Pierre-Henri Chavanis

New predictions from the logotropic model

Keywords: numbers: 95.30.Sf, 95.35.+d, 95.36.+x, 98.62.Gq, 98.80.-k

In a previous paper [P.H. Chavanis, Eur. Phys. J. Plus 130, 130 (2015)] we have introduced a new cosmological model that we called the logotropic model. This model involves a fundamental constant Λ which is the counterpart of Einstein's cosmological constant in the ΛCDM model. The logotropic model is able to account, without free parameter, for the constant surface density of the dark matter halos, for their mass-radius relation, and for the Tully-Fisher relation. In this paper, we explore other consequences of this model. By advocating a form of "strong cosmic coincidence" we predict that the present proportion of dark energy in the Universe is Ω de,0 = e/(1 + e) 0.731 which is close to the observed value. We also remark that the surface density of dark matter halos and the surface density of the Universe are of the same order as the surface density of the electron. This makes a curious connection between cosmological and atomic scales. Using these coincidences, we can relate the Hubble constant, the electron mass and the electron charge to the cosmological constant. We also suggest that the famous numbers 137 (fine-structure constant) and 123 (logotropic constant) may actually represent the same thing. This could unify microphysics and cosmophysics. We study the thermodynamics of the logotropic model and find a connection with the Bekenstein-Hawking entropy of black holes if we assume that the logotropic dark fluid is made of particles of mass mΛ ∼ √ Λ/c 2 = 2.08 × 10 -33 eV/c 2 (cosmons). In that case, the universality of the surface density of the dark matter halos may be related to a form of holographic principle (the fact that their entropy scales like their area). We use similar arguments to explain why the surface density of the electron and the surface density of the Universe are of the same order and justify the empirical Weinberg relation. Finally, we combine the results of our approach with the quantum Jeans instability theory to predict the order of magnitude of the mass of ultralight axions m ∼ 10 -23 eV/c 2 in the Bose-Einstein condensate dark matter paradigm.

I. INTRODUCTION

The nature of dark matter and dark energy remains one of the greatest mysteries of modern cosmology. Dark matter is responsible for the flat rotation curves of the galaxies and dark energy is responsible for the accelerated expansion of the Universe. It is found that dark energy represents about 70% of the energy content of the present Universe while the proportions of dark matter and baryonic matter are 25% and 5% respectively.

In a previous paper [1] (see also [2,3]) we have introduced a new cosmological model that we called the logotropic model. In this model, there is no dark matter and no dark energy. There is just a single dark fluid. What we call "dark matter" actually corresponds to its rest-mass energy and what we call "dark energy" corresponds to its internal energy. 1 Our model does not contain any arbitrary parameter so that it is totally constrained. It involves a fundamental constant Λ which is the counterpart of Einstein's cosmological constant [4] in the ΛCDM (cold dark matter) model and which turns out to have the same value. Still the logotropic model is fundamentally different from the 1 Many models try to unify dark matter and dark energy. They are called unified dark energy and dark matter (UDE/M) models. However, the interpretation of dark matter and dark energy that we give in Refs. [1,2] is new and original.

ΛCDM model. On the large (cosmological) scales, the logotropic model is indistinguishable from the ΛCDM model up to the present epoch [1][2][3]. The two models will differ in the far future, in about 25 Gyrs years, after which the logotropic model will become phantom (the energy density will increase as the Universe expands) and present a Little Rip (the energy density and the scale factor will become infinite in infinite time) contrary to the ΛCDM model in which the energy density tends towards a constant (de Sitter era).

On the small (galactic) scales, the logotropic model is able to solve some of the problems encountered by the ΛCDM model [1,2]. In particular, it is able to account, without free parameter, for the constant surface density of the dark matter halos, for their mass-radius relation, and for the Tully-Fisher relation.

In this paper, we explore other consequences of this model. By advocating a form of "strong cosmic coincidence", stating that the present value of the dark energy density ρ de,0 is equal to the fundamental constant ρ Λ appearing in the logotropic model, we predict that the present proportion of dark energy in the Universe is Ω de,0 = e/(1 + e) = 0.731 which is close to the observed value 0.691 [5]. The consequences of this result, which implies that our epoch is very special in the history of the Universe, are intriguing and related to a form of anthropic cosmological principle [START_REF] Barrow | The Anthropic Cosmological Principle[END_REF].

We also remark that the universal surface density of dark matter halos (found from the observations [START_REF] Donato | [END_REF] and predicted by our model [1,2]) and the surface density of the Universe are of the same order of magnitude as the surface density of the electron. This makes a curious connection between cosmological and atomic scales. Exploiting this coincidence, we can relate the Hubble constant, the electron mass and the electron charge to the cosmological constant Λ. We also argue that the famous numbers 137 (fine-structure constant) and 123 (logotropic constant) may actually represents the same thing. This may be a hint for a theory of unification of microphysics and cosmophysics. Speculations are made in the Appendices to try to relate these interconnections to a form of holographic principle [8] stating that the entropy of the electron, the entropy of dark matter halos, and the entropy of the Universe scales like their area as in the case of the entropy of black holes [9,10].

II. THE LOGOTROPIC MODEL A. Unification of dark matter and dark energy

The Friedmann equations for a flat universe without cosmological constant are [START_REF] Weinberg | Gravitation and Cosmology[END_REF]:

d dt + 3 ȧ a ( + P ) = 0, H 2 = ȧ a 2 = 8πG 3c 2 , (1)
where (t) is the energy density of the Universe, P (t) is the pressure, a(t) is the scale factor, and H = ȧ/a is the Hubble parameter. For a relativistic fluid experiencing an adiabatic evolution such that T d(s/ρ) = 0, the first law of thermodynamics reduces to [START_REF] Weinberg | Gravitation and Cosmology[END_REF]:

d = P + ρ dρ, (2)
where ρ(t) is the rest-mass density of the Universe. Combined with the equation of continuity (1), we get

dρ dt + 3 ȧ a ρ = 0 ⇒ ρ = ρ 0 a 3 , (3)
where ρ 0 is the present value of the rest-mass density (the present value of the scale factor is taken to be a 0 = 1). This equation, which expresses the conservation of the rest-mass, is valid for an arbitrary equation of state.

For an equation of state specified under the form P = P (ρ), Eq. ( 2) can be integrated to obtain the relation between the energy density and the rest-mass density ρ. We obtain [1]:

= ρc 2 + ρ ρ P (ρ ) ρ 2 dρ = ρc 2 + u(ρ). (4)
We note that u(ρ) can be interpreted as an internal energy density [1]. Therefore, the energy density is the sum of the rest-mass energy ρc 2 and the internal energy u(ρ).

B. The logotropic dark fluid

We assume that the Universe is filled with a single dark fluid described by the logotropic equation of state [1]:

P = A ln ρ ρ P , (5)
where ρ P = c 5 / G 2 = 5.16 × 10 99 g m -3 is the Planck density and A is a new fundamental constant of physics, with the dimension of an energy density, which is the counterpart of the cosmological constant Λ in the ΛCDM model (see below). Using Eqs. ( 4) and ( 5), the relation between the energy density and the rest-mass density is

= ρc 2 -A ln ρ ρ P -A = ρc 2 + u(ρ). (6)
The energy density is the sum of two terms: a restmass energy term ρc 2 = ρ 0 c 2 /a 3 that mimics the energy density m of dark matter and an internal energy term u(ρ) = -A ln (ρ/ρ P ) -A = -P (ρ) -A = 3A ln a -A ln(ρ 0 /ρ P ) -A that mimics the energy density de of dark energy. This decomposition leads to a natural, and physical, unification of dark matter and dark energy and elucidates their mysterious nature.

Since, in our model, the rest-mass energy of the dark fluid mimics dark matter, we identify ρ 0 c 2 with the present energy density of dark matter. We thus set ρ 0 c 2 = Ω m,0 0 , where 0 /c 2 = 3H 2 0 /8πG is the present energy density of the Universe and Ω m,0 is the present fraction of dark matter (we also include baryonic matter). As a result, the present internal energy of the dark fluid, u 0 = 0 -ρ 0 c 2 , is identified with the present dark energy density de,0 = Ω de,0 0 where Ω de,0 = 1 -Ω m,0 is the present fraction of dark energy. Applying Eq. ( 6) at the present epoch (a 0 = 1), we obtain the identity A = de,0

ln ρ P c 2 de,0 + ln Ω de,0 1-Ω de,0 -1 . ( 7
)
At that stage, we can have two points of view. We can consider that this equation determines the constant A as a function of 0 and Ω de,0 that are both obtained from the observations [5]. This allows us to determine the value of A. This is the point of view that we have adopted in our previous papers [1,2] and that we adopt in Sec. II D below. However, in the following section, we present another point of view leading to an intriguing result.

C. Strong cosmic coincidence and prediction of Ω de,0

Let us recall that, in our model, A is considered as a fundamental constant whose value is fixed by Nature. As a result, Eq. [START_REF] Donato | [END_REF] relates Ω de,0 to 0 for a given value of A. A priori, we have two unknowns for just one equation. However, we can obtain the value of Ω de,0 by the following argument.

We can always write the constant A under the form

A = ρ Λ c 2 ln ρ P ρΛ . (8)
This is just a change of notation. Eq. ( 8) defines a new constant, the cosmological density ρ Λ , in place of A. From the cosmological density ρ Λ , we can define an effective cosmological constant Λ by2

ρ Λ = Λ 8πG . ( 9
)
Again this is just a change of notation. Therefore, the fundamental constant of our model is either A, ρ Λ or Λ (equivalently). We now advocate a form of "strong cosmic coincidence". We assume that the present value of the dark energy density is equal to ρ Λ c 2 , i.e.,

de,0 = ρ Λ c 2 . ( 10
)
Since, in the ΛCDM model, de is a constant usually measured at the present epoch our postulate implies that ρ Λ c 2 coincides with the cosmological density in the ΛCDM model and that Λ, as defined by Eq. ( 9), coincides with the ordinary cosmological constant. This is why we have used the same notations. Now, comparing Eqs. ( 7), ( 8) and ( 10) we obtain ln [Ω de,0 /(1 -Ω de,0 )] -1 = 0 which determines Ω de,0 . We find that Ω th de,0 = e 1 + e 0.731 [START_REF] Weinberg | Gravitation and Cosmology[END_REF] which is close to the observed value Ω obs de,0 = 0.691 [5]. This agreement is puzzling. It relies on the "strong cosmic coincidence" of Eq. (10) implying that our epoch is very special. This is a form of anthropic cosmological principle [START_REF] Barrow | The Anthropic Cosmological Principle[END_REF]. This may also correspond to a fixed point of our model. In order to avoid philosophical issues, in the following, we adopt the more conventional point of view discussed at the end of Sec. II B.

D. The logotropic constant B

We can rewrite Eq. ( 8) as

A = Bρ Λ c 2 with B = 1 ln (ρ P /ρ Λ ) . ( 12
)
Again, this is just a change of notation defining the dimensionless number B. We shall call it the logotropic constant since it is equal to the inverse of the logarithm of the cosmological density normalized by the Planck density (see Appendix A). We note that A can be expressed in terms of B (see below) so that the fundamental constant of our model is either A, ρ Λ , Λ, or B. In the following, we shall express all the results in terms of B. For example, the relation (6) between the energy density and the scale factor can be rewritten as

0 = Ω m,0 a 3 + (1 -Ω m,0 )(1 + 3B ln a). (13)
Combined with the Friedmann equation (1) this equation determines the evolution of the scale factor a(t) of the Universe in the logotropic model. This evolution has been studied in detail in [1][2][3].

Remark: Considering Eq. ( 13), we see that the ΛCDM model is recovered for B = 0. According to Eq. ( 12) this implies that ρ P → +∞, i.e., → 0. Therefore, the ΛCDM model corresponds to the semiclassical limit of the logotropic model. The fact that B is intrinsically nonzero implies that quantum mechanics ( = 0) plays some role in our model in addition to general relativity. This may suggest a link with a theory of quantum gravity.

E. The value of B from the observations

The fundamental constant (A, ρ Λ , Λ, or B) appearing in our model can be determined from the observations by using Eq. [START_REF] Donato | [END_REF]. We take Ω de,0 = 0.6911 and H 0 = 2.195 × 10 -18 s -1 [5]. This implies 0 /c 2 = 3H 2 0 /8πG = 8.62 × 10 -24 g m -3 and de,0 /c 2 = Ω de,0 0 /c 2 = 5.96 × 10 -24 g m -3 . Since ln [Ω de,0 /(1 -Ω de,0 )] -1 = -0.195 is small as compared to ln(ρ P c 2 / de,0 ) = 283, we can write in very good approximation A as in Eq. ( 8) with ρ Λ de,0 /c 2 as in Eq. (10). Therefore,

ρ Λ = 3Ω de,0 H 2 0 8πG = 5.96 × 10 -24 g m -3 (14)
and

Λ = 3Ω de,0 H 2 0 = 1.00 × 10 -35 s -2 (15)
are approximately equal to the cosmological density and to the cosmological constant in the ΛCDM model. From Eq. ( 12) we get

B = 1 ln(ρ P /ρ Λ ) 1 123 ln(10) 3.53 × 10 -3 . ( 16
)
As discussed in our previous papers [1][2][3], B is essentially the inverse of the famous number 123 (see Appendix A). Finally,

A = B ρ Λ c 2 = 1.89 × 10 -9 g m -1 s -2 . ( 17
)
From now on, we shall view B given by Eq. ( 16) as the fundamental constant of the theory. Therefore, everything should be expressed in terms of B and the other fundamental constants of physics defining the Planck scales. First, we have

ρ Λ ρ P = G Λ 8πc 5 = e -1/B = 1.16 × 10 -123 . (18)
Then,

A ρ P c 2 = Be -1/B = 4.08 × 10 -126 . ( 19
)
The logotropic equation of state (5) can be written as P/ρ P c 2 = Be -1/B ln(ρ/ρ P ). Using Eq. ( 10) and de,0 = Ω de,0 0 , we get

0 ρ P c 2 = 1 Ω de,0 e -1/B = 1.67 × 10 -123 . (20)
Finally, using Eq. ( 1),

t P H 0 = 8π 3Ω de,0 1/2 e -1/2B = 1.18 × 10 -61 , (21)
where t P = ( G/c 5 ) 1/2 = 5.391 × 10 -44 s is the Planck time. In the last two expressions, we can either consider that Ω de,0 is "predicted" by Eq. ( 11) or take its measured value. To the order of accuracy that we consider, this does not change the numerical values.

F. Generalized logotropic models

Recently, the original logotropic model corresponding to a purely logarithmic equation of state P = A ln(ρ/ρ * ) [1][2][3] has been generalized to an expression of the form P = A(ρ/ρ * ) -n ln(ρ/ρ * ) where n is a free parameter [START_REF] Capozziello | [END_REF][13][14]. Interestingly, this equation of state is similar to the Anton-Schmidt [15] equation of state for crystalline solids in the Debye approximation [16]. In that case, the index n can be written as n = -1/6 -γ G where γ G is the so-called Grüneisen [17] parameter. The original logotropic model is recovered for n = 0. The authors of [START_REF] Capozziello | [END_REF] compared the predictions of the generalized logotropic model with cosmological observations and found that B

3.54 × 10 -3 and n = -0.147 +0.113 -0.107 . This confirms the robustness of the value of the fundamental constant B = 3.53 × 10 -3 introduced in [1,2]. On the other hand, up to the error bars, the value of n is close to n = 0, corresponding to the logotropic model (see also [3]). This suggests that the logotropic model tends to be selected among more general families of models containing additional parameters {n}. What is crucial in the logotropic model is the logarithmic factor ln(ρ/ρ * ). Very generally, we could consider an equation of state of the form P = A n (ρ) ln(ρ/ρ * ) with A n (ρ) → cte when n → 0. In the present paper, we stick to the original logotropic model [1][2][3] but it would be interesting to see how the results are modified in more general circumstances [START_REF] Capozziello | [END_REF][13][14].

III. PREVIOUS PREDICTIONS OF THE LOGOTROPIC MODEL

The interest of the logotropic model becomes apparent when it is applied to dark matter halos [1,2]. We assume that dark matter halos are described by the logotropic equation of state of Eq. ( 5) with A = 1.89 × 10 -9 g m -1 s -2 (or B = 3.53 × 10 -3 ). At the galactic scale, we can use Newtonian gravity.

A. Surface density of dark matter halos

It is an empirical evidence that the surface density of galaxies has the same value

Σ obs 0 ≡ ρ 0 r h 295 g m -2 141 M /pc 2 (22)
even if their sizes and masses vary by several orders of magnitude (up to 14 orders of magnitude in luminosity) [START_REF] Donato | [END_REF]. Here ρ 0 is the central density and r h is the halo radius at which the density has decreased by a factor of 4. The logotropic model predicts that the surface density of the dark matter halos is the same for all the halos (because A is a universal constant) and that it is given by [1, 2]

Σ th 0 = A 4πG 1/2 ξ h = B 32 1/2 ξ h π c √ Λ G , (23)
where ξ h = 5.8458... is a pure number arising from the Lane-Emden equation of index n = -1 expressing the condition of hydrostatic equilibrium of logotropic spheres. 3 Numerically,

Σ th 0 = 278 g m -2 133 M /pc 2 , (24)
which is very close to the observational value (22). The fact that the surface density of dark matter halos is determined by the effective cosmological constant Λ (usually related to the dark energy) tends to confirm that dark matter and dark energy are just two manifestations of the same dark fluid, as we have assumed in our model.

Remark:

The dimensional term c √ Λ/G in Eq. ( 23) can be interpreted as representing the surface density of the Universe (see Appendix B). We note that this term alone, c √ Λ/G = 14200 g m -2 = 6800M /pc 2 , is too large to account precisely for the surface density of dark matter halos so that the prefactor (B/32) 1/2 (ξ h /π) = 0.01955 is necessary to reduce this number. It is interesting to remark that the term c √ Λ/G arises from classical general relativity while the prefactor ∝ B 1/2 has a quantum origin as discussed at the end of Sec. II D. Actually, we will see that it is related to the fine-structure constant α [see Eq. ( 33) below].

B. Mass-radius relation

There are interesting consequences of the preceding result. For logotropic halos, the mass of the halos calculated at the halo radius r h is given by [1,2]

M h = 1.49 Σ 0 r 2 h . (25)
This determines the mass-radius relation of dark matterhalos. On the other hand, the circular velocity at the halo radius is v 2 h = GM h /r h = 1.49Σ 0 Gr h . Since the surface density of the dark matter halos is constant, we obtain

M h M = 198 r h pc 2 , v h km s -1 2 = 0.852 r h pc . (26)
The scalings M h ∝ r 2 h and v 2 h ∝ r h (and also the prefactors) are consistent with the observations.

C. The Tully-Fisher relation

Combining the previous equations, the logotropic model leads to the Tully-Fisher [23]

relation v 4 h ∝ M h or, more precisely, M b v 4 h th = f b 1.49Σ th 0 G 2 = 46.4 M km -4 s 4 , (27)
where f b = M b /M h ∼ 0.17 is the cosmic baryon fraction [24]. The predicted value from Eq. ( 27) is close to the observed one M b /v4 h obs = 47 ± 6 M km -4 s 4 [24].

Remark: The Tully-Fisher relation is sometimes justified by the MOND (Modification of Newtonian dynamics) theory [25] which predicts a relation of the form v 4 h = Ga 0 M b between the asymptotic circular velocity and the baryon mass, where a 0 is a critical acceleration.

Our results imply a th 0 = 1.62 × 10 -10 m s -2 which is close to the value a obs 0 = (1.3±0.3)×10 -10 m s -2 obtained from the observations [24]. Combining Eqs. ( 25) and ( 27), we first get

a th 0 = (1.49/f b )GΣ th 0 = GM h /(f b r 2 h
) which shows that a 0 can be interpreted as the surface gravity of the galaxies GΣ 0 (which corresponds to Newton's acceleration GM h /r 2 h ) or as the surface gravity of the Universe (see Appendix C). Then, using Eqs. ( 15) and ( 23), we obtain

a th 0 = (1.49/f b )(B/32) 1/2 (ξ h /π)c
√ Λ H 0 c/4 which explains why a 0 is of the order of H 0 c [2]. We emphasize, however, that we do not use the MOND theory in our approach and that the logotropic model assumes the existence of a dark fluid.

D. The mass M300

The logotropic equation of state also explains the observation of Strigari et al. [26] that all the dwarf spheroidals (dSphs) of the Milky Way have the same total dark matter mass M 300 contained within a radius r u = 300 pc, namely M obs 300 10 7 M The logotropic model predicts the value [1, 2]

M th 300 = 4πΣ th 0 r 2 u ξ h √ 2 = 1.82 × 10 7 M , (28)
which is in very good agreement with the observational value.

IV. A CURIOUS CONNECTION BETWEEN ATOMIC AND COSMOLOGICAL SCALES

A. The surface density of the electron

The classical radius of the electron r e can be obtained qualitatively by writing that the electrostatic energy of the electron, e 2 /r e , is equal to its rest-mass energy m e c 2 . Recalling the value of the charge of the electron e = 4.80 × 10 -13 g 1/2 m 3/2 s -1 and its mass m e = 9.11 × 10 -28 g, we obtain r e = e 2 /m e c 2 = 2.82 × 10 -15 m. As a result, the surface density of the electron is 4

Σ e = m e r 2 e = m 3 e c 4 e 4 = 115 g/m 2 = 54.9 M /pc 2 , ( 29
)
which is of the same order of magnitude as the surface density of dark matter halos from Eq. ( 22). This coincidence is amazing in view of the different scales (atomic versus cosmological) involved. More precisely, we find Σ e = σΣ th 0 with σ 0.413. Of course, the value of σ depends on the precise manner used to define the surface density of the electron, or its radius, but the important point is that this number is of order unity.

B. Relation between α and B

By matching the two formulae ( 23) and ( 29), writing Σ e = σΣ th 0 , we get

Λ = 32π 2 Bξ 2 h σ 2 m 6 e c 6 G 2 e 8 = 32π 2 Bξ 2 h σ 2 α 4 m 6 e c 2 G 2 4 , (30)
where we have introduced the fine-structure constant α in the second equality (see Appendix A). This expression provides a curious relation between the cosmological constant, the mass of the electron, and its charge. This relation is similar to Weinberg's empirical relation (see Appendix B) which can be written as [combining Eqs. ( 15) and (B3)]

Λ = 192π 2 µ 2 Ω de,0 m 6 e c 6 G 2 e 8 , (31)
where µ 3.42. Note that in our formula (30), Λ appears two times: on the left hand side and in B (which depends logarithmically on Λ). This will have important consequences in the following.

Böhmer and Harko [27], by a completely different approach, found a similar relation5

Λ = ν 2 G 2 m 6 e c 8 e 12 = ν α 6 G 2 m 6 e c 2 4 , (32)
where ν 0.816 is of order unity. Their result can be obtained as follows. They first introduced a minimum mass m Λ ∼ √ Λ/c 2 = 2.08 × 10 -33 eV/c 2 interpreted as being the mass of the elementary particle of dark energy, called the cosmon. Then, they defined a radius R by the relation m Λ ∼ ρ Λ R 3 where ρ Λ = Λ/8πG is the cosmological density considered as being the lowest density in the Universe. Finally, they remarked that R ∼ 8.54 × 10 -15 m has typically the same value as the classical radius of the electron r e = e 2 /m e c 2 = 2.82×10 -15 m. Matching R and r e leads to the scaling of Eq. ( 32). We have then added a prefactor ν and adjusted its value in order to exactly obtain the measured value of the cosmological constant [5]. Since the approach of Böhmer and Harko [27] is essentially qualitative, and depends on the precise manner used to define the radius of the electron, their result can be at best valid up to a constant of order unity.

We would like now to compare the estimates from Eqs. (30) and [START_REF] Eddington | Fundamental Theory[END_REF]. At that stage, we can have two points of view. If we consider that comparing the prefactors is meaningless because our approach can only provide "rough" orders of magnitude, we conclude that Eqs. (30) and [START_REF] Eddington | Fundamental Theory[END_REF] are equivalent, and that they are also equivalent to Weinberg's empirical relation (B3). Alternatively, if we take the prefactors seriously into account (in particular the presence of B which depends on Λ) and match the formulae (30) and [START_REF] Eddington | Fundamental Theory[END_REF], we find an interesting relation between the fine-structure constant α and the logotropic constant B:

α = ν 32 1/2 ξ h σ π √ B 0.123 √ B. (33)
Therefore, the fine-structure constant (electron charge normalized by the Planck charge) is determined by the logotropic constant B (logarithm of the cosmological density normalized by the Planck density) by a relation of the form α ∝ B 1/2 . This makes a connection between atomic scales and cosmological scales. This also suggests that the famous numbers 137 and 123 (see Appendix A) are related to each other, or may even represent the same thing. From Eq. ( 33), we have 6

137 12.3 √ 123. ( 34
)
Remark: the logotropic constant B is related to the effective cosmological constant Λ by [see Eq. ( 18)]

B = 1 ln 8πc 5 G Λ . ( 35
)
Using Eqs. ( 33) and ( 35), we can express the finestructure constant α as a function of the effective cosmological constant Λ or, using Eq. ( 21), as a function of the age of the Universe t Λ = 1/H 0 . We get

α = 0.123 ln 8πc 5 G Λ 1/2 = 0.123 √ 2 ln 8π 3Ω de,0 1/2 tΛ t P 1/2 . ( 36
)
We emphasize the scaling 1/α ∝ (ln t Λ ) 1/2 . It is interesting to note that similar relations have been introduced in the past from pure numerology (see [29], P. 428). These relations suggest that the fundamental constants may change with time as argued by Dirac [30,31].

C. The mass and the charge of the electron in terms of B Using Eqs. ( 32), [START_REF] Feynman | QED: The Strange Theory of Light and Matter[END_REF] and (35), we find that the mass and the charge of the electron (or the fine-structure constant) are determined by the cosmological constant Λ according to

m e = 1 ν 1/6 α Λ 4 G 2 c 2 1/6 = 1.03 α Λ 4 G 2 c 2 1/6 = 9.11 × 10 -28 g, (37)
α = e 2 c = ν 32
1/2 ξ h σ π 1 ln 8πc 5 G Λ 1/2 = 0.123 ln 8πc 5 G Λ 1/2 = 1 137 . (38)
Alternatively, using Eqs. ( 9), ( 18), ( 30) and ( 33), the normalized mass and the normalized charge of the electron 6 We note that the prefactors in Eqs. [START_REF] Feynman | QED: The Strange Theory of Light and Matter[END_REF] and [START_REF] Weinberg | [END_REF] appear to be close to 123/1000 and 123/10, where the number 123 appears again (!). We do not know whether this coincidence is fortuitous or if it bears a deeper significance than is apparent at first sight.

are determined by the logotropic constant B according to

m e M P = 8π ν 1/6 ν 32 1/2 ξ h σ π √ Be -1/(6B) = 0.217 √ Be -1/(6B) = 4.18 × 10 -23 , (39)
e 2 q 2 P = ν 32

1/2 ξ h σ π √ B = 0.123 √ B = 7.29 × 10 -3 , ( 40
)
where M P = ( c/G) 1/2 = 2.18 × 10 -5 g is the Planck mass and q P = ( c) 1/2 = 5.62×10 -12 g 1/2 m 3/2 s -1 is the Planck charge. These relations suggest that the mass and the charge of the electron (atomic scales) are determined by the effective cosmological constant Λ or B (cosmological scales). We emphasize the presence of the exponential factor e -1/(6B) in Eq. ( 39) explaining why the electron mass is much smaller than the Planck mass while the electron charge is comparable to the Planck charge [see Eq. ( 40)].

D. A prediction of B

If we match Eqs. ( 23) and (B2), or equivalently Eqs. ( 30) and (31), we obtain

B app = 1 6λ 2 ξ 2 h Ω de,0 . (41)
Taking λ app = 1 (since we cannot predict its value) and Ω th de,0 = e/(1+e) [see Eq. ( 11)], we get B app = 6.67×10 -3 instead of B = 3.53 × 10 -3 . We recall that the value of B was obtained in Sec. II E from the observations. On the other hand, Eq. ( 41) gives the correct order of magnitude of B without any reference to observations, up to a dimensionless constant λ 1.41 of order unity. Considering that B is predicted by Eq. ( 41) implies that we can predict the values of Λ, H 0 , α, m e and e without reference to observations, up to dimensionless constants λ 1.41, ν 0.816 and σ 0.413 of order unity. We note, however, that even if these dimensionless constants (λ, ν, σ) are of order unity, their precise values are of importance since B usually appears in exponentials like in Eqs. ( 18), ( 21) and (39).

V. CONCLUSION

In this paper, we have developed the logotropic model introduced in [1,2]. In this model, dark matter corresponds to the rest mass energy of a dark fluid and dark energy corresponds to its internal energy. The ΛCDM model may be interpreted as the semiclassical limit → 0 of the logotropic model. We have first recalled that the logotropic model is able to predict (without free parameter) the universal value of the surface density of dark matter halos Σ 0 , their mass-radius relation M h -r h , the Tully-Fisher relation M b ∼ v 4 h and the value of the mass M 300 of dSphs. Then, we have argued that it also predicts the value of the present fraction of dark energy Ω de,0 . This arises from a sort of "strong cosmic coincidence" but this could also correspond to a fixed point of the model. Finally, we have observed that the surface density of the dark matter halos Σ 0 is of the same order as the surface density of the Universe Σ Λ and the surface density of the electron Σ e . This makes an empirical connection between atomic physics and cosmology. From this connection, we have obtained a relation between the fine-structure constant α ∼ 1/137 and the logotropic constant B ∼ 1/123. We have also expressed the mass m e and the charge -e of the electron as a function of B (or as a function of the effective cosmological constant Λ). Finally, we have obtained a prediction of the order of magnitude of B independent from the observations. In a sense, our approach which expresses the mass and the charge of the electron in terms of the cosmological constant is a continuation of the program initiated by Eddington [START_REF] Eddington | Fundamental Theory[END_REF] in his quest for a 'Fundamental Theory' of the physical world in which the basic interaction strengths and elementary particle masses would be prediced entirely combinatorically by simple counting processes [START_REF] Barrow | The Anthropic Cosmological Principle[END_REF]. In the Appendices, we try to relate these interconnections to a form of holographic principle [8] (of course not known at the time of Eddington) stating that the entropy of the electron, the dark matter halos, and the Universe scales like their area as in the case of black holes [9,10].

This paper has demonstrated that physics is full of "magic" and mysterious relations that are still not fully understood (one of them being the empirical Weinberg relation). Hopefully, a contribution of this paper is to reveal these "mysteries" and propose some tracks so as to induce further research towards their elucidation.

Appendix A: The constants α and B

There are two famous numbers in physics, 137 and 123, which respectively apply to atomic and cosmological scales.

At the atomic level, the fine-structure constant α, also known as Sommerfeld's constant [29], is a dimensionless physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. Its value is

α = e 2 c = e 2 q 2 P 1 137 7.30 × 10 -3 . (A1)
It can be seen as the square of the charge e = 4.80 × 10 -13 g 1/2 m 3/2 s -1 of the electron normalized by the Planck charge q P = ( c) 1/2 = 5.62 × 10 -12 g 1/2 m 3/2 s -1 . The quantum theory does not predict its value. The number 1/α 137 intrigued a lot of famous researchers including Born, Dirac, Eddington, Feynman, Hawking, Heisenberg, Landé, and Pauli among others [29]. Feynman writes [START_REF] Feynman | QED: The Strange Theory of Light and Matter[END_REF]: It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." At the cosmological level, there is another famous number B = 1 ln(ρ P /ρ Λ ) 1 123 ln (10) 3.53 × 10 -3 .

(A2)

It can be seen as the inverse of the logarithm of the cosmological -or dark energy -density ρ Λ = Λ/8πG = 5.96 × 10 -24 g m -3 (where Λ = 1.00 × 10 -35 s -2 is the cosmological constant), normalized by the Planck density ρ P = c 5 / G 2 = 5.16×10 99 g m -3 . This number appeared in connection to the so-called cosmological constant problem [START_REF] Weinberg | [END_REF]35], i.e., the fact that there is a difference of 123 orders of magnitude (ρ P /ρ Λ ∼ 10 123 ) between the Planck density ρ P and the cosmological density ρ Λ interpreted as the vacuum energy.

We have suggested in this paper that the two dimensionless constants α and B, or the two numbers 137 and 123, are related to each other [see Eqs. ( 33) and [START_REF] Weinberg | [END_REF]] and that, in some sense, they correspond to the same thing. If this idea is correct, it would yield a fascinating connection between atomic and cosmic physics.

Σ Λ = M Λ 4πR 2 Λ = cH 0 8πG = 392 g m -2 = 188 M /pc 2 . (B1)
It can be written as Σ Λ = cH 0 /κc 4 where κ = 8πG/c 4 is Einstein's gravitational constant (which includes the 8π factor). Using Eq. ( 15), we obtain

Σ Λ = 1 8π 3Ω de,0 c √ Λ G . ( B2
)
This relation shows that the surface density of the Universe provides the correct scale for the surface density of dark matter halos [see Eq. ( 23)]. We have Σ Λ = λΣ th 0 with λ 1.41.

Therefore, the surface density of the Universe is of the same order as the surface density of the dark matter halos which is also of the same order as the surface density of the electron (as we have previously observed). We have Σ Λ = µΣ e with µ = λ/σ 3.42. Matching Eqs. ( 29) and (B1), we get

m e = e 4 H 0 8πµGc 3 1/3 . ( B3
)
This relation expresses the mass of the electron as a function of its charge and the Hubble constant. This mysterious relation is mentioned in the book of Weinberg [START_REF] Weinberg | Gravitation and Cosmology[END_REF] where it is obtained from purely dimensional arguments. 7 He observed that the term in the right hand side of Eq. (B3) has the dimension of a mass and that this mass, 1.37 × 10 -27 g (with µ app = 1), is of the order of the mass of the electron. The fact that relation (B3) expresses the commensurability of the surface density of the Universe and the surface density of the electron, as we observe here, may help elucidating its physical meaning (see Appendix C 4).

Remark: If the dark matter halos resulted from the balance between the gravitational attraction and the repulsion due to the dark energy, they would have a typical density M h /r 3 h ∼ ρ Λ . Actually, such an equilibrium is unstable as is well-known in the case of the Einstein static Universe. Therefore, the radius of dark matter halos must satisfy the constraint r h < (M h /ρ Λ ) 1/3 . Now, we have seen that their mass-radius relation scales as

M h ∼ (c √ Λ/G)r 2 h . We then find that the constraint r h < (M h /ρ Λ ) 1/3 is satisfied provided that M h < c 3 /G √ Λ.
Since the upper bound is of the order of the mass of the Universe, M Λ ∼ c 3 /G √ Λ, we conclude that the size of the dark matter halos is always much smaller than the critical size (r h ) crit = (M h /ρ Λ ) 1/3 as required for stability reasons.

where A = 4πR 2 is the area of the event horizon of the black hole and l P = (G /c 3 ) 1/2 = 1.62 × 10 35 m is the Planck length. The radius of a Schwarzschild black hole is connected to its mass by

R = 2GM c 2 . (C2)
The Hawking temperature [10] of a Schwarzschild black hole is

k B T = c 3 8πGM = c 4πR . ( C3
)
The black hole entropy (C1) can be obtained from the Hawking temperature (C3) by using the thermodynamic relation

T -1 = dS BH /d(M c 2 ).
If we consider a Planck black hole of radius l P and mass M P , we find that its temperature is of the order of the Planck temperature T P = M P c 2 /k B = 1.42 × 10 32 K and its entropy S BH /k B ∼ 1.

Analogy between the Universe and a black hole

Using the results of Appendix B, we note that the radius of the Universe is related to its mass by

R Λ = 2GM Λ c 2 . (C4)
This expression coincides with the mass-radius relation (C2) of a Schwarzschild black hole. This coincidence has sometimes led people to say that the Universe is a black hole, or that we live in a black hole, although this analogy is probably too naive. Nevertheless, at least on a purely dimensional basis, we can use the analogy with black holes to define the entropy and the temperature of the Universe. In this manner, we get a temperature scale (temperature on the horizon)

k B T Λ = c 4πR Λ = H 0 4π ∼ √ Λ. ( C5
)
Its value is T Λ ∼ 2.41 × 10 -29 K. The temperature can be written as

k B T Λ = 2 a Λ c , (C6)
where

a Λ = GΣ Λ = GM Λ 4πR 2 Λ = c 2 8πR Λ = cH 0 8π ∼ c √ Λ (C7)
is the surface gravity of the Universe (similar relations apply to black holes). We note that Σ Λ = c 2 /8πGR Λ . We can also write

k B T Λ = m Λ c 2 , (C8) with m Λ ∼ √ Λ c 2 = 2.08 × 10 -33 eV/c 2 . (C9)
This mass scale is often interpreted as the smallest mass of the bosons predicted by string theory [37] or as the upper bound on the mass of the graviton [38]. 8 It can be contrasted from the mass scale

M Λ ∼ c 3 G √ Λ = 7.16 × 10 88 eV/c 2 , (C10)
which is usually interpreted as the mass of the Universe. Thus m Λ and M Λ represent fundamental lower and upper mass scales. Their ratio is

M Λ m Λ ∼ c 5 G Λ ∼ ρ P ρ Λ ∼ e 1/B ∼ 10 123 , (C11)
which exhibits the famous number 123 (see Appendix A).

On the other hand, our analogy between the Universe and a black hole leads to an entropy scale (entropy on the Hubble horizon):

S Λ = k B πc 3 R 2 Λ G = k B πc 5 G H 2 0 ∼ k B c 5 G Λ . ( C12
)
We note that the entropy of the Universe can be written as

S Λ /k B ∼ M Λ m Λ ∼ Σ Λ R 2 Λ m Λ ∼ e 1/B ∼ 10 123 . (C13)
This entropy may be identified with the total entropy of the logotropic dark fluid (see the Appendix of [3] and Appendix E). It can be compared to the entropy of radiation [39]:

S rad /k B = 4 3 3Ω rad,0 8π 3/4 π 2 15 1/4 1 (H 0 t P ) 3/2 = 5.64 × 10 87 , (C14)
obtained by using Eq. (E2) with P rad = rad /3, rad = σT 4 with σ = π 2 k 4 B /15c 3 3 (Stefan-Boltzmann constant), rad = Ω rad,0 0 /a 4 and Ω rad,0 = 9.24×10 -5 . They differ by about 36 orders of magnitude.

Remarks: We note that T Λ S Λ = (1/2)M Λ c 2 so the free energy of the Universe is

F Λ = M Λ c 2 -T Λ S Λ = (1/2)M Λ c 2 .
On the other hand, using Eqs. ( 18) and (39), we obtain the relations

m Λ M P ∼ e -1/(2B) = 3.40 × 10 -62 , (C15)
m e m Λ ∼ √ Be 1/(3B) = 5.66 × 10 39 . (C16) Since m Λ ∼ ρ Λ r 3 e (see Sec. IV B) we have m e /m Λ ∼ ρ e /ρ Λ . The gravitational radius of the cosmon is r Λ = 2Gm Λ /c 2 ∼ G √ Λ/c 4 = 2.75 × 10 -96 m.

Entropy of logotropic dark matter halos

Let us define the entropy of a logotropic dark matter halo by

S ∼ k B N ∼ k B M h m Λ , (C17)
where M h is the halo mass and m Λ is the mass of the hypothetical particle composing the logotropic dark fluid.

Using the mass-radius relation M h ∼ Σ 0 r 2 h of a logotropic dark matter halo, where Σ 0 ∼ c √ Λ/G is the universal surface density given by Eq. ( 23), we get

S ∼ k B c √ Λr 2 h Gm Λ . (C18)
Interestingly, the entropy given by Eq. (C18) scales like the surface r 2 h of the object, similarly to the black hole entropy (C1). 9 This may be connected to a form of holographic principle [8]. Matching the formulae (C1) and (C18), we find that m Λ corresponds to the mass given by Eq. (C9). Inversely, if we assume from the start that the logotropic dark fluid is composed of particles of that mass (cosmons), we find that the entropy of dark matter halos coincides with the entropy of black holes. 10 On the other hand, since the surface density of the Universe is of the same order as the surface density of dark matter halos, the previous formulae also apply to the Universe as a whole and return the results of Appendix C 2. This may be a form of justification, for reasons of self-consistency, of Eq. (C17).

Remark: If we alternatively define the entropy of dark matter halos by S ∼ k B M h /m e where m e is the electron mass and use M h ∼ Σ 0 r 2 h with Σ 0 ∼ Σ e , where Σ e is the surface density of the electron given by Eq. ( 29), we obtain

S ∼ k B r 2 h r 2 e , (C19)
which is similar to the black hole entropy formula (C1) where the Planck length l P is replaced by the classical radius of the electron r e . It is not clear, however, if this formula is physically relevant.

Postulates: entropic principles

We can find a form of explanation of the different relations found in this paper by making the following two 9 Inversely, a manner to understand why the surface density of the dark matter halos has a universal value is to argue that their entropy given by Eq. (C17) should scale like r 2 h (see Appendix C 4). 10 Of course, we are not claiming that dark matter halos are black holes since they obviously do not fulfill the Schwarzschild relation (C2). However, they may have the same entropy as black holes expressed in terms of r h [see Eq. (C1)].

postulates.

Postulate 1: We postulate that the entropy of the electron, the entropy of dark matter halos and the entropy of the Universe (and possibly other objects) is given by

S ∼ k B c 3 R 2 G , (C20)
like the Bekenstein-Hawking [9,10] entropy of black holes (C1), where R is the radius of the corresponding object. This may be connected to a form of holographic principle [8] stating that the entropy is proportional to the area (instead of the volume). Therefore,

S e /k B ∼ c 3 r 2 e G (electron) (C21) S/k B ∼ c 3 r 2 h G (dark matter) (C22) S Λ /k B ∼ c 3 R 2 Λ G (Universe) (C23)
Postulate 2: We postulate that the entropy of the electron, the entropy of dark matter halos and the entropy of the Universe (and possibly other objects) is also given by11

S ∼ k B M m Λ , ( C24
)
where M is the mass of the corresponding object and m Λ is the mass defined by Eq. (C9). Therefore,

S e /k B ∼ m e m Λ ∼ Σ e r 2 e m Λ ∼ 10 39 (electron) (C25) S/k B ∼ M h m Λ ∼ Σ 0 r 2 h m Λ (dark matter) (C26) S Λ /k B ∼ M Λ m Λ ∼ Σ Λ R 2 Λ m Λ ∼ 10 123 (Universe) (C27)
The comparison of Eqs. (C20) and (C24) directly implies that the surface density of the electron, the surface density of all the dark matter halos, and the surface density of the Universe is (approximately) the same and has the typical value

Σ ∼ M R 2 ∼ m Λ c 3 G ∼ m Λ M P Σ P ∼ c √ Λ G , (C28)
where Σ P = (c 7 / G Remark: we have introduced the entropy of an electron [see Eqs. (C21) and (C25)] by analogy with the black hole entropy. If these ideas are physically relevant, a notion of thermodynamics for the electron (assuming that it is made of 10 39 subparticles of mass m Λ ) should be developed. Again, the analogy with black holes (although, of course, an electron is not a black hole) might be useful. In particular, from the entropy S e ∼ k B m e /m Λ , we may define the temperature of the electron by 1/T e = dS e /d(m e c 2 ) and get

k B T e ∼ m Λ c 2 ∼ √ Λ ∼ k B T Λ .
In this sense, the temperature of the electron turns out to coincide with the temperature of the Universe. The same argument applies to dark matter halos (with S ∼ k B M h /m Λ and 1/T = dS/d(M h c 2 )) leading again to T ∼ T Λ . Interestingly, similar relations are obtained in Appendix E from different arguments.

Appendix D: Large numbers and coincidences

The ratio between the electric radius of the electron r e = e 2 /m e c 2 and its gravitational radius r g = 2Gm e /c 2 is of the order of e 2 /Gm 2 e = 4.17 × 10 42 . This dimensionless number was computed by Weyl in 1919 [START_REF] Barrow | The Anthropic Cosmological Principle[END_REF][START_REF] Weyl | Raum-Zeit-Materie[END_REF]. He was the first to notice the presence of large dimensionless numbers in Nature. This led Eddington [START_REF] Eddington | Fundamental Theory[END_REF] and others to try to relate such large numbers to cosmological quantities. In particular, Eddington evaluated the total number of particles in the Universe and found N p ∼ 10 79 . He then tried to relate the basic interaction strenghts and elementary particle masses to this number. For example, it was observed by different authors that the following quantities are of the same order of magnitude (see Ref. which is one of the "coincidences" pointed out by Chandrasekhar [START_REF] Chandrasekhar | [END_REF].

In a sense, these results arise from the Weinberg relation (B3) that has been found by different authors (see footnote 7). Nevertheless we believe that our approach is original and may bring new light on the subject. In particular, we have proposed a form of common explanation of these different "coincidences" in terms of entropic principles (see Appendix C 4). Let us try to relate the results of the previous Appendices to the thermodynamics of the logotropic dark fluid.

We assume that the Universe is filled with a dark fluid at temperature T . From the first principle of thermodynamics, one can derive the thermodynamic equation [START_REF] Weinberg | Gravitation and Cosmology[END_REF]:

dP dT = 1 T ( + P ). (E1)
If the dark fluid is described by a barotropic equation of state of the form P = P ( ), Eq. (E1) can be integrated 12 The Eddington number corresponds typically to the number of protons in the Universe, N ∼ M Λ /mp, where mp is the proton mass (in Eq. (D2) we have replaced mp by me since we have not introduced the proton mass previously). Note that the Eddington number was introduced before dark matter and dark energy were discovered. If the dark fluid is made of cosmons of mass m Λ , the number of particles in the Universe is to obtain the relation T = T ( ) between the temperature and the energy density. On the other hand, the entropy of the dark fluid in a volume a 3 is given by [START_REF] Weinberg | Gravitation and Cosmology[END_REF] S = a 3 T (P + ). (E2)

N Λ = M Λ /m Λ ∼
From the Friedmann equations, one can show that the entropy of the Universe is conserved: dS/dt = 0 [START_REF] Weinberg | Gravitation and Cosmology[END_REF].

The previous results are general. Let us now apply them to the logotropic dark fluid. According to Eqs. ( 5) and ( 6), the equation of state P = P ( ) of the logotropic dark fluid is given by the inverse of [2,3] = ρ P e P/A c 2 -P -A.

(E3)

Eq. (E1) with Eq. (E3) can be easily integrated to give

T = ρ P c 2 K 1 - A ρc 2 , ( E4
)
where K is a constant of integration and we have used Eq. ( 5). Substituting Eqs. (E3) and (E4) into Eq. (E2), and using Eqs. ( 3) and ( 5), we find that

S = K ρ 0 ρ P . ( E5
)
We explicitly check on this expression that the entropy of the Universe is conserved. Furthermore, since the entropy is positive, we must have K > 0. Considering Eq. (E4), we note that the temperature is positive when ρ > ρ M = A/c 2 and negative when ρ < ρ M = A/c 2 , that is to say when the Universe becomes phantom [1,2]. 13 We can determine the constant K by assuming that the entropy of the logotropic dark fluid is given by

S ∼ k B M Λ m Λ ∼ 10 123 k B (E6)
as in Appendix C. Noting that the "true" entropy is obtained by multiplying Eq. (E2) by R 3 Λ (since we have taken a 0 = 1 at the present time), and comparing Eqs. (E5) and (E6), we obtain

K ∼ k B ρ P m Λ . ( E7
)
As a result, the temperature of the logotropic dark fluid is given by

k B T ∼ m Λ c 2 1 - Bρ Λ ρ , ( E8
)
13 This is a general result [42] which can be obtained from Eq. (E2) using the fact that the entropy is constant and positive. We see on Eq. (E2) that the sign of the temperature coincides with the sign of P + . As a result, the temperature is positive in a normal Universe (P > -) and negative in a phantom Universe (P < -).

where we have used Eq. ( 12). In the "early" Universe ρ ρ Λ we find that 14

T m Λ c 2 /k B = 2.41 × 10 -29 K. (E9)
In the late Universe ρ ρ Λ we find that

k B T ∼ -m Λ c 2 Bρ Λ /ρ ∝ -a 3 . (E10)
Remark: In Ref. [3] we have shown that the logotropic constant B could be interpreted as a dimensionless logotropic temperature

B = k B T L m Λ c 2 (E11)
in a generalized thermodynamical framework [1,2]. This shows that at least two temperatures exist for the logotropic dark fluid, a time-varying temperature T and a constant temperature T L . They become equal when

ρ * ρ Λ ∼ B 1 -B ∼ 3.54 × 10 -3 , (E12) corresponding to a * ∼ Ω m,0 Ω de,0 1 -B B 1/3 ∼ 5.01. ( E13
)
Appendix F: The mass of the bosonic dark matter particle

It has been suggested that dark matter may be made of bosons (like ultralight axions) in the form of Bose-Einstein condensates (BECs). 15 We can use the results of the present paper to predict the mass m of the bosonic dark matter particle in terms of the cosmological constant Λ. We assume that the smallest and most compact dark matter halo that is observed corresponds to the ground state of a self-gravitating BEC (to fix the ideas we assume that this halo is the dSphs Fornax with a mass M ∼ 10 8 M and a radius R ∼ 1 kpc). For noninteracting bosons, it can be shown by solving the Gross-Pitaevskii-Poisson equations (see, e.g., Sec. III.B.1 of 14 We recall that the logotropic model, which is a unification of dark matter and dark energy, is not valid in the very early Universe corresponding to the big bang, the inflation era, and the radiation era. Therefore, the temperature m Λ c 2 corresponds to the temperature of the dark fluid in the matter era, i.e., when the rest-mass energy of the dark fluid overcomes its internal energy (see Sec. II). We emphasize that the temperature T of the logotropic dark fluid is different from the temperature of radiation and of any other standard temperature. We also note that the corresponding temperature in the ΛCDM model is not defined since Eq. (E1) breaks down when P = 0. 15 See, e.g., the bibliography of Refs. [43,44] for an exhaustive list of references. The possible connections between the BECDM model and the logotropic model will be investigated in a future paper [45]. [46]) that the mass (M h ) min , the radius (r h ) min and the central density (ρ 0 ) max of this ultracompact halo (ground state) are related to each other by the relations (F2)

On the other hand, the minimum mass of dark matter halos may be obtained from a quantum Jeans instability theory (see, e.g., Ref. [44]) giving the result

M J = 1 6 π π 3 2 ρ 1/3 dm,0 Gm 2 3/4 . ( F3
)
For usually considered values of the boson mass, of the order of m ∼ 10 -22 eV/c 2 , the Jeans mass M J ∼ 10 7 M from Eq. (F3) is indeed of the order of magnitude of the minimum mass (M h ) min ∼ 10 8 M of observed dark matter halos. There may be, however, a numerical factor of order 10 between M J and (M h ) min . For that reason, we write (M h ) min = χM J with χ ∼ 10. Using Eq. (F3) with ρ dm,0 = (Ω dm,0 /Ω de,0 )ρ Λ = (Ω dm,0 /Ω de,0 )(Λ/8πG), we get (M h ) min = χ π 3 6 Ω dm,0 8Ω de,0 Comparing this expression with Eq. ( 23), we predict that the mass of the bosonic particle is given by m = χ 6 0.0234π 20 1458Bξ 2

h Ω dm,0 8Ω de,0

3/2 √ Λ c 2 = 15397χ 6 √ Λ c 2 . (F6)
We see that the mass of the bosonic dark matter particle is equal to the mass scale m Λ ∼ 10 -33 eV/c 2 given by Eq. (C9) multiplied by a huge numerical factor of order 10 10 (for χ ∼ 10). This gives m ∼ 10 -23 eV/c 2 which is the correct order of magnitude of the mass of ultralight axions usually advocated [47]. We note that this result has been obtained independently from the observations, except for the value of Λ and the other fundamental constants (Planck scales).

Appendix G: The proton mass

The maximum mass of neutron stars due to general relativity has been obtained by Oppenheimer and Volkoff [48] using the equation of state of an ideal Fermi gas at T = 0. It is given by

M OV max = 0.384 c G 3/2 1 m 2 n = 0.710 M , (G1)
where m n = 1.675 × 10 -24 g is the neutron mass. It is not well-known 16 that, at the same period, Zwicky [50,51] Interestingly, this equation provides a relation between the fine-structure constant α and the ratio m e /m p between the electron mass and the proton mass (we have used m n m p ). This type of relationships has been proposed in the past by several authors [29] using heuristic arguments or pure numerology. Since we have related the electron mass and the electron charge (hence the finestructure constant) to the cosmological constant (see Sec. IV C), using Eq. (G3) we can also relate the proton mass to the cosmological constant. Using Eqs. (37), (38) and (G3), we find that the mass of the proton is determined by the cosmological constant Λ according to

  Appendix B: Surface density of the Universe, surface density of the electron and Weinberg's empirical relation Using qualitative arguments, let us determine the surface density of the Universe. The Hubble time (∼ age of the Universe) is t Λ = 1/H 0 = 14.4 billion years. The Hubble radius (∼ radius of the visible Universe) is R Λ = ct Λ = c/H 0 = 1.37×10 26 m. The present density of the Universe is 0 /c 2 = 3H 2 0 /8πG = 8.62 × 10 -24 g m -3 . The Hubble mass (∼ mass of the Universe) is M Λ = (4/3)π( 0 /c 2 )R 3 Λ = c 3 /2GH 0 = 9.20 × 10 55 g. Combining these relations, we find that the surface density of the Universe is

2 e∼ N p ∼ M P m e 2 ∼ 4 ∼

 224 10 40 . (D1) These coincidences can be easily understood from our results(39) and[START_REF] Weyl | Raum-Zeit-Materie[END_REF] which express the mass and the charge of the electron in terms of the cosmological constant. For convenience, we shall replace the Eddington number by12 N e = M Λ m e ∼ e 2/(3B) ∼ 10 80 . 10 80 ∼ N e , (D6)

  Appendix E: Thermodynamics of the logotropic dark fluid

M h = 1 .91 ρ 0 r 3 h and M h r h = 1 .85 2 Gm 2 . 3 h 4 .

 112234 (F1)As a result, its surface density is given byΣ 0 = 0.153 G 2 m 4 M

1/4 3 / 2 Λ 1 / 4 Gm 3

 32143

= k m n e 2 Gm p m e 3 / 2 =

 232 also attempted to determine the maximum mass of neutron stars interpreted as Schwarzschild's mass. He used heuristic arguments and obtained an expression of the formM Zwicky max 91 k M ,(G2)where m p = 1.673 × 10 -24 g is the proton mass and k is a dimensionless number assumed to be of order unity. Matching these two expressions (G1) and (G2), we get k = 7.80 × 10 -3 and

6 = 1 .= 23 . 7 e

 61237 67 × 10 -24 g. (G4) Alternatively, using Eqs. (39), (40) and (G3), the normalized mass of the proton is determined by the logotropic constant B according to -1/(6B) = 7.68 × 10 -20 .(G5)

  3 ) 1/2 = 8.33 × 10 64 g m -2 is the Planck surface density. Then, comparing this universal value with the surface density of the electron [see Eq.

	(29)], we obtain the Weinberg relation
	Λ ∼	m 6 e G 2 c 6 e 8 .	(C29)

  10 123 giving another interpretation to the famous number 123 (see Appendix A). This number N Λ should supersede the Eddington number Np.

We stress that our model is different from the ΛCDM model so that Λ is fundamentally different from Einstein's cosmological constant[4]. However, it is always possible to introduce from the constant A an effective cosmological density ρ Λ and an effective cosmological constant Λ by Eqs. (8) and(9).

The logotropic spheres[1,2], like the isothermal spheres and some polytropic spheres[START_REF] Chandrasekhar | An Introduction to the Study of Stellar Structure[END_REF], have an infinite mass. This implies that the logotropic equation of state cannot describe dark matter halos at infinitely large distances. Nevertheless, it may describe the inner region of dark matter halos and this is sufficient to determine their surface density. The stability of bounded logotropic spheres has been studied in[START_REF] Chavanis | [END_REF] by analogy with the stability of bounded isothermal and polytropic spheres[20][21][22] and similar results have been obtained. In particular, bounded logotropic spheres are stable provided that the density contrast is not too large.

We note that the Thomson cross-section σ = (8π/3)(e 2 /mec 2 ) 2 can be written as σ = (8π/3)r 2 e giving a physical meaning to the classical electron radius re. We also note that re can be written as re = α /mec = αλ C where λ C = /mec is the Compton wavelength of the electron and α is the fine-structure constant α [see Eq. (A1)]. Similarly, we can write Σe = (1/α 2 )m 3 e c 2 / 2 .

A closely related formula, involving the Hubble constant instead of the cosmological constant, was first found by Stewart[28] in 1931 by trial and error.

It is simply obtained by equating the Compton wavelength of the particle λc = /mc with the Hubble radius R Λ = c/H 0 (the typical size of the visible Universe) giving m Λ = H 0 /c 2 . Using Eq. (15), we obtain Eq. (C9). By comparison, if we identify the Compton wavelength λc = /mc with the Schwarzschild radius r S ∼ Gm/c 2 we get the Planck mass M P = ( c/G) 1/2 .

Note that this relation is not satisfied by black holes since M BH ∝ R while S BH ∝ R 2 .

Appendix C: Analogy with black hole thermodynamics

Black hole entropy

The Bekenstein-Hawking [9,10] entropy of a Schwarzschild black hole is given by

C1) 7 Weinberg considers this relation as "so far unexplained" and having "a real though mysterious significance". Similar relations have been obtained in the past by Stewart [28], Eddington [START_REF] Eddington | Fundamental Theory[END_REF] and others from purely heuristic arguments or from dimensional analysis [START_REF] Barrow | The Anthropic Cosmological Principle[END_REF]29,36]. Their goal was to express the mass of the elementary particles in terms of the fundamental constants of Nature.


Benoît Pelopidas

Sandra Kemp

Jenny Andersson

The Birth of Nuclear Eternity

Keywords: nuclear weapons, future, arms control, democracy, futures-making

This chapter reconnects modes of futures-making with the requirements of democracy by focusing on the naturalization of nuclear weapons and their removal from the realm of de mocratic choice at a particular point in time. The chapter revolves around the concept of 'nuclear eternity' as a means of reducing public choices about the use of nuclear weapons. It critiques the idea that nuclear weapons have always been perceived as 'here to stay' and reassesses the dominant narrative about the 1960s as an emancipatory decade by arguing that the decade actually witnessed a significant shrinking of future po litical possibilities. Finally, the chapter identifies three shapes of the future which pro duce 'nuclear eternity'-an absent post-nuclear future, an inconsistent post-nuclear future, and a disconnected post-nuclear future-and illustrates them with historical exam ples

national nuclear history and political theory. For instance, in his classic book A Perpetual Menace, William Walker acknowledges the importance of 'discourses about future timethe manner in which expectations, visions, commitments, imaginaries, fantasies or what you will [say] about the future affect debates about order and the decisions relating to them'. And, one page before the end of the book, he articulates what I am trying to turn into a question: 'states and people cannot be resigned to the permanent, active presence in the world of nuclear weapons and weapon programmes'. 4 The literature on nuclear normalization, which offers fruitful insights on such a process, is mostly psychological or linguistic but is not focused on historicizing the phenomena it identifies, let alone investi gating the production of a particular image of the future. 5 The literature on nuclear fu tures and nuclear apocalypse does not engage with the making of nuclear weapons as a permanent feature of any possible future. For instance, in his insightful study of the US nuclear imaginary, Joseph Masco called for 'figuring a post-nuclear post-terror security studies' but he does not investigate how nuclear eternity came into being. 6 By nuclear eternity, I do not mean an endless time in which nuclear weapons and the pos sibility of a nuclear explosion will remain, but rather the idea that no future without them is conceivable. This is a good reminder that nuclear eternity might not be very long at all, since escalation to global nuclear war may end human life on the planet: it only names the time mankind has left, while acknowledging that the men and women I will be refer ring to considered a large scale nuclear war as a possibility. Nuclear eternity should also be acknowledged as a metaphor: Martin Amis's essay on the thinkability of a nuclear fu ture, for example, reminds us that if we assume that mankind is an earthbound species, then nuclear eternity is limited by 'the death of the sun' which, for sure, will end life on Earth but may not be the end of time. 7 Imaginaries of nuclear eternity perfectly coexist with schemes that attempt to manage or miniaturize the macro imaginary of eternity through calls to restraint, rollback, or reversal. This essay focuses on nuclear weapons and does not engage with nuclear energy, while fully recognizing that the distinction between the two cannot be grounded in (p. 486) physics. 8 The radioactive effects of fissile materials alone project us into a future of tens of thousands of years which, by human standards, is tantamount to eternity. In that re spect, with or without nuclear weapons, the issue of nuclear eternity would be at stake by the mere existence of radioactive material in the world. 9 It is quite telling that a Finnish documentary about the long-term repository of radioactive waste at the Onkalo site is en titled Into Eternity (directed by Michael Madsen, 2010). Let us consider the representa tions and metaphors that bring about an imagined future in which nuclear explosions are a continued possibility.

I would like to divide the original puzzle into three. First, the practice of nuclear deter rence was originally conceived as a temporary solution to the strategic and policy prob lems of the day. 10 However, the caveat about its temporary nature has been progressively removed. How did that become possible? This is all the more puzzling as most policies that are being perpetuated remain labelled as temporary, which is not the case for nu clear deterrence.

Second, a mainstream grand narrative of the nuclear age claims that the early attempts at finding a supranational solution to the nuclear weapons problem were doomed to fail and that Atoms for Peace and the Non-Proliferation Treaty illustrate a return to intergov ernmental reason. 11 This narrative of inevitability misses a crucial question. Whether or not one calls it a return to reason, when does this sense that the nuclear weapons prob lem can only be solved by classic Westphalian international politics emerge and how? In other words, as I will show in the first section of this essay, this narrative overlooks the fact that other forms of transformative engagement with the problem have been tried for at least fifteen years before being abandoned. 12 The most interesting and largely unad dressed question is why did such efforts not stop sooner or later than they did? Third, how did nuclear weapons become acceptable in a democratic context in spite of their anti-democratic nature? 13 To articulate this puzzle, I rely on Daniel Deudney and Robert Dahl. Deudney writes: 'Nuclear explosives are intrinsically (p. 487) despotic for three related reasons: the speed of nuclear use decisions, the concentration of the nuclear use decision in the hands of one individual, and the lack of accountability stemming from the inability of affected groups to have their interests represented at the moment of nu clear use.' 14 Interestingly, in 1985, Dahl had already noted: 'The democratic process has clearly failed to function in controlling what may well be the most important decisions that will ever be made on this earth.' 15 In this chapter, which attempts to address this puzzle before I do so in the form of a fulllength monograph, I would like to make three arguments. First, I will identify three fig ures of the future which lead us to imagine nuclear eternity as the only conceivable shape of it, and I will illustrate them with historical examples. Second, I will argue that the be lief in nuclear eternity as the only possible shape of the future is not co-terminus with the invention of nuclear weapons. Third, and consequently, I will argue that the 1960s-often conceived of as the decade of the opening of political futures, material and ideational changes through decolonization, and civil rights movements alongside other emancipato ry practices-also contributed significantly to the entrenchment of nuclear eternity in nu clear weapons states to such an extent that it would not be challenged until the 1980s.

Three Modes of Perpetualization

Nuclear eternity derives from at least three representations of a post-nuclear future. These are a disconnected post-nuclear future in which nuclear weapons no longer are, but which is depicted without any effort to connect it to present conditions, or (p. 488) which posits that the connection can only be made through unprecedented and massive change; an absent post-nuclear future in a managerialist form of presentism that literally does not engage with it and, in doing so, slowly but surely entrenches nuclear weapons in the world; and an inconsistent post-nuclear future which is claimed to be the outcome of a se ries of steps that are advocated to reach it, but can demonstrably be shown not to be the actual outcome of the planned steps.

First, the disconnected post-nuclear future is the most common shape of the future in the speeches of US Presidents and leaders of nuclear-armed states. 16 Here, I will only give two examples and two exceptions, from four different constituencies and eras: Barack Obama's 2009 Prague Speech, Winston Churchill's last speech before the House of Com mons in 1955, Rajiv Gandhi's 1988 address before the United Nations General Assembly, and the movement to eliminate nuclear weapons via a process of banning them and rais ing awareness of the humanitarian consequences of their use in the 2010s. Barack Obama's Prague Speech of 5 April 2009 explicitly displays the disconnect be tween the hoped-for post-nuclear future and the present. In it, he stated that the goal of a world free of nuclear weapons 'will not be reached quickly-perhaps not in my lifetime'. 17

Several public declarations of Obama administration officials focus on this idea of a very long-term goal. Robert Gates had done so after his reappointment as Secretary of De fense in 2008. The most telling statement may be the July 2010 thank you speech of Sec retary of State Hillary Clinton addressed to the staff involved in the negotiation of the New START treaty: 'I am personally very grateful for everything you have done to move us toward our goal of a world some day, in some century, free of nuclear weapons.' 18 One can find such hopeful pronouncements with no concern for their plausible implementa tion as early and eloquently as in Churchill's famous speech to the House of Commons on 1 March 1955. It ends with these words, which not only disconnect the post-nuclear fu ture from the present but also suggest that it requires no less than a change in human na ture as we know it:

To conclude: mercifully, there is time and hope if we combine patience and courage. All deterrents will improve and gain authority during the next ten years. By that time, the deterrent may well reach its acme and reap its final reward. The day may dawn when fair play, love for one's fellow-men, respect for justice and freedom, will enable (p. 489) tormented generations to march forth serene and tri umphant from the hideous epoch in which we have to dwell. Meanwhile, never flinch, never weary, never despair. 19 Implicitly in the speeches from the Obama era and explicitly in Churchill's, a post-nuclear nuclear future requires a world different from the one we live in so that the policy pre scription is to create such a world, which would carry a set of preconditions enabling a post-nuclear future.

In contrast, the speech by Indian Prime Minister Gandhi before the General Assembly of the United Nations on 9 June 1988, and the process of eliminating nuclear weapons start ing with a treaty prohibiting them, avoid this disconnect because the post-nuclear future they are calling for is not a future world different from ours; it is our present world with out nuclear weapons, i.e. a post-nuclear present. Gandhi's speech breaks out of the nu clear eternity frame: it states the goal of elimination of nuclear weapons and connects it to time-bound steps that could possibly lead to it, and are described in an action plan which is immediately implementable and expected to reach its final goal within twentytwo years. 20 Similarly, key proponents of the humanitarian initiative calling for nuclear disarmament regard the 2017 treaty on the prohibition of nuclear weapons as a milestone in a contingent and contested process of invention of nuclear disarmament in our world, which does not depend on a radically different future. [START_REF] Nick | Inventing Nuclear Disarmament[END_REF] However, such remarkable calls come from India, a country which had not weaponized its nuclear programme in spite of the 1974 test and from a coalition of 122 non-nuclear weapons states who voted in favour of adopting a Treaty on the Prohibition of Nuclear Weapons in the UN General Assembly in 2017, which suggests how widespread the representation of a disconnected post-nu clear future is in the discourse of the policymaking elites of nuclear weapons states.

Second, the absent post-nuclear future is widespread in nuclear weapons scholarship, in the policy world in nuclear weapons states, and in science fiction. Nuclear eternity fol lows: a post-nuclear future is either painted as undesirable or as impossible to achieve or both. However, this mode of nuclear eternity is quite diverse and compatible with the physical presence or absence of nuclear weapons, regardless of whether they explode. In either of those worlds, though, nuclear weapons do not disappear before mankind does.

I have shown elsewhere that most nuclear analyses operate under this frame, which does not make space for a post-nuclear future. [START_REF] Pelopidas | Nuclear Weapons Scholarship as a Case of Self-Censorship in Secu rity Studies[END_REF] This does not mean that (p. 490) analysts ac tively believe in a nuclear eternity. They act and write as though they believe in it and, if they do not, they produce it by reproducing the nuclear present. In expert as well as poli cy circles, no one actually speaks about 'nuclear eternity' and many would probably re ject the notion but its unspeakability may well be a condition of its continued reproduc tion. For policymaking elites, as well as mainstream analysts in nuclear weapons states, future horizons do not extend that far; they are limited to the current term in office or the current generation (see Obama's Prague Speech above). Their strategy therefore consists in postponing the moment of nuclear detonation or radical nuclear change beyond such a horizon, which is enough to reproduce nuclear eternity whether or not one acknowledges it. Persistent calls for 'strategic stability' as the best possible goal illustrate that situation.

One way of projecting such an absent post-nuclear future comes from the invocation of expected veto positions in discourses about nuclear policy, whatever nuclear or non-nu clear future the speaker would sincerely wish for. This invocation assumes that one actor or group who believes in the nuclear orthodoxy has to be taken into account because this actor will veto the proposed transformation. As a consequence, convincing people of the fallacies of part of the nuclear conversation might not be enough to change their prac tices. As long as they anticipate that powerful others believe in the orthodoxy and act up on this anticipation, their sincere belief does not actually affect what they do. [START_REF] Slavoj | The Sublime Object of Ideology[END_REF] For exam ple, between 1961 and 1964 there was a constant disconnect between what Secretary of Defense Robert McNamara claimed to be the nuclear force requirements for the national security of the United States and the actual weapons that he requested, which can be ex plained by what he regarded as the expected veto player. As early as February 1961, he claimed that the missile gap was 'an illusion' and, three years later, in spite of a signifi cant increase in Soviet nuclear weapons capabilities, he argued in a memo for President Johnson that four hundred 1 mt weapons would be enough to achieve 'assured destruc tion'. 24 However, he asked Congress for many more weapons than that. On 28 March 1961, he announced the construction of six hundred ground-to-ground Minuteman mis siles and twenty-four submarines carrying Polaris missiles. On 10 October of that year, Deputy Secretary of Defense Roswell Gilpatric stated publicly that the supposed missile gap which was driving the shape and size of US nuclear forces did not exist and that the US were ahead in terms of nuclear weapons capabilities. 25 (p. 491) In December 1961, McNamara argued before Congress in favour of even more weapons than he had before: one thousand Minuteman missiles and forty-one submarines. Later on, he would explain that he thought he would not be credible in front of Congress had he asked for fewer. [START_REF] Le | Armes nucléaires: Mythes et réalités[END_REF] In this case, the constituency producing his expected loss of credibility was the expected ve to player. One has to note, here, that in the early 1960s, where there was disagreement between McNamara's suggestions backed by systems analysis and procurement options proposed by the military, the US Congress often found McNamara's perspective more convincing. The questioning of systems analysis by Congress did not start before the mid-1960s. [START_REF] See | Rationalizing Capitalist Democracy: The Cold War Origins of Ra tional Choice Liberalism[END_REF] Nuclear history offers many other examples of the use of expected veto play ers but the interesting result is that they allow for a form of discourse in which no future is properly discussed, and certainly not a post-nuclear one. [START_REF]I offer two others in 'A Bet Portrayed as a Certainty: Reassessing the Added Deterrent Value of Nuclear Weapons[END_REF] One needs to observe, though, that the representation of an absent post-nuclear future does not necessarily coincide with a stance opposing the physical elimination of nuclear weapons. In his bestselling book The Abolition, Jonathan Schell crafted the notion of 'weaponless deterrence' in which it is the possibility of rebuilding nuclear weapons that makes a world without their physical presence achievable and desirable. In other words, the author calls for a world without nuclear weapons while making the case that the idea of such weapons can never be forgotten so that the world will remain nuclear in the mind in perpetuity. [START_REF] Schell | The Abolition[END_REF] In such a world, the author hopes, the weapons may not be detonated.

Science fiction as a genre offers other modalities of the absent post-nuclear future. This includes the continued physical presence of nuclear weapons as well as their detonation. On the one hand, post-World War Two apocalyptic science fiction depicts the end of the species in a nuclear war. The endings of Stanley Kramer's 1959 On the Beach and Stanley Kubrick's 1963 Dr Strangelove are probably the most famous instances. On the other hand, a few other science fiction stories depict a society from a very distant future in which almost every aspect of technology and social life has changed, except for the pres ence of nuclear weapons in the arsenals of the political entities that exist. In both cases, nuclear weapons are the least changeable feature of our world, therefore leaning towards nuclear eternity. The multi-award-winning (p. 492) TV show Battlestar Galactica (2004-9) can serve as an eloquent example of the latter. Even though the characters are looking for an illusory planet Earth and not living on it, they are still 'humans'. 30 So, if one ac cepts them as such, it is striking that cyborgs that can look and feel like humans have been invented but nuclear weapons are still playing the role of the most destructive weapons system available. This is all the more striking as, in the show, they have repeat edly been used to produce holocausts. Even that was not enough to undo nuclear eternity as the only conceivable horizon. The cyclical temporality that is one of the signature fea tures of the show is only another mode of nuclear eternity: humans are destined to repeat the destruction of their planet by nuclear weapons. As a few survivors escape into space, they take nuclear weapons on board their ships. [START_REF] Fey | The Nuclear Taboo, Battlestar Galacti ca and the Real World: Illustrations from a Science Fiction Universe[END_REF] Third, the four horsemen initiative, of George Shultz, William Perry, Henry Kissinger, and Sam Nunn, from 2007 to 2013, which claimed to seek a world free of nuclear weapons, is a good example of an inconsistent post-nuclear future. Their image of a world free of nu clear weapons is a mountain top that cannot be seen yet. [START_REF] Nunn | Council on Foreign Relations[END_REF] The merit of the initiative had to do with their attempt at connecting the steps with the goal and presenting both. Their post-nuclear future is neither absent nor disconnected but it does require the creation of a different world more amenable to nuclear disarmament. They write:

Reassertion of the vision of a world free of nuclear weapons and practical mea sures toward achieving that goal would be, and would be perceived as, a bold ini tiative…Without the bold vision, the actions will not be perceived as fair or urgent. Without the actions, the vision will not be perceived as realistic or possible. [START_REF] Shultz | A World Free of Nu clear Weapons[END_REF] However, in spite of the title of their first editorial, 'A World Free of Nuclear Weapons', it quickly became clear that this initiative was essentially an incrementalist effort. It em braced proliferation as the macro-dynamic of world history, extended it to the problem of nuclear terrorism, and could not produce disarmament. 34 (p. 493) They argued that interstate nuclear deterrence was decreasingly relevant and increasingly dangerous but did so within a framework in which those weapons remained security providers for the US. It could therefore be read as an attempt to re-establish the credibility of the United States' incredible pledge to disarm, according to the requirements of Article VI of the Non-Prolif eration Treaty. It ultimately produced the nuclear security summit process (2010-16) that led to more control over nuclear materials but nothing more transformative.

The 1960s as a Decade of Entrenchment of Nu clear Eternity

The triumph of nuclear eternity does not coincide with the invention of the weapons. In the first twenty years of the nuclear age, radical alternatives existed and were seriously considered, beyond the global anti-nuclear movement, whose efforts have been well docu mented. [START_REF] Sylvest | Nuclear Realism: Global Political Thought dur ing the Thermonuclear Revolution[END_REF] This is not incompatible with the well-established findings that there was bad faith in the early years of US-Soviet nuclear disarmament diplomacy and that the argu ment of technological irreversibility was applied to nuclear weapons early among US de fence intellectuals. [START_REF] Tal | The US Nuclear Disarmament Dilemma 1945-1963[END_REF] As Daniel Ellsberg, a war planner at RAND in the 1950s, recollected in his memoirs: '"you can't uninvent nuclear weapons". That has been a widespread and effective argument against a total unilateral abolition over the past seventy years.' [START_REF] Ellsberg | The Doomsday Machine: Confessions of a Nuclear War Planner[END_REF] This was confirmed by Freeman Dyson, a physicist from the Manhattan Project, speaking about the Federation of American Scientists: it only suggests that policy officials and de fence intellectuals from the US and the Soviet Union spoke as though they accepted nu clear eternity early on. However, the thermonuclear revolution triggered an interesting and rare level of agreement among prominent intellectuals of multiple traditions about its fundamental implications. Even the US strategic community originated alternative ideas about possible futures. Dyson underlined that some proponents of the Orion (p. 494) Project were hoping to use and do away with the stockpile of US nuclear weapons to build a spaceship to go to Mars. [START_REF] Dyson | [END_REF] Even if the belief in nuclear eternity had already started spreading in the 1950s, the core narrative of the 1960s as a decade of emancipation and protest against nuclear weapons can be challenged. [START_REF]Une Autre Histoire des Trente Glorieuses[END_REF] It is claimed that in the 1960s humanity tamed the nuclear dragon practically and legally: the peak of the US nuclear arsenal is said to have been reached in 1966-7, and testing, horizontal proliferation, and the arms race between the US and the Soviet Union are said to have been contained. The following treaties are often considered instrumental in the legal restriction and containment of nuclear weapons in the 1960s: the 1963 Partial Test Ban Treaty; the Outer Space Treaty; the 1967 Treaty of Tlatelolco (the first treaty instituting a nuclear weapons-free zone in a densely populated area, i.e. Latin America and the Caribbean); the 1968 Non-Proliferation Treaty; and the beginning of the Strategic Arms Limitation Talks process in 1969, which would lead to the SALT and ABM treaties in 1972. [START_REF] Norris | as the year with the most weapons on the US side ('Global Nuclear Weapons Inventories 1945-2013[END_REF] It is true that 1966 is the year when the size of the US nuclear arsenal reaches its peak if one counts the number of warheads deployed. But even if one counts US weapons only and not the world's stockpile, a focus on the lethality of the US arsenal shows that it does not peak until decades later. [START_REF]Lynn Eden shows the lethality of the US nuclear arsenal peaking in 2002[END_REF] This is because the technology of MIRVing, developed in the 1970s, allowed multiple warheads which could be targeted independently to be placed on a single missile; it is also explained by the shift from a quantitative to a qualita tive arms race after 1966. If one focuses on the total number of warheads in the world, the peak is in 1986, i.e. long after the end of the 1960s. 42 The task of measuring the ag gregated lethality of the world's nuclear arsenals remains to be done. Even though one has to recognize the crucial role of non-material factors in this dynamic, in 1969 there were almost three times as many nuclear warheads in the world as a decade earlier. This is a total of more than (p. 495) 38,000 nuclear warheads in the world, or 25,000 built in a decade. [START_REF] Norris | Global Nuclear Weapons Inventories 1945-2013[END_REF] It was also in the early years of the 1960s that intercontinental ballistic mis siles, coupled with thermonuclear warheads, entered service in silos and on submarines. Their destructive capability is in the order of magnitude of a thousand times more than the first generation of atomic bombs. Subsequent technological innovations do not funda mentally modify this. In addition, the most recent scholarship has convincingly chal lenged the conventional wisdom about parity and arms control as the overarching fea tures of the late 1960s and 1970s, up to the Reagan administration. Instead, it is now es tablished that as early as 1969 US policymakers sought to transcend parity rather than only manage it. [START_REF] Petrelli | Nuclear Superiority in the Age of Parity: US Planning, Intelligence Analysis, Weapons Innovation and the Search for a Qualitative Edge 1969-1976[END_REF] Moreover, in the 1960s nuclear weapons started to spread beyond the original partici pants of the Manhattan Project and the Soviet Union. In only a decade three additional nuclear-armed states appear on the map, a larger number than in any other decade of the nuclear age: France (the first A-bomb test in 1960 and the first H-bomb test in 1968), Chi na (1964), and Israel (1967). Of course, one does not want to fall for a teleological narra tive; it is important to observe, for instance, that the 1960s witnesses the reversal of Swedish plans to build nuclear weapons. [START_REF] Jonter | The Key to Nuclear Restraint[END_REF] At the same time, there were signs of attempts to acquire a latent/hedging nuclear weapons capability in Italy and Australia. The Ger man case is still debated. This is where the materiality of those weapons matters. [START_REF] Pouliot | The Materials of Practice: Nuclear Warheads, Rhetorical Common places and Committee Meetings in Russian-Atlantic Relations[END_REF] The endless reproduction of a nuclear(ized) present will be enough to enact a nuclear eternity because a substantive effort to dismantle them would be required. They will not just rot away. [START_REF]One would need to elaborate on this, as it is true that the perpetual physical presence of the weapons requires constant curation and reinvention. See Mike Bourne, 'Invention and Uninvention in Nuclear Weapons Politics[END_REF] For instance, the US disarmament movement lost its ability to put pressure on governments when the Partial Test Ban Treaty, which entered into force in 1963, led many activists to decrease their involvement, at the same time as the Vietnam War was becoming a priority for them. [START_REF] Wittner | The Struggle against the Bomb[END_REF] The absence of a post-nuclear future also has a social grounding: when a powerful coalition is lobbying for nuclear weapons acquisition or for the projection of the existing nuclear infrastructure into a distant future, not putting pres sure on the government is de facto entrenching nuclear eternity or, more precisely, mak ing its undoing more difficult.

(p. 496) The signing of the Treaty on the Non-Proliferation of Nuclear Weapons in 1968 manifests the entrenchment of nuclear eternity earlier if one focuses on the negotiation strategies that led to it. It is true that the parties gesture towards an intention to disarm in the preamble and Article VI of the Treaty. 49 However, Swedish diplomat and disarma ment negotiator Alva Myrdal wrote, early on, that the Treaty was intended to stabilize the US-Soviet hegemony over world politics. As suggested above, both countries were then engaged in a nuclear arms race and were definitely living the nuclear eternity. Interest ingly, the US delegation made sure that Articles I and II of the Treaty would not interfere with the previous arrangements to share US nuclear weapons with NATO allies. Washing ton and London also made sure that the restrictions in nuclear weapons trade would not affect their cooperation in nuclear warhead design. Those gestures are additional signs that in the 1960s, from the perspective of the two leading supporters of the Treaty, the US and the Soviet Union, its purpose was to stabilize their hegemony over a perpetual nu clear future rather than change it. [START_REF] Ruzicka | Behind the Veil of Good Intentions: Power Analysis of the Nonprolifera tion Regime[END_REF] When the Non-Proliferation Treaty entered into force in 1970, its Article III.1 assigned the Atomic Energy Agency with the mission of conclud ing safeguard agreements with state parties and verifying the adequate implementation of such agreements. Technical practices of safeguard agreements and verification by the associated International Atomic Energy Agency have naturalized the eternal nuclear present and contributed to the entrenchment of nuclear eternity. [START_REF] Weichselbraun | Constituting the International Nuclear Order: Bureaucratic Ob jectivity at the IAEA[END_REF] Finally, after the surprise Chinese nuclear test in 1964, suspicion of nuclear proliferation in US intelligence circles expanded beyond industrially developed countries. [START_REF] Yusuf | Predicting Proliferation: A History of the Future of Nuclear Weapons[END_REF] This obvi ously does not change the fact that most countries have not explored nuclear weapons op tions and that the imaginary of many policy elites in non-nuclear weapons states has re mained non-nuclear. It only expands the scope of US suspicion.

Beyond those material, strategic, and legal forms of entrenchment of nuclear eternity, a series of intellectual shifts that took place in the 1960s made it possible for the presence of nuclear weapons in the world to be thought of as an irreversible (p. 497) and defining feature of world politics. [START_REF]This is a fundamentally different claim from the existential one that Gunther Anders made in the 1950s. He suggested that the invention of nuclear weapons has opened and defines a new era in human history which is distinct from any previous one because it will be the last one (Anders, 'Theses for the Atomic Age[END_REF] In his February 1960 article in the Bulletin of the Atomic Sci entists entitled 'How to Live with the Bomb and Survive', Leo Szilard embodies the rise of what I call a disconnected post-nuclear future among nuclear scientists: no explicit state ment of the impossibility of a different future but a decrease of the sense of urgency for nuclear disarmament and an acceptance of postponing action towards it and transferring the associated responsibility to those who will come after. He wrote: 'I believe the time has come to face up to this situation and to ask in all seriousness whether the world could learn to live for a while with the bomb.' 54 More specifically, significant intellectual shifts producing nuclear eternity include the rise of 'arms control' and its triumph over 'disar mament' as a framing category of the nuclear weapons problem, and the widespread adoption of the metaphor of 'proliferation' as a descriptor of the direction of global nu clear history. The latter also led to a widening of the suspicion of proliferation within US intelligence circles after the Chinese nuclear test of 1964, as suggested above. [START_REF] Sims | Icarus Restrained: An Intellectual History of Nuclear Arms Control 1945-1960[END_REF] The shift from the language of 'non-dissemination' used by the Irish in their 1958 resolution, which evokes a prohibitive injunction, to that of 'non-proliferation', which includes a metaphori cal reference to self-begetting contagion, also illustrates the entrenchment of nuclear eternity. [START_REF][END_REF] The point of this essay is not to naturalize nuclear eternity as irreversible once it has been entrenched. Of course I recognize that movements proposing the abolition of nu clear weapons did exist after the 1960s and that some policymakers have proclaimed sup port for that goal in the five decades that followed. In fact, every American President since Harry Truman, with the exception of Richard Nixon, has proclaimed at least once that the long-term goal of abolishing nuclear weapons should be taken seriously. [START_REF] Pearl | Forecasting Zero: US Nuclear History and the Low Probability of Disarmament[END_REF] Similarly, the 1980s witnessed a new wave of anti-nuclear (p. 498) activism and anti-nu clear movements which have remained active and gained recognition in 1985, 1995, and 2017 with the International Physicians for the Prevention of Nuclear War (IPPNW), the Pugwash Conferences on Science and World Affairs, and the International Campaign to Abolish Nuclear Weapons (ICAN) being awarded the Nobel Peace Prize. 58 The issue to be investigated further is whether the efforts to create alternative imaginaries of the future after the 1960s came from different communities than the ones I am focusing on and whether and how they managed to set themselves free from the macro-frame of nuclear eternity.

In other words, the global history of the 1960s needs to be revisited beyond the produc tion of emancipatory futures of all kinds: the triumph of civil rights, decolonization and in dependence, and a consolidated civil rights movement.

Conclusion

This chapter is the beginning of an effort to reconnect modes of futures-making with the requirements of democracy by focusing on the naturalization of nuclear weapons and their removal from the realm of democratic choice. In particular, it has focused on one

The Birth of Nuclear Eternity way of shrinking the realm of choice about nuclear weapons by creating a sense that they will be with us forever.

It has made three interventions in existing debates. First, it has denaturalized the idea that nuclear weapons have always been perceived as 'here to stay' from the moment of their invention. This insight is present within the literature but its significance has been underplayed so far. Second, it is an invitation to rethink the 1960s as a decade of signifi cant shrinking of future political possibilities as opposed to the dominant narrative of it as the emancipatory decade. Third, it has identified three shapes of the future that produce nuclear eternity: a disconnected post-nuclear future in which nuclear weapons no longer exist, but which is depicted without any effort to connect it to present conditions, or which posits that the connection can only be made through unprecedented massive change; an absent post-nuclear future which appears in the speeches of heads of nuclear weapons states, in scholarship about nuclear weapons, and even in science fiction; and an inconsistent post-nuclear future which cannot be reached by the steps that are advocated as steps towards it. This is only a beginning and an invitation. Ideas and modes of futurity are treated here as though they were causal forces in their own right and this needs to be modified in at least two ways. First, the self-contradictory nature of nuclear discourse has long been recog nized, so the account of the rhetorical power of nuclear ideas (p. 499) should treat the force of contradiction as a tool towards unfalsifiability as opposed to the current assump tion that logical consistency always works as an asset. Second, assessing the entrench ment of nuclear eternity requires us to think about the embeddedness of those ideas in broader epistemic, affective, and institutional structures of nuclear authority. [START_REF] Masco | s work on the 'national security affect' is of primary importance in this respect[END_REF] Those structures of authority can be the unanimously adopted governance model of nuclear technocracy or 'guardianship' itself, the institutions mandated to produce official nuclear truths, which have been insufficiently studied, or the transnational mechanisms of profes sional recognition for the different professions that grant an authority to speak on nu clear matters.

At the empirical level, this chapter calls for additional questions. First, if the 1960s is such a decade of entrenchment of nuclear eternity, how does this modify the status and role of subsequent nuclear weapons politics? For instance, in regimes aspiring to be de mocratic, are the justifications of nuclear guardianship as the best possible governance structure for nuclear weapons validated by the attitudes and preferences of the citizens of today? Or do they manifest the ability of a generation of elites to institutionalize its preferences? 60 Second, how does time affect the entrenchment of nuclear eternity? Is it a generational effect and, if so, what are its manifestations? Third, how can we account for the processes through which this entrenchment is reversed?

( 3 ) I refer to the Thatcher/Reagan/Deng Xiaoping/Mitterrand turn to neoliberalism and Thatcher's famous claim that 'there is no alternative' to neoliberalism. See The Birth of Nuclear Eternity count, see Timothy Mitchell, Carbon Democracy (London: Verso, 2011). However, democ racies in the second half of the twentieth century did not seem to picture their own end as a possible future, so I am asking the question from their own perspective.

( 14 ) Deudney, Bounding Power, 255-6.

( 15 ) Robert Dahl, Controlling Nuclear Weapons: Democracy vs Guardianship (New York: Syracuse University Press, 1985), 16. Beyond Deudney's point about the absence of ac countability at the moment of nuclear use, there is an absence of accountability during the creation of these weapons and an expansion of the realm of secrecy; this will be inves tigated further in a paper with Fabricio Fialho, 'Probing the Priesthood', currently under review, and in Thomas Fraise's PhD dissertation. Avner Cohen summarizes the problem very well when he writes: 'In regard to procedure, nuclear weapons necessitate a level of secrecy never known before to liberal democracies. Such secrecy obviates fundamental democratic values like transparency, oversight, public deliberations and the public's right to know' (The Worst-Kept Secret: Israel's Bargain with the Bomb [New York: Columbia University Press, 2010, 147]). Studies of the US case reach similar conclusions. For in stance, Joseph Masco's studies of US nuclear history also suggest that nuclear weapons make possible an anti-democratic mode of governance by a secretive national security state: Joseph P. Masco ( 20 ) The statement can be read here: http://fissilematerials.org/library/gan98.pdf.

weapons enterprise or from anti-nuclear activists in order to problematize conflicts of interest and their effect on knowledge production.