# Long gradient mode and large-scale structure observables I: linear order

Alireza Allahyari  Department of Physics, Sharif University of Technology,
Tehran, Iran<EMAIL_ADDRESS>Javad T. Firouzjaee  School of
Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box
19395-5531, Tehran, Iran<EMAIL_ADDRESS>
###### Abstract

We study the effect of long gradient modes on large scale observables. When
defined correctly, genuine observables should not only be gauge invariant but
also devoid of any gauge artifacts. One such gauge artifact is a pure gradient
mode. Using the relativistic formulation of large scale observables, we
confirm that a long gradient mode which is still outside observer’s horizon
leaves no imprint on the large scale observables at first order. These include
the cosmic rulers and the number counts. This confirms the existing method for
relativistically defined observables. The general relativistic bias relation
for the halos and galaxies is also invariant under the presence of a long
gradient mode perturbation. The observed power spectrum is not affected by
this long mode.

##  I introduction

Latest developments in the cosmology are provided by deep redshift surveys for
cosmological structures which are the result of the evolution of cosmological
perturbations generated during inflation. Quantum fluctuations of the
inflation field set the seeds of curvature perturbations at primordial epochs.
The nature of these fluctuations are well studied in Cosmic Microwave
Background (CMB) which is closely related to the inflationary early universe
model. The fingerprints of these fluctuations at large scale structures (LSS)
can be interesting since the current and future galaxy surveys will give high
precision measurements of galaxy clustering, providing big statistical power
to solve the issues in the standard model of cosmology since LSS surveys are
three dimensional whereas CMB is two dimensional.

In galaxy surveys measurements the light propagates at the finite speed
through the Universe and is modified by the inhomogeneity and the curvature of
the matter. Therefore, we need to use proper general relativistic calculations
to relate the observables which we observe from the light to the physical
quantities of source galaxies and the inhomogeneities properties that change
the photon propagation. Since the exact relativistic calculation for
cosmological structure observables has complexity exactsolution , usually
first or second order perturbation of the Einstein equation are used to
calculate these observables. Some of these effects to the observed
fluctuations of galaxies are the dark matter density fluctuations, the
redshift-space distortions (peculiar velocity effect in the redshift space of
light) and the magnification bias where studied in the Jeong:2014ufa ;
Yoo:09-10 ; Bonvin:11-14 ; Challinor:2011bk . The standard way for calculating
the full relativistic formula of galaxy clustering is done with the relation
to the inhomogeneities and the source galaxy population, by tracing back the
photon path given the observed redshift and the angular position of the source
galaxies. Jeong et al. (for review see Jeong:2014ufa ) obtained a fully
general relativist expression for the observed galaxy density contrast at
linear order, a fundamental galaxy clustering observable, including the volume
distortions due to the light deflection, evolving number density, galaxy
density bias, as well as the magnification bias generalized to evolving
luminosity function.

Generally, there are different wavelengths for the gravitational potential
which come from the early universe fluctuations. The long wavelength potential
can produce a superhorizon perturbation that might have effects on the CMB map
or in the large scale structure observables. The power asymmetry in CMB map as
observed by Planck satellite planck13  (for earlier reports of hemispherical
asymmetry in WMAP data see wmap ) can be the generic property of the early
universe model. The observations show that the power spectrum in northern
hemisphere is different than the the power spectrum in southern hemisphere.
Erickcek et al. Erickcek:2008sm  have proposed a superhorizon perturbation
would introduce a preferred direction that generates the power asymmetry.
Along this way, the predictions of inflationary models with long mode
modulation of large scale structures are presented in Namjoo:2014nra . As a
matter of fact, it was shown that a long constant Weinberg  and a long
gradient mode Hinterbichler:2012nm  can be gauge artifacts of the perturbation
theory which does not leave any effect on the cosmological observables
Creminelli:2011sq ; Mirbabayi:2014hda . Our goal in this paper is to
investigate the long gradient mode effects on the large scale observables in
the relativistic formulation.

The plan of the paper is as follows. In section II we will review the long
mode effect in the perturbation theory of the large scale structure. Then, We
calculate the three large-scale observable in the presence of long gradient
mode in section III. The section IV is devoted to investigating the galaxy
clustering quantities such as galaxy number counts and halo bias in the
presence of the long gradient mode. Finally, we conclude in section V. The
Latin indices indicate the space components and Greek indices indicate the
space time components and we have set $c=1$.

##  II long gradient mode

The perturbation theory defines gauge invariant quantities on spatial
hypersurfaces with the assumption that perturbations fall of at the infinity.
This allows to decompose perturbations as scalar, vector, and tensor (SVT). In
the perturbation theory zero momentum modes will modulate the power spectrum
in case of the single field slow roll inflation as derived in Maldacena:2002vr
, called the consistency relations. It is argued that the consistency
relations are true in models of inflation in which the only dynamical field is
the inflaton field Creminelli:2004yq . This will induce a local type non-
Gaussianity. Although this type of non-Gassainity is small and proportional to
$n_{s}-1$, this is not clear how physics outside our horizon can influence the
local physics. Its is crucial to know if our large scale observations are
really contaminated by such effects.

It has been shown that even after fixing the gauge zero momentum
transformations are allowed Weinberg . This remaining gauge freedom is shown
to be the source of IR divergences Urakawa:2010it . This is attributed to the
fact that we will need boundary conditions to uniquely solve for the lapse
function after imposing the conventional comoving gauge condition. One may
also say that constant modes can be removed by a coordinate transformation, so
they should have no observable effect as they induce a relative shift in the
expansion history. Let us suppose that we have a long mode perturbation that
in a gradient expansion only the term $\frac{k}{aH}$ is dominant, where $a$,
$H$ and $k$ are scale factor, Hubble parameter and Fourier mode for the long
mode respectively. This potential could appear in an expansion of the
primordial perturbation of the form $\sin k.x$. The average of such
perturbations should vanish in the whole universe in order to keep homogeneity
and isotropy. We suppose that the effect of such perturbation does not vanish
in our Hubble patch. Such a perturbation does not fall off at infinity as
required by the perturbation theory. One may expect that such a pure gradient
mode in single field slow roll inflations must have no observable imprint on
the cosmological observables, since as the long mode freezes outside the
Hubble radius it turns into an adiabatic mode. The effect of a pure gradient
mode is then an acceleration which is not observable locally.
This seems to be justified in the separate universe approach in which the
universe can be locally approximated by a curved FRW on patches smaller than
the long mode as long as we are not measuring correlation between patches
Wands:2000dp . Here this patch is well beyond our observable universe. The
physical effects of a long mode start at $(\frac{k}{aH})^{2}$ order, giving
rise to an effective local curvature in the case of scalar perturbations. Long
tensor modes at order $(\frac{k}{aH})^{2}$ will imprint a quadrapole in
density fluctuations on small scale structures before tensor modes are damped
away Dai:2013kra . By extension of Weinberg adiabatic modes it is shown that a
pure gradient mode can be removed from our entire horizon Hinterbichler:2012nm
. We may require that true observables are invariant under both normalizable
and non-normalizable transformations Urakawa . The vanishing effect of a
constant mode in the power spectrum is emphasized in Dai:2013kra . In this
case it is crucial to note that the power spectrum should be scale invariant.
For the CMB, we can see this vanishing contribution if change of last
scattering surface position is taken into account Creminelli:2011sq .

Now we are interested in the case of a pure gradient mode. Using the conformal
Fermi coordinates one can show that time independent gradient modes are
removed in patches smaller than the gradient mode wavelength and the tidal
effects vanish identically Dai:2015rda . We are justified to say that pure
gradient modes leave no imprint on dynamics locally. However, because we
observe on our past light cone, gradient modes appear in relations between the
physical quantities and apparent observed ones. Thus, the vanishing effect of
such long modes in observables is a consistency check of the proposed
formulation for large scale observables.

To this end, we need to define observables. Observables should be gauge
invariant as a priori and made of information on our past light cone. They
should be devoid of gauge artifacts when written in any gauge. They are
described by the standard clocks, the standard rulers and the number counts
Jeong:2014ufa . One example of such a clock is the CMB on large scales. In the
case of the CMB it is shown that a gradient mode leaves no imprint at first
order using suitable coordinate transformations and averages when temperature
is physically defined for the CMB and the observer velocity is taken into
accountMirbabayi:2014hda . A consistent general relativistic formulation of
galaxy clustering should naturally take into account effects such as
displacements, velocities and other new relativistic effects. The other
cosmological observable is galaxy number counts. Future surveys will probe
cosmos on large volumes with unprecedented precision. Because of their large
statistical power, they will be able to discriminate between inflationary
models by the level of non-Gaussianity. It is crucial to consistently
formulate all effects in order to correctly interpret the data. A fully
relativistic formulation has recently emerged Jeong:2014ufa ; Yoo:2014kpa ;
Bonvin:11-14 . The authors in Challinor:2011bk  use the evolution equation for
the Jacobi map to derive the number counts. As a consistency check we show
that a pure gradient mode must have no observable imprint on observed galaxy
number counts for comoving sources and observers.

##  III large-scale observables

Kaiser formula defines the relation between the observed power spectrum in the
redshift space and the physical power spectrum Kaiser . However, for large sky
surveys this formula is hindered by the fact that this formula does not
consistently take into account all relativistic effects. In addition, on large
scales, gauge effects become important which should be taken into account and
we will need a gauge invariant formulation. For example galaxies are thought
to trace the underlying dark matter overdensity and are biased by a scale
independent factor $b_{g}$ with respect to the dark matter overdensity. This
definition is obscured because changing the gauge will introduce scale
dependencies. We also need to define observables which should be gauge
invariant in prior and physically defined as they will be explained. Since
photons are traveling through inhomogeneities, their path will deviate and
these inhomogeneities change the observed size of our rulers and our apparent
survey volume.In this section we investigate the photon propagation effects
from inhomogeneities.

An observer sees the cosmos images by projecting on his own instantaneous
screen space ellis book  spanned by orthonormal basis $\textbf{e}_{1}$ and
$\textbf{e}_{2}$ which are orthogonal to the observer’s velocity. His velocity
defines instantaneous volume in which he measures. When a photon is observed
at a redshift $z$ (Observed redshift), its apparent position is attributed as
$\tilde{x}^{\mu}(z)$ 111$\tilde{}$ refers to apparent quantities.. The photon
trajectory at the observed redshift $z$ is deviated from a similar trajectory
in a homogeneous universe. The actual position of a source is obtained by
integrating the geodesic equations for photons. These deviations are gauge
dependent and unobservable. Because null geodesics are conformally invariant
we can use conformally transformed metric and affine parameter for the
photons. The source position is given by

$x^{\mu}=\bar{x}^{\mu}(z)+\Delta x^{\mu}=\bar{x}^{\mu}(z)+\delta
x^{\mu}+\frac{d{\bar{x}}^{\mu}}{d\chi}\delta\chi,$

where $\chi$ is the affine parameter perturbation and $\bar{x}^{\mu}(z)$ is
the background position evaluated at the observed redshift. The first term is
given by the geodesic equation and the second term is given by the redshift
matching as it will be explained. The perturbed conformal photon four-vector
is given by $k^{\mu}=(-1+\delta\nu,n^{i}+\delta n^{i})$. When the observer
fixes the scale factor at his observation time $a(t_{p})=1$, the global scale
factor will be $a(\tau_{o})\neq 1$ where
$t_{p}=\int_{0}^{\tau_{0}}\sqrt{-g_{00}(x_{o},\tau)}a(\tau)d\tau$ is the
observer’s proper time and the subscript $o$ refers to the observer. The
metric in a general gauge is written as

$ds^{2}=a^{2}(-(1+2A)d\tau^{2}-2B_{i}d\tau
dx^{i}+(\delta_{ij}+h_{ij})dx^{i}dx^{j}).$

We work in the conformal Newtonian gauge in which the metric is given by

$\displaystyle ds^{2}=a^{2}(-(1+2\varphi)d\tau^{2}+(1-2\psi)dx^{i}dx_{i}).$
(1)

In the matter dominated Einstein-de Sitter (EdS) universe $\varphi=\psi$. The
scale factor difference between the global scale factor $a(\tau_{o})$ and the
$a(t_{p})$ is $\delta
a=a(\tau_{o})-a(t_{p})=-H_{o}\int_{0}^{\tau_{o}}\varphi(x_{o},\tau)a(\tau)d\tau$.
To solve the geodesic equations we need the initial conditions which are set
by requiring that in observer’s frame the frequency and direction of the
photons are given by

$\displaystyle 1=(a^{-2}g_{\mu\nu}e^{\mu}_{0}k^{\nu})_{o}$ (2) $\displaystyle
n_{i}=(a^{-2}g_{\mu\nu}e^{\mu}_{i}k^{\nu})_{o},$

where $e^{\mu}_{\nu}$ are orthonormal tetrads given by
$e^{\mu}_{0}=u^{\mu}=a^{-1}(1-\varphi,v^{i})$ and
$e^{u}_{i}=(v_{i},\delta_{i}^{j}+\psi\delta_{i}^{j})$. These will fix the
perturbation at observer’s position as

$\displaystyle\delta\nu_{o}=-\delta a+\varphi_{o}+v_{\lVert}$
$\displaystyle\delta n^{i}_{o}=n^{i}\delta a-v^{i}_{o}+\psi n^{i},$ (3) 222For
any vector $v^{i}$ we have defined $v_{\lVert}=v^{i}n_{i}$ and
$v^{i}_{\perp}=\mathcal{P}^{ij}v_{j}$ in which
$\mathcal{P}^{ij}=\delta^{ij}-n^{i}n^{j}$ is the projection operator.

. Integrating geodesic equation and using the initial conditions give

$\displaystyle\delta x^{0}=(-\delta
a-\varphi_{o}-v_{||o})\chi+\int\left(2\varphi+(\chi-\chi^{\prime})(\dot{\varphi}+\dot{\psi})\right)d\chi^{\prime}-\int_{0}^{\tau_{o}}\varphi(x_{o},\tau)ad\tau$
(4) $\displaystyle\delta x^{i}=(\delta
an^{i}-\psi_{o}+v_{o}^{i})\chi+\int\left(\psi
n^{i}+(\chi-\chi^{\prime})(-\partial_{i}\varphi-\partial_{i}\psi)\right)d\chi^{\prime}.$
(5)

On the other side, the redshift for the photons is defined by

$\displaystyle
1+z\equiv\frac{1}{\tilde{a}}=\frac{(k_{\mu}u^{\mu})_{e}}{(k_{\mu}u^{\mu})_{o}}=\frac{(1+\varphi+v_{\lVert}-\delta\nu)_{o}}{a(x^{0})}.$
(6)

The scale factor perturbation with respect to the apparent scale factor
$\tilde{a}$ is defined as $\Delta\ln a=\frac{a}{\tilde{a}}-1$. We can write
this as

$\displaystyle\Delta\ln a=\frac{\partial\ln
a}{\partial\tau}(x^{0}-\tilde{x}^{0}).$ (7)

Using (6) we get

$\displaystyle\delta\chi=\delta x^{0}-\frac{1+z}{H}\Delta\ln a.$ (8)

###  III.1 Cosmic rulers

Cosmic rulers are cosmological observables that their spatial scale $r_{0}$ is
known like CMB on large scales or a known scale in the matter power spectrum
like baryon acoustic oscillation (BAO) Fabian . Their observed value is
related to the cosmic candle perturbations which are related to luminosity
distance perturbations. These are used for studying the nature of dark energy
Barausse:2005nf . Cosmic ruler observed scale changes from $r_{0}$ to the
observed apparent scale $\tilde{r}$. Note that the size of the rulers may
change with time by a term proportional to cosmic clock perturbations. The
cosmic clock perturbations are not affected by pure gradient modes as we will
explain. Hence we consider non evolving rulers. To define their length we need
to define the observer. Their length is defined as the length measured in
instantaneous frame of the comoving observers whose velocity is given by
$u^{i}=\frac{T^{i}_{0}}{\rho+p}$ and his projected metric is given by
$(g_{\mu\nu}+u_{\mu}u_{\nu})$. If the size of the ruler is small we can
approximate its apparent length as

$\displaystyle\tilde{r}^{2}=\tilde{a}^{2}(z)\left(-(\delta\tilde{x}^{0})^{2}+\delta_{ij}\delta\tilde{x}^{i}\delta\tilde{x}^{j}\right),$
(9)

whereas its physical scale is given by

$\displaystyle
r_{0}^{2}=(g_{\mu\nu}+u_{\mu}u_{\nu})\left(\delta\tilde{x}^{\mu}+\Delta
x^{\mu}-\Delta x^{\prime\mu}\right)\left(\delta\tilde{x}^{\nu}+\Delta
x^{\nu}-\Delta x^{\prime\nu}\right),$ (10)

where $\delta\tilde{x}^{u}=x^{\mu}-x^{\prime\mu}.$ At first order using again
the small ruler approximation, $\Delta x^{i}-\Delta x^{\prime
i}\simeq\delta\tilde{x}^{\beta}\frac{\partial}{\partial\tilde{x}^{\beta}}\Delta
x^{i}$, and small angel approximation,
$\delta\tilde{x}^{0}=-\delta\tilde{x}_{\lVert}$, the relative ruler
perturbation is

$\displaystyle\frac{\tilde{r}-r_{0}}{\tilde{r}}=\mathcal{C}\frac{(\delta\tilde{x}_{\lVert})^{2}}{\tilde{r}^{2}_{c}}+\mathcal{B}_{i}\frac{\delta\tilde{x}_{\lVert}\delta\tilde{x}^{i}_{\perp}}{\tilde{r}_{c}^{2}}+\mathcal{A}_{ij}\frac{\delta\tilde{x}^{i}_{\perp}\delta\tilde{x}^{j}_{\perp}}{\tilde{r}^{2}_{c}},$
(11)

where $\tilde{r}_{c}\equiv\frac{\tilde{r}}{a}$ Fabian . First, we will
introduce each of these geometric factors and then we calculate the effect of
a pure gradient mode on these observables.

####  III.1.1 2-scalar C

The function $\mathcal{C}$ gives the line of sight perturbations of cosmic
rulers and causes perturbations in redshift space. It also encompasses the
redshift space distortion term which is the dominant term on small scales. In
the case of non-evolving rulers, the function $\mathcal{C}$ is given by

$\displaystyle\mathcal{C}=-\Delta\ln
a-\frac{1}{2}h_{\lVert}-v_{\lVert}-\partial_{\chi}\Delta x\lVert,$ (12)

where $h_{\lVert}=h_{ij}n^{i}n^{j}$. The $\mathcal{C}$ function should be
devoid of any gauge artifact since it is an observable. To check the
formulation let us calculate the effect of a pure gradient mode on
$\mathcal{C}$. In the EdS universe a pure gradient mode is given by
$\psi=\phi=k.x$ where $\frac{k}{aH}<1$. In the Newtonian gauge $\mathcal{C}$
is given by

$\displaystyle\mathcal{C}=-\Delta\ln
a\left(1-H(z)\frac{\partial}{\partial_{z}}(\frac{1+z}{H})\right)-\varphi-
v_{\lVert}+\frac{1+z}{H(z)}(-\partial_{\lVert}\varphi-\dot{v}_{\lVert})=\left(+\frac{2}{3}\chi(\frac{3}{2})-\chi+\sqrt{a}(\frac{1}{H_{o}}-\frac{3}{2})\right)k_{\lVert}=0,$
(13)

where $v_{\lVert}=-\frac{2}{3}\frac{\sqrt{a}k_{\lVert}}{H_{o}}$,
$k_{\lVert}=k_{i}n^{i}$ and $\chi=\frac{2}{H_{o}}(1-\sqrt{a}).$ We used the
fact that $\partial_{\lVert}v^{i}=n^{j}\partial_{j}v^{i}=0$. We also have

$\Delta\ln a=v_{\lVert}-v_{o}-\varphi=-\frac{2}{3}\chi k_{\lVert}.$

Note that neglecting first order perturbations we have $H_{o}=\frac{3}{2}$. As
a result, the line of site perturbations are not affected by the presence of
the pure gradient mode.

####  III.1.2 2-vector $\mathcal{B}$

The vectorial term which is a projected vector can be written as

$\displaystyle\mathcal{B}_{i}=-v_{\perp i}+\frac{1+z}{H(z)}\partial_{\perp
i}\Delta\ln a,$ (14)

where $\partial_{\perp
i}={P}_{i}^{j}\partial_{j}=(\delta_{i}^{j}-n_{i}n^{j})\partial_{j}$. This term
produces perturbations both in the line of sight and perpendicular to the line
of sight components. Any spin one quantity such as $\mathcal{B}_{i}$ term can
be written in the spin basis as

${}_{\pm
1}\mathcal{B}=m^{i}_{\mp}\mathcal{B}_{i}=-v_{\pm}+\frac{1+z}{H}\partial_{\pm}\Delta\ln
a$ (15)

where $\textbf{m}_{\pm}=\frac{\textbf{e}_{1}\mp i\textbf{e}_{2}}{\sqrt{2}}$
and $\textbf{e}_{1}$ and $\textbf{e}_{2}$ are vector basis on the sphere. For
a gradient perturbation we have
$\mathcal{B}_{i}=(\frac{2}{3H_{0}}-1)\sqrt{a}k_{\perp i}=0$. Again a pure
gradient gives vanishing contribution to $\mathcal{B}_{i}$. Similar to the
former case, the $\mathcal{B}$ term is not changed by the pure gradient mode.

####  III.1.3 Magnification and shear

The most crucial term is $\mathcal{A}_{ij}$ which is a transverse tensor which
can be written as

$\displaystyle\mathcal{A}_{ij}=-\Delta\ln
a\mathcal{P}^{i}_{j}-\frac{1}{2}\mathcal{P}_{i}^{k}\mathcal{P}_{j}^{l}h_{kl}-\partial_{\perp(i}\Delta
x_{\perp j)}-\frac{1}{\tilde{\chi}}\Delta x_{\lVert}\mathcal{P}_{ij}.$ (16)

Magnification which is the trace part of $\mathcal{A}_{ij}$ is

$\mathcal{M}=\mathcal{P}^{ij}\mathcal{A}_{ij}=-2\Delta\ln
a-\frac{1}{2}(-4\varphi)+2\hat{k}-\frac{2}{\chi}\Delta x_{\lVert}$

where $\hat{k}=-\frac{1}{2}\partial_{\perp i}\Delta x^{i}_{\perp}$. Note that
lensing convergence ($\hat{k}$) is not gauge invariant. Hence, it is not
observable. It has been shown that magnification is not perturbed by gradient
modes Fabian . The trace free part of the $\mathcal{A}_{ij}$ term is called
shear, $\gamma_{ij}$, which is a spin two quantity. The shear term can be
written in the spin basis as ${}_{\pm
2}\gamma=m^{i}_{\mp}m^{j}_{\mp}\gamma_{ij}$. In the Newtonian gauge in the
spin basis the shear term is given by

${}_{\pm
2}\gamma=\int(\tilde{\chi}-\chi)\frac{\chi}{\tilde{\chi}}m^{i}_{\mp}m^{j}_{\mp}\partial_{i}\partial_{j}(2\varphi)d\chi.$
(17)

Since the shear is related to the second derivatives of the metric, it is not
affected by gradient modes. Consequently surveys like weak lensing surveys
which measure the shear are not affected by the gradient modes.

####  III.1.4 Luminosity distance

The magnification which produces area perturbations is measured in the lensing
surveys and surveys which probe the luminosity like supernova surveys. It is
related to angular and luminosity distance by $\frac{\Delta
D_{A}}{D_{A}}=\frac{\Delta D_{L}}{D_{L}}=-\frac{1}{2}\mathcal{M}$
Bonvin:2005ps . Therefore, a pure gradient should induce no asymmetry in our
cosmic candle observations.

##  IV galaxy number counts

One of the most important observable quantities in the large scale structure
surveys is the galaxy number counts. We are interested in the effect of the
long gradient modes in the number count surveys. Since we observe tracers and
not the underlying matter field we will need to define a general relativistic
bias relation applicable for long modes. It is argued that the bias relation
in the synchronous gauge is appropriate Donghui . This is in contrast to the
advocated gauge in Yoo:09-10  that chooses constant redshift gauge for the
bias relation. The synchronous gauge is proposed for a second order
calculation of the galaxy clustering Yoo:2014vta . Suppose we have a comoving
source with four velocity $u^{\mu}$. In the instantaneous source rest frame a
volume element is defined by
$dV_{\mu}=\varepsilon_{\mu\nu\alpha\beta}dx^{\nu}dx^{\alpha}dx^{\beta}$. The
number of galaxies defined in the observed coordinates is given by

$\displaystyle
N(x)=\int\sqrt{-g}n_{g}u^{\mu}\varepsilon_{\mu\nu\alpha\beta}\frac{\partial
x^{\nu}}{\partial{{\tilde{x}}^{1}}}\frac{\partial
x^{\alpha}}{\partial{{\tilde{x}}^{2}}}\frac{\partial
x^{\beta}}{\partial{{\tilde{x}}^{3}}}d^{3}{\tilde{x}}.$ (18)

where $n_{g}$ is the physical number density of the galaxies. The number of
the galaxies can be written as

$\displaystyle
N(x)=\int(1+\varphi-3\psi)a^{3}\bar{n}_{g}(z)(1+\delta_{g}(z,x))[(1-\varphi)\arrowvert\frac{\partial
x^{i}}{\partial\tilde{x}^{j}}\arrowvert+v_{\lVert}]d^{3}\tilde{x}=\int\tilde{a}^{3}\tilde{n}_{g}(z,\tilde{x})d^{3}\tilde{x},$
(19)

where $\delta_{g}$ is number density perturbation and
$x^{i}=\tilde{x}^{i}(z)+\Delta x^{i}$. Thus, the observed number of the
galaxies in the observed coordinates is given by

$\displaystyle\delta_{g}(z,\tilde{x})=\delta_{g}(z,x)+\delta
V=\delta_{g}-\psi+v_{\lVert}+\partial_{\lVert}\Delta
x_{\lVert}+\frac{2}{\chi}\Delta x_{\lVert}-2\hat{k},$ (20)

where $\hat{k}=-\frac{1}{2}\partial_{\perp i}\Delta x^{i}_{\perp}$. This
formula is the relativistic generalization of Kaiser formula which includes
the new relativistic corrections. These corrections will become important on
the large scales where general relativity and other models may deviate from
each other. These corrections should be considered in the future surveys. On
these scales we are constrained by the cosmic variance. Detectability of these
new relativistic corrections is discussed in Yoo-test . The method proposed to
overcome cosmic variance is given by multi tracer method which uses the fact
that different biased tracers trace the same underlying density field Seljak .

We check how a pure gradient mode affects the $\delta V$ in the relativistic
formulation. Consider the case of a matter dominated universe with sources and
the observers comoving with the cosmic fluid. In the presence of the long mode
we have $v^{i}=-\frac{2}{3}\frac{\sqrt{a}k^{i}}{H_{o}}$ and
$v_{o}^{i}=-\frac{2}{3}\frac{1}{H_{o}}$. In the Newtonian gauge each term in
(20) is given by

$\displaystyle\partial_{\lVert}\Delta
x_{\lVert}=\frac{5}{3}\chi-\frac{3}{2}\sqrt{a}(-1+\frac{2}{3H_{o}}),$ (21)
$\displaystyle\frac{2}{\chi}\Delta
x_{\lVert}=2\chi+2\sqrt{a}k_{\lVert}=(\frac{2}{3H_{o}}-\chi)k_{\lVert}.$ (22)

Using (20) we find that

$\displaystyle\delta V=0.$ (23)

Consequently the number count observations are not changed by this long mode.
Since we observe galaxies and not the underlying dark matter density field, we
need a bias relation to relate $\delta_{g}^{t_{p}}$ in the synchronous gauge
to the dark matter perturbation $\delta^{t_{p}}$. As we observe on constant
redshift surfaces, this bias relation picks up a perturbation given by
$\delta_{g}^{t_{p}}=b\delta^{t_{p}}+\frac{d\ln n_{g}}{d\ln a}\delta\ln
a=b\delta^{t_{p}}+\frac{d\ln n_{g}}{d\ln a}\mathcal{T}(\boldmath n)$. The
$\mathcal{T}(\boldmath n)$ is the cosmic clock perturbation Jeong:2013psa .
The cosmic clock perturbation by the gradient mode vanishes as

$\displaystyle\mathcal{T}(\boldmath
n)=-\frac{1}{3}\varphi+v_{\lVert}-v_{o}=0,$ (24)

where it is assumed that observers are comoving with the cosmic fluid
$v_{\lVert}=-\frac{2}{3}\frac{\sqrt{a}k_{\lVert}}{H_{o}}$.

###  IV.1 Halo bias

The number density of halos, $n_{h}$, is biased by the presence of long mode
perturbations. In Press-Schechter method number density of small scale
perturbations in presence of long mode perturbations $(\nu_{b})$ is

$\displaystyle\mathcal{N}_{pk}(\nu_{s}|\nu_{b})d\nu_{s}=\frac{\mathcal{N}_{pk}(\nu_{s},\nu_{b})d\nu_{s}d\nu_{b}}{\mathcal{P}(\nu_{b})d\nu_{b}},$
(25)

where $\nu=\frac{\delta}{\sigma}$ in which $\sigma$ is the variance of matter
fluctuations, $\mathcal{N}_{pk}(\nu_{s},\nu_{b})$ is the number density of
peaks where the long mode has an amplitude $\nu_{b}\pm d\nu_{b}/2$ and
$\mathcal{P}$ is the distribution function. Long mode perturbations locally
act as a background. For long mode fluctuations equation (25) simplifies to

$\mathcal{N}_{pk}(\nu_{s}|\nu_{b})\backsimeq\mathcal{N}_{pk}(\nu_{p})$

where $\nu_{p}=\frac{\delta_{s}-\delta_{b}}{\sigma_{s}}$ simon . This will
lead to the bias relation for the fluctuations of peak number density given by
$\delta_{h}=b_{h}\delta_{b}$. This bias relation depends on $\sigma_{8}$ which
is the matter variance at $8$ Mpc. In this formulation long modes change the
threshold of the halo formation, $\delta_{c}$, which is obtained by the
spherical collapse model to $\delta_{c}-\delta_{b}$ where $\delta_{b}$ is the
long mode over density. We are interested in the effect of the pure gradient
modes on the halo bias. A long gradient mode does not change the bias relation
$b_{h}$ because $\delta_{b}=0$. However, on large scales gauge effects become
important. Changing the gauge will produce scale dependencies in the bias
relation. Thus, one needs to find a relativistic definition for the bias
relation. The synchronous gauge is assumed to be the right gauge. Since halo
formation is a local phenomena, it only depends on the dark matter
perturbations in this gauge. This gauge has been implemented for writing the
relativistic galaxy clustering. The long gradient By going to the Fermi normal
coordinates we can separate the local effects of long modes. These coordinates
can capture local effects for patches smaller than the Hubble length but fail
to describe the short mode dynamics when the initial conditions need to be
imposed. In this coordinate corrections start at quadratic order,
$h_{ij}x^{i}x^{j}$, which produce the tidal effects. It is shown that for the
long perturbations with spherical symmetry $h_{ij}$s are functions of second
derivatives of the metric perturbations. The bias relation defined in the
local Fermi normal coordinates depends on the second derivatives of the global
metric and reduces to the usual bias relation on the small scales Tobias . As
a result number density of halos is not affected by pure gradient modes.

The other complexity is that we observe on the slices of constant redshift.
This will produce another perturbation

$\delta_{h}(z)=b\delta^{t_{p}}+\frac{d\ln n_{h}}{d\ln a}\delta\ln
a=b\delta_{t_{p}}+\frac{d\ln n_{h}}{d\ln a}\mathcal{T}(\boldmath n)$

in the bias relation. As stated, cosmic clocks are not perturbed by pure
gradient modes. Hence, the observed bias relation is not changed.

###  IV.2 Galaxy power spectrum

Finally, we compute the effect of a pure gradient mode on the observed power
spectrum. We can compute the intrinsic power spectrum in a chosen gauge and
transform it back to the redshift space using the equations in Pajer:2013ana .
This relation is given by

$\displaystyle\tilde{\xi}(\tilde{r},z)=\left(1-a_{ij}\tilde{x}^{i}\partial^{j}+\mathcal{T}\partial_{\tilde{z}}\right)\xi(\tilde{r},\tilde{\tau}),$
(26)

where

$\displaystyle
a_{ij}=\mathcal{C}\hat{n}_{i}\hat{n}_{j}+\hat{n}_{(i}\mathcal{P}_{j)k}\mathcal{B}^{k}+\mathcal{P}_{ik}\mathcal{P}_{jl}\mathcal{A}^{kl}.$
(27)

since $\mathcal{C},\mathcal{B}^{k},\mathcal{A}^{kl}$ and $\mathcal{T}$ do not
change by the pure gradient presence, the observed large scale power spectrum
is not affected.

##  V discussion and conclusion

The vanishing effect of gradient modes can not be deducted from the
equivalence principle. It is shown that the local physical effects of long
mode perturbations start at $(k_{L}x)^{2}$ ( called tidal terms) if one uses
conformal Fermi coordinates as implied by the equivalence principle
Dai:2015rda . As a result the pure gradient mode does not contribute to the
tidal term at linear order. However, to calculate the observables at late
time, we have to transform from the conformal Fermi coordinates to the
observed coordinates. Observed quantities change when they are mapped to the
redshift space by terms which are proportional to the cosmic clocks and cosmic
rulers Pajer:2013ana . The equivalence principle does not imply that cosmic
clocks and rulers are not affected by the presence of a pure gradient mode.

We have studied that the effect of a pure gradient mode on the large scale
observables. First, we have shown that the contributions of gradient mode to
all cosmic ruler observables vanishes. This shows that the existing
relativistic formulation of observables encompasses all the effects including
observer’s velocity and displacement. We have shown that different projection
effects cancel each other in the relativistic galaxy clustering. Finally, we
confirmed that the observed power spectrum does not change by the presence of
the long mode.

Acknowledgments:

I would like to thank Hassan Firouzjahi, Ali Akbar Abolhasani and Reza
Mansouri for useful discussions and comments.

## References

  * (1) G. F. R. Ellis, Class. Quant. Grav. 28, 164001 (2011); T. Buchert and S. Räsänen, Ann. Rev. Nucl. Part. Sci. 62, 57 (2012); M. Parsi Mood, Javad T. Firouzjaee and Reza Mansouri, Phys. Rev. D 88, 083011 (2013); Rahim Moradi, Javad T. Firouzjaee, Reza Mansouri, Class. Quant. Grav. 32, no. 21, 215001 (2015); Reza Javadinezhad, Javad T. Firouzjaee and Reza Mansouri, arXiv:1510.04429 [gr-qc].
  * (2) D. Jeong and F. Schmidt, Class. Quant. Grav. 32 (2015) 4, 044001 [arXiv:1407.7979 [astro-ph.CO]].
  * (3) J. Yoo, A. L. Fitzpatrick and M. Zaldarriaga, Phys. Rev. D 80, 083514 (2009) [arXiv:0907.0707 [astro-ph.CO]]; J. Yoo, Phys. Rev. D 82, 083508 (2010) [arXiv:1009.3021 [astro-ph.CO]].
  * (4) C. Bonvin and R. Durrer, Phys. Rev. D 84, 063505 (2011) [arXiv:1105.5280 [astro-ph.CO]]; C. Bonvin, Class. Quant. Grav. 31, no. 23, 234002 (2014) [arXiv:1409.2224 [astro-ph.CO]].
  * (5) A. Challinor and A. Lewis, Phys. Rev. D 84, 043516 (2011) [arXiv:1105.5292 [astro-ph.CO]].
  * (6) P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 571, A23 (2014) [arXiv:1303.5083 [astro-ph.CO]].
  * (7) J. Hoftuft, H. K. Eriksen, A. J. Banday, K. M. Gorski, F. K. Hansen and P. B. Lilje, Astrophys. J. 699, 985 (2009) [arXiv:0903.1229 [astro-ph.CO]].
  * (8) A. L. Erickcek, M. Kamionkowski and S. M. Carroll, Phys. Rev. D 78, 123520 (2008) [arXiv:0806.0377 [astro-ph]].
  * (9) M. H. Namjoo, A. A. Abolhasani, S. Baghram and H. Firouzjahi, JCAP 1408, 002 (2014) [arXiv:1405.7317 [astro-ph.CO]].
  * (10) S. Weinberg, Phys. Rev. D 67, 123504 (2003) [astro-ph/0302326].
  * (11) K. Hinterbichler, L. Hui and J. Khoury, JCAP 1208 (2012) 017 [arXiv:1203.6351 [hep-th]].
  * (12) P. Creminelli, C. Pitrou and F. Vernizzi, JCAP 1111, 025 (2011) [arXiv:1109.1822 [astro-ph.CO]].
  * (13) M. Mirbabayi and M. Zaldarriaga, JCAP 1503 (2015) 03, 056 [arXiv:1409.4777 [astro-ph.CO]].
  * (14) P. Creminelli and M. Zaldarriaga, JCAP 0410, 006 (2004) [astro-ph/0407059].
  * (15) N. Kaiser, Mon. Not. R. Astron. Soc. 227, 1 (1987).
  * (16) J. Yoo, Class. Quant. Grav. 31 (2014) 234001 [arXiv:1409.3223 [astro-ph.CO]].
  * (17) J. M. Maldacena, JHEP 0305 (2003) 013 [astro-ph/0210603].
  * (18) E. Barausse, S. Matarrese and A. Riotto, Phys. Rev. D 71, 063537 (2005) [astro-ph/0501152].
  * (19) J. Yoo, Phys. Rev. D 90, no. 12, 123507 (2014) [arXiv:1408.5137 [astro-ph.CO]].
  * (20) T. Tanaka and Y. Urakawa, JCAP 05 (2011) 014.
  * (21) D. Wands, K. A. Malik, D. H. Lyth and A. R. Liddle, Phys. Rev. D 62, 043527 (2000) [astro-ph/0003278].
  * (22) L. Dai, E. Pajer and F. Schmidt, arXiv:1502.02011 [gr-qc].
  * (23) T. Baldauf, U. Seljak, L. Senatore and M. Zaldarriaga, JCAP 1110, 031 (2011) [arXiv:1106.5507 [astro-ph.CO]].
  * (24) D. Jeong and F. Schmidt, Phys. Rev. D 89 (2014) 4, 043519 [arXiv:1305.1299 [astro-ph.CO]].
  * (25) G F R Ellis, R Maartens and M A H MacCallum, _Relativistic Cosmology_ (Cambridge University Press).
  * (26) D. Jeong, F. Schmidt and C. Hirata, Phys. Rev. D 85 (2012), 023504
  * (27) E. Pajer, F. Schmidt and M. Zaldarriaga, Phys. Rev. D 88 (2013) 8, 083502 [arXiv:1305.0824 [astro-ph.CO]].
  * (28) L. Dai, D. Jeong and M. Kamionkowski, Phys. Rev. D 88 (2013) 4, 043507 [arXiv:1306.3985 [astro-ph.CO]].
  * (29) Y. Urakawa and T. Tanaka, Phys. Rev. D 82, 121301 (2010) [arXiv:1007.0468 [hep-th]].
  * (30) F. Schmidt and D. Jeong, Phys. Rev. D 86, 083527.
  * (31) J. Yoo, N. Hamaus, U. Seljak and M. Zaldarriaga, Phys. Rev. D 86, 063514,(2012).
  * (32) U. Seljak, Phys. Rev. Lett. 102, 021302 (2009).
  * (33) C. Bonvin, R. Durrer and M. A. Gasparini, Phys. Rev. D 73, 023523 (2006) [Phys. Rev. D 85, 029901 (2012)] [astro-ph/0511183].
  * (34) Mo H, van den Bosch FC, White S. 2010. Galaxy Formation and Evolution (Cambridge University Press)