Quantum Gravity without General Relativity

Takehisa Fujita111e-mail<EMAIL_ADDRESS>
Department of Physics, Faculty of Science and Technology,

Nihon University, Tokyo, Japan

Abstract

The quantum field theory of gravitation is constructed in terms of Lagrangian
density of Dirac fields which couple to the electromagnetic field $A_{\mu}$ as
well as the gravitational field $\cal G$. The gravity appears in the mass term
as $m(1+g{\cal G})\bar{\psi}\psi$ with the coupling constant of $g$. In
addition to the gravitational force between fermions, the electromagnetic
field $A_{\mu}$ interacts with the gravity as the fourth order effects and its
strength amounts to $\alpha$ times the gravitational force. Therefore, the
interaction of photon with gravity is not originated from Einstein’s general
relativity which is entirely dependent on the unphysical assumption of the
principle of equivalence. Further, we present a renormalization scheme for the
gravity and show that the graviton stays massless.

##  1  Introduction $-$ Problems of General Relativity

The motion of the earth is governed by the gravitational force between the
earth and the sun, and the Newton equation is written as

$m\ddot{\bm{r}}=-G_{0}mM{\bm{r}\over r^{3}}$ $None$

where $G_{0}$, $m$ and $M$ denote the gravitational constant, the mass of the
earth and the mass of the sun, respectively. This is the classical mechanics
which works quite well. Further, Einstein generalizes the Newton equation to
the relativistic equation of motion which can be valid even for the curved
space [1, 2]. However, this was achieved before the discovery of quantum
mechanics, and therefore it is natural that the general relativity cannot be
quantized properly. Indeed, the quantization of the general relativity has
intrinsic problems which are related to the invariance of the general
coordinate transformation. On the other hand, the first quantization is only
possible for the Cartesian coordinates [3]. This indicates that any attempt to
quantize the general relativity is not a proper starting point, but rather one
should try to make a field theory simply to include the gravity. This is
closely connected to the understanding of the first quantization
($[x_{i},p_{j}]=i\hbar\delta_{ij}$), and since this quantization procedure is
not a fundamental principle, we should try to make a field theory which
includes the gravitational interaction [3, 4].

Before constructing a theory that can describe the field equation under the
gravity, we discuss the fundamental problems in the theory of general
relativity. Basically, there are two serious problems in the general
relativity, the lack of field equation under the gravity and the assumption of
the principle of equivalence.

###  1.1 Field Equation of Gravity

When one wishes to write the Dirac equation for a particle under the
gravitational interaction, then one faces to the difficulty. Since the Dirac
equation for a hydrogen-like atom can be written as

$\left(-i\bm{\nabla}\cdot\bm{\alpha}+m\beta-{Ze^{2}\over r}\right)\Psi=E\Psi$
$None$

one may write the Dirac equation for the gravitational potential
$V(r)=-{G_{0}mM\over r}$ as

$\left(-i\bm{\nabla}\cdot\bm{\alpha}+m\beta-{G_{0}mM\over
r}\right)\Psi=E\Psi.$ $None$

But there is no foundation for this equation. At least, one cannot write the
Lagrangian density which can describe the Dirac equation for the gravitational
interaction. This is clear since one does not know whether the interaction can
be put into the zero-th component of a vector type or a simple scalar type in
the Dirac equation. That is, it may be of the following type

$\left[-i\bm{\nabla}\cdot\bm{\alpha}+\left(m-{G_{0}mM\over
r}\right)\beta\right]\Psi=E\Psi.$ $None$

This is a well known problem, but it is rarely discussed, and people seem to
be reluctant to treating this problem up on the table.

###  1.2 Principle of Equivalence

The theory of general relativity is entirely based on the principle of
equivalence. Namely, Einstein started from the assumption that physics of the
two systems (a system under the uniform external gravity and a system that
moves with a constant acceleration) must be the same. This looks plausible
from the experience on the earth. However, one can easily convince oneself
that the system that moves with a constant acceleration cannot be defined
properly since there is no such an isolated system in a physical world. The
basic problem is that the assumption of the principle of equivalence is
concerned with the two systems which specify space and time, not just the
numbers in connection with the acceleration of a particle. Note that the
acceleration of a particle is indeed connected to the gravitational
acceleration, $\ddot{z}=-g$, but this is, of course, just the Newton equation.
Therefore, the principle of equivalence inevitably leads Einstein to the space
deformation. It is clear that physics must be the same between two inertia
systems, and any assumption which contradicts this basic principle cannot be
justified at all.

Besides, this problem can be viewed differently in terms of Lagrangian. For
the system under the uniform external gravity, one can write the corresponding
Lagrangian. On the other hand, there is no way to construct any Lagrangian for
the system that moves with a constant acceleration. One can define a
Lagrangian for a particle that moves with a constant acceleration, but one
cannot write the system (or space and time) that moves with a constant
acceleration. Therefore, it is very hard to accept the assumption of the
principle of equivalence even with the most modest physical intuition.

###  1.3 General Relativity

Einstein generalized the classical mechanics to the relativistic equation of
motion where he started from the principle of equivalence. Therefore, he had
to introduce the new concept that space may not be uniform, and the general
relativity is the equation for the metric tensor $g_{\mu\nu}$. However, this
picture is still based on the particle mechanics which is governed by the
equations for coordinates of a point particle. This is, of course, natural for
Einstein since quantum mechanics was not discovered at that time. Since
quantum mechanics is a field theory, though with the non-relativistic
kinematics, it is essentially different from Newton’s classical mechanics but
is rather similar to the Maxwell equations. Newton equation can certainly
describe the dynamics of particles for the certain region of kinematics such
as the motion of the earth around the sun. However, it is a useless theory for
the description of electron motion in atoms. One should give up the idea of
particle picture and should accept the concept of field theory. At the time of
invention of the general relativity, Einstein knew quite well the Maxwell
equations which are indeed field theory equations. However, the Maxwell
equations are not realized as the basis equations for quantum mechanics [3].

##  2 Lagrangian Density for Gravity

It is by now clear that one should start from constructing the quantum
mechanics of the gravitation. In other words, one should find the Dirac
equation for electron when it moves in the gravitational potential. In this
paper, we present a model Lagrangian density which can describe electrons
interacting with the electromagnetic field $A_{\mu}$ as well as the
gravitational field $\cal G$.

###  2.1 Lagrangian Density for QED

We first write the well established Lagrangian density for electrons
interacting with the electromagnetic field $A_{\mu}$

${\cal
L}_{el}=i\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi-e\bar{\psi}\gamma^{\mu}A_{\mu}\psi-m\bar{\psi}\psi-{1\over
4}F_{\mu\nu}F^{\mu\nu}$ $None$

where

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$

This Lagrangian density of QED is best studied and is most reliable in many
respects. In particular, the renormalization scheme of QED is theoretically
well understood and is experimentally well examined, and there is no problem
at all in the perturbative treatment of QED. All the physical observables can
be described in terms of the free Fock space terminology after the
renormalization, and therefore one can compare any prediction of the physical
quantities with experiment. However, it should be noted that QED is the only
field theory model in four dimensions which works perfectly well without any
conceptual difficulties.

###  2.2 Lagrangian Density for QED plus Gravity

Now, we propose to write the Lagrangian density for electrons interacting with
the electromagnetic field as well as the gravitational field $\cal G$

${\cal
L}=i\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi-e\bar{\psi}\gamma^{\mu}A_{\mu}\psi-m(1+g{\cal
G})\bar{\psi}\psi-{1\over 4}F_{\mu\nu}F^{\mu\nu}+{1\over 2}\partial_{\mu}{\cal
G}\ \partial^{\mu}{\cal G}$ $None$

where the gravitational field ${\cal G}$ is assumed to be a massless scalar
field. It is easy to prove that the new Lagrangian density is invariant under
the local gauge transformation

$A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\chi,\ \ \ \psi\rightarrow
e^{-ie\chi}\psi.$ $None$

This is, of course, quite important since the introduction of the
gravitational field does not change the most important local symmetry.

###  2.3 Dirac Equation with Gravitational Interactions

Now, one can easily obtain the Dirac equation for electrons from the new
Lagrangian density

$i\gamma^{\mu}{\partial}_{\mu}\psi-e\gamma^{\mu}A_{\mu}\psi-m(1+g{\cal
G})\psi=0.$ $None$

Also, one can write the equation of motion of gravitational field

$\partial_{\mu}\partial^{\mu}{\cal G}=-mg\bar{\psi}\psi.$ $None$

The symmetry property of the new Lagrangian density can be easily examined,
and one can confirm that it has a right symmetry property under the time
reversal transformation, parity transformation and the charge conjugation [3].

###  2.4 Total Hamiltonian for QED plus Gravity

The Hamiltonian can be constructed from the Lagrangian density in eq.(2.2)

$H=\int\left\\{\bar{\psi}\left(-i\bm{\gamma}\cdot\bm{\nabla}+m(1+g{\cal
G})\right)\psi-e\bm{j}\cdot\bm{A}\right\\}d^{3}r+{e^{2}\over
8\pi}\int{j_{0}(\bm{r}^{\prime})j_{0}(\bm{r})d^{3}rd^{3}r^{\prime}\over{|\bm{r}^{\prime}-\bm{r}|}}$
$+{1\over
2}\int\left(\dot{\bm{A}}^{2}+(\bm{\nabla\times\bm{A}})^{2}\right)d^{3}r+{1\over
2}\int\left(\dot{{\cal G}}^{2}+(\bm{\nabla}{\cal G})^{2}\right)d^{3}r$ $None$

where $j_{\mu}$ is defined as $j_{\mu}=\bar{\psi}\gamma_{\mu}\psi$. In this
expression of the Hamiltonian, the gravitational energy is still written
without making use of the equation of motion. In the next section, we will
treat the gravitational energy and rewrite it into an expression which should
enable us to easily understood the structure of gravitational force between
fermions.

##  3 Static-dominance Ansatz for Gravity

In eq.(2.2), the gravitational field ${\cal G}$ is introduced as a real scalar
field, and therefore it cannot be a physical observable as a classical field
[5]. In this case, since the real part of the right hand side in eq.(2.5)
should be mostly time independent, it may be reasonable to assume that the
gravitational field ${\cal G}$ can be written as the sum of the static and
time-dependent terms and that the static part should carry the information of
diagonal term in the external source term. Thus, the gravitational field
${\cal G}$ is assumed to be written as

${\cal G}={\cal G}_{0}(\bm{r})+\bar{\cal G}(x)$ $None$

where ${\cal G}_{0}(\bm{r})$ does not depend on time. This ansatz is only a
sufficient condition, and its validity cannot be verified mathematically, but
it can be examined experimentally.

The equations of motion for ${\cal G}_{0}(\bm{r})$ and $\bar{\cal G}(x)$
become

$\bm{\nabla}^{2}{\cal G}_{0}=mg\rho_{g}$ $None$
$\partial_{\mu}\partial^{\mu}\bar{\cal G}(x)=-mg\\{(\bar{\psi}\psi)_{\rm[non-
diagonal]}+(\bar{\psi}\psi)_{\rm[diagonal\ rest]}\\}$ $None$

where $\rho_{g}$ is defined as

$\rho_{g}\equiv(\bar{\psi}\psi)_{\rm[diagonal]}$ $None$

where $(\bar{\psi}\psi)_{\rm[diagonal]}$ denotes the diagonal part of the
$\bar{\psi}\psi$, that is, the terms proportional to
$[a^{\dagger(s)}_{\bm{k}}a^{(s)}_{\bm{k}^{\prime}}-b^{\dagger(s)}_{\bm{k}}b^{(s)}_{\bm{k}^{\prime}}]$
of the fermion operators which will be defined in eq.(4.2). Further,
$(\bar{\psi}\psi)_{\rm[non-diagonal]}$ term is a non-diagonal part which is
connected to the creation and annihilation of fermion pairs, that is,
$[a^{\dagger(s)}_{\bm{k}}b^{\dagger(s)}_{-\bm{k}^{\prime}}+b^{(s)}_{-\bm{k}^{\prime}}a^{(s)}_{\bm{k}}]$
of the fermion operators. In addition, the term
$(\bar{\psi}\psi)_{\rm[diagonal\ rest]}$ denotes time dependent parts of the
diagonal term in the fermion density, and this may also have some effects when
the gravity is quantized. In this case, we can solve eq.(3.2) exactly and find
a solution

${\cal G}_{0}(\bm{r})=-{mg\over
4\pi}\int{\rho_{g}(\bm{r}^{\prime})\over{|\bm{r}^{\prime}-\bm{r}|}}d^{3}r^{\prime}$
$None$

which is a special solution that satisfies eq.(2.5), but not the general
solution. Clearly as long as the solution can satisfy the equation of motion
of eq.(2.5), it is physically sufficient. The solution of eq.(3.5) is quite
important for the gravitational interaction since this is practically a
dominant gravitational force in nature.

Here, we assume that the diagonal term of $(\bar{\psi}\psi)_{\rm[diagonal]}$
is mostly time independent, and in this case, the static gravitational energy
which we call $H_{G}^{S}$ can be written as

$H_{G}^{S}=mg\int\rho_{g}{\cal G}_{0}d^{3}r+{1\over 2}\int(\bm{\nabla}{\cal
G}_{0})^{2}d^{3}r$ $=-{m^{2}G_{0}\over
2}\int{\rho_{g}(\bm{r}^{\prime})\rho_{g}(\bm{r})\over{|\bm{r}^{\prime}-\bm{r}|}}d^{3}rd^{3}r^{\prime}$
$None$

where the gravitational constant $G_{0}$ is related to the coupling constant
$g$ as

$G_{0}={g^{2}\over 4\pi}.$ $None$

This static gravitational energy can be written in the momentum representation
as

$H_{G}^{S}=-{m^{2}G_{0}\over
4\pi^{2}}\sum_{\bm{p},\bm{p}^{\prime}}\int{\bar{u}(\bm{p}+\bm{q})u(\bm{p})\bar{u}(\bm{p}^{\prime}-\bm{q})u(\bm{p}^{\prime})\over{q^{2}}}d^{3}q.$
$None$

Eq.(3.6) is just the gravitational interaction energy for the matter fields,
and one sees that the gravitational interaction between electrons is always
attractive. This is clear since the gravitational field is assumed to be a
massless scalar. It may also be important to note that the $H_{G}^{S}$ of
eq.(3.6) is obtained without making use of the perturbation theory, and it is
indeed exact, apart from the static ansatz of the field ${\cal
G}_{0}(\bm{r})$.

##  4 Quantization of Gravitational Field

In quantum field theory, we should quantize fields. For fermion fields, we
should quantize the Dirac field by the anti-commutation relations of fermion
operators. This is required from the experiment in terms of the Pauli
principle, that is, a fermion can occupy only one quantum state. In order to
accommodate this experimental fact, we should always quantize the fermion
fields with the anti-commutation relations. On the other hand, for gauge
fields, we must quantize the vector field in terms of the commutation relation
which is also required from the experimental observation that one photon is
emitted by the transition between $2p-$state and $1s-$state in hydrogen atoms.
That is, a photon is created from the vacuum of the electromagnetic field, and
therefore the field quantization is an absolutely necessary procedure.
However, it is not very clear whether the gravitational field ${\cal G}$
should be quantized according to the bosonic commutation relation or not. In
fact, there must be two choices concerning the quantization of the
gravitational field ${\cal G}$.

###  4.1 No Quantization of Gravitational Field $\bar{\cal G}$

As the first choice, we may take a standpoint that the gravitational field
${\cal G}$ should not be quantized since there is no requirement from
experiments. In this sense, there is no definite reason that we have to
quantize the scalar field and therefore the gravitational field ${\cal G}$
should remain to be a classical field. In this case, we do not have to worry
about the renormalization of the graviton propagator, and we obtain the
gravitational interaction between fermions as we saw it in eq.(3.6) which is
always attractive, and this is consistent with the experimental requirement.

###  4.2 Quantization Procedure

Now, we take the second choice and should quantize the gravitational field
$\bar{\cal G}$. This can be done just in the same way as usual scalar fields

$\bar{\cal
G}(x)=\sum_{\bm{k}}{1\over{\sqrt{2V\omega_{k}}}}\left[d_{k}e^{-i\omega_{k}t+i\bm{k}\cdot\bm{r}}+d_{k}^{\dagger}e^{i\omega_{k}t-i\bm{k}\cdot\bm{r}}\right]$
$None$

where $\omega_{k}=|\bm{k}|$. The annihilation and creation operators $d_{k}$
and $d_{k}^{\dagger}$ are assumed to satisfy the following commutation
relations

$[d_{\bm{k}},d_{\bm{k}^{\prime}}^{\dagger}]=\delta_{\bm{k},\bm{k}^{\prime}}$
$None$

and all other commutation relations should vanish. Since the graviton can
couple to the time dependent external field which is connected to the creation
or annihilation of the fermion pairs, the graviton propagator should be
affected from the vacuum polarization of fermions. Therefore, we should carry
out the renormalization procedure of the graviton propagator such that it can
stay massless. We will discuss the renormalization procedure in the later
chapter.

###  4.3 Graviton

Once the gravitational field ${\cal G}$ is quantized, then the graviton should
appear. From eq.(4.1), one can see that the graviton can indeed propagate as a
free massless particle after it is quantized, and this situation is just the
same as the gauge field case in QED, namely, photon after the quantization
becomes a physical observable. However, it should be noted that the gauge
field has a special feature in the sense that the classical gauge field
($\bm{A}$) is gauge dependent and therefore it is not a physical observable.
After the gauge fixing, the gauge field can be quantized since one can
uniquely determine the gauge field from the equation of motion, and therefore
its quantization is possible.

On the other hand, the gravitational field is assumed to be a real scalar
field, and therefore it cannot be a physical observable as a classical field
[5]. Only after the quantization, it becomes a physical observable as a
graviton, and this can be seen from eq.(4.1) since the creation of the
graviton should be made through the second term of eq.(4.1). In this case, the
graviton field is a complex field which is an eigenstate of the momentum and
thus it is a free graviton state, which can propagate as a free particle.

##  5 Interaction of Photon with Gravity

From the Lagrangian density of eq.(2.2), one sees that photon should interact
with the gravity in the fourth order Feynman diagrams as shown in Fig. 1. The
interaction Hamiltonian $H_{I}$ can be written as

$H_{I}=\int\left(mg{\cal
G}\bar{\psi}\psi-e\bar{\psi}\bm{\gamma}\psi\cdot\bm{A}\right)d^{3}r$ $None$

where the fermion field $\psi$ is quantized in the normal way

$\psi(\bm{r},t)=\sum_{\bm{p},s}{1\over{\sqrt{L^{3}}}}\left(a_{{\bm{p}}}^{(s)}u^{(s)}_{{\bm{p}}}e^{i\bm{p}\cdot\bm{r}-iE_{\bm{p}}t}+{b}^{\dagger(s)}_{{\bm{p}}}v^{(s)}_{{\bm{p}}}e^{-i\bm{p}\cdot\bm{r}+iE_{\bm{p}}t}\right)$
$None$

where $u^{(s)}_{{\bm{p}}}$ and $v^{(s)}_{{\bm{p}}}$ denote the spinor part of
the plane wave solutions of the free Dirac equation. $a^{(s)}_{{\bm{p}}}$ and
$b^{(s)}_{{\bm{p}}}$ are annihilation operators for particle and anti-particle
states, and they should satisfy the following anti-commutation relations,

$\\{a^{(s)}_{{\bm{p}}},{a^{\dagger}}_{{\bm{p}}^{\prime}}^{(s^{\prime})}\\}=\delta_{s,s^{\prime}}\delta_{{{\bm{p}}},{{\bm{p}}^{\prime}}},\
\
\\{b^{(s)}_{{\bm{p}}},{b^{\dagger}}_{{\bm{p}}^{\prime}}^{(s^{\prime})}\\}=\delta_{s,s^{\prime}}\delta_{{{\bm{p}}},{{\bm{p}}^{\prime}}}$
$None$

and all other anticommutation relations should vanish. The gauge field
$\bm{A}$ can be quantized in the same way

$\bm{A}(x)=\sum_{\bm{k}}\sum_{\lambda=1}^{2}{1\over{\sqrt{2V\omega_{\bm{k}}}}}\bm{\epsilon}^{\lambda}(\bm{k})\left[c_{\bm{k},\lambda}e^{-ikx}+c^{\dagger}_{\bm{k},\lambda}e^{ikx}\right]$
$None$

where $\omega_{\bm{k}}=|\bm{k}|$. The polarization vector
$\bm{\epsilon}^{\lambda}(\bm{k})$ should satisfy the following relations

$\bm{\epsilon}^{\lambda}(\bm{k})\cdot\bm{k}=0,\ \ \ \
\bm{\epsilon}^{\lambda}(\bm{k})\cdot\bm{\epsilon}^{\lambda^{\prime}}(\bm{k})=\delta_{\lambda,\lambda^{\prime}}.$
$None$

The annihilation and creation operators $c_{\bm{k},\lambda}$,
$c_{\bm{k},\lambda}^{\dagger}$ should satisfy the following commutation
relations

$[c_{\bm{k},\lambda},\
c_{\bm{k}^{\prime},\lambda^{\prime}}^{\dagger}]=\delta_{\bm{k},\bm{k}^{\prime}}\delta_{\lambda,\lambda^{\prime}}$
$None$

and all other commutation relations should vanish.

$\bm{q}$$\bm{p}$$\bm{k}$$\bm{k}^{\prime}$$\bm{p}^{\prime}$Fig. 1: The fourth
order Feynman diagram

The calculation of the S-matrix can be carried out in a straightforward way
[6, 7, 8], and we can write

$S=(ie)^{2}\epsilon_{\mu}^{\lambda}(k)\epsilon_{\nu}^{\lambda^{\prime}}(k^{\prime})\left({mm^{\prime}g^{2}\over
q^{2}}\right)\bar{u}(p^{\prime})u(p)$ $\times\int{d^{4}a\over{(2\pi)^{4}}}{\rm
Tr}\left[\gamma_{\mu}{i\over{a\hbox
to0.0pt{\hss/}-m+i\epsilon}}\gamma_{\nu}{i\over{b\hbox
to0.0pt{\hss/}-m+i\epsilon}}{i\over{c\hbox
to0.0pt{\hss/}-m+i\epsilon}}\right]$ $None$

where $k$ and $k^{\prime}$ denote the four momenta of the initial and final
photons while $p$ and $p^{\prime}$ denote the four momenta of the initial and
final fermions, respectively. $m$ and $m^{\prime}$ denote the mass of the
fermion for the vacuum polarization and the mass of the external fermion. $a$,
$b$, $c$ and $q$ can be written in terms of $k$ and $p$ as

$q=p^{\prime}-p,\ \ \ \ k=a-b,\ \ \ \ k^{\prime}=a-c,\ \ \ \ q=k-k^{\prime}.$

Therefore, the S-matrix can be written as

$S=ie^{2}mm^{\prime}g^{2}\epsilon_{\mu}^{\lambda}(k)\epsilon_{\nu}^{\lambda^{\prime}}(k^{\prime}){1\over
q^{2}}\bar{u}(p^{\prime})u(p)\int{d^{4}a\over{(2\pi)^{4}}}{1\over
a^{2}-m^{2}}{1\over(a-k)^{2}-m^{2}}{1\over(a-k^{\prime})^{2}-m^{2}}$
$\times{\rm Tr}\left[\gamma_{\mu}(a\hbox to0.0pt{\hss/}+m)\gamma_{\nu}((a\hbox
to0.0pt{\hss/}-k\hbox to0.0pt{\hss/}+m)((a\hbox to0.0pt{\hss/}-k\hbox
to0.0pt{\hss/}^{\prime}+m)\right].$ $None$

Since the term proportional to $q$ does not contribute to the interaction, we
can safely approximate in the evaluation of the trace and the $a$ integration
as

$k^{\prime}\approx k.$

Now, we define the trace part as

$N_{\mu\nu}={\rm Tr}\left[\gamma_{\mu}(a\hbox
to0.0pt{\hss/}+m)\gamma_{\nu}((a\hbox to0.0pt{\hss/}-k\hbox
to0.0pt{\hss/}+m)((a\hbox to0.0pt{\hss/}-k\hbox
to0.0pt{\hss/}^{\prime}+m)\right]$ $None$

which can be evaluated as

$N_{\mu\nu}=4m[(k^{2}-a^{2}+m^{2})g_{\mu\nu}+4a_{\mu}a_{\nu}-2a_{\mu}k_{\nu}-2a_{\nu}k_{\mu}].$
$None$

Defining the integral by

$I_{\mu\nu}\equiv\int{d^{4}a\over{(2\pi)^{4}}}{N_{\mu\nu}\over{(a^{2}-m^{2})\left[(a-k)^{2}-m^{2}\right]\left[(a-k^{\prime})^{2}-m^{2}\right]}}$
$None$

we can rewrite it using Feynman integral

$I_{\mu\nu}=2\int{d^{4}a\over{(2\pi)^{4}}}\int_{0}^{1}zdz{N_{\mu\nu}\over{[(a-kz)^{2}-m^{2}+z(1-z)k^{2}]^{3}}}.$
$None$

Therefore, introducing the variable $w=a-kz$ we obtain the S-matrix as

$S=8ie^{2}m^{2}m^{\prime}g^{2}\epsilon_{\mu}^{\lambda}(k)\epsilon_{\nu}^{\lambda^{\prime}}(k^{\prime}){1\over
q^{2}}\bar{u}(p^{\prime})u(p)\times$
$\int{d^{4}w\over{(2\pi)^{4}}}\left[{(-w^{2}g_{\mu\nu}+4w_{\mu}w_{\nu})\over{[w^{2}-m^{2}+z(1-z)k^{2}]^{3}}}+{\left\\{m^{2}+k^{2}(1-z^{2})\right\\}g_{\mu\nu}+4k_{\mu}k_{\nu}z(1-z)\over{[w^{2}-m^{2}+z(1-z)k^{2}]^{3}}}\right].$
$None$

The first part of the integration can be carried out in a straightforward way
using the dimensional regularization, and we find

$\int{d^{4}w\over{(2\pi)^{4}}}{(-w^{2}g_{\mu\nu}+4w_{\mu}w_{\nu})\over{[w^{2}-m^{2}+z(1-z)k^{2}]^{3}}}={i\pi^{2}\Gamma(0)g_{\mu\nu}\over
2\Gamma(3)}(4-4)=0.$

Thus, the two divergent parts just cancel with each other, and the
cancellation here is not due to the regularization as employed in the vacuum
polarization in QED, but it is a kinematical and thus rigorous result. The
finite part can be easily evaluated [8], and therefore we obtain the S-matrix
as

$S={e^{2}\over
8\pi}m^{2}m^{\prime}g^{2}(\epsilon^{\lambda}\epsilon^{\lambda^{\prime}}){1\over
q^{2}}\bar{u}(p^{\prime})u(p)$ $None$

where we made use of the relation $k^{2}=0$ for free photon at the end of the
calculation.

##  6 Renormalization Scheme for Gravity

At the present stage, it is difficult to judge whether we should quantize the
gravitational field or not. At least, there is no experiment which shows any
necessity of the quantization of the gravity. Nevertheless, it should be worth
checking whether the gravitational interaction with fermions can be
renormalizable or not. We know that the interaction of the gravity with
fermions is extremely small, but we need to examine whether the graviton can
stay massless or not within the perturbation scheme.

Here, we present a renormalization scheme for the scalar field theory which
couples to fermion fields. The renormalization scheme for scalar fields is
formulated just in the same way as the QED scheme since QED is most
successful.

###  6.1 Vacuum Polarization of Gravity

First, we write the vacuum polarization for QED with the dimensional
regularization, and the divergent contributions to the self-energy of photon
can be described in terms of the vacuum polarization $\Pi^{\mu\nu}_{QED}(k)$
as

$\Pi^{\mu\nu}_{QED}(k)=i\lambda^{4-D}e^{2}\int{d^{D}p\over(2\pi)^{D}}{\rm
Tr}\left[\gamma^{\mu}{1\over p\hbox to0.0pt{\hss/}-m}\gamma^{\nu}{1\over
p\hbox to0.0pt{\hss/}-k\hbox to0.0pt{\hss/}-m}\right]$ $={e^{2}\over
6\pi^{2}\epsilon}(k^{\mu}k^{\nu}-g^{\mu\nu}k^{2})+{\rm finite\ terms}$ $None$

where $D$ is taken to be $D=4-\epsilon$. It is interesting to note that the
apparent quadratic divergence disappears due to the gauge invariant
dimensional regularization when evaluating the momentum integrations. This is
important since, if there were any quadratic divergence terms present, then it
would have caused serious troubles for the mass terms which cannot keep the
gauge invariance in QED. The fact that the quadratic divergence terms can be
erased by the proper dimensional regularization in QED is indeed related to
the success of QED renormalization scheme.

On the other hand, the vacuum polarization for the gravity becomes

$\Pi(k)=i\lambda^{4-D}m^{2}g^{2}\int{d^{D}p\over(2\pi)^{D}}{\rm
Tr}\left[{1\over p\hbox to0.0pt{\hss/}-m}{1\over p\hbox to0.0pt{\hss/}-k\hbox
to0.0pt{\hss/}-m}\right]$ $={m^{2}g^{2}\over
12\pi^{2}}\left\\{3\Gamma(-1+{\epsilon\over 2})\left(m^{2}-{1\over
6}k^{2}\right)+\Gamma({\epsilon})k^{2}\right\\}$ $None$

This can be rewritten as

$\Pi(k)=-{1\over 2}C_{1}k^{2}-{1\over 2}C_{2}m^{2}$ $None$

where

$C_{1}=-{m^{2}g^{2}\over 2\pi^{2}}\left({1\over\epsilon}-{\gamma\over
2}+{1\over 6}\right)$ $None$ $C_{2}={m^{2}g^{2}\over
2\pi^{2}}\left({2\over\epsilon}-{\gamma}+1\right).$ $None$

As can be seen, the second term in eq.(6.3) should correspond to the quadratic
divergence term, and this time it cannot be erased by the dimensional
regularization. However, this term can be safely eliminated by the counter
term. Therefore, we add the following Lagrangian density as the mass counter-
terms to the original Lagrangian density

$\delta{\cal L}={1\over 2}C_{1}\partial_{\mu}{\cal G}\ \partial^{\mu}{\cal
G}-{1\over 2}\delta M{\cal G}^{2}$ $None$

where the constant $\delta M$ is defined as

$\delta M\equiv C_{2}m^{2}.$ $None$

Therefore, the total Lagrangian density of the gravity ${\cal L}_{G}$ becomes

${\cal L}_{G}={1\over 2}(1+C_{1})\partial_{\mu}{\cal G}\ \partial^{\mu}{\cal
G}-{1\over 2}\delta M{\cal G}^{2}={1\over 2}\partial_{\mu}{\cal G}_{r}\
\partial^{\mu}{\cal G}_{r}-{1\over 2}\delta M{\cal G}_{r}^{2}$ $None$

where ${\cal G}_{r}$ is the renormalized gravity field. This shows that the
mass counter term cannot be renormalized into the wave function ${\cal G}$.
However, the mass counter term in eq.(6.7) has a proper symmetry property of
the gravity Lagrangian density, in contrast to the QED case where the mass
term violates the gauge invariance. Therefore, the introduction of the mass
counter term in the scalar field theory does not break the renormalization
scheme of the present formulation.

###  6.2 Fermion Self Energy from Gravity

The fermion self energy term in QED is calculated to be

$\Sigma_{QED}(p)=-ie^{2}\int{d^{4}k\over(2\pi)^{4}}\gamma_{\mu}{1\over p\hbox
to0.0pt{\hss/}-k\hbox to0.0pt{\hss/}-m}\gamma^{\mu}{1\over k^{2}}={e^{2}\over
8\pi^{2}\epsilon}(-p\hbox to0.0pt{\hss/}+4m)+\textrm{finite terms}.$ $None$

In the same way, we can calculate the fermion self-energy due to the gravity

$\Sigma_{G}(p)=-im^{2}g^{2}\lambda^{4-D}\int{d^{D}k\over(2\pi)^{D}}{1\over
p\hbox to0.0pt{\hss/}-k\hbox to0.0pt{\hss/}-m}{1\over k^{2}}={m^{2}g^{2}\over
8\pi^{2}\epsilon}(-p\hbox to0.0pt{\hss/}+4m)+\textrm{finite terms}$ $None$

which is just the same as the QED case, apart from the factor in front.
Therefore, the renormalization procedure can be carried out just in the same
way as the QED case since the total fermion self energy term within the
present model becomes

$\Sigma(p)={1\over 8\pi^{2}\epsilon}(e^{2}+m^{2}g^{2})(-p\hbox
to0.0pt{\hss/}+4m)+\textrm{finite terms}.$ $None$

###  6.3 Vertex Correction from Gravity

Concerning the vertex corrections which arise from the gravitational
interaction and electromagnetic interaction with fermions, it may well be that
the vertex corrections do not become physically very important. It is
obviously too small to measure any effects of the higher order terms from the
gravity and electromagnetic interactions. However, we should examine the
renormalizability of the vertex corrections and can show that they are indeed
well renormalized into the coupling constant. The vertex corrections from the
electromagnetic interaction and the gravity can be evaluated as

$\Lambda_{QED}(k,q)=i\lambda^{4-D}mge^{2}\int{d^{D}p\over(2\pi)^{D}}{\rm
Tr}\left[\gamma_{\mu}{1\over(k\hbox to0.0pt{\hss/}-p\hbox
to0.0pt{\hss/}-m)(k\hbox to0.0pt{\hss/}-p\hbox to0.0pt{\hss/}-q\hbox
to0.0pt{\hss/}-m)p^{2}}\gamma^{\mu}\right]$ $None$
$\Lambda_{G}(k,q)=i\lambda^{4-D}m^{3}g^{3}\int{d^{D}p\over(2\pi)^{D}}{\rm
Tr}\left[{1\over(k\hbox to0.0pt{\hss/}-p\hbox to0.0pt{\hss/}-m)(k\hbox
to0.0pt{\hss/}-p\hbox to0.0pt{\hss/}-q\hbox to0.0pt{\hss/}-m)p^{2}}\right].$
$None$

We can easily calculate the integrations and obtain the total vertex
corrections for the zero momentum case of $q=0$ as

$\Lambda(k,0)=\Lambda_{QED}(k,0)+\Lambda_{G}(k,0)={mg\over\pi^{2}\epsilon}(e^{2}+m^{2}g^{2})+{\rm
finite\ \ terms}$ $None$

which is logarithmic divergence and is indeed renormalizable just in the same
way as the QED case.

###  6.4 Renormalization Procedure

Since the infinite contributions to the fermion self-energy and to the vertex
corrections in the second order diagrams are just the same as the QED case,
one can carry out the renormalization procedure just in the same way as the
QED case. There is only one difference between QED and the gravity cases, that
is, the treatment of the quadratic divergence in the vacuum polarization. In
the QED case, the quadratic divergence terms should be eliminated by the
dimensional regularization since the mass term violates the gauge invariance
and thus one cannot consider the mass counter term in the QED Lagrangian
density. On the other hand, in the gravity case, the quadratic divergence
terms in the vacuum polarization can be canceled out by a mass counter term
since the gravity is not the gauge field theory, and thus, there is no problem
to introduce the mass counter term in the Lagrangian density. Further, the
graviton is never bound and always in the free state, and therefore, the mass
counter term in the gravity cancels the quadratic divergence contribution in a
rigorous way DIn this way, we can achieve a successful renormalization scheme
for the gravity, even though we do not know any occasions in which the higher
order contributions may become physically important.

##  7 Gravitational Interaction of Photon with Matter

From eq.(5.14), one finds that the gravitational potential $V(r)$ for photon
with matter field can be written as

$V(r)=-{G_{0}\alpha m^{2}_{t}M\over 2}{1\over r}$ $None$

where $m_{t}$ and $M$ denote the sum of all the fermion masses and the mass of
matter field, respectively. $\alpha$ denotes the fine structure constant
$\alpha={1\over 137}$. In this case, the equation of motion for photon
$\bm{A}_{\lambda}$ under the gravitational field becomes

$\left({\partial^{2}\over{\partial t^{2}}}-\bm{\nabla}^{2}-{G_{0}\alpha
m^{2}_{t}M\over 2}{1\over r}\right)\bm{A}_{\lambda}=0.$ $None$

Assuming the time dependence of the photon field $\bm{A}_{\lambda}$ as

$\bm{A}_{\lambda}=\bm{\epsilon}_{\lambda}e^{-i\omega t}A_{0}(\bm{r})$ $None$

we obtain

$\left(-\bm{\nabla}^{2}-{G_{0}\alpha m^{2}_{t}M\over 2}{1\over
r}\right)A_{0}(\bm{r})=\omega^{2}A_{0}(\bm{r}).$ $None$

This equation shows that there is no bound state for photon even for the
strong coupling limit of $G_{0}\rightarrow\infty$.

##  8  Conclusions

We have presented a new scheme of treating the gravitational interactions
between fermions in terms of the Lagrangian density. The gravitational
interaction appears always as the mass term and induces always the attractive
force between fermions. In addition, there is an interaction between photon
and the gravity as the fourth order Feynman diagrams. The behavior of photon
under the gravitational field may have some similarity with the result of the
general relativity, but the solution of eq.(7.4) is still to be studied in
detail.

Also, we have presented a renormalization procedure which is essentially the
same as the QED renormalization scheme. There is one important difference
between the QED and the gravity cases, that is, the treatment of the quadratic
divergence in the vacuum polarization. In QED, one has to eliminate the
quadratic divergence terms by the regularization so as to keep the gauge
invariance of the Lagrangian density. On the other hand, in the gravity case,
the quadratic divergence terms can be canceled out by the mass counter term
since it does not contradict with any important symmetry of the Lagrangian
density. Therefore, the renormalization scheme of the gravity interaction is
well justified, and thus the propagator of the gravity stays massless.
Clearly, this is the most important point in the whole renormalization
procedure.

In this paper, we have not decided whether the gravitational field should be
quantized or not since there is no definite requirement from experiment for
the quantization. At the present stage, both of the evaluation of the
gravitational interactions with fermions should be equally reasonable.
However, for the quantized theory of gravitational field, one may ask as to
whether there is any method to observe a graviton or not. The graviton should
be created through the fermion pair annihilation. Since this graviton can
propagate as a free graviton like a photon, one may certainly have some chance
to observe it through the creation of the fermion pair. But this probability
must be extremely small since the coupling constant is very small, and there
is no enhancement in this process unless a strong gravitational field like a
neutron star may rapidly change as a function of time.

## ACKNOWLEDGEMENTS

We thank Prof. K. Nishijima for encouragements and helpful comments. In
particular, the photon-gravity vertex part and the renormalization procedure
for the scalar field are clarified a great deal through discussions.

## References

  * [1] A. Einstein, ”The foundation of the general theory of relativity”, Annalen Phys. 49 (1916), 769
  * [2] S. Weinberg, ”Gravitation and Cosmology”, (Wiley & Sons, New York, 1972)
  * [3] T. Fujita, ”Symmetry and Its Breaking in Quantum Field Theory”,
(Nova Science Publishers, 2007)

  * [4] T. Fujita, S. Kanemaki and S. Oshima, ” New Concept of First Quantization”, hep-th/0601102.
  * [5] S. Kanemaki, A. Kusaka, S. Oshima and T. Fujita, ”Problems of scalar bosons”, in New Fundamentals in Fields and Particles, (Research Signpost, 2008)
  * [6] J.D. Bjorken and S.D. Drell, ”Relativistic Quantum Mechanics”,
(McGraw-Hill Book Company, 1964)

  * [7] F. Mandl and G. Shaw, ”Quantum field theory”, (John Wiley & Sons, 1993)
  * [8] K. Nishijima, “Fields and Particles”, (W.A. Benjamin, INC, 1969)


11institutetext: IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette,
France 22institutetext: Université de Paris, CNRS, Astroparticule et
Cosmologie, F-75006 Paris, France 33institutetext: Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, California, USA

# Impact of the calibration of the Halo Mass Function on galaxy cluster number
count cosmology

Emmanuel Artis 11 Jean-Baptiste Melin 11 James G. Bartlett 2233 Calum Murray
22 ———-

(Received ———-/ Accepted ————)

The halo mass function (HMF) is a critical element in cosmological analyses of
galaxy cluster catalogs. We quantify the impact of uncertainties in HMF
parameters on cosmological constraints from cluster catalogs similar to those
from Planck, those expected from the Euclid, Roman and Rubin surveys, and from
a hypothetical larger future survey. We analyse simulated catalogs in each
case, gradually loosening priors on HMF parameters to evaluate the degradation
in cosmological constraints. While current uncertainties on HMF parameters do
not substantially impact Planck-like surveys, we find that they can
significantly degrade the cosmological constraints for a Euclid-like survey.
Consequently, the current precision on the HMF will not be sufficient for
Euclid (or Roman or Rubin) and possible larger surveys. Future experiments
will have to properly account for uncertainties in HMF parameters, and it will
be necessary to improve precision of HMF fits to avoid weakening constraints
on cosmological parameters.

###### Key Words.:

cosmological parameters – cosmology: forecasts – galaxies: clusters: general –
large scale structures of the universe

††offprints: ———-,

##  1 Introduction

Galaxy clusters are unique tools for astrophysical and cosmological studies
ranging from galaxy formation to fundamental physics (Voit 2005; Allen et al.
2011; Kravtsov & Borgani 2012). In particular, it is now well established that
their number density $n(M,z)$ is a powerful cosmological probe, notably of the
matter density, $\Omega_{\textrm{m}}$, the root mean square density
fluctuation, $\sigma_{\textrm{8}}$, the dark energy equation-of-state and the
sum of the neutrino masses (Weinberg et al. 2013). For this reason, cluster
number counts figure among the four primary dark energy probes.

Surveys across different wavebands have used cluster number counts to derive
cosmological constraints (e.g., Vikhlinin et al. 2009; Mantz et al. 2010; Rozo
et al. 2010; Hasselfield et al. 2013; Planck Collaboration et al. 2016;
Bocquet et al. 2019; Costanzi et al. 2019, 2020). These analyses are based on
the halo mass function (HMF hereafter) originating in the Press & Schechter
(Press & Schechter 1974) formalism. The HMF provides the key link between
cosmological parameters and the number density of clusters; it essentially
bundles the complicated non-linear physics of halo formation into a simple
analytical formula involving only the linear power spectrum and other linear-
theory quantities. This is essential because it avoids the running of costly
N-body simulations at each point in parameter space when establishing
cosmological constraints.

Current cluster count analyses use catalogs of hundreds (in millimeter and
X-ray data) to thousands (in optical data) of clusters, and they are not
limited by our knowledge of the HMF. But uncertainties on the HMF could become
important as future catalogs grow by factors of 10 to 100. Euclid will detect
tens of thousands of clusters (Sartoris et al. 2016) as will eROSITA (Hofmann
et al. 2017), Simons Observatory (Ade et al. 2019), Rubin (Ivezić et al. 2019)
and CMB-S4 (Abazajian et al. 2019). In the longer term, Roman (Gehrels et al.
2015), PICO (Hanany et al. 2019) and a Voyage 2050 Backlight mission (Basu et
al. 2019) could detect from hundreds of thousands to one million clusters.

As statistical errors on cluster counts dramatically drop, effects now
considered as second order will begin to noticeably contribute to the
uncertainty budget of cosmological analyses. In this paper, we examine the
potential importance of uncertainties related to modeling errors in the HMF.
Popular HMF formulas like Tinker et al. (2008) predict cluster counts in
$\Lambda$CDM to a precision of several percent ($\sim 5$ to 10%) (compared to
full N-body simulations). The recent Despali et al. (2015) HMF provides an
expression with parameters and associated errors fitted on $\Lambda$CDM N-body
simulations. Fig. 1 presents the relative difference of the Tinker and Despali
mass functions at z=1 as a function of the virial mass. The difference is
below $5\%$ for masses less than a few $10^{14}\,M_{\odot}$, so within the
quoted error bars for the Tinker mass function. The red band delineates the
uncertainty from the HMF parameters provided by Despali et al. (2015). They
are much smaller than the difference between the two mass functions, bringing
to light the difference between the precision of a given HMF formula and its
potential bias (a question of accuracy). The origin of this difference is
beyond the scope of our study.

Instead, we take the Despali mass function and its given uncertainty, shown by
the red band, as representative of our knowledge of the mass function. This is
clearly optimistic faced with the importance of the difference between the
Tinker and other mass functions. Nevertheless, we will see that even in the
absence of any bias, these uncertainties may have to be accounted for in
future cosmological analyses of large galaxy cluster surveys.

Different approaches can be chosen to model the uncertainties related to the
HMF (see e.g. Costanzi et al. 2019). We investigate them by allowing the
parameters of the HMF to vary according to priors set by its fit to
simulations, such as provided by Despali et al. By modifying the size of these
priors, we forecast the level of precision required to avoid significant
degradation of cosmological parameter estimation. We require that the HMF
parameters be known to sufficient precision so that:

  * •

The precision on cosmological parameters is degraded by less than 5%, and

  * •

The precision on the Figure-of-Merit (FoM, the inverse of the area of the
confidence ellipse) is degraded by less than 10%.

Figure 1: Relative difference of the Tinker et al. (2008) and the Despali et
al. (2015) mass functions at z=1 for a $\Lambda$CDM cosmology. In red, the
convex envelope of all the scenarios covered by the error bars provided by
Despali et al. (2015). The mass is defined with respect to the virial radius.

We emphasize that this approach assumes that the fitted HMF yields unbiased
cosmological constraints. The uncertainties on the HMF parameters depend
primarily on the number of halos in the simulation used to fit (or
”calibrate”) the HMF, a number that potentially far exceeds the number of
clusters in the universe. But we see from the example of Figure 1 that the
difference between the HMF forms from Tinker et al. (2008) and Despali et al.
(2015) largely exceeds the statistical uncertainties shown by the red band.
The statistical uncertainties do not account for the accuracy, or potential
bias, of the HMF fitting formula. We will however consider only the
statistical uncertainties in the present study, returning briefly to the issue
of bias in Sect. 5.

The article is organized as follows. In Sect. 2, we give a brief overview of
the mass function, describe our likelihood, explain how we compute our Fisher
matrices, and finally list the cosmological models and cluster catalogs that
we examine. We present results in Sect. 3 for each cosmological model, and
then summarize in Sect. 4 before a concluding discussion in Sect. 5.

##  2 Methods

###  2.1 Mass function and cluster likelihood

The mass function formalism was originally introduced by Press and Schechter
(Press & Schechter 1974). It predicts the comoving number density of halos,
$n(M,z)$, for a given set of cosmological parameters as

$\frac{\textrm{d}n}{\textrm{d}\ln
M}(M,z)=\frac{\bar{\rho}}{M}\left|\frac{\textrm{d}\ln(\sigma(M,z))}{\textrm{d}\ln
M}\right|f(\sigma(M,z))$ (1)

where $\bar{\rho}$ is the mean density of the universe today, $f(\sigma)$ is
the multiplicity function (discussed hereafter) and

$\sigma^{2}(M,z)=\int{\frac{1}{2\pi^{2}}P_{\mathrm{lin}}(k,z)W^{2}[kR(M)]k^{2}\mathrm{d}k},$
(2)

with $P_{\mathrm{lin}}(k,z)$ the linear power spectrum at redshift $z$,
$R(M)=[M/(4\pi\bar{\rho}/3)]^{1/3}$ and
$W(kR)=[3/(kR)^{2}][\sin(kR)/(kR)-\cos(kR)]$ is the spherical top-hat window
function. Although the classical Press and Schechter approach provides a good
approximation under the assumption of a simple Gaussian process for
$f(\sigma)$, numerous recent studies (based on numerical simulations or
solving theoretical issues) have devised more accurate parametrizations of the
multiplicity function (e.g. Bond et al. 1991; Lacey & Cole 1994; Sheth &
Tormen 1999; Jenkins et al. 2001; Crocce et al. 2010; Corasaniti & Achitouv
2011; Despali et al. 2015).

Relying on the Le SBARBINE simulations, the main interest of the Despali mass
function (hereafter DMF), compared with other widely used HMF like Tinker
(Tinker et al. 2008), is the claim of universality, i.e., the parameters used
to describe the multiplicity function $f(\sigma)$ do not explictly depend on
either the redshift of the clusters or the cosmology. This is important
because it means that when performing a cosmological analysis, the universal
shape can be used at any redshift and in any cosmological scenario. This
universality, however, requires use of $M_{\mathrm{vir}}$, the cluster mass
defined with respect to the virial overdensity $\Delta_{c}$ (e.g. Lahav et al.
1991).

The shape of the function is the following, originally from Sheth & Tormen
(1999):

$\nu f(\nu)=A_{0}\left(1+\frac{1}{\nu^{\prime
p}}\right)\left(\frac{\nu^{\prime}}{2\pi}\right)^{1/2}\mathrm{e}^{-\nu^{\prime}/2},$
(3)

where $\nu^{\prime}=a\nu$. The parameter $\nu$ is related to the root mean
square density fluctuation via

$\nu=\delta_{c}^{2}(z)/\sigma^{2}(M,z),$ (4)

where $\delta_{c}(z)$ is the critical linear theory overdensity, which can be
approximated by Kitayama & Suto (1996)

$\delta_{c}(z)\approx\frac{3}{20}(12\pi)^{2/3}[1+0.0123\log(\Omega_{\textrm{m}}(z))],$
(5)

where $\log$ refers to logarithm in base 10.

We are then left with three parameters $(a,p,A_{0})$ describing, respectively,
the high-mass cutoff, the shape at lower masses, and the normalization of the
curve. Once again, if the virial overdensity is used, no change in these
parameters is expected with respect to either redshift or cosmology.

We compute the expected number density of objects per unit of mass, redshift,
and area on the sky $\Omega$ as

$\partial_{\small{{z,\ln{M},\Omega}}}N(M,z)=\frac{\bar{\rho}}{M}\left|\frac{\textrm{d}\ln(\sigma(M,z))}{\textrm{d}\ln
M}\right|f(\sigma(M,z))\partial_{z,\Omega}V$ (6)

where $\partial_{z,\Omega}V$ is the comoving volume per unit redshift and
solid angle at the considered redshift. The quality of the cosmological
parameter estimation with cluster number counts primarily depends on the
number of clusters in the survey. Since we focus on the effect of the HMF
parameterization, we do not consider any observable-mass relation in this
work. We also assume that we can directly measure cluster mass and that the
selection function can be modeled by a simple mass threshold $M_{\rm thres}$.
We present the survey cases studied (defined by $M_{\rm thres}$, $\Omega$,
redshift range $[z_{\mathrm{min}},z_{\mathrm{max}}]$) in Sect. 2.3.

For each survey-type, we estimate the cosmological parameters

$\Theta=\\{\Omega_{\textrm{m}},\sigma_{\textrm{8}},w_{\textrm{0}},...\\}$

and simultaneously constrain the Despali HMF parameters,

$\Xi=\\{a,p,A_{0}\\}.$

We only consider shot noise, so cosmic variance is neglected. Instead of using
the Cash (1979) statistics in bins of mass and redshift, we compute the
unbinned Poisson likelihood of the two dimensional point process:

$\ln\mathcal{L}(\Theta,\Xi)=\quad-\quad\int_{M_{\mathrm{thres}}}^{+\infty}\int_{z_{\mathrm{min}}}^{z_{\mathrm{max}}}\partial_{z,\ln{M}}N(M,z)\mathrm{d}\ln{M}\mathrm{d}z\\\
+\quad\sum_{i=1}^{N_{\mathrm{clusters}}}\ln\left(\partial_{z,\ln{M}}N(M_{i},z_{i})\right)$
(7)

where the sum runs over individual clusters in the catalog and
$N_{\mathrm{clusters}}$ is the total number of clusters.

###  2.2 Fisher matrix

The aim of our Fisher analysis is to quantify the precision on the HMF
parameters, $\Xi$, required to limit the impact of their uncertainty on the
cosmological constraints.

Figure 2: The $68$ and $95\%$ confidence contours on the parameters $a$ and
$A_{0}$, from the best-fit Gaussian to the MCMC chains provided by Despali et
al. (2015). The black dots delineate the original $68\%$ contour from the MCMC
chains.

Using the MCMC chains provided by Despali et al. (2015), we fit a 2-D Gaussian
on the HMF parameter distribution, for each pair of parameters. We show an
example in Figure 2 for the parameters $a$ and $A_{0}$. This enables us to
reconstruct the full covariance matrix $C_{\mathrm{HMF}}$ for the HMF
parameters, which we provide in Table 1.

Table 1: Covariance matrix, $C_{\mathrm{HMF}}$, reconstructed for parameters $(a,p,A_{0})$, all dimensionless. | $a$ | $p$ | $A_{0}$
---|---|---|---
$a$ | $5.56\times 10^{-7}$ | $-1.01\times 10^{-6}$ | $9.00\times 10^{-8}$
$p$ | $-1.01\times 10^{-6}$ | $3.19\times 10^{-6}$ | $-1.93\times 10^{-7}$
$A_{0}$ | $9.00\times 10^{-8}$ | $-1.93\times 10^{-7}$ | $4.37\times 10^{-8}$

From the likelihood described in the previous section, one can compute the
associated Fisher matrix for the clusters:

$\quad\quad\quad\quad\quad\quad\quad\quad
F_{\mathrm{clusters},{i,j}}=-\left\langle\frac{\partial^{2}\ln\mathcal{L}}{\partial
p_{i}\partial p_{j}}\right\rangle$ (8)

where $p_{i}\in\Theta\cup\Xi$ and the angle brackets indicate the average over
the data ensemble corresponding to the fiducial parameter values. The
covariance matrix of the parameters $p_{i}$ is the inverse of the Fisher
matrix.

We build the total Fisher information matrix as the sum of three components:

  * •

The information obtained from cluster number counts, $F_{\mathrm{clusters}}$,
given in Eq. (8),

  * •

The cosmological priors from Planck cosmic microwave background measurements,
$F_{\mathrm{cmb}}$ (Planck Collaboration et al. 2018),

  * •

The HMF priors, $F_{\mathrm{HMF}}$.

The full Fisher matrix is then given by

$F_{\mathrm{tot}}=F_{\mathrm{clusters}}+F_{\mathrm{cmb}}+F_{\mathrm{HMF}},$
(9)

where $F_{\mathrm{cmb}}$ (resp. $F_{\mathrm{HMF}}$) is non zero only in the
$\Theta$ (resp. $\Xi$) quadrant. The non zero quadrant of $F_{\mathrm{HMF}}$
is the inverse of $C_{\mathrm{HMF}}$.

In this work, we examine the impact of $F_{\mathrm{HMF}}$ on cosmological
parameter determination. In order to do so, we introduce the parameter
$\alpha$, and we modify the priors on the HMF in Eq. (9) as follows:

$F_{\mathrm{HMF}}\leftarrow\frac{1}{\alpha}F_{\mathrm{HMF}}.$

The covariance matrix being the inverse of the Fisher information, increasing
$\alpha$ degrades the constraints on the HMF. On the other hand,
$\alpha\rightarrow 0$ describes the situation where the HMF parameters are
perfectly known. For $p\in\Xi$, this yields

$\sigma_{p}^{2}\leftarrow\alpha\sigma_{p}^{2},$

where $\sigma_{p}$ is the error on one parameter of the HMF.

Figure 3: The $68$ and $95\%$ percent contours on the parameters $a,p$ and
$A_{0}$. The blue ellipses are obtained when $\alpha=1$, i.e., the current
precision provided by Despali et al. (2015). The black dashed curves are
obtained for $\alpha=0.3$, and the dotted curves for $\alpha=2$.

Figure 3 illustrates how two values of $\alpha$ impact the confidence ellipses
for the HMF parameters.

Finally, we obtain

$F_{\mathrm{tot}}(\alpha)=F_{\mathrm{clusters}}(\alpha=1)+F_{\mathrm{cmb}}+\left(\frac{1}{\alpha}-1\right)F_{\mathrm{HMF}}.$
(10)

Here, the last term comes from the fact that the cosmological parameters are
degenerate with the HMF for cluster cosmology. To get $F_{\mathrm{clusters}}$,
we need to set a value for $\alpha$ (here $\alpha=1$) and remove it
afterwards. In the following sections, we change the value of $\alpha$ to
measure its impact on $\sigma_{p}$ for any $p\in\Theta$, the set of the
cosmological parameters, and to identify the value necessary to avoid
degrading the cosmological parameter constraints by more than $5\%$, and the
total FoM by more than $10\%$.

###  2.3 Cosmological models and mock catalogs

We generate catalogs for three different cosmological scenarios. Each model
assumes a flat cosmology:

  * •

A base $\mathrm{\Lambda\mathrm{CDM}}$ cosmology (5 cosmological
parameters111We do not consider the optical depth to reionization, $\tau$, as
a parameter, since it has no impact on cluster counts, the normalization of
the primordial power spectrum being set with $\sigma_{\textrm{8}}$.,
$\Omega_{\textrm{m}}$, $\sigma_{\textrm{8}}$, $H_{\textrm{0}}$,
$\Omega_{\textrm{b}}$, $n_{\textrm{s}}$);

  * •

A $w_{0}w_{a}$CDM cosmology, where we assume a dynamical dark energy model (7
parameters, the parameters from the base flat $\mathrm{\Lambda\mathrm{CDM}}$
and $w_{\textrm{0}}$, $w_{\textrm{a}}$ describing the dark energy equation-of-
state (Chevallier & Polarski 2001): $w(a)=w_{\textrm{0}}+w_{\textrm{a}}(1-a)$,
where $a$ is the expansion scale factor normalized to unity at $z=0$);

  * •

A $\nu$CDM cosmology, in which we consider the physics of massive neutrinos.
In this case, the parameters are the same as the ones presented in the base
$\mathrm{\Lambda\mathrm{CDM}}$, but we need our computation of the power
spectrum, as described in section 3.3 (6 parameters, the parameters from the
base flat $\mathrm{\Lambda\mathrm{CDM}}$ and $\sum{m_{\nu}}$).

The expected cluster counts associated with these models are computed with the
python package HMF from Murray et al. (2013), which we have modified for the
two latter cosmological scenarios in a way described below. The values adopted
for the cosmological parameters for each scenario can be found in Table 2.

Table 2: Fiducial cosmological parameters adopted to generate the mock catalogs in the three cosmological scenarios. We adopt $T_{\textrm{CMB}}=2.725\,\mathrm{K}$, $N_{\textrm{eff}}=3.04$ and $\sum{m_{\nu}}=0.06\,\mathrm{eV}$ in the three cases. The parameters in black are the mean values of the chains provided by Planck Collaboration et al. (2018). In red, the dark energy equation of state parameters, which are fixed for a $\Lambda$CDM cosmology. $\Omega_{\textrm{m}}$ | $\sigma_{\textrm{8}}$ | $H_{\textrm{0}}\leavevmode\nobreak\ (\mathrm{km.s^{-1}.Mpc^{-1}})$ | $\Omega_{\textrm{b}}$ | $n_{\textrm{s}}$ | $w_{\textrm{0}}$ | $w_{\textrm{a}}$
---|---|---|---|---|---|---
$\Lambda$CDM (Planck2018) & $\nu$CDM
0.3153 | 0.8111 | 67.36 | 0.04931 | 0.9649 | -1 | 0
$w_{0}w_{a}$CDM (Planck2018)
0.2976 | 0.836 | 69.39 | 0.04645 | 0.9646 | -0.99 | -0.29

They are the mean values of the chains provided by Planck Collaboration et al.
(2018): PklimHM-TT-TE-EE-lowL-lowE-lensing chains in the $\Lambda$CDM and
$\nu$CDM cases and PklimHMTT-TE-EE-lowL-lowE-BA0-Riess18-Panthe for
$w_{0}w_{a}$. For the case of the $\nu$CDM scenario, we adopt the lower limit
provided by Abe et al. (2008), $\sum{m_{\nu}}=0.06\,{\rm eV}$, because the
best fit value provided by Planck is lower.

The current lack of a precise understanding of the selection function of
future optical surveys leads us to consider simple mass and redshift
thresholds for the completeness of the catalog:

$\chi(M,z)=\left\\{\begin{array}[]{ll}1&\mbox{if
}M\in[M_{\mathrm{thres}},+\infty]\leavevmode\nobreak\
\mbox{and}\leavevmode\nobreak\ z\in[0,2]\\\
0&\mbox{otherwise}\end{array}\right.$ (11)

The total expected number of objects in the catalog is then

$N_{tot}=\int_{M_{\rm
thres}}^{\infty}\int_{z=0}^{2}\partial_{\small{{z,\ln{M}}}}N(M,z)\mathrm{d}\ln
M\mathrm{d}z.$ (12)

We set the mass threshold so that $N_{tot}$ is close to the number of clusters
expected with a survey like Euclid, i.e., $\sim$$60,000$ clusters over
$15,000\,\mathrm{deg}^{2}$ (Sartoris et al. 2016). For the DMF, this mass
threshold is $\log(M_{\rm thres}/M_{\odot})=14.18$. We then reduce/increase
the area of the survey to obtain $N_{tot}$ of $\sim$$500$/$\sim$$1.5\times
10^{5}$ clusters (Planck-like and Future projects).

We generate five cluster catalogues. The first three
($\mathrm{C}_{\mathrm{Euclid}}$, $\mathrm{C}_{\mathrm{Planck}}$,
$\mathrm{C}_{\mathrm{Future}}$) fix the sum of the neutrino masses to its
minimal value of $\sum m_{\nu}=0.06$ and are used for analyses in the
$\Lambda$CDM and $w_{0}w_{a}$CDM scenarios. Since constraining the masses of
the neutrinos requires a special treatment for the power spectrum (see Sect.
3.3 for details), we generate two additional catalogs for the $\nu$CDM
scenario corresponding to Euclid-like and Future surveys. Since we are
interested in forecasting the precision on the measurement of the neutrino
mass scale, we do not generate a Planck-like catalog for the $\nu$CDM
scenario. The mocks are obtained from a Poisson draw of the DMF. The
characteristics of each catalog are given in Table 3.

Table 3: Characteristics of the mock catalogs. The first column is the survey area. Note that varying neutrino mass in the analysis changes the power spectrum. Thus, we generate a special set of two catalogs when letting the neutrino mass free. The two following columns are the number of clusters present in the catalog for a $\Lambda$CDM and a $w_{0}w_{a}$CDM cosmology, and the last one is the same for $\nu$CDM (The cosmological parameters correspond to Table 2). For all the catalogs, $[z_{min},z_{max}]=[0,2]$ and $M_{\mathrm{thres}}(\log(M/M_{\odot}))=14.18$. |  $\Omega$($\mathrm{deg}^{2}$) | $N_{\mathrm{clusters}}$ | $N_{\mathrm{clusters}}$ | $N_{\mathrm{clusters}}$
---|---|---|---|---
|  |  $\Lambda$CDM |  $w_{0}w_{a}$CDM |  $\nu$CDM
$\mathrm{C}_{\mathrm{Future}}$ | 34,503 | 153,891 | 191,408 | 151,427
$\mathrm{C}_{\mathrm{Euclid}}$ | 15,000 | 66,252 | 82,694 | 65,994
$\mathrm{C}_{\mathrm{Planck}}$ | 114 | 513 | 617 | -

##  3 Results

We provide here the results for the three cosmological scenarios
($\Lambda$CDM, $w_{0}w_{a}$CDM, $\nu$CDM) and the three catalogs (Planck-like,
Euclid-like and Future) whose characteristics are given in the previous
section. For each scenario and each catalog, we run the same analysis, and we
give the precision required on the halo mass function.

Parameters which are poorly constrainted with cluster counts such as
$\Omega_{\textrm{b}}$, $n_{\textrm{s}}$, $H_{\textrm{0}}$ make the computation
of the coefficients of the cluster Fisher matrix $F_{\mathrm{clusters}}$
difficult. Indeed, since the log-likelihood function is almost constant, we
are more subject to numerical instabilities when we compute the second order
derivatives. However, our analysis requires that we are able to capture
sufficiently the correlations between the different parameters. We thus
decided to run first an MCMC analysis and fit a Gaussian on the contours as we
did for the HMF parameters in Sect. 2.2. The Fisher matrix is then deduced as
the inverse of the covariance matrix. The MCMC analysis is performed with the
emcee python package from Foreman-Mackey et al. (2013). Large flat priors are
applied for the cosmological parameters of interest. They are summarized in
Table 4.

Table 4: Priors applied on the flat $\Lambda$CDM cosmology Parameter | Prior | Range
---|---|---
$\Omega_{m}$ | Flat | $[0.1,0.7]$
$\sigma_{8}$ | Flat | $[0.3,0.99]$
$H_{0}$ | Flat | $[40,250]$
$\Omega_{b}$ | Flat | $[0.03,0.06]$
$n_{s}$ | Flat | $[0.5,1.7]$
$a$ | Gaussian | from $\alpha\times C_{\mathrm{HMF}}$
$p$ | Gaussian | from $\alpha\times C_{\mathrm{HMF}}$
$A_{0}$ | Gaussian | from $\alpha\times C_{\mathrm{HMF}}$

Convergence is checked with the integrated autocorrelation time, as suggested
by Goodman & Weare (2010).

###  3.1 $\Lambda$CDM cosmology

We present the results for the clusters-only case,
$F_{\mathrm{clusters}}(\alpha=1)+\left(\frac{1}{\alpha}-1\right)F_{\mathrm{HMF}}$,
in Sect. 3.1.1 and the clusters + Planck CMB-case,
$F_{\mathrm{clusters}}(\alpha=1)+F_{\mathrm{cmb}}+\left(\frac{1}{\alpha}-1\right)F_{\mathrm{HMF}}$,
in Sect. 3.1.2.

####  3.1.1 Cosmological constrains from clusters only

Since galaxy cluster number counts are particularly sensitive to
$\Omega_{\textrm{m}}$, and $\sigma_{\textrm{8}}$, we naturally focus on these
parameters for our analysis. In Figure 4, we show the cosmological constraints
for the Euclid-like catalog with $\alpha=1$, as well as our Gaussian
approximation. As expected, $\Omega_{\textrm{b}}$ is poorly constrained as the
clusters primarily probe the total matter content of the universe.

Figure 4: In light blue, MCMC $68\%\leavevmode\nobreak\
\textrm{and}\leavevmode\nobreak\ 95\%$ contours for the cosmological
parameters and the halo mass function parameters for a Euclid-like catalog. We
fixed $\alpha=1$. The dark blue contours are obtained when adding Planck
priors. The dashed lines are the Gaussian fit from which the covariance matrix
and subsequently the Fisher matrix $F_{\mathrm{clusters}}(\alpha=1)$ is
computed.

We now vary the parameter $\alpha$ encapsulating the uncertainty on the HMF
parameters to see if we reach the required precision on the cosmological
parameters. Recall that we require $5\%$ precision on individual cosmological
parameters (here, $\Omega_{\textrm{m}}$ and $\sigma_{\textrm{8}}$), and $10\%$
precision on the total FoM (here, in the
($\Omega_{\textrm{m}}$,$\sigma_{\textrm{8}}$)-plane).

Figure 5: Relative variation of the error on the matter density parameter
$\epsilon(\Omega_{\textrm{m}})$ for the three different sizes of catalogs. For
all catalogs, the value of $\alpha$ required to avoid degrading the constraint
by more than 5% is much greater than 1. The current knowledge of the mass
function ($\alpha=1$) is sufficient. This is the result for clusters only; the
results for clusters plus Planck 2018 priors are summarized in Table 5.

.

Figure 5 shows that the precision on the parameter $\Omega_{\textrm{m}}$ is
not strongly affected by the variation of the parameters of the mass function.
This is expected since Figure 4 shows that this parameters is poorly
correlated with the HMF parameters.

On the other hand, $\sigma_{\textrm{8}}$ is correlated with $a,p$ and $A_{0}$.
Figure 6 shows that, for this parameter and a Future cluster catalog, we need
better constraints (20% improvement) on the halo mass function parameters than
those currently available.

Figure 6: Relative variation of the error on $\sigma_{\textrm{8}}$ for the
three catalogs. The level of precision on the HMF reaches the limit of current
knowledge ($\alpha\rightarrow 1$) for a Euclid-like catalog (Blue). For the
largest catalog (Green), the constraints on the HMF need to be $20\%$ better.
This is the result for clusters only; the results for clusters plus Planck
2018 priors are summarized in Table 5.

.

The last issue concerns the FoM. Figure 7 demonstrates that, for clusters
only, for a catalog of the size of Euclid and for a Future catalog, the
current precision is sufficient to match our objective of $10\%$.

Figure 7: Relative variation of the FoM (inverse of the area within the error
ellipse in the two-dimensional parameter space) in the
$\Omega_{\textrm{m}}-\sigma_{\textrm{8}}$ plane. For the two large catalogs,
$\alpha$ is close to the limit value 1. This is the result for clusters only;
the results for clusters plus Planck 2018 priors are summarized in Table 5.

.

####  3.1.2 Cosmological constrains from clusters with Planck 2018 priors

Since clusters only poorly constrain $\Omega_{\textrm{b}},H_{\textrm{0}}$ and
$n_{\textrm{s}}$, we will combine them with external constraints. The most
natural complementary probe is the Planck primary CMB (Planck Collaboration et
al. 2018). Consequently, we include priors from Planck as $F_{\mathrm{cmb}}$
(see Sect. 2.2) in our total Fisher matrix, and we run our analysis a second
time. The priors are the same as in the previous section (see Table 4). For
clarity, we do not add figures similar to those presented in the previous
section, but instead summarize our results in Table 5. Corner plots for
cosmological and HMF parameters are given in Appendix A.

For $\Omega_{\textrm{m}}$ the impact is more important because the constraints
are tighter, but the current level of precision on the HMF is nevertheless
good enough to avoid degrading the constraints beyond our threshold of 5%.
However, this time $\sigma_{8}$ and the FoM are strongly affected, and
improvements in HFM precision are needed to avoid degrading these constraints.
All the results are provided in Table 5. We highlighted in red when the
precision on the HMF needs to be improved. The precision on the HMF parameters
needs to be improved for Euclid-like and Future catalogs (up to $\sim 70\%$
for the FoM for Future catalogs) if one wants to fully exploit their
cosmological potential.

Table 5: Maximum value allowed for the $\alpha$ parameter in Flat $\Lambda$CDM. Critical cases requiring improved precision on HMF parameters correspond to a value lower than 1 and are highlighted in red. | Clusters only | Clusters+Planck2018
---|---|---
Planck-like |  |
(513 objects) |  |
$\Omega_{\textrm{m}}$ | $\gg 1$ | $\gg 1$
$\sigma_{\textrm{8}}$ | $\gg 1$ | $\gg 1$
FoM | $\gg 1$ | $\gg 1$
Euclid-like |  |
(66,252 objects) |  |
$\Omega_{\textrm{m}}$ | $\gg 1$ | $\gg 1$
$\sigma_{\textrm{8}}$ | $1.2$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.8}$
FoM | $1.6$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.67}$
Future |  |
(153,891 objects) |  |
$\Omega_{\textrm{m}}$ | $\gg 1$ | $3.6$
$\sigma_{\textrm{8}}$ | 0.8 | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.32}$
FoM | $1$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.31}$

###  3.2 $w_{0}w_{a}$CDM

The most common way of modeling the accelerated expansion of the universe is
to consider a fluid with negative pressure, which is characterized by its
equation-of-state $p=w\rho$. A cosmological constant corresponds to $w=-1$ and
$\rho=\mathrm{cst}$. Even though the cosmological constant is consistent with
current observations, many different dark energy models predict a dynamical
equation-of-state. One of the most popular parametrizations was proposed by
Chevallier & Polarski (2001) and Linder (2003),

$w(a)=w_{0}+w_{a}(1-a),$

which, as a two parameter expansion, adds a layer of complexity.

In this model, the HMF parameters have a different effect. First, leaving
$w_{0}$ and $w_{a}$ free increases the size of the confidence contours. In
addition, the shape of the $68\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ 95\%$ regions changes (see Fig. 14 and 15).
Corner plots for cosmological and HMF parameters are provided in Appendix B.

For these reasons, the impact of the HMF parameters is smaller than in the
$\Lambda$CDM case. For all three catalogs, we never reached a loss of relative
precision comparable to the $\Lambda$CDM case, not only for
$\Omega_{\textrm{m}}$ and $\sigma_{\textrm{8}}$, but also for the dark energy
equation-of-state parameters $w_{0}$ and $w_{a}$. This should not obscure the
fact that these results are highly dependant on the HMF uncertainties.
Moreover, future experiments (CMB, BAO, etc) will constrain cosmology better
than the Planck priors used here. These two aspects are discussed further in
Sect. 5.

###  3.3 $\nu$CDM

For our final study we vary the the sum of the neutrino masses. The presence
of massive neutrinos induces a scale dependence of the growth factor, and
slightly changes the model. Following Costanzi et al. (2013), we model the
power spectrum as

$P_{\mathrm{cdm}}(k,z)=P_{\mathrm{m}}(k,z)\left(\frac{\Omega_{\textrm{m}}T_{\mathrm{cdm}}(k,z)+\Omega_{\textrm{b}}T_{\mathrm{b}}(k,z)}{T_{\mathrm{m}}(k,z)(\Omega_{\mathrm{cdm}}+\Omega_{\textrm{b}})}\right)^{2},$
(13)

and we also change the collapse threshold (Hagstotz et al. 2019)

$\delta_{c}^{\nu}=\frac{\sigma(z)}{\sigma_{\mathrm{cdm}}(z)}\delta_{c}.$ (14)

With this formalism, we forecast cosmological constraints from the two mock
catalogs generated to study varying neutrino mass, as described in Sect. 2.3.
As we have already noted, the HMF uncertainty is not significant compared to
the statistical uncertainties for a Planck-like catalog.

For the primary CMB prior $F_{\rm cmb}$, we take the Planck chains
$\mathrm{PklimHM}+\mathrm{TT}+\mathrm{TE}+\mathrm{EE}+\mathrm{lowL}+\mathrm{lowE}+\mathrm{lensing}+\mathrm{BAO}$
with neutrino mass as a free parameter. Since the best fit value for the total
neutrino mass provided by these chains is lower than the minimal value
required by neutrino oscillation experiments (Abe et al. 2008), we center the
Planck Fisher contours on the fiducial value of $0.06\,{\rm eV}$, the minimal
mass value.

Figure 8: Cosmological parameters obtained for a catalog of 151,427 objects
(Future). In orange, the Planck (2018) contours, and in green, the clusters
combined with Planck priors. We expect, for the largest catalog, a $2\sigma$
measurement of the neutrino mass scale. The full corner plot with all
parameters is given in Appendix C.

Figure 8 shows the results for the largest catalog. We expect to achieve a
$2\sigma$ measurement of the neutrino mass scale. Full corner plots for both
cosmological and HMF parameters are provided in Appendix C. We see that
clusters significantly improve the constraints on neutrino mass when added to
the current Planck priors, in particular by constraining the lower mass range.
And if neutrinos are slightly more massive than the minimal value, the
confidence limits would improve: The green contour in the $\Sigma
m_{\nu}-\sigma_{\textrm{8}}$ plane would move to higher values, but since the
CMB would still constrain the the upper value of the neutrino mass, we would
obtain notably tighter constrains. In this sense, the plot is pessimistic.
Moreover, the result shown here marginalizes over the Planck uncertainty on
the optical depth to reionization, which can be expected to improve in the
coming years and hence yield even tighter constraints on neutrino mass.

Figure 9: Evolution of the uncertainty in the
$\Omega_{\textrm{m}}-\sum{m_{\nu}}$ plane. In blue, the Euclid-like case, and
in green a larger Future catalog.

We perform the same analysis as the previous section, focusing only on the
Clusters+Planck2018 case. Results are summarized in Table 6. We observe the
same effect as for the $w_{0}w_{a}$CDM scenario: since we allow the chains to
explore a larger area of parameter space, it appears that the impact of our
knowledge of the HMF is less important than in the case of $\Lambda$CDM. For a
Euclid-like catalog, the current precision on the HMF parameter is sufficient,
but not for the case of a Future catalog, which will require improvements by
$\sim 40\%$ in order to not degrade the constraint on $\sigma_{\textrm{8}}$ by
more than $5\%$.

We must emphasize, however, that the best fit values of the HMF parameters
$(a,p,A_{0})$ (and associated uncertainties) are provided for a $\Lambda$CDM
scenario. We have assumed these to be valid in the $\nu$CDM case, although we
could reasonably expect the errors to be larger for a $\nu$CDM cosmology. We
will come back to this point in the discussion.

Table 6: The maximum value allowed for $\alpha$ in a $\nu$CDM cosmology. Critical cases requiring improved precision on HMF parameters correspond to a value lower than 1, and are highlighted in red. | Clusters+Planck2018
---|---
Euclid-like |
(65,994 objects) |
$\Omega_{\textrm{m}}$ | $7.7$
$\sigma_{\textrm{8}}$ | 1.03
$\sum{m_{\nu}}$ | $\gg 1$
FoM($\Omega_{\textrm{m}}-\sigma_{\textrm{8}}$) | 1.57
FoM($\sigma_{\textrm{8}}-\sum{m_{\nu}}$) | 1.45
Future |
(151,427 objects) |
$\Omega_{\textrm{m}}$ | $3.86$
$\sigma_{\textrm{8}}$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.61}$
$\sum{m_{\nu}}$ | $\gg 1$
FoM($\Omega_{\textrm{m}}-\sigma_{\textrm{8}}$) | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.85}$
FoM($\sigma_{\textrm{8}}-\sum{m_{\nu}}$) | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0.73}$

##  4 Summary of Results

Our study focuses on cluster counts as a cosmological probe. The wider and
deeper surveys of the future will demand every increasing attention to
systematics. As the primary theoretical link between cosmological quantities
and the observed counts, the HMF is the mainstay of every past and future
analysis. The most important aspects of the HMF in this context are:

  * •

Universality, or the concept that the HMF functional parameters are not
explicitly dependant on redshift nor cosmology.

  * •

Accuracy, the ability to recover unbiased cosmological parameters.

  * •

Precision in the fit of the HMF functional parameters when fit on numerical
simulations.

State-of-the-art simulations are improving our understanding of the first two
aspects (Despali et al. 2015). In this paper, we focus on the last point,
estimating the required precision on a given set of parameters in order to
avoid degrading cosmological constraints. We discuss the first two aspects in
Section 5.

For our purposes, we set a threshold of $5\%$ for allowed degradation in the
precision for any single cosmological parameter, and $10\%$ for the FoM. We
find that, for the $\Lambda$CDM scenario, and for a Euclid-like catalog when
Planck priors are applied, one of the principal parameter of interest,
$\sigma_{\textrm{8}}$, as well as the FoM are violate this threshold with the
present precision: the precision on the HMF parameters must be improved by
$\sim 30\%$ (Table 5). For a Future catalog, the situation is more critical:
an improvement of $\sim 70\%$ is required to avoid degrading the FoM on
$(\Omega_{\textrm{m}},\sigma_{\textrm{8}})$ by more than 10%.

For $w_{0}w_{a}$CDM, any degradation in the dark energy equation-of-state
parameters remains under a percent. This is simply because allowing these
parameters to vary increases the size of the confidence ellipses, thereby
reducing the impact of the HMF parameters. A look at the corner plots in
Appendix B shows that there is in fact little correlation between
($w_{0},w_{a}$) and ($a,p,A_{0}$) in this case.

Lastly, we examined constraints on the sum of neutrino masses. Once again for
a future catalog, the current precision on HMF parameters does not meet our
threshold requirement, and one needs to improve the constraints on the HMF
parameters by $\sim 30\%$ (Table 6). For a Euclid-like catalog, the current
precision appears to be sufficient.

##  5 Discussion & Conclusion

The HMF is the foundation of cosmology with cluster counts. It promises to
provide a robust, simple, analytical form for the abundance of dark matter
halos across cosmological models as a function of linear-theory quantities
only. Universality is the key desired property: a functional form whose
parameters do not explicitly depend on the cosmological scenario. Cosmological
dependence only enters through the linear-theory power spectrum and growth
factor. In this way, a universal HMF enables exploration of cosmological
parameter space without the need for expensive numerical simulations at every
step. In most analyses to date, the HMF parameters are fitted on a set of
numerical simulations, and then fixed in the cosmological analysis, for
example Planck Collaboration et al. (2016).

Cluster catalogs will vastly increase in size with planned experiments, such
as eROSITA (now operating), Simons Observatory, CMB-S4, Euclid, Roman and
Rubin. As statistical power increases, we must quantify both the accuracy and
the precision of the HMF fitting form, and evaluate their impact on
cosmological constraints. The former tells us to what extent the best-fit HMF
functional form reproduces the halo abundance in simulations; the latter
refers to the precision of the HMF parameters around their best-fit values.
This has not been an issue at the times when cluster catalogs were containing
only a few hundreds objects. With Euclid, we expect $10^{5}$ clusters, so
these errors can no longer be ignored (Costanzi et al. 2019).

Our study provides a framework to estimate the impact of the precision of HMF
parameters on cosmological parameter estimation. We do this by freeing the
parameters of the HMF, applying priors from the fit and quantifying the impact
on the cosmological inference. We found that improvements of up to 70% in the
precision of HMF parameters is required to prevent degrading cosmological
constraints by more than our threshold of 5% on a single parameter and 10% on
the FoM.

Our results are reliable only for the mass function from Despali et al.
(2015). The results could change for other HMFs depending on the correlations
between HMF parameters and cosmological quantities. Thus, our main result is
that a simlar discussion should be made when presenting error bars extracted
from simulations. It will not be possible in future analyses to simply fix the
best-fit parameters of the HMF when running a cosmological analysis. One will
have to adjust at the same time cosmology, and halo mass function parameters,
with priors from simulation for these latter.

No perfect functional form exists for the HMF, and in fact Fig. 1 shows that
the error bar can not provide an overlap between two of the most widely used
HMFs. This is the issue of accuracy. It is therefore essential to also verify
that the adopted HMF provides unbiased cosmological constraints (see Salvati
et al. (2020)). Differences in HMFs can arise from numerous sources, including
the effect of baryons (Castro et al. 2021), which will be a major issue in the
upcoming years.

A related issue is whether different cosmological scenarios provide different
uncertainties on HMF parameters (apart from the best-fit values). Indeed, the
HMF formalism is based on the idea of universality, i.e., the claim that the
HMF parameterization does not depend on the cosmology. Despali et al. (2015)
demonstrated that this is nearly the case over a broad range of cosmological
scenarios. But this only means that the best-fit is the same for the HMF
parameters in different scenarios. Nothing is said about the uncertainties.
There is no reason why the errors should not be affected when the cosmology
changes, and this should be taken into account.

To summarize, we have to be sure that, in a given mass and redshift interval,
the error bars provided encapsulate all the uncertainties related to
simulation and halo finder (see Knebe et al. (2011) for a review).

In the near future, we will have discovered almost all dark matter halos
hosting massive clusters of galaxies. But as statistics increases, effort must
be made to not only provide a good description of what is seen in the
simulations, but also to provide a framework capable of yielding unbiased
cosmological constraints. We have focused on this latter point, in the special
case of halo mass function parameters. Our conclusion is that it is important
to construct a likelihood where

  * •

Cosmology
($\Omega_{\textrm{m}},\sigma_{\textrm{8}},w_{\textrm{0}},w_{\textrm{a}},\sum{m_{\nu}}$,…),

  * •

Selection function with priors from detection algorithm (see e.g. Euclid
Collaboration et al. 2019),

  * •

Halo mass function with priors from simulation,

  * •

Observable/mass relation with priors from simulation (see e.g. Bellagamba et
al. 2019),

  * •

Large scale bias of dark matter halos $b(M,z)$ with priors from simulation
(see e.g. Tinker et al. 2010)

are adjusted simultaneously.

###### Acknowledgements.

A portion of the research described in this paper was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration.

## References

  * Abazajian et al. (2019) Abazajian, K., Addison, G., Adshead, P., et al. 2019, CMB-S4 Science Case, Reference Design, and Project Plan
  * Abe et al. (2008) Abe, S., Ebihara, T., Enomoto, S., et al. 2008, Physical Review Letters, 100
  * Ade et al. (2019) Ade, P., Aguirre, J., Ahmed, Z., et al. 2019, Journal of Cosmology and Astroparticle Physics, 2019, 056
  * Allen et al. (2011) Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409
  * Basu et al. (2019) Basu, K., Remazeilles, M., Melin, J.-B., et al. 2019, arXiv e-prints, arXiv:1909.01592
  * Bellagamba et al. (2019) Bellagamba, F., Sereno, M., Roncarelli, M., et al. 2019, MNRAS, 484, 1598
  * Bocquet et al. (2019) Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55
  * Bond et al. (1991) Bond, J. R., Cole, S., Efstathiou, G., & Kaiser, N. 1991, ApJ, 379, 440
  * Cash (1979) Cash, W. 1979, ApJ, 228, 939
  * Castro et al. (2021) Castro, T., Borgani, S., Dolag, K., et al. 2021, MNRAS, 500, 2316
  * Chevallier & Polarski (2001) Chevallier, M. & Polarski, D. 2001, International Journal of Modern Physics D, 10, 213
  * Corasaniti & Achitouv (2011) Corasaniti, P. S. & Achitouv, I. 2011, Phys. Rev. D, 84, 023009
  * Costanzi et al. (2019) Costanzi, M., Collaboration, D., Rozo, E., et al. 2019, Monthly Notices of the Royal Astronomical Society, 488, 4779
  * Costanzi et al. (2020) Costanzi, M., Saro, A., Bocquet, S., et al. 2020, arXiv e-prints, arXiv:2010.13800
  * Costanzi et al. (2013) Costanzi, M., Villaescusa-Navarro, F., Viel, M., et al. 2013, Journal of Cosmology and Astroparticle Physics, 2013, 012
  * Crocce et al. (2010) Crocce, M., Fosalba, P., Castand er, F. J., & Gaztañaga, E. 2010, MNRAS, 403, 1353
  * Despali et al. (2015) Despali, G., Giocoli, C., Angulo, R. E., et al. 2015, Monthly Notices of the Royal Astronomical Society, 456, 2486
  * Euclid Collaboration et al. (2019) Euclid Collaboration, Adam, R., Vannier, M., et al. 2019, A&A, 627, A23
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, Publications of the Astronomical Society of the Pacific, 125, 306
  * Gehrels et al. (2015) Gehrels, N., Spergel, D., & WFIRST SDT Project. 2015, in Journal of Physics Conference Series, Vol. 610, Journal of Physics Conference Series, 012007
  * Goodman & Weare (2010) Goodman, J. & Weare, J. 2010, Communications in Applied Mathematics and Computational Science, Vol. 5, No. 1, p. 65-80, 2010, 5, 65
  * Hagstotz et al. (2019) Hagstotz, S., Costanzi, M., Baldi, M., & Weller, J. 2019, Monthly Notices of the Royal Astronomical Society, 486, 3927
  * Hanany et al. (2019) Hanany, S., Alvarez, M., Artis, E., et al. 2019, arXiv e-prints, arXiv:1908.07495
  * Hasselfield et al. (2013) Hasselfield, M., Hilton, M., Marriage, T. A., et al. 2013, J. Cosmology Astropart. Phys., 2013, 008
  * Hofmann et al. (2017) Hofmann, F., Sanders, J. S., Clerc, N., et al. 2017, A&A, 606, A118
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111
  * Jenkins et al. (2001) Jenkins, A., Frenk, C. S., White, S. D. M., et al. 2001, MNRAS, 321, 372
  * Kitayama & Suto (1996) Kitayama, T. & Suto, Y. 1996, MNRAS, 280, 638
  * Knebe et al. (2011) Knebe, A., Knollmann, S. R., Muldrew, S. I., et al. 2011, MNRAS, 415, 2293
  * Kravtsov & Borgani (2012) Kravtsov, A. V. & Borgani, S. 2012, ARA&A, 50, 353
  * Lacey & Cole (1994) Lacey, C. & Cole, S. 1994, Monthly Notices of the Royal Astronomical Society, 271, 676
  * Lahav et al. (1991) Lahav, O., Lilje, P. B., Primack, J. R., & Rees, M. J. 1991, MNRAS, 251, 128
  * Linder (2003) Linder, E. V. 2003, Phys. Rev. Lett., 90, 091301
  * Mantz et al. (2010) Mantz, A., Allen, S. W., Rapetti, D., & Ebeling, H. 2010, MNRAS, 406, 1759
  * Murray et al. (2013) Murray, S., Power, C., & Robotham, A. 2013, HMFcalc: An Online Tool for Calculating Dark Matter Halo Mass Functions
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A24
  * Planck Collaboration et al. (2018) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2018, arXiv e-prints, arXiv:1807.06209
  * Press & Schechter (1974) Press, W. H. & Schechter, P. 1974, ApJ, 187, 425
  * Rozo et al. (2010) Rozo, E., Wechsler, R. H., Rykoff, E. S., et al. 2010, ApJ, 708, 645
  * Salvati et al. (2020) Salvati, L., Douspis, M., & Aghanim, N. 2020, A&A, 643, A20
  * Sartoris et al. (2016) Sartoris, B., Biviano, A., Fedeli, C., et al. 2016, MNRAS, 459, 1764
  * Sheth & Tormen (1999) Sheth, R. K. & Tormen, G. 1999, MNRAS, 308, 119
  * Tinker et al. (2008) Tinker, J., Kravtsov, A. V., Klypin, A., et al. 2008, ApJ, 688, 709
  * Tinker et al. (2010) Tinker, J. L., Robertson, B. E., Kravtsov, A. V., et al. 2010, ApJ, 724, 878
  * Vikhlinin et al. (2009) Vikhlinin, A., Kravtsov, A. V., Burenin, R. A., et al. 2009, ApJ, 692, 1060
  * Voit (2005) Voit, G. M. 2005, Reviews of Modern Physics, 77, 207
  * Weinberg et al. (2013) Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., et al. 2013, Phys. Rep, 530, 87

##  Appendix A Cosmological corner plots $\Lambda$CDM

Figure 10: Cosmological results (clusters only in light shading, Planck2018
with the black dotted line and clusters+Planck2018 in dark shading) for a a
Planck-like catalog of 513 objects in a $\Lambda$CDM cosmology Figure 11:
Cosmological results (clusters only in light shading, Planck2018 with the
black dotted line and clusters+Planck2018 in dark shading) for a Euclid-like
catalog of 66,252 objects in a $\Lambda$CDM cosmology Figure 12: Cosmological
results (clusters only in light shading, Planck2018 with the black dotted line
and clusters+Planck2018 in dark shading) for a Future catalog of 153,891
objects in a $\Lambda$CDM cosmology

skippage

skippage

skippage

##  Appendix B Cosmological corner plots $w_{0}w_{a}$CDM

Figure 13: Cosmological results (clusters only in light shading, Planck2018
with the black dotted line and clusters+Planck2018 in dark shading) for a
Planck-like catalog of 513 objects in a $w_{0}w_{a}$CDM cosmology Figure 14:
Cosmological results (clusters only in light shading, Planck2018 with the
black dotted line and clusters+Planck2018 in dark shading) for a Euclid-like
catalog of 66,252 objects in a $w_{0}w_{a}$CDM cosmology Figure 15:
Cosmological results (clusters only in light shading, Planck2018 with the
black dotted line and clusters+Planck2018 in dark shading) for a Future
catalog of 153,891 objects in a $w_{0}w_{a}$CDM cosmology

skippage

skippage

skippage

skippage

skippage

skippage

##  Appendix C Corner plots $\nu\mathrm{CDM}$

Figure 16: Cosmological results (clusters only in light shading, Planck2018
with the black dotted line, clusters+Planck2018 in dark shading) for a Euclid-
like catalog of 65,994 objects in a $\nu$CDM cosmology Figure 17: Cosmological
results (clusters only in light shading, Planck2018 with the black dotted line
and clusters+Planck2018 in dark shading) for a catalog of 151,427 objects in a
$\nu$CDM cosmology