Laura Bekere

Nina Larina

Vladimir Lokshin

Arkady Ellern

Mark Sigalov

Vladimir Khodorkovsky

A New Type of Spirocyclic Photochromes Reacting with Light of both UV and Visible Ranges †

Condensation of bindone with cynnamic aldehydes produces cyclic isomers instead of the expected colored merocyanine dyes. These compounds exhibit unprecedented photochromic behavior: the formation of the colored open forms is observed upon irradiation by light of any wavelength between 254 and 642 nm.

During the past decades, merocyanine (MC) dyes 1 are the focus of numerous researches owing to the importance of this type of organic dyes in different fields of materials science, such as nonlinear optics, 2 solar cells, 3 thin-film transistors 4 and biological applications. 5 A special group of merocyanines is related to photochromism. Thus, such important classes of known spiroheterocyclic photochromes [START_REF]Organic Photochromic and Thermochromic Compounds[END_REF] as spiropyrans, spirooxazines, chromenes and others form merocyanines as a result of the UV light induced cleavage of the C-O, or more rarely C-C, bonds. 7,[START_REF] Brede | [END_REF] These merocyanines are usually less stable than the initial cyclic forms and undergo more or less rapid cyclization and the respective color loss. Depending on the structural factors, temperature or media, the equilibrium between the cyclic and merocyanine forms can be shifted toward the latter partially or completely. 9 The stable MC can be isolated and characterized. 10 Some of them can exhibit negative (reverse) photochromism, 11 that is the reversible cyclization upon irradiation by visible light. Recently, the important for the biomedical applications ability of photochromes to react with visible light was demonstrated on the example of substituted azobenzenes. 12 More than a decade ago, we found that the product of the reaction between bindone (1) and p-dimethylaminocinnamic aldehyde (2a) is not the expected colored merocyanine 4a, but its cyclic isomer 3a. [START_REF] Zinger | SPIE Proceedings[END_REF] In solution or polymer matrix, irradiation of 3a at 480 nm generates the colored open form 4a that reverts back to the cyclic form 3a in the dark. [START_REF] Zinger | SPIE Proceedings[END_REF] The formation of the cyclic isomers is the unprecedented phenomenon for 1,3-indandione derived merocyanines, which behave as typical donor -acceptor conjugated systems. [START_REF] Meshulam | [END_REF] Moreover, we find that the photochemical properties of this compound and its analogs strikingly differ from the behavior of the hitherto known photochromic compounds. Here, we report on the synthesis, photochromic behavior, and X-ray structures of a series of derivatives of type 3.

Derivatives 3 were prepared by condensation of bindone 1 with the commercially available p-substituted cinnamic aldehydes 2 in acetic anhydride or ethanol in 42 -90% yields as reddish (3a, b) or yellow-orange (3c) crystals. Scheme 1. Synthesis and photochromic conversions of derivatives 3.

The X-ray crystal structures ‡ were determined for derivatives 3a -3c (Fig. 1) and confirmed their cyclic forms. There [START_REF] Desiraju | The Weak Hydrogen Bond in Structural Chemistry and Biology[END_REF] . Each of the three carbonyl groups within 3a -3c participate in the formation of HBs, but although the C-H counterparts are specific for each derivative, the phthaloyl C-H atoms of the 1,3-indandione moieties (C16, C19 of 3a, C16 of 3b, C17, C45 of 3c) are always involved (Fig. 1). The presence of numerous C-H…O intermolecular HBs in the solid nonconjugated indan-1,3-dione derivatives giving rise to supramolecular aggregates is typical for these derivatives. 16 We also showed that the presence of intra-and intermolecular HBs of this type can explain anomalies in the NMR spectra of bindone 1 and its analogs and of a series of isobenzofuranones also in solution. 17 In solution, equilibrated in the dark for several hours, we detected the following isomer distribution for 3a, 4a and 4'a, respectively: 83, 10 and 7% in toluened8, 89, 8 and 3% in CD2Cl2, 96, 3 and 1% in CDCl3 and 98, 2, 0% in acetoned6. The presence of both E,Zand Z,Z-isomers in smaller quantities cannot be excluded, but their detection is limited by relatively low solubility of 3. Derivative c exists exclusively as cyclic isomer 3.

O O O O R O O O O O R O O O R R CHO Ac 2 O or EtOH a) R = NMe 2 b) R = NEt 2 c) R = OMe
Electronic absorption spectra of derivatives a -c in the visible range exhibit broad low intensity (e < 800 M -1 cm -1 ) bands corresponding to absorption of 3 overlapped with absorption of some amounts of 4 in the case of a and b (Fig. 2). Solutions of 3a -c are photochromic and sensitive even to ambient light. Initially, we assumed that the photochromic behaviour of 3 is analogous to that of the known spiroheterocyclic photochromes [START_REF]Organic Photochromic and Thermochromic Compounds[END_REF] . However, the expected acceleration of discoloration upon irradiation by green or red light did not occur and, moreover, led to the increase of the absorption of the photo-generated isomers. Thus, a rapid color change (Fig. 2) is observed upon irradiation of their solutions at any wavelength between 254 and 610 nm (3a), 254 and 642 nm (3b) and 254 and 532 nm (3c). The rate of discoloration of photo-induced isomers 4' and 4 strongly depends on the temperature and solvent polarity. Thus, at room temperature, the discoloration rate constant of 4a changes by two orders of magnitude (2.55x10 4 s -1 in cyclohexane vs. 744x10 4 s -1 in methylene chloride) (Fig 3a). Replacement of the dimethylamino group by the diethylamino group (4a vs 4b) brings about not only a considerable bathochromic shift (20 nm in toluene), but also increases the Please do not adjust margins Please do not adjust margins discoloration rate by another order of magnitude (Fig. 3b). In protic solvents coloration can be observed only under irradiation and the colored forms disappear within few seconds after the light source is off. Complete restoration of the 4c spectrum takes about 6 h (in methylene chloride) and several days (in toluene). The colored forms rapidly disappear at elevated temperatures: in toluene, for instance, discoloration of 4a is almost instantaneous at 90 -100°C.

The MC forms are not photosensitive. Thus, 4a cannot be generated by irradiation of 3a at 642 nm in n-octane, since 3a practically does not absorb at this wavelength. On the contrary, absorption of 4a is still considerable here. Discoloration of 4a, generated at 405 nm, under permanent irradiation at 642 nm, yielded the same discoloration rate constant as the dark measurements. The discoloration rate of 4c in methylene chloride was not affected by light at 642 nm as well. The photochemical inertness of the photo-induced isomers 4 is not unexpected, taking into account their behaviour as typical donor -acceptor conjugated systems with intramolecular charge transfer. Thus, strong positive solvatochromism of 4 (Fig. 4) confirms the presence of the efficient conjugation observed also with their nonphotochromic analogs [START_REF] Meshulam | [END_REF] . The longest wavelength absorption band of derivative 4a undergoes a 62 nm red shift in methylethyl ketone compared to the octane solution, confirming its nature as a charge transfer band. The electronic transitions of this kind are photochemically inert. For instance, the efficient conjugation between the electron donating and accepting moieties leads, in particular, to the loss of photochromism in fulgides. Repeating irradiation at 532 nm and thermal discoloration of 3a -c for 10 -15 times did not reveal any noticeable fatigue features.

Quantum mechanical calculations † on the cyclic derivative 3a and all four isomers of the MC form using B3LYP/6-31G(d,p)//B3LYP/6-31G(d,p) model chemistry showed that the most stable isomers are 4a (Z,E), by 10.4 kcal/mol and 4'a (E,E), by 7.0 kcal/mol (ZPE corrected) more stable than cyclic 3a. Z,Zand E,Z-isomers are the least stable, but still more stable than 3a by 4.7 and 3.3 kcal/mol, respectively. Calculations in methylene chloride increased the energy difference by about 1 kcal/mol in favor of the MC isomers. Using the B3LYP/6-311G+(2d,p) basis set gave essentially the same results.

TD DFT calculations at the same model chemistry predicted that the first electronic transition within derivatives 3 (581 nm, f = 0.009) is predominantly the HOMO -LUMO transition of the p,p*-type (Fig. 5). Upon excitation, charge transfer occurs from the electron donating aryl groups toward the neighboring electron accepting fluorenonyl and 1,3-indandionyl moieties. The low intensity longest wavelength absorption bands observed for derivatives 3 are thus through-space charge transfer bands in contrast to the MC-type isomers of 4 exhibiting intense through-bond charge transfer bands (549 nm, f = 0.7). The degree of charge transfer can be roughly evaluated by the change in the NBO charges of the C=O groups. Thus, excitation leads to the increase in the sum of negative charges on the C=O groups by 0.19 e -(3a) and by 0.04 e -for (4a).

The mechanism of the photochemical reaction is currently under investigation in our lab. Our preliminary results strongly suggest that the photochromic behavior of derivatives 3 may This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins stem from at least partial retaining of the HBs, observed in solid, also in solution. Derivatives 3a -c, as new photochromes, exhibit thus a number of unique features, such as sensitivity to both UV and visible light, the formation of the insensitive to light colored forms and a wide span of their thermal discoloration rates. The ability of these compounds to react with green and red light and strong dependence of the discoloration rates on the polarity of surroundings suggest their possible use for biomedical studies.

2 |

 2 J. Name., 2012, 00, 1-3 This journal is © The Royal Society of Chemistry 20xx Please do not adjust margins Please do not adjust margins are several short C-H...O intermolecular contacts in the range of 3.137 -3.802 Å (C...O distances), seven for 3a, five for 3b and 15 for 3c, † that correspond to the definition of the C-H...O hydrogen bond

Fig. 1 .

 1 Fig. 1. ORTEP representations of the molecular structures of 3a (a), 3b (b) and 3c (c) with thermal ellipsoids (50% probability) showing short C-H…O intermolecular contacts. The shortest intermolecular distances (Å), 3a: 3.137(12), O3…C28A,3.216(11), O2…C3A; 3b: 3.183(4), O2…C16A; 3c:3.1953(18), O7…C45A, 3.2107(18), O3…C17, 3.2263(19), O4…C32.

Fig. 2 .

 2 Fig. 2. a) Irradiation of 3a at 532 nm in n-octane; b) thermal discoloration of 4c in CH2Cl2 after irradiation at 532 nm.

Fig. 3 .

 3 Fig. 3. Discoloration kinetics at 25°C of a) 4a in different solvents (k x 10 4 s -1 ): 1cyclohexane (2.55), 2 -n-octane (3.33), 3 -toluene (10.8), 4 -THF (47), 5 -methylethyl ketone (258), 6 -CH2Cl2 (744); b) 4a and 4b (k x 10 4 s -1 ): 1 -4a in THF (47), 2 -4b in THF (130), 3 -4a in CH2Cl2 (744), 4 -4b in CH2Cl2 (4810).

18

 18

Fig. 4 .

 4 Fig. 4. Normalized absorption spectra of 4a in n-octane (black), toluene (blue), THF (green) and methylethyl ketone (red).

Fig. 5 .

 5 Fig. 5. HOMO and LUMO plots of 3a.

Acknowledgements

This work was supported in part by the Euripides/COSEC-ID project (No 102930210) from the Ministère de l'économie de l'industrie et de l'emploi.

Notes and references


domain: phys sdu

text:
Alessandro D A M Spallicci
email: spallicci@cnrs-orleans.fr

Micol Benetti
email: micol.benetti@unina.it

Salvatore Capozziello
email: salvatore.capozziello@unina.it

The Heisenberg limit at cosmological scales

For an observation time equal to the universe age, the Heisenberg principle fixes the value of the smallest measurable mass at mH = 1.35 × 10 -69 kg and prevents to probe the masslessness for any particle using a balance. The corresponding reduced Compton length to mH is λH, and represents the length limit beyond which masslessness cannot be proved using a metre ruler. In turns, λH is equated to the luminosity distance dH which corresponds to a red shift zH. When using the Concordance-Model parameters, we get dH = 8.4 Gpc and zH = 1.3. Remarkably, dH falls quite short to the radius of the observable universe. According to this result, tensions in cosmological parameters could be nothing else but due to comparing data inside and beyond zH. Finally, in terms of quantum quantities, the expansion constant H0 reveals to be one order of magnitude above the smallest measurable energy, divided by the Planck constant.

The Heisenberg principle, considered at cosmological scales, has been adopted [1] to address the so-called Hubble tension [2][3][4][5][START_REF] Perivolaropoulos | Challenges for λCDM: an update[END_REF][START_REF] Yang | [END_REF][8][9][START_REF] Petronikolou | Alleviating H0 tension in Horndeski gravity[END_REF][START_REF] Riess | A comprehensive measurement of the local value of the Hubble constant with 1 km/s/Mpc uncertainty from the Hubble Space Telescope and the SH0ES Team[END_REF] taking into account the differences between kinematic and dynamical measurements of the Hubble-Lemaître constant 1 H 0 . Here, we want to investigate if the Heisenberg principle has other roles at cosmological scales and if there are implications for fundamental physics. In other words, we intend to broadly explore the issue of measurability and the relations between quantum mechanics and cosmology. Are measurements always possible in an expanding universe? Does the Heisenberg principle prevent the possibility of measurements at some given cosmological scale for those quantities related to the principle? Since most of cosmological observations are performed with light, awareness of the implications of photon properties is essential.

The Compton length represents the distance at which the concept of a single point-like particle breaks down since the particle cannot be localised within this distance. In the reduced form, it is given by

λ C = mc . ( 1
)
where is the reduced Planck constant, m the mass of the particle and c the speed of light. Thus, the Compton length is the quantum counterpart of mass. It means that defining a Compton length means defining masses of particles as well as any effective mass. When we deal with Bosons, the Compton length assumes the further meaning of scale of the interaction range. Gravitation and electromagnetism are interactions without a Yukawa term [16]. As long as photons and gravitons are massless, λ C remains infinite. However, a tiny mass would reduce the interaction at a distance r by a factor e -r/λC . For photons, this observation was made by Schrödinger [17].

The group velocity dispersion and the deviations from the Gauss law on the electric field divergence or from the Ampère-Maxwell law on the magnetic field curl, caused by a photon mass and thereby to a non-infinite Compton length, were first written by de Broglie [18,[START_REF] De Broglie | Nouvelles Recherches sur la Lumière[END_REF].

Tinier and tinier photon mass upper limits may be sought through testing at larger and larger distances [16]. Precision local experiments on mass might be replaced by exploring the possible finitude of the range of the interaction. Indeed, experimentally, a demanding precision measurement on a Boson mass may be circumvented if looking for deviations of the potential from the 1/r inverse law.

Turning to the Heisenberg indeterminacy or uncertainty principle

2 ∆p∆x = ∆E∆t > /2 , (2)
as E = mc2 , a lower limit on the measurability of the rest mass of any particle is implied. The Heisenberg principle in the energy-time formulation already appeared in his earliest work [START_REF] Heisenberg | [END_REF], and it states that there is a smallest measurable quantity of mass m H (the letter H as lower index stands for Heisenberg). Today estimate is 1.35 × 10 -69 kg for an observation time equal to the universe age3 of about 13.8 Gy.

The principle implies an uncertainty in measuring massive particles but also the intrinsic indeterminacy or impossibility of asserting masslessness. The same principle indicates the minimum time necessary to achieve a given precision. Nevertheless, tests might be carried out observing a classical effect by an ensemble of Bosons, thereby partly circumventing the constraints of a limited observation time. For discussions on the Heisenberg principle in the energy-time form, see [21][22][23][24][25][START_REF] Busch | Time in quantum mechanics[END_REF].

In [1], we portrayed the Compton length as the concept relating the luminosity distance with the Heisenberg principle. Indeed, inserting m H in Eq. ( 1), the reduced Compton length λ C becomes λ H and it assumes the meaning of mass limit expressed in metres. In other words, the photon masslessness cannot be proved also through the corresponding Compton length. As there is not a balance measuring below m H , there is not a test, at a given distance r, that can measure an attenuation factor smaller than e -r/λH for the electromagnetic interaction.

Before dealing with cosmology, we remark that testing of massive electromagnetism4 and massive gravitation 5 has led to significant mass upper limits [16].

Photons are the only massless particles in the Standard-Model (SM), but acquire an effective mass in the SM Extension (SME) [40][41][42][43]. For massive photons, gauge invariance, renormalisability and charge conservation may hold [1].

We now turn to cosmology and recall that H 0 is valued 74.03 ± 1.42 km/s per Mpc [44] by current Supernova measurements. On the other hand, such a constant is constrained at 66.88 ± 0.92 km/s per Mpc [45] using the Cosmic Microwave Background and the Baryonic Acoustic Oscillations data. The difference between the two values, named Hubble tension, is more than 4σ (actually 5σ [START_REF] Riess | A comprehensive measurement of the local value of the Hubble constant with 1 km/s/Mpc uncertainty from the Hubble Space Telescope and the SH0ES Team[END_REF]) and can be compared to the Heisenberg uncertainty [1]. Herein, for our purposes, we take an average value of H 0 = 70 km/s per Mpc.

The m H value corresponds to a reduced Compton length λ H of 2.6×10 26 m that is 8.4 Gpc, Eq (1). Incidentally, the today official upper limit on photon mass corresponds to a reduced Compton length of 1.9 × 10 11 m [START_REF] Zyla | [END_REF], somewhere between Jupiter and Saturn, thanks to the Voyager and Pioneer missions.

We stated that the Compton length for a Boson is a measure of the interaction range. The latter appears best corresponded by the distance crossed by the photon from the source at emission time to the Earth at reception time. In an expanding universe such distance may be best corresponded by the luminosity distance, though such distance is defined through bolometric quantities.

FIG. 1: Luminosity distance (Mpc) as function of z plotted against SN Union 2.1 data [46]. For the (2,1) Padé approximant, Eq. ( 5), using the Concordance-Model parameters (see footnote 7): H 0 = 70 km/s per Mpc, Ω m = 0.3 and Ω Λ = 0.7, we get q 0 = -0.29, j 0 = 0.55 [47,48]. The horizontal line represents the reduced Compton length λ H = 8.4 Gpc, corresponding to the smallest measurable mass, 1.35 × 10 -69 kg. The vertical line at z H = 1.35, represents the Heisenberg limit.

We aim to identify the red shift z H corresponding to λ H , without relying on a specific cosmological model. We thereby compute the luminosity distance through a cosmographic parametrisation. Specifically, we can use Padé approximants as they have proven to be efficient in parametrising cosmic distances whilst reducing error propagation for z > 1 [47,48]. The idea consists in expressing the luminosity distance as a ratio of Taylor series and probe its convergence against data. In particular, the luminosity distance computed out of the (2,1) Padé approximant 6 reads

d L = c H 0 6(q 0 -1) -z[5 + 2j 0 -q 0 (8 + 3q 0 )] 2q 0 (3 + z + 3q 0 z) -2(3 + z + j 0 z) z = c H 0 P 2,1 z , (5)
where q 0 and j 0 are the deceleration and jerk parameters, respectively. As shown in [48], this approximant well fits data at high and low red shifts. The (2,1) Padé approximant can be checked by Supernova (SN) data from the Union 2.1 catalogue [46], Fig. 1.

The check being successful 7 , we deduce that the z, corresponding to λ H = 8.4 Gpc is 1.35, Fig. 1, for the Concordance-Model parameters. Our aim is now to interpret such a result. 6 The Taylor expansion of a generic function f (z) is

f (z) = ∞ i=0 c i z i , (3)
where c i = f (i) (0)/i!, whereas the (n, m) Padé approximant of f (z) is defined as rational polynomial by

Pn,m(z) = n i=0 a i z i 1 + m j=1 b j z j . ( 4
)
The distance of 8.4 Gpc falls into the radius of the observable universe 8 , that is the cosmological region causally connected by light. We can now reinterpret λ H as the Heisenberg radius, the cosmological length beyond which the indeterminacy principle prevails 9 .

Combining Eqs. ( 1) and ( 5), the measurable mass as function of z is given by, Fig. 2 m = cλ C λC=dL [START_REF] Perivolaropoulos | Challenges for λCDM: an update[END_REF] Equation ( 6) shows a peculiar situation, mentioned before: any measurement beyond λ H will not lower the mass limit m H . This occurs well within the observable universe. 6). For the (2,1) Padé approximant, Eq. ( 5), using the Concordance-Model parameters (see footnote 7): H 0 = 70 km/s per Mpc, Ω m = 0.3 and Ω Λ = 0.7, we get q 0 = -0.29, j 0 = 0.55 [47,48]. The horizontal line represents the smallest measurable mass, 1.35 × 10 -69 kg. The vertical line at z H = 1.35, represents the Heisenberg limit.

Perhaps, not surprisingly at this stage, we can relate the expansion constant H 0 to quantum quantities. The H 0 averaged and indicative value of 70 km/s per Mpc corresponds to 2.3 × 10 -18 m/s per metre. Such value may be derived from the Heisenberg principle, Eq. ( 2), in the energy-time form, recalling that the H 0 constant has the dimension of time -1 . Indeed

H 0 = 1 ∆t = 2∆E = 2m H c 2 = 4πm H c 2 h = 2.3 × 10 -18 m/s per metre . (7)
Equation ( 7) may suggest that the Hubble tension [1] is not accidental, but simply one facet of a possibly emerging evidence: the observation of the expansion reveals being a quantum measurement. Notably, H 0 reveals to be just one order of magnitude (4π) above the smallest measurable energy, divided the Planck constant h: 10% of H 0 is at the same time the Heisenberg limit and the magnitude of the Hubble tension.

According to our analysis, we can draw the following conclusions.

1. Any measurement on photon mass through its reduced Compton length beyond λ H will provide at best the Heisenberg limit, 1.35 × 10 -69 kg, observed distances with analytical models. The latter include a set of cosmological parameters which determine the coefficients of the approximants. In Fig. 1, we have used Concordance-Model parameters (Ωm = 0.3 and Ω Λ = 0.7), and thereby we have identified the coefficients q 0 and j 0 . We could have departed from the Concordance-Model parameters and then obtained other values of the Heisenberg radius, but without relevant consequences for our work. 8 The radius of the observable universe is usually expressed as comoving distance, 14.3 Gpc, roughly at z = 1100. For a flat universe, Ω k = 0, the luminosity distance is (1 + z) times the comoving distance, achieving thereby extremely large values for increasing z. 9 McCrea [49] used the term uncertainty for pointing out the consequences of the finitude of the speed of light on the observability of the universe and did not address the Heisenberg principle in any manner. In [50][START_REF] Sahlén | The philosophy of cosmology[END_REF][START_REF] Beltrame | Scienza e fede. se i quanti gettano luce sulla teologia[END_REF], the origin of probability in cosmology, the relations to inflation, the boundaries of knowledge are addressed. The latter issue is closest to our work. Further discussions on the limits of knowledge, possibly in connection with Gödel incompleteness theorem, go well beyond the aims of this paper.

2. Masslessness cannot be proved due to the Heisenberg principle, anywhere, either with a balance or with a metre ruler at any scale length. If there is no evidence of a particle being massive at a distance smaller than λ H , there is no way to ascertain its masslessness beyond, through its reduced Compton length.

3. A basic question ensues: can we interpret light coming from cosmological sources as composed by massless photons, when we cannot prove their masslessness? The answer lies on whether computations and observations are consistent with such an assumption or not. This is applicable before and after λ H .

4. One of the tenets of multi-messenger astronomy is comparing gravitational and electromagnetic signals, assuming that the latter travel at the speed of light c. This is done in view of discriminating gravitational waves from general relativity (speed = c), from gravitational waves of other gravitation theories (speed = c). Multimessenger astronomy might need to take into account that gravitons and photons cannot be proved massless and thereby their speed is not c. Incidentally, at λ H , the delay of a massive photon, supposedly 10 -54 kgactual upper limit [START_REF] Zyla | [END_REF] -with respect to a massless particle, reaches 10 ns at 1 GHz, Fast Radio Burst typical frequency [32,33,36].

5.

Assuming the universe as a standard physical (quantum) system, cosmological measurements before or after λ H , must be read with care. Issues as the H 0 tension point out that measurements inside and beyond λ H could be not consistent. Searching in the current state of observations an unequivocal trend between multiple different measurements at low red shift and multiple different measurements at high red shift appears the following step.

Being the state of the art of measuring photon mass quite coarser than the Heisenberg uncertainty, rendering the items 1-2 beyond reach, our interpretation of the Hubble tension [1] stands as the only indication of this trend, we are actually aware of.

6. The Hubble-Lemaître constant H 0 is related to the Heisenberg principle and thereby determined by quantum quantities.

FIG. 2 :

 2 FIG.2: Measurable mass (kg) as function of z, Eq. (6). For the (2,1) Padé approximant, Eq. (5), using the Concordance-Model parameters (see footnote 7): H 0 = 70 km/s per Mpc, Ω m = 0.3 and Ω Λ = 0.7, we get q 0 = -0.29, j 0 = 0.55[47,48]. The horizontal line represents the smallest measurable mass, 1.35 × 10 -69 kg. The vertical line at z H = 1.35, represents the Heisenberg limit.

Herein, we use indeterminacy when we address massiveness as opposed to masslessness and uncertainty when we refer to measurement accuracy. Principle is used when referring to the general concept rather than an estimate.

Photons were emitted at decoupling about 370.000 years after the Big Bang; photons by galaxies some hundred million years later.

De Broglie estimated the photon mass below 10 -53 kg through group velocity dispersion[27,[START_REF] De Broglie | Recherches sur la théorie des quanta[END_REF]. After one century of painstaking experiments, the Particle Data Group (PDG)[START_REF] Zyla | [END_REF] sets the upper limit just one order of magnitude below, at m < 10 -18 eV c -2 or 1.8 × 10 -54 kg in the solar wind, but see[30] for a critical assessment. Recently, limits up to

5.1 × 10 -51 kg were established estimating dispersion in Fast Radio Bursts (FRBs)[31], more stringent than other attempts[32,33]; for a review, see[34]. The best laboratory test -based on Coulomb's law -showed an upper limit of 1.6 × 10 -50 kg[35], while a space mission was conceived targeting dispersion in the very-low radio frequencies[36].5 Massive gravity was first proposed by Fierz and Pauli[37], and later in different contexts including alternatives to dark matter[38]. For gravitons, the PDG[START_REF] Zyla | [END_REF] sets the upper limit at 6 × 10 -32 eV or 1.1 × 10 -67 kg, obtained from gravitational lensing[39].

The check consists in fitting observational data with the Padé approximants. The Padé-approximant based cosmography reproduces

Acknowledgements

ADAMS acknowledges the Erasmus+ programme for visiting the Università di Napoli, SC the Université d'Orléans and Campus France for the hospitality. SC and MB acknowledge the Istituto Nazionale di Fisica Nucleare (INFN), sezione di Napoli, iniziative specifiche MOONLIGHT2 and QGSKY. The authors thank O. Luongo (Frascati) for discussions. Finally, the authors are indebted to the referee for the detailed comments.