# Is Cosmological Tuning Fine or Coarse?

Daniel Andrés Díaz-Pachón11footnotetext: Corresponding author Ola Hössjer
Robert J. Marks II

###### Abstract

The fine-tuning of the universe for life, the idea that the constants of
nature (or ratios between them) must belong to very small intervals in order
for life to exist, has been debated by scientists for several decades. Several
criticisms have emerged concerning probabilistic measurement of life-
permitting intervals. Herein, a Bayesian statistical approach is used to
assign an upper bound for the probability of tuning, which is invariant with
respect to change of physical units, and under certain assumptions it is small
whenever the life-permitting interval is small on a relative scale. The
computation of the upper bound of the tuning probability is achieved by first
assuming that the prior is chosen by the principle of maximum entropy
(MaxEnt). The unknown parameters of this MaxEnt distribution are then handled
in such a way that the weak anthropic principle is not violated. The MaxEnt
assumption is “maximally noncommittal with regard to missing information.”
This approach is sufficiently general to be applied to constants of current
cosmological models, or to other constants possibly under different models.
Application of the MaxEnt model reveals, for example, that the ratio of the
universal gravitational constant to the square of the Hubble constant is
finely tuned in some cases, whereas the amplitude of primordial fluctuations
is not.

##  1 Introduction

The fine-tuning of the universe for life is a claim that the constants in the
laws of nature or the ratios thereof, and/or the boundary conditions, both in
particle physics and in the standard cosmological model, belong to minuscule
life-permitting intervals such that outside them life could not exist. The
modern version of fine-tuning was introduced by Carter [1]. Since inception,
fine-tuning remains a hot topic among scientists and popularizers [2, 4, 3,
5]. Perhaps fine-tuning’s biggest claim to fame came with the book The
Anthropic Cosmological Principle, by Barrow and Tipler [6]. Subsequently the
argument has been scrutinized in physics and cosmology. See, for instance, [9,
8, 7] and all the references therein.

Even though the definition of fine-tuning appears to be simple, there are
variations in the particulars. On one hand, researchers have differing
opinions inasmuch as the constants of nature that must be considered. Just to
mention one instance out of many, Adams [7] and Rees [10] consider the
gravitational constant, but Tegmark et al [9] and Barnes [11] ignore it,
limiting themselves to the constants in the Standard Models. On the other
hand, when talking about life, consensus is illusive largely because there is
not even a consistent definition of what life is. As an illustration, Adams
extends his argument to any conjectured form of life, not necessarily carbon-
based [7]; Sandora focuses on complex intelligent life [12, 13, 14, 15]; and
philosopher Robin Collins talks about embodied moral agents [16].

However, in spite of fine-tuning ado and the passions elicited in critics and
defenders, the degree of tuning, either fine or coarse, remains unsettled. To
emphasize, the claim is not that the tuning is fine —or coarse. The claim is
that, up to this point, the degree of tuning has yet to be determined. In
order to explain this assertion, let’s examine in more detail how the tuning
should be measured. Assuming the constants to be considered are agreed upon,
the process can be summarized in two steps.

  1. 1.

The bounds for the life-permitting intervals of constants or of their ratios
must be identified.

  2. 2.

Probabilities over life-permitting intervals must be calculated.

Both steps require elaboration. The main thing to notice is that the first
step belongs exclusively to the realm of physics, whereas the second step
belongs to the realm of mathematics.

For step 1 physicists have proposed useful boundaries of the life-permitting
intervals for many of the constants or the ratios of constants [8]. We must
add, however, that in the absence of a theory of everything the task is far
from complete. Many of these numbers could change in future research in at
least three ways: (i) by changing the limits of the life-permitting intervals,
(ii) by removing constants because they have lost relevance, or (iii) by
identification of constants hitherto not considered.

We focus on step 2 —finding the degree of tuning. More specifically, in order
to determine the probability of having a constant of nature or the ratio
between two constants of nature within a life-permitting interval, a Bayesian
approach is used. Either the prior distribution of the constant is chosen by
the principle of maximum entropy (MaxEnt), or two constants forming a ratio
are chosen to have two separate MaxEnt distributions. Such an approach allows
finding an upper bound, $P_{\mbox{\scriptsize max}}$, for the probability of
tuning. This upper bound is dimensionless, i.e. invariant with respect to
changing physical units. It is very small for some (but not all) examples, and
we refer to tuning as being fine or coarse depending on whether
$P_{\mbox{\scriptsize max}}$ is small or not. Our proposed approach is
sufficiently general to be applied to the constants in the current (standard)
models or to different constants if they change by any of the three
aforementioned circumstances.

##  2 Probabilities and Tuning

We do not consider the first step, finding the bounds of life-permitting
intervals, relying instead on what has already been identified. Our analysis
is restricted to the second step —how to calculate the probabilities of tuning
given that the life-permitting intervals have been determined. It appears that
this problem is still open. For instance, Adams halts his presentation at the
point of finding the life-permitting intervals, explaining that not much can
be said of the probability distributions that must govern the behavior of the
constants of nature [7, p. 6]:

> A full assessment of fine-tuning requires knowledge of these fundamental
> probability distributions, one for each [constant of nature] of interest
> (although they are not necessarily independent). These probability
> distributions, however, are not currently assessable.

In the absence of these probabilities, not much can be said about the degree
of tuning. As surprising as a small interval might intuitively seem, the
interval size matters less than the probability over the interval. For
instance, for a standard normal distribution, the intervals $(-\infty,0)$ and
$(-0.6745,0.6745)$ each have probability 0.5 even though the first has
infinite length and the second has length 1.349. In contrast, under this
distribution, the interval $\left(-\infty,-10^{9}\right)$, has infinite length
but almost zero probability.

In addition, there is the normalization problem. Normalization imposes
limitations to Bernoulli’s Principle of Insufficient Reason (PrOIR), otherwise
known as the Principle of Indifference [17, 18, 19]. As originally conceived,
the PrOIR states that in the absence of any prior knowledge events must be
uniformly distributed. The unspoken assumption is that the space must be
finite. Normalization posits a general warning against using the PrOIR beyond
finite spaces. Placing a uniform distribution over the whole real line, for
example, is untenable. McGrew, McGrew, and Vestrup [20]; as well as Colyvan,
Garfield, and Priest [21]; raise the normalization objection in the context of
fine-tuning. Since the space where the constants of nature or their ratios
could take values has infinite length, attempts to find the probabilities
using a uniform distribution cannot be successful. This criticism against the
PrOIR in the fine-tuning literature is legitimate.

Yet some persist, for instance, borrowing an idea from quantum field theory in
his analysis of the initial entropy of the universe, Roger Penrose reduces an
infinite-dimensional phase space $\mathcal{P}_{\mathcal{U}}$ to a new finite-
dimensional space in which each dimension has finite size.222With respect to
converting the space to a finite-dimensional one Penrose writes “In fact
$\mathcal{P}_{\mathcal{U}}$ will be infinite dimensional… This causes some
technical problems for the definition of entropy, since each required phase
space region $\mathcal{V}$ will have infinite volume. It is usual to deal with
this problem by borrowing ideas from quantum (field) theory, which enables a
finite answer to be obtained for the phase-space volumes which refer to
systems that are appropriately bounded in energy and spatial dimension…
Although there is no fully satisfactory way of dealing with these issues in
the case of gravity—owing to a lack of a satisfactory theory of quantum
gravity—I am going to regard these as technicalities that do not affect the
general discussion of the issues raised by the second law.” [22, pp. 700-701].
See also the more formal result in [23]. And regarding making each dimension
to have finite size he adds: “We shall use the phase space
$\mathcal{P}_{\mathcal{U}}$ appropriate to the entire universe, so the
evolution of the universe as a whole is described by the point $x$ moving
along a curve $\xi$ in $\mathcal{P}_{\mathcal{U}}$. The curve $\xi$ is
parametrized by the time coordinate $t$, and we can expect that, from the
second law, $\xi$ enters immensely larger and larger coarse-graining boxes as
$t$ increases. We suppose that some ‘reasonable’ coarse graining has been
applied to $\mathcal{P}_{\mathcal{U}}$, but if we wish to obtain finite values
for the entropies that $x$ encounters, we would want the volumes of these
boxes to be finite.” [22, pp. 701-702].

Tegmark et al have also addressed the second step of tuning [9]. They propose
to calculate the probability that a constant of nature $x$ belongs to its
life-permitting interval by decomposing the density function of $x$ as a
product of a prior density and a selection probability ($f(x)\propto
f_{prior}(x)f_{sel}(x)$). Here the prior is a theoretically predicted
distribution at some random point during inflation and the selection
distribution obtains the probability of observing that point. This second term
is related to the weak anthropic principle. Observers in a universe are bound
to measure constants of nature for which a habitable universe is possible with
a nonzero $f_{sel}$. As an example, Tegmark et al. consider the probability
distribution of the vacuum energy density assuming uniformity. However, other
distributions are possible [24].

Recently, Barnes has fleshed out these probabilities using a Bayesian approach
[25, 26, 27, 8]. Barnes, as Collins [16], does what McGrew, McGrew, and
Vestrup criticize in [20] by assuming finite sample spaces, therefore using
uniform distributions to calculate probabilities. To our knowledge, Barnes has
been the first to justify his use of a continuous uniform (or what physicists
call a “flat” distribution which is the MaxEnt distribution in a space of
finite size absent all other knowledge), following Edwin Jaynes’s
recommendations [28].

Throughout this article we also apply a Bayesian approach [29]. Using
informational maximum entropy, a concept also due to Jaynes [30, 31], we
directly assume an infinite sample space for constants of nature, or ratios of
such constants, using a class of prior distributions over this space. The
maximum entropy principle generalizes the PrOIR. Maximum entropy over an
infinite domain is no longer uniform but is still applicable under the
appropriate MaxEnt prior distribution, given some restrictions.

A clarification should be made that entropy as treated here refers to
information entropy, not to thermodynamics entropy. In thermodynamics, one
uses entropy in a slightly different way than here: Randomness refers to the
degree of disorder of a physical system, and the second law of thermodynamics
states that the entropy of a closed system, without external influences, will
increase towards an equilibrium state of maximal disorder. In this paper
randomness refers to epistemic uncertainty regarding the value of a constant
of nature, or the ratio between two such constants. In particular, entropy in
information is an inherent property of the distribution function that
describes this epistemic uncertainty, and it corresponds to our degree of
ignorance. This perspective allows to extend entropy even to distributions of
random variables whose domain is not of finite (counting or Lebesgue) measure.

Then, with appropriate restrictions applied to the moments of distributions in
non-compact spaces, it is possible to find even in these settings MaxEnt
distributions. In particular, for our case of interest, maximum entropy for
the distribution of the constants of nature or ratios then refers to a maximum
degree of ignorance about their values, given the imposed restrictions. As
Jaynes saw it, entropy in statistical mechanics is but an application of the
more general information concept [30].

Even though our main focus in this article is epistemic uncertainty, it should
be added that MaxEnt distributions for physical systems are observed in
nature. As we saw, the second law of thermodynamics famously states that the
entropy of a gas in a closed room reaches maximum entropy in pressure in
accordance to a uniform PrOIR. What if, on the other hand, there is only a
single boundary? The barometric pressure measured from the surface of the
earth to space follows the MaxEnt distribution of Exp(1/$\mu$). An example of
MaxEnt that is unbounded for negative and positive values is the Maxwell-
Boltzmann distribution describing the velocity of ideal gas particles. The
projection of the velocity vector along each direction is unbounded with a
$\mathcal{N}(0,\sigma^{2})$ distribution, which is MaxEnt for domains bounded
on neither side (see Table 1 below).

##  3 Tuning and MaxEnt

Fine-tuning asserts that the constants of nature and the boundary conditions
of the universe must live in narrow intervals of low probability in order to
make life possible and that if such constants would not have had the actual
values they possess, life would have never been possible. Therefore, fine-
tuning arguments assume that the value of a constant of nature, or the ratio
between two constants, let’s call it $x=X_{\text{obs}}$, is an observation of
a random variable $X$. Then, for $X$, a probability of the life-permitting
interval is calculated.

As seen in the previous section, the interval width of the life-permitting
interval is less important than the corresponding probability. Denote the
life-permitting interval of $X$ as $\ell_{X}$, and its length as $|\ell_{X}|$.
In order to properly assess this probability, three steps (I-III) need to be
followed:

I) Determine the right sample space $\Omega$ for $X$. The incorrect
determination of $\Omega$ is what McGrew, McGrew, and Vestrup criticized, and
ad-hoc attempts to force a finite sample space from an infinite one do not
seem convincing [32]. Proceeding is not possible without first providing the
right sample space.

II) Find a probability distribution $F$ of $X$. $F$ must be such that it best
represents the current knowledge of the behavior of a constant of nature or of
the ratio between two constants in the most noncommittal way. In this step we
apply, instead of the uniform assumption of the PrOIR, either the MaxEnt
principle to $X$ itself or the MaxEnt principle applied to each of the two
constants whose ratio is $X$. Jaynes summarizes the advantages of MaxEnt over
the PrOIR in this way [30, p. 623]:

> The principle of [MaxEnt] may be regarded as an extension of the [PrOIR] (to
> which it reduces in case no information is given except enumeration of
> possibilities $x_{i}$), with the following essential difference. The
> [MaxEnt] distribution may be asserted for the positive reason that it is
> uniquely determined as the one which is maximally noncommittal with regard
> to missing information, instead of the negative one that there was no reason
> to think otherwise. Thus the concept of entropy supplies the missing
> criterion of choice which Laplace needed to remove the apparent
> arbitrariness of the [PrOIR], and in addition shows precisely how this
> principle is to be modified in case there are reasons for “thinking
> otherwise.” [Emphasis added.]

The MaxEnt principle can be applied directly to $\Omega$ for the case of
consideration of a single constant. Another option is to assume that in the
ratio of two physical constants, each constant is MaxEnt. MaxEnt can still
find a reference distribution for unbounded domains [33, 34], whereas the
PrOIR cannot. Table 1 below illustrates the best-known cases of MaxEnt
distributions that can be applied to constants or ratios of constants living
in a unidimensional space. (A more comprehensive table can be found in [35].)

To introduce the MaxEnt Bayesian approach, first consider a problematic
application to fine-tuning. Let $x$ be an observation of a random variable $X$
that belongs to $\Omega$. Since the sample size is 1, one may first assume $x$
was chosen as the observed value of $X$, and, as such, $x=X_{\text{obs}}$
would become the average of the sample. Being a sample of size one, this value
serves as a sufficient statistic for the expected value $\mu=E(X)$ of $X$ [36,
Ch. 6]. In usual statistical notation, $\hat{\mu}=x$. From the first step the
space is known and, from the second step, an estimated value for the mean can
be obtained.

However, this approach has a weakness due to the weak anthropic principle.
Since we live in a habitable universe, $x$ is not an unbiased observation of a
random variable $X$ with distribution $F$. It is rather an observation of a
random variable with a truncated distribution

$F_{\text{trunc}}=F|X\in\ell_{X}.$

Then $\hat{\mu}=x$ is not an unbiased estimate of $\mu=E_{F}(X)$ but rather an
unbiased estimate of $\mu^{*}=E_{F_{\text{trunc}}}(X)$. In particular, if
$\ell_{X}$ is narrow, so that a uniform distribution approximates
$F_{\text{trunc}}$, then $\mu^{*}$ approximately equals the mid point of
$\ell_{X}$. On the other hand, $F_{\text{trunc}}$ may differ substantially
from a uniform distribution for constants of nature with a wider $\ell_{X}$.
In such a case $x$ is not an unbiased estimate of the mid point of $\ell_{X}$.

These considerations lead to the proper approach: although $x$ is not an
observation of $X\sim F$, we can assume that $X$ belongs to a class of MaxEnt
distributions $F=F(\;\cdot\ ;\theta)$ taken from Table 1, with
$\theta=(\theta_{1},\ldots,\theta_{d})$ a finite-dimensional unknown parameter
that due to the weak anthropic principle cannot be estimated easily. In
Bayesian statistics such a parameter on the prior distribution $F$ is referred
to as a hyperparameter [29]. Alternatively, when $X=G/D$ is the ratio of two
constants of nature $G$ and $D$, we can also reasonably assume that $G$ and
$D$ are independent with distributions $F_{G}=F_{G}(\;\cdot\ ;\theta_{G})$ and
$F_{D}=F_{D}(\;\cdot\ ;\theta_{D})$ chosen as MaxEnt distributions from Table
1. We need then to find the distribution $F(\;\cdot\ ;\theta)$ of $X$, where
$F$ and the hyperparameter $\theta$ are functions of $F_{G}$, $\theta_{G}$,
$F_{D}$, and $\theta_{D}$. In both cases, whether a MaxEnt distribution is
used for $X$ itself, or for $G$ and $D$, the MaxEnt principle helps to reduce
the class of possible $F$ from an infinite class of distributions to a finite-
dimensional class of distributions.

Space | Restrictions/Knowledge | MaxEnt distribution
---|---|---
Finite | None | Equiprobability
Finite interval $[a,b]$ | None | $\mathcal{U}(a,b)$
Finite interval $[a,b]$ |  $\textbf{E}X=\mu_{T}$; $\textbf{E}(X-\mu_{T})^{2}=\sigma_{T}^{2}$ | Truncated normal
$\mathbb{N}$ | $\textbf{E}X=\mu$ | Geom($1/\mu$)
$\mathbb{R}^{+}$ | $\textbf{E}X=\mu$ | Exp($1/\mu$)
$\mathbb{R}$ |  $\textbf{E}X=\mu$; $\textbf{E}(X-\mu)^{2}=\sigma^{2}$ | $\mathcal{N}(\mu,\sigma^{2})$
Table 1: Maximum entropy distributions over some relevant spaces under
different restrictions.

III) Calculate the maximum probability of the life-permitting interval
$\ell_{X}$ under the class of distributions
$\mathcal{F}=\\{F(\cdot;\theta);\,\theta\in\Theta)\\}$. Following Thorvaldsen
and Hössjer [37], let $A$ be the event “We observe a universe that exists and
permits life,” and let us now regard $x$ as a parameter for a model $P(A|x)$
that gives the probability of observing a life-permitting universe. The tuning
probability of the event $A$ is then obtained by regarding $x$ as an
observation of $X\sim F$, and averaging $P(A|x)$ with respect to $x$, i.e.,

$P(A;\theta)=\int_{\Omega}P(A|x)dF(x;\theta),$ (3.1)

where, provided the prior density $f(x;\theta)$ exists,
$dF(x;\theta)=f(x;\theta)dx$; and $P(A|x)$ is the likelihood. If $A=\ell_{X}$,
then $P(A|x)=1$ if $x\in A$, and $P(A|x)=0$ if $x\notin A$. Then (3.1) reduces
to the tuning probability

$P(A;\theta)=F(\ell_{X};\theta)$ (3.2)

considered here. Notice in particular that we might know of $\ell_{X}$ through
the experiments or research that led to $\ell_{X}$ being determined. However,
in any case, the event $A$ is well defined whether we, as observers, know of
$\ell_{X}$ or not.

In order to finalize the third step and determine the degree of tuning, we
also need to maximize (3.2) with respect to the hyperparameter $\theta$, when
$\theta$ varies over a finite-dimensional space $\Theta$. That is, our final
degree of tuning equals

$P_{\mbox{\scriptsize max}}=\max_{\theta\in\Theta}F(\ell_{X};\theta).$ (3.3)

Notice in particular that the degree of tuning (3.3) can be calculated without
violating the weak anthropic principle. This is so, since we did not assume
that $x$ is an observer’s value of $X$ (since this would have enforced $X\sim
F_{\text{trunc}}$), nor did we estimate the hyperparameter $\theta$ from the
single observation $x$, but rather maximized the tuning probability (3.2) with
respect to $\theta$.

Let us from now on assume that $x$ is the mid point of the life permitting
interval. In more detail, suppose the life permitting interval

$\ell_{X}=x[1-\varepsilon,1+\varepsilon]=[a,b]$ (3.4)

is centered around $x$, with a relative half size $\varepsilon$. In the next
section, it will be shown that whenever $\ell_{X}$ is small, the upper bound
(3) of the tuning probability is proportional to $\varepsilon$, i.e.

$P_{\text{max}}(\varepsilon)=C\varepsilon,$ (3.5)

for a constant of proportionality $C$ that depends on the size of the family
of prior distributions $\mathcal{F}=\\{F(\cdot;\theta);\,\theta\in\Theta\\}$.
Since $\varepsilon$ is a dimensionless constant, it follows that the upper
bound of the tuning probability is dimensionless as well.

Now, in practice the life-permitting interval $\ell_{X}=[a,b]$ will never be
centered around the actual value of the physical parameter. It is possible to
solve this inconvenience by retaining our assumption $x=(a+b)/2$, but not
requiring $x=X_{\text{obs}}$. However, for simplicity of exposition we will
not make any distinction between $x$ and $X_{\text{obs}}$ in the text.

In the next two sections the proposed methodology is applied to the
gravitational constant and the primordial fluctuations using values of the
width of life-permitting intervals found in the literature. In both cases we
assume no additional knowledge beyond the sample space $\Omega$ of $X$, and
one or two moments of $X$. This assumption will simplify the calculation of
the respective probabilities. It does not mean, however, that additional
justified restrictions on the distribution of $X$, based on various types of
knowledge, cannot be incorporated. In Section 6.3 we argue that in order to
determine the parametric class $\mathcal{F}$ of possible distributions of $X$,
it is always possible to apply the maximum entropy principle conditionally on
the restrictions that were imposed on $X$.

##  4 Example 1: The Gravitational Constant

The quantity of interest here is the gravitational constant
$G_{\text{obs}}=6.67408\times
10^{-11}\text{m}^{3}\text{kg}^{-1}\text{s}^{-2}$. Thus $G_{\text{obs}}$ is an
observation of the random variable $G$. However, in this example
$G_{\text{obs}}$ itself is not considered, but rather a ratio
$x=G_{\text{obs}}/d$ between $G_{\text{obs}}$ and some other constant of
nature $d=D_{\text{obs}}$. This ratio is then an observation of the random
quantity $X=G/D$.

Several possible life-permitting intervals of the ratio $x=G_{\text{obs}}/d$
between $G_{\text{obs}}$ and some other constant of nature $d$ are now
illustrated. In all of these cases the life-permitting interval has the form
defined in (3.4); i.e.,

$\ell_{X}=[x-\delta,x+\delta]=x\cdot[1-\varepsilon,1+\varepsilon],$ (4.1)

where $\delta$ is a positive number usually small, and $\varepsilon=\delta/x$,
half of the relative size, is a dimensionless small number. As we will see
below, our tuning results will confirm (3.5) and be expressed in terms of this
dimensionless $\varepsilon$.

When $D=H^{2}$ is the random variable corresponding to the Hubble’s constant
squared, with observed value $d=H^{2}_{\text{obs}}$, the first Friedmann
equation (assuming $\Lambda_{\text{bare}}$, the cosmological constant, to be
0) is $x=G_{\text{obs}}/H^{2}_{\text{obs}}=3/(8\pi\rho_{\text{crit}})$, where
$\rho_{\text{crit}}$ is the critical density of the universe. According to
Davies, it then follows that $\varepsilon=10^{-60}$ [2, pp. 88-89].

Let $d=\Lambda_{\text{vac}}$, the dynamical contribution from vacuum energy to
the cosmological constant. Under the Weinberg-Salam electroweak theory,
$G_{\text{obs}}/\Lambda_{\text{vac}}=-\sqrt{2}c^{4}g_{w}/\pi m_{\phi}^{2}$,
where $c$ is the speed of light, $g_{w}$ is the weak force constant and
$m_{\phi}$ is the mass of the scalar particles. In this scenario, Davies
suggests $\varepsilon=10^{-50}$ or even $10^{-100}$ in the case of grand
unified theories [2, p. 107]. In fact, taking $\Lambda_{\text{bare}}\neq 0$,
we arrive at the cosmological constant problem in which the observed total
cosmological constant,
$\Lambda_{\text{tot}}=\Lambda_{\text{bare}}+\Lambda_{\text{vac}}$ is
$10^{120}$ times smaller than the predicted value of $\Lambda_{\text{vac}}$
[8, Ch. 5].

Other ratios are possible. For instance, the ratio of constant of gravity to
the constant of electromagnetic force is common in the fine-tuning literature
[8, Ch. 4], [10, Ch. 3], [38, Sect. IV]. Whatever the ratio, or the constant,
the general three-step method can be applied. Once the life-permitting
interval $\ell_{X}$ has been determined, (3.3) can be used to obtain the
tuning probability.

Remark: Notice that, even though we are using $G_{\text{obs}}$ here, we can
equivalently evaluate the tuning of the gravitational fine-structure constant
$\alpha_{G}=G_{\text{obs}}m_{P}^{2}/(\hbar c)\approx 5.9\times 10^{-39}$,
where $m_{P}$ is the Planck mass, $\hbar$ is the reduced Planck constant, and
$c$ is the speed of light. In contrast to $G_{\text{obs}}$, $\alpha_{G}$ has
the advantage of being dimensionless. Whereas $G_{\mbox{\scriptsize obs}}$ has
only been accurately measured to a relative precision of
$\varepsilon=10^{-15}$ [39], the fine-tuning assertions of $\alpha_{G}$
correspond to relative precision of $\varepsilon=10^{-60}$. In any case,
$\varepsilon$ is dimensionless for both of $G$ and $\alpha_{G}$.

In the next subsections we study the tuning status of the gravitational
constant. We do so under different assumptions of the relevant space, which in
turn reverberates in the election of the underlying MaxEnt distribution.

###  4.1 The maximum entropy principle applied to a ratio $G/D$ in
$\mathbb{R}^{+}$.

According to the first step, in order to determine the appropriate sample
space, gravity is assumed to be an attraction force, i.e., $G_{\text{obs}}>0$,
and the other constant of nature $d$ is non-negative. Therefore $x$ must be a
non-negative real number as well. Thus the sample space to be considered is
$\Omega=\mathbb{R}^{+}$.

For the second step —determining the right distribution— recall that the
untruncated distribution $F$ of $X$ is of interest. Since $X$ takes on values
in $\mathbb{R}^{+}$,

$P(X\leq z)=F(z;\mu)=F\left(\frac{z}{\mu};1\right)$ (4.2)

for some distribution $F$ (chosen below) with scale parameter $\theta=\mu$.
Thus, we choose the distribution of $X$ according to the MaxEnt principle.
Since the sample space is $\Omega=\mathbb{R}^{+}$, it follows from Table 1
that $F\sim\mbox{Exp}(1/\mu)$.

As for the third step, the probability of tuning is:

$\displaystyle P(X\in\ell_{X})$
$\displaystyle=\exp(-(x-\delta)/\mu)-\exp(-(x+\delta)/\mu)$
$\displaystyle=\exp(-(x-\delta)/\mu)[1-\exp(-2\delta/\mu)]$
$\displaystyle=2e^{-x/\mu}\sinh(\delta/\mu)$
$\displaystyle=P(\mu,\varepsilon),$ (4.3)

where $\varepsilon=\delta/x$. Although $P(\mu,\varepsilon)$ depends on $\mu$,
it is uniformly small in $\mu$ (while keeping $\varepsilon$ fixed). That is,
according to (3.3), and in agreement with (3.5), the degree of tuning,

$P_{\text{max}}(\varepsilon)=\sup_{\mu>0}P(\mu,\varepsilon),$ (4.4)

is of the same order as $\varepsilon$. $P_{\text{max}}(\varepsilon)$ in (4.4)
can be calculated analytically by first solving for
$dP(\mu,\varepsilon)/d\mu=0$, and then inserting this value of $\mu$ into
(4.1). In fact, $P(\cdot;\varepsilon)$ has a maximum at

$\mu=\frac{2x\varepsilon}{\log((1+\varepsilon)/(1-\varepsilon))}.$

A Taylor expansion of the denominator of the maximizing $\mu$ gives

$\mu\approx x(1-\varepsilon)\approx x,$ (4.5)

so that

$\displaystyle P_{\text{max}}(\varepsilon)$ $\displaystyle\approx
P(x;\varepsilon)$ $\displaystyle=2e^{-1}\sinh(\varepsilon)$
$\displaystyle\approx 0.7358\cdot\varepsilon,$ (4.6)

where $\varepsilon$ equals $10^{-50}$, $10^{-60}$, $10^{-100}$ or other values
depending, respectively, on whether $d$ is the Hubble constant squared, the
energy of the quantum vacuum, or any other relevant ratio under consideration.
Since $\varepsilon\ll 1$, there is extreme fine tuning in all these cases.

###  4.2 The maximum entropy principle applied to each of two random variables
$G$ and $D$ in $\mathbb{R}^{+}$ that form a ratio.

As a second application of (3.3), assume that $X=G/D$, where $G>0$ and $D>0$
are independent random variables, with distributions chosen according to the
MaxEnt principle. Assuming that the first moments of $G$ and $D$ are known,
then $G\sim\mbox{Exp}(1/\mu_{G})$ and $D\sim\mbox{Exp}(1/\mu_{D})$, where
$\mu_{G}=E(G)$ and $\mu_{D}=E(D)$ are two parameters that vary independently.
Then $X$ has a ratio distribution with density

$\displaystyle f_{X}(y;\mu)$
$\displaystyle=\int_{0}^{\infty}uf_{D}(u;\mu_{D})f_{G}(uy;\mu_{G})du$
$\displaystyle=\frac{1}{\mu}\cdot\frac{1}{(1+y/\mu)^{2}}$

for $y>0$ and $\mu=\mu_{G}/\mu_{D}$. This gives a tuning probability

$\displaystyle P(X\in\ell_{X})$
$\displaystyle=\frac{1}{\mu}\int_{x-\delta}^{x+\delta}\frac{1}{(1+y/\mu)^{2}}dy$
$\displaystyle=2\delta\cdot\frac{\mu}{(\mu+x-\delta)(\mu+x+\delta)}$ (4.7)
$\displaystyle=P(\varepsilon;\mu),$

where $\varepsilon=\delta/x$. Since the tuning probability in (4.2) only
depends on $\mu_{G}$ and $\mu_{D}$ through their ratio $\mu=\mu_{G}/\mu_{D}$,
according to (3.3), $P_{\mbox{\scriptsize max}}$ is obtained by maximizing
(4.2) with respect to $\mu>0$. This maximum is

$\mu=\sqrt{(x-\delta)(x+\delta)}=x\sqrt{1-\varepsilon^{2}}\approx x.$ (4.8)

Inserting (4.8) into (4.2) we thus obtain the upper bound of the tuning
probability,

$\displaystyle P_{\mbox{\scriptsize max}}(\varepsilon)$
$\displaystyle=P(x\sqrt{1-\varepsilon^{2}};\varepsilon)$ $\displaystyle\approx
2x\varepsilon\cdot\frac{x}{4x^{2}}$ (4.9)
$\displaystyle=\frac{\varepsilon}{2}.$

Note that (4.2) is minuscule being slightly smaller than the corresponding
upper bound in (4.1), where the MaxEnt principle was applied directly to the
distribution of the ratio $X$. Again, extreme fine tuning is seen whenever
$\varepsilon$ is small.

###  4.3 The maximum entropy principle applied to a ratio $X=G/D$ in
$\mathbb{R}$.

In the previous examples the only possibility considered was for gravity to be
a nonnegative number. Were $G$ allowed to be negative, both repulsion and
attraction would be possible. Were $G$ allowed to be also 0, gravity would be
a neutral force. In this scenario, according to the first step,
$G_{\text{obs}}$ must be thought of as a constant whose possible values
constitute the whole real line $\mathbb{R}$. The same is true for the ratio
$x=G_{\text{obs}}/d$ whether $d$ is negative or positive. In this example the
MaxEnt principle will be applied to the distribution of $X$.

In this section the MaxEnt principle is applied to $X$. Table 1 says that, if
the first two moments are known, the distribution that best explains $X$ is a
normal $\mathcal{N}(\mu,\sigma^{2})$. As in the previous example, $x$ can be
regarded as the midpoint of $\ell_{X}$. This leads to a probability of the
event $\\{X\in\ell_{X}\\}$ (step III):

$\displaystyle P(X\in\ell_{X})$
$\displaystyle=\Phi[(x+\delta-\mu)/\sigma)]-\Phi[(x-\delta-\mu)/\sigma]=P(\mu,\sigma;\varepsilon),$
(4.10)

where $\Phi(\cdot)$ is the standard normal distribution and
$\varepsilon=\delta/x$. Generally, (4.10) is small for large $\sigma$
regardless of the value of $\mu$. However, by choosing $\mu=x$ in (4.10) and
then letting $\sigma\to 0$, (3.3) attains the largest possible value

$P_{\text{max}}=\sup_{\mu\in\mathbb{R},\sigma>0}P(\mu,\sigma;\varepsilon)=1.$
(4.11)

In this case, since $\mu/\sigma$ is unrestricted, the constant of
proportionality $C$ in (3.5) is unbounded. This makes the tuning coarse.

In order to have a smaller value of $P_{\mbox{\scriptsize max}}$, additional
restrictions to the class of prior distributions are necessary. For instance,
under the assumption that neither positive nor negative values of $X$ should
be favored a priori, enforcing $\mu=0$ is natural. Then (4.10) takes the form

$P(\sigma;\varepsilon)\approx\phi(x/\sigma)\frac{2\delta}{\sigma},$ (4.12)

where $\phi(\cdot)=\Phi^{\prime}(\cdot)$ is the density function of a standard
normal. This expression will have a very small upper bound, uniformly in
$\sigma$, since

$\displaystyle P_{\max}(\varepsilon)$
$\displaystyle=\sup_{\sigma>0}P(\sigma;\varepsilon)$
$\displaystyle\approx\frac{2\delta}{x}\max_{\sigma>0}\left\\{\phi(x/\sigma)\frac{x}{\sigma}\right\\}$
$\displaystyle=\frac{2\delta\phi(1)}{x}$ (4.13)
$\displaystyle\approx\frac{2\delta\cdot 0.242}{x}$
$\displaystyle=0.484\cdot\varepsilon,$

where the second equality is obtained by maximizing $\phi(x/\sigma)(x/\sigma)$
with respect to $\sigma>0$. This is equivalent to maximizing $\phi(z)z$ with
respect to $z>0$, which achieves a maximum at
$\phi(1)=e^{-1/2}/\sqrt{2\pi}\approx 0.242$. See Figure 1 below, for an
illustration of the bounds in (4.11) and (4.3). Again, we see extreme fine-
tuning provided $0\notin\ell_{G}$ and $\varepsilon$ is small.

Figure 1: When the variance $\sigma^{2}$ of the prior distribution of $X$
approaches 0, the normal distribution approaches a Dirac delta measure at
$\mu$; thus $\mu\in\ell_{X}$, implies $P_{\mbox{\scriptsize max}}=1$ (left).
On the other hand, when $\mu\notin\ell_{X}$, where $\ell_{X}$ is the life-
permitting interval (LPI), $P[\ell_{G}]$ will go to zero either when
$\sigma\rightarrow 0$ or when $\sigma\rightarrow\infty$. Therefore
$P_{\mbox{\scriptsize max}}$ is strictly less than 1 (right). For the figure
at the left, the tuning is coarse, whereas for the figure at the right it is
fine.

###  4.4 The maximum entropy principle applied to each of two random variables
$G$ and $D$ in $\mathbb{R}$ that form a ratio.

We now turn to the second approach of choosing the distribution $F$ of
$X=G/D$, by using the MaxEnt principle to $G$ and $D$ separately, where $G$
and $D$ are both allowed to take on negative and positive values. In order not
to favor either positive or negative values of $G$ or $D$, assume that
$G\sim\mathcal{N}(0,\sigma_{G}^{2})$ and $D\sim\mathcal{N}(0,\sigma_{D}^{2})$.
Then $X$ has a Cauchy distribution with density

$\displaystyle f_{X}(y;\sigma)$
$\displaystyle=\int_{-\infty}^{\infty}|z|f_{D}(z;\sigma_{D})f_{G}(yz;\sigma_{G})dz$
$\displaystyle=\frac{1}{\sigma\pi}\frac{1}{(1+(y/\sigma)^{2})},$

for $y\in\mathbb{R}$, with $\sigma=\sigma_{G}/\sigma_{D}$. This gives a tuning
probability

$\displaystyle P(\sigma;\varepsilon)$
$\displaystyle=\int_{x-\delta}^{x+\delta}f_{X}(z;\sigma)dz$
$\displaystyle\approx 2\delta f_{X}(x;\sigma)$ (4.14)
$\displaystyle=\frac{2x\varepsilon}{\sigma}\frac{1}{\pi(1+(x/\sigma)^{2})}$

that only depends on $\sigma_{G}$ and $\sigma_{D}$ through their ratio
$\sigma$. Maximizing (4.4) with respect to $\sigma>0$ is equivalent to taking
the maximum with respect to $z=x/\sigma>0$. Consequently

$\displaystyle P_{\text{max}}(\varepsilon)$
$\displaystyle\approx\frac{2\varepsilon}{\pi}\max_{z>0}\frac{z}{1+z^{2}}$
$\displaystyle=\frac{\varepsilon}{\pi}.$

Interestingly, the Cauchy distribution itself (with scale parameter $\sigma$)
is MaxEnt over $\mathbb{R}$ under the restriction that
$E(\ln(1+X^{2}/\sigma^{2}))=2\ln 2$ [35].

###  4.5 The maximum entropy principle applied to each of two constants $G$ in
$\mathbb{R}$ and $D$ in $\mathbb{R}^{+}$, that form a ratio.

Another possibility is when $G\in\mathbb{R}$ and $D>0$ are chosen to have
MaxEnt distributions $G\sim\mathcal{N}(\mu_{G},\sigma_{G}^{2})$ and
$D\sim\mbox{Exp}(1/\mu_{D})$. As before, we assume $\mu_{G}=0$. Then $X=G/D$
will have a symmetric ratio distribution with scale parameter
$\sigma=\sigma_{G}/\mu_{D}$. The tuning probability is obtained similarly as
in (4.4), although the distribution of $X$ is no longer Cauchy.

###  4.6 The maximum entropy principle with an upper bound on prior
information

Recall from Section 4.3 that a normal MaxEnt prior distribution
$N(\mu,\sigma^{2})$ was assumed for a real-valued ratio $X=G/D$ of the
constant of gravity $G$ and another constant of nature $D$. The associated
tuning probability $P_{\text{max}}$ differs a lot depending on whether $\mu=0$
is enforced or not. The signal-to-noise ratio $\text{SNR}=\mu^{2}/\sigma^{2}$
quantifies the amount of information that the prior distribution carries about
$X$, on a relative scale. With no restriction on $\mu$, the prior distribution
could have an arbitrarily high SNR, resulting in $P_{\text{max}}=1$, whereas
$\mu=0$ corresponds to a requirement $\text{SNR}=0$. In between these two
extremes, we will assume that the signal-to-noise ratio has a finite upper
bound

$\displaystyle\text{SNR}\leq\text{SNR}_{\text{max}}\ll\varepsilon^{-2},$
(4.15)

where $\varepsilon$, the half length of the life-permitting interval
$\ell_{X}$ divided by the modulus of its mid point, is a small number. Then it
can be shown that the degree of tuning for a real-valued constant of nature,
or a real-valued ratio of two such constants, is given by

$P_{\text{max}}=\max\left\\{P(\mu,\sigma;\varepsilon);\,\,\frac{\mu^{2}}{\sigma^{2}}\leq\text{SNR}_{\text{max}}\right\\}\approx
2\varepsilon\sqrt{\text{SNR}_{\text{max}}}\phi(0)\ll 1,$ (4.16)

where $P(\mu,\sigma;\varepsilon)$ is the tuning probability defined in (4.10),
with $\delta=x\varepsilon$ and $x$ the observed value of $X$.

##  5 Example 2: Amplitude of primordial fluctuations

The amplitude of primordial fluctuations, $q=Q_{\text{obs}}\approx 2\times
10^{-5}$, is here the dimensionless value of interest [25]. Notice that in
contrast to Example 1, $q$ is here a constant of nature rather than a ratio
between two constants of nature. Rees writes [10, p. 128]:

> “If [$Q_{\text{obs}}$] were smaller than $10^{-6}$, gas would never condense
> into gravitationally bound structures at all, and such a universe would
> remain forever dark and featureless, even if its initial ‘mix’ of atoms,
> dark energy and radiation were the same as our own.
>
> On the other hand, a universe where [$Q_{\text{obs}}$] were substantially
> larger than $10^{-5}$ – where the initial ‘ripples’ were replaced by large-
> amplitude waves – would be a turbulent and violent place. Regions far bigger
> than galaxies would condense early in its history. They wouldn’t fragment
> into stars but would instead collapse into vast black holes, each much
> heavier than an entire cluster of galaxies in our universe… Stars would be
> packed too close together and buffeted too frequently to retain stable
> planetary systems.”

Adams gives an even larger life-permitting interval than Rees: $10^{-6}\leq
Q_{\text{obs}}\leq 10^{-2}$ [7]. But as we will see, there is little
difference in our overall conclusion that the amplitude of primordial
fluctuations is coarsely tuned. Following the three-step procedure for
calculating $P_{\text{max}}$, the first step is to determine the right sample
space.

The amplitude of the primordial fluctuations could be any nonnegative number;
in other words, there seems to be no mathematical restriction to impose that
the sample space of $Q$ must be bounded above. On the other hand, were $Q$
large, the FRW assumption (e.g., homogeneity and isotropy) would be violated,
which implies that the universe obtained would be so different to ours that
life as we know it would not be possible [see, e.g., 10, pp. 118, 128].
Therefore such universe would not be tuned for life. As such, large values of
$Q$ impose a boundary on the life-permitting interval, not on the sample space
of $Q$. Since we already have an upper bound of the life-permitting interval,
the FRW assumption imposes no further restriction on this interval. The
conclusion is that the sample space of $Q$ is $\mathbb{R}^{+}$.

Thus, from Table 1, we see that $Q\sim\text{Exp}(1/\mu)$, provided the mean
$\mu$ is given. The third step is to calculate the probability of the life-
permitting interval, so let’s take $\ell_{Q}=(10^{-6},10^{-5})$

$\displaystyle
P(Q\in\ell_{Q})=\exp\left(-10^{-6}/\mu\right)-\exp\left(-10^{-5}/\mu\right)=P(\mu),$
(5.1)

that depends on $\mu$. In this case, according to (3.3), the maximal value is
$P_{\text{max}}=\max_{\mu>0}P(\mu)$. This maximization is similar to the one
given in Section 4.1. Putting $a=10^{-6}$ and $b=10^{-5}$, the maximum tuning
probability is obtained for

$\displaystyle\mu=\frac{b-a}{\log(b/a)}=\frac{9\times 10^{-6}}{\log(10)}.$
(5.2)

Inserting (5.2) into (5.1) we obtain a rather large probability.

$\displaystyle P_{\mbox{\scriptsize max}}$
$\displaystyle=10^{-1/9}-10^{-10/9}$ $\displaystyle\approx 0.697.$ (5.3)

This is surprising given that there are claims of a much higher degree of
tuning [40]. In fact, our choice of $\ell_{Q}$ is extremely generous in favor
of fine-tuning, taking into account that $Q_{\text{obs}}\approx 2\times
10^{-5}\notin(10^{-6},10^{-5})=\ell_{Q}$! This reveals that there looks to be
a coarse tuning on the amplitude of the primordial fluctuations according to
the MaxEnt model. From a mathematical viewpoint, the coarse tuning of $Q$ is
not too surprising though, since the upper limit of $\ell_{Q}$ is one order of
magnitude larger than its lower limit.

##  6 Discussion

###  6.1 Summary

In this paper a Bayesian statistical procedure has been devised for
calculating an upper bound $P_{\mbox{\scriptsize max}}$ for the probability
that a constant of nature, or the ratio of two constants of nature, belongs to
a life permitting interval. This upper bound is invariant with respect to
change of physical units, and under certain assumptions it is small whenever
the life-permitting interval is small on a relative scale, corresponding to a
small value of $\varepsilon$. We obtain $P_{\mbox{\scriptsize max}}$ through a
three-steps procedure, where I) the sample space is determined, II) the
finite-dimensional class of distributions, of the constant of nature or the
ratio between constants of nature, is found for this sample space by applying
the maximum entropy principle, and III) the tuning probability is maximized
over this finite-dimensional class of distributions. The overall conclusion is
that the probability is proportional to the relative width of the life-
permitting interval, which will be small for intervals of narrow width, unless
the prior distribution carries a lot of information and is concentrated within
this region.

###  6.2 Choice of sample space

Recall that the first step of our tuning approach is to choose a sample space
$\Omega$ for a constant of nature or a ratio of such constants. This raises
the question of what should influence the choice of $\Omega$. For instance,
for the gravitational constant $G$ of Section 4, and the choice the whole real
line as the set of possible values of $G$, as in Sections 4.3 and 4.4, some
concern may appear when $G<0$ taking into account that in this case flat space
is explosively unstable. However this does not rule out the possibility of
$G<0$, it only rules out the possibility of a universe of the kind we know. In
particular for the tuning problem, it would exclude the possibility of carbon-
based life. But this does not entail that a negative gravitational constant
cannot be possible. To see this more clearly, Cline, Jeon, and Moore argue
that under our current theories certain constraints must be imposed on the
physical parameters [41]. This is true when we are evaluating our theories or
applying them to study the natural world within, not the natural world as
such. For the latter, it is not theories about the constants that we are
studying, but the constants themselves. Therefore, in the most strict sense,
such theories do not determine the support of the random variables nor their
realization in the constants of nature as we know them. Cosmological tuning
—why the physical constants have the values they have— is about the world as
such, not about the world within. Were theories in focus, fine-tuning would be
studied for theories, not for natural constants.

Physical parameters can be measured and quantified, whereas theories are
abstractions. For this reason, by inductive reasoning, it is physical
parameters that determine which theories are acceptable. It is also important
to have in mind that natural constants are actually constants, not
observations of random variables. Nevertheless, since all the Bayesian fine
tuning enterprise dwells on the assumption of random variables whose
observations are the natural constants we observe, it might be artificial to
restrain the values that such random variables can take for reasons other than
mathematical. This is to say that if natural constants are more fundamental
than physical theories, it seems reasonable that a priori constraints on such
constants are dictated by mathematics rather than theories.

Finally, it is not only physical theories but also observations that possibly
could restrict a priori assumptions on the constants of nature. The latter is
in fact an instantiation of the weak anthropic principle. As we argued in
Section 3, attempts to estimate hyperparameters based on observations are not
the right approaches to measure the probabilities of tuning, since they are
adding bias to the measurements of probabilities of life-permitting intervals.

###  6.3 Choice of prior and MaxEnt approach

It is important to notice that our approach goes beyond total ignorance and
can incorporate whatever knowledge is at hand, provided a MaxEnt distribution
exists under the relevant restrictions. In practice this means that if we are
using Lagrange optimization to obtain the MaxEnt distribution, such knowledge
will take the form of constraints — additional terms with their respective
Lagrange multipliers. Thus the addition of partial knowledge can be
incorporated into our MaxEnt approach.

In fact, for the sample spaces as considered in the examples, other
distributions are possible under different restrictions. For instance, Park
and Bera list eight MaxEnt distributions for $(-\infty,\infty)$, and six
MaxEnt distributions for $[0,\infty)$, corresponding to different restrictions
[35]. Of course, there are many other ways of choosing restrictions on the
prior distributions. This highlights the fact that the MaxEnt principle does
not imply that the distribution of the parameter needs to be totally unknown.
Rather, it means that the distribution is maximally unbiased given whatever
knowledge there is at hand. Indeed, if the distribution is known, it must
correspond to the MaxEnt distribution, conditioned on whatever is making it
known.

In order to describe in more detail how additional knowledge of constants of
nature influences the MaxEnt approach, consider equation (3.3): Additional
knowledge may on one hand increase the dimensionality $d$ of the parameter
vector $\theta=(\theta_{1},\ldots,\theta_{d})$, if, for instance, each
component of $\theta$ corresponds to a different aspect of the distribution of
the constant of nature $X$ (or the distribution of the ratio $X$ of two
constants of nature), such as the mean, variance, third moment, truncation
limits etc. Secondly, the more information that is available about the
distribution of $X$, the more the $d$-dimensional parameter space $\Theta$ is
shrunk. In particular, total knowledge of the distribution of $X$ implies that
$\Theta$ has only one element, say $\theta^{\ast}$, in which case
$P^{*}=F(\ell_{X};\theta^{*})$ is obtained from the only permissible
distribution of $X$; the one that corresponds to the known hyperparameter
$\theta^{*}$.

Now, if the "known" fine tuning probability $P^{*}$ does not correspond to
$P_{\text{max}}$, that was obtained in (3.3) with less knowledge about the
hyperparameter $\theta$, the bias produced by $P^{*}$ must be accounted for. A
very efficient way to keep accountability is by calculating the active
information $I^{+}=\log(P^{*}/P_{\text{max}})$ [42, 43, 44]. $I^{+}$ measures
the amount of information added or subtracted when $P^{*}$ is used instead of
$P_{\text{max}}$. If $I^{+}$ is positive, $P^{*}$ is overfitting for
$\ell_{X}$ with respect to $P_{\text{max}}$; if $I^{+}$ is negative, $P^{*}$
is underfitting the life-permitting interval with respect to $P_{\text{max}}$;
and if $I^{+}=0$, $P^{*}$ the extra assumptions used to calculate $P^{\ast}$
do not contribute anything in terms of cosmological tuning.

###  6.4 Outlook

The purpose of the examples was to illustrate as simply as posible the
mathematical approach to find expressions for the upper bound
$P_{\mbox{\scriptsize max}}$ of the tuning probability. However, these upper
bounds are accurate, given the assumed class $\mathcal{F}$ of prior
distributions and the observed life permitting interval $\ell_{X}$. In
particular we think that, if the constants are really fundamental, they cannot
be determined by theories, and it is theories that have to accommodate to
fundamental constants. Only other constant could constrain the existing ones
(however, we do not have a theory of everything that allows us to do so). For
these reasons we regard our approach as promising for future research.

On the other hand, a more epistemological approach can limit the sample spaces
of constants of nature, or ratios of constants of nature, according to
restrictions set by theories. Therefore, this paper sets the stage for a
fruitful research project in several aspects:

  1. 1.

To calculate the level of tuning for other parameters, constants of nature,
and boundary conditions. For instance, from Example 1, the ratio
$x=\Lambda_{\text{vac}}/\Lambda_{\text{bare}}$ of the two terms involved in
the cosmological constant is contained within an interval $\ell_{X}$ that is
centered around -1, with a relative half length of $\varepsilon=10^{-120}$.

  2. 2.

To study tuning probabilities under different assumptions that may be imposed
by theories.

  3. 3.

To find upper bounds for the joint tuning probability of several constants of
nature. This would in turn provide a lower bound for the expected number of
parallel universes needed in order to obtain one that permits life.

  4. 4.

To estimate the tuning probability by means of the signal-to-noise ratio, in
order to obtain an upper bound on the amount of information that the prior
distribution carries about a constant of nature, or a ratio of such constants.
In Section 4.6 this was done in a context where the first two moments of the
prior distribution are specified. This approach can be generalized, using
other measures of information of the prior distribution.

Finally, as mentioned in Section 1, the actual lengths of the life-permitting
intervals might change in the future, and thereby affect the values of
$P_{\text{max}}$. This is not a weakness of our method, since it will serve to
quantify the amount of tuning for a specific interval. The very nature of the
scientific enterprise will also produce new constants of nature, with
associated life-permitting intervals. In fact, as we saw in the examples,
different theories do produce different intervals. When new intervals or new
parameters arrive, our method will give an accurate upper bound to the
probability of tuning, conditionally on this new and updated knowledge.

##  7 Acknowledgements

The authors are thankful to Luke Barnes for his suggestions, as well as to
Aron Wall for his observations on the first version of this paper. We are also
grateful to an anonymous reviewer for excellent comments which greatly
improved this paper.

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# Constraining branon dark matter from observations of the dwarf spheroidal
galaxies with the MAGIC telescopes

T. Miener1 D. Nieto1 V. Gammaldi2 D. Kerszberg3 and J. Rico3<EMAIL_ADDRESS>1Instituto de Física de Partículas y del Cosmos and Department of EMFTEL,
Universidad Complutense de Madrid, E-28040 Madrid, Spain;
2Departamento de Física Teórica, Universidad Autónoma de Madrid, Madrid, Spain
$\&$ Instituto de Física Teórica, UAM/CSIC, E-28049 Madrid, Spain
3Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology (BIST), E-08193 Bellaterra (Barcelona), Spain

###### Abstract

We present the first branon dark matter (DM) search in the very high-energy
gamma-ray band with observations of the dwarf spheroidal galaxy Segue 1
carried out by the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescope
system. Branons are new degrees of freedom that appear in flexible brane-world
models corresponding to brane fluctuations. They behave as weakly interacting
massive particles, which are natural DM candidates. In the absence of a gamma-
ray signal in the Segue 1 data, we place constraints on the branon DM
parameter space by using a binned likelihood analysis. Our most constraining
limit to the thermally-averaged annihilation cross-section (at $95\%$
confidence level) corresponds to $\langle\sigma v\rangle\simeq 1.4\times
10^{-23}\,\mathrm{cm^{3}s^{-1}}$ at a branon DM mass of $\sim
0.7\,\mathrm{TeV}$.

##  1 Introduction

Astrophysical and cosmological evidences propose that non-baryonic cold dark
matter (DM) accounts for $84\%$ of the matter density of the Universe [1].
Nevertheless, the nature of DM is still an open question for modern physics,
as this elusive kind of matter can not be made of any of the Standard Model
(SM) particles. Brane-world theory has been put forward as a prospective
framework for DM candidates [2]. The characteristics of the suggested massive
brane fluctuations (branons) in this theory match the ones of weakly
interacting massive particles (WIMPs), which are one of the most favored
candidates of cold DM [3].

Dwarf spheroidal galaxies (dSphs) are preferred targets for indirect DM
searches, due to their close proximities and high mass-to-light ratios. In
this work, we are focusing on the ultra-faint dSph Segue 1 located in the
Northern Hemisphere and outside of the Galactic plane
($\text{RA}=10.12\,\mathrm{h}$, $\text{DEC}=16.08^{\circ}$) only $23\pm
2\,\mathrm{kpc}$ away from us [4]. The Segue 1 observational campaign was
carried out by the ground-based gamma-ray telescope MAGIC between 2011 and
2013 and is with 157.9 hours the deepest observation of any dSph by a
Cherenkov telescope to date [5].

##  2 Expected branon dark matter gamma-ray flux

The expected gamma-ray flux produced by branon DM annihilation is composed of
the astrophysical factor (J-factor), which depends on both the distance $l$
and the DM distribution at the source region $\rho_{\text{DM}}$, and the
branon DM annihilation photon yield. It can be expressed in a given region of
the sky, $\Delta\Omega$, and observed at Earth as

$\frac{\text{d}\Phi}{\text{d}E}\left(\langle\sigma
v\rangle\right)=\underbrace{\int_{\Delta\Omega}d\Omega^{\prime}\int_{\text{l.o.s.}}dl\,\rho_{\text{DM}}^{2}(l,\Omega^{\prime})}_{\text{Astrophysics}}\cdot\underbrace{\frac{1}{4\pi}\frac{\langle\sigma
v\rangle}{2m^{2}_{\chi}}\sum_{i=1}^{n}\text{Br}_{i}\frac{\text{d}N_{i}}{\text{d}E}}_{\text{Particle
physics}},$ (1)

where $\langle\sigma v\rangle$ is the thermally-averaged annihilation cross
section, $m_{\chi}$ is the mass of the branon DM particle and l.o.s. stands
for line-of-sight. The left panel of Fig. 1 shows the branon branching ratios
$\text{Br}_{i}$ as a function of $m_{\chi}$. The differential photon yield per
branon annihilation $\text{d}N/\text{d}E$ is depicted for a set of branon DM
masses (from light to dark: 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50 and
$100\,\mathrm{TeV}$) in the right panel of Fig. 1.

Figure 1: Taken from JCAP05(2022)005. See text for more details.

In our analysis, we are using the J-factor and its statistical uncertainty for
Segue 1 from [6] with the value of
$\mathrm{log}_{10}\left(J\left(\theta\right)[\mathrm{GeV^{2}cm^{-5}}]\right)=19.02_{-0.35}^{+0.32}$.
The differential gamma-ray yields per annihilation $\text{d}N_{i}/\text{d}E$
(see Eq. 1) are taken from the PPPC 4 DM ID distribution [7].

##  3 Likelihood analysis method

The data reduction of the Segue 1 observation have been published by the MAGIC
Collaboration [5, 8]. We re-analyse those high-level data in the context of
brane-world extra-dimensional theories using the likelihood analysis described
in Aleksić, Rico and Martinez [9]. The binned ($N_{\text{bins}}=30$)
likelihood function of the dataset $\bm{\mathcal{D}}$ with nuisance parameters
$\bm{\nu}$ reads as:

$\begin{split}\mathcal{L}_{\text{bin}}\left(\langle\sigma
v\rangle;\bm{\nu}\mid\bm{\mathcal{D}}\right)&=\mathcal{L}_{\text{bin}}(\langle\sigma
v\rangle;\\{b_{i}\\}_{i=1,\ldots,N_{\text{bins}}},\tau\mid\\{N_{\text{ON},i},N_{\text{OFF},i}\\}_{i=1,\ldots,N_{\text{bins}}})\\\
&=\prod_{i=1}^{N_{\text{bins}}}\Big{[}\mathcal{P}(s_{i}(\langle\sigma
v\rangle)+b_{i}\mid N_{\text{ON},i})\cdot\mathcal{P}(\tau b_{i}\mid
N_{\text{OFF},i})\Big{]}\times\mathcal{T}\left(\tau\mid\tau_{\text{o}},\sigma_{\tau}\right)\end{split}$
(2)

where $\mathcal{P}(x|N)$ is the Poisson distribution of mean $x$ and measured
value $N$ and $s_{i}(\langle\sigma v\rangle)$ and $b_{i}$ are the expected
numbers of signal and background events in the $i$-th energy bin. Besides
$b_{i}$, the normalization between background and signal regions $\tau$,
described by the likelihood function $\mathcal{T}$, is also a nuisance
parameter in the analysis [10]. The total number of observed events in a given
energy bin $i$ in the signal (ON) and background (OFF) regions are
$N_{\text{ON},i}$, $N_{\text{OFF},i}$, respectively.

The Segue 1 observational campaign $\bm{\mathcal{D}}_{\mathrm{Segue1}}$
results in $N=8$ distinct datasets with an individual set of instrument
response functions (IRFs). The joint likelihood function

$\mathcal{L}\left(\langle\sigma
v\rangle;J,\bm{\nu}\mid\bm{\mathcal{D}}_{\text{Segue1}}\right)=\prod_{k=1}^{N}\Big{[}\mathcal{L}_{\text{bin},k}\left(\langle\sigma
v\rangle,\bm{\nu_{k}}\mid\bm{\mathcal{D}_{k}}\right)\Big{]}\times\mathcal{J}\left(J\mid
J_{\text{o}},\sigma_{\log_{10}J}\right)$ (3)

is the product of the likelihood function of each dataset. We treat the
J-factor as a nuisance parameter and include the likelihood $\mathcal{J}$ for
the J-factor [11]. $\bm{\nu_{k}}$ represents the set of nuisance parameters
different from the J-factor affecting the analysis of the $k$-th dataset
$\bm{\mathcal{D}_{k}}$.

The same analysis technique, implemented in the open-source DM analysis
software packages [12] gLike [13] and LklCom [14], has been used by the MAGIC
Collaboration [10, 15] and other gamma-ray observatories in several DM
searches [16, 17].

##  4 Results

As reported in [18], we present the observational $95\%$ confidence level (CL)
upper limits on the thermally-averaged cross-section $\langle\sigma v\rangle$
(see left panel of Fig. 2) and on the brane tension $f$ (see right panel of
Fig. 2) for branon DM annihilation obtained with almost 160 hours of Segue 1
observation with the MAGIC telescopes. We perform a model dependent search for
branon DM particles of masses between $100\,\mathrm{GeV}$ and
$100\,\mathrm{TeV}$. Previous branon DM limits from CMS [19] (blue) and AMS-02
[20] (orange) as well as the prospects of the future CTA [21] (purple) and SKA
[22] (yellow) are depicted in Fig. 2.

As expected from the no significant gamma-ray excess in the Segue 1 data [5],
our constraints for branon DM annihilation are located within the $68\%$
containment band, which is consistent with the no-detection scenario. We
obtain our strongest limit $\langle\sigma v\rangle\simeq 1.4\times
10^{-23}\,\mathrm{cm^{3}s^{-1}}$ for a $\sim 0.7\,\mathrm{TeV}$ branon DM
particle mass.

Figure 2: Taken from JCAP05(2022)005. See text for more details.

##  5 Conclusion and outlook

This work leads to the most constraining branon DM limits in the multi-TeV
mass range, superseding previous constraints by CMS [19] and AMS-02 [20] above
$\sim 1\,\mathrm{TeV}$. We can achieve even more stringent exclusion limits by
adding further dSph observations of the MAGIC telescopes [10] or other gamma-
ray telescopes [16, 17] to this analysis.

## Acknowledgments

The authors thank the MAGIC Collaboration for providing private data.
TM acknowledges support from PID2019-104114RB-C32. DN and TM acknowledge
support from the former Spanish Ministry of Economy, Industry, and
Competitiveness / European Regional Development Fund grant FPA2015-73913-JIN.
VG’s contribution to this work has been supported by Juan de la Cierva-
Incorporación IJC2019-040315-I grant, and by the PGC2018-095161-B-I00 and
CEX2020-001007-S projects, both funded by MCIN/AEI/10.13039/501100011033 and
by ”ERDF A way of making Europe”. VG thanks J.A.R. Cembranos for useful
discussions.
DK is supported by the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 754510. DK and
JR acknowledge the support from MCIN/AEI/ 10.13039/501100011033 under grants
PID2019-107847RB-C41 and SEV-2016-0588 (”Centro de Excelencia Severo Ochoa”),
and from the CERCA institution of the Generalitat de Catalunya.

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