#  Time Non-locality in Dark Matter and LSS

Arhum Ansari  Arka Banerjee  Sachin Jain  and Shaunak Padhyegurjar

###### Abstract

We explore the intriguing phenomenon of time non-locality in the evolution of
dark matter and Large Scale Structure (LSS). Recently in [1], it was shown
that time non-locality emerges in bias tracer fluctuations, which are $SO(3)$
scalars in real space, at fifth order in the perturbation expansion in dark
matter overdensity. We demonstrate that by breaking the symmetry down to
$SO(2)$, which is the case whenever line-of-sight effects become important,
such as for flux fluctuations in the Lyman $\alpha$ forest, the temporal non-
locality appears at the third order in expansion. Additionally, within the
framework of EFTofLSS, we demonstrate that time non-locality manifests in the
effective stress tensor of dark matter, which is a second rank tensor under
$SO(3)$ transformations, again at the third order in dark matter overdensity.
Furthermore, we highlight the effectiveness of the standard $\Pi$ basis [2] in
handling time non-local operators.

##  1 Introduction

The Standard Model of cosmology includes an inflationary phase of accelerated
expansion, leading to homogeneity and isotropy on large scales, while at the
same time giving rise to small inhomogeneities from quantum fluctuations.
These inhomogeneities are reflected in the anisotropy of the cosmic microwave
background (CMB) [3, 4, 5] , and also serve as initial density perturbations
that grow during matter domination era and give rise to large scale structures
(LSS). The evolution of LSS perturbations, therefore, contains information
about the initial spectrum of fluctuations but is also sensitive to the
relative ratios and properties of energy components that drive the subsequent
expansion, including Dark Energy (DE) and Dark Matter (DM). Understanding the
distribution of LSS is fundamental to our understanding of very early Universe
physics, as well to discern the ultimate fate of the Universe.

Due to the three-dimensional nature, the number of observable modes in LSS is
larger compared to the CMB and therefore is, in principle, statistically more
constraining on the parameters of interest. However, since gravitational
evolution drives the perturbations into the nonlinear regime on small scales,
accurate theoretical modeling down to these scales remains a challenge. The
equations governing the dynamics of dark matter are the Boltzmann equations in
an expanding background. One approach has been to use computationally
intensive $N$-body simulations [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18] of the dark matter phase space, under the assumption that DM is the major
clustering component, subsequently driving the evolution of the baryon
clustering. These studies have been extended beyond $\Lambda$CDM, see for e.g.
[19, 20, 21]. In a (hypothetical) Universe with DM as the only clustering
component, these simulations are accurate down to the scales much smaller than
the highly nonlinear, virialized halo structures.

The second approach to treating gravitational clustering beyond linear order
is to use higher-order perturbation theory. In the Standard Perturbation
Theory [22, 23, 24, 25, 26, 27], the relevant equations are solved order by
order using a series ansatz. However, even for quasilinear wavenumbers where
the perturbation expansion should be technically valid, this approach is
unable to reproduce the matter power spectrum computed through simulations.
Therefore, principles of Effective Field Theories (EFT) have been applied to
the context of the LSS perturbations to model the quasi-linear scales [28, 29,
30, 31, 32, 33, 34, 35, 36, 37]. In this approach, termed the EFT of LSS
(EFTofLSS), we integrate out the short scale modes in SPT. The process of
integrating out short modes generates an effective stress tensor in the
equations of motion. The stress tensor incorporates all the short-scale
physics which is needed to describe the distribution of dark matter at quasi-
linear scales. Various correlation functions of the density fluctuations can
then be calculated within the framework. Predictions of EFTofLSS have been
benchmarked against full simulations to a high degree of accuracy on
quasilinear scales, and these techniques have now been applied to various
datasets to extract state-of-the-art constraints on various cosmological
parameters [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52].

In the perturbative approach, baryonic matter such as galaxies are assumed to
be tracers of the underlying dark matter field. The overdensity for tracers is
given as a “bias expansion" [53], in terms of the dark matter fields. In most
cases, the expansion is assumed to be local in time - i.e. the overdensity of
tracers at some time $t$ is written in terms of dark matter operators at the
same time. However, it has been argued that since galaxies are not formed
instantaneously — the formation time is of order $(1/10)^{\text{th}}$ of the
evolution timescales of the dark matter field itself — the bias expansion
should be non-local in time [54]. Recent investigations [1] have shown that,
indeed, time non-locality is an essential feature of the bias expansion at the
fifth order. Technically, it was shown that when writing a bias expansion
consisting of zero derivative $SO(3)$ invariant operators of dark matter
fields in real space, the local and the non-local basis differ at the fifth
order in the perturbative expansion. This indicates that the physics of biased
tracers cannot be described accurately by local expansion beyond the fourth
order in perturbation theory.

While the bias expansion for the clustering of tracers in real space is
written in terms of $SO(3)$ invariant operators, there are observationally
relevant cases in cosmology when additional $SO(2)$ invariant terms must also
be included in the expansion. Such cases typically arise when the line-of-
sight (LoS) direction becomes special, therefore breaking the $SO(3)$
symmetry. This is exactly what happens while considering the clustering of
tracers in redshift space [55] — the main observable of spectroscopic surveys
such as DESI [56]. Another case where such operators are essential is in the
bias expansion of the flux fluctuations in the Lyman-$\alpha$ forest [57]. It
is, therefore, interesting to investigate the issue of time non-locality for
such $SO(2)$ invariant operators. This is the first such study, and we find
that time non-locality appears at the third order for these operators, as
opposed to the the fifth order for the $SO(3)$ invariant operators. This
implies that time non-locality may be detected in the 1-loop power spectrum
for observables for which selection operators need to be included. In
contrast, for the $SO(3)$ bias expansion, time non-locality can only be
detected at the two-loop power spectrum of the observables.

Another question that arises is whether the issue of time non-locality plays
any role in the evolution of the dark matter field itself within the context
of EFTofLSS. For the EFT approach to work, a separation of scales is needed
between the scales at which we want predictions (i.e. the quasilinear scales)
and the scales at which new (nonlinear) physics kicks in. The wavenumber at
which the density perturbations become of $\mathcal{O}$(1) is known as the
nonlinear wavenumber ($k_{\text{NL}}$). This, therefore, defines a natural
scale in the space domain and makes it possible to define an effective field
theory (EFTofLSS), by integrating out the modes above the non-linear scale.
The presence of a scale separation in space ensures that integrating out the
short modes gives a space local effective theory. The local EFT has a
derivative expansion that is regulated by the non-linear scale
($k_{\text{NL}}$). However, the typical timescales associated with the
evolution of both the large scales and the small scales, are similar
$\mathcal{O}(H^{-1})$. Therefore, interestingly, there is no scale separation
in the time domain. Therefore, integrating out the short modes generically
yields a time non-local EFT. This time non-locality manifests itself in the
governing equations of LSS through the effective stress tensor in the
equations of motion for dark matter [32, 58, 33, 34, 36]. The non-locality in
stress tensor is because there is no hierarchy in evolution timescales between
the long and the short modes. We have investigated the structure of the time
non-local stress tensor beyond linear order. We have found that it differs
from the time local stress tensor at third order. This is due to the tensorial
nature of the stress tensor $\tau_{ij}$ —- for scalars such as
$\partial_{i}\partial_{j}\tau_{ij}$ the difference between the local-in-time
and non-local-in-time expressions is at the fourth order.

The paper is organised as follows. In Sec. 2, we review the basic equations of
SPT and biased tracers relevant to our discussion. We describe an alternate
system ($\Pi$-system) of writing operators in the bias expansion and also
discuss the structure of selection operators. We use this section to also set
up the conventions and notations that will be used throughout the paper. In
Sec. 3, we discuss the EFTofLSS and the effective stress tensor. Then, we
discuss how time non-locality appears in dark matter and lay out the details
of getting the non-local stress tensor up to third order. Then, in Sec. 4, we
discuss time non-locality in the bias expansion of $SO(3)$ invariant
operators. We also discuss the invertible mapping between the non-local and
$\Pi$-basis. In Sec. 5, we discuss bias expansion with the $SO(2)$ invariant
(selection) operators and the time non-local basis upto fourth order. Finally,
in Sec. 6, we conclude by stating our results and giving some future
directions. In appendices A and B we discuss the vector operator structure for
dark matter and renormalisation of the flux field in Lyman $\alpha$
respectively.

##  2 Background details

In this section, we will discuss the fluid equations governing the dynamics of
dark matter at large scales. We will also discuss the SPT approach to solve
the fluid equations (Sec.2.1). We will then discuss how tracers (galaxies) are
modelled and will describe the bias expansion for $SO(3)$ (Sec.2.2) and
$SO(2)$ (Sec.2.3) invariant operators. In the end of this section, we will
describe an alternate system ($\Pi$ system) for writing operators in the bias
expansion which will be relevant for later discussion on time non-locality.

###  2.1 Dark-Matter: Standard Perturbation Theory (SPT)

Dark matter at large scales has been shown to behave as an effective fluid
following the Vlasov-Poisson equation which is the dynamical equation
describing collisionless matter coupled with gravity [28, 29]. In the single
stream approximation i.e. for a vanishing velocity dispersion, the Boltzmann
hierarchy can be truncated after the first moment and we end up with the
continuity and Euler equation for the dark matter overdensity and velocity
field. The continuity and Euler equation for collisionless fluid is given as

$\displaystyle\delta^{\prime}(\vec{x},\eta)$
$\displaystyle+\nabla\cdot[\vec{v}(\vec{x},\eta)(\delta(\vec{x},\eta)+1)]=0\;,$
$\displaystyle v_{i}^{\prime}(\vec{x},\eta)+$
$\displaystyle\mathcal{H}v_{i}(\vec{x},\eta)+(\vec{v}(\vec{x},\eta).\nabla)v_{i}(\vec{x},\eta)=-\nabla_{i}\Phi(\vec{x},\eta)\;,$
(2.1)

where ′ denotes derivative w.r.t the conformal time $\eta$ and
$\delta\equiv(\rho-\bar{\rho})/\bar{\rho}$ is the dark matter overdensity and
$\vec{v}$ is the velocity field. Here $\rho$ and $\bar{\rho}$ denote the
matter density and it’s average respectively. The system of equations given in
Eq.(2.1) along with the Poisson equation for gravitational potential

$\displaystyle\nabla^{2}\Phi=\frac{3}{2}\Omega_{m}(\eta)\mathcal{H}^{2}(\eta)\delta(\vec{x},\eta)\;,$
(2.2)

closes the system. Here $\mathcal{H}$ is the Hubble parameter expressed in
conformal time. In the standard perturbation theory (SPT), the system of
equations in Eq.(2.1) is solved by a series ansatz in momentum space as,

$\displaystyle\delta(k,\eta)=\sum_{n=1}^{\infty}D^{n}(\eta)\delta^{(n)}(k)\;,$
$\displaystyle\theta(k,\eta)=-\mathcal{H}f\sum_{n=1}^{\infty}D^{n}(\eta)\theta^{(n)}(k)\;,$
(2.3)

where $D(\eta)$ is the solution of the linear equation for $\delta$ given in
Eq.(2.1) and $f\equiv(d\ln D/d\ln a)$ is the logarithmic growth factor. At
each order $\delta^{(n)}(k)$ and $\theta^{(n)}(k)$ are defined as a
convolution over linear overdensity in momentum space,

$\displaystyle\delta^{(n)}(\vec{k})=\int_{\vec{q}_{1}...\vec{q}_{n}}(2\pi)^{3}\delta^{3}(\vec{k}-\vec{q}_{1,n})F_{n}(\vec{q}_{1},\vec{q}_{2},...\vec{q}_{n})\delta^{(1)}_{\vec{q}_{1}}\delta^{(1)}_{\vec{q}_{2}}...\delta^{(1)}_{\vec{q}_{n}}\;,$
$\displaystyle\theta^{(n)}(\vec{k})=\int_{\vec{q}_{1}...\vec{q}_{n}}(2\pi)^{3}\delta^{3}(\vec{k}-\vec{q}_{1,n})G_{n}(\vec{q}_{1},\vec{q}_{2},...\vec{q}_{n})\delta^{(1)}_{\vec{q}_{1}}\delta^{(1)}_{\vec{q}_{2}}...\delta^{(1)}_{\vec{q}_{n}}\;,$
(2.4)

where $\vec{q}_{1,n}=\sum_{i=1}^{n}\vec{q}_{i}$ and $F_{n},G_{n}$ are kernel
derived recursively from Eq.(2.1) order by order. Formally, SPT might be
expected to work as long as the perturbations $\delta\lesssim\mathcal{O}(1)$.
The scale, or wavenumber at which typical fluctuations are $\mathcal{O}(1)$ is
referred to as the non-linear scale and denoted by $k_{\rm NL}$. However, by
comparing to simulations, it has been shown that the SPT treatment breaks down
for wavenumbers well below $k_{\rm NL}$. This led to the development of
EFTofLSS which correctly incorporates the effective treatment of short-scale
physics and matches well with data. However, for what follows for the next
couple of sections, we do not need to consider EFTofLSS for the evolution of
Dark Matter. The SPT kernels introduced here will be sufficient for the
analysis and we will discuss the modifications of the Dark Matter equations in
the context of EFTofLSS in Sec.3.1.

###  2.2 Bias-Expansion

Baryonic matter, such as galaxies, traces the underlying dark matter field as
a result of which the number density of these "tracers" is correlated with the
dark matter overdensity. Due to the absence of an equation of motion for
tracers, there is no first principle way to get the tracer overdensity.
However, we know that the tracer overdensity can be affected by dark matter
fields only locally and therefore, can only depend on the locally measurable
quantities constructed out of the dark matter fields. Due to rotational
invariance and the equivalence principle, the locally measurable quantities
are the tidal field ($\partial_{i}\partial_{j}\Phi$) and velocity gradient
($\partial_{i}v_{j}$). Tracer overdensity is a scalar quantity, therefore it
can depend only on scalars constructed out of the tidal field and velocity
gradient. In general, the relation between galaxy number overdensity and dark
matter overdensity can be expressed with the help of a function $G$ such that
the galaxy overdensity ($\delta_{g}(\vec{x},t)$) is given as [54, 59],

$\displaystyle\delta_{g}(\vec{x},t)\equiv\frac{n_{g}-\bar{n}_{g}}{\bar{n}_{g}}=G\Bigg{(}\partial_{i}\partial_{j}\Phi(\vec{x},t),\partial_{i}v_{j}(\vec{x},t)\Bigg{)}\;,$
(2.5)

where $n_{g}$ is the number density of galaxies and the bar represents the
mean values. The function $G$ is such that it vanishes when all the arguments
vanish. It contains higher derivative as well as stochastic terms, which we
have suppressed. We can Taylor expand Eq.(2.5) about zero and get the
following expansion,

$\displaystyle\delta_{g}(\vec{x},t)=b_{\partial^{2}\Phi}(t)\partial^{2}\Phi(\vec{x},t)+b_{\partial_{i}v_{i}}(t)\partial_{i}v_{i}(\vec{x},t)+b_{\partial_{i}\partial_{j}\Phi\partial_{i}v_{j}}(t)\partial_{i}\partial_{j}\Phi\partial_{i}v_{j}(\vec{x},t)+...\;,$
(2.6)

where $b$’s are time-dependent coefficients also called "bias" coefficients
which can be measured from simulation or galaxy data by galaxy-galaxy
correlation or galaxy-dark matter cross correlations. We have just written the
leading order terms in Eq.(2.6) and not the terms containing higher
derivatives which are also present111In this work we will not be considering
higher derivative and stochastic terms in the bias expansion for tracers.. We
can write Eq.(2.6) in a more familiar and compact form as,

$\displaystyle\delta_{g}(\vec{x},t)=\sum_{\mathcal{O}_{a}}b_{\mathcal{O}_{a}}(t)\mathcal{O}_{a}(\vec{x},t)\;,$
(2.7)

where $b_{\mathcal{O}_{a}}(t)$ are time-dependent bias coefficients and
$\mathcal{O}_{a}$ are scalar operators made of long wavelength dark matter
fields that are allowed by the symmetry i.e. $SO(3)$ invariance and the
equivalence principle. Therefore, the operators $\mathcal{O}_{a}$ are $SO(3)$
scalars constructed from dimensionless tidal field and velocity gradients
defined as,

$\displaystyle
r_{ij}\equiv\frac{2\partial_{i}\partial_{j}\Phi}{3\Omega_{m}a^{2}H^{2}}\hskip
56.9055ptp_{ij}\equiv-\frac{D}{a\dot{D}}\partial_{i}v_{j}\;,$ (2.8)

where $\Omega_{m}$, $a$, and $H$ are matter fraction, scale factor, and Hubble
parameter respectively. $D$ is the linear growth factor defined below
Eq.(2.1). We can construct $SO(3)$ invariant operators by taking products of
the operators given in Eq.(2.8) contracting all indices to make an $SO(3)$
scalar as,

$\displaystyle\delta\equiv\delta_{ij}^{K}r_{ij},\hskip
14.22636pt\theta\equiv\delta_{ij}^{K}p_{ij},\hskip 14.22636ptr^{2}\equiv
r_{ij}r_{ji}\;.$ (2.9)

At each order in dark matter overdensity, the allowed set of operators are all
possible $SO(3)$ scalars constructed from $r_{ij}$ and $p_{ij}$ in the manner
shown in Eq.(2.9) which are relevant at that order. Both the tidal field and
velocity gradient start at linear order, so an operator such as $r^{2}$ or
$rp$ will start at second order in dark matter overdensity and hence will be
relevant at second order and not at lower orders. This implies that, at each
order, there will be a finite number of operators appearing in the bias
expansion.

An operator $\mathcal{O}_{a}$ can contribute to any given order $n$ if it
starts at an order greater than or equal to $n$. For example, the operator
$rp$ can be evaluated at order three as,

$\displaystyle(rp)^{(3)}=r^{(1)}p^{(2)}+r^{(2)}p^{(1)}\;,$ (2.10)

where each operator is evaluated at the same point $\vec{x}$. We can express
Eq.(2.10) in momentum space and use Eq.(2.8) and Eq.(2.1) to write the kernel
for $(rp)^{(3)}$. Similarly, in momentum space, any operator $\mathcal{O}_{a}$
that occurs in Eq.(2.7) at $n^{th}$ order can be written as convolution over
linear dark matter fields with a kernel $K^{\mathcal{O},a}_{n}$ as,222Let us
note that to compute the kernels for operators we have used (2.1) which are
the solutions of SPT.

$\displaystyle\mathcal{O}_{a}^{n}(\vec{k})=\int_{\vec{q}_{1}...\vec{q}_{n}}(2\pi)^{3}\delta^{3}(\vec{k}-\vec{q}_{1,n})K^{\mathcal{O},a}_{n}(\vec{q}_{1},\vec{q}_{2},...\vec{q}_{n})\delta^{(1)}_{\vec{q}_{1}}\delta^{(1)}_{\vec{q}_{2}}...\delta^{(1)}_{\vec{q}_{n}}\;,$
(2.11)

where $\vec{q}_{1,n}=\sum_{i=1}^{n}\vec{q}_{i}$ and
$K^{\mathcal{O},a}_{n}(\vec{q}_{1},\vec{q}_{2},...\vec{q}_{n})$ are rational
functions of the momenta $\vec{q}_{i}$. So essentially, in momentum space the
difference in the operators appearing in the bias expansion as given in
Eq.(2.7), comes through their kernels. However, it can be shown that not all
operators at a given order are independent. For example at the second order,
we have the following set of operators that contribute to the bias expansion,

$\displaystyle\\{\delta^{(2)},\theta^{(2)},(\delta^{2})^{(2)},(\delta\theta)^{(2)},(\theta^{2})^{(2)},(r^{2})^{(2)},(rp)^{(2)},(p^{2})^{(2)}\\}\;.$
(2.12)

However, in momentum space, there exists a linear relation satisfied by the
following four operators,

$\displaystyle\delta^{(2)}-\theta^{(2)}-\frac{2}{7}(\delta^{2}-r^{2})^{(2)}=0\;,$
(2.13)

which implies that one of these operators is degenerate and its kernel can be
written in terms of the other three. Hence, it should not appear in the bias
expansion given in Eq.(2.7) with an independent bias coefficient. In this way,
we can remove all the degenerate operators from the bias expansion if a linear
relation exists among subsets of the operators. This will lead us to an
independent set of operators that constitutes the "basis" of the operator
space at any given order.

###  2.3 Selection operators

In this section we introduce some new $SO(2)$ invariant operators, which we
call selection operators, in the bias expansion as given in Eq.(2.7). We will
explain the mathematical structure for these operators and discuss physical
situations in which they may naturally arise.

We will show below that selection operators have distinct structures compared
to the usual $SO(3)$ operators considered in Eq.(2.7). This implies that
selection operators appear with independent coefficients in the bias expansion
of tracers. So now the bias expansion takes the following form,

$\displaystyle\delta_{g}(\vec{x},t)=\sum_{a}b_{\mathcal{O}_{a}}(t)\mathcal{O}_{a}(\vec{x},t)+\sum_{m}c_{\mathcal{P}_{m}}(t)\mathcal{P}_{m}(\vec{x},t)\;,$
(2.14)

where $c_{\mathcal{P}_{m}}$ are new bias coefficients and
$\mathcal{P}_{m}(\vec{x},t)$ are new operators that are invariant under
rotation about a vector, which is generally taken to be the direction of line-
of-sight (LoS). Due to the presence of a LoS, the symmetry is reduced to
$SO(2)$ invariance along the LoS. $SO(2)$ invariance and the equivalence
principle ensures that $\partial_{i}\partial_{j}\Phi$ and $\partial_{i}v_{j}$
and their projections along the line of sight are also locally observable
quantities. Therefore, selection operators are constructed from the same
building blocks as the $SO(3)$ operators i.e. the tidal field and the velocity
gradient as given in Eq.(2.8). From these we construct two indexed objects
which are then contracted with unit vectors ($\hat{z}$) to make $SO(2)$
scalars such as,

$\displaystyle\delta_{z}\equiv r_{ij}\hat{z}^{i}\hat{z}^{j},\hskip
14.22636pt\eta\equiv p_{ij}\hat{z}^{i}\hat{z}^{j},\hskip
14.22636pt(rp)_{\parallel}\equiv r_{ij}p_{jk}\hat{z}^{i}\hat{z}^{k}\;,$ (2.15)

where $\hat{z}$ is the unit vector along the line of sight. The operators that
appear in bias expansion will consist of all such operators as given in
Eq.(2.15) and their products.

Note that the $SO(2)$ scalars, being a projection of $SO(3)$ tensors allow for
more operators to be independent of each other which enhances the operator
space. For example, consider $r_{ij}$ and $\delta^{K}_{ij}\delta$ which are
same operator, $\delta$ if we try to construct $SO(3)$ scalars by taking
trace. However, if we construct $SO(2)$ scalars by taking projections as
$r_{ij}\hat{z}^{i}\hat{z}^{j}$ and
$(\delta_{ij}^{K}\delta)\hat{z}^{i}\hat{z}^{j}$, then they are independent
operators and occur with independent bias coefficients as given in Eq.(2.14).
Due to this fact, one might expect the time non-locality to appear at lower
order. We have found that this is indeed the case and we have discussed this
in detail in Sec. 5.

Physically, these selection operators are crucial while comparing the
theoretical predictions with observations. This is due to the fact that we
observe galaxies through the light that reaches us. The probability that we
will observe a photon emitted from a galaxy in a particular direction depends
on the line-of-sight of observation. These effects can be incorporated in the
bias expansion by including new line of sight dependent operators [55] which
are precisely the selection operators given in Eq.(2.14). In the context of
galaxies, a subset of these selection operators appear when we move to galaxy
clustering in redshift space.

However, in other contexts such as when writing an EFT for the flux field from
Lyman-$\alpha$ forest, we need to include all possible selection operators
[57] in the bias expansion of the flux fluctuations. This happens due to the
non-linear mapping between the flux field ($F$) and the optical depth
($\tau$), which is a biased tracer of the underlying dark matter field. The
non-linear mapping is given by an exponential as $F\propto e^{-\tau}$. Owing
to this exponential map, a "renormalised" EFT of flux overdensity field,
$\delta_{F}=(F-\bar{F})/\bar{F}$, with the optical depth taken to be in
redshift space, contains all possible selection operators. We have commented
on this in a bit more detail in Appendix. B. This way of generating all
possible selection operators is termed as the "top-down" approach to writing
EFTs as opposed to the "bottom-up" approach that we have discussed so far.

###  2.4 $\Pi$ basis

In this section, we describe an alternate system of writing operators
constructed out of long wavelength dark matter fields, the $\Pi$ system [2].
However, the effectiveness of $\Pi$ system will be shown in capturing the time
non-locality in biased tracers or dark matter as discussed later in Secs. 4
and 3 respectively. The usual method to compute the set of independent non-
local operators for biased tracers is discussed in [60, 1]. This method is
very tedious and contains a lot of redundancies. However, we point out that
when working with the $\Pi$ system [2], the calculation of determining the set
of independent non-local operators is much more direct and efficient.

For biased tracers of the dark matter field, such as halos and galaxies, [2]
proposed a system for writing the relevant operators under the assumption that
galaxy overdensity field follows the same continuity equation as the dark
matter overdensity. We will use this system to write down the EFT operators
appearing in the effective stress tensor relevant for the dark matter field
itself and also to write the operators appearing in the bias expansion for
tracers. The reason to do this is that they have a nice connection with time
non-locality which will be discussed in later sections. In this system we
define

$\Pi_{ij}^{[1]}\equiv r_{ij}.$ (2.16)

We also define [2],

$\displaystyle\Pi_{ij}^{[n]}=\frac{1}{(n-1)!}\Bigg{[}(\mathcal{H}f)^{-1}\frac{D}{D\eta}\Pi_{ij}^{[n-1]}-(n-1)\Pi_{ij}^{[n-1]}\Bigg{]}\;,$
(2.17)

where $f\equiv d\ln D/d\ln a$ and $\mathcal{H}$ are the logarithmic growth
factor and Hubble parameter (expressed in conformal time) respectively and
convective derivative is defined as

$\displaystyle\frac{D}{D\eta}\equiv\frac{\partial}{\partial\eta}+v^{i}\partial_{i}\;.$
(2.18)

Here the basic idea is, a convective derivative of a locally measurable
quantity is also locally measurable so we can act with convective derivative
on the tidal field and get another locally measurable operator333 Locally
measurable quantities are tidal field ($\partial_{i}\partial_{j}\Phi$) and
velocity gradient ($\partial_{i}v_{j}$) as a uniform acceleration given by
$\partial_{i}\Phi$ is not locally observable due to the equivalence
principle.. So in this system, instead of tidal field and velocity gradient as
given in Eq.(2.8), the building blocks for constructing $SO(3)$ scalars are
$\Pi_{ij}^{[n]}$ operators. Using (2.17), we can construct operators by taking
trace of $\Pi_{ij}^{[n]}$ as Tr$[\Pi_{ij}^{[1]}]=\Pi_{ii}^{[1]}$, or other two
indexed objects made from $\Pi^{[n]}$ as,
Tr$[\Pi^{[2]}\Pi^{[1]}]\equiv$Tr$[\Pi_{ij}^{[2]}\Pi_{jk}^{[1]}]=\Pi_{ij}^{[2]}\Pi_{ji}^{[1]}$
and products of the traces of individual $\Pi_{ij}^{[n]}$ appropriately.

##  3 Non-locality in dark matter

SPT has been sucessful in describing LSS at very large scales but as we
approach shorter distances i.e. higher wavenumber, linear SPT starts to
diverge from the results of fully nonlinear $N$-body simulations. Even for
$k<k_{\rm NL}$, i.e. quasilinear scales where $\delta<\mathcal{O}(1)$, higher
loop calculations of the SPT power spectrum do not converge to the correct
result [32]. To cure this, the correct treatment is to smooth out the
equations of motion over a length scale above the non-linear scale. The effect
of this smoothing out procedure is to include “effective" contributions from
non-linear scale which is the subject of EFTofLSS [29]. In this section, we
will discuss the governing equations for EFTofLSS and the effective stress
tensor which comes as result of defining the EFT. We will also discuss
significance of this effetcive stress tensor in the context of time non-
locality.

###  3.1 EFT of LSS

In this sub section, we continue the discussion outlined in Sec. 2.1. In order
to define an EFT pertaining to the equations of motion given in Eq.(2.1), we
need to integrate the short scale modes. After integrating out the short modes
in the fluid equations, we get,

$\displaystyle\delta_{l}^{\prime}(\vec{x},\eta)$
$\displaystyle+\nabla\cdot[\vec{v}_{l}(\vec{x},\eta)(\delta_{l}(\vec{x},\eta)+1)]=0\;,$
$\displaystyle v_{i,l}^{\prime}(\vec{x},\eta)+$
$\displaystyle\mathcal{H}v_{i,l}(\vec{x},\eta)+(\vec{v}_{l}(\vec{x},\eta).\nabla)v_{i,l}(\vec{x},\eta)=-\nabla_{i}\Phi_{l}(\vec{x},\eta)-\frac{1}{\rho_{l}}\partial_{j}\tau_{ij}(\vec{x},t)\;,$
(3.1)

where the subscript "$l$" indicates the long wavelength quantities and
$\tau_{ij}$ is an effective stress tensor generated from the smoothing and
contains all the information about short scales. The stress tensor regulates
the loops in SPT which access the non-linear scales through integrals in
Eq.(2.1) and keeps the theory finite. A generic expansion for the effective
stress tensor in terms of the long wavelength dark matter fields is given as,

$\displaystyle\tau_{ij}(\vec{x},t)=\sum_{\mathcal{O}^{m}_{ij}}c_{\mathcal{O}^{m}_{ij}}(t)\mathcal{O}^{m}_{ij}(\vec{x},t)\;,$
(3.2)

where $c_{\mathcal{O}^{m}_{ij}}(t)$ are EFT coefficients that are fixed by
data or simulations and $\mathcal{O}^{m}_{ij}$ are operators constructed from
locally observable long wavelength dark matter fields. Again for the case of
dark matter, locally observable fields are tidal fields
($\partial_{i}\partial_{j}\Phi$) and the velocity gradient
($\partial_{i}v^{j}$). So the operators appearing in Eq.(3.2) are constructed
by contracting these long wavelength local observable. For example, at lowest
order

$\displaystyle\tau_{ij}^{(1)}=c_{1}(t)\partial_{i}\partial_{j}\Phi^{(1)}+c_{2}(t)\delta^{K}_{ij}\partial_{k}\partial^{k}\Phi^{(1)}\;,$
(3.3)

where $c_{i}(t)$ are EFT coefficients and $\delta_{ij}^{K}$ is the Kronecker
delta function respectively. We can see from Eq.(3.2), that it is local in
time as both r.h.s and l.h.s are evaluated at the same time. To write down a
local effective theory for the long mode, it is important to note that there
should be a separation of length scale. However, for dark matter, even though
there is a separation of length scale in the spatial domain but there is no
separation of scale in temporal domain. This leads to time non-locality in the
effective description of the dark matter after integrating out the short
modes. To understand the absence of scale separation in the time domain, we
can also look at the linear form of Eq.(2.1). If we linearise the equations
given in Eq.(2.1) and substitute the Euler equation in the continuity
equation, we get the following linear growth equation for $\delta$,

$\displaystyle\delta^{\prime\prime}(\vec{k},\eta)+\mathcal{H}(\eta)\delta^{\prime}(\vec{k},\eta)-\frac{3}{2}\Omega_{m}(\eta)\mathcal{H}^{2}(\eta)\delta(\vec{k},\eta)=0\;,$
(3.4)

where ′ denotes a derivative w.r.t the conformal time. From Eq.(3.4), we can
see that there is no explicit $\vec{k}$ dependence and hence both the long and
short modes follow the same dynamics. This is another manifestation of the
fact that there is no scale separation in the time domain even though there is
a scale separation in the momentum domain through $k_{\text{NL}}$444 This can
also be seen from the discussion above Eq.(2.1), where the dark matter is
modelled as a collisionless (pressureless) fluid. Due to the absence of
pressure, the speed of sound vanishes in such a fluid. Hence, there does not
exist any dispersion relation which relates the wavenumber ($\vec{k}$) with
the frequency. However, after integrating out the short modes we get an
effective stress tensor on the r.h.s of Eq.(3.4). This introduces an effective
pressure and the wavenumber and frequency are related through a dispersion
relation.. Therefore, as mentioned earlier, when we smooth out the short modes
to write an EFT, the physics is local in space domain but non-locality is
introduced in the time domain. This time non-locality is incorporated in the
effective stress tensor which can be written as [32, 60],

$\displaystyle\tau_{ij}(x,t)=\sum_{\mathcal{O}^{m}_{ij}}\int^{t}dt^{\prime}c_{\mathcal{O}^{m}_{ij}}(t,t^{\prime})\mathcal{O}^{m}_{ij}(\vec{x}_{\text{fl}}(\vec{x},t,t^{\prime}),t^{\prime})\;,$
(3.5)

where $c_{\mathcal{O}^{m}_{ij}}(t,t^{\prime})$ are time dependent kernels and
$\vec{x}_{\text{fl}}$ is defined implicitly through the following relation,

$\displaystyle\vec{x}_{\text{fl}}(\vec{x},t,t^{\prime})=\vec{x}-\int_{t^{\prime}}^{t}dt^{\prime\prime}\vec{v}(t^{\prime\prime},\vec{x}_{\text{fl}}(\vec{x},t,t^{\prime\prime}))\;.$
(3.6)

where $\vec{x}_{\text{fl}}$ is the coordinate labelling the past trajectory of
the fluid element that ended up at position $\vec{x}$ at time $t$ and
$\vec{v}$ is the velocity of the fluid element along the past trajectory. Note
that Eq.(3.5) reduces to expansion given in Eq.(3.2) when
$c_{\mathcal{O}^{m}_{ij}}(t,t^{\prime})=c_{\mathcal{O}^{m}_{ij}}(t)\delta(t-t^{\prime})$.

As we can see from Eq.(3.5), that due to the time integration, Eq.(3.5) is
non-local in time as compared to Eq.(3.2) which is local in time. However, we
can use Eq.(3.6) iteratively in Eq.(3.5) and Taylor expand it around $\vec{x}$
to get $\mathcal{O}_{i}$ up to any arbitrary order. After expanding about
$\vec{x}$, we will still be left with the integral given in Eq.(3.6). Assuming
no vorticity i.e. $v^{i}\propto\partial^{i}\theta$, we can perform the
integral given in Eq.(3.6) and get an expression arranged in powers of
$\frac{D(t^{\prime})}{D(t)}$ [60, 1]. This process removes the time non-
locality but introduces space non-locality and the structure of operators
appearing in the expansion given in Eq.(3.5) changes [60].

Restricting ourselves upto order $n$ in dark matter overdensity, we get [1,
60],

$\displaystyle[\mathcal{O}^{m}_{ij}(\vec{x}_{\text{fl}}(\vec{x},t,t^{\prime}),t^{\prime})]^{(n)}=\sum_{\alpha=1}^{n-m+1}\left(\frac{D(t^{\prime})}{D(t)}\right)^{\alpha+m-1}\mathbb{C}_{\mathcal{O}^{m}_{ij},\alpha}^{(n)}(\vec{x},t)\
.$ (3.7)

where $\mathbb{C}_{\mathcal{O}^{m}_{ij},\alpha}^{(n)}$ are the new operators
stemming from each operator $\mathcal{O}^{m}_{ij}$ upon the Taylor expansion
along fluid trajectory. Substituting the form of
$(\mathcal{O}^{m}_{ij})^{(n)}$ from Eq.(3.7) into Eq.(3.5) we get the
following expansion that is local in time,

$\displaystyle\tau_{ij}(\vec{x},t)=\sum_{\mathcal{O}_{i}}\sum_{\alpha=1}^{n-m+1}c_{\mathcal{O}^{m}_{ij},\alpha}(t)\,\mathbb{C}^{(n)}_{\mathcal{O}^{m}_{ij},\alpha}(\vec{x},t)\;,$
(3.8)

where,

$\displaystyle
c_{\mathcal{O}^{m}_{ij},\alpha}(t)\equiv\int^{t}dt^{\prime}c_{\mathcal{O}^{m}_{ij}}(t,t^{\prime})\left(\frac{D(t^{\prime})}{D(t)}\right)^{\alpha+m-1}\;,$
(3.9)

are the time dependent bias coefficients similar to what is given in the local
expansion in Eq.(3.2).

|  |  |
---|---|---|---
| Local Basis | Non-local basis |  $\Pi$-basis
|  |  |
|  |  |
$1^{st}$ order | $r_{ij},\delta_{ij}^{K}\delta$ | $\mathbb{C}^{(1)}_{r_{ij},1},\mathbb{C}^{(1)}_{\delta_{ij}^{K}\delta,1}$ | $\Pi_{ij}^{[1]},\delta_{ij}^{K}\text{Tr}[\Pi^{[1]}]$
| (2) | (2) | (2)
|  |  |
$2^{nd}$ order | $r_{ij},p_{ij},r_{ij}\delta,(r^{2})_{ij}$ | $\mathbb{C}^{(2)}_{p_{ij},1},\mathbb{C}^{(2)}_{p_{ij},2},\mathbb{C}^{(2)}_{r_{ij},2},\mathbb{C}^{(2)}_{p_{ij}\delta,2}$ | $\Pi_{ij}^{[1]},\Pi_{ij}^{[1]}\text{Tr}[\Pi^{[1]}],\Pi_{ik}^{[1]}\Pi_{kj}^{[1]},\Pi_{ij}^{[2]}$
|  |  |
| $\delta_{ij}^{K}(\delta,\delta^{2},r^{2})$ | $\delta_{ij}^{K}(\mathbb{C}^{(2)}_{\delta,1},\mathbb{C}^{(2)}_{\delta,2},\mathbb{C}^{(2)}_{\theta,2})$ | $\delta_{ij}^{K}(\text{Tr}[\Pi^{[1]}],\text{Tr}[\Pi^{[1]}]^{2},$
|  |  | $\text{Tr}[\Pi^{[1]}\Pi^{[1]}])$
| (7) | (7) | (7)
|  |  |
$3^{rd}$ order | $r_{ij},p_{ij},p_{ij}\delta,p_{ij}\theta,r_{ij}\delta$ | $\mathbb{C}^{(3)}_{r_{ij},1},\mathbb{C}^{(3)}_{r_{ij},2},\mathbb{C}^{(3)}_{r_{ij},3},\mathbb{C}^{(3)}_{p_{ij},2}$ | $\Pi_{ij}^{[1]},\text{Tr}[\Pi^{[1]}]\Pi_{ij}^{[1]},\Pi_{ik}^{[1]}\Pi_{kj}^{[1]},\Pi_{ij}^{[2]},$
|  |  |
| $(r^{2})_{ij},(p^{2})_{ij},p_{ij}\delta^{2},(r^{3})_{ij},$ | $\mathbb{C}^{(3)}_{p_{ij},3},\mathbb{C}^{(3)}_{p_{ij}\delta,1},\mathbb{C}^{(3)}_{p_{ij}\delta,2},\mathbb{C}^{(3)}_{p_{ij}\theta,2},$ | $\Pi_{ij}^{[1]}\text{Tr}[\Pi^{[1]}]^{2},[\Pi^{[1]}\Pi^{[1]}\Pi^{[1]}]_{ij},$
| $(r^{2})_{ij}\delta$ | $\mathbb{C}^{(3)}_{p_{ij}\delta^{2},1}$ | $\text{Tr}[\Pi^{[1]}\Pi^{[1]}]\Pi_{ij}^{[1]}$
|  |  |
| $\delta_{ij}^{K}(\delta,\delta^{2},\delta\theta,r^{2},rp,\delta^{3},r^{3})$ | $\mathbb{C}^{(3)}_{r_{ij}\delta,2},\mathbb{C}^{(3)}_{(r^{2})_{ij},2},$ | $[\Pi^{[1]}\Pi^{[1]}]_{ij}\text{Tr}[\Pi^{[1]}],\text{Tr}[\Pi^{[1]}]\Pi_{ij}^{[2]},$
|  |  |
| (17) | $\delta_{ij}^{K}(\mathbb{C}^{(3)}_{\delta,1},\mathbb{C}^{(3)}_{\delta,2},\mathbb{C}^{(3)}_{\delta,3}$ | $[\Pi^{[1]}\Pi^{[2]}]_{ij},\Pi_{ij}^{[3]},\delta_{ij}^{K}(\text{Tr}[\Pi^{[1]}],$
|  |  |
|  | $\mathbb{C}^{(3)}_{\theta,2},\mathbb{C}^{(3)}_{\theta,3},\mathbb{C}^{(3)}_{\delta^{2},2},\mathbb{C}^{(3)}_{r^{2},2})$ | $\text{Tr}[\Pi^{[1]}]^{2},\text{Tr}[\Pi^{[1]}\Pi^{[1]}],\text{Tr}[\Pi^{[2]}],$
|  |  |
|  | (18) | $\text{Tr}[\Pi^{[1]}]^{3},\text{Tr}[\Pi^{[1]}\Pi^{[1]}\Pi^{[1]}],$
|  |  | $\text{Tr}[\Pi^{[1]}\Pi^{[1]}]\text{Tr}[\Pi^{[1]}])$
|  |  | (18)
|  |  |
Table 1: List of the local, non-local and $\Pi$ basis for the effective stress
tensor is shown. We can see that the local and non-local basis differ from
each other at third order. In writing local operators, we have used the
notation given in Eq.(2.8) and (2.9). The notation for writing non-local
operators has been taken from [1, 60], and is explained in Eqs.(3.5) and
(3.7).

Time non-locality enters the dynamics of long wavelength dark matter fields
through the stress tensor in Eq.(3.1). The effective stress tensor has the
local in time expansion as given in Eq.(3.2) and the non-local in time
expansion in Eq.(3.5). At each order, only a finite number of independent
operators appear in the expansion. These form the basis of operators at each
order. We have found that up to second order, the time local and time non-
local stress tensor as given in Eq.(3.2) and Eq.(3.5) respectively, are
equivalent. However, they differ from each other at third order which we
discuss in the next section. We have analysed the structure of
$\partial_{j}\tau_{ij}$ as well, that appears in Eq.(3.1), and which we refer
to as vector structure. We have found that time non-locality appears in the
vector structure at third order as well. We have discussed this in bit more
detail in Appendix. A.

###  3.2 Time non-locality in stress tensor ($\tau_{ij}$)

In this section, we discuss the disparity between the time local and time non-
local basis of operators that appear in the effective stress tensor. We have
found that the time local basis differs from the time non-local basis at third
order. This result is summarised in Table. 1 where we have listed the
independent set of operators in the time local and time non-local basis up to
third order.

We can see from Table. 1, that up to second order the time local and non-local
basis are equivalent to each other. However, at third order, the local basis
consists of 17 operators while the non-local basis contains 18 independent
operators. Hence, we find time non-locality in the effective stress tensor at
third order. As a crosscheck of non-local structures, we have computed the
independent set of operators in the $\Pi$ system as well. We have found that
the $\Pi$ basis also contains 18 independent operators at third order which is
same as the non-local basis. We have checked that both the non-local and $\Pi$
bases are related by an invertible map. The list of independent $\Pi$
operators is also given in Table. 1.555It’s worth mentioning that despite our
focused examination on the non-locality in stress tensor, a similar rationale
can be applied to any generic second-rank tensor that the non-locality will
shift to third order. Let us note that, at third order,
$(\Pi^{[2]}_{ij})^{(3)}$ which can be treated as a non-local operator (due to
the invertible map), can’t be written in terms of the local operators listed
in Table. 1.

The dynamical equations, however, does not contain $\tau_{ij}$ directly but
it’s first derivative in the form $\partial_{j}\tau_{ij}$ which we have called
as vector structure. We have analysed the structure of local as well as non-
local expansion of $\partial_{j}\tau_{ij}$. We have found that non-locality
shows up in vector structure also at third order. Please see Appendix. A for
details.

##  4 Non-locality in biased tracers in real space

Having discussed the local bias expansion in Sec. 2.2, we can discuss time
non-locality in biased tracers. In this section, we will layout the physics of
biased tracers and discuss the origin of time non-locality among biased
tracers. We will mainly be reviewing ideas presented in [54, 60, 1]. We also
discuss that writing operators in the $\Pi$ system, which is described in Sec.
2.4, makes the task of finding the non-local basis much more efficient.

As mentioned earlier, tracers such as galaxies evolve over the background of
dark matter fields. However, it has been shown that the time of virialisation
for galaxies or halo is around one-tenth of the time of evolution of dark
matter fields i.e. the time taken by a long mode to reach the non-linear scale
[54]. This implies that galaxy formation does not occur instantaneously but
takes a small but finite time period666If galaxies are formed instantaneously,
the background dark matter fields do not change significantly during the
formation time. Due to this, the galaxy distribution and hence, the number
density will be affected by dark matter fields at the same time. For such a
scenario, the bias expansion given in Eq.(2.7) is appropriate as it is local
in time in the sense that both r.h.s and l.h.s is evaluated at the same time..
Since, galaxy formation is not instantaneous, the tracers are affected by the
dark matter fields not only at the same time but also through the whole past
history along the fluid trajectory. Therefore, it has been argued [54], that
the actual bias expansion should be non-local in time and the r.h.s of
Eq.(2.7) should be generalized to an integral over the fluid trajectory of
dark matter as follows,

$\displaystyle\delta_{g}(\vec{x},t)=\sum_{\mathcal{O}_{i}}\int^{t}dt^{\prime}c_{\mathcal{O}_{i}}(t,t^{\prime})\mathcal{O}_{i}(\vec{x}_{\text{fl}}(\vec{x},t,t^{\prime}),t^{\prime})\;,$
(4.1)

where $c_{\mathcal{O}_{i}}(t,t^{\prime})$ are kernels having support over one
Hubble time and $\vec{x}_{\text{fl}}$ is defined in Eq.(3.6). Note that taking
$c_{\mathcal{O}_{i}}(t,t^{\prime})=c_{\mathcal{O}_{i}}(t)\delta(t-t^{\prime})$
gives us the local-in-time bias expansion given in Eq.(2.7).

We can use Eq.(3.6) iteratively to Taylor expand the operator
$\mathcal{O}_{i}$ in Eq.(4.1). Restricting ourselves upto order $n$ in dark
matter overdensity, we get [1, 60],

$\displaystyle[\mathcal{O}_{i}(\vec{x}_{\text{fl}}(\vec{x},t,t^{\prime}),t^{\prime})]^{(n)}=\sum_{\alpha=1}^{n-i+1}\left(\frac{D(t^{\prime})}{D(t)}\right)^{\alpha+i-1}\mathbb{C}_{\mathcal{O}_{i},\alpha}^{(n)}(\vec{x},t)\
.$ (4.2)

where $\mathbb{C}_{\mathcal{O}_{i},\alpha}^{(n)}$ are the new operators
stemming from each operator $\mathcal{O}_{i}$ upon the Taylor expansion along
fluid trajectory. Substituting the form of $\mathcal{O}_{i}^{(n)}$ from
Eq.(4.2) into Eq.(4.1) we get the following bias expansion that is local in
time,

$\displaystyle\delta^{(n)}_{g}(\vec{x},t)=\sum_{\mathcal{O}_{i}}\sum_{\alpha=1}^{n-i+1}c_{\mathcal{O}_{i},\alpha}(t)\,\mathbb{C}^{(n)}_{\mathcal{O}_{i},\alpha}(\vec{x},t)\;,$
(4.3)

where

$\displaystyle
c_{\mathcal{O}_{i},\alpha}(t)\equiv\int^{t}dt^{\prime}c_{\mathcal{O}_{i}}(t,t^{\prime})\left(\frac{D(t^{\prime})}{D(t)}\right)^{\alpha+i-1}\;,$
(4.4)

are the time dependent bias coefficients similar to what is given in the local
expansion in Eq.(2.7)777In [1], it has been shown that if the kernels have
support over a time scale $1/\omega$ then the non-local in time bias
parameters go like $H/\omega$. Therefore, if the bias parameters are order
unity, then it implies that the galaxy sample under observation was formed
over a timescale of $1/H$.. From Eq.(4.3), it is clear that
$\mathbb{C}^{(n)}_{\mathcal{O}_{i},\alpha}(\vec{x},t)$ are the local in time
operators that affect tracer over density $\delta_{g}(\vec{x},t)$ at location
$\vec{x}$ and time $t$.

Again we can look for linear relations among different operators appearing in
Eq.(4.3) to get a basis of operators at a given order. Due to Taylor expansion
along the fluid trajectory, the structure of operators that appear in (4.3) is
qualitatively different from those that appear in Eq.(2.7). Therefore, in
principle, the time local and time non-local expansion can have different
basis at each order.

Since the virialisation timescale is small as compared to the timescale of
evolution of long modes, we expect that to a certain accuracy, the bias
expansion can be taken to be local-in-time [54]. However, if we go to
sufficiently higher order in overdensity, the local in time and the non-local
in time bias expansion must differ in their structure. It has been pointed out
[60, 1], that up to fourth order in overdensity, non-local expansion can be
written in terms of local expansion i.e the non-local basis and local basis
are related by a linear invertible map.

However, at fifth order, the local and the non-local basis are different and
hence, there does not exists a linear map between them [1]. It has been shown
that at the fifth order, the time-local basis contains 26 independent
operators, while the time non-local basis has 29 independent operators. In
fact, at this order the total number of non-local operators that appear in
Eq.(4.3) are of $\mathcal{O}(100)$ but the operator space has a lot of
degeneracies. After removing degeneracies, the number of independent operators
turn out to be only 29 [1]. Out of the 29 non-local operators, 26 of them can
be written as linear combinations of the time-local basis and this map is
invertible. This implies that the non-local basis contains three extra
operators that cannot be written as a linear combinations of operators in the
time-local basis. The three non-local terms are pointed out in [1] and are
listed below,

$\displaystyle\mathbb{C}^{(5)}_{\delta,5},\hskip
14.22636pt\mathbb{C}^{(5)}_{r^{2},4},\hskip
14.22636pt\mathbb{C}^{(5)}_{p^{3},3}\;.$ (4.5)

The presence of these extra non-local terms point towards time non-locality in
the physics of structure formation, and it should be possible to detect
through data or simulations. In the next sub-section, we discuss the use of a
different basis to write non-local expansion, which makes calculation much
more efficient and easy. In particular, number of possible structures as well
as degeneracies reduces significantly in the new basis as discussed below.

###  4.1 $\mathbf{\Pi}$ operators: An efficient way to get the non-local basis

In this section, we will show that if we use a different system of writing
operators, then the task of finding the complete time non-local basis at any
give order becomes swift and easy. In Sec. 2.4, we have described an alternate
way of writing operators in the bias expansion [2]. Taking trace over the
contraction of $\Pi_{ij}^{[n]}$’s and their product, we can construct scalars
at any give order. We can now perform the same exercise of writing all
possible operators relevant at a given order. Then, we can look for
degeneracies among them to get the independent set of operators that will form
the basis at that order. We have found that up to fifth order, the basis
obtained by $\Pi$-system maps directly to the non-local basis through an
invertible linear map. At fifth order, we found that there are a total of 52
scalar operators out of which only 29 are independent. The list of 29
independent operators at order five is given as follows,

$\displaystyle m=1$ $\displaystyle\text{Tr}[\Pi^{[1]}]$ (4.6) $\displaystyle
m=2$ $\displaystyle\text{Tr}[(\Pi^{[1]})^{2}],\;(\text{Tr}[\Pi^{[1]}])^{2}$
$\displaystyle m=3$
$\displaystyle\text{Tr}[(\Pi^{[1]})^{3}],\;\text{Tr}[(\Pi^{[1]})^{2}]\text{Tr}[\Pi^{[1]}],\;(\text{Tr}[\Pi^{[1]}])^{3},\;\text{Tr}[\Pi^{[2]}\Pi^{[1]}]$
$\displaystyle m=4$
$\displaystyle\text{Tr}[(\Pi^{[1]})^{4}],\;\text{Tr}[(\Pi^{[1]})^{3}]\text{Tr}[\Pi^{[1]}],\;\left(\text{Tr}[(\Pi^{[1]})^{2}]\right)^{2},\;(\text{Tr}[\Pi^{[1]}])^{4},$
$\displaystyle\text{Tr}[\Pi^{[2]}\Pi^{[1]}\Pi^{[1]}],\;\text{Tr}[\Pi^{[2]}\Pi^{[1]}]\text{Tr}[\Pi^{[1]}],\;\text{Tr}[\Pi^{[2]}\Pi^{[2]}],\;\text{Tr}[\Pi^{[3]}\Pi^{[1]}]$
$\displaystyle m=5$
$\displaystyle\text{Tr}[(\Pi^{[1]})^{5}],\;\text{Tr}[(\Pi^{[1]})^{4}]\text{Tr}[\Pi^{[1]}],\;\text{Tr}[(\Pi^{[1]})^{3}](\text{Tr}[\Pi^{[1]}])^{2},\;\text{Tr}[(\Pi^{[1]})^{3}]\text{Tr}[(\Pi^{[1]})^{2}],$
$\displaystyle(\text{Tr}[\Pi^{[1]}])^{5},\;\text{Tr}[\Pi^{[2]}\Pi^{[1]}\Pi^{[1]}\Pi^{[1]}],\;\text{Tr}[\Pi^{[2]}\Pi^{[1]}\Pi^{[1]}]\text{Tr}[\Pi^{[1]}],$
$\displaystyle\text{Tr}[\Pi^{[2]}\Pi^{[1]}](\text{Tr}[\Pi^{[1]}])^{2},\;\text{Tr}[\Pi^{[2]}\Pi^{[2]}\Pi^{[1]}],\;\text{Tr}[\Pi^{[2]}\Pi^{[2]}]\text{Tr}[\Pi^{[1]}],\;\text{Tr}[\Pi^{[3]}\Pi^{[1]}\Pi^{[1]}],$
$\displaystyle\text{Tr}[\Pi^{[3]}\Pi^{[1]}]\text{Tr}[\Pi^{[1]}],\;\text{Tr}[\Pi^{[3]}\Pi^{[2]}],\;\text{Tr}[\Pi^{[4]}\Pi^{[1]}]\;,$

where we have arranged the set of operators by the order at which they start
and we are denoting it by index m. Also we are using the notation
$\text{Tr}[\Pi^{[3]}\Pi^{[1]}]\equiv\Pi^{[3]}_{ij}\Pi^{[1]}_{ij}$. As
mentioned earlier, the basis that we get from the $\Pi$ system given in
Eq.(4.6) and from the non-local expansion which is given in Eq.(4.3) are
related by an invertible map up to fifth order. We have shown the explicit
mapping in the Mathematica file attached with the manuscript. Here we will
just write the three "non-local" operators as pointed in [1] as a linear
combination of $\Pi$ operators,

$\displaystyle\mathbb{C}^{(5)}_{\delta,5}$
$\displaystyle=\frac{2071}{11466}(\text{Tr}[\Pi^{[1]}])^{5}-\frac{4022}{9555}\text{Tr}[(\Pi^{[1]})^{5}]+\frac{591}{3185}\text{Tr}[(\Pi^{[1]})^{4}]\text{Tr}[\Pi^{[1]}]$
(4.7)
$\displaystyle\quad+\frac{1303}{5733}\text{Tr}[(\Pi^{[1]})^{3}](\text{Tr}[\Pi^{[1]}])^{2}+\frac{5}{13}\text{Tr}[(\Pi^{[1]})^{3}]\text{Tr}[(\Pi^{[1]})^{2}]+\frac{2}{91}\text{Tr}[\Pi^{[2]}\Pi^{[2]}\Pi^{[1]}]$
$\displaystyle\quad+\frac{3}{130}\text{Tr}[\Pi^{[2]}\Pi^{[2]}]\text{Tr}[\Pi^{[1]}]+\frac{166}{1365}\text{Tr}[\Pi^{[2]}\Pi^{[1]}\Pi^{[1]}\Pi^{[1]}]$
$\displaystyle\quad-\frac{8}{1365}\text{Tr}[\Pi^{[2]}\Pi^{[1]}\Pi^{[1]}]\text{Tr}[\Pi^{[1]}]+\frac{2}{13}\text{Tr}[\Pi^{[2]}\Pi^{[1]}](\text{Tr}[\Pi^{[1]}])^{2}+\frac{9}{455}\text{Tr}[\Pi^{[3]}\Pi^{[2]}]$
$\displaystyle\quad+\frac{38}{1365}\text{Tr}[\Pi^{[3]}\Pi^{[1]}\Pi^{[1]}]+\frac{5}{91}\text{Tr}[\Pi^{[3]}\Pi^{[1]}]\text{Tr}[\Pi^{[1]}]+\frac{2}{39}\text{Tr}[\Pi^{[4]}\Pi^{[1]}]$
$\displaystyle\mathbb{C}^{(5)}_{r^{2},4}$
$\displaystyle=2\text{Tr}[\Pi^{[3]}\Pi^{[2]}]+4\text{Tr}[\Pi^{[4]}\Pi^{[1]}]$
$\displaystyle\mathbb{C}^{(5)}_{p^{3},3}$
$\displaystyle=\frac{204}{175}\text{Tr}[(\Pi^{[1]})^{5}]+\frac{27}{25}\text{Tr}[\Pi^{[2]}\Pi^{[2]}\Pi^{[1]}]+\frac{432}{175}\text{Tr}[\Pi^{[2]}\Pi^{[1]}\Pi^{[1]}\Pi^{[1]}]$
$\displaystyle\quad+\frac{9}{7}\text{Tr}[\Pi^{[3]}\Pi^{[1]}\Pi^{[1]}]\;.$

As mentioned earlier (Eq.(4.5)), the specific non-local operators cannot be
written in terms of the time-local basis but we have shown in Eq.(4.7), they
can indeed be written as a linear combination of operators given in the
alternative $\Pi$ basis, Eq.(4.6). We would like to point out that the $\Pi$
system is more efficient in getting the non-local basis as we just get 52
number of scalar operators which is much less in contrast to around 130 scalar
operators we get from the usual Taylor expansion in Eq.(4.3). As a result, at
fifth order, it has less number of degenerate operators888Only 23 degenerate
operators are present at fifth order in the $\Pi$ system. as compared to the
non-local expansion which contains $\mathcal{O}(100)$ degenerate operators and
hence a lot of redundancies. This reduces the task of finding degeneracy
equations which is one of the most challenging aspect of this calculation.
Note that Eq.(4.6) does not contain operators involving velocity gradient. We
have checked that the operators constructed from velocity gradient and it’s
convective derivatives are degenerate with the basis given in Eq.(4.6) up to
fifth order. This was pointed out in [2].

In the usual bias expansion, we only consider $SO(3)$ scalars. However, let’s
(hypothetically) allow for tensors in the bias expansion. Due to the tensorial
nature, this will allow for more structure in the operator space. This may
lead to time non-locality being shifted to lower orders. One such scenario
occurs when we include selection operators in the bias expansion. Selection
operators, being a projection of a tensorial object along the line-of-sight,
have essentially this tensorial structure which we will discuss in the next
section.

##  5 Non-locality in selection operators in the bias expansion

In this section, we discuss the time non-locality in selection operators which
were discussed in Sec. 2.3. We have found that including selection operators
in the bias expansion shifts the time non-locality to third order instead of
fifth order as for the case of $SO(3)$ invariant operators. It has been
pointed out in Sec.2.3, the need for introducing new line of sight dependent
operators in the bias expansion especially when writing an EFT of the flux
field in Lyman-$\alpha$ forest. We have found that including selection
operators in the bias expansion (Eq.(2.14)) the non-locality appears at third
order in tracer overdensity. In this section, we will discuss in more detail
about this non-locality and show that the time local and time non-local basis
differ at third order.

###  5.1 Non-locality shifts to $3^{{\rm rd}}$ order

As mentioned earlier, selection operators are needed to take into the line of
sight dependent effects that can affect the detection of tracer overdensity.
In this section, we discuss the set of independent selection operators
appearing at third order. Note that the numbers we quote for independent
operators is specific to selection operators only. The full bias expansion,
however, will consist of both the $SO(3)$ as well as the selection operators.
For the case of Lyman-$\alpha$ forest, all such operators are generated by
loops through the exponential map even if they are not present at tree level
[57]. So we need to include all possible operators with the structure given in
Eq.(2.15). Constructing all possible contractions we get the set of local
operators. After removing the degenerate operators we find that at third order
in tracer overdensity, there are 15 independent local selection operators999A
subset of the operators shown in Eq.(5.1) are generated when writing the bias
expansion in redshift space. However, we have checked that up to third order,
the local and the non-local basis are equivalent.. The basis of selection
operators at third order is given below,

$\displaystyle\begin{aligned}
\left\\{\eta,\;\delta_{z},\;\eta^{2},\;\delta_{z}^{2},\;\eta\delta,\;\delta_{z}\delta,\;(r^{2})_{\parallel},\;(p^{2})_{\parallel},\;\eta^{3},\;\eta^{2}\delta,\;\eta\delta^{2},\;(r^{3})_{\parallel},\;(r^{2})_{\parallel}\eta,\;(r^{2})_{\parallel}\delta,\;r^{2}\eta\right\\}\end{aligned}\;,$
(5.1)

where the symbols $\eta$ and $\delta_{z}$ have been defined in Eq.(2.15).

Based on the same reasoning given in Sec. 4, we can argue that the bias
expansion in Eq.(2.14) should be generalised to the non-local bias expansion
and hence should have integration along the full past trajectory of dark
matter fields. Performing the Taylor expansion along the fluid trajectory
gives the non-local selection operators with a structure similar to that given
in Eq.(4.3). After removing the degenerate operators, we find that there are
16 independent non-local selection operators. The list of independent non-
local selection operators is

$\displaystyle\begin{aligned}
&\left\\{\mathbb{C}^{(3)}_{\eta,1},\;\mathbb{C}^{(3)}_{\eta,2},\;\mathbb{C}^{(3)}_{\eta,3},\;\mathbb{C}^{(3)}_{\eta^{2},1},\;\mathbb{C}^{(3)}_{\eta^{2},2},\;\mathbb{C}^{(3)}_{\delta_{z}^{2},2},\;\mathbb{C}^{(3)}_{\eta\delta,1},\;\mathbb{C}^{(3)}_{\eta\delta,2},\;\mathbb{C}^{(3)}_{\delta_{z}\delta,2},\;\mathbb{C}^{(3)}_{(r^{2})_{\parallel},1},\;\mathbb{C}^{(3)}_{(r^{2})_{\parallel},2},\;\mathbb{C}^{(3)}_{\eta^{3},1},\right.\\\
\phantom{\mathrel{=}}&\;\;\left.\mathbb{C}^{(3)}_{\eta^{2}\delta,1},\;\mathbb{C}^{(3)}_{\eta\delta^{2},1},\;\mathbb{C}^{(3)}_{(r^{3})_{\parallel},1},\mathbb{C}^{(3)}_{(r^{2})_{\parallel}\delta,1}\right\\}\;.\end{aligned}$
(5.2)

From Eq.(5.1) and (5.2) we can see that the time local basis contains 15
operators while the time non local basis contains 16 operators. This implies
that the time non-local basis has one extra operator as compared to the time
local basis. Hence, we conclude that including line of sight dependent
selection effects enables us to detect non-locality at third order which can
be measured in one-loop power spectrum of the $\delta_{F}$ (Sec.2.3).

We can also write selection operators in the $\Pi$ system used in Sec.2.4. For
that we need to make two indexed objects from $\Pi_{ij}^{[n]}$ defined in
Eq.(2.17) and take projection with the unit vectors along the line of sight
such as
$(\Pi^{[1]}\Pi^{[2]})_{\parallel}\equiv\Pi_{ik}^{[1]}\Pi_{kj}^{[1]}\hat{z}^{i}\hat{z}^{j}$.
At third order, we find only 16 independent operators. The list of independent
selection operators up to third order in dark matter over density is as
follows,

$\displaystyle\begin{aligned} m=1&&&\Pi_{\parallel}^{[1]}\\\
m=2&&&\text{Tr}[\Pi^{[1]}]\Pi_{\parallel}^{[1]},\;[(\Pi^{[1]})^{2}]_{\parallel},\;(\Pi^{[1]}_{\parallel})^{2},\;\Pi^{[2]}_{\parallel}\\\
m=3&&&[(\Pi^{[1]})^{3}]_{\parallel},\;[(\Pi^{[1]})^{2}]_{\parallel}\Pi^{[1]}_{\parallel},\;[(\Pi^{[1]})^{2}]_{\parallel}\text{Tr}[\Pi^{[1]}],\;\Pi^{[1]}_{\parallel}\text{Tr}[(\Pi^{[1]})^{2}],\;\Pi^{[1]}_{\parallel}(\text{Tr}[\Pi^{[1]}])^{2},\\\
&&&(\Pi^{[1]}_{\parallel})^{2}\text{Tr}[\Pi^{[1]}],\;(\Pi^{[1]}_{\parallel})^{3},\;[\Pi^{[2]}\Pi^{[1]}]_{\parallel},\;\text{Tr}[\Pi^{[1]}]\Pi^{[2]}_{\parallel},\;\Pi^{[2]}_{\parallel}\Pi^{[1]}_{\parallel},\;\Pi^{[3]}_{\parallel}\;,\end{aligned}$
(5.3)

where we have arranged the operators by the index “$m$” that denotes the order
in dark matter overdensity at which a particular operator starts. As was the
case for $SO(3)$ operators, we find that here also there exists an invertible
map between the non-local basis given in Eq.(5.2) and $\Pi$ basis as given in
Eq.(5.3). We have shown the explicit mapping in the Mathematica file attached
with the manuscript. We can see from Eq.(5.3), that it does not contain
operators involving velocity gradient. This is due to the fact that the
operators constructed from velocity gradient ($\partial_{i}v_{j}$) and its
convective derivatives are degenerate with the $\Pi$ basis given in Eq.(5.3).
This was pointed out in [2, 61] and we have explicitly verified this.

The relation between non-local expansion and the $\Pi$ basis holds true at
fourth order as well. We will explore this in a bit more detail ahead. As for
the case of third order, we can write the time-local selection operators up to
fourth order by constructing $SO(2)$ scalars. We construct 67 such operators
out of which only 41 are independent and form the basis at fourth order. The
independent operators are listed below,

$\displaystyle\\{$
$\displaystyle\delta_{z},\;\eta,\;\delta_{z}^{2},\;\delta_{z}\eta,\;\eta^{2},\;\delta_{z}\delta,\;\delta_{z}\theta,\;\eta\delta,\;\eta\theta,\;(r^{2})_{\parallel},\;(rp)_{\parallel},\;(p^{2})_{\parallel},\delta_{z}^{3},\;\delta_{z}^{2}\eta,\;\delta_{z}^{2}\delta,\;\delta_{z}^{2}\theta,\;\delta_{z}\delta^{2},$
$\displaystyle\delta_{z}\delta\theta,\;\delta_{z}\eta\delta,\;\eta\delta^{2},\;(r^{3})_{\parallel},\;(r^{2}p)_{\parallel},\;(rp^{2})_{\parallel},\;(r^{2})_{\parallel}\delta_{z},\;(r^{2})_{\parallel}\eta,\;(r^{2})_{\parallel}\delta,\;(r^{2})_{\parallel}\theta,\;(rp)_{\parallel}\delta_{z},\;$
$\displaystyle(rp)_{\parallel}\delta,\;r^{2}\delta_{z},\;rp\delta_{z},\;\eta^{4},\;\eta^{3}\delta,\;\eta^{2}\delta^{2},\;\eta\delta^{3},\;(r^{4})_{\parallel},\;(r^{3})_{\parallel}\eta,\;(r^{2})_{\parallel}\eta^{2},\;(r^{2})_{\parallel}\eta\delta,\;(r^{2})_{\parallel}\delta^{2},\;((r^{2})_{\parallel})^{2}\\}\;,$
(5.4)

where we have followed the notation given in Eq.(2.9) and Eq.(2.15) for
writing operators in a compact form. To get the time non-local basis we need
to Taylor expand the operators in Eq.(5.1) along the fluid trajectory as we
did for the case of $SO(3)$ operators given in Eq.(4.3). At fourth order, we
have found 133 such non-local operators, out of which 87 are degenerate and
therefore only 46 operators are independent which we list out here,

$\displaystyle\begin{aligned}
&\left\\{\mathbb{C}^{(4)}_{\eta,1},\;\mathbb{C}^{(4)}_{\eta^{2},1},\;\mathbb{C}^{(4)}_{\eta\delta,1},\;\mathbb{C}^{(4)}_{(r^{2})_{\parallel},1},\;\mathbb{C}^{(4)}_{\eta^{3},1},\;\mathbb{C}^{(4)}_{\eta^{2}\delta,1},\;\mathbb{C}^{(4)}_{\eta\delta^{2},1},\;\mathbb{C}^{(4)}_{(r^{3})_{\parallel},1},\;\mathbb{C}^{(4)}_{(r^{2})_{\parallel}\eta,1},\;\mathbb{C}^{(4)}_{(r^{2})_{\parallel}\delta,1},\;\mathbb{C}^{(4)}_{r^{2}\eta,1},\;\mathbb{C}^{(4)}_{(p^{3})_{\parallel}),2},\right.\\\
\phantom{\mathrel{=}}&\;\;\left.\mathbb{C}^{(4)}_{(prp)_{\parallel},2},\;\mathbb{C}^{(4)}_{(p^{2})_{\parallel},2},\;\mathbb{C}^{(4)}_{(p^{2})_{\parallel},3},\;\mathbb{C}^{(4)}_{(p^{2})_{\parallel}\delta,2},\;\mathbb{C}^{(4)}_{(p^{2})_{\parallel}\delta_{z},2},\;\mathbb{C}^{(4)}_{(p^{2})_{\parallel}\eta,2},\;\mathbb{C}^{(4)}_{(p^{2})_{\parallel}\theta,2},\;\mathbb{C}^{(4)}_{p^{2}\delta_{z},2},\;\mathbb{C}^{(4)}_{p^{2}\eta,2},\;\mathbb{C}^{(4)}_{(r^{3})_{\parallel},2},\right.\\\
\phantom{\mathrel{=}}&\;\;\left.\mathbb{C}^{(4)}_{(r^{3})_{\parallel}\delta,1},\;\mathbb{C}^{(4)}_{(r^{3})_{\parallel}\eta,1},\;\mathbb{C}^{(4)}_{r^{3}\eta,1},\;\mathbb{C}^{(4)}_{(rp)_{\parallel}\delta,2},\;\mathbb{C}^{(4)}_{(rp)_{\parallel}\delta_{z},2},\;\mathbb{C}^{(4)}_{(r^{2})_{\parallel},3},\;\mathbb{C}^{(4)}_{(r^{2})_{\parallel}\eta^{2},1},\;\mathbb{C}^{(4)}_{(r^{2})_{\parallel}\eta\delta,1},\;\mathbb{C}^{(4)}_{r^{2}\eta^{2},1},\right.\\\
\phantom{\mathrel{=}}&\;\;\left.\mathbb{C}^{(4)}_{r^{2}\eta\delta,1},\;\mathbb{C}^{(4)}_{\delta_{z},2},\;\mathbb{C}^{(4)}_{\delta_{z},3},\;\mathbb{C}^{(4)}_{\delta_{z},4},\;\mathbb{C}^{(4)}_{\delta_{z}^{2},2},\;\mathbb{C}^{(4)}_{\delta_{z}^{2},3},\;\mathbb{C}^{(4)}_{\delta_{z}^{2}\delta,2},\;\mathbb{C}^{(4)}_{\delta_{z}^{2}\eta,2},\;\mathbb{C}^{(4)}_{\delta_{z}^{2}\theta,2},\right.\\\
\phantom{\mathrel{=}}&\;\;\left.\mathbb{C}^{(4)}_{\delta_{z}\delta,2},\;\mathbb{C}^{(4)}_{\delta_{z}\delta,3},\;\mathbb{C}^{(4)}_{\delta_{z}\delta^{2},2},\;\mathbb{C}^{(4)}_{\delta_{z}\eta,3},\;\mathbb{C}^{(4)}_{\eta^{3}\delta,1},\;\mathbb{C}^{(4)}_{\eta^{4},1}\right\\}\;,\end{aligned}$
(5.5)

where we have followed the same notation to write the operators as we used in
Eq.(4.3).

Now we write the selection operators in the $\Pi$ system. We notice that the
16 operators listed at order three in Eq.(5.3) also contribute at fourth
order. But now certain new operators appear which start at fourth order in
matter overdensity itself. The full list of independent operators at fourth
order is given as,

$\displaystyle\begin{aligned} m=1&&&\Pi_{\parallel}^{[1]}\\\
m=2&&&\text{Tr}[\Pi^{[1]}]\Pi_{\parallel}^{[1]},\;[(\Pi^{[1]})^{2}]_{\parallel},\;(\Pi^{[1]}_{\parallel})^{2},\;\Pi^{[2]}_{\parallel}\\\
m=3&&&[(\Pi^{[1]})^{3}]_{\parallel},\;[(\Pi^{[1]})^{2}]_{\parallel}\Pi^{[1]}_{\parallel},\;[(\Pi^{[1]})^{2}]_{\parallel}\text{Tr}[\Pi^{[1]}],\;\Pi^{[1]}_{\parallel}\text{Tr}[(\Pi^{[1]})^{2}],\;\Pi^{[1]}_{\parallel}(\text{Tr}[\Pi^{[1]}])^{2},\\\
&&&(\Pi^{[1]}_{\parallel})^{2}\text{Tr}[\Pi^{[1]}],\;(\Pi^{[1]}_{\parallel})^{3},\;[\Pi^{[2]}\Pi^{[1]}]_{\parallel},\;\text{Tr}[\Pi^{[1]}]\Pi^{[2]}_{\parallel},\;\Pi^{[2]}_{\parallel}\Pi^{[1]}_{\parallel},\;\Pi^{[3]}_{\parallel}\\\
m=4&&&[(\Pi^{[1]})^{4}]_{\parallel},\;[(\Pi^{[1]})^{3}]_{\parallel}\Pi^{[1]}_{\parallel},\;\Pi^{[1]}\text{Tr}[(\Pi^{[1]})^{3}],\;[(\Pi^{[1]})^{2}]_{\parallel}(\text{Tr}[\Pi^{[1]}])^{2},\;\text{Tr}[\Pi^{[1]}\Pi^{[1]}][\Pi^{[1]}\Pi^{[1]}]_{\parallel}\\\
&&&[(\Pi^{[1]})^{2}]_{\parallel}\Pi^{[1]}_{\parallel}\text{Tr}[\Pi^{[1]}],\;[(\Pi^{[1]})^{2}]_{\parallel}(\Pi^{[1]}_{\parallel})^{2},\;(\Pi^{[1]}_{\parallel})^{2}\text{Tr}[(\Pi^{[1]})^{2}],\;(\Pi^{[1]}_{\parallel})^{4},\\\
&&&(\Pi^{[1]}_{\parallel})^{3}\text{Tr}[\Pi^{[1]}],\;(\Pi^{[1]}_{\parallel})^{2}(\text{Tr}[\Pi^{[1]}])^{2},\;\Pi^{[1]}_{\parallel}(\text{Tr}[\Pi^{[1]}])^{3},\;[(\Pi^{[1]})^{2}]_{\parallel}\text{Tr}[(\Pi^{[1]})^{2}],\\\
&&&([(\Pi^{[1]})^{2}]_{\parallel})^{2},\;[\Pi^{[2]}\Pi^{[1]}\Pi^{[1]}]_{\parallel},\;[\Pi^{[1]}\Pi^{[2]}\Pi^{[1]}]_{\parallel},\;[\Pi^{[2]}\Pi^{[1]}]_{\parallel}\Pi^{[1]}_{\parallel},\;[\Pi^{[2]}\Pi^{[1]}]_{\parallel}\text{Tr}[\Pi^{[1]}],\\\
&&&\Pi^{[1]}_{\parallel}\text{Tr}[\Pi^{[2]}\Pi^{[1]}],\;\Pi^{[2]}_{\parallel}[(\Pi^{[1]})^{2}]_{\parallel},\;\Pi^{[2]}_{\parallel}\text{Tr}[(\Pi^{[1]})^{2}],\;\Pi^{[2]}_{\parallel}(\Pi^{[1]}_{\parallel})^{2},\;\Pi^{[2]}_{\parallel}\Pi^{[1]}_{\parallel}\text{Tr}[\Pi^{[1]}],\\\
&&&\Pi^{[2]}_{\parallel}(\text{Tr}[\Pi^{[1]}])^{2},\;[\Pi^{[2]}\Pi^{[2]}]_{\parallel},\;(\Pi^{[2]}_{\parallel})^{2},\;[\Pi^{[3]}\Pi^{[1]}]_{\parallel},\;\Pi^{[3]}_{\parallel}\text{Tr}[\Pi^{[1]}],\;\Pi^{[3]}_{\parallel}\Pi^{[1]}_{\parallel},\;\Pi^{[4]}_{\parallel}\;.\end{aligned}$
(5.6)

So Eq.(5.6) together gives the total number of selection operators at fourth
order. From Eq.(5.6) we can see that there are 30 new operators that appear at
fourth order coming from $m=4$ sector, which makes the total number of
selection operators at order four to be 46.

The $\Pi$ operators give in Eq.(5.6) are related to the time non-local basis
with an invertible map. We have explicitly shown the mapping in the
Mathematica file attached with the manuscript.101010Note that Eq.(5.6) does
not contain operators with velocity gradient ($\partial_{i}v_{j}$) and it’s
convective derivatives. This is because at fourth order also, all such
operators are degenerate with the operators given in Eq.(5.6). At fourth
order, we can write 53 operators involving velocity gradient and their
convective derivatives. We have explicitly checked that each of them is
degenerate with the $\Pi$ operators given in Eq.(5.6).

###  5.2 Connection with observation

We have found through Eq.(5.1) and (5.2), that the time non-local basis at
third order contains one more operator than the time local basis. This implies
that there exists one operator in the non-local basis that cannot be written
in terms of the local basis. We have also shown that the time non-local basis
is mathematically equivalent to the $\Pi$ basis given in Eq.(5.3). The extra
non-local operator, is easily extracted through the $\Pi$ basis which is
$\Pi^{[2]}_{\parallel}$. The fact that time non-locality manifests itself at
third order implies that it can be measured in one-loop power spectrum of
$\delta_{F}$ through a contribution like
$\langle\delta_{F}^{(3)}(\bm{k}_{1})\delta_{F}^{(1)}(\bm{k}_{2})\rangle$ or in
the tree level four point function
$\langle\delta_{F}^{(3)}(\bm{k}_{1})\delta_{F}^{(1)}(\bm{k}_{2})\delta_{F}^{(1)}(\bm{k}_{3})\delta_{F}^{(1)}(\bm{k}_{4})\rangle$.

##  6 Discussion and Conclusion

Galaxy formation time is of order $(1/10)^{\text{th}}$ the evolution
timescales of the Universe as a whole and hence is inherently a time non-local
process. Indeed, in the context of the EFTofLSS, [1] has recently shown that
time non-locality arises explicitly at the fifth order in bias expansion.
However, a natural question is that if galaxy formation is intrinsically time-
non-local in nature, why does this aspect show up only at the fifth order in
the expansion? Our analysis reveals that delving beyond $SO(3)$ scalar
observables that were considered in [1] can unveil non-locality even at lower
orders. For instance, in the context of the effective stress tensor within the
framework of EFTofLSS — $\tau_{ij}$ — which transforms as a second-rank tensor
under $SO(3)$ transformations, temporal non-locality manifests at the third
order111111This non-locality in dark matter is anticipated due to the lack of
separation between the time scales of long and short modes.. Another instance
arises when incorporating selection operators in the bias expansion. These
operators, acting as projections of tensorial objects along the line-of-sight,
inherently possess the structures necessary to exhibit temporal non-locality
at lower orders. Notably, in the case of Lyman-$\alpha$ flux fluctuations,
where the symmetry breaks down to SO(2) along the line of sight, temporal non-
locality emerges again at the third order in the bias expansion121212Let us
note that in [1] and in the current paper, we have only used SPT kernels which
corresponds to non-renormalised matter overdensity and velocity divergence as
given in (2.1).. This suggests that temporal non-locality might be observable
in the 1-loop power spectrum for phenomena necessitating the inclusion of
selection operators. This is in contrast to the $SO(3)$ bias expansion, where
non-locality is only detectable at the two-loop power spectrum level for
observables.

In addition to exploring these physics concepts, our paper delved into several
intriguing technical aspects. Typically, non-locality emerges at significantly
higher orders in perturbative expansions. For instance, in the case of the
$SO(3)$ bias expansion, it arises at the 5th order. Analyzing non-locality at
these higher orders presents considerable technical challenges. Therefore,
gaining insight into more efficient methods for investigating non-locality is
crucial. In this study, we discovered that employing the $\Pi$ basis [2]
offers a more efficient approach to handling non-locality. Specifically,
concerning biased tracers, we demonstrated that the correspondence between the
$\Pi$ basis and the time non-local basis extends up to the fifth order for
$SO(3)$ invariant operators. It’s noteworthy that the $\Pi$ system proves more
effective in establishing the non-local basis, requiring only 52 scalar
operators, a significant reduction compared to the approximately 130 scalar
operators in the conventional basis. We have also established efficacy of
$\Pi$ basis when dealing with selection operators.

In deriving these results, we have employed the usual method of writing
selection operators in the bias expansion as given in Eq.(2.15). In this
method we write all possible $SO(2)$ invariant operators at a given order.
This is termed as the bottom-up approach to writing operators in the bias
expansion. However, for the case of Lyman-$\alpha$ forest, we can generate the
same set of operators in the bias expansion of flux fluctuation ($\delta_{F}$)
through renormalisation of the flux field. Due to the non-linear exponential
map between the optical depth ($\tau$) and the total flux ($F$), given as
$F=e^{-\tau}$, $\delta_{F}$ contains operators with products of
$\delta_{\tau}$ evaluated at the same point in space. This requires
renormalisation of these composite operators. Taking the optical depth
fluctuation in redshift space and renormalising the $\delta_{F}$ field, in
principle, should generate all possible operators that appear in the bias
expansion in Eq.(2.15). For more details see Appendix. B.

Below we list out some of the future directions which we think would be very
interesting to pursue.

  * •

In [62], it has been pointed out that renormalisation generates all operators
that are allowed by the symmetries. It would be intriguing to check, whether
renormalisation generates the time local or time non-local operators for the
case of both $SO(3)$ and $SO(2)$ invariant operators.

  * •

In order to derive the non-local basis for biased tracers, no assumption is
made regarding whether tracers should follow continuity equation or not [1].
$\Pi$ system, on the other hand, is derived by assuming that the tracers
satisfy a continuity equation. However, we have shown that the non-local basis
is related to the $\Pi$ basis by an invertible map up to fifth for $SO(3)$
invariant operators and up to fourth order for $SO(2)$ invariant operators. It
would be nice to have a physical understanding of this mathematical
equivalence.

  * •

Presence of time non-locality in the effective stress tensor as given in
Table. 1, implies that the structure of kernels for dark matter overdensity as
given in Eq.(2.1) should also change after renormalisation as these new non-
local terms will be added as counterterms to the dark matter overdensity field
[60]. As a result, if we write the operators in bias expansion with the
renormalised kernels of EFT rather than using SPT kernels as given in
Eq.(2.1), it would be interesting to see how the analysis of Secs. 4 and 5
will be modified. We leave this investigation for future exploration. Also in
this study, we have not considered higher derivative operators in the bias
expansion. It would be interesting to see how the results are modified when
higher derivative operators are also considered for both $SO(3)$ and selection
operators. Given the fact that time non-locality appears at fourth order in
$\partial_{i}\partial_{j}\tau_{ij}$, which is a two derivative operator, one
would expect the same for higher derivative operators. Exploring this aspect
of bias expansion is left for future work.

  * •

Recently, bootstrap ideas have been applied to write the kernels for the
biased tracers [63]. Just using symmetry properties of the equations of
motion, it has been shown that the kernels for biased tracers can be
bootstrapped up to third order. It would be interesting to investigate time
non-locality from a bootstrap perspective. Preliminary investigation along
these lines suggests one interesting technical point, which is the presence of
an even more efficient basis than the $\Pi$ basis that we have employed in
this paper.

  * •

One can also do a full $N$-body simulation and measure the contribution of
time non-local operators found in this study and the studies preceding this
[1]. An exhaustive $N$-body simulation with generic initial conditions maybe
computationally challenging. However, with simpler initial conditions, see for
e.g. [64], such computations maybe tractable. Measuring the contributions due
to time non-local operators in simulations will establish time non-locality as
a generic feature of LSS.

## Acknowledgments

AA acknowledges the support from a Senior Research Fellowship, granted by the
Human Resource Development Group, Council of Scientific and Industrial
Research, Government of India. AB acknowledges support from Science and
Engineering Research Board (SERB) India via the Startup Research Grant
SRG/2023/000378. SJ would like to thank participant of Cosmology Journal club
of iiser pune and DTP, TIFR seminar series for their probing questions and
which helped clarify many important physics issues. SJ also would like to
thank S. Minwalla, B. Dasgupta for their insightful discussions. We would also
like to thank B. Dasgupta for useful comments on an earlier version of the
draft. The authors would like to thank M. Lewandowski for pointing out an
error and useful comments on an earlier version of the manuscript. The authors
would also like to acknowledge their debt to the people of India for their
steady support of research in basic sciences.

##  Appendix A Basis of vector operators ($\partial_{j}\tau_{ij}$) up to third
order

In Sec. 3, we have mentioned about time non-locality appearing in tensor
($\tau_{ij}$) and vector ($\partial_{j}\tau_{ij}$) operators. We have given
the local, non-local and $\Pi$ basis appearing in the tensor sector upto third
order in Table. 1. Here we provide local, non-local and $\Pi$ basis for the
vector operators.

At linear order in matter overdensity there is only one independent operator
in the local, non-local as well as $\Pi$ system. We list here the independent
set of operators,

Local basis: $\displaystyle\hskip 14.22636pt\partial_{j}r_{ij},\hskip
14.22636pt$ Non-local basis: $\displaystyle\hskip
14.22636pt\mathbb{C}^{(1)}_{\partial_{j}r_{ij},1},\hskip 14.22636pt$
$\displaystyle\Pi\text{-basis:}$ $\displaystyle\hskip
8.5359pt\partial_{j}\Pi_{ij}^{[1]}$ (A.1)

where we have used the same notation as is used in writing the operators in
Table. 1.

Second Order

At second order also, the local, non-local and $\Pi$ basis each have four
operators which are listed below,

Local basis: $\displaystyle\hskip
8.5359pt\\{\partial_{j}(r_{ij}\delta),\partial_{i}(\delta,\delta^{2},r^{2})\\}$
Non-local basis: $\displaystyle\hskip
8.5359pt\\{\mathbb{C}^{(2)}_{\partial_{j}p_{ij},1},\mathbb{C}^{(2)}_{\partial_{i}\delta,1},\mathbb{C}^{(2)}_{\partial_{i}\delta,2},\mathbb{C}^{(2)}_{\partial_{i}\theta,2}\\}$
$\displaystyle\Pi\text{-basis:}$ $\displaystyle\hskip
8.5359pt\\{\partial_{j}(\Pi_{ij}^{[1]}\text{Tr}[\Pi^{[1]}]),\partial_{i}(\text{Tr}[\Pi^{[1]}],\text{Tr}[\Pi^{[1]}]^{2},\text{Tr}[\Pi^{[1]}\Pi^{[1]}])\\}\;.$
(A.2)

Third order

At third order, the local basis consists of 12 operators while the non-local
basis contains 13 independent operators. Hence, we find that the local and
non-local basis differ from each other. As expected, we find 13 independent
operators in the $\Pi$ basis. We list out the set of operators that appear in
the three basis below,

Local basis:
$\displaystyle\\{\partial_{j}(p_{ij}\delta,p_{ij}\theta,r_{ij}\delta,p_{ij}\delta^{2},(r^{3})_{ij}),\partial_{i}(\delta,\theta,\delta^{2},\delta\theta,r^{2},rp,\delta^{3})\\}$
Non-local basis:
$\displaystyle\\{\mathbb{C}^{(3)}_{\partial_{j}r_{ij},1},\mathbb{C}^{(3)}_{\partial_{j}r_{ij},2},\mathbb{C}^{(3)}_{\partial_{j}p_{ij},2},\mathbb{C}^{(3)}_{\partial_{j}(p_{ij}\delta),1},\mathbb{C}^{(3)}_{\partial_{j}(p_{ij}\delta),2},\mathbb{C}^{(3)}_{\partial_{j}(r_{ij}\delta),2},$
$\displaystyle\mathbb{C}^{(3)}_{\partial_{i}\delta,1},\mathbb{C}^{(3)}_{\partial_{i}\delta,2},\mathbb{C}^{(3)}_{\partial_{i}\delta,3},\mathbb{C}^{(3)}_{\partial_{i}\theta,2},\mathbb{C}^{(3)}_{\partial_{i}\theta,3},\mathbb{C}^{(3)}_{\partial_{i}\delta^{2},2},\mathbb{C}^{(3)}_{\partial_{i}r^{2},2}\\}$
$\displaystyle\Pi\text{-basis:}\hskip 8.5359pt$
$\displaystyle\\{\partial_{j}(\text{Tr}[\Pi^{[1]}]\Pi_{ij}^{[1]},\Pi_{ij}^{[2]},\Pi_{ij}^{[1]}\text{Tr}[\Pi^{[1]}]^{2},[\Pi^{[1]}\Pi^{[1]}\Pi^{[1]}]_{ij},[\Pi^{[1]}\Pi^{[1]}]_{ij}\text{Tr}[\Pi^{[1]}],$
$\displaystyle\text{Tr}[\Pi^{[1]}]\Pi_{ij}^{[2]}),$
$\displaystyle\partial_{i}(\text{Tr}[\Pi^{[1]}],\text{Tr}[\Pi^{[1]}]^{2},\text{Tr}[\Pi^{[1]}\Pi^{[1]}],\text{Tr}[\Pi^{[2]}],\text{Tr}[\Pi^{[1]}]^{3},\text{Tr}[\Pi^{[1]}\Pi^{[1]}\Pi^{[1]}],$
$\displaystyle\text{Tr}[\Pi^{[1]}\Pi^{[1]}]\text{Tr}[\Pi^{[1]}])\\}$ (A.3)

From eq.(A), we can see that the non-local basis contains 13 while the local
basis contains only 12 operators. Hence, there cannot exist an invertible map
between them. Also we see that the $\Pi$ basis contains the same number of
operators as the non-local basis. We have explicitly checked that there is an
invertible map between the non-local and $\Pi$ basis.

The extra term in the non-local basis coming in the $\partial_{j}\tau_{ij}$
will contribute in the the equation of motion given in Eq.(3.1) [60]. Hence,
the kernels for $v^{i}$ will change as compared to the case when the local
vector operator is considered.

##  Appendix B Comments on Renormalisation

In a generic bias expansion for tracers given in Eq.(2.7), we can see that we
have product of operators evaluated at the same point in space. This leads to
divergences when computing an observable at higher order. To remove these
divergences we need to renormalise such operators by adding appropriate
counterterms. This needs to be done with all operators to ensure all
observables are finite.

In this section, we briefly discuss how all the selection operators are
generated as counterterms by renormalisation of the flux field $\delta_{F}$ in
Lyman-$\alpha$. This is the "top-down" approach to get the same bias expansion
as given in Eq.(2.14), as opposed to the "bottom-up" approach that we have
discussed in Sec. 2.3. Along the way we will review some of the equation
relevant in the discussion of Lyman alpha forest.

For the case of Lyman alpha forest, we measure flux of photons, $F$ coming
towards our line of sight. The fluctuation over the mean value of the flux
field, $\bar{F}$, is defined as,

$\displaystyle\delta_{F}=\frac{F-\bar{F}}{\bar{F}}\;.$ (B.1)

The total flux $F$ received depends on the optical depth ($\tau$) of neutral
hydrogen along the line of sight of observation in an inverse manner. Optical
depth is proportional to the number density of neutral hydrogen. So if the
number density is high then there will be more absorption and hence flux will
be reduced. The number density of neutral hydrogen is a direct tracer of the
underlying dark matter. Being a direct tracer of the underlying dark matter,
the optical depth $\delta_{\tau}$ which is proportional to the number density
has the same bias expansion as given in Eq.(2.7). Mathematically, the total
flux $F$ is related to the optical depth by an exponential map as follows [65,
66],

$\displaystyle F=e^{-\tau_{0}(1+\delta_{\tau}^{r})}\;,$ (B.2)

where $\tau_{0}$ is the mean optical depth and $\delta_{\tau}^{r}$ is the
fluctuation in the optical depth in redshift space. Now, using Eq.(B.2), we
can write how the flux fluctuation in the Lyman alpha field is related to the
fluctuation in the optical depth which is given as [57],

$\displaystyle\delta_{F}=-\tau_{0}\delta_{\tau}^{r}+\frac{1}{2}\tau_{0}^{2}(\delta_{\tau}^{2,r}-\langle\delta_{\tau}^{2,r}\rangle)-\frac{1}{6}\tau_{0}^{3}\delta_{\tau}^{3,r}\;.$
(B.3)

In order to get an expansion of flux fluctuation field $\delta_{F}$, we should
have the expression for optical depth fluctuation in redshift space.

So, let’s look at the structure of operators that arise when we go to redshift
space. Redshift space is related to the real space by a coordinate
transformation given as [67, 68],

$\displaystyle\vec{x}_{r}=\vec{x}+\frac{\vec{v}\cdot\hat{z}}{aH}\hat{z}\;,$
(B.4)

where $\vec{x}_{r}$ and $\vec{x}$ are the redshift space and real coordinates
respectively and $\hat{z}$ is the unit vector along the line of sight. Under
the coordinate change given in Eq.(B.4) and imposing mass conservation, the
optical depth fluctuation is expressed in momentum space as,

$\displaystyle\delta_{\tau}^{r}(\bm{k})=\delta_{\tau}(\bm{k})+\int
d^{3}xe^{-i\bm{k}\cdot\bm{x}}\Bigg{(}\Bigg{[}\exp{-i\frac{\vec{k}\cdot\hat{z}}{aH}\vec{v}(\vec{x})\cdot\hat{z}}-1\Bigg{]}(1+\delta_{\tau}(\vec{x}))\Bigg{)}\;,$
(B.5)

where $\delta_{\tau}(\vec{k})$ has the same bias expansion as given in
Eq.(2.7). We can expand Eq.(B.5) up to desired order in overdensity to get the
bias expansion in redshift space. For example up to third order, we have,

$\displaystyle\delta_{\tau}^{r}(\bm{k})$
$\displaystyle=\delta_{\tau}(\bm{k})-\frac{ik_{z}}{aH}[v_{z}]_{\bm{k}}+\frac{1}{2}\Bigg{(}\frac{ik_{z}}{aH}\Bigg{)}^{2}[v_{z}^{2}]_{\bm{k}}-\frac{1}{6}\Bigg{(}\frac{ik_{z}}{aH}\Bigg{)}^{3}[v_{z}^{3}]_{\bm{k}}$
$\displaystyle-\frac{ik_{z}}{aH}[v_{z}\delta_{\tau}]_{\bm{k}}+\frac{1}{2}\Bigg{(}\frac{ik_{z}}{aH}\Bigg{)}^{2}[v_{z}^{2}\delta_{\tau}]_{\bm{k}}\;,$
(B.6)

where $[f]_{\bm{k}}\equiv\int d^{3}xe^{-i\bm{k}\cdot\bm{x}}f(\vec{x})$ and we
have used $v_{z}\equiv\vec{v}.\hat{z},k_{z}\equiv\vec{k}.\hat{z}$ for brevity.
From Eq.(B), we can see that a subset of selection operators are generated by
going to redshift space.

We can do a time non-local expansion of Eq.(B) to get the set of non-local
operators in redshift space. We have checked that up to third order, the non-
local operators can be written in terms of the local operators. Hence, time
non-locality does not appear in redshift space up to third order.

We can substitute the expression for $\delta_{\tau}^{r}$ from Eq.(B) in
Eq.(B.3) to get the expression for flux fluctuation in redshift space. The
second and third terms in Eq.(B.3) are composite operators. Therefore, it has
been pointed out that the second and the third term in Eq.(B.3) need to be
renormalised [57] appropriately. Upon renormalisation we expect that all
operators allowed by the symmetry are generated as counterterms. Therefore, we
get the same bias expansion as in Eq.(5.1). To generate all selection
operators upto third order, we need third order tree level counterterms. The
systematic method of renormalisation for local Eulerian halo biasing in the
context of $SO(3)$ invariant operators is developed in [62]. From that study,
it can be observed that to generate all counterterm at third order, we need to
evaluate the quantity,

$\displaystyle\langle\delta_{F,\bm{q}}^{(5)}\delta_{\bm{q}_{1}}^{(1)}\delta_{\bm{q}_{2}}^{(1)}\delta_{\bm{q}_{3}}^{(1)}\rangle\;,$
(B.7)

where $\delta_{\bm{q}_{1}}^{(1)}$ are linear overdensity in dark matter. Now,
$\delta_{F}$ has the expansion given in Eq.(B.3) which contains
$\delta_{\tau}^{r}$. The kernel for $\delta_{F}$ upto fifth order in momentum
space is too big to deal with. Furthermore, identifying individual selection
operators is impractical, therefore we leave that exercise for future and
assume that all selection operators are generated when $\delta_{F}$ is
renormalised to all loops.

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