Espen Gaarder
email: espenhaug@mac.com

Eugene Terry Tatum

The Hawking Hubble temperature as a minimum temperature, the Planck temperature as a maximum temperature and the CMB temperature as their geometric mean temperature

Keywords: Hawking temperature, Planck temperature, CMB temperature, geometric mean, Compton wavelength, Hubble sphere, Cosmological models

We illustrate how the Hawking Hubble temperature could simply be the minimum temperature above zero anywhere in the Hubble sphere. It perhaps defines the "temperature gap," so to speak -the minimum temperature above zero -and therefore, also the "energy gap." Of course, this does not exclude that Hawking radiation from any black hole of any size may also occur. Secondly, the Planck temperature is likely the maximum temperature at any localized subatomic point in the Hubble sphere. Additionally, we demonstrate how the Cosmic Microwave Background (CMB) temperature could simply be a form of geometric mean temperature between these minimum and maximum temperatures over the course of cosmic time. This mathematical discovery suggests a re-consideration of R h = ct cosmological models, including black hole cosmological models, even if it possibly could also be consistent with the ⇤-CDM model. Most importantly, this paper contributes to the growing literature in recent years asserting a tightly-constrained relationship between the CMB temperature and the Hubble constant, as well as other global parameters of the Hubble sphere.

1 All electromagnetic energy can be expressed through the Compton wavelength Arthur Holly Compton [START_REF] Compton | A quantum theory of the scattering of x-rays by light elements[END_REF] in 1923 gave us a formula for the Compton wavelength of a particle

1 = h mc (1)
Furthermore, the reduced Compton wavelength is equal to ¯ = mc , where ~is the reduced Planck constant ~= h 2⇡ . Since pure energy can be expressed as equivalent rest-mass energy m = E c 2 then we can also write

¯ = mc = Ẽ c 2 c = ~c E (2)
Haug [START_REF] Haug | The Compton wavelength is the true matter wavelength, linked to the photon wavelength, while the de Broglie wavelength is simply a mathematical derivative[END_REF] has recently at length discussed and demonstrated how the Compton wavelength likely is the true matter wavelength and how the de Broglie wavelength likely only is a mathematical derivative of this. We ask the reader to go to that paper for an in-depth discussion on this. Even if not agreeing on this, one simply needs to understand that the formulas just represented are valid. [START_REF] Haug | The Compton wavelength is the true matter wavelength, linked to the photon wavelength, while the de Broglie wavelength is simply a mathematical derivative[END_REF] The maximum and minimum temperatures and their links to the shortest and longest reduced Compton wavelengths

The Planck [START_REF] Planck | Der Königlich Preussischen Akademie Der Wissenschaften[END_REF][START_REF] Planck | Vorlesungen über die Theorie der Wärmestrahlung[END_REF] temperature is given by

T p = 1 k b r ~c5 G = E p k b = ~fp 1 kb = ~c l p 1 k b ⇡ 1.42 ⇥ 10 32 k (3)
This means that the Planck temperature is the Planck frequency times the Planck constant, which is the Planck energy expressed as a temperature by dividing the Planck energy by the Boltzmann constant.

The Planck length is assumed by many physicists [START_REF] Sivaram | The Planck length as a cosmological constraint. astrophysics and space science[END_REF][START_REF] Adler | Six easy roads to the Planck scale[END_REF][START_REF] Hossenfelder | Can we measure structures to a precision better than the Planck length? Classical and Quantum Gravity[END_REF][START_REF] Hossenfelder | Minimal length scale scenarios for quantum gravity[END_REF] to be the shortest meaningful length. This also means that the Planck frequency is the highest possible frequency and this again indicates that the Planck energy is the highest possible energy for a photon: E p = m p c 2 = ~c lp = ~fp , where f p is the Planck frequency. Furthermore, since the Planck temperature simply is the Planck energy converted to the temperature scale of Kelvin by dividing it by the Boltzmann constant, this implies that the Planck temperature is likely the highest possible temperature, as also suggested by multiple researchers [START_REF] Unnikrishnan | Standard and derived Planck quantities: selected analysis and derivations[END_REF], even if there is still some question as to whether the maximum temperature could be somewhat lower [START_REF] Dai | Maximal temperature in a simple thermodynamical system[END_REF] or somewhat higher [START_REF]Beyond the Planck temperature[END_REF] than the Planck temperature. It has recently been demonstrated that the Planck length is closely related to gravity, as the Planck length and other Planck units can be extracted directly from gravitational observations without knowledge of G or ~, see [START_REF] Haug | Finding the Planck length multiplied by the speed of light without any knowledge of G, c, or h, using a newton force spring[END_REF][START_REF] Haug | Planck units measured totally independently of big G[END_REF].

The reduced Compton wavelength of the Planck energy is

¯ = mp c = ~c E p = l p (4)
Alternatively, we could modify the Planck temperature slightly by assuming that the shortest possible wavelength is related to the radius of a Planck mass micro black hole, which has a Schwarzschild radius of R s = 2Gmp c 2 = 2l p and a circumference of 2⇡R s = 4⇡l p with a corresponding (circumference) Planck temperature of:

T p,BH = ~c 4⇡l p 1 k b ⇡ 1.13 ⇥ 10 31 k (5)
This is exactly half of the Hawking temperature of a Planck mass micro black hole:

T Hw = c 3 kb 8⇡Gm p = kb 4⇡ 2Gmp c 2 = ~c 4⇡R s 1 kb = 1 2 T p,BH (6)
If the maximum temperature is linked to the shortest possible reduced Compton wavelength, and if the Planck energy is the highest localized energy possible, then the lowest possible energy must be linked to the longest possible wavelength in the universe. We assert that the longest possible wavelength is the diameter of the Hubble sphere. This implies that the minimum temperature is related to what we can call the "Hubble frequency," and it must be given by:

T min = ~c 2R H 1 k b ⇡ 8.3 ⇥ 10 30 k (7)
The term c 2R H can be referred to as the Hubble frequency f H ⇡ 1.09 ⇥ 10 18 per second. Consequently, the minimum temperature is simply given by T min = ~fH 1 k b . This represents the lowest frequency above zero that one can observe, as no wavelength can be longer than the diameter of the universe or certainly no longer than the circumference of the universe as we soon will look at also.

In the ⇤-CDM model, the diameter of the universe extends far beyond the Hubble radius R H due to the assumption of the expansion of space, including accelerated expansion. However, there is a series of alternative cosmological models within the linear R H = ct category, as illustrated in, for example, [START_REF] John | Generalized chen-wu type cosmological model[END_REF][START_REF] John | Comparison of cosmological models using bayesian theory[END_REF][START_REF] Melia | The r h = ct universe[END_REF][START_REF] Tatum | How a realistic linear R h = ct model of cosmology could present the illusion of late cosmic acceleration[END_REF][START_REF] John | R h = ct and the eternal coasting cosmological model[END_REF]. Yet another concept to consider is the idea that the Hubble sphere is a black hole with an event horizon at the the Hubble radius. In this scenario, no wavelength can be longer than the diameter of the Hubble sphere or, at the very most, the circumference of the Hubble sphere. This imposes a maximum limit on the wavelength of electromagnetic radiation and, consequently, a minimum energy above zero -a sort of "energy gap," and the lowest even theoretical measurable energy above zero. The notion that the observable universe could be inside a black hole is not a recent idea. It was proposed as early as 1972 by Pathria [START_REF] Pathria | The universe as a black hole[END_REF] and later in 1994 by Stuckey [20]. This idea, despite being in conflict with the ⇤-CDM model, continues to be a topic of ongoing discussion, as evidenced by recent publications such as [START_REF] Pop | The universe in a black hole in Einstein-Cartan gravity[END_REF][START_REF] Akhavan | The universe creation by electron quantum black holes[END_REF][START_REF] Lineweaver | All objects and some questions[END_REF].

We observe that if we multiply this minimum temperature by 1 2⇡ , it is surprisingly identical to the Hawking temperature [START_REF] Hawking | Black hole explosions[END_REF] when the mass in the Hawking temperature is the critical Friedmann mass M c = c 3 2GH 0 :

T Hw = ~c3 k b 8⇡GM c = ~c 2R H 1 k b 1 2⇡ = ~fH 1 k b 1 2⇡ ⇡ 1.32 ⇥ 10 30 k (8)
The di↵erence of 1 2⇡ between the prediction from equation ( 7) and the Hawking temperature formula could have multiple reasons. Even if not ideal, it is not abnormal to adjust the end result with a factor like 2⇡. For example, Adler et al. [START_REF] Adler | The generalized uncertainty principle and black hole remnants[END_REF] did so and simply called it a "calibration factor." The maximum reduced Compton wavelength could also be seen as the circumference of the Hubble sphere instead of the Hubble diameter, making the di↵erence only a factor of 2 relative to Hawking temperature equation (8) as we then get a minimum temperature of:

T min = ~c 2⇡R H 1 k b ⇡ 2.64 ⇥ 10 30 k (9)
So, we will assume, by Hawking temperature equation [START_REF] Hossenfelder | Minimal length scale scenarios for quantum gravity[END_REF], that the minimum temperature is the Hawking temperature.

The small di↵erence mentioned above could be because the Hawking radiation is being derived from the Schwarzschild metric. Other metrics, such as the Kerr [START_REF] Kerr | Gravitational field of a spinning mass as an example of algebraically special metrics[END_REF] and Kerr-Newman [START_REF] Newman | Note on the Kerr spinning-particle metric[END_REF][START_REF] Newman | Metric of a rotating, charged mass[END_REF] metric as well as recent Haug and Spavieri metric [START_REF] Haug | Mass-charge metric in curved spacetime[END_REF], can likely also be used to derive similar temperatures; but, likely, there will only be small di↵erences from the Hawking temperature. We do not have the final answer as to why there is a small di↵erence of 2⇡ or ⇡ in the formulas, so we will, like Adler et al, for the moment, call whatever the small di↵erence may be a calibration factor.

For now, let's assume that the Hawking temperature and the minimum temperature calculated from this alternative method, based on the Compton wavelength and the Compton frequency, indeed represents the minimum temperature inside a black hole. This would mean that the Hawking temperature possibly does not solely represent radiation emitted from a black hole, but could also be the minimum temperature (above zero) that could potentially be observed at any point inside the Hubble sphere. We can call this the Hawking Hubble temperature.

3 The CMB temperature as a geometric mean of the maximum and minimum temperature in the Hubble sphere

Since the discoveries by Hubble [START_REF] Hubble | Extragalactic nebulae[END_REF] and Lemaitre [START_REF] Lemaître | Un univers homogétne de masse constante et de rayon croissant rendant compte de la vitesse radiale des nétbuleuses extra-galactiques[END_REF], as well as the theoretical work of Friedmann, there has been great progress in cosmology, both experimental and theoretical. However, the standard ⇤-CDM model, as well as most other cosmological models, has no way to predict the current CMB temperature.

Mean temperature plays an important role in various fields, including climate science, fluid dynamics, and biophysics [START_REF] Allahverdyan | Thermodynamic definition of mean temperature[END_REF]. To our knowledge, mean temperatures have not been linked to the CMB temperature, except that the 2.725k measurement can by words be called a type of mean (i.e., average) temperature in all of the empty space of the Hubble sphere. To establish a more solid theoretical connection to the CMB as a mean temperature, a solid mathematical and physical foundation is necessary. It is not su cient to simply take the mean of some temperatures and call it the mean temperature; as we will suggest, one must comprehend the Planck temperature, cosmic temperatures, and their inter-relationships in terms of the only variable that di↵erentiates di↵erent energy levels-namely, the electromagnetic wavelength. Additionally, we know from mathematics and statistics that various types of means exist.

The geometric mean traces back to the Pythagoreans, who defined the three most commonly used means even of our time-namely, arithmetic mean, geometric mean, and harmonic mean (see [START_REF] Stević | Geometric Mean In[END_REF]). The geometric mean, indicating a central tendency of a finite set of real numbers by using the product of their values, has wide applications across diverse fields, from economics, finance, engineering, nuclear medicine, informatics, ecology, surface and groundwater hydrology, geoscience, geomechanics, machine learning, and chemical engineering, see [START_REF] Vogel | The geometric mean? Communications in Statistics -Theory and Methods[END_REF]. The geometric mean is utilized for various applications and challenges also in physics. For instance, Henderson [START_REF] Henderson | The theorem of the arithmetic and geometric means and its application to a problem in gas dynamics[END_REF] demonstrated the use of the geometric mean for problems in gas dynamics, Zhang et al. [START_REF] Zhang | The geometric mean density of states and its application to one-dimensional nonuniform systems[END_REF] employed the geometric mean density of states in one-dimensional nonuniform systems, and Yamagami [START_REF] Yamagami | Geometric mean of states and transition amplitudes[END_REF] relied on the geometric mean of states and transition amplitudes, see also [START_REF] Kocic | Geometric mean of bimetric spacetimes[END_REF]. The potential role of geometric means in average temperatures should not be surprising. The reason that maximum and minimum temperatures in the Hubble sphere have not previously been linked to the CMB as a geometric mean temperature is likely to be that it has only recently been discovered that the reduced Compton wavelength plays a much more central role in energy, matter, and even gravity than previously thought.

Assume that the measured CMB temperature is somehow related to some type of mean value between the maximum and minimum allowed temperature in the Hubble sphere. The longest possible reduced Compton wavelengh in the universe, or at least in a R H = ct universe as well as a black hole Hubble sphere, is the diameter of the Hubble sphere, so we have

¯ Maximum = 2R H . (10)
Furthermore, the shortest possible reduced Compton wavelength is assumed to be the Planck length so we have

¯ Minimum = l p . (11)
The geometric mean of the shortest and longest reduced Compton wavelength is given by

¯ gm = p ¯ Maximum ¯ Minimum = p 2R H l p (12)
we can call this wavelength: ¯ gm , the geometric mean reduced Compton wavelength of the observable universe. The temperature correlating with this wavelength we find by taking the energy of its frequency and simply dividing that by the Boltzmann constant. This gives

T gm = ~c ¯ gm 1 k b = ~fm 1 k b ⇡ 34.27k (13)
If we divide this by a calibration factor of 4⇡, and use the recent Hubble constant value given by Kelly et al. [START_REF] Kelly | Constraints on the Hubble constant from supernova Refsdal's reappearance[END_REF] of 66.6 +4.1 3.3 (km/s)/M pc, we obtain the CMB temperature of

T gm = ~c p ¯ max ¯ min 1 k b 4⇡ = ~c ¯ gm 1 k b 4⇡ = ~fm 1 k b 4⇡ ⇡ 2.72 +0.082 0.069 k (14)
Or, if we use instead the circumference of the Hubble sphere instead of the diameter as the longest possible wavelength, and the Planck circumference of a micro black hole instead of the Planck length as the minimum wavelength, then we get:

T gm = ~c p ¯ max ¯ min 1 k b 2 = ~c ¯ gm 1 k b 2 ⇡ 2.72 +0.082 0.069 k (15)
So here we only need to use a calibration factor of 1 2 to get the Hubble temperature very close to measured CMB temperature values; see, for example, [START_REF] Fixsen | The temperature of the cosmic microwave background at 10 ghz[END_REF][START_REF] Fixsen | The temperature of the cosmic microwave bacground[END_REF][START_REF] Noterdaeme | The evolution of the cosmic microwave background temperature[END_REF][START_REF] Dhal | Calculation of cosmic microwave background radiation parameters using cobe/firas dataset[END_REF] (see also the appendix). We are convinced that this is not simply a coincidence. The CMB temperature seems indeed to simply be related to the geometric mean of the shortest and longest wavelengths possible within the observable universe. Since the only thing that di↵erentiates the di↵erent energy levels of electromagnetic radiation (a single beam of photons) is the wavelength, an energy that corresponds to the geometric mean of the shortest and longest wavelengths, expressed as temperature (simply by dividing it by the Boltzmann constant), can be termed the geometric mean temperature. This suggests that the CMB temperature is simply the geometric mean temperature between the lowest and highest possible temperatures in the Hubble sphere, but through our reduced Compton wavelength approach. In our view, this is quite revolutionary, as it also points in the direction that R H = ct cosmological models as well as black hole cosmological models seem to be supported over the ⇤-CDM model. However, as we will also show, the ⇤-CDM model should not yet be excluded.

Our new discovery also seems to be related to recent breakthroughs in the theoretical foundation of the CMB temperature linked to the Planck scale. In their black hole cosmological model, Tatum et al. [START_REF] Tatum | The basics of flat space cosmology[END_REF][START_REF] Tatum | Temperature scaling in flat space cosmology in comparison to standard cosmology[END_REF] proposed the following formula for the CMB temperature in 2015:

T CMB = ~c3 k b 8⇡G p M c m p (16)
In direct comparison to the Hawking temperature formula, the M in the denominator is simply changed to p M c m p , where M c is the critical mass in the Friedmann [START_REF] Friedmann | Über die krüng des raumes[END_REF] universe. Making use of the Schwarzschild formula, Tatum et al also re-wrote their formula in the form of

T CMB = ~c k b 4⇡ p R h R pl (17)
Their use of geometric means in these formulae is now obvious. Recently, it has been proven that these Tatum et al formulae are derivable also from the Stefan-Boltzmann law [START_REF] Haug | How to predict the temperature of the cmb directly using the hubble parameter and the planck scale using the stefan-boltzmann law[END_REF]. Furthermore, it has been demonstrated that rewriting them to find the Hubble constant (by substituting H 0 for c/R 0 ) and then deriving the Hubble constant from published CMB temperature studies dramatically increases the accuracy in predicting the Hubble constant, see [START_REF] Tatum | High precision hubble constant determinations based upon a new theoretical relationship between cmb temperature and h 0 . Hal archive[END_REF]. Tatum et al define R pl above as the Schwarzschild radius of the Planck mass, so this means

R pl = R s = 2Gmp c 2
= 2l p which also means that their formula easily can be re-written as

T CMB = ~c k b 4⇡ p 2R h l p = ~fm 1 k b 1 2 ⇡ 2.725k (18)
Where f m is the reduced Compton frequency of an energy with a wavelength consisting of the geometric mean wavelength from the lowest and highest possible temperatures in the Hubble sphere. The Tatum et al formula immediately above gives an identical result to that given in our last geometric mean temperature formula. One should remember that the use of the Schwarzschild formula for substitutions between mass and radius is allowable in a black hole cosmology model. One can even use such a substitution in the Hawking black hole temperature formula to arrive at the following equation (by realizing that the mass in a Schwarzschild black hole must be M = c 2 R 2G ):

T Hw = ~c3 k b 8⇡GM = ~c3 2G k b 8⇡Gc 2 R = ~c 2⇡R 1 2k b ( 19
)
This result is identical to equation ( 6), so no calibration factor is needed.

Our new geometric mean methodology in this paper arrives at the same formulae using a very di↵erent approach in comparison to Tatum et al: by focusing on the Compton wavelength and Compton frequency in matter and energy, and by taking the geometric mean between the maximum and minimum reduced Compton wavelengths, we then examine the temperature to which this leads. This provides a new perspective on the Hawking temperature, potentially being the minimum temperature in a black hole, and the Planck temperature the maximum temperature, and the average temperature inside the black hole simply as the geometric mean temperature in the way described in detail above.

Figure 1 demonstrates the Hubble sphere, where we show the minimum temperature that likely is in the Hubble sphere surface and the CMB temperature that is the geometric mean temperature in the Hubble sphere. The CMB temperature varies according to which cosmic epoch we are considering. As with the Hubble sphere minimum temperature, the CMB temperature was a higher temperature in the past than it is now. Cosmic mass and radius values were presumably smaller the earlier the cosmic epoch. This figure illustrates one possible geometric interpretation of our approach. The Hawking temperature, besides representing Hawking radiation, also seems to be consistent with a minimum temperature in the Hubble sphere, which could be at the very surface of the Hubble sphere. The current CMB temperature is the geometric mean temperature related to the minimum possible wavelength (the Planck length) and the maximum possible wavelength, which is related to the Hubble diameter. This seems to be able to precisely measure the CMB temperature now and in the past. Such a geometric picture can likely be consistent with multiple cosmological models, in particular, black hole cosmological models, something that will be discussed in the following section. [START_REF] Planck | Vorlesungen über die Theorie der Wärmestrahlung[END_REF] Hubble sphere in growing black hole models Two of the most interesting cosmological models likely to benefit from using a geometric mean CMB temperature are growing black hole models and R h = ct models. A growing black hole model is illustrated in Figure 2.

The Tatum et al. [START_REF] Tatum | The basics of flat space cosmology[END_REF] model appears to have been the first cosmological model that accurately predicted the Hubble constant value using only the CMB temperature as an input value. This model basically assumes that the current Hubble sphere started out as a Schwarzschild black hole continually growing in mass and radius in a way consistent with a type of R h = ct model, where the Hubble radius is equal to the Schwarzschild radius (R h = R s ) at any cosmic time t. On the other hand, the Lambda-CDM model does not have the mathematics or geometry that can predict today's CMB temperature from a known Hubble constant value, or vice-versa. That said, we do not exclude yet that the geometric mean approach between minimum and maximum temperatures, which we introduce herein, might also be incorporated into the Lambda-CDM model. However, if so, we suspect that it might be an overly-complicated process, something we leave up to future research to find out.

The growing black hole model of Tatum et al. is interesting in several other respects, not the least of which is the 4-axis log graph it implies with respect to growth and expansion of the cosmological black hole. See Figure 3.

In such a model, expansion is commensurate with mass growth over cosmic time. This is very di↵erent from the ⇤-CDM model, but it may be worth considering that there could be ongoing matter production coming from the cooling cosmic vacuum under continuing Geometric methods are very general, as they rely on core geometric principles, specifically the geometric mean in this case. The same approach can therefore likely be applied to a series of di↵erent black hole models, such as Kerr [START_REF] Kerr | Gravitational field of a spinning mass as an example of algebraically special metrics[END_REF] black holes, Kerr-Newman [START_REF] Newman | Note on the Kerr spinning-particle metric[END_REF] black holes, and Haug-Spavieri [START_REF] Haug | Mass-charge metric in curved spacetime[END_REF] metric black holes. Most, if not all, of these could be extended to growing black hole models somewhat similar to that of Tatum et al. While this must be carefully investigated, the main framework for how to incorporate CMB temperature in them based on our geometric mean approach is already laid out in this paper. The general principles described here should still hold, but the exact values and interpretations of di↵erent black hole universes will naturally vary to some extent. We leave that for further work and investigation.

Information horizon universes and steady-state black hole type universes

As the geometric mean approach is mostly based on core principles in geometry and energy (such as simply E = hf ), it is very likely that our approach can be applied to a series of di↵erent cosmological models. Each model must naturally be carefully investigated and checked against many observations, as well as other models such as ⇤-CDM, before one can come to any conclusions and a consensus view. However, we would like to mention a few more models that are likely consistent with the geometric mean temperature approach, just to demonstrate its possible broad application for cosmological models.

Above, we briefly described growing black hole models, where one model, the Tatum et al model, has already been linked to CMB and Hubble constant theoretical predictions that can also be explained by our geometric mean approach, both in the present and as a dynamic model wherein the Hubble sphere keeps expanding. Another perspective would be a steadystate black hole type universe where every point in the universe is ultimately linked to Planck mass particles popping into and out of existence (see [START_REF] Haug | Unified quantum gravity field equation describing the universe from the smallest to the cosmological scales[END_REF]), and where the Hubble radius is more of a type of information horizon. It is important to be aware that any universe with infinite spatial extension (extending infinitely in space), if it has even a minimum energy density, will, at least from a mathematical point of view, have an event horizon as seen from any point in space. Therefore, the observable universe will be di↵erent from the total universe.

The exact event horizon and its interpretations can naturally vary between di↵erent metric solutions to Einstein's [START_REF] Einstein | Näherungsweise integration der feldgleichungen der gravitation[END_REF] field equation. Such an event horizon could, under certain assumptions, simply also be interpreted as an information horizon, preventing any information from reaching us beyond such a radius.

Such a steady-state model of the universe, wherein the center is everywhere and there is an event horizon at the Hubble radius everywhere, simply due to the energy density of the universe, is illustrated in figure 5. Such models would then need another form of dynamics to explain them, for example, as to why the predicted CMB temperature would vary as one looks farther out. One possibility could be that the energy density actually varies through the Hubble sphere, but other models of cosmological red-shift might also suggest an alternative explanation. We just point this out to illustrate that our geometric mean approach could potentially be consistent with multiple cosmological models. This figure illustrates a possible steady-state universe where the center of the universe is everywhere, and all matter is ultimately linked to Planck mass particles. In this model, the Hubble radius serves as an information horizon because the universe has a minimum energy density everywhere. We mention this to demonstrate that our geometric mean CMB temperature approach could potentially be applied to multiple types of cosmological models. [START_REF] Adler | Six easy roads to the Planck scale[END_REF] Multiple possible mechanisms by which the CMB temperature could be linked to the Planck temperature.

Adler, Chen, and Santiago [START_REF] Adler | The generalized uncertainty principle and black hole remnants[END_REF] claim that, "In the current standard viewpoint, small black holes are believed to emit black body radiation at the Hawking temperature." This is something we agree on, and Haug [START_REF] Haug | Unified quantum gravity field equation describing the universe from the smallest to the cosmological scales[END_REF][START_REF] Haug | Di↵erent mass definitions and their pluses and minuses related to gravity: The kilogram mass is likely incomplete[END_REF] has recently indicated, based upon quantum gravity theory, that all matter may ultimately be derived from "virtual" Planck mass particles popping into and out of existence, lasting only the Planck time. These Planck mass particles should have the mathematical properties of micro black holes, and their energy could be the currency of such interchanges. Perhaps they are largely responsible for quantum fluctuations in vacuum energy. This speculation provides us with a possible underlying deeper theory of how the Cosmic Microwave Background (CMB) temperature could be related to the Planck temperature.

The Planck temperature, the maximum temperature which could exist in space, could be present only in Planck-sized areas of space, popping into and out of existence. Even an electron or proton is enormous in terms of spatial dimensions compared to the radius of a Planck mass particle micro black hole. All of our measurements of the CMB temperature might be of photons radiating from Planck mass vacuum particles popping into and out of existence. Our approach appears to be consistent with a new way to quantize general relativity theory; Einstein's field equation can be re-written as [START_REF] Haug | Di↵erent mass definitions and their pluses and minuses related to gravity: The kilogram mass is likely incomplete[END_REF]:

R µv 1 2 g µv R + ⇤g µv = 8⇡l 2 p ~c T µv . (20)
This re-written form of the field equation gives all the same results as before, but it leads, for example, to a re-written Schwarzschild metric of the following form

ds 2 = ✓ 1 2GM c 2 r ◆ c 2 dt 2 + ✓ 1 2GM c 2 r ◆ 1 dr 2 + r 2 g⌦ 2 ds 2 = ✓ 1 2l p r l p ¯ M ◆ c 2 dt 2 + ✓ 1 2l p r l p ¯ M ◆ 1 dr 2 + r 2 g⌦ 2 (21)
where ¯ M is the reduced Compton wavelength of the mass M , and g⌦ 2 = (d✓ 2 + sin 2 ✓d 2 ). The term lp ¯ M is the reduced Compton frequency per Planck time that represents the quantization of gravity. This new way to rewrite the Schwarzschild metric provides exactly the same predictions as the standard Schwarzschild solution, but it o↵ers a deeper insight, in our view. It shows that gravity is ultimately linked to the Planck scale and establishes a connection between gravitational objects such as the Earth, the Sun, and even the mass or energy of the Hubble sphere and the Planck scale.

Again it is important to be aware that the Planck length can be found from gravity observations without any knowledge of G or even ~, see [START_REF] Haug | Finding the Planck length multiplied by the speed of light without any knowledge of G, c, or h, using a newton force spring[END_REF]. Already in 1984 Cahill [START_REF] Cahill | The gravitational constant[END_REF][START_REF] Cahill | Tetrads, broken symmetries, and the gravitational constant[END_REF] suggested that the Planck units could be more fundamental than the gravity constant; he simply rearranged the Planck mass formula m p = q ~c G with respect to G and got G = ~c

m 2 p .
However, Cohen [START_REF] Cohen | Fundamental Physical Constants[END_REF] correctly pointed out that this just led to a circular problem, as no one at that time had demonstrated a way to find Planck units independent of G. This view has been held until recently, see for example the interesting paper by McCulloch [START_REF] Mcculloch | Quantised inertia from relativity and the uncertainty principle[END_REF]. Now it is fully possible to find Planck units from gravity observations with no knowledge of G and ~. In our view this means that the Planck scale now indirectly has been detected. This also indirectly may explain why we can predict the CMB temperature from theoretical considerations combined with a practical knowledge of the shortest and longest possible wavelengths in the Hubble sphere.

Another view more consistent with the ⇤-CDM model is that Planck mass particles (micro black holes) existed only just after the Big Bang, and that these particles decayed into today's known particles. Lloyd Motz [START_REF] Motz | Gauge invariance and the structure of charged particles[END_REF][START_REF] Motz | A gravitational theory of the mu meson and leptons in general[END_REF][START_REF] Motz | Gauge invariance and the quantization of mass (of gravitational charge)[END_REF] was likely the first to suggest the existence of a very fundamental particle with a mass equal to the Planck mass. However, he knew the Planck mass was way too massive in comparison to observed particles, such as protons and electrons. Motz tried to overcome this challenge by claiming that the Planck mass particles created just after the Big Bang had radiated most of their energy away and that this energy could have been the origin of the creation of particles such as protons and electrons. Others have suggested a similar idea that there were plenty of Planck mass-type particles just after the Big Bang, see De [START_REF] De | Quantum creation of highly massive particles in the very early universe[END_REF], but that such super-heavy particles have radiated away most of their energy. One possibility is therefore that the Cosmic Microwave Background (CMB) is in some way a remnant of that epoch. Only further scrutinizing of the CMB temperature can provide a final conclusion to these answers. What is most important is that both the Stefan-Boltzmann law and this new geometric mean temperature approach essentially lead to the same conclusion and to the formulae suggested by Tatum et al. Now the CMB temperature can clearly be predicted and described theoretically and not only simply measured. This theoretical relationship, unknown to most astrophysics at the time of this writing, seems to bind the di↵erent properties of the observable universe more closely together than before. It could be that our recent theoretical breakthroughs concerning the CMB temperature might lead to improved cosmological models.

Consistent with this view remains the possibility that the universe is a black hole-like object wherein the Hubble horizon has been, and continues to be, expanding at light speed. This would be a distinct class of R H = ct growing black hole cosmological models. Our new theoretical understanding of the Cosmic Microwave Background should naturally be carefully scrutinized by multiple researchers over time to check which cosmological models, as well as quantum gravity models, it best conforms with, or if even our new theoretical CMB framework needs further modification. Surprisingly, there now appears to be a close theoretical link between CMB temperature, the Hubble constant, and other global parameters of the universe; see [START_REF] Tatum | The basics of flat space cosmology[END_REF][START_REF] Tatum | High precision hubble constant determinations based upon a new theoretical relationship between cmb temperature and h 0 . Hal archive[END_REF].

Outstanding issues

Our geometric approach explains the likely relationship between the CMB temperature, the Planck temperature, and the Hawking temperature and even between the Hubble constant and the CMB temperature. Despite our current geometric approach, a similar relation has also been proven by the Stefan-Boltzmann law. However, there is likely still more to understand here. The Hubble tension [START_REF] Valentino | In the realm of the hubble tension -a review of solutions[END_REF][START_REF] Krishnan | Does hubble tension signal a breakdown in flrw cosmology? Classical and Quantum Gravity[END_REF] implies an unexplained di↵erence in the Hubble constant when inferred from local measurements, such as supernovae standardized candles, and when inferred from the cosmic microwave background (CMB). It could potentially be that our new and deeper theoretical geometrical understanding of the CMB temperature can help solve this issue, or alternatively, that a future solution to the Hubble tension will slightly alter our CMB model. We have already pointed out that more metrics should be investigated based on this approach. But for now, we know at least that using the Planck temperature and the Hawking temperature corresponds to both a maximum and minimum temperature in the Hubble sphere, even when relying on just the Schwarzschild metric.

We are also naturally aware of the empirical color temperature T r of the CMB as a function of redshift, T r ⇡ T CMB (1 + z). No deviations from the expected (1 + z) scaling behavior of the CMB temperature have been observed. However, it should be noted that the uncertainty in measurements is extremely large for a CMB linked to high z measurements. For example, in [START_REF] Riechers | Microwave background temperature at a redshift of 6.34 from h 2 o absorption[END_REF], the CMB temperature was measured for the cosmic epoch at z = 6.34, yielding a microwave background temperature of 16.4-30.2k within one standard deviation uncertainty. This uncertainty is notably extremely high compared to most other properties of the cosmos. Therefore, it is not unthinkable that our new methodology and theoretical understanding could also lead to new insights in this regard, but to find that out would likely require a great deal more research.

Conclusion

The Planck temperature is likely the maximum possible temperature anywhere in the observable universe. We propose that it is linked to the shortest possible reduced Compton wavelength, the Planck length. We have also explained how the minimum possible temperature can be considered to be associated with a reduced Compton wavelength corresponding to a photon wavelength the diameter of the Hubble sphere. Interestingly and surprisingly, the CMB temperature could well be connected to an energy level that is simply the geometric mean between these minimum and maximum energy levels, or more precisely, to an energy connected to the geometric mean of the shortest and longest possible reduced Compton wavelengths in the universe. This would seem to imply that the Hawking Hubble temperature is, in fact, a minimum temperature (above zero) anywhere in the Hubble sphere. Our approach suggests a solid theoretical framework for predicting and understanding the CMB

Figure 1 :

 1 Figure1: This figure illustrates one possible geometric interpretation of our approach. The Hawking temperature, besides representing Hawking radiation, also seems to be consistent with a minimum temperature in the Hubble sphere, which could be at the very surface of the Hubble sphere. The current CMB temperature is the geometric mean temperature related to the minimum possible wavelength (the Planck length) and the maximum possible wavelength, which is related to the Hubble diameter. This seems to be able to precisely measure the CMB temperature now and in the past. Such a geometric picture can likely be consistent with multiple cosmological models, in particular, black hole cosmological models, something that will be discussed in the following section.

Figure 2 :

 2 Figure 2: This figure illustrates a universe that started as a Planck mass black hole and then grew into today's Hubble sphere. The black hole radius, which is also the Hubble radius, grows at speed of light c.

Figure 3 adiabatic

 3 Figure 3

Figure 4

 4 illustrates such a growing black hole model. The universe in this model starts as a Planck mass black hole at the Planck temperature, and presumably arises from what we might call a "primordial Planck mass soup" at the Planck density. The radius of the black hole universe grows at speed of light c, as one might expect for a R h = ct model. The CMB temperature, which is the temperature inside the black hole is always simply the geometric mean between the Planck temperature (maximum temperature) and the Hawking Hubble minimum temperature, which again is directly linked to the Hawking temperature at the continually-expanding cosmic horizon surface of the Hubble sphere.

Figure 4

 4 Figure 4

Figure 5 :

 5 Figure5: This figure illustrates a possible steady-state universe where the center of the universe is everywhere, and all matter is ultimately linked to Planck mass particles. In this model, the Hubble radius serves as an information horizon because the universe has a minimum energy density everywhere. We mention this to demonstrate that our geometric mean CMB temperature approach could potentially be applied to multiple types of cosmological models.

temperature, rather than solely observing it.

Conflict of interest

The authors declare no conflict of interest.

Appendix: Wien's Law versus Planck's law for getting to the CMB temperature

The CMB temperature has not been measured directly. Instead, it is the radiation frequency that is measured. There is a whole spectrum of radiation frequencies, but the peak frequency can be easily transformed into temperature by utilizing Wien's (approximation) law or the more exact Planck's law. The transformation from the measured peak wavelength to CMB temperature can be achieved by utilizing Wien's law. According to Wien's law, we can determine that b = hc 5k b [START_REF] Akhavan | The universe creation by electron quantum black holes[END_REF] and the CMB temperature is then given as

For a peak wavelength of 1.0634 mm this gives a CMB temperature of T CMB = 2.706k From Planck's-Law we get b = ch k b (5 + W 0 ( 5/e 5 ))

⇡ 0.002897773 [START_REF] Hawking | Black hole explosions[END_REF] where W 0 is the Labert W function. If the peak wavelength is 1.0634 mm this gives a CMB temperature of approximately

This means that transforming measured peak wavelength from the CMB spectrum to CMB temperature using Wien's law will underestimate the CMB temperature by approximately 0.019k compared to using the more exact Planck's law.


Guillaume Calvet
email: calvet@irit.fr

Julien Kahn
email: julien.kahn@rd.francetelecom.com

Pascal Salembier
email: salembier@irit.fr

Moustapha Zouinar
email: zouinar@irit.fr

"In the pocket" : An empirical study of multimodal devices for mobile activities

This paper presents some methodological issues and some of the first findings that have arisen from our work on the « Multimodal mobile devices » joint project at GRIC-IRIT, CLIPS-IMAG, and FT R&D department. Our main research and design interest is in the development of novel interactive devices for the mobile user that exploit classical (i.e. existing) and innovative input modalities. Two different studies are briefly presented and some results are commented. The first experimental study aims at providing some empirical results in order to discuss the functional roles play by multimodal interfaces in the context of a real task. The second study explores the role of coordinating artefacts in mobile cooperation, and addresses the problem of contextual constraints which may restrict the use of some modalities.

Introduction

Multimodal interfaces combine many simultaneous input modalities and may present information using different output modalities. The input modalities can be quite simple (for example two different pointing devices) ; but they can also include more sophisticated techniques such as speech and gestures recognition.

The growing interest in multimodal interface design is inspired largely by the goal of supporting more flexible, natural and robust performance in the context of real-world situations (ever changing factors, continuous shifting from acceptable to less than perfect conditions, social context of use,…). Unfortunately so far few studies have given an account of the way multimodal systems are actually used by people in the context of real tasks. Moreover most of the previous studies have considered a limited number of modalities.

In the same time, rapid and major technological advances lead to the use of many electronic devices (laptops, mobile phones, PDA,…) that allow users to interact with people and information in different geographical locations [START_REF] Harper | The Wireless World[END_REF]. Mobile usage of PDAs offers even more challenges, as not only do the problems arising from the diminishing size of the PDA screens but also the completely different user environments. The use of mobile devices gives opportunities to act in contextually enriched environments, but natural field environments also are challenging because of the large number of contextual sources of disturbance ; for example physical and social constraints imposed on the interaction with the device may vary rapidly. Multimodality can be seen as a candidate to help the user dealing with this variation [START_REF] Oviatt | Multimodal System Processing in Mobile Environments[END_REF]; but so far few empirical data have been produced to explore the theme of whether multimodal interfaces can reliably support adverse usage environments.

Exploring multimodal interaction

This first study was focused on the assesment of a device integrating different input modalities such as gesture, speech and physical manipulation of the device itself ("embedded physicality").

The main objectives were :

• To tackle the problem of multimodality in "meaningful" situations of interaction, i.e. contexts of tasks/activities the users are familiar with. • To study the processes of appropriation of the multimodality along several sessions of interaction. • To explore the use of innovative modalities in order to empirically check out the role they may play in the man-machine interaction, the challenges they pose individually (monomodality) and in relation to the use of other modalities (multimodality).

More precisely this study aimed at addressing some specific issues: how is multimodality carried out and what are the factors that determine the choice of modality ? Is there any shift from a modality to another ? Do the properties of modalities play any role in this choice? What are the criteria that lead a user to adopt or give up a modality of interaction ? How do the users combine the different modalities ?

Method

Setting

The strategy adopted for addressing those problems was first to use empirical Wizard-of-Oz (WOZ) simulation to test a partially simulated multimodal system1 . The purpose of this WoZ experiment was to enable the users to interact in a natural way while limiting the cost of development of the system and to avoid the current technological limitations at this exploratory step of the study of innovative modalities (gestual and tangible modalities).

Subjects

10 subjects (regular users of Internet but not computer specialists) took part in this experimental phase. In order to put the subjects in a familiar situation, they were asked to use an e-mail application implemented on a pocket PC which was connected to their personal electronic mailbox.

Procedure

At the beginning of each session, the subjects were briefly introduced to the task to be performed and to the setup including the messenger software, the input and output modalities. The actions performed by the subjects during the sessions (from 4 to 10 sessions of 15 minutes average duration each) were recorded by means of various sensors (video cameras, system events recorder). The experimental sessions were immediately followed with a self-confrontation session (interview of the user based on the video recording of his/her activity).

Data

Different data were collected. During the interaction, the commands and the windows manipulated by the subjects were indexed temporally and automatically stored in a file. The actions of the subjects, the modalities used, the duration and the contents of the commands were collected a posteriori from the analysis of the video recordings of the experimental sessions. All those data were encoded using the Actogram software2 which offers quantitative analysis and graphical display functions.

Recording of the self-confrontation sessions was carried out in order to allow qualitative analysis focused on the identification of the strategies applied by the subjects when using the modalities.

The modalities : between innovation and classicism

Four modalities were implemented on the PDA (see figure 1) :

• the vocal modality (unconstrained natural language interaction).

• the tactile modality, (with a stylet to point the command buttons)

• the gestural modality, (which relies on a set of predefined gestures)

• the "tangible" modality (which associates commands with predefined handling modes of the artifact : changes of orientation, various movements, etc.)

Results

Global use

One of the first results obtained about the use of the modalities, indicates that all the modalities were used by the subjects during the different sessions. In addition, one can observe an evolution in the use of multimodality during sessions:

• During the first sessions there was a homogeneous distribution in the use of the modalities which corresponds to a phase of an informal test of the system. This fact is corroborated by the data resulting from the self-confrontation sessions ;

• but quickly some preferences in the use of the modalities appear in the following sessions ;

• these preferences vary according to the subjects.

Intra-individual specialization

As mentioned above, many subjects exhibit a tendency to preferential uses of modalities. These tendencies may be local (they appear only during one session of interaction) or general (they appear in other sessions), specialized (the same modality is used to carry out only one and only one command) or plural (the same modality is used to carry out several commands).

Influencing factors in the choice of modality

The activity graphs and the verbal reports put into evidence several factors which are likely to direct the choice and the change of use of the modalities. The four principal ones are listed below :

• Context of course of action (recurring operational procedure) :

We could identify that the tendencies to preferential use generally result in sequences of recurring actions done by the subjects. These sequences are strongly related to the context of the course of action.

• Change of goal :

The changes of modality are often associated with phases of local reorientation of the activity.

• Properties of the particuliar implementation of the modalities :

During the evaluation of a multimodal device, it is necessary to distinguish the inherent characteristics of a given modality from the particuliar implementation of this modality. The fact that a modality is not used does not mean that it is unsuited but that its technical realization can be inadequate or constraining.

• Error recovering Many shift between modalities happens in situations perceived by the user as dysfunctioning ones ; the shift from a modality to another is then seen as a possible ressource to recover from errors.

Cooperative mobile activities in real-world settings

The second study we briefly present here is a first attempt to identify the contextual constraints users may have to face when using a mobile device for coordination purpose in a real world situation.

Two subjects (teachers) were asked to explore the two levels of a museum in order to prepare a visit with young children. They had to select a set of objects and places of interest in respect with pedagogical issues and to define a convenient pathway for the visit. As the subjects were not located at the same floor, they were given a simple communication device3 (a multimedia mobile phone with which they could send each other vocal messages, shorttext messages, and pictures) in order to enable them to coordinate their joint activities. The task had to be fulfilled in real-time as the subjects were told there was no opportunity to exchange information outside of the museum. Each subject was briefly introduced to the scenario and to the MMS mobile phone. They were not encouraged to follow any particular strategy, such as to coordinate step-by-step while moving in the rooms or to wait until the end of the parallel exploration of the museum. The activity of each subject was video-recorded all along the session, and used as a support for verbal reports gathering. In addition, one of the user was equipped with a wearable microcamera embedded in her spectacles to give a subjective view of the interaction with the environment and the device (see figure 2). Despite some identified limitations [START_REF] Mark | A Methodology Using a Microcamera for Studying Mobile IT Usage and Person Mobility[END_REF], the microcamera technology satisfies many requirements in terms of capture of the mobile device usage from the viewpoint of the users and broder aspects of the surrounding environment. The collected data outline the different forms of interaction performed by the users with the mobile device and provide information on the acceptable practice in social environments. They give an account of the strategies the subjects use when facing dynamic contextual constraints.

Conclusion and proposals for new research

In this article we presented two exploratory studies focused on the use of a portable multimodal system in a experimental setting and on the use of a mobile device for coordination purpose in a real world environment. The results of the first study generalize earlier findings : effective use of multimodality (all the modalities were used), interindividual differences (existence of preferential tendencies), change of modalities in the dysfunctioning situations. Beyond this convergence, an original result of this study is that the subjects get used to nonusual modalities rather easily ("tangible" modality for example). This kind of device is likely to be used in variable contexts. Thus difficulties may arise when the modalities of interaction will not be equally adapted. The multimodality takes a crucial importance then as it offers to the users a ressource to cope with the contextual variations of the environment (physical, social, etc…).

In the next step of this work we intend to link the multimodal and mobile dimensions in real world settings observation. We will made available to the users a multimodal device they will be able to use in different contexts. The purpose of this step is to situate our inquiry of innovative input modalities in the daily activities of the users over a longer period of time.

Figure 1 :

 1 Figure 1 : The tactile, gestural and tangible modalities Only the tactile modality was really implemented ; the other modalities were simulated by the WoZ setup.

Figure 2 :

 2 Figure 2 : Subjective view of the use of the MMS device

WoZ setup has been used in other empirical studies of multimodality (see for exampleBernsen & Dybkjaer,

2001)

http://www.actogram.net/

For a presentation of a more sophisticated scenario of electronic tourist guide see for example the HIPS-I 3 EC project


Pascal Salembier
email: salembier@irit.fr

Julien Kahn
email: julien.kahn@rd.francetelecom.com

Guillaume Calvet
email: calvet@irit.fr

Moustapha Zouinar
email: moustapha.zouinar@rd.francetelecom.com

Marc Relieu
email: marc.relieu@rd.francetelecom.com

"Just follow me". Examining the use of a multimodal mobile device in natural settings

This paper describes a method and a piece of equipment for capturing data about the use of a mobile multimodal device in natural settings . The equipment includes an instrument worn by the user to capture his or her perspective while moving, a means of capturing a wider view of the context and a multimodal data logger.

Introduction

The growth of mobile systems (e.g. Personal Data Assistants, mobile handsets, etc.) sets new methodological and technical problems for empirically studying their use and their usability in natural settings. The mobility of a user implies not only a change of environment, but also physical motion and rapid shifts of contextual features that make both the interaction with the system more complex for the user and the data collection process harder for the analyst. Among the empirical studies devoted to the use of mobile systems, there is a well-established distinction between two main approaches (laboratory studies and field studies),that are based on different epistemological assumptions (see [START_REF] Kjeldskov | Is it worth the hassle? Exploring the added value of evaluating the usability of context-aware mobile systems in the field[END_REF].

The first approach consists of studying the interaction in a laboratory. This focuses on controlling the variables which can affect the interaction, and allows the construction of artificial settings to put the user in a mobility situation. For example, in some studies, users are asked to walk on a running machine [START_REF] Beck | Experimental evaluation of techniques for usability testing of mobile systems in a laboratory setting[END_REF][START_REF] Pirhonen | Gestural and audio metaphors as a means of control for mobile devices[END_REF]. In general, with this experimental approach, the assumption is that it is possible to recreate in a laboratory a "natural" situation of user mobility. From this point of view, mobility is primarily considered under its physical and awareness aspects. Such reductionism must address the traditional problem of the ecological validity of the collected data. To an actor, the act of movement implies more or less important context changes, not only from a physical point of view (lights, sounds, objects, etc.) but also from a socio-cultural one (places, people, situations, etc). Even if they do not have an automatic effect on the activity, these variations are potentially relevant and may have unpredictable effects on the users' course of action and consequently on the usage of the device. Moreover, the human-machine interaction itself generates its own context of achievement by a moment by moment dynamical management, focused on the achievement of a given goal (e.g. a search for information, writing messages, etc.)

To address the limitations of laboratory experiments when evaluating mobile devices a second approach has been explored that focuses on naturalistic studies. With this approach, the natural character of mobility, and consequently its context, is preserved as much as possible and what is lost in the control of the variables is gained in terms of ecological validity.

Some recent research has questioned the added value of naturalistic studies in the identification of usability problems compared to laboratory experiments, and has managed to address this issue in a systematic way (Kjeldskov et al., 2004). Some authors have found a trade-off between the two approaches, using the notions of "field experiments" [START_REF] Goodman | Using field experiments to evaluate mobile guides[END_REF]) and "quasi-experiments" [START_REF] Roto | Examining mobile phone use in the wild with quasi-experimentation[END_REF]. They attempt to keep a strict experimental control of the factors whilst at the same time preserving the realism of the usage.

Their approach is essentially quantitative and aims at explaining the causal relationships between factors. This follows the orthodox tradition of experimental psychology applied to HCI (testing hypotheses, controlling variables, and performing statistical tests).

However, one of the main challenges of the naturalistic approach is to define methods and equipment that enables to study empirically the use and the usability of mobile technologies in the wild. The purpose of this paper is to describe the approach that we adopted in a study held during summer 2004 at La Cité des Sciences of Paris-La Villette, which aimed to evaluate the usage of a mobile multimodal device1 . A PDA with a multimodal interface and a 802.11 wireless network card was used in a collaborative context where two users had to play a kind of quiz game (the goal was to discover clues in order to answer to a set of questions connected with different thematic spaces of the museum).

General principles

Our objective is to perform empirical studies of interaction with mobile multimodal devices , in natural settings, with an emphasis on the observation of real practices and on the analysis of meaningful action. Different relevant features of the situation have to be recorded: the user's actions and reactions, the interaction with the multimodal device, and the relevant aspects of the surrounding environment. Apart from the physical effects of motion on the activity, one of the main consequences of mobility is to induce rapid shifts in the context, that may have direct effects on the usage of a device. Thus, it is necessary to collect data related to the evolution of context, be it physical or social. However, all of the different contextual factors cannot be defined a priori, especially when users operate in a open environment. The collected data must therefore enable the experimenter to perform the analysis afterwards, based on a rich recording of activities.

This methodological orientation requires the equipment to be portable and to be sufficiently autonomous to enable the experimenter to follow the user in the different environments. In addition, the equipment should be as unobstrusive as possible (physically, psychologically, socially) for the user. A possible solution is to follow the user as closely as possible with a video recorder. Unfortunately, this method has many shortcomings. First, it may induce a feeling of discomfort, as it is not always socially pleasant to be followed and video-recorded in a public space. A second problem comes from the difficulties associated with recording simultaneously the user's actions and the multimodal device; the user can rapidly shift from one input modality to another, and in the same time it is necessary to keep a trace of the output modality. Therefore, the experimenter would need to switch continuously from the user to the display, selecting in realtime what is the meaningful information at a given moment. A third problem comes from the fact that it is not possible to define a priori the important factors in the dynamic surrounding context that can explain the user's actions. The experimenter has to make a choice between focusing on the manipulation of the device and losing the contextual aspects, or taking a broader view, which for instance could include interactions with other people [START_REF] Vom Lehn | Exhibiting interaction: conduct and collaboration in museums and galleries[END_REF], and missing the fine-grained sequences of actions on the device.

This particular issue exemplifies from a technical point of view the difference between two general traditions of observation: in one hand an approach drawing inspiration from sociology and ethnosciences which focuses on interactions with other actors in the same physical environment; and in another one the classical HCI approach, which mainly focuses on the interaction between a single user and a device.

Description of the recording set up

These different constraints led us to design a mobile recording unit which was physically unobtrusive and discreet and that enabled us to collect data on the user's actions and interaction with the device, and the context. This equipment includes three modules (figure 1):

∑ A micro-camera fixed on spectacles providing a so-called "subjective" view since it captures part of user's visual field. This view also captures a part of the interaction with the device and provides some information on the current context. ∑ A contextual view captured by an experimenter which accounts for the user's global context. ∑ A automatic logging tool, which captures all of the system's events and thus avoids a tiresome manual coding. These data are then enriched with the verbalisations of the users from the self-confrontation interviews (see section 3.4.); this arrangement of data allows a fine-grained analysis of the user's activity.

3.1

Contextual view

In order to identify afterwards the contextual features that make the user's actions meaningful to the analyst, it is crucial to capture the contextually relevant information from the user's point of view. This can be achieved by using cameras distributed in the environment at judicious locations (when the user moves in a restricted or known environment) or by using a video camera held by an experimenter.

A limitation with the fixed cameras is their lack of flexibility: there is one single viewpoint on the situation and it cannot be dynamically changed. On the other hand an experimenter following the user with a video camera can adjust in real-time the centring in order to have the more relevant viewpoint at a given time. A drawback with this approach is that different experimenters will not behave uniformly when following and recording the users. Ideally, it would be better to have the same experimenter recording all of the experimental sessions throughout the study. We selected the option of having an experimenter who follows the user at a distance in order to have a wide viewpoint of the situation. Contrary to other approaches (for example, [START_REF] Kjeldskov | New techniques for usability evaluation of mobile systems[END_REF]) the presence of the experimenter remains discreet and unobtrusive since there is no need to record the user's interaction with the device as these data are available through the subjective view and the automatic logging application (see section 3.3).

Subjective view

The capture of contextual information in the surrounding environment of the user is not sufficient, since it does not provide access to the user's viewpoint, noticeably when he/she pays attention to artefacts or manipulates them. Therefore we used a wearable micro-camera attached to spectacles [START_REF] Calvet | In the pocket": An empirical study of multimodal devices for mobile activities[END_REF] that enabled the collection of data about the immediate environment in the user's visual field, the interaction with the multimodal device, and more especially the manipulation of this device. This information completes the data collected via the automatic logger, and facilitates the analysis of the raw data in the log files. The micro-camera contains a microphone that provides sound information about the surrounding environment, specifically: the:vocal interaction with the speech recognition module of the multimodal application, the interaction with other agents, and the ambient environment in which the activity takes place. The audio-video data are directly recorded and compressed in DivX format in a blow on a portable hard-disk (Archos TM ). The user carries the hard disk and a set of batteries in a backpack.In a variation of this basic configuration, the audio-video stream is directly transmitted via a wireless emitter to a remote receiver (figure 4). This set-up was actually not used during the LUTIN experiments because of the ubiquitous metallic structures in the environment that generates interferences with the wireless system.

Logger

The subjective view is not always sufficient to capture all the events that happen on the display of the device. This may be because of mundane causes such as light reflections on the screen or because the user is looking at something or someone in the environment and subsequently the device is out of the scope of the camera. Moreover, coding the data related to the interaction with the device from the video-tapes is difficult and time-consuming. To cope with those limitations a logging application has been designed that automatically records the traces of the interaction with the device on a server in a nearby room over a 802.11 wireless local area network [START_REF] Merlin | Plateforme générique d'observation des interactions (Rapport Technique IRIT-DIAMANT)[END_REF].

The system also includes filtering and export functions that facilitate further data processing, facilities for selecting and formatting data, and functions for combining logged data with the audio and video data.

Replay and self-confrontation sessions

The contextual and subjective views provide an audiovisual support in order to conduct selfconfrontation interviews with the users (figure 2).

Figure 5. A self-confrontation session

The general idea behind the confrontation approach is to provide a subject or a group of subjects with traces of an activity (e.g. writings, schemas, annotations, data from the automatic logging of actions on a computer system, and more usually audio and video recordings) in order to collect verbal comments. In the same time external traces enable the analyst to match the verbal report with the traces of activity. The so-called self-confrontation term applies when a subject is shown a recording of his/her own activity in order to put him/her in the context of a past experience.

In the LUTIN experiment, the digital videotape was shown to the subjects on a desktop computer running a customized version of the Actogram TM data analysis software2 [START_REF] Kerguelen | Quels outils concevoir pour aider au relevé d'observation sur le terrain? In Actes du XXXIIIe Congrès de la SELF[END_REF] which features video editing facilities.

Analysis

The audio-video data were coded by hand on the Actogram TM tool which enables the simultaneous viewing of multiple synchronized videotapes. The software provides a dedicated window that supports on-the-fly coding of information based on the categories of contextual information previously identified to perform the analysis (figure 6).

Different taxonomies of contextual information have been used in recent works on usability testing for mobile and handheld devices (see for example [START_REF] Hoyoung | An empirical study of use contexts and usability problems in mobile internet[END_REF][START_REF] Roto | Examining mobile phone use in the wild with quasi-experimentation[END_REF]. Most of them are essentially empirical and are not grounded in a core of theoretically defined concepts. The different events (including the users' actions) were coded in three categories in order to describe the contextual dimensions of the situated activities. We draw inspiration here from the distinction introduced by Quéré between "environment", "context" and "situation" [START_REF] Quéré | La situation toujours négligée?[END_REF].

In our study the "environment" refers to the surrounding physical properties (light, noise, physical arrangement of the space,) that may play a role in the selection and use of the available interactive modalities. These properties are identified and coded by the experimenter when repeatedly viewing the videotapes (contextual view). The "context" refers to the features that can give sense to a particular action: the type of activity that the user is engaged in at a given moment (e.g. interaction with the device, information acquisition, motion management, etc.), the historical dynamics of the user's course of action (e.g. incidental sequence, interaction episode with the other player, interaction with other actors, etc;). The viewing of the tapes, combined with information from the log files, is used to code these contextual elements. The "situation" addresses directly the experience of the user and subsequently the dynamics of his/her commitment in action at a given moment. This level of analysis, which requires a description of the activity from the user's point of view, is based on the self-confrontation reports.

Conclusion

A big challenge for empirical studies of mobile activities is to go beyond the punctual usability evaluation of a technological device conducted during a limited period of time. Even though some usability problems can be quickly identified in a one hour experiment, a real understanding of the usage of a technological device and how it reshapes an activity requires covering a wide range of user experiences in real world situations. However, in order to address this appropriation process i.e. the process through which the artefact is adopted and transformed [START_REF] Carroll | Identity, power and fragmentation in cyberspace: technology appropriation by young people[END_REF], it is necessary to conduct longitudinal studies. Therefore, as it becomes impossible in practice to track and observe users continuously during long periods of time, we need to extend the scope of methods that we use to study mobile activities. Methods such as diary reports [START_REF] Palen | Welcome to the Wireless World: Problems Using & Understanding Mobile Telephony[END_REF] and interviews supported by the traces of the user's activity (provided for example by automatic logging) are natural candidates. Nevertheless it is still important to combine these approaches with the direct observation of the real activities of users.

Figure 1 .

 1 Figure 1. Mobile recording set up.

Figure 2 .

 2 Figure 2. Unobtrusively shadowing the user : the experimenter follows the user with a camcorder.

Figure 3 .

 3 Figure 3. User's "subjective" viewpoint.

Figure 4 .

 4 Figure 4. The wireless video-recording system (which actually still requires some wires between the handset and the wireless emitter…).

Figure 6 .

 6 Figure 6. The Actogram TM data analysis tool.

This study was conducted in the context of the LUTIN project (http://www.lutin.utc.fr)

http://www.actogram.net/

Acknowledgments

This work was funded by RNRT through the LUTIN project (usability lab at the Cité des Sciences) and France Telecom R&D through the MOTIVE project (multimodal interaction and mobility). Thanks to Julie Dugdale for helpul comments.