Espen Gaarder
email: espenhaug@mac.com

Eugene Terry Tatum

Solving the Hubble Tension by Extracting Current CMB Temperature from the Union2 Supernova Database

Keywords: Hubble tension, Hubble constant, Cosmological red-shift, z, CMB temperature

We will demonstrate that the formula proposed by Tatum et al. for the Cosmic Microwave Background (CMB) temperature: T 0 = ~c 4⇡k b p R H 2lp in relation to observed (and theoretical) redshift fits the well-known formula T t = T 0 (1 + z) , but only when we have = 0 or = 1 2 . If one one chooses = 0, it seems to imply that one must also accept the redshift scaling of: z = p R H p Rt 1 = Tt T 0 1. On the other hand, if one chooses = 1 2 , then it allows for the scaling of z = R H Rt 1 = T 2 t T 2 0 1. Only careful further empirical and theoretical studies can likely clarify whether it is best to set = 0 or = 1

2 , especially as one should investigate this for di↵erent R H = ct models, particularly those related to growing black hole models. Additionally, it is important to consider the compatibility of these models with the ⇤-CDM model.

 in combination with redshifts from all 580 supernovae. Furthermore, we get an estimated Hubble constant value of H 0 = 66.8945 when starting out with the CMB temperature measured by Fixen et al. (2009) [2]. In summary, our simple redshift model essentially perfectly matches all 580 type Ia supernova redshifts while extracting H 0 or T 0 values. The Hubble constant we extract by this method fits very well with the Hubble constant extracted from the CMB temperature by other studies. We believe that our new approach likely solves the Hubble tension problem.

1 The CMB temperature prediction formula Tatum et al. [START_REF] Tatum | The basics of flat space cosmology[END_REF] presented the following formula for the Cosmic Microwave Background (CMB) temperature in 2015:

T CMB,0 = ~c3 k b 8⇡G p M H m p = ~c k b 4⇡ p R H 2l p ⇡ 2.72 +0.082 0.069 , K (1)
We can calculate this CMB temperature when using the current 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. Symbols m p and l p are respectively the Planck mass and the Planck length [START_REF] Planck | Der Königlich Preussischen Akademie Der Wissenschaften[END_REF][START_REF] Planck | Vorlesungen über die Theorie der Wärmestrahlung[END_REF], k b is the Boltzmann constant, ~is the reduced Planck constant and G is Newton's gravitational constant. M H and R H represent the current Hubble mass and current Hubble radius, respectively. The current value R H = c Ht one can get by using the current CMB-determined Hubble constant H 0 , and the current mass inside the Hubble sphere of the model we will describe as the Friedmann critical mass M H = c 2 R H 2G ; see [START_REF] Tatum | The basics of flat space cosmology[END_REF] for more in-depth discussion. This formula is quite similar to the Hawking [START_REF] Hawking | Black hole explosions[END_REF][START_REF] Hawking | Black holes and thermodynamics[END_REF] temperature formula: T Hw = ~c3 k b 8⇡GM , except that one replaces M with p M H m p . Despite its likely great potential significance, this formula has received little attention from the wider astrophysics community. The likely reasons are that it has not been published in one of the more prestigious established journals, and secondly, until recently, there have been no papers providing strong mathematical or other proofs of its foundation.

However, recently, Haug and Wojnow [START_REF] Haug | Planck, and Hubble scale relations consistent with recent quantization of general relativity theory[END_REF][START_REF] Haug | How to predict the temperature of the CMB directly using the Hubble parameter and the Planck scale using the Stefan-Boltzman law[END_REF] have shown that formula (1) can be derived from the Stefan-Boltzmann [START_REF] Stefan | Über die beziehung zwischen der wärmestrahlung und der temperatur[END_REF][START_REF] Boltzmann | Ableitung des Stefanschen gesetzes, betre↵end die abhängigkeit der wärmestrahlung von der temperatur aus der electromagnetischen lichttheori[END_REF] law. Furthermore, Haug and Tatum [START_REF] Haug | The Hawking Hubble temperature as a minimum temperature, the Planck temperature as a maximum temperature and the CMB temperature as their geometric mean temperature[END_REF] have derived the same formula from a geometric-mathematical approach, demonstrating its consistency with a geometric mean temperature between the lowest and highest possible current temperatures in the Hubble sphere. All of these approaches seem to create a consistent and interesting framework, particularly in line with growing black hole models such as the FSC Schwarzschild metric model, but likely also other growing black hole models that can be built around other metrics, such as the Kerr [START_REF] Kerr | Gravitational field of a spinning mass as an example of algebraically special metrics[END_REF], 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] or the recent Haug-Spavieri [START_REF] Haug | Mass-charge metric in curved spacetime[END_REF] metric.

An actively discussed class of cosmological models, in comparison to the ⇤-CDM model, is the so-called R H = ct models. See, for example, [START_REF] John | r h = ct and the eternal coasting cosmological model[END_REF][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 without inflation[END_REF][START_REF] Melia | The linear growth of structure in the r h = ct universe[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]. The FSC model, originating with the referenced Tatum et al. 2015 paper, falls within the category of growing black hole R H = ct models, which is a subclass of R H = ct models. Thus, there is also thermodynamics related to the current and past CMB temperature in this cosmological model. A generalized version of formula (1) can be:

T t = T CMB (t) = ~c k b 4⇡ p R t 2l p (2)
where R t is the black hole radius at any stage in the growing black hole universe. In this model the universe starts out with a Planck mass Schwarzschild radius of R s = 2Gmp c 2 = 2l p and then expands one Planck length in radius per Planck time. Thus, one can also say that it increases by one-half a Planck mass per Planck time.

Cosmological red-shift from CMB temperatures

In general, for many cosmological models, we have:

z = R(t 0 ) R(t) 1 (3)
where R(t) is the scale factor dependent on the cosmological model, and t 0 is the reference time, which is now. In the ⇤-CDM model, light is red-shifted based on the idea of the expansion of space and cosmic time. The redshifted wavelength can be treated as stretching as space-time expands. In R H = ct models, there is expansion of space and space-time corresponding to a constant horizon speed of light c. In such models, one can possibly also mathematically treat the redshift as a relativistic Doppler redshift in a pre-existing space, as outlined in reference [START_REF] Tatum | The basics of flat space cosmology[END_REF]. In the FSC Tatum et al. model, we have

R 0 = R(t 0 ) = c H 0 = R H , which is simply the Hubble radius R H = c
H 0 at present. The Hubble radius is the distance light has traveled since the beginning of the black hole universe, which in the FSC model started as a Planck mass black hole and today is the Hubble sphere with mass equal to the mass in the critical Friedmann [START_REF] Friedmann | Über die krüng des raumes[END_REF] universe

M H = M c = c 2 R H 2G .
This means that, in R H = ct cosmological models, and particularly growing black hole models, we can have:

1 + z = a now a then = R H R t = ✓ ~c k b 4⇡ p Rt2lp ◆ 2 ✓ ~c k b 4⇡ p R H 2lp ◆ 2 = T 2 t T 2 0 ( 4
)
where R H = c H 0 and R t = c Ht . Alternatively, we could write this as:

z = observed emitted emitted = R H R t R t = R H R t 1 = ✓ ~c k b 4⇡ p Rt2lp ◆ 2 ✓ ~c k b 4⇡ p R H 2lp ◆ 2 1 = T 2 t T 2 0 1 (5)
Solving for T t gives:

T t = T 0 (1 + z) = T 0 (1 + z) 1 2 (6)
In the current paper we provide some detail about its derivation and additional discussion. Tatum and Seshavatharam have been aware that the formula T t = T 0 (1 + z) is also likely valid, and have used it in a recent paper [START_REF] Tatum | How the flat space cosmology model correlates the recombination CMB temperature of 3000K with a redshift of 1100[END_REF]. But then this has some potential implications, as discussed in the section below.

It is also worth looking at the R t distance as a function of z and H 0 in the following way:

z = R H R t 1 R t = c H 0 (1 + z) (7)
So, the distance to the observed redshift is then

R H R t = R H c H 0 (1 + z) R H R t = c H 0 c H 0 (1 + z) R H R t = c H 0 ✓ 1 1 (1 + z) ◆ (8)
and solved for H 0 we have:

H 0 = c d ✓ 1 1 (1 + z) ◆ (9)
where d = R H R t is the estimated distance to the object emitting the photons.

When z ⌧ 1 we can use the first term of the Taylor series expansion which is:

R H R t ⇡ cz H 0 (10)
and naturally

z ⇡ dH 0 c (11)
where d = R H R t is the distance to the object emitting the photon, that is identical to the standard ⇤-CDM cosmological redshift prediction formula typically used for low z values .

3 Comparison of T t = T 0 p 1 + z versus T t = T 0 (1 + z)
The ⇤-CDM model has, for many years, been using the following redshift formula:

T t = T 0 (1 + z) (12)
As there has been considerable uncertainty as to whether this really is the best model of CMB temperature versus cosmological red shift z, Lima et al. [START_REF] Lima | Is the radiation temperature±redshift relation of the standard cosmology in accordance with the data[END_REF] in 2000 suggested the following generalization of the formula:

T t = T 0 (1 + z) 1 (13)
where is an unknown constant that, if set to zero, yields the standard formula T t = T 0 (1+z); however, it has also been suggested that can be other than zero. Chluba [START_REF] Chluba | Tests of the CMB temperature-redshift relation, CMB spectral distortions and why adiabatic photon production is hard[END_REF] has suggested that: "decay of vacuum energy leads to 'adiabatic' photon production (or destruction), such that the CMB temperature scales like T t = T 0 (1 + z) 1 ." Research based on observations suggests that should be close to zero. For example, see [START_REF] Riechers | Microwave background temperature at a redshift of 6.34 from H 2 O absorption[END_REF]. However, this is still uncertain, as measuring CMB temperatures at high redshifts (z) is subject to significant uncertainties. Additionally, measuring cosmological redshift can be challenging because there are no direct observations of the emitted photons; we can only observe the received photons. This means that cosmological redshift is always seen through the lens of a mathematical model, and it is never entirely free of assumptions. On the other hand, measuring phenomena such as Doppler redshift or gravitational redshift on Earth provides much better control because we can directly compare the wavelengths of the emitted and received photons with minimal reliance on model assumptions. We mention this because one can then easily imagine how much more di cult it must be to know the exact cosmological redshift, as one can never have a direct observation of the emitted photon wavelength. Based on knowledge of 'standardzed' objects as well as decades of impressive research in cosmology, one can make an educated guess; nevertheless, there could still remain some uncertainties.

If we set = 0 then we can "only" 1 make T t = T 0 (1 + z) compatible with CMB equations (1) and (2) in the following manner:

z = observed emitted emitted = p R H p R t p R t = r R H R t 1 = ✓ ~c k b 4⇡ p Rt2lp ◆ ✓ ~c k b 4⇡ p R H 2lp ◆ 1 = T t T 0 1 (14)
In that case T t = T 0 (1 + z) would give the same result as in the previous sections, except we now also must have z

= p R H p Rt
1 which is an alternative possibility that should be carefully investigated, as this potentially could have an impact on how one interprets the need for space expansion in accordance with the ⇤-CDM and R H = ct models.

Riechers et al. [START_REF] Riechers | Microwave background temperature at a redshift of 6.34 from H 2 O absorption[END_REF] have reported the Cosmic Microwave Background (CMB) temperature from the cosmic epoch at z = 6.34, resulting in a temperature range of 16.4 30.2K within one standard deviation uncertainty. In other words, even the one standard deviation, which represents only about 68% probability for the CMB temperature to be inside that range, is very wide. The two standard deviation CMB temperature range is much broader. Therefore, we can conclude that the formula T t = T 0 (1 + z) is not su ciently well-tested, given the very large uncertainty. The formula T t = T 0 p 1 + z predicts a CMB temperature of T t = 2.725K p 1 + 6.34 ⇡ 7.4K which is well inside the 95% confidence interval of the above report (using two standard deviations); however, as we have also shown, the FSC framework at present also appears to be consistent with T t = T 0 (1 + z). Only further investigation can help us to decide on the optimal choice, even if observations appear to currently favor = 0.

It is also worth looking at the R t distance as a function of z and H 0 in the following way:

z = r R H R t 1 (z + 1) 2 = R H R t R H = (1 + z) 2 R t R t = R H (1 + z) 2 R t = c H 0 (1 + z) 2 (15)
So, the distance to the observed redshift is then

R H R t = R H c H 0 (1 + z) 2 R H R t = c H 0 c H 0 (1 + z) 2 R H R t = c H 0 ✓ 1 1 (1 + z) 2 ◆ ( 16
)
and naturally:

H 0 = c ⇣ 1 1 (1+z) 2 ⌘ D ( 17
)
where D = R H R t , which is the distance between us and the object emitting the observed photons. When z ⌧ 1 we can use the first term of the first order Taylor series expansion and we get

D = R H R t ⇡ 2cz H 0 (18)
that naturally gives:

z ⇡ DH 0 2c (19)
and naturally

H 0 ⇡ 2zc D (20)
This is indeed di↵erent from the standard redshift formulation. However, in this case, the distance D will also di↵er from the distance predicted by the standard formula. Therefore, only careful further investigation can determine whether it is superior to the standard formula z ⇡ dH 0 c or not. First of all, it is important to be aware that we have:

H 0 ⇡ 2zc D = zc d , z ⇡ DH 0 2c = dH 0 c (21)
These two formulas give identical predictions for H 0 and z when dealing with short distances and low z. The '2' in the numerator is o↵set exactly by the fact that D = 2d in the denominator. This holds true when the distance is short (z ⌧ 1). Thus, when used to extract the Hubble constant from, for example, nearby supernovae, the estimated H 0 or estimated z are identical. It is the distances that di↵er in the two approaches. However, when dealing with supernovae or other objects far from us, the estimated Hubble constant values will be di↵erent, as we then need to use the exact formulas, rather than convenient approximations. This implies that in the standard ⇤-CDM model, the value of the Hubble constant based on redshifts far away is likely overestimated.

For redshift and Hubble constant analysis at short distances, we can say that the ⇤-CDM model likely has two errors that cancel each other out: it has the wrong distance and also the wrong formula, which o↵sets the error in the wrong distance. Therefore, its predictions for redshift z and H 0 at short distances will be correct and the same as in our model. However, at much larger distances, their model will not be fully accurate, which we believe has led to the Hubble tension problem. This issue is addressed with our new way of looking at redshifts, as will become clear from the remainder of our paper, particularly in the sections where we aim to resolve the Hubble tension, namely sections 4 and 5.

If this latter redshift formula is correct, then distances to astronomical objects based on redshift are likely o↵ by as much as a factor of 2. This could explain why the ⇤-CDM model must have an expansion of space faster than R h = ct models. Once again, we believe that our new approach and explanation likely resolves the Hubble tension problem.

If one is not deeply entrenched in the topic, it may be easy to assume that our redshift equation ( 18) must be incorrect, given the highly precise measurements of distances to various astronomical objects through independent methods, such as parallax. However, parallax is not applicable to supernovae. Type Ia supernovae represent some of our most reliable standardized candles. Therefore, the most accurate method of determining their distance is likely through cosmological redshift, albeit this approach is naturally model-dependent. While our math, which suggests that the distance to the more remote supernovae is approximately twice that predicted by the ⇤-CDM model, may initially seem unfeasible, based on our current limited knowledge, it should not be immediately discounted. We would welcome any compelling arguments against our findings, in particular after studying the rest of our paper.

Figure 1 shows the predicted distances for 580 supernovas from the Union2 database. We have sorted the supernova database based on redshift, so supernova 1 has the lowest redshift and supernova 580 has the highest. Then we have predicted the distances based on the two approximation formulas. The two approximation formulas are based on the first term of the Taylor series expansion and are, in reality, only good approximations when z ⌧ 1. This is why these approximations strongly overestimate the distance for high-z supernovae. The two exact solutions, based on the assumptions given in this paper, yield di↵erent values. The one consistent with T t = T 0 (1 + z) predicts that the more remote supernovae are about twice as far away from us as the ⇤-CDM model predicts. The one consistent with T t = T 0 (1 + z) 1/2 gives the same distance predictions as the standard model, at least for z ⌧ 1, but then the relationship between the current CMB temperature and that in past cosmic epochs is di↵erent from that in the ⇤-CDM model. We need the current value of H 0 to make the distance approximations, which we take from the very recent study of Sneppen et. al [START_REF] Sneppen | Measuring the hubble constant with kilonovae using the expanding photosphere method[END_REF], H 0 = 67 ± 3.6 km/s/M pc. In Figure 2 we have removed the approximation models and we now focus on those models which are relatively more accurate at higher redshifts. This makes more obvious the di↵erence between the model consistent with T t = T 0 (1 + z) 1/2 and the model consistent with T t = T 0 (1 + z). Be aware that the ⇤-CDM model predicts the same distances as the blue line, at least for low z values. The red line prediction is what likely is correct, however, as it is consistent with T t = T 0 (1 + z), which we will show that observations seem to favor. This will become more clear in the next section. The green line represents D/d. We can see that, for low z quasars, the standard model is o↵ by a factor of almost 2, which also appears to be true for the higher observed z supernovae in the Union2 database. There is more explanation to follow. 4 Extracting the current CMB temperature from 580 type Ia supernovae Here we use the observed redshifts from 580 supernova data points in the Union2 database to determine the current CMB temperature (z = 0). The methodology employed is as follows: from the cosmological redshifts, we first predict the CMB temperatures for each observed z value, by using the standard and well-known relation T t = T 0 (1 + z). Since our goal is to find T 0 a priori, we start with a wild guess. For instance, we might start with T 0 = 4K, or even T 0 = 10K, which is naturally far o↵ from the currently observed CMB temperature of approximately 2.72K. We start with a wild guess because we will later use optimization to determine if there exists a T 0 that leads to an estimation error near zero, using the redshift prediction formula z = q R h Rt 1. This can be achieved using optimization algorithms such as the Newton-Raphson method or the bisection method.

Next, we calculate the radius of the Hubble sphere going back in time for each z value. This is done by assuming a FSC-like R h = ct cosmology and solving Eq. ( 2) for R t . This gives:

R t = ✓ ~c T t k b 4⇡ ◆ 2 1 2l p (22)
and since we also have T t = T 0 (1 + z) we can replace T t with this and get:

R t = ✓ ~c T 0 (1 + z)k b 4⇡ ◆ 2 1 2l p (23)
Because we are assuming that we do not know T 0 , we will not rely upon measured CMB temperatures. However we can now input this expression for R t into our red-shift formula (Eq. 14) and we get:

z pre = r R h R t 1 = v u u t R h ⇣ ~c T 0 (1+z obs )k b 4⇡ ⌘ 2 1 2lp 1 ( 24
)
We can now minimize the errors between z pred and z obs by adjusting the unknown T 0 value. This can be accomplished through pure trial-and-error, or more e ciently by using "intelligent" trial-and-error systems such as the Newton-Raphson method or the bisection method. The results from these approaches are the same, except that, by naive trial-and-error, one will waste more time finding the optimal CMB temperature. To do this, we also need R h = c H 0 , which is the current Hubble radius, and therefore, we also need the current value of H 0 , which we take from the very recent study of Sneppen et. al [START_REF] Sneppen | Measuring the hubble constant with kilonovae using the expanding photosphere method[END_REF], H 0 = 67 ± 3.6 km/s/M pc. Thus, H 0 a↵ects the value; it is actually this relatively new theoretical relationship between H 0 and CMB, first implied by Tatum et al. [START_REF] Tatum | The basics of flat space cosmology[END_REF] in 2015 and later proved to be derivable from the Stefan-Boltzmann law [START_REF] Haug | Planck, and Hubble scale relations consistent with recent quantization of general relativity theory[END_REF][START_REF] Haug | How to predict the temperature of the CMB directly using the Hubble parameter and the Planck scale using the Stefan-Boltzman law[END_REF], that makes this method possible. We also need the Planck length in our formula, so we have used the NIST CODATA (2018) value of l p = 1.616255 ⇥ 10 35 m (with standard uncertainty of 0.000018 ⇥ 10 35 m).

Figure 3 best illustrates the trial-and-error procedure. Assume that we initially have guessed a CMB temperature of 4K. This is the green line in the figure, that we see is far above the observed redshift represented by the blue line. However, at least it looks like it correlates well. Still, the 4K prediction is way o↵. Given that the redshift predictions are proportional to the current CMB temperature, we must guess a lower CMB temperature. Assume that we now guess T 0 = 1.5K; we then get the predictions presented by the purple line. It becomes obvious that our 1.5K prediction is now too low compared to the observed redshifts. We now know that the CMB temperature needed in order to minimize the prediction errors must be between 4K and 1.5K. We, therefore, now guess 3K, and the redshift predictions we get from this are much closer to observed, but still too high.

We can continue 'manually' like this, or we can resort to e cient search algorithms that are used for similar statistical problems in many scientific fields. Among the most commonly used algorithms are the Newton-Raphson method or the bisection method. One can also use the goal seek function in Excel, which is likely based on the bisection method. These trialand-error methods are simply a form of calibration method. The question is whether there exists a single CMB temperature, denoted as T 0 , for the current epoch of the cosmos that can make our "CMB redshift prediction formula" match the observed redshifts with precision.

E↵ectively, we are calibrating our new cosmological redshift prediction equation relative to the observed 580 supernova redshifts by finding what value of T 0 minimizes our prediction error. Only one parameter is adjusted (optimized), namely the "unknown" current CMB temperature, so that the errors indicated by the sum of P n i (z pred,i z obs,i ) are minimized. This approach results in a predicted current CMB temperature T 0 ⇡ 2.72763155978204 K from the 580 supernova observations in the Union2 database. We are not claiming that the current CMB temperature is exactly this (although it could be); rather, this is what it appears to be, based on the observed supernova redshifts in combination with the H 0 = 67 km/s/Mpc.

When we also take into account the uncertainty in the value of H 0 (67 ± 3.6 km/s/Mpc), as reported in the study of Sneppen et. al, we get a one standard deviation (STD) confidence interval of 2.6533 K to 2.7800 K for the current CMB temperature.

As di↵erent Hubble constant measurement studies and methods yield considerable uncertainty in H 0 , this uncertainty could be even larger. Moreover, there is the unexplained Hubble tension problem; see for example [START_REF] Reiss | Observational evidence from supernovae for an accelerating universe and a cosmological constant[END_REF][START_REF] Capozziello | Addressing the cosmological H 0 tension by the Heisenberg uncertainty[END_REF][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]. Nevertheless, using our new radically-di↵erent theoretical approach, we can closely approximate the recent CMB temperature observations [START_REF] Dhal | Calculation of cosmic microwave background radiation parameters using cobe/firas dataset[END_REF][START_REF] Fixsen | The temperature of the cosmic microwave bacground[END_REF][START_REF] Fixsen G | The temperature of the cosmic microwave background at 10 GHz[END_REF][START_REF] Noterdaeme | The evolution of the cosmic microwave background temperature[END_REF].

By using a current CMB temperature T 0 ⇡ 2.72763 ± 0.072 0.074 K, the predictions from Eq. ( 24) are now perfectly aligned with the observational blue line in Figure 1. In other words, the redshift formula we have presented is now capable of matching observed cosmological redshifts. See Figure 4. In addition, it shows the predicted redshifts for various qualified guesses on the current CMB temperature. We note that the predictions and observations are highly correlated, but they do not match. The predicted redshifts are too high when using CMB temperatures 4K and 3K and too low when using 1.5K. However, by using e cient trial-and-error algorithms, we can quickly find the current CMB temperature that gives the best fit, as shown in the next figure. See Figure 4.

It is interesting that, by incorporating only the current CMB temperature, our redshift prediction function can match observations. It is important to note that we are using the same epochal T 0 for all supernovae, rather than di↵erent T 0 values for di↵erent supernovae. This basically demonstrates that our framework is consistent and robust. Our findings are fully consistent with the empirically-tested relation T t = T 0 (1 + z) and also with R h = ct and assuming a current CMB temperature of 2.72763. We find that there is a match between observed and predicted redshifts for this particular current CMB temperature. cosmology. It is our recommendation that the astrophysics community pay attention to this and investigate to what degree it is or is not consistent with the ⇤-CDM model. Could it be that the ⇤-CDM model needs further adjustments? Or could it be that R h = ct models are actually more realistic in some respects? Only further research can settle these questions. In either case, researchers should be made aware of the recent progress in our theoretical understanding of CMB temperature and its relationship to cosmological redshift and the Hubble constant.

Why we think that we may have solved the Hubble tension problem

In the section above, where we found the CMB temperature that optimally fits with the observed supernova redshifts, we had to know the Planck-derived Hubble constant value. Alternatively, one can use the observed CMB temperature to optimally fit a Hubble constant value for the same 580 supernova redshifts. The CMB temperature has been extremely accurately measured in recent years, for example, by [START_REF] Dhal | Calculation of cosmic microwave background radiation parameters using cobe/firas dataset[END_REF][START_REF] Fixsen | The temperature of the cosmic microwave bacground[END_REF][START_REF] Fixsen G | The temperature of the cosmic microwave background at 10 GHz[END_REF][START_REF] Noterdaeme | The evolution of the cosmic microwave background temperature[END_REF].

First we will use the most recent measurement by Dhal et. al of T 0 ⇡ 2.725007K±0.0043K. We then start out by assuming that we know very little about the Hubble constant. All we know is that many di↵erent studies have arrived at di↵erent values. So we start with a qualified, but wild, guess that it must be in a range from 50 to 80 km/s/M pc. We then ask if there is a single Hubble constant (H 0 ) value that, when used in our new CMB redshift prediction formula, matches all of the observed supernova redshifts. So, we start with a guess of H 0 = 50 km/s/M pc and use the same formula as in the previous section:

z pre = r R h R t 1 = v u u t R h ⇣ ~c T 0 (1+z obs )k b 4⇡ ⌘ 2 1 2lp 1 (25)
The only di↵erence in our approach (compared to in the previous section) is that we now substitute the measured CMB temperature by Dhal et. al for T 0 . In the previous example, we assumed that we knew H 0 and tried to find T 0 . We can now use R h = c H 0 to solve for H 0 . The result is shown in Figure 5 as the red line. Once again, the actual observed redshifts for all 580 supernovae in the Union2 database are represented as the blue line. As we can see, the red line indicates that this was an underestimated Hubble constant value, as our redshift prediction formula is inversely related to H 0 . Accordingly, we next guess a much higher Hubble constant value, for example H 0 = 80 km/s/M pc, which gives redshift predictions correlating to the green line. Now the redshift predictions are too low. By simple trial-anderror, or by "intelligent" trial-and-error methods, such as the Newton-Raphson method or the bisection method, we can find the H 0 that minimizes the errors between predicted and observed redshifts. Figure 6 shows the end-result. We obtain H 0 = 66.8713 km/s/M pc (and a 1STD range of 66.859 to 66.882 km/s/M pc) extracted from all of the supernovae when linked to the Dahl et. al current CMB temperature. A similar approach is used for the Fixen et al (2009) CMB temperature of 2.72548K. See Figure 7.

Here, it is important to be aware that we are able to achieve essentially a perfect fit with all of the Union2 data base supernovae using either a single Hubble constant value or a single CMB temperature. This appears to us to solve the Hubble tension problem in favor of the Planck Collaboration Hubble constant determination. The Hubble constant value cannot be measured directly; it is the redshift that is measured in the type Ia supernova studies. The Hubble constant is estimated, and therefore relies on a model-based definition of redshift. In this paper, we use a new redshift model rooted in our new understanding of the inter-relationships between the current CMB temperature, the current Hubble constant, and cosmological redshifts; see [START_REF] Tatum | The basics of flat space cosmology[END_REF][START_REF] Haug | Planck, and Hubble scale relations consistent with recent quantization of general relativity theory[END_REF][START_REF] Tatum | Upsilon constants and their usefulness in Planck scale quantum cosmology[END_REF]. This approach is clearly consistent with R h = ct cosmological models. Whether it can also be made compatible with the ⇤-CDM model is too early to say. If so, we believe that the ⇤-CDM model, at minimum, would need some adjustments. Alternatively, it may be that R h = ct models are actually more realistic.

Our results are also fully in line with the recent findings of Tatum et al [START_REF] Tatum | High precision hubble constant determinations based upon a new theoretical relationship between cmb temperature and h 0 . Hal archive[END_REF] that one can accurately find H 0 from the current CMB temperature, something that the ⇤-CDM model cannot do at present, as far as we know. Here, we have taken an additional step forward and linked these two cosmological parameters to cosmological redshift. These inter-relationships are consistent with core principles in R h = ct cosmology in general and FSC in particular. 6 Has solving the Hubble tension provided us with new universal laws of cosmology? Figures 4, 6 and 7 almost seem too good to be true. There may be a tendency to react in such a way, but the math appears to indicate otherwise. One even wonders if there are 'new universal laws of cosmology' revealed by such an approach, because there appear to be exact direct relationships between the Hubble constant, cosmological redshift and the CMB temperature, without the need for concepts such as dark energy or space expanding faster than light.

Naturally, it is not for us to say whether we have discovered 'new universal laws.' But we do wonder about this. Clearly, we have not simply found broad statistical correlations between these parameters, but rather very precise mathematical inter-relationships between them. If one knows any two of these cosmological parameters with high precision, one can automatically solve for the other with high precision. To take but one example, if one knows the CMB temperature and the Hubble constant, one can precisely estimate the cosmological redshift for a given cosmological distance. There is no wiggle room. We find that a convenient way to illustrate this is in the figure below.

The formula in the middle of the triangle demonstrates that the Hubble constant and the CMB temperature are simply connected by a composite constant that we have called upsilon (the Latin symbol for upsilon2 ):

⌦ = k 2 b 32⇡ 2 G 1/2 c 5/2 ~3/2 .
The equations above and their direct relations with each other can explain much of the cosmos; they all seem to be consistent with a new way to quantize general relativity theory [START_REF] Tatum | Upsilon constants and their usefulness in Planck scale quantum 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] and therefore fully consistent with Einstein's general relativity theory, see [START_REF] Haug | Planck, and Hubble scale relations consistent with recent quantization of general relativity theory[END_REF]. When we ponder these inter-relationships, we are left asking if the Lambda-CDM model can accommodate them, or whether they are appropriate for a new paradigm represented by R H = ct models in general, and the FSC model in particular. We leave this question for others to ponder as well.

Conclusion

In the context of growing black hole cosmology models, such as R H = ct, we now have a robust theoretical framework for predicting the Cosmic Microwave Background (CMB) temperature, using a known Hubble constant value [START_REF] Haug | How to predict the temperature of the CMB directly using the Hubble parameter and the Planck scale using the Stefan-Boltzman law[END_REF], or vice versa [START_REF] Tatum | The basics of flat space cosmology[END_REF]. This can be done both for present and past cosmic epochs (presumably), as well as in relation to cosmological redshifts. This framework initially emerged from the FSC cosmological model proposed by Tatum et al. However, it is worth exploring whether it can also be applied to other black hole R H = ct models. The theory appears to be compatible with the relationship T t = T 0 (1 + z) , but only when = 0 or = 1 2 . Although it appears that we are gaining a new perspective on understanding and investigating cosmological redshift, further theoretical and observational studies are needed in order to reach a final consensus. There is still much to be discovered.

In this paper, we have demonstrated, from the observed redshifts of 580 type Ia supernovae (the Union2 database), how one can predict a CMB temperature of 2.72763 ± 0.072 0.074 K. Furthermore, we predict that future type Ia supernovae will show a similar excellent fit with a current CMB temperature of 2.72763K. Given its derivation, our CMB redshift prediction formula is, of course, also fully consistent with the T t = T 0 (1 + z) formula, so long as is zero.

We have also demonstrated how to achieve a perfect match between particular Hubble constant values and observed supernova redshifts, simply by incorporating the observed CMB temperatures from two well-known CMB studies. We refined our predictions by minimizing errors in our redshift prediction formula relative to the observed redshifts from the same 580 type Ia supernovae. Utilizing the CMB temperature from Dhal et. al (2023) of T 0 = 2.725007K, we derive a Hubble constant value from the full supernova database of H 0 = 66.8713 km/s/M pc. Similarly, utilizing the CMB temperature from Fixen et al. (2009) of T 0 = 2.72548K, we obtain a Hubble constant value from the full supernova database of H 0 = 66.8945 km/s/M pc. Remarkably, our redshift formula can accurately predict the redshift of every supernova in the Union2 supernova database after calibration for distance. While this process resembles curve-fitting, the only free parameters are either the Hubble constant or the CMB temperature, with one being derived from measurement and the other automatically determined as a result. These results are consistent with the predictive power mentioned in references [START_REF] Tatum | Upsilon constants and their usefulness in Planck scale quantum 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]. Accordingly, we believe that our new approach has possibly resolved the Hubble tension problem in favor of the Planck-based CMB-derived Hubble constant value. Nevertheless, our work should undergo careful scrutiny by others before drawing any firm conclusions.

Figure 1 :

 1 Figure 1: This figure shows the predicted supernova distances for di↵erent redshift models. Two of the models employ first series Taylor aproximations and only give reasonably accurate solutions when z ⌧ 1.

Figure 2 :

 2 Figure 2: This figure also shows the predicted supernova distances for the red and blue models in somewhat greater detail, and in comparison to the D/d ratio.

Figure 3 :

 3 Figure 3: This figure shows observed redshift values from 580 type Ia supernovae, sorted by redshift.In addition, it shows the predicted redshifts for various qualified guesses on the current CMB temperature. We note that the predictions and observations are highly correlated, but they do not match. The predicted redshifts are too high when using CMB temperatures 4K and 3K and too low when using 1.5K. However, by using e cient trial-and-error algorithms, we can quickly find the current CMB temperature that gives the best fit, as shown in the next figure. See Figure4.

Figure 4 :

 4 Figure 4: This figure shows observed redshift values from 580 type Ia supernovae, sorted by redshift, and assuming a current CMB temperature of 2.72763. We find that there is a match between observed and predicted redshifts for this particular current CMB temperature.

Figure 5 :

 5 Figure 5: This figure shows observed redshift values from 580 type Ia supernovae, sorted by redshift, and based on the measured CMB temperature by Dhal et. al (2023) of 2.72507K. The blue line represents observed redshifts, the red line represents predictions based on a wild guess of H 0 = 50 km/s/M pc, and the green line represents redshift predictions based on a wild guess of H 0 = 80 km/s/M pc.

Figure 6 :

 6 Figure 6: This figure shows observed redshift values from 580 type Ia supernovae, sorted by redshift, and based on the measured CMB temperature by Dhal et. al (2023) of 2.72507K. The blue line represents observed redshifts, and the red line represents predictions based on H 0 = 66.8713 km/s/M pc, which we extracted from the data using our trial-and-error approach.

Figure 7 :

 7 Figure 7: This figure shows observed redshift values from 580 type Ia supernovae, sorted by redshift, and based on the measured CMB temperature by Fixen et al (2009) of 2.72548K. The blue line represents observed redshifts, and the red line represents predictions based on H 0 = 66.8945 km/s/M pc, which we extracted from the data using our trial-and-error approach. Remarkably, this H 0 value was predicted in 2015 by Tatum et al. [3].

There could naturally be additional methods not discovered here, for example based on new cosmology or other metrics.

In the first paper about this formula, we used the Greek symbol for upsilon: ⌥

Data availability statements

The supernova Union-2 database that we have used can be found here: https://supernova.lbl.gov/Union/figures/SCPUnion2.1_mu_vs_z.txt.

Conflict of interest

The authors declare no conflict of interest.


Tom Pégeot
email: tom.pegeot@cea.fr

Inna Kucher
email: inna.kucher@cea.fr

Adrian Popescu
email: adrian.popescu@cea.fr

A Comprehensive Study of Transfer Learning under Constraints

Pre-training on an upstream task is widely used in deep learning to boost performance of downstream tasks. Recent studies analyzed pre-training with large datasets and large deep neural network architectures. However, pretraining is very useful in practice when downstream tasks have scarce data and are trained under computational constraints. To assess pre-training performance in this setting, we train different deep architectures with 1M parameters. We create different subsets of ImageNet to study the influence of upstream dataset in detail by varying the total size, but also the ratio between number of classes and samples per class for a constant total size. Then, we use the resulting models in transfer toward six diversified downstream tasks using linear probing and full fine tuning for downstream training. Experimental results confirm previous ones regarding performance saturation in downstream tasks, but we find that saturation occurs faster for compact deep architectures. The use of different ImageNet subsets leads to globally similar performance when enough data is included, regardless of the dataset structure. The comparison of downstream training strategies shows that linear probing can be competitive, particularly for few-shot settings. This is at odds with previous reports, which assert the superiority of full fine tuning. Finally, we observe that the type of deep architecture has a significant effect on results, but that their relative performance varies depending on the downstream training strategy.

Introduction

Deep neural networks are known to be data hungry [START_REF] Marcus | Deep learning: A critical appraisal[END_REF], particularly when they include a large number of parameters. Transfer learning alleviates this problem by pre-training an upstream dataset to improve performance in downstream task, or accelerate the training process [29,[START_REF] Evci | Head2Toe: Utilizing intermediate representations for better transfer learning[END_REF]. Pre-training is also useful when the target domain data are not sufficient to learn an effective model from scratch, and the gain obtained from the upstream model is larger than the loss of representativeness due to domain shift [START_REF] Sinno | A survey on transfer learning[END_REF]. The importance of pre-training grew with the advent of deep neural networks, whose learned representations are transferable [29]. Recent studies of transfer learning [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF]5,[START_REF] Goyal | Self-supervised pretraining of visual features in the wild[END_REF][START_REF] Vincent Dumoulin | Comparing transfer and meta learning approaches on a unified few-shot classification benchmark[END_REF][START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF] focused on pre-training models with increasingly large number of parameters and amounts of data. They conclude that increasing the size of models and of data improves the performance in target tasks, at least until saturation is reached [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF].

While interesting, these studies disregard the fact that transfer learning is often useful when training and inference capacity are limited [13]. In this work, we investigate transferability under constraints by analyzing the effects of core factors which drive this process. During pre-training, we notably test the influence of: (1) pre-training for compact deep architectures, which are likely to be used in transfer learning for constrained environments [START_REF] Xu | Scaling for edge inference of deep neural networks[END_REF]; (2) deep neural network architectures since they are known to influence both the upstream and downstream performance [14];

(3) the amount of available training data, as well the ratio between number of classes and samples per class for a fixed-size upstream dataset, since downstream accuracy saturation was already analyzed for large deep architectures [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF][START_REF] El-Nouby | Are large-scale datasets necessary for self-supervised pre-training?[END_REF][START_REF] Kornblith | Do better imagenet models transfer better[END_REF], but not for compact ones. During inference, we analyze the influence of: [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF] the type of downstream training strategy, with the deployment of linear probing and full fine tuning, since the depth of the fine tuning process leads determines the degree to which features are adapted to the downstream task or preserved from the upstream model [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF]; (2) the number of images per class in the target datasets to assess transferability in four few-shot settings [5] and full dataset availability scenario since they are all important in practice.

We run experiments using different subsets of Ima-geNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] as upstream dataset, four deep architectures, and with six downstream datasets designed for diverse visual tasks. The empirical study reported here concludes that:

• Downstream performance saturation is reached much faster with the compact deep architectures compared to the large architectures analyzed in previous studies [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF][START_REF] Kornblith | Do better imagenet models transfer better[END_REF]. This finding indicates that very large pretraining datasets are not needed to obtain good downstream performance with compact deep architectures.

• The structure of the upstream dataset (number of classes, samples per class) has a small influence on downstream accuracy once there are enough data in it.

• The type of architecture makes a difference, particularly when linear probing is used for downstream training. In this setting, architectures with higherdimensional output features are clearly a better choice.

• The performance of the full fine tuning and of linear probing depends on the downstream configuration. The latter strategy is competitive when the domain shift between upstream and downstream tasks is small and/or in case of low-shot settings. Since linear probing training is much simpler, it should be considered for deployment in these cases.

As a whole, the reported results give a comprehensive view of transfer learning under constraints. They provide a sound baseline for future work performed by both researchers and engineers.

Related Work

Transfer learning is important for practical applications of deep learning, and is the subject of a large number of existing studies. We discuss the most relevant studies for transfer learning under constraints, which is in focus here. Prior works are further put into perspective when analyzing the results of the different experiments.

Past examination of pre-training tend to show that increasing the size of the upstream dataset has a positive effect on downstream accuracy [20,[START_REF] Sun | Revisiting unreasonable effectiveness of data in deep learning era[END_REF]38]. However, recent studies, such as [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF][START_REF] El-Nouby | Are large-scale datasets necessary for self-supervised pre-training?[END_REF], find that the improvement tends to saturate, and this phenomenon occurs faster for selfsupervised pre-training. The works cited above focus on the total size of the dataset in terms of samples, and they give less importance to the structure of the dataset in terms of the number of classes and of samples per class. The importance of the dataset structure was highlighted for domain adaptive transfer learning [START_REF] Ngiam | Domain adaptive transfer learning with specialist models[END_REF]. The authors of this study conclude that adding more data, including more classes, can have a deleterious effect on downstream performance. In this study, the pre-training has the prior knowledge about the target task. In contrast, we pre-train models without any assumption regarding the content of downstream tasks in order to avoid meta-overfitting [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF].

The strategy used for downstream training has a strong influence on performance. Past studies [START_REF] Kornblith | Do better imagenet models transfer better[END_REF][START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF] tested pretraining for full downstream datasets, but also in few-shot learning settings. They showed that full fine tuning of downstream models is better than linear probing, which consists in retraining only the final fully-connected layer of the model. This finding seems intuitive since fine tuning adapts the features of downstream models to the characteristics of the downstream tasks. A nuance was brought by [START_REF] Kumar | Fine-tuning can distort pretrained features and underperform out-of-distribution[END_REF], a study which shows that linear probing is actually better than fine tuning when testing with out-of-distribution data for downstream tasks. However, past results were reported for the pre-training with large deep models. It is interesting to study whether they hold for smaller models, which are in focus here. Importantly, we run a more systematic study of few-shot settings compared to [START_REF] Kornblith | Do better imagenet models transfer better[END_REF][START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF] in order to have a fine-grained analysis of the merits and limitations of the two strategies. We note that there exist more refined transfer strategies. Image-level adaptation of the strategy is proposed in [START_REF] Guo | Spottune: transfer learning through adaptive fine-tuning[END_REF], adaptive fine tuning is explored in [START_REF] Guo | Adafilter: Adaptive filter fine-tuning for deep transfer learning[END_REF], while a combination of features from different layers is used in [START_REF] Evci | Head2Toe: Utilizing intermediate representations for better transfer learning[END_REF]. While interesting, they are out of the immediate scope of this work, which focuses on two opposite strategies.

Previous works focused on transfer learning for computationally-constrained devices showed the benefits of freezing part of the networks [33,[START_REF] Xie | Deep transferring quantization[END_REF]. However, they focused on hardware optimization [33] in order to reduce the overall energetic footprint of the implemented deep models, or architecture quantization [START_REF] Xie | Deep transferring quantization[END_REF] to reduce their parametric footprint. Here, we take a complementary approach and pay more attention to the upstream and downstream data, and use network scaling to preserve the precision of downstream representations.

Study Setup

Datasets

Pre-training datasets. Following the common practice [START_REF] Evci | Head2Toe: Utilizing intermediate representations for better transfer learning[END_REF][START_REF] Kornblith | Do better imagenet models transfer better[END_REF][START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF], we transfer data from a single upstream dataset to all downstream tasks in order to assess the generalization capacity of the upstream model. The authors of [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF] underline the importance of mitigating metaoverfitting when transferring knowledge. They advise to create the upstream model independently of any knowledge about downstream data. Therefore, we generate different versions of pre-training datasets by sampling Ima-geNet21k [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF]. The classes included in these datasets are selected randomly from the set of leaves classes that have enough samples per classes. A first series of tests use a variable number of classes from 100 to 6000 and fixes the number of 500 samples per class. These subsets are used to assess if downstream performance continues to increase or saturates when adding new classes. A second series of tests simultaneously vary the number of classes and samples to keep the total number of samples in the dataset constant. The size of the dataset is 1M images and the number of classes varies from 1000 to 6000. This experiment could not be carried out with fewer classes since ImageNet does not contain enough richly-represented leaf classes to reach the target dataset size. This setting corresponds to an upstream training on a fixed budget. These subsets are used to assess whether class diversity or individual class representations are more important. Note that there is no assumption made regarding the similarity between the upstream dataset and the downstream ones. This is important in order to simulate a situation in which pre-training is done without knowledge of the downstream tasks, and thus ensure the generalization of the proposed transfer scheme. Downstream datasets. A thorough evaluation of the usefulness of pre-training requires the use of multiple and diversified downstream datasets [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF]. We follow this observation and transfer upstream models toward six downstream tasks: Oxford-IIIT Pet [26] is designed for pet race recognition, Describable Textures Dataset (DTD) [2] provides different types of textures as perceived by humans, GTSRB [START_REF] Stallkamp | Man vs. computer: Benchmarking machine learning algorithms for traffic sign recognition[END_REF], Street View House Numbers (SVHN) [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF] includes house number images, FGVC-Aircraft (FGVC) [START_REF] Maji | Fine-grained visual classification of aircraft[END_REF] is designed for aircraft model recognition and Cifar100 [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] includes commonsense-level classes [START_REF] Rosch | Principles of categorization[END_REF]. These datasets cover a wide range of visual tasks, and the conclusions drawn from a study of pre-training involving all of them are robust. Their main statistics are presented in Table 1. Images are resized to match the input size used during pre-training which is 224x224. Standard data augmentation [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF][START_REF] He | Deep residual learning for image recognition[END_REF] which includes random cropping and random horizontal flipping is applied for four datasets out of six. Horizontal flipping is deactivated for GTSRB and SVHN because they mainly represent classes that depends on the orientation.

Downstream data availability

It is important to study the influence of the amount of data available for downstream tasks since pre-training is most needed when downstream data are scarce. We first run experiments with the full datasets, and then test with four few-shot learning regimes. For this we limit the number of samples per class in the downstream task to 1, 5, 10 and 25. This is a finer-grained investigation of the influence of data availability compared to [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF], where a single few-shot learning setting were used. To mitigate data selection bias, we follow a standard procedure in few-shot learning and sample training images five times for each regime.

Downstream training strategies.

Following [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF][START_REF] Kumar | Fine-tuning can distort pretrained features and underperform out-of-distribution[END_REF], we run experiments with fine tuning and linear probing, two opposite strategies. Fine tuning retrains all the layers of downstream models, while linear probing only retrains the final layer. Fine tuning is usually preferred [START_REF] Kornblith | Do better imagenet models transfer better[END_REF][START_REF] Kumar | Fine-tuning can distort pretrained features and underperform out-of-distribution[END_REF][START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF] since the full retraining of the downstream model adapts it to the domain of the downstream task. Linear probing [29] is less adaptive since it exploits pre-trained features as such. The latter can be interesting if the computational capacity of the device is limited [33] and/or when the amount of available training data is insufficient to learn a full model in an efficient manner. We note the existence of the other downstream training strategies [START_REF] Guo | Adafilter: Adaptive filter fine-tuning for deep transfer learning[END_REF][START_REF] Guo | Spottune: transfer learning through adaptive fine-tuning[END_REF][START_REF] Evci | Head2Toe: Utilizing intermediate representations for better transfer learning[END_REF], but their usage is out of the immediate scope here.

Training details

Training parametrization is done using a procedure which is inspired by the lightweight sweep mode proposed in [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF]. Fixed values are used for most hyperparameters across network architectures and tasks. While not fully optimized for each task, this mode allows a fair comparison in a constrained environment.

The resolution of the input images used for training was classically set to 224x224 [START_REF] He | Deep residual learning for image recognition[END_REF].

Upstream training. To make the results comparable we used the same hyper-parameters for each training. All of them were trained during 110 epochs using the "1cycle" learning rate scheduler [START_REF] Leslie | Super-convergence: Very fast training of neural networks using large learning rates[END_REF]. We chose this scheduler because it allows fast training [START_REF] Leslie | Super-convergence: Very fast training of neural networks using large learning rates[END_REF], which was important in order to reduce the time needed to pre-train all the networks on all the different subsets of ImageNet21K. The batch-size is set to 128, and the maximum learning rate for the OneCy-cleLR is set to 0.005. Also even if recent works demonstrate that increasing the weight-decay on the head of the network lead to better downstream performances [38], we use a constant weight-decay of 5e-4 for all the layers to avoid any side effects.

Downstream training for full dataset. Two common training strategies for transfer are considered here. Linear probing is deployed because it is well adapted for constrained environments [33]. We use a fully-connected layer for classification, which receives the features provided by the upstream model. This final layer is trained for 100 epochs, with a ReduceLrOnPlateau 1 learning rate scheduler based on the loss metrics with a patience of 5. This allow us to stop the training if the learning rate reaches 10 -8 . The Full fine tuning adapts all the layers of the architecture during downstream training, and past studies indicate that it outperforms linear probing, even in few-shot learning scenarios [START_REF] Kornblith | Do better imagenet models transfer better[END_REF][START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF]. While it requires more computational power than linear probing, it can be implemented on edge devices after optimization [33]. During this training we used the same parameters as for linear probing except for the initial learning rate which is set to 0.001 to avoid damaging the pre-trained features in the first steps.

Downstream training in a few-shot setting. A recent work pointed out that training for a large number of epochs can be beneficial if downstream datasets are small [START_REF] El-Nouby | Are large-scale datasets necessary for self-supervised pre-training?[END_REF]. However, overfitting sometimes occurs if this process is run until its end. To accommodate these two observations, we fine-tune for a large number of epochs (2500 for single shot), but stop the process if the learning rate value is too low (10 -7 ). Following [START_REF] El-Nouby | Are large-scale datasets necessary for self-supervised pre-training?[END_REF], when increasing the number of samples per class we divide the number of epochs by the number of samples per class to keep the same number of updates during the different training. The learning rate is again reduced on plateau.

Deep Network Architectures

We choose MobileNetv2 [28], ShuffleNetv2 [START_REF] Ma | Shufflenet v2: Practical guidelines for efficient cnn architecture design[END_REF], Resnet18 [START_REF] He | Deep residual learning for image recognition[END_REF] architectures for our experiments. To make the results obtained with these architectures comparable they are all downscaled to reach a size of 1M parameters. When scaling strategies are presented in the original papers, as it is the case for MobileNetv2 [28] and Shuf-fleNetv2 [START_REF] Ma | Shufflenet v2: Practical guidelines for efficient cnn architecture design[END_REF], we follow them here. We use a similar strategy for ResNet18, and also create a second version of Mo-bileNetv2 to study the effect of embedding sizes.

MobileNetv2. We selected MobileNetv2 [28] because it was designed for computationally constrained environments. We test two version of the model scaling. The first version, MobileNet 868 , is scaled using the scaling method from the original paper [28, 12] with a width multiplier of 0.678. The number of channels in the output of the inverted residual blocks are 11, 16, 22, 43, 65, 108 and 217 and the size of the vector in output of the feature extractor is 868. The second version, MobileNet 151 , is downscaled by fixing the embedding size of the extractor to 151 and before using the strategy from [28] to adapt the rest of the network. We created MobileNet 151 to match the embedding size of ResNet 151 , and also test the influence of the embedding size against MobileNet 868 .

ShuffleNetv2. We used the scaling method proposed in the original paper [START_REF] Ma | Shufflenet v2: Practical guidelines for efficient cnn architecture design[END_REF], to downsize this architecture to 1M parameters. The output channels of each stage are multiplied by a subunit factor (0.866), while leaving the first and the last convolution unchanged. Since the output of the extractor is 1024, we will refer to the downsized architecture as ShuffleNet 1024 .

ResNet18. ResNet18 [START_REF] He | Deep residual learning for image recognition[END_REF] is a generic architecture which is often used in literature. It has over 11M parameters in its full version and we downscale it to reach 1M parameters. The number of channels in each residual block is reduced uniformly, using a 0.295 width multiplication factor.

Experiments

Effect of a larger pre-training dataset

Past studies of pre-training [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF][START_REF] Goyal | Self-supervised pretraining of visual features in the wild[END_REF]20] showed that larger upstream datasets translated into higher downstream performance. However, it was noted that saturation occurs beyond a certain point, and adding supplementary data is not useful anymore [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF]. Given that past studies were focused on large deep neural networks, it is interesting to analyze the behavior of smaller models with respect to the number of upstream classes. We keep the number of images per class constant at 500, regardless of the total number of classes included in the pre-training dataset.

We present the results obtained with different architectures in Figures 1a and1b with a linear probing and full fine tuning of downstream tasks, respectively. Performance increases a lot when the total number of classes used for pre-training is small. An important gain is observed between 100 and 500 classes, particularly for linear probing. The relative gain starts to decline between 500 and 1000 classes, and even more between 1000 and 2000 classes. Then, performance starts to saturate beyond 2000 classes. Some performance variability is observed in the 1000 to 6000 classes range for all tested architectures and both training strategies when increasing the number of classes, but they do not exceed 3 accuracy points between the lowest and highest points. This finding is important insofar it indicates that increasing the number of classes is not useful for deep architectures designed for constrained environments. Performance saturation occurs for much smaller volumes of data compared to previous studies [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF][START_REF] Goyal | Self-supervised pretraining of visual features in the wild[END_REF]20], which fo- An interesting observation is that fine-tuning-based training is clearly better than a direct use of features learned upstream via linear probing. The accuracy gain when using the first strategy is over 15 points for all tested numbers of classes of the upstream dataset, and all backbone architectures. A similar finding was already reported in literature [START_REF] Kornblith | Do better imagenet models transfer better[END_REF][START_REF] Kumar | Fine-tuning can distort pretrained features and underperform out-of-distribution[END_REF] for larger deep architectures, and is confirmed here for compact architectures, which are adapted for computationally-constrained environments. We also note that the gain offered by fine tuning over linear probing is larger when upstream training is done with a low number of classes (up to 1000). This is explained by the stronger sensitivity of linear probing to the quality of the upstream features, due to the direct use of features versus an adaptation of them for downstream tasks during fine tuning.

The performance obtained with the four tested architectures varies for both downstream training strategies (linear probing in Figure 1a, fine tuning in Figure 1b). Globally, MobileNetv2 and ShuffleNetv2 behave better than ResNet after scaling to 1M parameters. This is somewhat expected since the first two types of architectures were designed purposely for computationally-constrained environments. Interestingly, the difference between MobileNet 868 and MobileNet 151 is much smaller when the upstream models are fine tuned (Figure 1b) compared to linear probing (Figure 1a). This indicates that models which have a wider output are more adequate for linear probing if the overall number of parameters is equivalent. An explanation resides in the higher dimensionality of the frozen features produced by MobileNet 868 , which favors the separability of classes downstream. This finding is in line with the well-known result reported for wide residual networks [37].

We propose a per-dataset view of results obtained with linear probing and with fine tuning in Figure 2. These results are reported with a MobileNet 868 , which provides the best overall results in Figures 1a and1b. Fine tuning is better than linear probing for five datasets out of six and number of classes included in the upstream datasets. The differences are much stronger for downstream tasks whose domain shift compared to the upstream task is larger. This is the case of FGVC, SVHN and GTSRB, three datasets focused on aircrafts, house number plates and street signs. The domains are not well represented in ImageNet and the retraining of all weights during fine tuning is clearly needed. Linear probing is better than fine tuning only for DTD . This result might be explained by the low total size of this dataset, which includes only 1600 training images, combined with the large domain shift between ImageNet and this texture-focused dataset.

We also report performance with downstream tasks without pre-training on average and per dataset to assess the overall effect of pre-training. The difference between the best and worst of the four tested architectures is 4 points in Figure 1c, but there are strong differences between individual datasets (Figure 2c). The global comparison shows that the use of pre-training for fine tuning brings a significant improvement compared to training from scratch. The dataset-level analysis (Figure 2) gives more insight into the merits and limitations of pre-training with the two downstream training strategies. Linear probing is effective for small domain shifts between upstream and downstream tasks (Oxford-IIIT Pet) or when the dataset size is small (DTD), but provides lower performance in the other cases. This is expected since the features are not adapted to each task. Fine tuning provides similar performance to that of training from scratch for easy tasks, such as GT-SRB and SVHN, and brings important improvements for FGVC, Oxford-IIIT Pet and DTD.

Effect of pre-training with a constant-size dataset

We complement the analysis from Subsection 4.1 with experiments run with a dataset which total size is kept fixed at 1M images. Here, the number of images per class decreases when the number of selected classes increases. The dataset is balanced, meaning the samples are distributed evenly between classes. This corresponds to an upstream training with a fixed sample budget.

The figures 3a and 3b show the mean accuracy on the six downstream task with linear probing and full fine-tuning. The global trends are similar to those observed in Figure 1, as is the accuracy obtained with linear probing and fine tuning in different configurations.

We observe a performance gain of up to two points when the number of classes in the dataset changes from 1000 to 2000. Beyond 2000 classes, performance seems to oscillate, and is even slightly decreasing for linear probing (Figure 3a). For this strategy, the obtained accuracy decreases in all tested configurations except one when the number of classes increases from 4000 to 6000. This result can be explained by 2 opposite phenomena. While a more diversified pre-training dataset is likely to lead to a better representation, a larger number of classes also makes the upstream task more difficult. Our finding is consistent with the saturation of downstream performance beyond a certain point, even when upstream performance is improved [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF]. Here, the improvement of the representation brought by the addition of new classes is degraded by the growth of the complexity of the task, and by the scarcer representation of each class when the total number of classes increases.

The comparison of the results for 4000 and 6000 classes from Figures 1 and3 is interesting because the total size of the upstream dataset is smaller in the latter configuration. There are 2M and 3M images for 4000 and 6000 classes in Figure 1, but only 1M in Figure 3. This finding shows that a representation of upstream classes with fewer images does not have a significant impact on downstream performance.

Effect of pre-training in few-shot scenarios

Pre-training is particularly useful when only few samples are available per class since deep neural networks are data hungry [5]. Performance is reported for pre-training with MobileNet 868 for 1M parameters, the configuration which works best for downstream training with all data. We plot accuracy for pre-training with 1000 and 6000 classes for each model to also assess the influence of this parameter. We investigate the performance on downstream tasks for different few-shot learning regimes. We again report results with two upstream pre-training strategies: the number of per class is constant in Figure 4 and the total size of the dataset is constant 5. The observation that performance is very similar in the two upstream dataset configurations remains valid for all tested few-shot settings. Interestingly, the obtained results indicate that linear probing is significantly better than full fine tuning up to 10 samples per class. The full training of the downstream models is difficult with very few samples due to the occurrence of overfitting [36]. The gap between the two training strategies narrows when the number of samples increases. Fine tuning becomes better than linear probing when 25 samples are available. Naturally, this tendency is even clearer when all samples are available for downstream tasks, as we discussed in Subsection 4.1. Our results are at odds with those reported previously [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF], regarding the superiority of fine tuning over linear probing even in a few-shot setting. The main difference comes from the scale of the networks, with much larger architectures being tested in [START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF]. The observations made here show that the downstream training strategy should be adapted depending on the quantity of data available for target datasets. In practice, the pre-trained model should be used as a frozen feature extractor if the number of samples in the downstream task is smaller.

Performance of few-shot learning is similar for upstream dataset variants with a constant number of images per class (Figure 4) and with a constant size dataset (Figure 5). This echoes the results obtained when using the full downstream datasets. The difference between pre-training with 1000 and 6000 datasets is larger for linear probing compared to fine tuning. The accuracy gain between these two variants of the upstream dataset reaches approximately 2, 2 and 1.5 accuracy points for 1-shot, 5-shots and 25-shots settings, respectively. This is expected since linear probing makes direct usage of upstream features, and thus benefits more from strong pre-trained representations.

The global comparison of linear probing and fine tuning in few shot scenarios, presented in Figures 4 and5, is refined with a presentation of the accuracy per dataset. Figures 6 and 7 illustrate results for MobileNet 868 pre-trained with 6000 and 1000 classes, respectively. Linear probing is clearly better than fine tuning for datasets which are semantically related to the content on the pre-trained models, such as Oxford-IIIT Pet and CIFAR-100. ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF], the dataset used for pre-training, includes a large number of classes which describe the natural world, which are also well represented in the three downstream datasets for which probing has good results in few-shot scenarios. Linear probing accuracy is also better for DTD, a texture dataset which does not benefit from fine tuning even when all its images are available (Figure 2). Fine tuning is the better option for FGVC, SVHN, and GTSRB, the three datasets with a larger domain shift compared to ImageNet.

Conclusion

We investigate transfer learning in image recognition under constraints through a comprehensive empirical study, which analyzes the roles of the dataset used for upstream training, the performance of different deep architectures, the results obtained with two opposite training strategies. Our experiments confirm findings reported in previous studies regarding performance saturation for large deep architectures [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF][START_REF] Goyal | Self-supervised pretraining of visual features in the wild[END_REF]20]. It shows that the phenomenon appears faster in terms of scale of the upstream dataset, due to the com-pactness of tested architectures. As a result, the conclusion to use increasingly larger pre-training datasets to improve performance [START_REF] Abnar | Exploring the limits of large scale pretraining[END_REF][START_REF] Goyal | Self-supervised pretraining of visual features in the wild[END_REF]20] does not seem justified for compact deep architectures. In contrast to past studies [START_REF] Kornblith | Do better imagenet models transfer better[END_REF][START_REF] Zhai | A large-scale study of representation learning with the visual task adaptation benchmark[END_REF] which assert that full fine tuning is preferable to linear probing, our result shows which strategy is better depending on number of images available. Linear probing is a good strategy when the domain shift is small and/or when the available number of samples per class in the downstream task is low. It is also interesting due to its lower computational complexity, which is important in constrained environments [33].

We used variants of a fully-supervised dataset for pretraining. It would be useful to extend it by testing weaklysupervised and unsupervised pre-training. It would be equally interesting to explore ways to predict an adapted downstream training strategy based on an analysis of the domain shift between the upstream and downstream tasks.

Figure 1 :

 1 Figure 1: Mean accuracy on the downstream tasks as a function of the number of classes, and using 500 images per class for all tested deep architectures.

Figure 2 :

 2 Figure 2: Detailed accuracy for each downstream task with MobileNet 868 .

Figure 3 :

 3 Figure 3: Mean accuracy on the downstream tasks when the total size of the dataset is constant (1M images) and the number of images per class decreases when the number of classes increases. The minimum number of classes is 1000 because ImageNet does not contain enough leaf classes with enough images to run experiments with 100 and 500 classes.

Figure 4 :

 4 Figure 4: Mean accuracy on the downstream tasks in low-shots four settings. The number of image per class is constant (500) in the upstream dataset. It includes 1000, 2000, 4000, 6000 classes, from left to right.

Figure 5 :

 5 Figure 5: Mean accuracy on the downstream tasks in low-shots four settings. The total size of the upstream dataset stays constant (1M images). It includes 1000, 2000, 4000, 6000 classes, from left to right.

Figure 6 :

 6 Figure 6: Downstream accuracy for each downstream task with 6000 classes and 500 images per classes for the pre-training.

Figure 7 :

 7 Figure 7: Downstream accuracy for each downstream task with 1000 classes and 500 images per classes for the pre-training.

Table 1 :

 1 Downstream datasets statistics. initial learning rate is set to 0.01. The weight decay is set to a constant 5e-4 over the whole network for the same reason as the upstream training.

	Dataset	Oxford-IIIT Pet [26] DTD [2] GTSRB [31] SVHN [23] FGVC [21] Cifar100 [17]
	# classes	37	47	43	10	100	100
	# training/class	99.432	39.979	619.512	7325.7	33.34	500
	stdev training/class	1.534	0.144	457.377	2800.661	0.474	0
	# test/class	99.135	39.979	293.721	2603.2	33.33	100

1 https://pytorch.org/docs/stable/generated/ torch.optim.lr_scheduler.ReduceLROnPlateau.html


T Leyssens
email: tom.leyssens@uclouvain.be

G Springuel

N Candoni

S Veesler

A THERMODYMANICALLY GUIDED APPROACH TO CO-CRYSTAL SCREENING: APPLICATION TO MALEIC ACID / CAFFEINE SYSTEM

Keywords: co-crystals, thermodynamics, process control tools

Introduction

Co-crystallization of pharmaceutically active molecules allows to enhance the physical properties of a drug substance, such as its solubility, bioavailability… Although dry grinding, solvent drop grinding, and melt crystallization have been reported as key preparative techniques, [START_REF] Schartmann | On the thermodynamics of cocrystal formation[END_REF] the development of a solvent crystallization often remains imperative from an industrial point of view.

Issues encountered during the development of an up-scaled co-crystallization process are often related to a limited understanding of the thermodynamic behavior of a ternary system consisting of two distinct compounds, as well as the solvent. [START_REF] Chiarella | Making co-crystals -The utility of ternary phase diagrams[END_REF] The distinct pairwise interactions between these components, lead to complex phase diagrams, as shown in Figure 1a. [START_REF] Guo | Co-crystallization in the caffeine/Maleic Acid System: Lessons from Phase equilibria[END_REF] The thermodynamic conditions under which a pure co-crystal phase can be identified in solution, are often difficult to identify by a trial and error approach and would ideally require the construction of ternary phase diagrams. Construction of such diagrams can, nevertheless, be quite time-consuming, and is often impractical from an industrial point of view. Screening for co-crystal phases in solution can, however, be simplified to a great extent, when using thermodynamic considerations to guide the screening trials. In this contribution, such an approach was applied to the caffeine/maleic acid co-crystal system. For this system, 1:1 and 2:1 co-crystals have been obtained through wet grinding, but only the former has been obtained in a reproducible manner in solution. [START_REF] Guo | Co-crystallization in the caffeine/Maleic Acid System: Lessons from Phase equilibria[END_REF][START_REF] Trask | Pharmaceutical cocrystallization: Engineering a remedy for caffeine hydration[END_REF] A single, non-repeatable, occurrence of a 2:1 co-crystallization in solution has been reported so far. The aim of this contribution is to show that the 2:1 phase can be obtained in solution in a repeatable manner. To achieve this goal, an approach is constructed based on thermodynamic arguments.

Approach

The 1:1 co-crystal phase can easily be obtained from solution crystallization in acetone. In this solvent, maleic acid is about 30 times more soluble compared to caffeine, leading to a twisted ternary phase diagram, as presented in Figure 1a. As shown by this diagram, the 1:1 phase is thermodynamically accessible, but the 2:1 phase is expected to be meta-stable in solution. Spontaneous crystallization of this phase in acetone would require substantial supersaturation levels. To render the 2:1 phase thermodynamically accessible in solution, the ternary phase diagram would have to be altered, as shown in Figure 1b. The most straightforward manner to achieve this transformation is by modifying the solvent used, ideally leading to a diagram as presented in Figure 1b. This would require using a solvent showing comparable solubility of both co-crystal components.

To show the importance of the solvent for the development of an in solution cocrystallization of the 2:1 crystal phase, four different solvents were tested, having respectively maleic acid vs. caffeine solubility ratios of 100, 40, 5, and 0.05. The solvent presenting a relative ratio of 5, is most likely to lead to a ternary phase diagram, as presented in Figure 1b, and is thus most likely of interest for the co-crystallization of the 2:1 phase. Ideally, the complete ternary phase diagram for each solvent should be determined. However, this often turns out to be time-consuming. Recently, alternative phase diagrams drawing have been developed, [START_REF] Horst | Discovering new cocrystals[END_REF][START_REF] Gagnière | Formation of co-crystals: kinetic and thermodynamic aspects[END_REF] which zoom-in onto the region of the ternary phase diagram where the co-crystal phases possibly appear. These phase diagrams are constructed by considering how the solubility of one component varies, if the amount of the other component is altered, leading to diagrams as presented in Figure 2a. With respect to classical screening methods, based on stoichiometric ratios of both components, the idea behind these diagrams is to consider the respective saturation concentrations of both components as a starting point. As shown in this diagram, a solution saturated in both components is likely to be supersaturated with respect to the co-crystal phase (point p). Figure 2a shows an ideal situation, for which the solubility of one component is independent on the presence of the other. This is, however, rarely the case, especially for co-crystal systems, as both components are assumed to interact with one another. To identify possible co-crystal phases, an experimental plan was therefore designed, centred around point p, as shown on Figure 2b.

As the phase diagram presented Figure 2 is temperature dependent, the experimental plan can be created at different temperatures, under the condition that the solubility of both components is known at these temperatures. At first, the solubility curves for maleic acid and caffeine were measured for the different solvents selected. A multiwell set-up with video microscopy, as shown in Figure 3, was used to determine the solubility curves. Standard 1ml HPLC glass vials were inserted into two blocks that were temperature controlled by Peltier elements. The whole assembly is mounted on a translation table to allow scanning of the different vials. [START_REF] Detoisien | A rapid method for screening crystallization conditions and phases of an active pharmaceutical ingredient[END_REF] Once the solubility curves was determined, experimental plan such as presented in Figure 2b, was created for each solvent, at two different reference temperatures. The Multiwell device was used, to check for spontaneous crystallization. Crystalline phases were identified by XRPD. Both 1:1 and 2:1 phases crystallized spontaneously in the solvent showing a maleic acid/caffeine solubility ratio of 5. Both forms show needle-shaped habits as shown in Figure 4. This solvent was then selected to determine conditions for which the 2:1 phase is kinetically favoured, as well as conditions for which this phase is thermodynamically favoured. The spontaneous crystallization of the 2:1 phase is achieved by extending the experimental plan presented in Figure 2b. Moreover, to study the thermodynamic stability, the experimental plan is repeated, and all samples are seeded with both the 1:1 and 2:1 phases, and left to equilibrate for over two weeks. Based on these findings, repeatable 2:1 co-crystallization conditions were determined, with the ultimate goal of growing a single 2:1 co-crystal for structure determination.

Conclusion

In this contribution, a thermodynamically guided approach is applied for the screening of co-crystals in solution. The methodology used, allowed us for a rapid screening of crystallization conditions of potential co-crystals. In less than 6 weeks, conditions for the co-crystallization of the caffeine/maleic acid 2:1 co-crystal in solution were defined, showing the interest of the developed approach, especially from an industrial and commercial point of view.

Figure 1 .

 1 Figure 1. a) Ternary Phase diagram caffeine/maleic acid/acetone at 25°C in mass % (Source: ref 3). b) Hypothetical phase diagram in a solvent showing comparable maleic acid and caffeine solubility.

Figure 2 .

 2 Figure 2. a) Schematic phase diagram for a system leading to a 1:1 co-crystal. b) Experimental design centred around point p.

Figure 3 .

 3 Figure 3. a) Multiwell set-up used b) translational stage c) blocks containing vials.[START_REF] Detoisien | A rapid method for screening crystallization conditions and phases of an active pharmaceutical ingredient[END_REF]

Figure 4 .

 4 Figure 4. Optical microscopy images (left 1:1 co-crystal, right 2:1 co-crystal).