#  Perturbations of Cosmological Redshift Drift

Pedro Bessa  Ruth Durrer and  Dennis Stock

###### Abstract

In this paper we calculate the linear perturbations of the cosmological
redshift drift. We show explicitly that our expressions are gauge-invariant
and compute the power spectrum of the redshift drift perturbations and its
correlations with galaxy number counts within linear perturbation theory. Our
findings show that the perturbations are small, and that the peculiar velocity
and acceleration terms are dominating and cannot be neglected when modeling
the full perturbative expression for the redshift drift. We also find that the
cross-correlations with galaxy number count fluctuations might increase the
detectability of the effect and can help to separate the perturbative effects
from the background cosmological redshift drift signal.

##  1 Introduction

The redshift of a far away source increases as time goes on. Typically in
about a Hubble time $1+z$ roughly doubles. Within short timescales of a few
years this ’redshift drift’ is very small. Nevertheless, this effect,
originally pointed out by Sandage and McVittie [1, 2], will probably be
measurable in the near future [3, 4]. Near future experiments with extremely
sensitive spectrography like the E-ELT or the SKAO, will reach the sensitivity
to measure the redshift drift, e.g. by monitoring Ly-$\alpha$ absorption lines
of distant quasars in the case of the E-ELT, within a time span of the order
of a decade.

In a perfect Friedmann-Lemaître (FL) spacetime the redshift drift of a source
at redshift $z$ is defined as $dz/d\tau_{o}$, where $\tau_{o}$ is the proper
time of the observer. Denoting comoving time by $t$, using $dt_{s}=dt_{o}$ and
$1+z=a_{o}/a_{s}$, so that $d\tau_{s}=a_{s}dt_{s}=(a_{s}/a_{o})d\tau_{o}$, one
obtains

$\displaystyle\frac{dz}{d\tau_{o}}$ $\displaystyle=$
$\displaystyle\frac{d}{d\tau_{o}}\left(\frac{a_{o}}{a_{s}}\right)=\frac{1}{d\tau_{o}}\left[\frac{a_{o}(1+H_{o}d\tau_{o})}{a_{s}(1+H_{s}(a_{s}/a_{o})d\tau_{o})}-\frac{a_{o}}{a_{s}}\right]=H_{o}(1+z)-H_{s}\,.$
(1.1)

Knowing the redshift drift of many sources at different redshifts can
therefore not only help us to measure the present Hubble parameter,
$H_{o}\equiv H_{0}$, but especially also to determine its redshift dependence,
$H(z)=H_{s}$. This is important, e.g. to constrain models of dark energy but
also in homogeneous cosmological models, it can give us a more direct
measurement of $H(z)$ which we presently mainly measure via Supernova data
where we in principle have access to the luminosity distance which is
proportional to an integral of $1/H(z)$.

In general, the angular dependence of the redshift drift can be used to test
its isotropy and it is sensitive to large fluctuations on small scales in a
statistically homogeneous and isotropic spacetime [5, 6]. In this work we
study the redshift drift within first order perturbation theory, where
backreaction from large perturbations on small scales are not considered.

Cosmic structure changes the redshift via the induced velocities of source and
observer and via the gravitational potential. To determine the precise
expressions, we consider the redshift drift in a generic spacetime as given in
[7], and apply it to a perturbed Friedmann universe.

Redshift drift in a perturbed FL universe has already been discussed before in
[8]. But in this paper the authors derived an expression for the redshift
drift valid for sources and observers with vanishing peculiar velocity in
longitudinal gauge. We shall derive a fully gauge invariant expression and
find that the velocity terms actually dominate the result by several orders of
magnitude.

Our paper is structured as follows: in the next section, we introduce the
redshift drift and give its general expression. In Section 3 we calculate
$dz/d\tau_{o}$ for a perturbed FL universe to first order in perturbation
theory. In Section 4 we derive the power spectrum of the redshift drift and
present numerical results using a modified version of class. We also compute
the cross-correlation between redshift drift and galaxy number counts. In
Section 5 we discuss our findings and conclude.

Notation: We work within a spatially flat FL spacetime with conformal time $t$
such that

$ds^{2}=a^{2}(t)[-dt^{2}+\delta_{ij}dx^{i}dx^{j}]\,.$

The conformal Hubble parameter is ${\cal H}=\dot{a}/a$, where an overdot
denotes the derivative wrt. conformal time $t$. Physical time is denoted by
$\tau$. In the FL background, for a comoving observer, $d\tau=a\,dt$ and the
physical Hubble parameter is

$H=\frac{1}{a}\frac{da}{d\tau}={\cal H}a^{-1}\,.$

##  2 Redshift Drift

To determine the redshift drift in a generic spacetime, let us consider the
worldline of a cosmic observer, in the following denoted by $o$, parametrised
by its proper time $\tau_{o}$. The observer measures the redshift $z$ to a
source, denoted by $s$. Recall that the redshift is given in terms of the
ratio of the observed and emitted photon energies, $E_{o}=-u\cdot
k\big{|}_{o}$ and $E_{s}=-u\cdot k\big{|}_{s}$, where $\big{|}_{o}$ denotes
evaluation at the observer of both 4-velocity $u^{\mu}$ and photon 4-momentum
$k^{\mu}$, respectively at the source:

$1+z=\frac{E_{s}}{E_{o}}\,.$ (2.1)

The change of the redshift $dz$ per proper time interval of the observer
$d\tau_{o}$ is called the redshift drift $\frac{dz}{d\tau_{o}}$, see Fig. 1.

Figure 1: The change $dz$ of the redshift to a source $s$ per proper time
interval $d\tau_{o}$ measured by the observer defines the redshift drift
$\frac{dz}{d\tau_{o}}$. Note that in general the proper time of source and
observer are different, $d\tau_{o}\neq d\tau_{s}$.

Because redshifts are observable and the proper time $\tau_{o}$ is the time
measured by the observer, the redshift drift is indeed a cosmological
observable. Note that in general there are several contributions to the
redshift which vary with time, for example cosmic expansion, or peculiar
velocities of observer and source. We will discuss all contributions in a
perturbed FL universe in detail in Section 3.2. The redshift drift in a FL
spacetime for an observer comoving with the fluid is derived in the
introduction, Eq. (1.1).

Before computing the redshift drift in a linearly perturbed FL spacetime, we
briefly review its general expression in an arbitrary spacetime in terms of
kinematic quantities. This was originally derived in [9] and we shall use it
later. Let us denote the null vector generating the geodesic connecting source
and observer along which the photon travels by $k$, and the (timelike)
4-velocities of source and observer by $u_{s}$ and $u_{o}$ respectively, which
are normalised to one: $u\cdot u=-1$. The source and the observer move with
the cosmic fluid to which we assign the 4-velocity field $u$. Along the photon
geodesic we decompose $k$ into a part parallel to $u$ and a vector $e$, which
is spacelike, orthogonal to $u$, and normalised to one, hence we have:

$k\cdot k=0\quad,\quad e\cdot e=1\quad,\quad u\cdot e=0\quad\text{with}\quad
k=E(u-e)\quad,$ (2.2)

where the projection of $k$ onto the timelike direction $u$ is minus the
photon energy measured by an observer moving with $u$, $E=-k\cdot u$. The
redshift can then be expressed as

$1+z=\frac{E_{s}}{E_{o}}=\frac{(u\cdot k)|_{s}}{(u\cdot
k)|_{o}}=\frac{d\tau_{o}}{d\tau_{s}}\quad,$ (2.3)

where for the last equality we use that for photons $E=h\nu$, the frequency
$\nu$ is the change of the phase $\varphi$ of the wave describing the photon
with respect to proper time measured by observer,
$\nu_{o}=\frac{d\varphi}{d\tau_{o}}$, respectively for the source, and that
the phase is constant along a null geodesic. The redshift drift can then be
written as

$\frac{dz}{d\tau_{o}}=\nabla_{u_{o}}\left(\frac{u\cdot k|_{s}}{u\cdot
k|_{o}}\right)\quad.$ (2.4)

To carry out the derivative in (2.4) explicitly, we use the kinematic
decomposition of $\nabla_{\mu}u_{\nu}$, see for example [10, 11],

$\nabla_{\mu}u_{\nu}=\omega_{\mu\nu}+\Theta_{\mu\nu}-a_{\mu}u_{\nu}.$ (2.5)

This decomposition splits into an antisymmetric part $\omega_{\mu\nu}$ which
is often referred to as vorticity, a symmetric part $\Theta_{\mu\nu}$, and a
part proportional to the 4-acceleration $a^{\mu}=(\nabla_{u}u)^{\mu}$
quantifying the deviation of $u$ from geodesic motion. The symmetric part
$\Theta_{\mu\nu}$ can be further decomposed into a trace $\theta$, the
expansion scalar, and a traceless part $\sigma_{\mu\nu}$, referred to as shear
tensor,

$\Theta_{\mu\nu}=\frac{1}{3}\theta h_{\mu\nu}+\sigma_{\mu\nu}\,,$ (2.6)

where $h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$ is the metric on the tangent
space normal to $u$. Eq. (2.3) implies that in full generality
$\nabla_{u_{o}}=\frac{1}{1+z}\nabla_{u_{s}}$. Using also that $k$ follows a
null geodesic $\nabla_{k}k=0$, we arrive at the expression derived in [9],

$\displaystyle\frac{dz}{d\tau_{o}}$
$\displaystyle=(1+z)\left[\mathfrak{h}-\frac{\nabla_{e}E}{E}\right]_{o}-\left[\mathfrak{h}-\frac{\nabla_{e}E}{E}\right]_{s}\,,~{}~{}\text{
with}$ (2.7) $\displaystyle\mathfrak{h}$
$\displaystyle:=\frac{1}{3}\theta+\sigma_{\mu\nu}e^{\mu}e^{\nu}-a_{\mu}e^{\mu}\,,$
(2.8)

where $[\;]_{o/s}$ denotes that all terms inside the bracket are evaluated at
the observer, respectively at the source and $\nabla_{e}$ is the covariant
derivative in direction of the spacelike vector $e$. Note that in an FL
spacetime $\mathfrak{h}=H$ and $\nabla_{e}E=0$ so that we recover (1.1).
Expression (2.7) will be used to compute the redshift drift in perturbed FL in
the next section.

##  3 Redshift Drift in Perturbed FL

In this section, we first briefly introduce linear perturbations of the
Friedmann metric and the energy-momentum tensor, before computing the redshift
drift (2.7) for linear perturbations about an FL background. We also comment
on how our results relate to previous findings in the literature.

###  3.1 Perturbations of Metric and Energy-momentum Tensor

We fix the background spacetime $\bar{g}_{\mu\nu}$ to be given by a FL
spacetime, whose line element in conformal time $t$ reads

$d\bar{s}^{2}=a^{2}(t)\left[-dt^{2}+\gamma_{ij}dx^{i}dx^{j}\right]\,,$ (3.1)

where $\gamma_{ij}$ is the spatial 3-metric of constant time hypersurfaces,
which we consider to be spatially flat, i.e., $\gamma_{ij}=\delta_{ij}$. Here
and in the following, overbars denote quantities evaluated in the background
spacetime. Adding linear perturbations to the background in a general gauge,
we obtain the metric

$ds^{2}=a^{2}(t)\left[-(1+2A)dt^{2}-2B_{i}dx^{i}dt+\left(\delta_{ij}+H_{ij}\right)dx^{i}dx^{j}\right]\,,$
(3.2)

see e.g. [11] for a more general and detailed account of cosmological linear
perturbation theory. In addition to time, the perturbations also depend on
position, because homogeneity and isotropy only hold for the background
spacetime. In this work, we are interested in scalar perturbations and neglect
vector and tensor perturbations. Vector perturbations decay in an FL
background, see [11] and the contribution from tensor perturbations is small
compared to scalar perturbations, see e.g. CMB data [12]. Scalar spatial
perturbations $H_{ij}$ can be decomposed into

$H_{ij}=H_{L}\delta_{ij}+H_{T\,ij}\qquad\text{with}\qquad
H_{T\,ij}=\partial_{i}\partial_{j}H_{T}-\frac{1}{3}\delta_{ij}\Delta H_{T}\,.$
(3.3)

where $\Delta=\sum\partial_{i}^{2}$ denotes the flat space Laplacian. Thus,
scalar, linear metric perturbations are encoded in four functions: $A$, $B$
with $B_{i}=\partial_{i}B$, $H_{L}$ and $H_{T}$. In addition, the 4-velocity
$u$ of the cosmic fluid is also perturbed,

$\left(u^{\mu}\right)=\frac{1}{a}\left(1-A,v^{i}\right)\qquad\text{with}\qquad
v_{i}=\partial_{i}v\,.$ (3.4)

The $u^{0}$ component is determined by the normalisation condition and the
peculiar velocity is given by the potential $v$.

###  3.2 Computation of the Redshift Drift in Perturbed FLRW

Next, we perturb all quantities appearing in (2.7) to linear order and only at
the end combine all terms into gauge invariant expressions. Because the final
result is an observable, it must be gauge invariant.

The perturbation of the 4-velocity field of source and observer is given in
(3.4). The perturbations of the photon 4-vector are determined by integrating
the geodesic equation $\nabla_{k}k=0$ in the perturbed spacetime. One obtains

$\displaystyle k^{\mu}$ $\displaystyle=\bar{k}^{\mu}+\delta
k^{\mu}\qquad\text{with}\qquad\bar{k}^{\mu}=\frac{1}{a^{2}}(1,\bar{n}^{i})\qquad\text{and}$
(3.5) $\displaystyle\delta k^{0}$ $\displaystyle=\frac{1}{a^{2}}\delta
n^{0}=\frac{1}{a^{2}}\left\\{\left[-2A-B_{j}\bar{n}^{j}\right]_{s}^{o}-\int_{s}^{o}d\lambda\;\left[\dot{H}_{L}-\dot{A}+\dot{H}_{Tab}\bar{n}^{a}\bar{n}^{b}-\dot{B}_{a}\bar{n}^{a}\right]\right\\}\,,$
(3.6) $\displaystyle\delta k^{i}$ $\displaystyle=\frac{1}{a^{2}}\delta
n^{i}=\frac{1}{a^{2}}\left\\{\left[-B^{i}+2H_{L}\bar{n}^{i}+H_{T}^{ij}\bar{n}_{j}\right]_{s}^{o}-\int_{s}^{o}d\lambda\;\left[\partial^{i}(H_{L}-A)+\bar{n}^{a}\bar{n}^{b}\partial^{i}H_{Tab}\right]\right\\}\,,$
(3.7)

where $\bar{n}^{i}$ is the unperturbed direction of the photon geodesic
satisfying $\bar{n}_{i}\bar{n}^{i}=1$. The integrals are performed along the
unperturbed photon geodesic in the background spacetime from source to
observer, and $\lambda$ denotes the affine parameter along the null geodesic.
The spacelike vector $e$ orthogonal to $u$ from (2.2) is then given by

$e^{\mu}=u^{\mu}-\frac{1}{E}k^{\mu}=\frac{1}{a}\left(\bar{n}^{i}(B_{i}-v_{i})\;,\;-\bar{n}^{i}-\delta
n^{i}+v^{i}+\bar{n}^{i}\left(A+\bar{n}^{j}(B_{j}-v_{j})+\delta
n^{0}\right)\right)\,,$ (3.8)

where we used the first order expression for the photon energy

$E=-k_{\mu}u^{\mu}=\frac{1}{a}\left[1+A+\bar{n}^{i}(B_{i}-v_{i})+\delta
n^{0}\right]\,.$ (3.9)

With that, we can determine the remaining quantities in (2.7) to linear order,

$\displaystyle a^{\mu}$
$\displaystyle=\left(0\;,\;-\frac{1}{a^{2}}\partial^{i}\left[\dot{v}-\dot{B}-A+\mathcal{H}(-B+v)\right]\right)$
(3.10) $\displaystyle\theta$
$\displaystyle=\frac{1}{a}\left[3\mathcal{H}(1-A)-\Delta
v+3\dot{H}_{L}\right]$ (3.11) $\displaystyle\sigma_{00}$
$\displaystyle=\sigma_{0i}=\sigma_{i0}=0$ (3.12) $\displaystyle\sigma_{ij}$
$\displaystyle=a\left(\partial_{i}\partial_{j}-\delta_{ij}\frac{1}{3}\Delta\right)(\dot{H}_{T}-v)$
(3.13) $\displaystyle\nabla_{e}E$
$\displaystyle=-\frac{1}{a^{2}}\left\\{\bar{n}^{i}\partial_{i}(A+\delta
n^{0})+\bar{n}^{i}\bar{n}^{j}\partial_{i}(B_{j}-v_{j})+\mathcal{H}\bar{n}^{i}(B_{i}-v_{i})\right\\}\,.$
(3.14)

While it is easy to verify that $a^{\mu}$, $\sigma_{\mu\nu}$ and $\nabla_{e}E$
are gauge invariant (as they should be according the the Stewart–Walker lemma,
since they vanish in the Friedmann background), $\theta$ is not. Therefore
$\mathfrak{h}$ is not gauge invariant. Putting together the individual terms
to obtain the generalised Hubble function $\mathfrak{h}$ (2.8), and using the
relations given in Appendix A to convert the expression to gauge-invariant
perturbation variables whenever possible, we find

$\mathfrak{h}(t)=\frac{1}{a}\left[\mathcal{H}(1-A)+\dot{\mathcal{R}}-\bar{n}^{i}\partial_{i}\left(\dot{V}+\mathcal{H}V-\Psi\right)-\bar{n}^{i}\bar{n}^{j}\partial_{i}\partial_{j}V\right]\quad,$
(3.15)

with the gauge-invariant velocity potential $V$, the spatial curvature
$\mathcal{R}$ of the spacelike hypersurface of constant time, and the Bardeen
potentials $\Phi$, $\Psi$. Note that $A$ and $\mathcal{R}$ are still gauge
dependent, hence (3.15) is not gauge invariant as expected.

In a next step, we express $\nabla_{e}E$ in terms of gauge-invariant
variables. In order to achieve that, we first express the integrand of (3.6)
in terms of the Bardeen potentials using the relations given in Appendix A:

$\dot{H}_{L}-\dot{A}+\dot{H}_{Tab}\bar{n}^{a}\bar{n}^{b}-\dot{B}_{a}\bar{n}^{a}=-\dot{\Phi}-\dot{\Psi}-\dddot{H}_{T}+\ddot{B}+\bar{n}^{i}\bar{n}^{j}\partial_{i}\partial_{j}\dot{H}_{T}+\bar{n}^{i}\partial_{i}\dot{B}\quad.$
(3.16)

Next, we use the partial integration formula for a first order quantity $f$

$\frac{df}{d\lambda}=\nabla_{n}f=\dot{f}+\bar{n}^{i}\partial_{i}f\quad\Rightarrow\quad\int_{s}^{o}d\lambda\,\bar{n}^{i}\partial_{i}f=f_{o}-f_{s}-\int_{s}^{o}d\lambda\,\dot{f}\quad,$
(3.17)

in order to turn the spatial derivatives in $\bar{n}^{i}\partial_{i}\dot{B}$
and $\bar{n}^{i}\bar{n}^{j}\partial_{i}\partial_{j}\dot{H}_{T}$ into time
derivatives, yielding

$\int_{s}^{o}d\lambda\;\left[\dot{H}_{L}-\dot{A}+\dot{H}_{Tab}\bar{n}^{a}\bar{n}^{b}-\dot{B}_{a}\bar{n}^{a}\right]=-\int_{s}^{o}d\lambda\,(\dot{\Phi}+\dot{\Psi})+(\dot{B}_{o}-\dot{B}_{s})+\bar{n}^{i}\partial_{i}(\dot{H}_{T}^{o}-\dot{H}_{T}^{s})-(\ddot{H}_{T}^{o}-\ddot{H}_{T}^{s})\quad.$
(3.18)

In a next step, we evaluate

$\delta
n^{0}|_{s}^{o}=\left[-2A-B_{j}\bar{n}^{j}\right]_{s}^{o}-\int_{s}^{o}d\lambda\;\left[\dot{H}_{L}-\dot{A}+\dot{H}_{Tab}\bar{n}^{a}\bar{n}^{b}-\dot{B}_{a}\bar{n}^{a}\right]\,.$
(3.19)

In what follows we neglect the terms evaluated at the observer, because they
give rise to an unmeasurable monopole, and to a dipole contribution due to the
observer velocity. This dipole is interesting as it provides an alternative
measure of the observer velocity. It has been studied in [13]. Neglecting the
observer terms in $\delta n^{0}$ we obtain

$\delta
n^{0}=-2A-\bar{n}^{i}\partial_{i}\sigma_{t}+\dot{\sigma}_{t}-\int_{s}^{o}d\lambda\,(\dot{\Phi}+\dot{\Psi})\quad,$
(3.20)

where we dropped the source label $s$, such that if not specified differently,
all terms are to be evaluated at the source. We have also introduced the shear
of the constant time hypersurface $\sigma_{t}$ which is not gauge invariant,
see Appendix A. Using again the relations given in Appendix A, we arrive at
the following expression for $\nabla_{e}E$ in terms of gauge-invariant
variables:

$\nabla_{e}E=\frac{1}{a^{2}}\bar{n}^{i}\partial_{i}\left[\Psi-\mathcal{H}V-\bar{n}^{j}\partial_{j}V+\int_{s}^{o}d\lambda\,(\dot{\Phi}+\dot{\Psi})\right]\quad.$
(3.21)

Not surprisingly, $\nabla_{e}E$ which vanishes on the background is a gauge
invariant quantity (a consequence of the Stewart Walker lemma).

However, $\mathfrak{h}$ given above depends on the chosen time-slicing. So
far, we evaluated all quantities at fixed time $t$ and as such evaluated them
on hypersurfaces of constant time. However, constant time hypersurfaces are
observationally not accessible, instead, we can observe hypersurfaces of
constant observed redshift $z$. Since time is related to the background
redshift, $t(\bar{z})$, and $z=\bar{z}+\delta z$, we have to convert a
quantity $f(t)$ evaluated on constant time hypersurfaces via

$f(t)\rightarrow f(t(\bar{z}))\equiv f(\bar{z})=f(z-\delta z)\approx
f(z)-\frac{d\bar{f}}{d\bar{z}}\,\delta z$ (3.22)

to first order. Note that within linear perturbation theory the redshift
correction term proportional to $\delta z$ is only relevant for background
quantities, because for first order terms, the correction would be of second
order and therefore it can be neglected. As $\nabla_{e}E$ in our expression
(2.7) for the redshift drift is already first order, its conversion gives no
first order contribution and only $\mathfrak{h}$ has to be converted.
According to (3.22)

$\mathfrak{h}(t)\approx\mathfrak{h}(z)-\frac{d\bar{\mathfrak{h}}}{d\bar{z}}\,\delta
z=\mathfrak{h}(z)+\left(\frac{\dot{\mathcal{H}}}{\mathcal{H}}-\mathcal{H}\right)\,\delta
z\quad.$ (3.23)

The redshift fluctuation $\delta z$ was already computed e.g. in [14] and
reads, neglecting again observer monopole and dipole terms,

$\delta
z=-(1+z)\left[H_{L}+\frac{1}{3}H_{T}+n^{i}\partial_{i}V+\Phi+\Psi+\int_{s}^{o}d\lambda\,(\dot{\Phi}+\dot{\Psi})\right]\,.$
(3.24)

Using the relations in Appendix A, $\mathfrak{h}(z)$ can be expressed in terms
of gauge-invariant variables,

$\displaystyle\mathfrak{h}(z)=$
$\displaystyle\frac{1}{a}\left\\{\mathcal{H}-\dot{\Phi}-\bar{n}^{i}\partial_{i}\left(\dot{V}+2\mathcal{H}V-\Psi\right)-\bar{n}^{i}\bar{n}^{j}\partial_{i}\partial_{j}V-\frac{\dot{\mathcal{H}}}{\mathcal{H}}\left(\Psi-\bar{n}^{i}\partial_{i}V\right)+\right.$
$\displaystyle\left.\qquad+\left(\mathcal{H}-\frac{\dot{\mathcal{H}}}{\mathcal{H}}\right)\int_{s}^{o}d\lambda\,(\dot{\Phi}+\dot{\Psi})\right\\}\,,$
(3.25)

where again observer monopole and dipole terms are neglected. Finally, putting
all contributions to (2.7) together and neglecting observer monopole and
dipole terms, we arrive at the following, manifestly gauge-invariant
expression for the redshift drift as function of the observed redshift:

$\frac{dz}{d\tau_{o}}=(1+z)\left[-\mathcal{H}+\dot{\Phi}+\bar{n}^{i}\partial_{i}(\dot{V}+\mathcal{H}V)+\frac{\dot{\mathcal{H}}}{\mathcal{H}}(\Psi-\bar{n}^{i}\partial_{i}V)+\left(\frac{\dot{\mathcal{H}}-\mathcal{H}^{2}}{\mathcal{H}}+\bar{n}^{i}\partial_{i}\right)\int_{s}^{o}d\lambda\left(\dot{\Phi}+\dot{\Psi}\right)\right]\,.$
(3.26)

Note that the zeroth order expression of this no longer coincides with (1.1),
both expressions differ by an observer monopole term, but they agree at zeroth
order after this observer term is dropped. We then get $-H_{s}=-(1+z){\cal
H}$. Here $\frac{dz}{d\tau_{o}}$ is considered as a function the observed
redshift $z$ and the observed incoming direction of the photon ${\mathbf{n}}$.

It is worth commenting on the individual contributions. As already mentioned,
the leading order contribution is given by the background Hubble function
$-H_{s}=-(1+z)\mathcal{H}$ due to the isotropic and homogeneous expansion of
the Universe, compare with (1.1) neglecting the observer term. The
contributions proportional to $\bar{n}^{i}\partial_{i}V$ are Doppler terms
arising from the source’s peculiar velocity which affects the redshift, and
$\bar{n}^{i}\partial_{i}\dot{V}$ is a peculiar acceleration term. The terms
involving the Bardeen potentials $\Phi$, $\Psi$ account for local fluctuations
in the gravitational potential at the source position, whereas
$\int_{s}^{o}d\lambda\left(\dot{\Phi}+\dot{\Psi}\right)$ is the integrated
Sachs-Wolfe effect accounting for the time variations of the Bardeen
potentials integrated along the line of sight. In fact, we will use a slightly
different expression in the next section, where the spatial derivative of the
integrated Sachs-Wolfe effect is rewritten via the partial integration formula
(3.17),

$\bar{n}^{i}\partial_{i}\left[\int_{s}^{o}d\lambda\,(\dot{\Phi}+\dot{\Psi})\right]=-(\dot{\Phi}+\dot{\Psi})-\int_{s}^{o}d\lambda\,(\ddot{\Phi}+\ddot{\Psi})\,.$
(3.27)

Observer monopole terms are again neglected. This brings (3.26) to the form

$\displaystyle\frac{dz}{d\tau_{o}}=(1+z)\bigg{[}$
$\displaystyle-\mathcal{H}-\dot{\Psi}+\bar{n}^{i}\partial_{i}(\dot{V}+\mathcal{H}V)+\frac{\dot{\mathcal{H}}}{\mathcal{H}}(\Psi-\bar{n}^{i}\partial_{i}V)$
$\displaystyle+\frac{\dot{\mathcal{H}}-\mathcal{H}^{2}}{\mathcal{H}}\int_{\mathcal{S}}^{\mathcal{O}}d\lambda\left(\dot{\Phi}+\dot{\Psi}\right)-\int_{s}^{o}d\lambda\,(\ddot{\Phi}+\ddot{\Psi})\bigg{]}\,.$
(3.28)

Even though this formula is a bit more cumbersome than (3.26), it is useful as
in the numerical analysis we shall neglect time derivatives of the Bardeen
potentials which are known to be very small.

There are previous results on the redshift drift in the literature, most
notably in [8], where the authors compute the redshift drift in a perturbed
Friedmann spacetime on constant time hypersurfaces, neglecting peculiar
velocities. It turns out that if we disregard their monopole and dipole terms
at the observer, we match their result when converting it from constant
conformal time hypersurfaces to constant redshift hypersurfaces. After
neglecting our velocity terms ($V=0$) in (3.26), both results agree. As it is
shown in the next section, however, the contributions coming from the source
velocity dominate the redshift drift perturbations and therefore they have to
be included.

##  4 Angular Power Spectra

Next, we study the angular power spectrum of redshift drift perturbations. In
fact, we are interested in the spectra of the relative fluctuations with
respect to the background value,

$\displaystyle\delta\dot{z}$
$\displaystyle:=\frac{\frac{dz}{d\tau_{o}}-\frac{d\bar{z}}{d\bar{\tau}_{o}}}{\frac{d\bar{z}}{d\bar{\tau}_{o}}}$
$\displaystyle=-\frac{1}{\mathcal{H}}\Bigg{[}-\dot{\Psi}+\bar{n}^{i}\partial_{i}\dot{V}+\frac{\dot{\mathcal{H}}}{\mathcal{H}}\Psi+\left(\frac{\dot{\mathcal{H}}-\mathcal{H}^{2}}{\mathcal{H}}\right)\left(\int_{\mathcal{S}}^{\mathcal{O}}d\lambda\left(\dot{\Phi}+\dot{\Psi}\right)-\bar{n}^{i}\partial_{i}V\right)-\int_{s}^{o}d\lambda\,(\ddot{\Phi}+\ddot{\Psi})\Bigg{]}\;,$
(4.1)

where $\frac{dz}{d\tau_{o}}$ is the redshift drift in the perturbed spacetime
(3.28), and $\frac{d\bar{z}}{d\bar{\tau}_{o}}=-(1+z)\mathcal{H}$ the redshift
drift in the background spacetime, neglecting the observer term, $H_{0}(1+z)$.

Because $\delta\dot{z}(\mathbf{n},z)$ is a function of directions on the
sphere, $\mathbf{n}$, we can decompose it into spherical harmonics, $Y_{\ell
m}(\mathbf{n})$,

$\delta\dot{z}(\mathbf{n},z)=\sum_{\ell,m}^{\infty}a_{\ell
m}^{\delta\dot{z}}(z)Y_{\ell m}(\mathbf{n})\,,$ (4.2)

where the expansion coefficients $a_{\ell m}^{\delta\dot{z}}$ are given by

$a_{\ell m}^{\delta\dot{z}}(z)=\int
d\Omega_{\mathbf{n}}\,\delta\dot{z}(\mathbf{n},z)Y^{*}_{\ell
m}(\mathbf{n})\,.$ (4.3)

The angular power spectrum $C_{\ell}(z)$ is then defined as

$C_{\ell}^{\delta\dot{z}}(z):=\langle|a_{\ell
m}^{\delta\dot{z}}|^{2}\rangle\,.$ (4.4)

Here $\langle\,\rangle$ denotes the average over a statistical ensemble. For
statistically homogeneous and isotropic distributions, these expectation
values are independent of $m$, see e.g. [11].

Using the Fourier decomposition of a general function $X$ of the spacetime
coordinates,

$X(\mathbf{x},t)=\frac{1}{(2\pi)^{3}}\int
X(\mathbf{k},t)e^{-i\mathbf{k}\cdot\mathbf{x}}d^{3}k\quad,$ (4.5)

we can rewrite the expansion coefficients $a_{\ell m}$ of the terms appearing
in (4.1), using (4.3), as

$\displaystyle a_{\ell m}^{\Psi}(z_{s})$
$\displaystyle=\frac{i^{\ell}}{2\pi^{2}}\int
d^{3}kj_{\ell}(kr_{s})\Psi(\mathbf{k},t_{s})Y^{*}_{\ell
m}(\mathbf{\hat{k}}),\qquad a_{\ell m}^{\dot{\Psi}}(z_{s})$
$\displaystyle=\frac{i^{\ell}}{2\pi^{2}}\int
d^{3}kj_{\ell}(kr_{s})\dot{\Psi}(\mathbf{k},t_{s})Y^{*}_{\ell
m}(\mathbf{\hat{k}}),$ $\displaystyle a_{\ell
m}^{\mathbf{V}\cdot\mathbf{n}}(z_{s})$
$\displaystyle=\frac{i^{\ell}}{2\pi^{2}}\int
d^{3}kj^{\prime}_{\ell}(kr_{s})V(\mathbf{k},t_{s})Y^{*}_{\ell
m}(\mathbf{\hat{k}}),\qquad a_{\ell
m}^{\dot{\mathbf{V}}\cdot\mathbf{n}}(z_{s})$
$\displaystyle=\frac{i^{\ell}}{2\pi^{2}}\int
d^{3}kj^{\prime}_{\ell}(kr_{s})\dot{V}(\mathbf{k},t_{s})Y^{*}_{\ell
m}(\mathbf{\hat{k}}),$

where $\mathbf{V}\cdot\mathbf{n}=n^{i}\partial_{i}V$, and ′ denotes derivation
with respect to the argument.

Next, we assume that each of these fluctuations at redshift $z$ is related to
the initial primordial perturbation $\Psi_{\mathrm{in}}$ via a respective
transfer funtion $T(k,z)$, which propagates the perturbation up to the desired
redshift and therefore contains information about the evolution history of the
Universe,

$\displaystyle\Psi(\mathbf{k},z)$
$\displaystyle=T_{\Psi}(k,z)\Psi_{\text{in}}(\mathbf{k})$ (4.7)
$\displaystyle\Phi(\mathbf{k},z)$
$\displaystyle=T_{\Phi}(k,z)\Psi_{\text{in}}(\mathbf{k})$ (4.8) $\displaystyle
V(\mathbf{k},z)$ $\displaystyle=T_{V}(k,z)\Psi_{\text{in}}(\mathbf{k}).$ (4.9)

This is certainly the case for adiabatic perturbations of standard single
field slow roll inflation. In this case the initial power spectrum is
characterized by a simple power law,

$k^{3}\langle\Psi_{\text{in}}(\mathbf{k})\Psi^{*}_{\text{in}}(\mathbf{k})\rangle=k^{3}P_{\Psi}\equiv(2\pi)^{3}\delta(\mathbf{k}-\mathbf{k^{\prime}})A\left(\frac{k}{k_{*}}\right)^{n_{S}-1}\,,$
(4.10)

with amplitude $A$ at a fixed pivot scale $k_{*}$ and a spectral index
$n_{S}$.

In the actual redshift drift perturbation power spectrum the amplitude $A$ is
the one of the dimensionful initial power spectrum of the Bardeen potential
(4.10) $P_{\Psi}$, which in a $\Lambda$CDM universe is related to the
dimensionless curvature power spectrum via

$\displaystyle k^{3}P_{\Psi}$
$\displaystyle=2\pi^{2}\left(\frac{3}{5}\right)^{2}{\cal P}_{\cal R}$ (4.11)
$\displaystyle A\left(\frac{k}{k_{*}}\right)^{n_{S}-1}$
$\displaystyle=2\pi^{2}\left(\frac{3}{5}\right)^{2}A_{S}\left(\frac{k}{k_{*}}\right)^{n_{S}-1}\,,$
(4.12)

where $A_{S}$ is the amplitude of the curvature power spectrum and $n_{S}$ is
the spectral index. Their values have been measured from CMB fluctuations, see
[15].

The transfer functions $T_{\Psi}$, $T_{\Phi}$, and $T_{V}$ are related via the
first order Einstein equations, and for the standard $\Lambda$CDM universe
with only one degree of freedom in the perturbations, they are given, e.g., in
terms of the transfer function of the Bardeen potential $T_{\Psi}$, [14]:

$\displaystyle T_{\Phi}$ $\displaystyle=T_{\Psi}\quad,$ (4.13a) $\displaystyle
T_{V}$
$\displaystyle=\frac{2a}{3\Omega_{m}}\frac{k}{\mathcal{H}_{0}^{2}}\left(\mathcal{H}T_{\Psi}+\dot{T}_{\Psi}\right)$
(4.13b) $\displaystyle kT_{\Psi}$ $\displaystyle=\dot{T}_{V}+{\cal H}T_{V}.$
(4.13c)

In terms of the time-independent transfer function $T(k)$, we can write the
transfer function for the Bardeen potential $T_{\Psi}$ as [16, 14]

$T_{\Psi}(k,z)=\frac{9}{10}\frac{D_{+}(a)}{a}T(k)\quad,$ (4.14)

where $D_{+}(a)$ is the growth rate for the given cosmology and $z=a_{0}/a-1$.

The transfer functions in (4.7)-(4.9) can be obtained in multiple ways: either
numerically through Boltzmann codes such as class [17], assuming an analytical
ansatz for the matter transfer function $T(k)$ such as the Eisenstein-Hu
formula [18], or using one of the more recent algebraic expressions obtained
through so called genetic algorithms [19]. We have checked that for our
cosmological parameters the Eisenstein-Hu (EH) transfer function agrees within
less than one percent with the numerical transfer function derived from the
CLASS code, which is precise enough for the numerical results presented in
this paper. However, for our final numerical plots we chose to modify the
CLASS transfer function module originally called CLASSgal and described in
detail in [20].

This allows us to write the power spectrum of the redshift drift fluctuation
(4.1) at redshift $z_{s}$ of the source, $C_{\ell}(z_{s})$ as

$C_{\ell}(z_{s})=\frac{2A}{\pi}\int\frac{dk}{k}\left(\frac{k}{k_{*}}\right)^{n_{S}-1}|F_{\ell}(k,z_{s})|^{2},$
(4.15)

where

$\displaystyle F_{\ell}(k,z_{s})=-\frac{1}{\mathcal{H}}\Bigg{[}$
$\displaystyle
j_{\ell}(kr_{s})\left(-\dot{T}_{\Psi}+\frac{\dot{\mathcal{H}}}{\mathcal{H}}T_{\Psi}\right)+j^{\prime}_{\ell}(kr_{s})\left(\dot{T}_{V}-\frac{\dot{\mathcal{H}}-\mathcal{H}^{2}}{\mathcal{H}}T_{V}\right)$
$\displaystyle+\frac{\dot{\mathcal{H}}-\mathcal{H}^{2}}{\mathcal{H}}\int_{s}^{o}d\lambda\,j_{\ell}(kr_{s})\left(\dot{T}_{\Phi}+\dot{T}_{\Psi}\right)-\int_{s}^{o}d\lambda\,j_{\ell}(kr_{s})\left(\ddot{T}_{\Phi}+\ddot{T}_{\Psi}\right)\Bigg{]}\quad.$
(4.16)

In the following paragraphs, we use a modified version of CLASS in order to
study the dependence of the redshift drift on redshift and angular scales, as
well as its cross correlations with galaxy number count fluctuations. To that
end, we apply a Gaussian window function in redshift of width $\Delta z=0.01$
in all figures below. The numerical values for the cosmological background
parameters underlying this analysis are based on the latest Planck results and
are summarised in Table 1, in particular the amplitude $A_{S}$ and spectral
index $n_{S}$ of the primordial power spectrum, as well as the Planck values
for $h$, $z_{\text{eq}}$, $k_{\text{eq}}$, $\Omega_{\text{b}}$ and
$\Omega_{\text{cdm}}$ which are needed for the transfer functions.

Parameter | Value | Units
---|---|---
$\Omega_{b}$ | $0.022383$ | $h^{-2}$
$\Omega_{\text{CDM}}$ | $0.12011$ | $h^{-2}$
$h$ | $0.7$ | $10^{-2}{\rm km/s/Mpc}$
$A_{S}$ | $2.100549\cdot 10^{-9}$ | -
$n_{S}$ | $0.9660499$ | -
$z_{\text{eq}}$ | $3387$ | -
$k_{\text{eq}}$ | $0.010339$ | $h/{\rm Mpc}$
Table 1: Cosmological parameters underlying the numerical results in Section
4.

###  4.1 Power spectra at a single redshift and correlated at diffent
redshifts

In Fig. 2 we plot the angular power spectrum as a function of redshift for
some fixed multipoles, $\ell$, neglecting the integrated terms in the second
line of (4.16). We have checked that the integrated terms $\dot{T}_{\Psi}$ and
$\ddot{T}_{\Psi}$ in (3.26) contribute less than $1\%$ to the total power
spectrum over the $\ell$ range of $2-300$, so we neglect them in the numerical
evaluations. At redshifts $z\gtrsim 1$, the universe is matter dominated and
the Bardeen potentials in (4.15) are nearly constant [11]. For low multipoles,
$\ell\leq 10$, after an initial increase up to $z\simeq 0.05$, the redshift
drift decays rapidly. For the higher multipoles, $\ell\geq 50$, after an
initial growth the redshift drift nearly levels off.

Figure 2: Evolution of the perturbed redshift drift power spectra
$C_{\ell}^{\delta\dot{z}}(z)$ as a function of redshift, for different
multipoles $\ell$.

We plot the full spectra for a set of redshifts and $\ell\leq 500$ in Fig. 3.
Clearly, the perturbations are small, with amplitudes less than $10^{-3}$ on
all scales. While the low redshift fluctuations dominate on large angular
scales $\ell\lesssim 30$, the higher redshift spectra $z\geq 1$ peak at
somewhat smaller angular scales, $\ell\simeq 100$.

Figure 3: Redshift drift perturbation power spectra for a given set of
redshifts $z_{s}$ and scales $\ell\leq 600$.

Figure 4: Redshift drift and independent components angular power spectra at
different redshifts as function of $\ell$. The plot is in log-log scale and we
use a Gaussian window function of width $\Delta z=0.01$. The cross
correlations $\langle\Psi|V\rangle$ are positive at lower redshifts
($z=0.1,0.5$), but turn negative at higher redshifts ($z=1,5$).

The dominating contribution from the peculiar velocity and acceleration terms,
especially at large scales and over the redshift ranges smaller than unity
highlights the importance of this term in the full gauge-invariant expression
for the redshift drift (3.26), corroborating the analytical estimates such as
the one in [21] and improving on works that neglect or discard the effect of
peculiar acceleration and velocity of sources in their expression for the
perturbed redshift drift [8, 22]. In Refs. [23] and [24] the authors discuss,
based on the use of hydrodynamical simulations, possible astrophysical sources
where the peculiar acceleration and velocity can be distinguished from the
background redshift drift, aiming for an increase in the detectability of the
signal. Fig. 4 indicates that the drift perturbations are dominated by the
velocity terms at all proposed observational redshift ranges. Nevertheless,
our results show that for the redshift drift signal from several sources
averaged over an angular patch of a degree or more at $z=1$ the background
signal will be about 1000 times larger than the perturbations from peculiar
velocities and acceleration, thus, linear perturbation theory can be trusted.
Furthermore, the contribution from the potential and cross correlated terms
are at least another order of magnitude smaller at all scales.

In Fig. 5 we show in detail the redshift dependence of each component. The $V$
terms keep dominating at all scales, while the potential and velocity cross-
correlation terms are somewhat larger at all scales for small redshifts. The
dip in both of these contributions at around $z\approx 1$ comes from the pre-
factor $\dot{\mathcal{H}}/{\cal H}$ for $F_{l}^{\Psi}$, see Table 2, which
changes sign around this redshift.

$F^{V}_{\ell}(z)$ | $\frac{1}{\mathcal{H}}\left(-\dot{T}_{V}(k,t_{s})+\frac{\mathcal{H}^{2}-\dot{\mathcal{H}}}{\mathcal{H}}T_{V}(k,t_{s})\right)j_{\ell}^{\prime}(kr_{s})$
---|---
$F^{\Psi}_{\ell}(z)$ | $\frac{1}{\mathcal{H}}\left(\dot{T}_{\Psi}(k,t_{s})-\frac{\dot{\mathcal{H}}}{\mathcal{H}}T_{\Psi}(k,t_{s})\right)j_{\ell}(kr_{s})$
$F^{V}_{\ell}(z)\cdot F^{\Psi}_{\ell}(z)$ | $\frac{1}{\mathcal{H}^{2}}\left(-\dot{T}_{V}(k,t_{s})+\frac{\mathcal{H}^{2}-\dot{\mathcal{H}}}{\mathcal{H}}T_{V}(k,t_{s})\right)\cdot\left(\dot{T}_{\Psi}(k,t_{s})-\frac{\dot{\mathcal{H}}}{\mathcal{H}}T_{\Psi}(k,t_{s})\right)j_{\ell}(kr_{s})j^{\prime}_{\ell}(kr_{s})$
Table 2: We label each component of the redshift drift power spectra in Fig. 4
and Fig. 5 according to the transfer functions inside each contribution to the
total redshift drift, compare (4.16).

We have also calculated the correlation between the redshift drift spectra at
different $z$ by slightly altering (4.15) to integrate over the kernels
$F_{\ell}(k,z)$ at different redshifts:

$C^{\delta\dot{z}}_{\ell}(z_{1},z_{2})=\frac{2A}{\pi}\int\frac{dk}{k}\left(\frac{k}{k_{*}}\right)^{n_{S}-1}F_{\ell}(k,z_{1})F_{\ell}(k,z_{2}),$
(4.17)

while the kernels are kept the same as in the original expression.

Figure 5: Redshift drift and components power spectra as a function of
redshift. We chose a set of $\ell$s to characterize the scale dependence found
in Fig. 4.

The correlations are at most of the same order of magnitude of the power
spectra measured at a single redshift signal, and in general several orders of
magnitudes smaller. This small effect for non-diagonal correlations does not
give any new information about the redshift dependence of the signal. In fact,
the small correlations at different redshifts give support to the
observational perspectives described for instance in [24, 21], where the
authors enforce that measurements of the drift at different redshifts, for
sources with similar spectra, should improve significantly the statistics of
the measurement, particularly over the peculiar acceleration of the sources.
As the correlation of perturbations from different redshifts can be safely
neglected, the constraining power for the evolution of $\dot{z}$ will improve
when more redshift bins are observed.

###  4.2 Cross-correlations with galaxy number counts

We also study the correlation between the redshift drift (4.15) and the
theoretical relativistic number counts of galaxies at linear order. We follow
the standard treatment of [14], which is also manifestly gauge-invariant and
therefore easily mapped to observables for checking our predictions with
forthcoming surveys and simulations.

The relativistic galaxy number count fluctuation spectrum can be expressed as

$C^{N}_{\ell}(z_{s})=\frac{2A}{\pi}\int\frac{dk}{k}(kt_{O})^{n-1}\left|F^{N}_{\ell}(k,z_{s})\right|^{2},$
(4.18)

where the kernel is given by [14]

$\displaystyle F^{N}_{\ell}(k,z_{s})$ $\displaystyle=$ $\displaystyle
j_{\ell}(kr_{s})\\!\left[\\!T_{D}+\left(\\!1\\!+\\!\frac{\dot{\cal H}}{{\cal
H}^{2}}\\!+\\!\frac{2}{r_{s}{\cal
H}}\\!\right)T_{\Psi}\\!+\\!T_{\Phi}\\!+\\!\frac{1}{{\cal
H}}\dot{T}_{\Phi}\\!\right]+j_{\ell}^{\prime}(kr_{s})\left(\\!\frac{\dot{\cal
H}}{{\cal H}^{2}}\\!+\\!\frac{2}{r_{s}{\cal
H}}\right)T_{V}\\!+\\!\frac{k}{{\cal H}}T_{V}j_{\ell}^{\prime\prime}(kr_{s})$
$\displaystyle+\frac{1}{r_{s}}\int_{0}^{r_{s}}j_{\ell}(k\lambda)\left(2+\frac{r_{s}-\lambda}{\lambda}\ell(\ell+1)\right)(T_{\Psi}+T_{\Phi})d\lambda+\left(\frac{\dot{\cal
H}}{{\cal H}^{2}}+\frac{2}{r_{s}{\cal
H}}\right)\int_{0}^{r_{s}}j_{\ell}(k\lambda)(\dot{T}_{\Psi}+\dot{T}_{\Phi})d\lambda,$
(4.19)

and the density contrast transfer function $T_{D}$ is related to the Bardeen
potential and velocity transfer functions (4.13b) through Einstein’s
equations:

$T_{D}=-\frac{2a}{3\Omega_{m}}\left(\frac{k}{{\cal
H}_{0}}\right)^{2}T_{\Psi}-3T_{\Psi}-3\frac{{\cal H}}{k}T_{V}.$ (4.20)

a complete derivation of this expression can be found in e.g. [14, 11]. By
denoting the redshift drift kernel (4.16) by $F^{\dot{z}}_{\ell}(k,z)$, the
cross-correlation power spectrum is given by

$C^{\delta\dot{z}N}_{\ell}(z)\equiv\frac{A}{\pi}\int\frac{dk}{k}\left(\frac{k}{k_{*}}\right)^{n_{S}-1}\left(F^{\delta\dot{z}}_{\ell}(k,z)F^{N}_{\ell}(k,z)\right).$
(4.21)

We plot this cross correlation power spectrum for different redshifts in Fig.
6.

Figure 6: The cross correlation $C_{\ell}^{\delta\dot{z}\Delta}$ of redshift
drift $\delta\dot{z}$ and relativistic number count perturbations $\Delta$ as
a function of $\ell$ for different values of redshift: $z=0.1$ in red, $z=0.5$
in blue, $z=1$ in black, and $z=5$ in yellow. The spectrum is negative for
$z=0.1$, and for $z=0.5$ at low $\ell$. For $z=0.5$, a sign change from
negative to positive happens around $\ell\approx 150$. All other spectra are
positive.

As we see in the figure, there is a similar behavior for all redshifts, with
the absolute value of the amplitude increasing from large to intermediary
scales and then flattening at smaller scales. For redshift values $z\lesssim
0.5$ however, the signals are anticorrelated for some ranges of $\ell$: for
$z=0.1$ the signal is strictly anticorrelated, while for $z=0.5$ the signal is
anticorrelated for scales $\ell\lesssim 150$, where it changes sign. For
higher values of $\ell$ the curves reach a maximum, respectively a minimum, at
$\ell_{\text{max}}(z)$ and have a slow decrease/increase up to highly
nonlinear scales $\ell\approx 1000$, where we see a plateau between
intermediate and small scales. This result is interesting because, as it has
been argued in [23], that peculiar acceleration and velocities are the
perturbations which are most difficult to distinguish from the cosmic
background signal in future observational probes of redshift drift. Therefore,
an observational probe that is correlated with the peculiar acceleration such
as the number counts could provide a way to distinguish these effects from the
true background signal. We notice that the maximum of the amplitude increases
with redshift, such that for redshifts $z>0.5$ the curves reach it at
$\ell_{\text{max}}>100$. This behavior, and the anticorrelation for smaller
redshifts, indicates that the $z\lesssim 0.5$ redshift range has a
characteristic cross-correlation spectra which could possibly help to
strenghten the detection of the $\delta\dot{z}$ signal through the
correlation. The measured drift from sources at different redshifts in this
redshift range can be correlated with the galaxy number count, and the change
of sign observed in Fig. 6 for the same scale and different redshifts in this
range could be a characteristic feature of the redshift drift signal,
distinguishing it from the $\delta\dot{z}$ self-correlation power spectrum,
which is strictly positive, see Fig. 3.

This redshift range has also been cited as promising in detecting the redshift
drift precisely due to its small dependence on other cosmological probes [21].
Furthermore, the correlation amplitude is about three orders of magnitude
higher than the amplitude of $\dot{z}$, as one can see by comparing Fig. 6
with Fig. 3.

Having different redshift ranges where the correlation is significant in the
$\Lambda$-dominated and matter-dominated eras as well as the change from anti-
correlation to correlation in the transition epoch can provide additional
constraints for the dark energy equation of state, as explored at the
background level in [21]. The curves in Fig. 6 show that small scales provide
the highest correlations for redshifts $z\gtrsim 0.5$, while at larger scales
number count perturbations are more strongly correlated with fluctuations of
the drift at lower redshifts, inside the $\Lambda$-dominated era. Furthermore,
there is a change from anti-correlation to correlation as the redshift
increases. This behavior may provide insight into the redshift dependence of
the drift, and particularly how possible tracers of the effect should be
chosen, as the sign and amplitude of the cross-correlation change, especially
for smaller redshifts.

In all the cases discussed, the cross-correlation signal has a higher
amplitude than the $\delta\dot{z}$ power spectra alone. On average this
difference is about three orders of magnitude for all redshifts. This
correlation may be useful to both strengthen the significance in the detection
of the signal and to constrain the cosmological model through the redshift
drift signal, such as in the case for the ISW effect [25].

The question whether this correlation signal can truly be measured in one of
the proposed surveys, and if yes with which significance, however, is a more
difficult question. This depends on the number density of observed sources and
on the accuracy of the measurement of their redshift drift. Nevertheless,
correlating the signal with galaxy number counts is boosting it by about a
factor $\sqrt{C^{\delta\dot{z}N}/C^{\delta\dot{z}}}\sim 100$. Furthermore, the
cross-correlation of number counts and redshift drift does not suffer from
shot noise and can provide a significant signal already with a lower number
density of galaxies. A detailed calculation of the signal to noise of such a
measurement for a given experiment goes beyond the present study.

##  5 Discussion and conclusion

In this paper we have derived the fully gauge-invariant perturbation of the
cosmological redshift drift and we have numerically calculated its power
spectra over a wide range of redshifts and scales within linear perturbation
theory. Our derivation is general and includes all terms up to first order,
expanding and generalizing previous results that relied on source or observer
dependent assumptions.

Our main results for the perturbed cosmic redshift drift, expressions (3.26)
and (3.28), are manifestly gauge-invariant. They generalize previous
expressions for the perturbed redshift drift, as found in e.g. [8, 23], and
make it straightforward to derive constraints on this quantity from survey and
simulation data. We expect the theoretical results to translate easily to
numerical codes and simulations, as has been done in this paper through the
use of class.

We find that within linear perturbation theory redshift drift fluctuations are
small, typically of order $10^{-3}$ or smaller. Of course, a single redshift
drift measurement might be strongly affected by peculiar acceleration of the
source. However, averaging the measured redshift drift at fixed redshift over
many widely separated directions, as proposed e.g. in [24], will yield a good
measure of the background redshift drift, as in this situation the
fluctuations of this mean are well-described by linear perturbation theory.
Nevertheless, planned measurements, e.g. with SKA2, will need $N\simeq 10^{7}$
galaxies in each redshift bin to reach their goal [26]. They assume that the
signals from the galaxies are uncorrelated so that the total error is given by
the shot noise term, $1/\sqrt{N}$. Our findings show that they have to add an
effect from the correlations which is of a similar order of magnitude. More
precisely, we obtain a variance from the correlations of

$\sigma_{\delta z}^{2}(z)=\frac{1}{4\pi}\sum_{\ell}(2\ell+1)C_{\ell}(z)\,.$
(5.1)

$z$ | $\sigma_{\delta z}$ | $1/\sqrt{N(z)}$
---|---|---
0.1 | $7\times 10^{-4}$ |  $1.8\times 10^{-4}$
0.5 | $1\times 10^{-3}$ |  $3.2\times 10^{-4}$
1 | $1.1\times 10^{-3}$ |  $1.8\times 10^{-3}$
5 | $9.5\times 10^{-4}$ | ?
Table 3:  The expected error from correlations for a full sky experiment of
the redshift drift at $z=0.1,~{}0.5,~{}1$ and $5$. The expected numbers of
galaxies for an SKA2 survey are taken from [26] (Fig. 2 lower panels). SKA2
only plans to measure redshift drift out to $z=1$.

In Table 3 we give the values of $\sigma_{\delta z}(z)$ for some redshifts.
Clearly, the correlation contribution to the error is comparable with shot
noise and has to be added. For the E-ELT the situation is different. They plan
to measure the redhift drift much more precisely, i.e. with much smaller
errors per measurement, but only for order ten lines of sight per redshift bin
[Cristiani:2023iwi]. The shot noise contribution is then much larger, such
that the correlation error is irrelevant.

The dominant contribution to first order perturbations comes from the peculiar
velocity and acceleration of sources: this term is more than two orders of
magnitude larger than the contribution from the gravitational potential for
most redshifts, as it is seen in Fig. 4. Furthermore, the acceleration and
potential terms are very weakly correlated. While the relevance of the
peculiar acceleration and velocity of sources as perturbative effects on the
cosmic redshift drift have been previously studied in the literature [23, 27],
our results show for the first time the dominance of this term in comparison
to other contributions to the perturbed redshift drift signal; moreover
previous results have all relied on gauge-dependent derivations of the
perturbed redshift drift. Here, the theoretical power spectra for the redshift
drift perturbations were calculated for the first time and its numerical
evaluation is presented in Fig. 3. The scale dependence of the effect has not
been previously studied and the power spectrum prediction quantifies this,
paving the way for further observational probes of the effect. We have also
studied the evolution of the signal with redshift in Fig. 2.

Finally, we have also investigated the cross-correlation with galaxy number
counts, and find significant correlations at all scales, which increases with
$\ell$ and peaks at intermediary scales for smaller redshifts and small scales
for high redshifts. In particular, for smaller redshifts $z\lesssim 0.5$,
there is a change of sign from anti-correlated to correlated as the redshift
increases. This correlation might help in the detection of the cosmic redshift
drift fluctuations by strengthening the power spectra signal. Furthermore, the
scale dependence of the cross-correlation power spectrum could point to
possible astrophysical probes of the effect, such as explored in [24].

## Acknowledgement

We thank Enea Di Dio for help with the classgal code and useful discussions.
PB acknowledges financial support from Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES) and Fundação de Apoio à Pesquisa do Espírito
Santo (FAPES) for his PhD fellowship and is very thankful for the hospitality
of the Université de Genève, where most of this project was worked out. RD and
DS acknowledge support by the Swiss National Science Foundation under grant No
200020 182044.

##  Appendix A Appendix: Gauge-invariant Perturbation Variables

This appendix summarises some of the relations between the perturbation
variables used in Section 3 needed to arrive at (3.26). For a more detailed
account we refer to [11], here, we restate the relations relevant for this
work. We first introduce scalar potentials for the velocity perturbation
$v^{i}$, the perturbations of the 0i-elements of the metric $B^{i}$, the
4-acceleration $a^{\mu}$, and the shear tensor $\sigma_{\mu\nu}$:

$\displaystyle v^{i}$ $\displaystyle=-\partial^{i}v\quad,$ (A.1)
$\displaystyle B^{i}$ $\displaystyle=-\partial^{i}B\quad,$ (A.2)
$\displaystyle a^{i}$ $\displaystyle=-\partial^{i}\mathcal{A}\quad,\quad
a^{0}=0\quad\text{with}\quad$ (A.3) $\displaystyle\mathcal{A}$
$\displaystyle=\frac{1}{a^{2}}\left(\dot{B}-\dot{v}+A+\mathcal{H}(B-v)\right)=\frac{1}{a^{2}}\left(-\dot{V}-{\cal
H}V+\Psi\right)\ \,,$ (A.4) $\displaystyle\sigma_{ij}$
$\displaystyle=a\left(\partial_{i}\partial_{j}-\frac{1}{3}\delta_{ij}\Delta\right)\sigma\quad,\quad\sigma_{00}=0=\sigma_{0i}=\sigma_{i0}\quad.$
(A.5)

The final result (3.26) is expressed in terms of the gauge-invariant velocity
perturbation $V$, the Bardeen potentials $\Phi$ and $\Psi$, and the spatial
curvature perturbation $\mathcal{R}$, which are related to the above mentioned
perturbation variables via:

$\displaystyle V$ $\displaystyle=v-\dot{H}_{T}\quad,$ (A.6)
$\displaystyle\Psi$
$\displaystyle=A-\mathcal{H}\sigma_{t}-\dot{\sigma}_{t}\quad,$ (A.7)
$\displaystyle\Phi$ $\displaystyle=-\mathcal{R}+\mathcal{H}\sigma_{t}\quad.$
(A.8)

The following relations between the perturbation variables are useful:

$\displaystyle\mathcal{R}$ $\displaystyle=H_{L}-\frac{1}{3}\Delta H_{T}\quad,$
(A.9) $\displaystyle\sigma$ $\displaystyle=-V=\dot{H}_{T}-v\quad,$ (A.10)
$\displaystyle\sigma_{t}$ $\displaystyle=\dot{H}_{T}-B\quad,$ (A.11)
$\displaystyle V$ $\displaystyle=v-B-\sigma_{t}\quad.$ (A.12)

Note that while $\sigma=-V$ is gauge invariant, $\mathcal{R}$ and $\sigma_{t}$
are not. The denote the spatial curvature and the shear on the $t=$ constant
hypersurface and therefore depend on the chosen time-slicing.

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# Expert System for Diagnosis of Chest Diseases Using Neural Networks

Ismail Kayali M.S.c Big Data Systems, Faculty Business and Management,
National Research University Higher School of Economics. Moscow, Russia.
E-mail<EMAIL_ADDRESS>
(Feb 14 2018)

###### Abstract

This article represents one of the contemporary trends in the application of
the latest methods of information and communication technology for medicine
through an expert system helps the doctor to diagnose some chest diseases
which is important because of the frequent spread of chest diseases nowadays
in addition to the overlap symptoms of these diseases, which is difficult to
right diagnose by doctors with several algorithms: Forward Chaining, Backward
Chaining, Neural Network(Back Propagation)

However, this system cannot replace the doctor function, but it can help the
doctor to avoid wrong diagnosis and treatments. It can also be developed in
such a way to help the novice doctors.

##  1 Introduction

Did you think beforehand when you were sick to ask your computer for medical
advice? Is it reasonable to leave your computer making decisions about your
health? Will we call our personal computers Dr. Computer?Well, you would
probably do more in the near future. because the ability of computer for store
data and processed it becomes bigger in order to get useful information.

This system can be used by the domain experts themselves in order to retrieve
diagnoses and treatments quickly. It can also be used by novices who have
recently graduated from the medical school in order to get the best diagnosis
for specific symptoms and then generate the treatments of this diagnosis. In
addition to those users, the system can be a very influential teaching tool in
the faculty of medicine and health institutions. The system assumes that users
have some medical knowledge about diagnosis of diseases and a good background
in order to use personal computers.

Diagnostic rules will be inspired by the experts, doctors, specialized in this
kind of diseases. The system performance can be increased by applying some
machine learning methods on these rules and expertise. Moreover, the system
can recommend treatments for the diseases in its specialism. These treatments
are suggested based on many conditions and constraints related to the patient
and the diagnosed disease.

Based on the above, the system has six major components: standard interfaces,
administration interfaces, knowledge-base editor, inference engine,
explanation subsystem, and management system. The first two components have
been clarified before. The knowledge-base editor allows the high-level users
to add, remove and edit rules in the knowledge-base. The inference engine is
used to reason with the expert knowledge, which is extracted from our experts
and data specific to the particular problem being solved. The inference engine
is the most important component in the system. The explanation subsystem
allows the program to explain its reasoning to the user. This component brings
the life to the system! The last component, management system, manages the
users in the system. It defines the permissions for every user and routes the
control when the user logs in. Moreover, all the permanent and temporary data
are stored in a database which is designed to support all the system
components.

Finally, the system has to support working over networks; either local network
or internet. This advantage will facilitate the access to the system from many
experts and knowledge engineers. Consequently, the knowledge-base can be
expanded quickly to cover many medical domains. This diversity in expertise
will make the system more robust and reliable.

##  2 Applicability Statement

Basic features will be included and simple inference approaches will be used.
Two inference algorithms, which are used in the rule-based systems, These
algorithms are backward and forward chaining algorithms. Moreover, providing
explanation facilities to interpret the system behavior and output. After
finishing the basic requirements, providing the system with some learning
capabilities using Data Mining approaches and algorithms.

##  3 Knowledge Representation

The knowledge base contains the domain-specific knowledge required to solve
the problem. The knowledge base is created by the knowledge engineer, who
conducts a series of interviews with the expert and organises the knowledge in
a form that can be directly used by the system. The knowledge engineer has to
have the knowledge of KBES technology and should know how to develop an expert
system using a development environment or an expert system development shell.
It is not necessary that the knowledge engineer be proficient in the domain in
which the expert system is being developed. But a general knowledge and
familiarity with the key terms used in the domain is always desirable, since
this will not only help in better understanding the domain knowledge but will
also reduce the communication gap between the knowledge engineer and the
expert. Before deciding on the structure of the knowledge base, the knowledge
engineer should have a clear idea of different knowledge representation
schemes and the suitability of each under different circumstances. [1]
Developing an expert system involves tasks such as acquiring knowledge from an
acknowledged domain expert, documenting it and organizing it, generating a
knowledge net to check the relationships between different knowledge sources,
checking for consistency in the knowledge and finally transforming the
knowledge net into a computer program using appropriate tools. The same piece
of knowledge can be represented using more than one formal scheme, but with
varying degrees of difficulty. The difficulty is not in the representation of
the knowledge, but in its usage. The decision on selection of a scheme
primarily depends on the type of application being built. Also, different
knowledge representation schemes can be adopted for developing one
application. [3] The knowledge engineer has to decide which portion of the
knowledge should be represented in what form, depending on the nature of the
knowledge and the efficiency of its use [5]. The most common methods of
knowledge representation are:
1\. Predicate logic
2\. Production rules
3\. Frames (objects) and semantic networks
4\. Conventional programs I have adopted the second mechanism, production
rules, to represent the medical expertise. This method is easy to formalize
the medical knowledge especially in the distinctive diagnosis which does not
require a very deep knowledge in medicine.

###  3.1 Production Rules

Production rules are simple but powerful forms of knowledge representation
providing the flexibility of combining declarative and procedural
representation for using them in a unified form. A production rule has a set
of antecedents and a set of consequences. The antecedents specify a set of
conditions and the consequences a set of actions. A value assigned to a
variable can be used to represent facts in the rules. This representation is
called a variable-operator-value triple. when the IF portion of a rule is
satisfied by the facts stored in the context or fact established by a user
input, the actions specified in the THEN portion are performed, and the rule
is then said to be fired [2]. It is a common practice in the development of
expert systems to logically divide the rules into smaller rule bases and to
control from a higher-level rule base which has knowledge about the different
rule bases in the knowledge base. If there are more than one rule bases, each
of them should have separate working memories. The higher-level rule base
having meta-rules will control the overall inference process and will have a
global context. The inference engine of the expert system shell should
properly handle different contexts for proper solution of the problem [6].

###  3.2 Inference Mechanisms

Inference mechanisms are control strategies or search techniques, which search
through the knowledge base to arrive at decisions. The knowledge base is the
state space and the inference mechanism is a search process. As expert systems
predominantly process symbols, the inference process manipulates symbols by
selection of rules, matching the symbols of facts and then firing the rules to
establish new facts. This process is continued like a chain until a specified
goal is arrived at. In an expert system, the inference can be done in a number
of ways. The two popular methods of inference are backward chaining and
forward chaining. Backward chaining is a goal-driven process, whereas forward
chaining is data-driven. The inference engine of our system uses backward
chaining algorithm to infer facts.[4]

###  3.3 Backward Chaining

It is the goal-driven reasoning. In backward chaining, an expert system has
the goal (a hypothetical solution) and the inference engine attempts to find
the evidence to prove it. First, the knowledge base is searched to find rules
that might have the desired solution. Such rules must have the goal in their
THEN (action) parts. If such a rule is found and its IF (condition) part
matches data in the database, then the rule is fired and the goal is proved.
However, this is rarely the case. The inference engine puts aside the rule it
is working with (the rule is said to stack) and sets up a new goal, a sub-
goal, to prove the IF part of this rule. Then the knowledge base is searched
again for rules that can prove the sub-goal. The inference engine repeats the
process of stacking the rules until no rules are found in the knowledge base
to prove the current sub-goal [7]. Then, the user is asked to provide a value
for this goal and the inference engine goes back recursively. There are three
basic data structures used to implement the backward chaining algorithm. They
are the working memory (context), rule stack and goal stack. When any one of
the actions of inference process i.e. select, match and execute occurs, one or
more of these data structures get modified [9].

###  3.4 Forward Chaining

It is the data-driven reasoning. The reasoning starts from the known data and
proceeds forward with that data. Each time only the topmost rule is executed.
When fired, the rule adds a new fact in the database. Any rule can be executed
only once. The match-fire the cycle stops when no further rules can be fired
[8]. Moreover, this method is used to gather information and then infer from
it whatever can be inferred. Hence, many rules may be executed that have
nothing to do with the established goal.

###  3.5 Hybrid Inference

The third inference strategy is the hybrid inference mechanism. It is a
combination of the backward and forward chaining process. Backward chaining is
suitable, if there are few goal states and many initial states. The sequence
in which the data are requested depends on the flow of the inference process.
In forward-chaining all the data that the user knows have to be given a prior
and the system does not prompt for any data. If the data are sufficient the
goal may be reached; otherwise it may exit stating that the goal is
unreachable with the available data. Hence, the forward chaining process is
suitable only when there are very few initial states and many goal states. In
large expert systems, many initial and goal states can exist. Also, the user
may not be able to give all the necessary data a prior, as he/she may not be
able to visualize the flow of the inference process. In this context, a hybrid
inference mechanism seems to be suitable. It starts in the forward chaining
mode with an empty context by trying to establish the facts as they appear in
the rule base and then backward chains to prove or disprove them. The overall
direction of the hybrid chaining process is forward with backward chaining
being invoked as and when facts are to be established. The user does not give
all the data a prior but gives data as and when prompted for [8].

##  4 Project Plan

The major work activities, which must be achieved in this project, are
summarized below in a hierarchical format in order to be readable. They are
numbered according to the Work Breakdown Structure (WBS) levels. The
activities are as following:

###  4.1 Writing Software Project Management Plan

1\. Preliminary Analysis of the System.
2\. Understanding sections of the SPMP document.
3\. Drawing the Block Diagram of the System.
4\. Defining the Project Activities and Deliverable.
5\. Drawing the activities and WBS diagrams.
6\. Generating SPMP Document.

###  4.2 Building a Prototype

1\. Specifying the system components
2\. Specifying the Standard Part.
3\. Specifying the Inference Engine.
4\. Specifying the Knowledge-base.
5\. Specifying the Knowledge-base Editor.
6\. Specifying the Explanation System.
7\. Specifying the Administrator Part.
8\. Specifying the Management System.
9\. Generating Business Requirement Specification Document.
10\. Designing the system components.
11\. Designing the Inference Engine.
12\. Designing the Knowledge-base.
13\. Designing the Knowledge-base Editor.
14\. Designing the Explanation System.
15\. Designing the Administrator Part.
16\. Designing the Management System.
17\. Generating Software Design Document (UML modeling)
18\. Implementing the system components.
19\. Implementing the Standard Part.
20\. Implementing the Inference Engine.
21\. Implementing the Knowledge-base.
22\. Implementing the Knowledge-base Editor.
23\. Implementing the Explanation System.
24\. Implementing the Administrator Part.
25\. Implementing the Management System.
26\. Generating the Technical Report
27\. Testing the system components
28\. Testing the Standard Part.
29\. Testing the Inference Engine.
30\. Testing the Knowledge-base.
31\. Testing the Knowledge-base Editor.
31\. Testing the Explanation System.
32\. Testing the Administrator Part.
33\. Testing the Management System.
34\. Generating the Testing Document.
35\. Integrating all the system components.

###  4.3 Building the Medical Dataset

1\. Defining the system specialism.
2\. Collecting adequate information.
3\. Manipulating the information and extracting rules.
4\. Populating the knowledge-base with this information.

###  4.4 Customizing the prototype

1\. Revising the specification document.
2\. Revising the design document.
3\. Revising the implementation.

###  4.5 Testing the system

1\. Testing using Black Box Test methods.
2\. Testing using White Box Test methods.
3\. Integration Testing.
4\. Acceptance Testing.

###  4.6 Releasing the final software product

1\. Packaging the product.
2\. Deployment the system on web server.

##  5 Conclusion and Future Work

The system aims to develop and produce highly interactive system for
diagnosing chest diseases and suggesting treatments. Therefore, the system can
be used in clinics, medical centers and hospitals. It enables the novices to
work in the cases where the professionals are absent. Moreover, the system can
be very influential teaching tool in the faculty of medicine and health
institutes. The experts can also use the system for generating solutions
quickly since there are a lot of similar cases can be diagnosed for one
patient.Also, they can use the system to archive the cases they encounter
during their career. Then, they use the data mining tool,supported by the
system to discover new medical cases. The ability of neural networks to
classify and separate convergent data is able to diagnose chest diseases,
despite the severe overlap between disease symptoms that have good results in
disease diagnosis based on the medical preview. We hope the future system will
develop to include all possible chest diseases by securing a larger range of
samples as they represent the biggest obstacle in artificial intelligence
applications.

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