On the asymptotic behaviour of cosmic density-fluctuation power spectra of
cold dark matter

Sara Konrad1,∗, Yonadav Barry Ginat2,† and Matthias Bartelmann1

1 Institute for Theoretical Physics, Heidelberg University, Germany

2 Faculty of Physics, Technion – Israel Institute of Technology, Haifa,
3200003, Israel

*<EMAIL_ADDRESS>$\dagger$<EMAIL_ADDRESS>
###### Abstract

We study the small-scale asymptotic behaviour of the cold dark matter density
fluctuation power spectrum in the Zel’dovich approximation, without
introducing an ultraviolet cut-off. Assuming an initially correlated Gaussian
random field and spectral index $0<n_{s}<1$, we derive the small-scale
asymptotic behaviour of the initial momentum-momentum correlations. This
result is then used to derive the asymptotics of the power spectrum in the
Zel’dovich approximation. Our main result is an asymptotic series, dominated
by a $k^{-3}$ tail at large wave-numbers, containing higher-order terms that
differ by integer powers of $k^{n_{s}-1}$ and logarithms of $k$. Furthermore,
we show that dark matter power spectra with an ultraviolet cut-off develop an
intermediate range of scales, where the power spectrum is accurately described
by the asymptotics of dark matter without a cut-off. These results reveal
information about the mathematical structure that underlies the perturbative
terms in kinetic field theory and thus the non-linear power spectrum. We also
discuss the sensitivity of the small-scale asymptotics to the spectral index
$n_{s}$.

###### Contents

  1. 1 Introduction
  2. 2 Small-scale asymptotics of the initial momentum correlations
  3. 3 Small-scale asymptotics of the Zel’dovich power spectrum
  4. 4 Late time evolution
  5. 5 Implications for dark matter
  6. 6 Summary and discussion
  7. A Mellin transform technique
  8. B Asymptotics of the initial momentum correlations
  9. C Neglecting other orders in the asymptotics
  10. D Large $m$

##  1 Introduction

The Zel’dovich approximation [1] is ubiquitous in the theory of structure
formation (see ref. [2] for a relatively recent review), as a key analytic
approximation that encapsulates many of the physical effects pertinent to
structure formation. One of its many advantages is that it allows for the
derivation of a simple formula for the two-point correlation function [3, 4],
and thus the power spectrum. While the Zel’dovich power spectrum accurately
describes structures on linear scales, it is a non-linear power spectrum since
it is a non-linear functional of the initial power spectrum.

In kinetic field theory (hereafter KFT), the Zel’dovich power spectrum plays
an important role and is key to the full non-linear power spectrum [5, 6, 7,
8]. It represents the leading order power spectrum of perturbation theory and
additionally is the starting point of a mean-field theory within KFT, which is
an alternative way to incorporate those particle interactions that are not
already captured by the Zel’dovich approximation [7]. To derive the Zel’dovich
power spectrum from KFT, three ingredients are necessary: (i) the
specification of the initial conditions in classical phase-space, (ii) the
assumption of straight particle trajectories111They are inertial if time is
transformed to a specific non-linear function of itself. that contain parts of
the gravitational interactions and (iii) the expanding space-time as
background [9, 5].

We consider Gaussian initial conditions, implying that two-point correlation
functions are sufficient to completely determine the initial probability
distribution in phase space. We assume that initial momenta are correlated,
while initial density-density and density-momentum correlations are neglected.
This is reasonable on large scales and at late times, as they grow much more
slowly than the initial momentum-momentum correlations. The initial velocity
field can be considered to be curl-free, such that it can be expressed in
terms of a gradient of a velocity potential $\psi$. Since the continuity
equation has to hold initially, the initial density contrast has to satisfy
Poisson’s equation

$\delta^{(i)}=-\nabla^{2}\psi\;.$ (1)

(All the constants that might appear are absorbed into $\psi$ here.)
Consequently, the initial velocity potential power spectrum is related to the
initial density fluctuation power spectrum via

$P_{\psi}^{\mathrm{(i)}}(k)=k^{-4}\,P_{\delta}^{\mathrm{(i)}}(k)\;.$ (2)

The initial momentum-momentum correlation matrix of two particles separated by
$\mathbb{q}$ is then given by

$\displaystyle\hat{C}_{pp}\left(\mathbf{q}\right)=\int_{k^{\prime}}\left(\mathbf{k}^{\prime}\otimes\mathbf{k}^{\prime}\,\right)P_{\psi}^{\mathrm{(i)}}(k^{\prime})\,\mathrm{e}^{\mathrm{i}\mathbf{k}^{\prime}\cdot\mathbf{q}}\;.$
(3)

The ansatz that the particle trajectories are straight is key to the
Zel’dovich approximation. In KFT, this ansatz is expressed by splitting up the
actual time dependent trajectory of a particle into the Zel’dovich part
$\mathbb{q}^{Z}(t)$, that is linear in the initial phase space coordinates
$\mathbb{q}^{(i)}$, $\mathbb{p}^{(i)}$, and an interaction part
$\mathbb{q}^{I}(t)$ that contains all the remaining interactions that are not
captured by the Zel’dovich approximation,

$\mathbb{q}(t)=\mathbb{q}^{(i)}+g^{Z}_{qp}(t,t^{(i)})\mathbb{p}^{(i)}+\mathbb{q}^{I}(t)\equiv\mathbb{q}^{Z}(t)+\mathbb{q}^{I}(t)\;,$
(4)

where $g_{qp}(t,t^{(i)})$ is the so-called propagator. Considering only
$\mathbb{q}^{Z}(t)$ and specifying $g^{Z}_{qp}(t,t^{(i)})\equiv
D_{+}(t)-D_{+}(t^{(i)})$ with the linear growth factor $D_{+}$, yields the
Zel’dovich approximation. In the following, we define the initial time
$t^{(i)}=0$ and then replace in our notation the propagator $g_{qp}(t,0)$ by
$t$. This not only simplifies the notation but also highlights that the
propagator serves as a time coordinate. We emphasise that Eq. (4) is exact.

When the mentioned assumptions are combined, the unperturbed zeroth-order
power spectrum of KFT is equivalent to the Zel’dovich power spectrum and is
given by

$\mathcal{P}(k,t)=\mathrm{e}^{-Q}\int\mathrm{d}^{3}q\left(\mathrm{e}^{t^{2}\,\mathbf{k}^{\top}\hat{C}_{pp}(\mathbf{q})\mathbf{k}}-1\right)\mathrm{e}^{\mathrm{i}\mathbf{k}\cdot\mathbf{q}}\;,$
(5)

as shown in [9, 5]. The exponent $Q$ is

$Q=\frac{\sigma_{1}^{2}}{3}\,k^{2}\,t^{2}\;,$ (6)

where $\sigma_{1}^{2}$ is one of the moments

$\sigma_{n}^{2}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k\,k^{2n+2}\,P_{\psi}^{\mathrm{(i)}}(k)$
(7)

of the power spectrum $P_{\psi}^{\mathrm{(i)}}$ of the initial velocity
potential $\psi$. It is important for the present derivation of our results
that $\sigma_{2}$ is finite, which is the case for $0<n_{s}<1$, but higher
moments are allowed to be infinite.

We emphasise that nothing in this paper relies on features peculiar to KFT –
indeed, all that we assume is that the power spectrum in the Zel’dovich
approximation is given by equation (5), which can be arrived at from, e.g.,
Lagrangian perturbation theory, too [4].

The small-scale limit of the power spectrum is of interest to cosmologists for
many reasons: it can be used to study the behaviour of dark matter inside
haloes [10], and thereby to gain some understanding on their universal density
profile (e.g. [11]). Besides, the rate of decay of the power spectrum with $k$
at large $k$ influences the degree of relativistic gravitational back-reaction
(e.g. [12, 13]). Another reason for investigating the power spectrum on small
scales is that, through the time dependence of the asymptotic expansion, it
might shed light on the process of virialisation in gravitational collapse
[14]. Even though we are deriving results here for ensembles of classical
particles following Zel’dovich trajectories in an expanding background space-
time, we should like to emphasise that our results are applicable without
change also to static space-time and inertial trajectories in a different time
coordinate. Our essential assumptions are that initial phase-space coordinates
sample a Gaussian random field whose power spectrum satisfies some fairly
general properties. Our results should therefore extend to wide classes of
classical particles streaming freely from Gaussian initial conditions.

Konrad & Bartelmann [15] proved that equation (5) may be expanded as

$\mathcal{P}(k,t)\sim\sum_{m=0}^{\infty}\frac{\mathcal{P}^{(m)}(t)}{k^{3+2m}}\;,$
(8)

for $k\rightarrow\infty$, where the time dependent coefficients
$\mathcal{P}^{(m)}$ depend on the initial power spectrum only through
$\sigma_{n}^{2}$ for $n\leq m$, and they derived explicit expressions for
these coefficients. Ref. [16] also derived a similar expansion, based on a
Lagrangian view-point. However, these results relied on the assumption, that
_all_ the moments (7) are finite, which implies that if one assumes dispersion
free cold dark matter, where the initial power spectrum has a flat tail, one
needs to introduce a regulator and that all results consequently depend on
this regulator. In this work, we derive the asymptotic behaviour for cold dark
matter power spectra, without introducing a regulator, which leads to markedly
different results.

We do so by applying the Mellin transform technique of asymptotic expansions,
which consists of computing the Mellin transform, continuing it analytically
to the entirety of the complex plane (except for various possible poles), and
then using this analytic continuation, together with Gauss’ residue theorem to
compute the inverse Mellin transform by shifting the contour of integration
(see, e.g. Chapter 4 of [17] or Appendix A of [18]). This technique has been
applied before in various works in astrophysical contexts (e.g., [19, 20], to
which we refer the readers for pedagogical utilisations). Its great advantage
over other expansion techniques is that it produces, automatically, the
correct expansion functions, even non-analytic ones: it can handle non-integer
powers in a power-series, logarithms, _etc_. For completeness, we summarise
the relevant theorem in Appendix A.

This paper is structured as follows: we start by deriving the asymptotic
expansions of the initial momentum correlation matrix $\hat{C}_{pp}$ both at
small and large $q$ in §2. We then use these results in conjunction with the
Mellin transform technique in §3 to expand equation (5) at large values of
$k$, without assuming an ultraviolet cut-off on the initial power spectrum. In
§4 we specialise the expansion to late times (we define what ‘late’ means
there), where some simplifications are possible. We then look at the time-
dependence of the expansion coefficients in §5, and summarise in §6.

##  2 Small-scale asymptotics of the initial momentum correlations

We start with the small-scale asymptotics of the initial momentum-momentum
correlations of two particles that are separated by a vector $\vec{q}$. As
shown in ref. [5], the initial correlation matrix can be written as

$\hat{C}_{pp}(\mathbf{q})=-\hat{\mathbf{q}}\otimes\hat{\mathbf{q}}\;a_{2}(q)-\mathbb{I}_{3}a_{1}(q)\,,$
(9)

with $\hat{\mathbf{q}}$ being the unit vector pointing in the direction of
$\mathbf{q}$ and

$\displaystyle a_{1}(q)$
$\displaystyle\equiv\frac{\xi^{\prime}_{\psi}(q)}{q}=-\frac{1}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k\
P^{\text{(i)}}_{\delta}(k)\frac{j_{1}(kq)}{kq}\;,$ (10) $\displaystyle
a_{2}(q)$
$\displaystyle\equiv\xi^{\prime\prime}_{\psi}(q)-\frac{\xi^{\prime}_{\psi}(q)}{q}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k\
P^{\text{(i)}}_{\delta}(k)j_{2}(kq)\;,$ (11)

where $\xi_{\psi}$ denotes the correlation function of the initial velocity
potential and $j_{n}$ denotes the spherical Bessel function of order $n$. We
take the initial power spectrum to be

$P_{\delta}^{(i)}(k)=\frac{8\pi^{2}A_{s}\left(\frac{c}{H_{0}}\right)^{4}k_{s}^{n_{s}}}{25\Omega_{m}^{2}k_{0}^{n_{s}-1}}\left[\frac{k}{k_{s}}\right]^{n_{s}}\tilde{T}_{D}^{2}\left(\frac{k}{k_{\rm
eq}}\right)\equiv
A\left[\frac{k}{k_{s}}\right]^{n_{s}}\tilde{T}_{D}^{2}\left(\frac{k}{k_{\rm
eq}}\right),$ (12)

where the wave number $k_{s}$ is defined below, and use a Dicus transfer
function [21, p. 307]

$\tilde{T}_{D}^{2}(\kappa)=\left(\frac{\ln\left((\tilde{c}_{1}\kappa)^{2}+1\right)}{(\tilde{c}_{1}\kappa)^{2}}\right)^{2}\frac{(\tilde{c}_{2}\kappa)^{2}+(\tilde{c}_{3}\kappa)^{4}+(\tilde{c}_{4}\kappa)^{6}+1}{(\tilde{c}_{5}\kappa)^{2}+(\tilde{c}_{6}\kappa)^{4}+(\tilde{c}_{7}\kappa)^{6}+1},$
(13)

with $\tilde{c}_{1}=0.124$, $\tilde{c}_{2}=1.257$, $\tilde{c}_{3}=0.4452$,
$\tilde{c}_{4}=0.2197$, $\tilde{c}_{5}=1.606$, $\tilde{c}_{6}=0.8568$, and
$\tilde{c}_{7}=0.3927$.

We use the Planck 2015 cosmological parameters [22], as given in table 1. We
stress that all results presented here are independent of the explicit choice
of a transfer function for the initial power spectrum. The only important
features are the asymptotics at small scales, i.e. at $k\to\infty$, that has
to agree with (15); and that $0<n_{s}<1$.

Parameter | $h$ | $\Omega_{m}$ | $z_{\rm eq}$ | $\Omega_{r}$ | $k_{0}$ | $A_{s}$ | $k_{\rm eq}$ | $n_{s}$
---|---|---|---|---|---|---|---|---
Value | $0.6774$ | $0.3089$ | $3371$ | $\frac{\Omega_{m}}{1+z_{\rm eq}}$ | $0.05~{}\textrm{Mpc}^{-1}$ | $2.142\times 10^{-9}$ | $\frac{\Omega_{m}H_{0}}{c\sqrt{\Omega_{r}}}$ | $0.9667$
Table 1: Cosmological parameters used in this paper [22].

To derive an asymptotic expansion, it is best to move from distances $q$ to a
dimension-less variable $x$, defined by $x=qk_{s}$, where

$k_{s}=\frac{k_{\rm eq}}{c_{1}},$ (14)

and switch from wave numbers $k$ to the dimension-less quantity
$\kappa=k/k_{s}$. This effectively re-scales $\tilde{c}_{1,\ldots,7}\mapsto
c_{1,\ldots,7}\equiv{\tilde{c}_{1,\ldots,7}}/{\tilde{c}_{1}}$, which is what
we shall do below. Likewise, we define $T_{D}(\kappa)$ in the same way as
$\tilde{T}_{D}$, but with $c_{1,\ldots,7}$ instead of
$\tilde{c}_{1,\ldots,7}$.

The initial power spectrum $P_{\delta}^{(i)}$ has an asymptotic expansion as
$\kappa\equiv\frac{k}{k_{s}}\to\infty$ that reads

$P_{\delta}^{(i)}\sim\kappa^{n_{s}-4}\sum_{m=0}^{\infty}\kappa^{-m}\sum_{n=0}^{2}c_{mn}\ln^{n}\kappa\;.$
(15)

Explicitly, one has

$c_{02}=\frac{4Ac_{4}^{6}}{c_{7}^{6}},$ (16)

and $c_{00}=c_{01}=0$. We will not need the higher-order terms in this paper.

Applying the Mellin transform technique (cf. Appendix A) to the definitions of
$a_{1,2}$, we find that, as $x=qk_{s}\to 0$,

$\displaystyle a_{1}(q)$
$\displaystyle\sim-\frac{1}{6\pi^{2}}\mathcal{M}[P_{\delta}^{(i)};1]+\frac{x^{2}}{60\pi^{2}}\mathcal{M}[P_{\delta}^{(i)};3]$
$\displaystyle-\frac{x^{3-n_{s}}}{2\pi^{2}}\frac{c_{02}}{A}\sum_{j=0}^{2}\binom{2}{j}\left(-\ln
x\right)^{j}\mathcal{M}^{(2-j)}[j_{1};n_{s}-4]+o(x^{3-n_{s}})$ (17)
$\displaystyle a_{2}(q)$
$\displaystyle\sim\frac{x^{2}}{30\pi^{2}}\mathcal{M}[P_{\delta}^{(i)};3]+\frac{x^{3-n_{s}}}{2\pi^{2}}\frac{c_{02}}{A}\sum_{n=0}^{2}\binom{2}{j}\left(-\ln
x\right)^{j}\mathcal{M}^{(2-j)}[j_{2};n_{s}-3]+o(x^{3-n_{s}}),$ (18)

where $\mathcal{M}^{(k)}[f;z]$ is the $k$-th derivative of the Mellin
transform of the function $f$, $\mathcal{M}[f;z]$, with respect to $z$. Note
that the functions $a_{1,2}$ have units of area. Here, we have defined

$\mathcal{M}[P_{\delta}^{(i)};z]\equiv
Ak_{s}\int_{0}^{\infty}\kappa^{n_{s}+z-1}T_{D}^{2}(\kappa)\mathrm{d}\kappa\;,$
(19)

so that it, too, has units of area.

The coefficient $\xi_{1}(x,n_{s})$ of $x^{3-n_{s}}$ in $a_{1}$ can be written
as

$\xi_{1}(x,n_{s})=-\frac{2}{\pi^{2}}\frac{c_{4}^{6}}{c_{7}^{4}}\left(\mathcal{M}^{\prime\prime}[j_{1};n_{s}-4]-2\ln
x\mathcal{M}^{\prime}[j_{1};n_{s}-4]+\ln^{2}x\mathcal{M}[j_{1};n_{s}-4]\right),$
(20)

and the coefficient $\xi_{2}(x,n_{s})$ of $x^{3-n_{s}}$ in $a_{2}$ is likewise

$\xi_{2}(x,n_{s})=\frac{2}{\pi^{2}}\frac{c_{4}^{6}}{c_{7}^{4}}\left(\mathcal{M}^{\prime\prime}[j_{2};n_{s}-3]-2\ln
x\mathcal{M}^{\prime}[j_{2};n_{s}-3]+\ln^{2}x\mathcal{M}[j_{2};n_{s}-3]\right).$
(21)

Figure 1 shows that considering only the order $q^{2}$ asymptotics (green
lines) is insufficient to describe the functions $a_{1,2}$ (purple lines)
accurately at cosmologically reasonable small scales. Since the next order,
the $\textrm{ord}\left(q^{3-n_{s}}\right)$ terms (dashed dark blue lines), are
negative and $\left|1-n_{s}\right|\ll 1$, they balance the order $q^{2}$
terms. The sum of both orders (light blue lines) describes the small scale
asymptotics of $a_{1,2}$ very well up to scales of a megaparsec. Let us recall
here, that our results (2) and (18) are only valid for $n_{s}<1$. In the limit
$n_{s}\rightarrow 1$, the analytical continuation of
$\mathcal{M}\left[P_{\delta}^{(i)};3\right]$, i.e. the second moment of the
initial velocity potential power spectrum, diverges (see Fig. 2). Thus, for
$n_{s}=1$, the resulting series contain terms with even powers of $q$ and
logarithms of higher order than 2 due to coinciding poles of
$\mathcal{M}\left[P_{\delta}^{(i)};z\right]$ and
$\mathcal{M}\left[j_{1,2};z\right]$. The coefficients may be computed by
applying the second strategy of Case IV of the Mellin transform technique
described in Appendix A.

Figure 1: Left: The function $a_{1}(q)+\frac{\sigma_{1}^{2}}{3}$ (purple line)
is shown with $a_{1}$ from (10) together with the order $q^{2}$ asymptotics
(green line) and the absolute value of the order $q^{3-n_{s}}$ asymptotics
including the both log terms (blue dashed line) from (2) as well as the sum of
those first two leading order asymptotics (light blue line). Right: Same as
left, but $a_{2}(q)$ from (10) with asymptotics from (18). The Dicus transfer
function [21] and the parameters in Table 1 are used. Figure 2: The moments of
the initial velocity potential as a function of the spectral index $n_{s}$ for
Planck parameters (Table 1). While the first moment $\sigma_{1}^{2}$ (left)
varies only little with the spectral index, the second moment $\sigma_{2}^{2}$
and the analytic continuation of the third moment $\sigma_{3}^{2}$ diverge as
$n_{s}\rightarrow 1$ (right). The Dicus transfer function [21] is used.

##  3 Small-scale asymptotics of the Zel’dovich power spectrum

The small-scale asymptotics of the Zel’dovich power spectrum as $k\to\infty$
are determined by the saddle points of the exponent of the integrand in (5).
The saddle point, that dominates the $k\to\infty$ asymptotics of the integral,
is located at the origin $q\to 0^{+}$.

To prove this claim, let us now show that any potential saddle point of the
form $q=\textrm{ord}\left(k^{\beta}\right)$, where $\beta>0$, gives an
exponentially-decreasing contribution to $\mathcal{P}(k,t)$. As $q\to\infty$,
the Mellin-transform technique may be used to show that $a_{1,2}\sim
q^{-1-n_{s}}$. (The explicit expression for the large-scale asymptotics of
$a_{1,2}$ is shown in Appendix B.2.) This result implies that for any saddle
point where $\displaystyle\lim_{k\to\infty}q=\infty$, we have
$\hat{C}_{pp}(q)=o(1)$, and therefore that the exponential,

$t^{2}k^{i}(\hat{C}_{pp})_{ij}k^{j}-Q\;,$ (22)

with $Q\sim k^{2}$ from (6) is dominated by $-Q$ in the large $k$ limit. That
is, the contribution to $\mathcal{P}$ from such a saddle is proportional to
the exponentially-decaying factor

$\exp\left[-k^{2}\frac{\sigma_{1}^{2}t^{2}}{3}+o\left(k^{2}\right)\right].$
(23)

In a similar fashion, when $\beta=0$ and $q\neq 0$ (i.e. $q$ has a constant
finite value), $\frac{t^{2}k^{i}(\hat{C}_{pp})_{ij}k^{j}-Q}{k^{2}}$ is a
negative, order unity quantity, and thus gives rise to an exponentially-
decaying contribution. Consequently, if a saddle point exists with $\beta<0$,
i.e. at $q\to 0^{+}$, and if this saddle point yields a contribution that
decays more slowly than exponentially, then it does dominate over all other
saddle points with $\beta\geq 0$.

We are now in a position to evaluate the dominant contribution to the power
spectrum, by inserting (2) and (18) into (9) and (5), and integrating over
$q$. Explicitly, upon defining
$\sigma^{2}\equiv\mathcal{M}[P_{\delta}^{(i)};3]$ (having dimensions of area;
see (19)), we obtain, as $k\rightarrow\infty$,

$\displaystyle\mathcal{P}(k,t)$
$\displaystyle\sim\int\mathrm{d}^{3}q~{}e^{-t^{2}(\mathbf{k}\cdot\hat{\mathbf{q}})^{2}(x^{2}\frac{\sigma^{2}}{15}+x^{3-n_{s}}\xi_{2})-t^{2}k^{2}(x^{2}\frac{\sigma^{2}}{30}-x^{3-n_{s}}\xi_{1})+\mathrm{i}\mathbf{k}\cdot\mathbf{q}}$
(24) $\displaystyle\sim 2\pi
k_{s}^{-3}\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}x~{}x^{2}e^{-t^{2}k^{2}(2\mu^{2}+1)\frac{\sigma^{2}}{30}x^{2}+\mathrm{i}\mathbf{k}\cdot\mathbf{q}}\exp\left(t^{2}k^{2}x^{3-n_{s}}(\xi_{1}-\mu^{2}\xi_{2})\right)\;,$

where we changed variables to the dimension-less radial coordinate $x=k_{s}q$.
Recall that the functions $\xi_{1,2}$ both are quadratic polynomials in
logarithms of $x$. Defining $\xi\equiv\xi_{1}-\mu^{2}\xi_{2}$ and using the
Taylor expansion of the last exponential to generate an asymptotic series
expression, we arrive at

$\displaystyle\mathcal{P}(k,t)$ $\displaystyle\sim 2\pi
k_{s}^{-3}\sum_{m=0}^{\infty}\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}x~{}x^{2}e^{-t^{2}k^{2}(2\mu^{2}+1)\frac{\sigma^{2}}{30}x^{2}}\cos\left(kq\mu\right)\frac{(tk)^{2m}x^{m(3-n_{s})}}{m!}\xi^{m}$
(25)
$\displaystyle=\frac{2\pi}{(tkk_{s}\sigma)^{3}}\sum_{m=0}^{\infty}\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}y~{}y^{2}e^{-(2\mu^{2}+1)y^{2}/30}\cos\left(\frac{y\mu}{t\sigma
k_{s}}\right)\frac{y^{m(3-n_{s})}}{m!(tk\sigma)^{m(1-n_{s})}}\left[\frac{\xi}{\sigma^{2}}\right]^{m}\;,$

as $k\to\infty$. Again, recall that $\frac{\xi}{\sigma^{2}}$ is a dimension-
less quantity. The (dimension-less) integration variable $y$ is defined as
$y=tx\sigma k$. So, as $k\to\infty$,

$\mathcal{P}(k,t)\sim\frac{2\pi}{(tkk_{s}\sigma)^{3}}\sum_{m=0}^{\infty}\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}y~{}y^{2}\cos\left(\frac{y\mu}{t\sigma
k_{s}}\right)\frac{e^{-(2\mu^{2}+1)y^{2}/30}y^{m(3-n_{s})}}{m!(tk\sigma)^{m(1-n_{s})}}\left[\frac{\xi}{\sigma^{2}}\right]^{m}+O\left(k^{-5}\right).$
(26)

One can summarise this expansion as

$\mathcal{P}(k,t)=\frac{2\pi}{(k\tau_{2})^{3}}\sum_{m=0}^{\infty}\frac{k_{s}^{m(1-n_{s})}\mathcal{P}_{m}\left(t,\ln(k\tau_{2}/k_{s})\right)}{(k\tau_{2})^{m(1-n_{s})}},$
(27)

with $\tau_{2}\equiv t\sigma k_{s}$ and222Recall that $\tau_{2}$ is simply a
monotonically increasing function of time.

$\mathcal{P}_{m}\left(t,\ln(k\tau_{2}/k_{s})\right)=\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}y~{}y^{2}\cos\left(\frac{y\mu}{\tau_{2}}\right)\frac{e^{-(2\mu^{2}+1)y^{2}/30}y^{m(3-n_{s})}}{m!}\left[\frac{\xi}{\sigma^{2}}\right]^{m}.$
(28)

We are now in a position to extract all of the $k$-dependence of the power
spectrum. Like $\xi_{1,2}$, $\xi$ is a quadratic polynomial in $\ln k$, and
therefore

$\mathcal{P}(k,t)\sim\frac{1}{(k\tau_{2})^{3}}\sum_{m=0}^{\infty}\left(\frac{k_{s}}{k\tau_{2}}\right)^{m(1-n_{s})}\sum_{n=0}^{2m}\mathcal{P}_{mn}(t)\ln^{n}\left(\frac{k\tau_{2}}{k_{s}}\right)+\mathcal{O}\left(k^{-5}\right)$
(29)

with

$\mathcal{P}_{mn}(t)=\frac{2\pi}{m!(\sigma_{2}^{2})^{m}}\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}y\;\mathrm{e}^{-\frac{1}{30}(2\mu^{2}+1)y^{2}}\mathrm{e}^{\mathrm{i}\mu\frac{y}{\tau_{2}}}y^{m(3-n_{s})+2}\xi_{m,n}\left(\mu^{2},\ln
y\right)\;,$ (30)

with $\xi_{m,0}\left(\mu^{2},\ln
y\right)\equiv\tilde{\xi}^{m}_{0}\left(\mu^{2},\ln y\right)$ and for $n\neq
0$, the functions $\xi_{m,n}$ are given recursively by

$\xi_{m,n}\left(\mu^{2},\ln
y\right)=\frac{1}{n\tilde{\xi}_{0}}\left[(m-n+1)\tilde{\xi}_{1}\xi_{m,n-1}+(2m-n+2)\tilde{\xi}_{2}\xi_{m,n-2}\right]\;,$
(31)

with

$\displaystyle\tilde{\xi}_{2}(\mu^{2},\ln y)$
$\displaystyle\equiv\frac{\xi_{2}}{\sigma_{2}^{2}}\left[(3-n_{s})\mu^{2}+1\right]\;,$
(32) $\displaystyle\tilde{\xi}_{1}(\mu^{2},\ln y)$
$\displaystyle\equiv-\frac{\xi_{1}}{\sigma_{2}^{2}}\left[\left(3-n_{s}+2\frac{\xi_{2}}{\xi_{1}}\right)\mu^{2}+1\right]-2\ln
y\frac{\xi_{2}}{\sigma_{2}^{2}}\left[(3-n_{s})\mu^{2}+1\right]$ (33)
$\displaystyle\tilde{\xi}_{0}(\mu^{2},\ln y)$
$\displaystyle\equiv\ln^{2}y\frac{\xi_{2}}{\sigma_{2}^{2}}\left[(3-n_{s})\mu^{2}+1\right]+\ln
y\frac{\xi_{1}}{\sigma_{2}^{2}}\left[\left(3-n_{s}+2\frac{\xi_{2}}{\xi_{1}}\right)\mu^{2}+1\right]$
$\displaystyle\quad+\frac{\xi_{0}}{\sigma_{2}^{2}}\left[\left(3-n_{s}+\frac{\xi_{1}}{\xi_{0}}\right)\mu^{2}+1\right]\;.$
(34)

Equation (29) is the main result of this paper: it shows the full dependence
of the Zel’dovich power spectrum on $k$, for large values $\frac{k}{k_{s}}$,
up to corrections of order $k^{-5}$ (for the Dicus transfer function), which
are in principle calculable using the Mellin transform technique. Here, as
opposed to earlier works [15, 16], higher moments of the initial power
spectrum needn’t be finite. It also shows why terms beyond the order $k^{-3}$
are necessary – the weak increase of logarithms in the numerator almost
precisely cancels the weak increase (for realistic $n_{s}$) of
$k^{m(1-n_{s})}$ in the denominator. The need for high orders can be seen in
Fig. 4, where we show the series computed up to a maximum value of $m\leq M$,
for various $M$, and will be discussed for redshift $z=0$ in more detail in
the next section.

Figure 3: Left: The Zel’dovich power spectrum (purple line) from (5) and the
power spectrum from linear theory (gray line)
$P_{\delta}^{\text{lin.}}=D_{+}^{2}(z)P_{\delta}^{ini}$ are shown at redshift
$z=20$. Right: Same power spectra at today’s redshift $z=0$. Parameters as
shown in Table 1.

##  4 Late time evolution

As $t$ grows, $\tau_{2}$ becomes quite large. The argument of the cosine in
(28) tends to $0$ at late times, thus

$\mathcal{P}_{m}\to\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}y~{}y^{2}\frac{e^{-(2\mu^{2}+1)y^{2}/30}y^{m(3-n_{s})}}{m!}\left[\frac{\xi}{\sigma^{2}}\right]^{m},$
(35)

which is time-independent and simpler to evaluate numerically. More precisely,
in the late-time limit $\tau_{2}\rightarrow\infty$, $\mathcal{P}_{mn}$ from
(30) can be expanded into the series

$\mathcal{P}_{mn}(t)\to\frac{2\pi}{m!(\sigma_{2}^{2})^{m}}\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}y\;\mathrm{e}^{-\frac{1}{30}(2\mu^{2}+1)y^{2}}\sum_{l=0}^{\infty}\frac{(-1)^{l}}{(2l)!}\left(\mu\frac{y}{\tau_{2}}\right)^{2l}y^{m(3-n_{s})+2}\xi_{m,n}\left(\mu^{2},\ln
y\right)\;.$ (36)

For the parameters in Table 1, $\tau_{2}\approx 259$ at redshift $z=0$, so one
can safely use the late-time expression. The results are shown in Fig. 4. One
can see excellent agreement with the asymptotic expansion at scales $k\geq
10^{3}$ Mpc-1 when going to high orders in the asymptotics: for low numbers of
terms ($M=15$, green line), the asymptotics underestimate the actual
Zel’dovich power spectrum by more than an order of magnitude. Summing more and
more orders improves the result significantly. Including terms up to order
$M=1000$ leads to an agreement of the asymptotics with the Zel’dovich power
spectrum at small scales. We note that the computation of the double integral
in (35) is more involved for large $m$, and we explain the method we used in
Appendix D. Furthermore, let us note that the order $k^{-5}$ term can be
neglected although we go to very high orders in $M$, which we discuss in
Appendix C.

Figure 4: The dimension-less cold dark matter power spectrum in the Zel’dovich
approximation (purple line) at today’s redshift $z=0$, compared with the
analytical prediction of equation (35), up to high orders $M$. The Dicus
transfer function [21] and the parameters in table 1 are used.

##  5 Implications for dark matter

If dark matter made of massive fermions, then the initial power spectrum does
not decay like $k^{n_{s}-4}\log^{2}k$. It may rather have a cutoff, determined
by the particle mass and the particle production mechanism, which is related
to the type(s) of interaction the particle undergoes [23]. The ultraviolet
cut-off $k_{D}$ is also typically related to a temperature of dark matter. In
this section, we discuss two aspects of the Zel’dovich power spectrum that are
related to dark matter with a finite temperature. In [15], Konrad & Bartelmann
have shown that the Zel’dovich power spectrum for large $k$ can be expanded in
a series with terms of order $k^{-3-2m}$, when the initial power spectrum has
a cut-off and all moments of the initial velocity potential power spectrum
exist. The first two coefficients in this series (8) are given by

$\displaystyle\mathcal{P}^{(0)}(t)$
$\displaystyle=3(4\pi)^{3/2}\left(\frac{5}{2\tau_{2}^{2}}\right)^{3/2}\mathrm{e}^{-\frac{5}{2\tau_{2}^{2}}}\;,$
(37) $\displaystyle\mathcal{P}^{(1)}(t)$
$\displaystyle=\frac{(4\pi)^{3/2}}{28}\frac{\sigma_{3}^{2}}{\sigma_{2}^{2}}\left(\frac{5}{2\tau_{2}^{2}}\right)^{5/2}\mathrm{e}^{-\frac{5}{2\tau_{2}^{2}}}\left[123-132\left(\frac{5}{2\tau_{2}^{2}}\right)+20\left(\frac{5}{2\tau_{2}^{2}}\right)^{2}\right]\;,$

where
$\mathcal{P}(k,t)\sim\mathcal{P}^{(0)}(t)k^{-3}+\mathcal{P}^{(1)}(t)k^{-5}$ as
$k\to\infty$. While the first term describes the asymptotic $k^{-3}$ tail very
well, the intersection with the second term, of order $k^{-5}$, engenders a
time-dependent scale
$k_{\text{asymp}}=\sqrt{\left|\frac{\mathcal{P}^{(0)}}{\mathcal{P}^{(1)}}\right|}$,
below which the asymptotics fail to reproduce the Zel’dovich power spectrum.
Thus, an analytical description of the latter between this scale and the
linear regime was missing.

In Fig. 5 we show the dimension-less Zel’dovich power spectra of strictly cold
dark matter, i.e. when dark matter is treated as a non-relativistic Vlasov
fluid at all scales (purple line), at redshift $z=0$, together with those of
three types of warm dark matter, where the initial (linear) power spectrum is
exponentially cut off at a scale $k_{D}=\\{10^{2},10^{4},10^{6}\\}$ Mpc-1 by a
multiplicative Gaussian $e^{-k^{2}/k_{D}^{2}}$ (yellow, blue and green line,
respectively). The latter corresponds to a power spectrum of WIMP dark matter
with particle mass of $\sim 100$ GeV [23]. For small and intermediate wave
numbers, the Zel’dovich power spectra align with the spectrum of the strictly
cold dark matter. At large wave numbers, the spectra become flat, reaching the
$k^{-3}$ asymptotics (coloured dashed lines). The wave number where the
spectra with finite initial temperature deviate from the strictly cold dark
matter spectrum is in all cases indicated very well by the intersection of the
$k^{-3}$ (coloured dashed lines) and the $k^{-5}$ (dotted dashed lines)
asymptotics. In case of the WIMP dark matter, the wave number regime just
below validity of the $k^{-3}$ asymptotics is now very well described by the
asymptotics of the strictly cold dark matter (thick dashed dark blue line)
that we derived in this paper up to order $M=1000$. This is remarkable, since
the latter contains information about the asymptotic tail of the initial power
spectrum, that is actually cut-off for WIMP dark matter, but ultimately
originates from the fact that
$\sigma_{3}^{2}/k_{s}^{4}\gg\left|\mathcal{M}[P_{\delta}^{(i)};5]\right|$,
where the former is computed with a cut-off, and the latter – without one.

It may be surprising, _prima facie_ that the asymptotic tails of the
Zel’dovich power spectra with initial ultraviolet cut-off have a higher
amplitude at redshift $z=0$ the stronger the ultraviolet cut-off is, i.e. the
smaller $k_{D}$. The reason for this is that the time evolution of the
asymptotic amplitude $\mathcal{P}^{(0)}$, (37), depends on the product
$\sigma_{2}^{2}t^{2}$. This implies that smaller values of $\sigma_{2}$ lead
to a slower evolution of the asymptotic tail. Since at redshift $z=0$, all of
these amplitudes are decreasing in time due to re-expansion, the spectra with
larger $k_{D}$, i.e. larger $\sigma_{2}^{2}$ already progressed further in re-
expansion. The time evolution of the amplitude is discussed in detail in [15].

Figure 5: The cold dark matter power spectrum in the Zel’dovich approximation
without small scale smoothing (purple line), compared with power spectra that
are initially smoothed on small scales. For larger smoothing wave numbers,
i.e. smaller smoothing scales, the initially smoothed power spectra align with
the unsmoothed spectrum up to higher wave numbers $k$. This implies that the
intermediate regime of initially smoothed Zel’dovich power spectra is well
approximated by the asymptotics of the exact cold dark matter power spectrum.
The Dicus transfer function [21] and the parameters in Table 1 are used.

##  6 Summary and discussion

In this paper we derived an asymptotic expansion for the cold dark matter
power spectrum in the Zel’dovich approximation, at large wave numbers $k$, by
using the Mellin transform technique; the result, Eq. (29), shows that in
addition to the expected $k^{-3}$, $k^{-5}$, _etc._ terms, there are terms
that go like negative integer powers $k^{1-n_{s}}$, accompanied by positive
powers of $\log k$. These are a _sine qua non_ in any practical use of
asymptotic expansions of the Zel’dovich power spectrum, since only by going to
high order, can a reasonable accuracy be achieved (see Fig. 4).

This differs from the case where the power spectrum contains an ultraviolet
cut-off, $k_{D}$ [15]. In this case, there exists a cut-off and time dependent
wave number $k_{\text{asymp}}$ above which the Zel’dovich power spectrum
aligns with the $k^{-3}$ asymptotics. However if $k_{D}\gg k_{s}$, there is an
intermediate regime, where $k_{s}\ll k\ll k_{\text{asymp}}$. There,
physically, the cut-off should have no effect, and for these values of $k$, we
indeed see in Fig. 5 that $\mathcal{P}(k,t)$ follows (29), rather than the
asymptotic expansion of [15, 16]. This already means that for such $k$ one
should not use the expansion with a cut-off, but rather treat the cut-off as
non-existent. The applicability of an expansion without a cut-off, well below
the cut-off in asymptotic expansions is known to exist elsewhere in cosmology,
e.g. in the distribution of the astrophysical gravitational-wave background
[20]. We therefore have shown that (29) is valid even for realistic models of
dark matter, which do include some ultraviolet cut-off, as long as one
considers scales below $k_{\text{asymp}}$.

While one has to include many terms in the series (29) for it to be accurate,
knowing the functional form of the $k$-dependence of the power spectrum is of
theoretical significance in itself. Evaluating the Zel’dovich power spectrum
can be achieved by solving the integral (5) numerically. This expansion will
help to find out more about the mathematical structure of the Zel’dovich power
spectrum in general; we anticipate that once this series is known, in the
future one would be able to re-sum it to arrive at more accurate results
without needing to go to very high orders. The Zel’dovich power spectrum
resembles the zeroth order power spectrum of KFT and is a fundamental
building-block of perturbative terms; the result of this paper will therefore
guide future work on KFT perturbation theory and thereby on analytical
expressions for the non-linear power spectrum of dark matter, and put them on
mathematically solid ground. In future work we will extend the calculation
here beyond the Zel’dovich approximation, and endeavour to find an asymptotic
series of the form (29) for the full cold dark matter power spectrum [24].

Moreover, as is visible from Fig. 2, the coefficients $\mathcal{P}_{m}(t,\ln
k)$ (or $\mathcal{P}_{mn}(t)$) are extremely sensitive to the spectral index
$n_{s}$. In future surveys, if they could be inferred experimentally, they
could furnish an additional probe of this cosmological parameter.

## Acknowledgements

S.K. and M.B. thank Manfred Salmhofer for numerous helpful discussions, and
Y.B.G. thanks Vincent Desjacques, Hagai Perets and Mor Rozner for useful
comments. S.K. thanks Robert Scheichl for helpful conversations. Y. B. G. is
grateful for the generous hospitality of the Institute of Theoretical Physics
at Heidelberg University, where some work on this research was done. This
research received funding from the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 -
390900948 (the Heidelberg STRUCTURES Excellence Cluster). Y. B. G. is also
supported by the Adams Fellowship Programme of the Israeli Academy of Sciences
and Humanities.

##  Appendix A Mellin transform technique

In this section, we briefly sketch the Mellin transform technique as described
in [17], in-so-far as it is used in this paper. This technique is a powerful
tool to derive the asymptotics of functions that are described by integral
transforms. The Mellin transform of a function $f(k)$ is defined as

$\mathcal{M}\left[f;z\right]\equiv\int_{0}^{\infty}\mathrm{d}k~{}k^{z-1}f(k)\;.$
(38)

The H-transform of a function $f$ with respect to a function $h$ is defined as

$\displaystyle H[f;\lambda]$
$\displaystyle\equiv\int_{0}^{\infty}\mathrm{d}k~{}f(k)h(\lambda k)$ (39)
$\displaystyle=\frac{1}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}\lambda^{-z}\mathcal{M}[h;z]\mathcal{M}[f;1-z]\mathrm{d}z\;,$
(40)

where $c\in\mathbb{R}$ lies in the strip of analyticity of $\mathcal{M}[h;z]$
and $\mathcal{M}[f;1-z]$. Upon defining

$G(z)\equiv\mathcal{M}[h;z]\mathcal{M}[f;1-z]\;,$ (41)

and using the residue theorem, the $H$-transform has the asymptotic expansion
as $\lambda\rightarrow\infty$

$H[f;\lambda]=-\sum_{c<\mathrm{Re}~{}z<R}\mathrm{Res}\left\\{\lambda^{-z}G(z)\right\\}+\frac{1}{2\pi\mathrm{i}}\int_{R-\mathrm{i}\infty}^{R+\mathrm{i}\infty}G(z)\mathrm{d}z\;,$
(42)

where $R$ is chosen such that $\lambda^{-z}G(z)$ has no poles for
$\mathrm{Re}~{}z=R$. The latter integral is $\mathcal{O}(\lambda^{-R})$ as
$\lambda\rightarrow\infty$, where $G$ is analytically continued as much as
possible, preferably to a meromorphic function of $z$.

To calculate the asymptotics as $\lambda\rightarrow\infty$ explicitly, we
assume that, as $t\rightarrow\infty$,

$h(t)\sim\mathrm{e}^{-dt^{\nu}}\sum_{m=0}^{\infty}t^{-r_{m}}\sum_{n=0}^{N(m)}c_{mn}\ln^{n}t\;,$
(43)

where $d\geq 0$, $\nu>0$, $\mathrm{Re}~{}r_{m}\nearrow\infty$, $0\leq N(m)$ is
finite. We also assume that as $t\rightarrow 0$,

$f(t)\sim\mathrm{e}^{-qt^{-\mu}}\sum_{m=0}^{\infty}t^{a_{m}}\sum_{n=0}^{\bar{N}(m)}p_{mn}\ln^{n}t\;,$
(44)

where $q\geq 0$, $\mu>0$, $\mathrm{Re}~{}a_{m}\nearrow\infty$, and
$0\leq\bar{N}(m)$ is finite. The relevant theorem, quoted here from ref. [17]
for completeness, considers four cases:

  * •

Case I: $d\neq 0\neq q$:

$H[f;\lambda]=o(\lambda^{-R})\quad\forall R>0\;.$ (45)

  * •

Case II: $d\neq 0$, $q=0$:

$H[f;\lambda]\sim\sum_{m=0}^{\infty}\lambda^{-1-a_{m}}\sum_{n=0}^{\bar{N}(m)}p_{mn}\sum_{j=0}^{n}\binom{n}{j}(-\ln\lambda)^{j}\left.\mathcal{M}^{(n-j)}[h;z]\right|_{z=1+a_{m}}\;.$
(46)

  * •

Case III: $d=0$, $q\neq 0$:

$H[f;\lambda]\sim\sum_{m=0}^{\infty}\lambda^{-r_{m}}\sum_{n=0}^{N(m)}c_{mn}\sum_{j=0}^{n}\binom{n}{j}\ln^{j}\lambda\left.\mathcal{M}^{(n-j)}[f;z]\right|_{z=1-r_{m}}\;.$
(47)

  * •

Case IV: $d=0$, $q=0$:

    * –

$r_{m}\neq a_{n}+1$ for all pairs $n,m$, then the resulting asymptotics is
simply given by the sum of Cases II and III.

    * –

When $r_{m}=a_{n}+1$ for one or several pairs $n,m$, then one has to consider
the residue

$\mathrm{Res}_{r_{m}}\left\\{\lambda^{-z}\mathcal{M}[h;z]\mathcal{M}[f;1-z]\right\\}\;,$
(48)

which follow from the equation below for the residue. Note that the
multiplicity (order) of the pole is increased, which produces logarithmic
terms even if there weren’t any in $f$ or $h$ to begin with.

In all the above cases, $\mathcal{M}$ denotes the Mellin transform, or its
analytical continuation. $\mathcal{M}^{(n-j)}$ denotes the $(n-j)$-th
derivative of the Mellin transform, or its analytical continuation.

##  Appendix B Asymptotics of the initial momentum correlations

###  B.1 Small Scale Asymptotics

We assume that the initial power spectrum $P_{\delta}^{(i)}(k)$ has an
asymptotic expansion as $k\rightarrow\infty$ that is given by

$\displaystyle P_{\delta}^{(i)}(k)$
$\displaystyle=A\left(\frac{k}{k_{s}}\right)^{n_{s}}T^{2}_{D}\left(\frac{k}{k_{s}}\right)$
(49)
$\displaystyle\sim\left(\frac{k}{k_{s}}\right)^{n_{s}-4}\sum_{m=0}^{\infty}\left(\frac{k}{k_{s}}\right)^{-m}\sum_{n=0}^{2}c_{mn}\ln^{n}\left(\frac{k}{k_{s}}\right)\;.$
(50)

Note that with $\kappa=k/k_{s}$ and $k_{s}$ as defined in (14) the argument is
actually dimension-less.

The functions $a_{1}$ and $a_{2}$ that characterise the initial momentum
correlations are given by

$\displaystyle a_{1}(q)$
$\displaystyle:=\frac{\xi^{\prime}_{\psi}(q)}{q}=-\frac{1}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k\
P^{\text{(i)}}_{\delta}(k)\frac{j_{1}(kq)}{kq}\;,$ (51) $\displaystyle
a_{2}(q)$
$\displaystyle:=\xi^{\prime\prime}(q)-\frac{\xi^{\prime}_{\psi}(q)}{q}=\frac{1}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k\
P^{\text{(i)}}_{\delta}(k)j_{2}(kq)\;.$ (52)

let us derive the asymptotics of these functions for small scales, i.e. when
$q\rightarrow 0$, by applying the Mellin transform technique.

Let’s commence with the function $a_{1}$ for the large parameter $x^{-1}\equiv
k_{s}^{-1}q^{-1}$. We start by transforming the integral expression for
$a_{1}$ by using $\kappa=k/k_{s}$

$\displaystyle a_{1}(q)$
$\displaystyle=-\frac{1}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k~{}P_{\delta}^{(i)}(k)\frac{j_{1}(\kappa
x)}{\kappa x}$ (53)
$\displaystyle=-\frac{1}{2\pi^{2}}x^{-1}k_{s}\int_{0}^{\infty}\mathrm{d}\kappa~{}P_{\delta}^{(i)}(k_{s}\kappa
x^{-1})\frac{j_{1}(\kappa)}{\kappa}$ (54)
$\displaystyle=-\frac{1}{2\pi^{2}}x^{-1}k_{s}\int_{0}^{\infty}\mathrm{d}\kappa~{}A\left(\frac{\kappa}{x}\right)^{n_{s}}T^{2}_{D}\left(\frac{\kappa}{x}\right)\frac{j_{1}(\kappa)}{\kappa}$
(55)

Since the spherical Bessel function $j_{1}$ has a convergent Taylor series at
$\kappa=0$,

$\frac{j_{1}(\kappa)}{\kappa}=\sum_{m=0}^{\infty}\kappa^{2m}\frac{(-1)^{m}}{(2m+3)(2m+1)!},$
(56)

and $T^{2}_{D}$ has a power law tail with logarithms (50), the resulting
asymptotics is given by Case IV of the appendix A. Since we assume
$0<n_{s}<1$, the exponents of the series of $P_{\delta}^{(i)}$ in (50) are not
integers, which implies that the asymptotic series is a sum of Cases II and
III:

$\displaystyle a_{1}(q)$
$\displaystyle\sim-\frac{1}{2\pi^{2}}\sum_{m=0}^{\infty}x^{2m}\frac{(-1)^{m}}{(2m+3)(2m+1)!}\mathcal{M}\left[P_{\delta}^{(i)};1+2m\right]$
(57)
$\displaystyle-\frac{1}{2\pi^{2}}x^{3-n_{s}}\sum_{m=0}^{\infty}x^{m}\sum_{n=0}^{2}\frac{c_{mn}}{A}\sum_{j=0}^{n}\binom{n}{j}(-\ln
x)^{j}\left.\mathcal{M}^{(n-j)}\left[j_{1};z\right]\right|_{z=n_{s}-4}\;,$
(58)

where we have defined

$\mathcal{M}\left[P_{\delta}^{(i)};z\right]\equiv
Ak_{s}\int_{0}^{\infty}\kappa^{n_{s}+z-1}T^{2}_{D}(\kappa)\;\mathrm{d}\kappa\;.$
(59)

Note that for $z=2m-1$, these terms are scaled versions of the analytical
continuation of the moments of the initial velocity power spectrum (7),

$\mathcal{M}\left[P_{\delta}^{(i)};2m-1\right]=2\pi^{2}\frac{\sigma_{m}^{2}}{k_{s}^{2m-2}}\;.$
(60)

Similarly, we get for $a_{2}$ as $q\rightarrow 0$

$\displaystyle a_{2}(q)$
$\displaystyle=\frac{1}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k~{}P_{\delta}^{(i)}(k)j_{2}(kq)$
(61)
$\displaystyle=\frac{1}{2\pi^{2}}x^{-1}k_{s}\int_{0}^{\infty}\mathrm{d}\kappa~{}P_{\delta}^{(i)}(k_{s}\kappa
x^{-1})j_{2}(\kappa)$ (62)
$\displaystyle\sim\frac{1}{2\pi^{2}}\sum_{m=0}^{\infty}x^{2m+2}\frac{(-1)^{m}(2m+2)}{(2m+5)(2m+3)!}\mathcal{M}\left[P_{\delta}^{(i)};3+2m\right]$
(63)
$\displaystyle+\frac{1}{2\pi^{2}}x^{3-n_{s}}\sum_{m=0}^{\infty}x^{m}\sum_{n=0}^{2}\frac{c_{mn}}{A}\sum_{j=0}^{n}\binom{n}{j}(-\ln
q)^{j}\left.\mathcal{M}^{(n-j)}\left[j_{2};z\right]\right|_{z=n_{s}-3}\;.$
(64)

###  B.2 Large Scale Asymptotics

We assume that the initial power spectrum $P_{\delta}^{\text{(i)}}$ has an
asymptotic expansion as $k\rightarrow 0$ that is given by

$P_{\delta}^{\text{(i)}}\sim\kappa^{n_{s}}\sum_{m=0}^{\infty}p_{m}\kappa^{m}\;,$
(65)

with $\kappa=k/k_{s}$ and coefficients $p_{m}$ that have units of volume. We
consider again the integral expressions of $a_{1}$ and $a_{2}$ in terms of the
dimensional integration variable $\kappa$ and $x=k_{s}q$,

$\displaystyle a_{1}(q)$
$\displaystyle=-\frac{1}{2\pi^{2}}k_{s}\int_{0}^{\infty}\mathrm{d}\kappa\,P_{\delta}^{\text{(i)}}(k_{s}\kappa)\frac{j_{1}(\kappa
x)}{\kappa x}\;,$ (66) $\displaystyle a_{2}(q)$
$\displaystyle=\frac{1}{2\pi^{2}}k_{s}\int_{0}^{\infty}\mathrm{d}\kappa\,P_{\delta}^{\text{(i)}}(k_{s}\kappa)j_{2}(\kappa
x)\;.$ (67)

Since the spherical Bessel functions $j_{n}(x)$ are oscillating for
$x\rightarrow\infty$, in order to compute the asymptotics of $a_{1}$ and
$a_{2}$ for $q\rightarrow\infty$, Case II of the Mellin transform technique
applies for both integrals. Thus, as $x=k_{s}q\rightarrow\infty$,

$\displaystyle a_{1}(q)$
$\displaystyle\sim-\frac{1}{2\pi^{2}}\sum_{m=0}^{\infty}x^{-1-n_{s}-m}k_{s}p_{m}\mathcal{M}\left[j_{1};n_{s}+m\right]$
$\displaystyle=-\frac{x^{-1-n_{s}}}{2\pi^{2}}k_{s}p_{0}\sqrt{\pi}\frac{2^{n_{s}-2}\Gamma\left(\frac{n_{s}+1}{2}\right)}{\Gamma\left(\frac{4-n_{s}}{2}\right)}+\mathcal{O}\left(q^{-2-n_{s}}\right)\;,$
(68) $\displaystyle a_{2}(q)$
$\displaystyle\sim\frac{1}{2\pi^{2}}\sum_{m=0}^{\infty}x^{-1-n_{s}-m}k_{s}p_{m}\mathcal{M}\left[j_{2};1+n_{s}+m\right]$
$\displaystyle=\frac{x^{-1-n_{s}}}{2\pi^{2}}k_{s}p_{0}\sqrt{\pi}\frac{2^{n_{s}-1}\Gamma\left(\frac{n_{s}+3}{2}\right)}{\Gamma\left(\frac{4-n_{s}}{2}\right)}+\mathcal{O}\left(q^{-2-n_{s}}\right)\;.$
(69)

###  B.3 Computing the analytic continuation of the Mellin transform

In order to actually evaluate the coefficients appearing in the small scale
asymptotics of $a_{1}$ and $a_{2}$, we need to compute the analytical
continuation of the moments of the initial power spectrum
$\mathcal{M}\left[P_{\delta}^{(i)};2m-1\right]$ into the right-hand half-plane
numerically. We assumed that $P_{\delta}^{(i)}$ has an asymptotic expansion as
$k\rightarrow\infty$ that is given by

$P_{\delta}^{(i)}\sim\sum_{m=0}^{\infty}\kappa^{-r_{m}}\sum_{n=0}^{2}c_{mn}\ln^{n}\kappa\;.$
(70)

The analytic continuation of its Mellin transform is calculated by first
defining the function

$h_{p}(\kappa):=\begin{cases}\sum_{\mathrm{Re}~{}r_{m}<p}\kappa^{-r_{m}}\sum_{n=0}^{N(m)}c_{mn}\ln^{n}\kappa\;,&\quad\text{for
}\kappa\geq 1\;,\\\ 0\;,&\quad\text{else}\;.\end{cases}$ (71)

Then, by defining

$\tilde{P}_{\delta}^{(i)}(\kappa):=A\kappa^{n_{s}}T_{D}^{2}\left(\kappa\right)=P_{\delta}^{(i)}(k_{s}\kappa)\;,$
(72)

we get for the following expression

$\displaystyle\mathcal{M}\left[P_{\delta}^{(i)},z\right]$
$\displaystyle=k_{s}\int_{0}^{\infty}\mathrm{d}\kappa~{}\kappa^{z-1}\tilde{P}_{\delta}^{(i)}(\kappa)$
(73)
$\displaystyle=k_{s}\int_{0}^{1}\mathrm{d}\kappa~{}\kappa^{z-1}\tilde{P}_{\delta}^{(i)}(\kappa)+k_{s}\int_{1}^{\infty}\mathrm{d}\kappa~{}\kappa^{z-1}\tilde{P}_{\delta}^{(i)}(\kappa)$
(74)
$\displaystyle=k_{s}\int_{0}^{1}\mathrm{d}\kappa~{}\kappa^{z-1}\tilde{P}_{\delta}^{(i)}(\kappa)+k_{s}\int_{1}^{\infty}\mathrm{d}\kappa~{}\kappa^{z-1}\left[\tilde{P}_{\delta}^{(i)}(\kappa)-h_{p}(\kappa)\right]+k_{s}\int_{1}^{\infty}\mathrm{d}\kappa~{}\kappa^{z-1}h_{p}(\kappa)\;,$
(75)

where the first two integrals can be calculated numerically and the last one
is given by its analytical continuation

$\int_{1}^{\infty}\mathrm{d}\kappa~{}\kappa^{z-1}h_{p}(\kappa)=\sum_{\mathrm{Re}~{}r_{m}<p}\sum_{n=0}^{N(m)}\frac{(-1)^{n+1}c_{mn}n!}{(z-r_{m})^{n+1}}\;.$
(76)

##  Appendix C Neglecting other orders in the asymptotics

While for dark matter with an initial ultraviolet cut-off the $k^{-5}$
asymptotics typically dominate the $k^{-3}$ asymptotics on small scales (see
figure 5), this is not the case for strictly cold dark matter. To compute the
asymptotics for the Zel’dovich power spectrum of strictly cold dark matter, we
therefore exclusively added terms from the sum (25), i.e. we did not add the
order $k^{-5}$ term, although $M=1000$ implies that we went up to order
$-M(1-n_{s})=-33.3$. Nevertheless, as can be seen in figure 6: at today’s
redshift $z=0$ this term is negligible at small scales, because the logarithms
in the numerator almost cancel the powers of $k$ in the denominator.

Figure 6: The cold dark matter power spectrum in the Zel’dovich approximation
at $z=0$ (solid purple line), compared with the analytical prediction of the
asymptotics (35), up to high orders. On the scales shown here, the $k^{-3}$
asymptotics (dashed purple line) underestimate the dimension-less power
spectrum by approximately an order of magnitude and the
$\textrm{ord}\left(k^{-5}\right)$ term is completely negligible. The Dicus
transfer function and the parameters in table 1 are used.

##  Appendix D Large $m$

The integral in equation $\eqref{eqn:late time Pm}$ is a double integral over
$y$ and $\mu$. Simply evaluating it numerically in `Mathematica` works only up
to $m\lesssim 50$. For larger $m$, one might presume naïvely that a simple
Laplace-type approximation would be appropriate, but this is true only for the
$y$ integral.

Let us start, therefore, with the $y$ integral, and write

$\mathcal{P}_{m}=\frac{1}{m!}\int_{0}^{\infty}\mathrm{d}y\int_{-1}^{1}\mathrm{d}\mu~{}y^{2}e^{m(1-n_{s})\ln
y-(2\mu^{2}+1)y^{2}/30}\left[\frac{\xi}{\sigma^{2}}\right]^{m}$ (77)

This is a moveable saddle problem, and therefore we need to re-scale $y$
appropriately before proceeding. The exponent

$m(3-n_{s})\ln y-\frac{2\mu^{2}+1}{30}y^{2}$ (78)

is stationary when

$y^{2}=m\frac{15(3-n_{s})}{2\mu^{2}+1},$ (79)

i.e. the stationary point is $\sim\sqrt{m}$. So, let us change variables to
$y=u\sqrt{m}$, and define

$u_{0}(\mu)\equiv\sqrt{\frac{15(3-n_{s})}{1+2\mu^{2}}}.$ (80)

$\mathcal{P}_{m}$ then becomes

$\displaystyle\mathcal{P}_{m}$
$\displaystyle\sim\frac{1}{m!}\int_{-1}^{1}\mathrm{d}\mu\int_{0}^{\infty}\mathrm{d}u~{}m^{(3+m(3-n_{s}))/2}u_{0}^{2}\left[\frac{\xi(y=\sqrt{m}u_{0}(\mu)/k\tau_{2})}{\sigma^{2}}\right]^{m}$
(81)
$\displaystyle\times\exp\left[\frac{m(3-n_{s})}{2}\left(\ln(u_{0}^{2})-1\right)\right]\exp\left[-m\frac{1+2\mu^{2}}{15}(u-u_{0})^{2}+\ldots\right]$
(82)
$\displaystyle\sim\frac{m^{\frac{3+m(3-n_{s})}{2}}e^{-m(3-n_{s})/2}}{m!}\int_{-1}^{1}\mathrm{d}\mu~{}u_{0}^{2+m(3-n_{s})}\left[\frac{\xi(y=\sqrt{m}u_{0}(\mu)/k\tau_{2})}{\sigma^{2}}\right]^{m}\int_{-\infty}^{\infty}\mathrm{d}u~{}e^{-\frac{m(1+2\mu^{2})}{15}(u-u_{0})^{2}}$
(83)
$\displaystyle=\frac{m^{1+\frac{m(3-n_{s})}{2}}e^{-m(3-n_{s})/2}\sqrt{15\pi}}{m!}\int_{-1}^{1}\mathrm{d}\mu~{}\frac{u_{0}^{2+m(3-n_{s})}}{\sqrt{1+2\mu^{2}}}\left[\frac{\xi(y=\sqrt{m}u_{0}(\mu)/k\tau_{2})}{\sigma^{2}}\right]^{m},$
(84)

where, in moving to the penultimate line from the one above it, we used
Laplace’s method for $m\gg 1$.

Equation (84) can be used to evaluate $\mathcal{P}_{m}$ numerically with
`Mathematica` up to $m\lesssim 90$ – after that, the power of $u_{0}$ and
$\xi$ in the integral becomes too large. Upon changing $\mu$ to $x$, equation
(84) may be written as

$\mathcal{P}_{m}\sim\frac{m^{1+\frac{m(3-n_{s})}{2}}e^{-m(3-n_{s})/2}\sqrt{15\pi}}{m!}\int_{-1}^{1}\mathrm{d}x~{}\frac{u_{0}^{2}(x)e^{m\varphi(x)}}{\sqrt{1+2x^{2}}},$
(85)

where

$\varphi(x)\equiv(3-n_{s})\ln
u_{0}(x)+\ln\left[\frac{\xi(y=\sqrt{m}u_{0}(x)/k\tau_{2},x)}{\sigma^{2}}\right].$
(86)

Laplace’s method works only if $\varphi$ has variations of order one about it
maximum $x=0$. However, here, $\varphi$ is almost constant, for the parameters
in table 1 and, e.g. $k\gtrsim 10$. For $80\lesssim m\lesssim 1000$,
$m\varphi$ is still almost constant for $x\in[-1,1]$ (i.e.
$m|\varphi^{\prime\prime}(0)|x^{2}\ll 1$), and therefore a Laplace-type
approximation would be inaccurate. Instead, we simply approximate
$\varphi(x)\approx\varphi(0)+\varphi^{\prime\prime}(0)x^{2}/2$, and write

$\displaystyle\mathcal{P}_{m}$
$\displaystyle\approx\frac{{}^{1+\frac{m(3-n_{s})}{2}}e^{-m(3-n_{s})/2+m\varphi(0)}\sqrt{15\pi}}{m!}\int_{-1}^{1}\mathrm{d}x~{}\frac{u_{0}^{2}(x)}{\sqrt{1+2x^{2}}}\left(1+m\varphi^{\prime\prime}(0)\frac{x^{2}}{2}\right)$
(87)
$\displaystyle=\frac{m^{1+\frac{m(3-n_{s})}{2}}e^{-m(3-n_{s})/2+m\varphi(0)}\sqrt{15\pi}}{m!}\left[15(3-n_{s})\right]\left[\frac{2}{\sqrt{3}}+\frac{m\varphi^{\prime\prime}(0)}{2}\left(\frac{\textrm{arcsinh}(\sqrt{2}}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)\right]$
(88)

For figure 4, we use equation (84) for $40<m\leq 85$, and for $m>85$ we use
equation (88). Finally, let us remark that for $n_{s}<1$, Stirling’s
approximation for $m!$ implies that the series (27) diverges very slowly, for
fixed $k$, but is of course still asymptotic as $k\to\infty$.

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# Modeling emergence of norms in multi-agent systems by applying tipping
points ideas

Francisco Lopez

School of Computer Science  University of Manchester  Manchester, United
Kingdom<EMAIL_ADDRESS>
###### Abstract

Norms are known to be a major factor determining humans behavior. It’s also
shown that norms can be quite effective tool for building agent-based
societies. Various normative architectures have been proposed for designing
normative multi-agent systems (NorMAS). Due to human nature of the concept
norms, many of these architectures are built based on theories in social
sciences. Tipping point theory, as is briefly discussed in this paper, seems
to have a great potential to be used for designing normative architectures.
This theory deals with the factors that affect social epidemics that arise in
human societies. In this paper, we try to apply the main concepts of this
theory to agent-based normative architectures. We show several ways to
implement these concepts, and study their effects in an agent-based normative
scenario.

###### keywords:

norms, multi-agent systems, tipping points

##  1 Introduction

Human societies are simultaneously frustratingly unchanging and yet
susceptible to “epidemics” that sweep across the social fabric causing people
to adopt previously rare practices. Tipping point theories attempt to explain
the subtle triggers behind these social processes. In 2000, Malcolm Gladwell
[25] produced a popular science book summarizing three key factors which
trigger tipping points: 1) scale-free networks (the Law of the Few); 2)
effective messaging (the Stickiness Factor) and 3) environmental influences
(the Power of Context). This paper relates tipping point theory to the process
of norm emergence in multi-agent systems; we propose that normative agent
architectures can serve an excellent computational model for expressing many
contagious social phenomena, including tipping points and information
cascades.

Social norms are known to be a major factor governing humans’ behavior;
unbeknownst to us, many of our everyday behaviors are influenced by these
implicit standards [10]. Various normative architectures have been proposed
for designing normative multi-agent systems (NorMAS) capable of reasoning
about norm adoption. Some of these systems have been grounded in social
science theory, but the aim of many architectures is simply to effectively
address standard multi-agent system challenges, including agreement formation,
coordination and conflict resolution [9].

Despite recent research progress in the area, the complete life-cycle of norms
is far from fully understood. The complex nature of human decision-making
makes comprehending the rationale behind social interactions difficult, since
people are notoriously bad at self-reporting their motivations [15]. The field
of agent-based modeling aims to create agents in the image of humans. These
agents typically have cognitively-inspired decision-making components, and are
situated in life-like scenarios. In both standard multi-agent systems and
cognitively-inspired models, existing social theories have been employed
toward the construction of normative models [5]. Various stages of the norm
life-cycle including recognition, adoption, compliance and emergence are often
modeled on similar concepts in social sciences.

This paper proposes a unified model of how norm emergence in networked agent
societies can be used to predict the effects of common tipping point triggers.
Previous work on norm emergence in networks has investigated the effects of
social network topology in static [34, 31] and also dynamic networks [29]. Yu
et al. [39] presented an evaluation of different learning methods on norm
emergence in networked systems. In our work, we simply employ network
structures as a medium to apply ideas from tipping point theory relating to
the Law of the Few. Therefore, the structure of agents’ network is not of
interest by itself, other than making it congruent with human social networks.
Our main contribution is showing the role and significance of tipping point
principles in normative agent systems, and evaluating the potential impact of
this model on NorMAS design. The next section provides an overview of related
work in this area.

##  2 Related Work

Hollander and Wu [24] refer to three categories of normative studies in the
social sciences: 1) social function of norms [15], 2) impact of social norms
[7], 3) mechanisms leading to the emergence and creation of norms [14]. In the
context of social function, norms are often concerned with the oughtness and
expectations of agent behavior; where oughtness refers to the condition where
an agent should or should not perform an action, and expectation refers to the
behavior that other agents expect to observe from that agent [13, 2]. An
example of work belonging to this category is Boella and Torre’s architecture
containing separate subsystems for counts-as conditionals, conditional
obligations, and conditional permissions [16].

Within the second category, social impact, norms are considered in terms of
cost provided to or imposed on the parties involved in a social interaction
[4]. For instance, punishment and sanctions are introduced as two enforcement
mechanisms used to achieve the necessary social control required to impose
social norms [33]. Here they demonstrate a normative agent that can punish and
sanction defectors and also dynamically choose the right amount of punishment
and sanction to impose [12].

As noted in [24], the third category is concerned with the how of norms more
than the why. Relevant work in NorMAS domain that falls into this category is
divided into the following groups [11]. The first group is composed of
research in which norms are practically hard-wired into the system or dictated
directly to the agents. In the second set, norms emerge from social
interactions among agents. Sen and Airiau’s work [32] in which agent
interactions are modeled using payoff matrices, focuses on norm emergence
through social learning in agent societies. Conte et al.’s EMIL was designed
as an architecture for modeling norm emergence [17]. The EMIL architecture
includes a dynamic cognitive model of norm emergence and innovation [6].

Much of the existing work in normative multi-agent systems explicitly or
implicitly relies on social science theories [8]. In a recent work, some of
the well-known theories of philosopher David Hume were evaluated using an
agent-based model called HUME2.0 [17]. This work demonstrates how social
justice concepts can even emerge from heterogeneous agents that are not
endowed with norm representations.

Self-determination theory is also referenced by some of the normative works
[3]. Here the focus is on the agents’ motivation and the extent to which the
motivation is intrinsic or extrinsic. Neumann studied existing normative
architectures to see how much they comply with self-determination theory [27].

Practice theory is an example drawn from anthropology; this theory describes
how changes in the society are based on the interactions between the human
agents and social structure. For instance, an agent-based model for energy
demand and supply social practices is presented in [1], which shows how energy
consumption norms form and evolve in urban societies. The next section
provides background on tipping point theory, primarily from a sociology
perspective.

##  3 Tipping Point Theory

The term, “tipping point” was initially coined in physics to describe the
situation in which the state of an object rapidly changes from one stable
equilibrium to another different equilibrium. Morton Grodzins was the first to
use term in social sciences to describe an interesting phenomenon he observed
in the US cities, known as white flight [23]. His observation was that in some
metropolitan areas the percentage of African-American people would increase up
to a certain point. After that point, those with white ethnicity immigrated
from those cities in large numbers. Thomas Schelling presented the general
theory of tipping, which describes how individuals’ micromotives and
microbehavior can aggregate in the big picture [30]. The model of collective
behavior introduced by Mark Granovetter [22] uses thresholds to determine the
path of social events. This model was initially used to describe how fads are
created.

In normative studies, tipping points are usually denoted as the point of
maximum return at which time the behavior has the highest level of
acceptability from the population. For instance, in a certain group of
friends, the number of times they shower in a week may vary, but a specific
value has the highest acceptability by group members as the conventional
pattern of behavior. In this paper, we study the impact of Gladwell’s three
factors on norm emergence in agent-based normative systems and demonstrate
practical ways to apply this versatile theory.

##  4 Experimental Setup

For our experiments, we employ the classic scenario, rules of the road, that
is frequently used to study normative behavior in multi-agent systems. In this
scenario, there exists a population of agents that do not have any preference
toward driving on the left or right side of a two-way road. No rules or higher
enforcement exist to determine the preferred side. This scenario represents a
two-action stage game that models the situation where agents need to agree on
one of several equally desirable alternatives. The societal norms that we
would like to evolve are either driving on the left or driving on the right
[32].

In this scenario agents receive a fixed value reward and punishment based on
the following payoff matrix shown in Table 1.

Table 1: Payoff matrix for rules of the road scenario

As Yu et al. [39] note, although this payoff matrix appears simple, the
coordination game poses a very challenging puzzle for human beings to solve
efficiently. The game has two pure Nash-equilibria: both agents drive left or
both agents drive right. Classical game theory, however, does not give a
coherent account of how people would play a game like this. The conundrum is
that there is nothing in the structure of the game itself that allows the
players (even purely rational players) to infer what they ought to do. In
reality, people can play such games because they can rely on some contextual
cues to agree on a particular equilibrium [38].

In similar studies on normative systems, usually the cumulative payoff
(reward) of the whole population of agents is used as a measure of comparing
various methods (see [32] and [39] for examples). Instead, we opt to use the
norm emergence time for each method as an evaluation metric. This is
functionally equivalent since the payoff received by all agents post norm
emergence is the same, hence a method which leads to faster norm emergence
will also yield the higher cumulative payoff.

##  5 Key few members

In this section, we study the effects of key members of an agent society on
the rate of norm emergence. These key members are selected using standard
heuristics for measuring influence within a network; we evaluate the
performance of three centrality measures: degree, closeness, and betweenness.
Degree centrality measures the number of edges connected to a node. Closeness
is calculated based on the total distance to all other nodes. Nodes with a
high betweenness centrality fall on a large proportion of the shortest paths
(geodesics) in the graph.

To model the characteristics of a real social network, we use an algorithm
introduced in [35] to create a synthetic network which follows power law
degree distribution and exhibits homophily, a greater number of link
connections between similar nodes.111Commonly described as “birds of a feather
flock together” [26] The network generator uses link density (ld) and
homophily (dh) to govern network formation. A simplified version of the
pseudo-code for this method is shown in Figure 1. For our model, we assumed
predefined values for ld and dh. The nodes of the graph represent the
individuals (agents) in the simulation, who can be considered as car drivers.

$G=\text{Null}$

repeat

sample $r$ from uniform distribution $U(0,1)$

if $r\vskip 6.0pt plus 2.0pt minus 2.0pt\leq\vskip 6.0pt plus 2.0pt minus
2.0ptld$ then

randomChooseSource($G$)

determineCandidateSink($dh$,$G$)

pickSink() $\triangleright$ based on power-law distribution

connect(source,sink)

else

add a new node to G

end if

until desired number of nodes added to the network

Figure 1: Synthetic friendship network generator

We use a weighted voting approach (also known as a structure based method) to
determine an agent’s decision with regard to its neighbors. The weight for
each of an agent’s neighbors is computed using a normalized value of that
neighbor’s centrality value as shown in Equation 1.

$\text{weight}_{i,j}=\dfrac{C_{j}}{\Sigma_{k=1}^{Deg_{i}}C_{k}}$ (1)

This equation shows the weight of the link connecting neighbor $j$ to node
$i$. $C$ refers to the corresponding centrality value (degree, betweenness and
closeness). Also, $Deg_{i}$ denotes the number of neighbors for node $i$. The
top 10 percent of the population of agents with the greatest centrality values
are assumed to be the key elements of a society. At the beginning of our
experiments, all of the agents follow a single norm; in other words, all of
them have learned (through social learning [32]) to always drive on one side
of the road. In our implementation, each agent has a utility value defined for
each of four possible cases: Up-Left, Up-Right, Down-Left and Down-Right,
where Up and Down determine the section of road, and Left and Right determine
the direction an agent drives. These values are updated while receiving
payoffs based on the matrix shown in Table 1.

In our experiments, we compare the penetration of norm changing behaviors that
emanate from key members of a society vs. other cases. We compare emanation
from the top to emanation from the middle and bottom 10 percent of the
population. At the beginning of the simulation, the agents (nodes) are ranked
based on their centrality value to determine the top, middle and bottom
agents. The utility value of these agents is kept fixed. Neighbors of these
agents continue updating their behavior until a new norm emerges in the
system. Figure 2, Figure 3, and Figure 4 show the number of iterations
required for each case to converge. The population of agents contained 100
agents, and the reported results show the average values over 20 runs.

Figure 2: Average number of iterations until the emergence of one norm, when
using degree centrality to determine key agents. Figure 3: Average number of
iterations until the emergence of one norm, when using betweenness centrality
to determine key agents. Figure 4: Average number of iterations until the
emergence of one norm, when using closeness centrality to determine key
agents.

The pattern observed in all of three cases was very similar. When the norm
propagation starts from the top 10% of the population, the norm emerges much
faster compared to the other cases. Moreover, there is a fairly sizable
difference among top, middle and bottom agents. The magnitude of difference
between the top and middle 10% is more than the difference between the middle
and bottom. These results are consistent with the role of connectors in
tipping point theory.

##  6 Stickiness Factor

According to the tipping point theory, the extent and rate of emerging social
norms in a society is not only related to the members of the society, but also
related to the content of the message. An effective message needs to be
interesting or “sticky” enough to remain in agents’ minds. This factor is
almost completely independent of the society and its structure, and is a
property of the idea.

As Gladwell [21] points out, it is potentially very complicated to determine
if a certain message has the necessary stickiness or not, but one
characteristic that is usually common to sticky ideas is that it frequently
returns to a person’s mind. This could be in the form of a desire to sit and
watch a popular TV show every night, or in a more extreme case, a clinical
addiction to smoking or gambling. Conventional marketing and advertising
domains refer to this phenomenon as rule of 27. According to this rule, a
message (advertisement) should be seen at least 27 times, if the message is
going to stick [28].

Figure 5: Average number of iterations until the emergence of one norm, when 2
out of 4 agents with fixed utility values play twice with each agent that they
encounter. Figure 6: Average number of iterations until the emergence of one
norm, when 2 out of 4 agents with fixed utility values go (drive) faster.

In order to model this property, we assume that the stickiness is represented
by the number of games that an agent plays with another agent. Therefore a
higher number of games will result in the same effect as a stickier belief. In
our experiments, this idea is evaluated in two different ways. The first way
is to increase the number of games that a certain set of agents play. The
second way is to have a certain number of agents driving faster than other
agents to be exposed to more cars.

Figures 5 and 6 show results related to these two cases. In both cases, we
have our original 100 agents plus a group of 2 agents which have a fixed
preference to drive on either the left or right. In the first scenario, one
group of agents plays two games each time it encounters another agent. In the
second scenario, one group of agents moves faster. Both of these scenarios
lead to the same effect: increasing the number of times that an agent is
exposed to an idea. This simulates the property of frequently returning to a
person’s mind. In both cases, when the stickiness factor is implemented, the
entire system converges to a single norm faster.

##  7 Power of Context

The third element of the tipping point theory refers to the power of context.
As Gladwell points out: it is possible to be a better person on a clean street
or in a clean subway, than in one littered with trash and graffiti [21]. The
idea is mostly based on what known in criminology as the theory of broken
windows [37]. According to this theory, slight changes in the environment
could result in tipping effects over the whole society.

In order to apply this part of the tipping point theory, we use ideas from the
methods for studying fads and cascading effects in networks [36]. First, we
build a network using the same approach described in Section 5. Then, we
assign a threshold value for each agent. Similar to the probabilistic
information cascade models, if the cumulative value of the perceived cascade
is less than the threshold, nothing will change. If it’s higher, the agent
will change its current behavior, which in our scenario would result in
driving on the other side of the road. Figure 7 shows the percentage of times
that a norm emerged in the system for a set of threshold values. The columns
show the average results over 20 runs. Agents were selected randomly as a
source of a small initial shock in the network, which results in negating the
current payoff values for driving on each side of the road. The frequency of
shocks is determined randomly. The system runs until it reaches some fixed
iteration number (50,000), unless a different norm is observed. This
experiment illustrates how minor shocks can shape a population fad, resulting
in a population-level behavior change. The shocks (pulses) in this model can
be viewed as any of the small changes that tipping point theory predicts can
result in large changes in the whole society. According to the results
presented in Figure 7, thresholds as small as 0.02 can lead to the emergence
of norms in the system in almost 5 percent of the experiments. The computed
values for each agent are compared to its tipping point value (normalized
between 0 and 1).

Figure 7: Percentage of times that a norm emerges in the population, when
agents have different threshold values for activating.

There is a second aspect to the power of context, which refers to the number
of people in groups. The Rule of 150 says that the size of groups is a subtle
contextual factor that makes a big difference. This number is referred as
Dunbar’s number [18], after the anthropologist who originally proposed the
idea. In groups with fewer than 150 members, people will cooperate relatively
easily and rapidly become infected with the community ethos. Once that
threshold is crossed people begin to behave very differently. 150 is our
social channel capacity as determined on the basis of personal loyalties and
1-on-1 contacts. Beyond the tipping point of 150 the group dynamics simply
become too complex. For the average person there are just too many
relationships to manage. The group then becomes divided and alienated, and
usually splits into two. Smaller groups have been shown to be more effective
at tasks than larger groups. This may be due to biological limitations of
humans which make it very difficult for them to handle a larger community.

With the growth of virtual social media sites and the spread of online groups,
there has been renewed interest in evaluating the importance of this limit on
Facebook [19], Myspace [20] and within massively multiplayer online role-
playing games (MMORPGs). The pivotal issue here is that a person cannot
maintain a close relationship with all of the members of a larger group which
ultimately sabotages its success. Having a direct connection with each member
of the group is a necessary component to having a positive social
relationship.

Figure 8: Average number of iterations until the emergence of one norm, when
the network structure of agents follows a power-law distribution and when the
network is a complete clique.

We propose using a clique structure to illustrate this idea. In a clique each
node has a direct edge to all of other nodes. There are $n*(n-1)$ edges in the
resulting graph. We opt to use a directed graph, as that seems to be the
general assumption for friendship networks. We compare the emergence of
driving norms in a network generated using the method described in Section 5.
It should be noted that having more edges does not result in faster
convergence. More connections makes the diffusion of ideas easier, while it
makes it harder for the agents to find an idea that all agents like. In a
clique structure the major voting approach and the weighted voting approach
(using the number of edges) are the effectively same, so neither of them
elicits earlier norm emergence. Figure 8 shows the number of iterations that
were required on average for the two cases to reach norm emergence. The
driving norm emerged faster in case of the clique structure than in the power-
law degree distribution network. This shows the potential benefit of such a
structure in constructing agent systems, at least for ideal cases.

##  8 Conclusion and Future Work

Norms are complex social behaviors that have been extensively studied in
sociology, psychology, and other related fields. Most normative architectures
draw upon theories from the social sciences. The theory of tipping points has
inspired much research in different disciplines. For this paper, we model some
of the well-known elements of this theory, as applied to networked agent
populations. We illustrate how three of principle ideas including key few
members, stickiness factor, and the role of environment can affect the process
of norm emergence. This paper is meant to be an initial step for convincing
the NorMAS community of the importance of tipping point theory concepts. For
future work, we are interested in mapping the performance of our normative
model to a real-dataset.

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