar H_{-1}}{2\pi}\,.$ (60)

Figure 9:  Cascade of horizons. In the actual universe (Fig. 7) depicted in
conformal time $\tau$ and comoving radius $r$, the Gibbons–Hawking radiation
at present ($\tau=0$) depends on a cascade (zigzag line) of radiation
generated by past cosmological horizons (red) curve. Depending on the relative
phase, radiation is created or annihilated. The multiple interference of all
creation processes gives rise to the effective Gibbons–Hawking temperature and
vacuum energy density.

But neither this nor the primary temperature (30) is the effective temperature
$T_{\mathrm{eff}}$ of the radiation in total, because the Hawking particles
interfere with their partners. We have worked out that they have the
logarithmic phase difference (46). Like in parametric amplification [55] the
phase of the incident radiation determines whether it gets amplified or
de–amplified, whether particles are created or annihilated. The Hawking
partners from the previous horizon may very well annihilate some of the
Gibbons–Hawking radiation at the present, depending on the relative phase.
Furthermore, the horizon before the previous horizon interferes with the
particle production as well, and so does the whole cascade of past
cosmological horizons (Fig. 9). Each horizon establishes the Bogoliubov
transformation

$\widehat{b}_{\pm\nu}=\widehat{a}_{\pm\nu}\cosh\zeta+\widehat{a}_{\mp\nu}^{\dagger}\sinh\zeta\quad\mbox{with}\quad\tanh\zeta=\mathrm{e}^{-\pi\nu}\,.$
(61)

Between horizons, the modes are phase shifted according to Eq. (46). As the
frequencies relevant to the vacuum energy much exceed the Hubble parameter, we
are in the regime of $\nu\gg 1$ where we get for the final $\widehat{b}_{\nu}$
in terms of the initial vacuum mode operators $\widehat{a}_{\pm\nu}$:

$\widehat{b}_{\nu}\sim\widehat{a}_{\nu}+\widehat{a}_{-\nu}^{\dagger}\,S\,\mathrm{e}^{-\pi\nu}$
(62)

with $S$ summing up the phase factors of the $m$–th previous horizons relative
to the present one:

$S=\sum_{m=1}^{\infty}\left(\frac{a_{-m}H_{-m}}{a_{0}H_{0}}\right)^{2\mathrm{i}\nu}\,.$
(63)

This sum is highly oscillatory, but we are interested in the net effect of the
interfering horizons, i.e. in the average. When averaged over $\delta\nu\sim
1$ only an exponentially small contribution will remain that, together with
the primary $\mathrm{e}^{-\pi\nu}$, turns the Bogoliubov transformation (62)
into

$\widehat{b}_{\nu}\sim\widehat{a}_{\nu}+\widehat{a}_{-\nu}^{\dagger}\,\mathrm{e}^{\mathrm{i}\Phi-\pi\omega/H_{\mathrm{eff}}}$
(64)

with some phase $\Phi$ that does not affect vacuum correlations. The exact
expression for $H_{\mathrm{eff}}$ we shall derive in the next section, but
here we can already draw some qualitative conclusions. Since
$H_{\mathrm{eff}}$ depends on the history of cosmic evolution, it will
introduce a memory effect in the cosmologically relevant vacuum energy. This
memory of the past should remove the oscillations that would otherwise plague
the cosmic dynamics. Destructive interference from past cosmological horizons
may also explain why first–order perturbation theory with the primary $H$
instead of the full $H_{\mathrm{eff}}$ agrees so remarkably well with
astronomical data [7].

It is also interesting to note that the cosmic vacuum energy vanishes within
one cosmic era and thrives in the transition periods between different eras.
By era we mean a period in the cosmic evolution dominated by one type of fluid
with a characteristic equation of state. In the radiation–dominated era [40]
the Hubble parameter $H$ goes with $a^{-2}$, in the matter–dominated era [40]
$H\propto a^{-3/2}$ and during vacuum domination $H$ would become constant.
Apart from the exponential expansion in the vacuum era, all other eras are
characterized by a power law:

$H=H_{0}\,a^{-\gamma}$ (65)

with constant $H_{0}$ and $\gamma>1$ (where $H_{0}$ denotes $H$ at $a=1$
here). The partner radiation arriving at time $t$ with scale factor $a$ and
Hubble parameter $H$ originates from the past cosmological horizon the
conformal time interval $\tau$ earlier, with

$\tau=\int\frac{\mathrm{d}a}{a^{2}H}=\frac{1}{\gamma-1}\left(\frac{1}{aH}-\frac{1}{a_{-1}H_{-1}}\right)=\frac{1}{a_{-1}H_{-1}}\,,$
(66)

which gives

$\frac{aH}{a_{-1}H_{-1}}=\frac{1}{\gamma}\,.$ (67)

This recurrence relation remains true for all the phases in the sum (63) such
that the sum forms a perfect harmonic Fourier series with vanishing
zero–frequency component. The cycle average of such a series vanishes: the
number of particles produced is exactly zero. For a power–law expansion,
creation and annihilation thus cancels out exactly as the result of multiple
interference between past horizons (Fig. 9).

##  7 Wigner function

The interferences in the cosmic cascade of creation and annihilation at
horizons (Fig. 9) are captured in the sum (63). Yet this sum is difficult to
evaluate and mathematically ill–defined. Let us therefore try to deduce a
better formula for the effective Gibbons–Hawking temperature. The principal
problem of our previous approach (Sec. 5) is the Fourier transformation. We
wish to deduce the radiation spectrum as it evolves in time, and there we are
interested in spectral features $\sim\exp(-2\pi/H)$ that would require an
integration time in the order of $1/H$ for their accurate resolution. However,
the universe also evolves on a time scale of $1/H$ and with it the
Gibbons–Hawking spectrum. The two aspects, spectral accuracy and temporal
resolution, appear to be mutually exclusive. Frequency and time are as
mutually exclusive as position and momentum in quantum mechanics (being
Fourier transforms of each other). Fortunately, there are good compromises. To
give a simple example, music sheets describe tones – frequencies — in time; to
give a sophisticated example, quantum quasiprobability distributions [55, 66,
67, 68] describe both position and momentum. Probably the best compromise is
the Wigner function [66]. In quantum mechanics, the Wigner function is a
partial Fourier transformation of the density matrix [55]. The density matrix
is a correlation function of two variables, for example two positions. The
Wigner function performs a Fourier transformation with respect to the position
difference as a function of the position average. In this way the Wigner
function captures the momentum spectrum as a function of position. The
marginal distributions (reduced probability distributions) all give the
correct probability distributions of either position or momentum, or of any
linear combination of the two, with perfect accuracy. This property defines
the Wigner function uniquely [69] and explains why the Wigner function
describes conjugate variables (position and momentum, time and frequency) with
the highest possible precision. Here we employ the time–frequency Wigner
function:

$W=\frac{1}{2\pi}\int_{-\infty}^{+\infty}K(t+\theta/2,t-\theta/2)\,\mathrm{e}^{\mathrm{i}\omega\theta}\,\mathrm{d}\theta$
(68)

of the two–time field correlation function $K$ defined as the vacuum
expectation value

$K=\langle\widehat{A}_{1}\widehat{A}_{2}+\widehat{A}_{2}\widehat{A}_{1}\rangle$
(69)

with the indices indicating the two times and positions
$\\{t_{1},\bm{r}_{1}\\}$ and $\\{t_{2},\bm{r}_{2}\\}$. In the conformal
vacuum, the electromagnetic field fluctuations propagate like in empty
Minkowski space of the conformal times $\tau$ and comoving positions $\bm{r}$.
We may thus use the well–known expression of the Minkowski vacuum correlations
[8]:

$K=\frac{1}{(2\pi)^{2}s^{2}}\,,\quad
s^{2}=a_{1}a_{2}\left[c^{2}(\tau_{2}-\tau_{1})^{2}-(\bm{r}_{2}-\bm{r}_{1})^{2}\right]$
(70)

in terms of the Minkowski metric $s$ with reciprocal conformal factor
$a_{1}a_{2}$. Here we are interested in the spectrum measured with respect to
the cosmological times $t_{1}$ and $t_{2}$ at a given point comoving with the
universe:

$\bm{r}_{2}=\bm{r}_{1}\,,\quad t_{2}=t+\theta/2\,,\quad t_{1}=t-\theta/2\,.$
(71)

There are several ways to derive Eq. (70) — we may expand the field in terms
of the plane–wave modes (34) and integrate, or we may use the fact that $K$ is
the real part of the analytic function
$\langle\widehat{A}_{1}\widehat{A}_{2}\rangle$ with imaginary part given by
the difference between retarded and advanced Green function, and derive $K$ in
one line from the Kramers–Kronig relation [5].555There is a sign error in Eq.
(50) of Ref. [5] and subsequent expressions, because the wrong half plane was
taken in closing the integration contour. Fortunately — thanks to another sign
error — the result (80) carries the correct sign. Note that these vacuum
correlations exist outside of the causal cone ($s<0$) as has been recently
measured in quantum optics [70]. The correlations peak at $s=0$, because
electromagnetic waves propagate along light cones, including electromagnetic
noise. Space–time points on the light cone ($s=0$) are thus strongly
correlated. Wave noise is organized (Fig. 1).

Cosmology adds one subtle complication to the definition (68) of the Wigner
function: there was a beginning of time (say at $t=0$). For a given
cosmological time $t$ the Fourier time $\theta$ runs only from $-2t$ to $+2t$
in the real world. Close to the beginning, the expansion factor $a$ develops a
branch point [41] such that $a$ becomes complex in the time before, which
explains [41] why there was nothing real before the beginning of reality. The
conformal time $\tau$, being defined as the integral of the inverse of $a$,
inherits the branch point and ceases to be real for $|\theta|>2t$ as well. The
branch points of $\tau$ are harmless in the integrand (70), but $a$ goes to
zero with some fractional power [41]. We might be inclined to run the integral
in the definition (68) of the Wigner function from $-2t$ to $+2t$, but the
branch points of $a_{1}$ or $a_{2}$ at $\pm 2t$ in the integrand (70) would
then create oscillations with period $\pi/t$ in the spectrum. The spectral
oscillations average out for frequencies $\omega$ much larger than the inverse
cosmic age, but like the oscillations in the cosmic cascade (Sec. 6) they
obscure the subtle thermal spectrum of Gibbons–Hawking radiation. It is
therefore wise to analytically continue $a$ around the beginning of time. If
we lead the integral (68) slightly above the branch points $\pm 2t$ for
$\omega>0$ and slightly below for $\omega<0$ the oscillations are gone,
because if we approximate $a(t)$ by some root for $t\sim 0$ we could close the
integration contour on the upper half plane for $\omega>0$ and on the lower
half plane for $\omega<0$ (due to the Fourier factor
$\mathrm{e}^{\mathrm{i}\omega t}$) and get zero.

Having cleared the way we are now ready to calculate the Wigner function. It
is wise not to use the explicit expression (70) of the correlation function,
but rather our experience with Rindler modes in expanding flat space (Sec. 4).
We expand [Eq. (47)] the radiation field $\widehat{A}$ in terms of the vacuum
modes (36) defined at some arbitrary time $t_{0}\geq t$. The value of $t_{0}$
is not important, as the modes capture the conformal vacuum for all times. In
the vacuum expectation value (69) of $K$ the $\langle\widehat{a}^{\dagger}$
and $\widehat{a}\rangle$ vanish while the $\langle\widehat{a}_{\nu}$ and
$\widehat{a}_{\nu^{\prime}}^{\dagger}\rangle$ produce delta functions
$\delta(\nu-\nu^{\prime})$. For the modes at $r=|\bm{r}_{2}-\bm{r}_{1}|=0$ we
apply Eqs. (48) and (49) with the normalization (16) and (17), note that the
negative–$\nu$ modes are reduced by the factor $\mathrm{e}^{-\pi\nu}$, and
obtain the expression

$K=\frac{a_{0}^{2}H_{0}^{2}}{(2\pi)^{2}c^{2}a_{1}\eta_{1}a_{2}\eta_{2}}\int_{0}^{\infty}2\nu\left(\frac{1}{2}+\frac{1}{\mathrm{e}^{2\pi\nu}-1}\right)\cos\left(\nu\ln\frac{\eta_{2}}{\eta_{1}}\right)\mathrm{d}\nu\,.$
(72)

Let us check that this formula agrees with the standard result (70) for $K$.
Formula (72) contains the typical Planck term
$\nu(\mathrm{e}^{2\pi\nu}-1)^{-1}$ plus the contribution $\nu/2$ of the vacuum
energy. We express these terms in a geometrical series:

$\frac{\nu}{2}+\nu\left(\frac{1}{2}+\frac{1}{\mathrm{e}^{2\pi\nu}-1}\right)=\nu{\sum_{m=0}^{\infty}}^{\prime}\mathrm{e}^{-2\pi
m\nu}$ (73)

where the prime should indicate that the zeroth term is meant to be divided by
2. As

$\frac{1}{4\sinh^{2}(z/2)}=\sum_{m=-\infty}^{+\infty}\frac{1}{(z-2\pi
m\mathrm{i})^{2}}=-\partial_{z}\sum_{m=-\infty}^{+\infty}\frac{1}{z-2\pi
m\mathrm{i}}$ (74)

we see that the term (73) is the Fourier transform of $[4\sinh^{2}(z/2)]^{-1}$
for $\nu>0$ where we can close the integration contour on the upper half
plane. Running through the pole at zero (instead of surrounding it) produces
the factor $1/2$ in the vacuum term. For $\nu<0$ we close the contour on the
lower half plane and get the same expression with $\nu$ replaced by $|\nu|$.
From the inverse Fourier transformation then follows

$\int_{0}^{\infty}\nu\left(\frac{1}{2}+\frac{1}{\mathrm{e}^{2\pi\nu}-1}\right)\cos\nu
z\,\mathrm{d}\nu=\frac{1}{4\sinh^{2}(z/2)}$ (75)

and from this — and definition (37) for $\eta$ — we obtain Eq. (70). We have
thus reproduced the known vacuum correlation, but only for times less than
$t_{0}$. In the Wigner function (68) we must integrate from $-\infty$ to
$+\infty$. Therefore we should move $t_{0}$ to $+\infty$. In the infinite
future the expansion $a_{0}$ goes to infinity and $H_{0}$ to a finite value,
and so [Eq. (37)] the ratio $\eta_{2}/\eta_{1}$ goes to
$(\tau_{\infty}-\tau_{2})/(\tau_{\infty}-\tau_{1})$ while the factors
$(a_{0}H_{0})/(a\eta)$ go to $1/(\tau_{\infty}-\tau)$. In Eq. (72) we may thus
replace $\eta$ by

$\eta=\tau_{\infty}-\tau=\int_{a}^{\infty}\frac{\mathrm{d}a}{a^{2}H}$ (76)

and remove the $a_{0}H_{0}$ altogether. We thus obtain for the thermal part of
the Wigner function

$W_{\mathrm{th}}=\frac{1}{(2\pi)^{2}c^{2}}\int_{0}^{\infty}\frac{\nu}{\mathrm{e}^{2\pi\nu}-1}\,\mathscrsfs{D}(\nu,\omega)\,\mathrm{d}\nu$
(77)

with the kernel

$\mathscrsfs{D}=\frac{1}{\pi}\,\mathrm{e}^{-\omega\sigma}\int_{-\infty}^{+\infty}\frac{1}{a_{1}\eta_{1}a_{2}\eta_{2}}\,\cos\left(\nu\ln\frac{\eta_{2}}{\eta_{1}}\right)\mathrm{e}^{\mathrm{i}\omega\vartheta}\,\mathrm{d}\vartheta$
(78)

where we have lifted the integration line in expression (68) by the constant
positive imaginary time $\sigma$, assuming positive frequencies where, as we
know, we should lead the integration contour above the branch points at the
origin of physical time:

$\theta=\vartheta+\mathrm{i}\sigma\quad\mbox{with}\quad\sigma>0\quad\mbox{for}\quad\omega>0\,.$
(79)

Consider de–Sitter space as a test case of our formula. In this case, $a$
grows exponentially with constant $H_{0}$, $\tau=-(H_{0}a)^{-1}$ with
$\tau_{\infty}=0$, and
$\ln(\tau_{2}/\tau_{1})=-H_{0}(\vartheta+\mathrm{i}\sigma)$. We get

$\mathscrsfs{D}=H_{0}^{2}\delta(\omega-H_{0}\nu)$ (80)

and hence a perfect Planck spectrum with Gibbons–Hawking temperature (30).
Formula (78) thus reproduces Gibbons’ and Hawking’s classic result [53].
Consider now the realistic case of cosmic evolution, which deviates from pure
exponential expansion. The kernel $\mathscrsfs{D}$ is of course independent of
the integration contour (unless singularities or branch cuts are crossed) but
for any given real time $t$ there will be only one imaginary time $\sigma$
when $\mathscrsfs{D}$ does approach the defining integral of a delta function
in the asymptotic limit of large frequencies $\omega$ (whereas for de–Sitter
space all $\sigma$ do). In the following we work out the condition when this
is the case.

But first we need to consider some realistic cosmology in order to estimate
the validity of the approximation we are going to make. In the spatially flat,
isotropic and homogeneous universe the square of the Hubble parameter is
proportional to the energy density (by Friedman’s equations [40, 41]). For
radiation (photons and neutrinos) the energy density goes with the inverse
fourth power of the expansion factor $a$, because the energy falls with the
inverse wavelength and hence with $a^{-1}$ and the density falls with
$a^{-3}$. For matter (baryonic and dark) the energy density is essentially the
rest–mass mass density multiplied by $c^{2}$ and $a^{-3}$. Dark energy
$\Lambda$ — being the cosmological constant — remains constant. This gives the
$\Lambda$ Cold Dark Matter ($\Lambda$CDM) model:

$H^{2}=H_{0}^{2}\left(\frac{\Omega_{\mathrm{R}}}{a^{4}}+\frac{\Omega_{\mathrm{M}}}{a^{3}}+\Omega_{\Lambda}\right)$
(81)

where $H_{0}$ denotes the Hubble constant at the present time ($a=1$) and the
$\Omega_{m}$ describe the weights of the various contributions to the energy
density with all $\Omega_{m}$ summing up to unity. The cosmic parameters are
retrieved from the fluctuations of the Cosmic Microwave Background [1] and are
listed in Ref. [40]. For $a\gg\Omega_{\mathrm{R}}/\Omega_{\mathrm{M}}\approx
0.3\times 10^{-3}$ we can ignore the radiation contribution and enter a stage
of cosmic evolution entirely dominated by matter and $\Lambda$. For describing
this matter–vacuum era in the simplest possible way we change the scale of $a$
and the units of time replacing
$(\Omega_{\mathrm{M}}/\Omega_{\Lambda})^{1/3}a\rightarrow a$ and
$H_{0}\sqrt{\Omega_{\Lambda}}t\rightarrow t$ such that

$H^{2}=a^{-3}+1\,.$ (82)

From $t$ being the integral of $1/(aH)$ with respect to $a$ we obtain

$a=\left(\sinh\frac{3t}{2}\right)^{2/3}\quad\mbox{and}\quad
H=\coth\frac{3t}{2}\,.$ (83)

We get the conformal time

$\tau=\int_{0}^{a}\frac{\mathrm{d}a}{a^{2}H}=2\sqrt{a}\,\,{}_{2}F_{1}\left({1}/{6},{1}/{2},{7}/{6},-a^{3}\right)\,,\quad\tau_{\infty}=\frac{\Gamma(1/3)\,\Gamma(7/6)}{\Gamma(3/2)}$
(84)

in terms of the hypergeometric function ${}_{2}F_{1}$ and the Gamma function
$\Gamma$, and from the relationship (e.6) [71] of the hypergeometric function:

$\eta=\tau_{\infty}-\tau=a^{-1}\,\,{}_{2}F_{1}\left({1}/{3},{1}/{2},{4}/{3},-a^{-3}\right)\,.$
(85)

Consider now the curves in the complex $a$–plane where the Hubble parameter is
real. For the $\Lambda$–matter model (82) we get three curves where $H^{2}$ is
real: straight lines going through the origin with angles
$\\{0,\pi/3,-\pi/3\\}$. The Hubble parameter itself is real for
$\infty>H^{2}>0$. So the curves come in from $\infty$ and end at the points
where $H=\infty$ or $H=0$, which is
$\\{0,\mathrm{e}^{\mathrm{i}\pi/3},\mathrm{e}^{-\mathrm{i}\pi/3}\\}$ for the
$\Lambda$–matter stage (82). The positive real axis corresponds to the real
world with real time $t$, the $\pi/3$–line in the upper half plane corresponds
to the line with positive imaginary part $\pi/3$ in the complex plane of
cosmological time. In terms of the time $t+\theta/2$ in the Wigner function
(68) it draws a line (79) parallel to the real axis with $\sigma=2\pi/3$. This
is the line we are going to need in our integral (78). The $\Lambda$CDM model
(81) has four roots of $H^{2}=0$ we can calculate from Ferrari’s formula for
the roots of quartic equations, two are real and negative, the other two
complex conjugate to each other; we take the root $a_{+}$ on the upper half
plane, for which $|a_{+}|=0.775$ and $\arg a_{+}=\pi/3-1.09\times 10^{-4}$.
Calculating $\eta$ according to Eq. (76) we find
$\arg\eta_{+}=-\pi/3+1.34\times 10^{-4}$. We see that $a_{+}\eta_{+}$ is real
to an accuracy in the order of $10^{-5}$.

This has consequences if we calculate the integral (78) in the saddle–point
approximation for large $\omega$, because we get for the first and second
derivatives of the phase $\ln(\eta_{2}/\eta_{1})$ in the cosine:

$\left.\partial_{\theta}\ln\frac{\eta_{2}}{\eta_{1}}\right|_{\mathrm{i}\sigma}=\left.-\mathrm{Re}\,\frac{1}{a\eta}\right|_{\mathrm{i}\sigma}\,,\quad\left.\partial_{\theta}^{2}\ln\frac{\eta_{2}}{\eta_{1}}\right|_{\mathrm{i}\sigma}=\left.\frac{1}{2}\,\mathrm{Im}\left\\{\frac{H}{a\eta}-\frac{1}{(a\eta)^{2}}\right\\}\right|_{\mathrm{i}\sigma}$
(86)

and so the second derivative vanishes: the integral (78) gives a delta
function. In fact, for large $\omega$ only $\vartheta\sim 0$ matters where we
may approximate
$\ln(\eta_{2}/\eta_{1})\sim-\mathrm{i}\widetilde{\sigma}-\widetilde{H}\vartheta$
and $(a_{1}\eta_{1}a_{2}\eta_{2})^{-1}\sim\widetilde{H}^{2}$ with the
definitions

$\left.\widetilde{H}=\frac{1}{a\eta}\right|_{\mathrm{i}\sigma}\,,\quad\widetilde{\sigma}=\left.2\arg
a\right|_{\mathrm{i}\sigma}\,.$ (87)

We thus obtain from Eq. (78):

$\mathscrsfs{D}=\mathrm{e}^{-(\omega\sigma-\nu\widetilde{\sigma})}\,\widetilde{H}^{2}\delta(\omega-\widetilde{H}\nu)\,.$
(88)

For the matter–vacuum universe in our scaled units we have in particular

$\widetilde{H}={}_{2}F_{1}\left[{1}/{3},{1}/{2},{4}/{3},\mathrm{sech}^{2}({3t}/{2})\right]\,,\quad\widetilde{\sigma}=\sigma=2\pi/3\,.$
(89)

From Eq. (88) follows that, in the full $\Lambda$CDM model, the thermal part
(77) of the Wigner function (68) approaches the high–frequency asymptotics:

$W_{\mathrm{th}}\sim\frac{\omega}{(2\pi)^{2}c^{2}}\,\mathrm{e}^{-2\pi\omega/H_{\mathrm{eff}}}\quad\mbox{for}\quad\omega\gg\widetilde{H}$
(90)

expressed in terms of the effective Hubble parameter

$H_{\mathrm{eff}}=\frac{\widetilde{H}}{1+\frac{1}{2\pi}(\sigma\widetilde{H}-\widetilde{\sigma})}\,.$
(91)

The problem is solved.

##  8 Summary and outlook

We have derived the Gibbons–Hawking temperature for the standard cosmological
model — the $\Lambda$ Cold Dark Matter model — from the physical picture of
wave noise (Fig. 1). The resulting temperature,

$k_{\mathrm{B}}T=\frac{\hbar H_{\mathrm{eff}}}{2\pi}\,,$ (92)

depends on the effective Hubble parameter $H_{\mathrm{eff}}$ of Eq. (91) with

$\frac{1}{H_{\mathrm{eff}}}=\frac{1}{\widetilde{H}}+\frac{\sigma}{2\pi}-\frac{\widetilde{\sigma}}{2\pi\widetilde{H}}\,.$
(93)

The effective Hubble parameter sums up the multiple interferences in the
cascade of creation and annihilation at cosmological horizons (Fig. 9). It
does it by analytic continuation of the cosmic dynamics to complex times. The
parameter $\widetilde{H}$ is given by

$\left.\widetilde{H}=\frac{1}{a(\tau_{\infty}-\tau)}\right|_{t+\mathrm{i}\sigma/2}$
(94)

in terms of the scale factor $a$ and the conformal time $\tau$ evaluated at
infinity and at a certain complex time $t+\mathrm{i}\sigma/2$ on the upper
half plane. The real part of this complex time is the cosmological time at
which Gibbons–Hawking radiation is acting at the moment, the imaginary part
$\sigma/2$ needs to be determined from the requirement

$\left.\mathrm{Im}\widetilde{H}\right|_{t+\mathrm{i}\sigma/2}=0\,.$ (95)

The parameter $\widetilde{\sigma}$ is given by twice the argument of $a$ at
the complex time:

$\widetilde{\sigma}=\left.2\arg a\right|_{t+{\mathrm{i}\sigma}/2}\,.$ (96)

Expression (94) generalizes the Gibbons–Hawking formula (30) for de–Sitter
space [53]. In de–Sitter space [58] the scale factor $a$ grows exponentially
as $a=\mathrm{e}^{H_{0}t}$ while the conformal time $\tau$ falls as
$-H_{0}^{-1}\mathrm{e}^{-H_{0}t}$ approaching $\tau_{\infty}=0$ in the
infinite future. The product $a(\tau_{\infty}-\tau)=H_{0}^{-1}$ clearly is
constant and real for all imaginary times. In a realistic cosmological model
$\sigma$ needs to be calculated. For example, in the most relevant case, the
matter–vacuum dominated period of cosmic evolution, we get $\sigma=2\pi/3$ for
all times $t$ (in appropriate units666Here time is measured in the inverse
units of $\sqrt{\Omega_{\Lambda}}H_{0}$ where $\Omega_{\Lambda}H_{0}^{2}$
describes the contribution of the cosmological constant $\Lambda$ to the
square of the Hubble parameter at the present time ($a=1$).).

The temperature (92) lies in the order of $10^{-29}\mathrm{K}$ (at the present
cosmological time) and so the particles of Gibbons–Hawking radiation are
completely negligible, but the amplitude fluctuations are not — according to
Lifshitz theory [5]. They are predicted to produce the contribution (1) to the
renormalized vacuum energy proportional to

$\Delta=\partial_{t}^{3}\frac{1}{H_{\mathrm{eff}}}\,.$ (97)

This contribution drives the cosmological term $\varepsilon_{\Lambda}$ [5, 6,
7] (but is not proportional to $\varepsilon_{\Lambda}$ itself). Expression
(97) with effective Hubble parameter (93) hopefully is the final formula in a
series of attempts [5, 7] to determine the correct vacuum energy of expanding
flat space. For the matter–vacuum dominated period we obtain in our units

$\Delta=\frac{1}{\widetilde{H}^{4}}\left[4-8H\widetilde{H}-\left(4-\frac{26}{3}H^{2}\right)\widetilde{H}^{2}+\left(6-\frac{20}{3}H^{2}\right)H\widetilde{H}^{3}\right]$
(98)

in terms of expression (94) at the complex time $t+\mathrm{i}\pi/3$ where
$\widetilde{H}$ is real — with $\widetilde{H}$ given by Eq. (89) — and the
Hubble parameter [Eq. (83)] that is real as well — with $H=\tanh(3t/2)$.
Figure 10 compares this result with the previous attempts for the vacuum
energy, Eqs. (2) and (3).

Figure 10:  Comparison. Black curve: $H_{\mathrm{eff}}$ for the matter–vacuum
dominated period of cosmic evolution in scaled units. We see that
$H_{\mathrm{eff}}$ gently falls from $1.24$ to unity for $t\rightarrow\infty$
(de Sitter space in the far future). Red curves: $\Delta$ (proportional to
$-\varepsilon_{\mathrm{vac}}$) in scaled units. Solid curve: result of this
paper, Eq. (98). Dashed curve: $\frac{1}{6}\Delta$ obtained from Eq. (3) and
used, in perturbation theory, in the comparison [7] with astronomical data.
Dashed–and–dotted curve: result of Eq. (2), ruled out by the data [7]. The
factor $\frac{1}{6}$ was chosen such that the curves have the same asymptotics
for $t\rightarrow\infty$. The curves are similar, but with a different
prefactor that would correspond to a different cutoff [5]. The cutoff is a
parameter of the theory, because it is not precisely known (only in its order
of magnitude). It remains to be seen how the solid curve compares with
astronomical data.

Formula (93) depends on the history of cosmic expansion — being determined by
analytic continuation of the entire expansion. As the vacuum energy acts back
on the cosmic evolution due to its gravity, it has the tendency of developing
oscillations in the Hubble parameter if $\Delta$ depends on just the local
values of $a$. It is hoped that the memory effect in the vacuum energy derived
here will eliminate such artefacts. The multiple interference of cosmic
creation and annihilation (Fig. 9) summed up in $H_{\mathrm{eff}}$ may also
explain why first–order perturbation theory is remarkably good at fitting the
cosmological data [7] while the full theory with the previous expressions
would fail.

We obtained our result (93) assuming that the electromagnetic vacuum noise
consists of modes oscillating with conformal time (22) whereas an observer at
rest with the universe counts time as cosmological time. Furthermore we
assumed, inspired by optical analogues of gravity [17, 18, 19, 20, 21, 22, 23,
24], that the “medium” of space behaves like a medium, comoving with the
universe like the observer at rest. We therefore required that the spectrum of
vacuum fluctuations perceived by space is the spectrum with respect to
cosmological time. As the two times differ the spectrum becomes nontrivial
and, as it turned out, thermal. We used the physical picture of wave noise
(Fig. 1) and the asymptology [34] of Wigner functions [68] to work out the
temperature.

We can draw another conclusion from the picture of wave noise (Fig. 1). Our
analysis has been entirely local: we picked an arbitrary point in the
spatially flat universe and considered the vacuum fluctuations at this point
evolving in time. Nevertheless, the quantum vacuum is arriving from long
distances away, in particular the noise of the Hawking partners. For
sustaining the correlations responsible for the Gibbons–Hawking effect and
hence the vacuum energy $\varepsilon_{\mathrm{vac}}$, perfect vacuum modes
need to be formed according to Eq. (36). These are superpositions of perfect,
non–dispersive plane waves (34) sustaining correlations across vast distances
in space. Such long–range correlations cannot exist in massive fields, even
for energies at the Planck scale where mass is almost irrelevant. Let us
estimate the requirement for maintaining correlations. A field with particles
of mass $m$ obeys the dispersion relation

$\hbar^{2}\omega^{2}=\hbar^{2}c^{2}k^{2}+m^{2}c^{4}\,.$ (99)

Assuming $\lambda=2\pi c/\omega=\ell_{\mathrm{p}}$ with Planck length
$\ell_{\mathrm{p}}$ we get for the deviation of the phase from the
cosmological horizon to the point of observation:

$\delta\varphi=r_{H}\delta k\sim-\pi
r_{H}\frac{\ell_{\mathrm{P}}}{\lambda_{\mathrm{C}}^{2}}\,,\quad\lambda_{\mathrm{C}}=\frac{2\pi\hbar}{mc}$
(100)

where $\lambda_{\mathrm{C}}$ denotes the Compton wavelength. We obtain that
for $r_{\mathrm{H}}\sim 10^{10}\mathrm{ly}$ the mass $m$ must not exceed
$10^{-2}\mathrm{eV}$ for not ruining the noise correlations. There is only one
field with particles of such low mass, the electromagnetic field. Gluons are
massless like the electromagnetic photons, but they are short–range due to
interactions with themselves. Neutrinos have masses $\lesssim 0.8\mathrm{eV}$
[72] and are therefore probably too heavy as well. Moreover, neutrinos are
fermions, and it seems questionable whether fermions can create vacuum forces.
The standard vacuum fluctuations acting in the Casimir or van der Waals forces
[42] are not fluctuations of particles and antiparticles, but field
fluctuations. What are the physically relevant field fluctuations of fermions
in the vacuum state? The need for massless bosonic fields to sustain wave
correlation across cosmological distances may thus explain why only the
electromagnetic field seems to contribute to the cosmological vacuum energy,
as the comparison with astronomical data suggests [7].

Finally, we found that for cosmological eras dominated by only one type of
matter or energy the effective Gibbons–Hawking temperature is strictly zero or
constant. These eras are the radiation–dominated era at the youth of the
universe ($a\ll 10^{-3}$), the matter–dominated era in its middle age, and the
vacuum–dominated era for the eternity to follow ($a\gg 1$). We found this by
summing up the cosmic interferences, but we also see it in one glance from our
analytic theory. Pure eras are described by power laws with
$H=H_{0}\,a^{-\gamma}$, $\gamma>1$ or exponential expansion with $\gamma=0$.
For a power law the conformal time $\tau$ grows with $a^{\gamma-1}$ and hence
is analytic. We may close the integration contour of the Wigner function (68)
of the vacuum correlations (70) at infinity, get the vacuum term by
integrating through the double pole at $\tau_{2}=\tau_{1}$ but zero thermal
contribution. Formula (94) also indicates that power laws in $H$ generate zero
Gibbons–Hawking temperature, because $\tau$ tends to infinity for
$a\rightarrow\infty$, but the formula requires a finite $\tau_{\infty}$. For
exponential expansion, the Gibbons–Hawking temperature (92) is constant, and
so its contribution (97) to the dynamical vacuum energy density (1) vanishes
as well. As the cosmological term is driven by the dynamical vacuum energy it
remains constant. The vacuum energy acts only in transitions.

This is a typical feature of the Casimir effect. In dielectrics [8, 9], the
Casimir energy thrives on differences in the dielectric properties of a medium
causing forces at interfaces and boundaries. In Casimir cosmology [40], the
vacuum energy arises in the transitions between cosmic eras, changing the
cosmological constant there [5, 6, 7]. The current era is such a transition
period — the transition from matter to vacuum domination — and so the
cosmological constant varies, which affects the Hubble constant (the Hubble
parameter at the present time). The predicted variation of the Hubble constant
[7] appears to agree with the astronomical data [36], giving some empirical
support to the theory presented here. Wave noise (Fig. 1) may thus not only
explain the mundane, the stickiness of the microworld, but perhaps also the
arcane, the force of the macroworld that drives the universe apart.

## Acknowledgements

Two and a half decades ago Michael Berry’s work on the optical Aharonov Bohm
effect inspired me to look for connections between quantum optics and general
relativity, and he has been an inspiration ever since. I am most grateful to
him and wish him a happy anniversary. I would also like to thank Dror
Berechya, David Bermudez, Nikolay Ebel, Jonathan Kogman, Amaury Micheli, and
Scott Robertson for discussions and comments on this paper. The paper has been
supported by the Israel Science Foundation and the Murray B. Koffler
Professorial Chair.

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