Effect of temperature on measurement of fundamental constants using white dwarfs in Gaia-EDR3 survey

Fundamental constants are crucial for comprehending physical mechanisms, but their measurements contain uncertainties due to experimental limitations. We investigate the impact of system temperature on these uncertainties using nearby white dwarfs observed in the Gaia Early Data Release 3 (EDR3) survey. Using the structures of these white dwarfs, we show that the variation in system temperature can affect the accuracy of measurements for fundamental parameters such as the fine-structure constant and the proton-to-electron mass ratio. This exploration emphasizes the importance of considering the energy of a system while putting bounds on the values of fundamental constants.


INTRODUCTION
Testing fundamental couplings that lead to the investigation of the universality of physical laws always remains one of the most interesting fields for researchers (Uzan 2011;Martins 2015).The spatial variation of the fine-structure constant was investigated using the high-resolution spectroscopic studies of absorption systems along the line of sight of bright quasars (Webb et al. 2011), and consequently, detailed further tests on this claim were discussed by Evans et al. (2014) and Kotuš et al. (2017) using different quasars.Similar tests for compact objects, such as solar-type main-sequence stars (Adams 2008;Vieira et al. 2012), population III stars (Coc et al. 2010), and neutron stars (Pérez-García & Martins 2012) were also carried out to put bounds on the fine-structure constant, the Newton gravitational constant, etc.It is worth mentioning that due to significant uncertainties associated with the nuclear processes, these tests are not as strong as the quasar tests.Moreover, as these two tests were carried out in different physical environments, they led to completely independent results.Nevertheless, the possible dependencies of fundamental couplings on the local environment, such as the strength of the local gravitational field in the presence of compact objects, should also be handled carefully.
More recently, researchers put constraints on the fine structure constant  as well as the proton-to-electron mass ratio  using white dwarf (WD) observations from different surveys (Magano et al. 2017;Kalita & Uniyal 2023).WDs are the end state of main-sequence stars possessing mass less than approximately (10 ± 2) M ⊙ (Lauffer et al. 2018).Earlier, Chandrasekhar (1935) showed that a non-rotating and ★ E-mail: akhil_uniyal@sjtu.edu.cn† E-mail: surajit.kalita@uct.ac.za; corresponding author ‡ E-mail: sayan.chakrabarti@iitg.ac.in non-magnetized WD can contain a maximum mass of 1.4 M ⊙ ; famously known as the Chandrasekhar mass-limit.In order to describe the phenomenology of a WD, one needs to understand its underlying equation of state (EoS) which describes the state of matter within it.
So far in the literature, there exist three major EoSs which describe the degenerate matter of the WDs known as the Chandrasekhar EoS (Chandrasekhar 1931), the Salpeter EoS (Salpeter 1961;Salpeter & van Horn 1969), and the relativistic Feynman-Metropolis-Teller (RFMT) EoS (Feynman et al. 1949;Rotondo et al. 2011a).Among them, the RFMT EoS takes care of the coulomb interactions and the local inhomogeneities of the relativistic electrons making it a more generalized form of the Chandrasekhar and Salpeter EoSs.The contributions from these terms decrease WD masses but enhance their radii as compared to those obtained from the Chandrasekhar and Salpeter EoSs.A detailed comparison of these EoSs was studied by Rotondo et al. (2011b) including their advantages and drawbacks.The widely used polytropic EoSs for WDs are the non-relativistic and relativistic limiting cases of the Chandrasekhar and Salpeter EoSs (Zel'dovich & Novikov 1966;Shapiro & Teukolsky 1983).It is worth mentioning that RFMT EoS shows significant deviation from the other EoSs only at relatively low densities, primarily below 10 4 g cm −3 (de Carvalho et al. 2013).In general, the observed WDs cannot possess such low core densities, which is responsible for their overall masses, and hence it does not significantly affect the overall structure of the observed WDs.Thus, we consider the Chandrasekhar EoS and its possible modification in the presence of temperature in this work.Our target is to incorporate the temperature effect into the EoS so that the mass-radius relation can be matched with the observed WD data and thereby look for the effects of temperature on the constraints of  and .
Moreover, it was shown that the standard WD mass-radius relation gets affected by the spacetime variation of fundamental couplings.Using the masses and radii from a simulated catalog of 100 WDs, Magano et al. (2017) found the bound to be Δ/ = (2.7 ± 9.1) × 10 −5 .It is worth mentioning that these 100 WDs are not the actual WD data; they are instead simulated ones within the mass range of 0.3 M ⊙ <  < 1.2 M ⊙ using the standard Chandrasekhar massradius relation.The actual masses and radii of WDs indeed deviate from this relation as confirmed by different astronomical surveys, such as Sloan Digital Sky Survey (SDSS), Gaia, etc.In our previous work, with the help of massive WDs observed in Gaia Data Release 2 survey, we obtained bounds on  and  under the Newtonian gravity as well as in modified gravity theory and thereby showed that these bounds are significantly affected by the underlying gravity theories (Kalita & Uniyal 2023).Through our exploration, it was firmly established that modified gravity has a significant impact on dense astrophysical scenarios.This has also led to the revelation of stronger bounds of fundamental parameters under alternate gravity theories.
In this paper, we consider the grand unified theory models for the variations of relevant couplings depending on the two dimensionless parameters R and S (Campbell & Olive 1995;Coc et al. 2007).Using WDs from Gaia Early Data Release 3 (EDR3) catalogue (Gentile Fusillo et al. 2021), we show that the variation in temperature can significantly affect the constraints on  and .This article is organized as follows.In Section 2, we first obtain the temperature-dependent EoS for WD matter and thereby the relations for the bounds of  and .In Section 3, we describe our selection of dataset using Gaia-EDR3 survey, which we use to constrain  and  for different temperatures.Finally, Section 4 discusses these results and provides some parting thoughts as conclusions in Section 5.

EFFECT OF TEMPERATURE ON THE WHITE DWARF STRUCTURE
As we are going to study the WD structures in the presence of temperature, let us first discuss the corresponding EoS, which can be determined through the pressure and energy density of the matter present inside.Generally, except the envelope region near the surface (which is comparatively much smaller than the total radius of WD), the pressure  inside a WD is dominated by the electron gas pressure  e ; the pressure due to the positively charged nuclei  N is insignificant.On the other hand, the energy density E is dominated by that of the nuclei E N and the energy density of the electrons E e is negligibly small.These approximations were considered by Chandrasekhar (1931) and mathematically they can be represented as and where  u = 1.6604 × 10 −24 g is the atomic mass unit,  is the speed of light,  e is the electron number density, and  e is the mean molecular weight per electron.Because electrons are fermions, they follow the Fermi-Dirac statistics, and the electron number density is given by (Boshkayev et al. 2016;Boshkayev 2018) where  B is the Boltzmann constant, ℎ is the Planck constant,  e is the rest mass of an electron, μe is the chemical potential,  is the temperature,  =  B /( e  2 ),  is the momentum of the electron, Ẽ ( ) = √︃  2  2 +  2 e  4 −  e  2 is the kinetic energy, and with  = Ẽ ( )/( B ) being a dimensionless parameter and  = μe /( B ) the normalised chemical potential energy (Timmes & Arnett 1999;de Carvalho et al. 2014).Consequently, the total pressure  and matter density  are respectively given by (de Carvalho et al. 2014) and It is evident from these equations that the EoS depends on the temperature of the matter.The complexity of the function   (, ) does not allow us to solve these two equations analytically, and therefore we solve them numerically.Using this EoS, we need to solve the pressure balance and mass estimate equations (together known as where  is the luminosity,  is the power produced per unit mass of WD matter,  is the opacity, and  is the Stefan-Boltzmann constant.
We can divide a WD into two parts: the core and the envelope.In the core, electron degeneracy pressure dominates over ideal gas pressure due to high density, whereas in the envelope region, it is the opposite.
Envelopes are typically only a few kilometers in radius, much smaller compared to the WD radius.Additionally, most of the WD mass accumulates in the core.Previous studies suggest that temperature does not vary significantly in the core region and drops steeply in the envelope (Bhattacharya et al. 2022).Hence, we can safely ignore the d/d and d/d equations, as they have negligible effects on the WD mass and radius.We keep the temperature contribution only through the WD EoS.
Our interest lies in finding out the bounds on  and .Considering that the Planck mass is fixed but particle mass and quantum chromodynamics (QCD) scale can vary, (Coc et al. 2007) provided the following uncertainties in electron and proton masses and where R and S are dimensionless phenomenological parameters.Thereby the uncertainty in  can be written as The values of R and S tend to vary based on different observations.For instance, the data from the Wilkinson Microwave Anisotropy Probe (WMAP) suggest that R ≈ 36 and S ≈ 160 (Coc et al. 2007), whereas a dilaton-type model gives R ≈ 109 and S ≈ 0 (Nakashima et al. 2010).However, in this paper, we consider values obtained from the astrophysical observations of a BL Lac object PKS 1413 + 135, which suggest R = 278 ± 24 and S = 742 ± 65 (Monteiro et al. 2014).We now insert these uncertainties of electron and proton masses in Equations ( 6) and ( 7) to further solve the hydrostatic balance equations for a fixed temperature and different Δ/.

EFFECT OF TEMPERATURE ON THE CONSTRAINT OF FINE-STRUCTURE CONSTANT
In this work, we analyze the effect of temperature on the bounds of  and  using Gaia-EDR3 data reported by Gentile Fusillo et al. (2021)1 .The dataset contains 1 280 266 objects, each associated with a probability of being a WD ( WD ) and geometric distance.Among them, masses are measured for 298 317 objects considering that they are comprised of pure hydrogen atmosphere.In this work, we limit our analysis to objects within a geometric distance of 50 pc and which are almost certain to be WD candidates with  WD ≥ 0.9.This reduces our sample size to 2094.We further use their masses and log  values with  being the surface gravity to obtain the radii.
Figure 1 shows the masses and radii of these WDs along with their respective error bars.Different lines in that plot show the theoretical mass-radius curves obtained by solving the hydrostatic balance equations together with Equations ( 6) and ( 7) for different temperatures.We assume  e = 2 Throughout this work indicating the cores of the WDs are comprised of carbon or oxygen or any other similar matter.It is evident that the effect of temperature is more prominent for less massive WDs and this significance gradually drops with the increase in mass.It is also noticeable from this figure that the masses and radii are more scattered towards the low mass region whereas they nearly follow the Chandrasekhar zero-temperature mass-radius curve as mass increases.The temperature of a WD is unequivocally influenced by its evolutionary history and surrounding environment.Because temperature exerts a greater impact on low densities, it leads to more sporadic data points toward the low mass regimes in the mass-radius plot.It is important to note that there are other physical effects such as rotation and magnetic fields which may potentially impact the shape and size of WDs.However, the data points in the high mass regimes suggest that they do not have discernible effects on the mass and size of WDs.As a result, we do not factor in these effects when deriving theoretical mass-radius curves for WDs.
We need to understand the goodness of the theoretical mass-radius curves over the data points, which eventually helps in estimating the quantity Δ/.In order to do so, one may define the following quantity (Jain et al. 2016) which needs to be minimized over Δ/.Here   and R  are respectively the observed mass and radius of each data point with    and  R  being their respective error bars, whereas R Th () denotes the theoretical radius for mass .However, in our case, the radii of these WDs are not independently measured from observations; we rather obtain them using the measured values of  and log .Thus, the statistical method demands that we are supposed to consider only one term in the aforementioned  2 function.For convenience, we define the  2 function as We now minimize this function for different Δ/ over each temperature.Figure 2(a) shows |Δ/| as well as 3- confidence intervals for which  2 is minimized at different temperatures.The detailed analysis of the statistical method is discussed in Appendix A. It is evident that until the temperature significantly affects the WD mass-radius curves, |Δ/| nearly remains constant.However, above 10 6 K it starts decreasing and reaches a minimum approximately at 9 × 10 6 K.A further increase in temperature increases the |Δ/| indefinitely because such high temperatures are not possible inside of a WD.
Substituting these values of |Δ/| in Equation ( 14), we calculate |Δ/| and their corresponding 3- error bars.which is depicted in Figure 2(b) to observe that the variation of |Δ/| with respect to temperature behaves the same way as |Δ/|.

DISCUSSION
This work delves into the impact of system temperature on the measurement of fundamental constants  and .Our methodology involves selecting nearby WDs from the Gaia-EDR3 survey, which are within a distance of 50 pc and with the probability of being a WD greater than or equal to 0.9.After comparing their structures with the theoretical mass-radius curves for different temperatures, we have observed that the effect of temperature is prominent for the less massive WDs.With the increase in WD mass, the influence of temperature on the overall structure of WDs gradually diminishes, and above 1 M ⊙ , it is almost insignificant.Since the uncertainties in electron and proton mass can be related to the uncertainties in , we have used these mass uncertainties in the finite temperature EoS assuming that the Planck mass is fixed but particle mass and QCD scale can vary (Coc et al. 2007).Through our analysis, we have determined the optimal value of Δ/ for each temperature by minimizing the  2 function.By repeating this process for various temperatures, we have established the bounds on  and .These values serve as the most stringent constraints on  and  at their respective temperatures for these WD data.Our findings reveal that the constraints on these parameters become tighter as the temperature rises, culminating the tightest constraints of |Δ/| = 4.203 × 10 −7 and |Δ/| = 3.233 × 10 −7 at approximately 9 × 10 6 K before they relax indefinitely.
We have made some justifiable assumptions in our work.Specifically, in Section 2, we have excluded d/d and d/d equations as they do not significantly impact the masses and radii of WDs.We have solely focused on the temperature contribution through the WD EoS.Note that we have only considered the nearby WDs.It is unlikely that choosing all WDs would significantly affect the final results of Figure 2.This is because the mass distribution of the selected WDs is similar to that of the complete sample of WDs.Moreover, the influence of magnetic fields and rotation, which can increase the mass of a WD, have not been taken into account in our study.The magnetic-togravitational (ME/GE) and kinetic-to-gravitational (KE/GE) energy ratios would need to be exceptionally high for these factors to significantly impact the WD's mass, potentially surpassing the proposed limits by Komatsu et al. (1989) and Braithwaite (2009).Therefore, the inferred masses and radii of WDs in the Gaia-EDR3 catalog would remain unchanged unless the ME/GE and KE/GE ratios are exceedingly high.Furthermore, we have chosen  e = 2 which indicates the presence of carbon, oxygen, neon, or other similar elements at the core of these WDs.This assumption is supported by Figure 1 where all the massive WDs nearly follow the mass-radius relation with  e = 2. Nevertheless, we do not rule out the possibility of the presence of other elements although they would not significantly alter the final outcome.

CONCLUSIONS
The significance of system temperature in determining fundamental constants and couplings is emphasized in this work.In the introduction, we have already mentioned some bounds on  and  from different observations.Note that these bounds are different while considering objects at different redshifts.The thermal history of the universe shows that the temperature gradually decreases after the Big Bang, and hence different redshifts indicate the different temperatures of our universe.In this way, these aforementioned results also indirectly indicate that the constraints on fundamental parameters are affected by the temperature.
In our work, by using WD data from an astronomical survey, we have explicitly shown that these constraints can vary as the temperature alters.Note that this statement is true in a broader sense, although the situation can be different for spectroscopic analyses.For instance, in the case of quasar absorption measurements of , which are discussed in the Introduction, it is true that the gas temperature is relevant and it can be treated as a free parameter of equal status to all other free parameters for which they are solved.However, if the quasar analysis is restricted to the some atomic species, e.g.[Fe II], the measurement is independent of temperature.It is to be noted that at a certain temperature, two different observations might yield two different constraints for the same parameter.In this scenario, one must consider the tightest one at that particular point.Our work stands out for providing some of the tightest constraints on  and  compared to existing works in literature, which are mentioned in the Introduction.This exploration highlights that the significance of taking into account the system energy when defining limits for fundamental constants is undeniable.As we move forward, conducting further analysis through physical observations can unlock a deeper understanding of these outcomes.de Carvalho S. M., Rotondo M., Rueda J. A., Ruffini R., 2013, Int.J. Mod.Phys.Conf.Ser., 23, 244 de Carvalho S. M., Rotondo M., Rueda J. A., Ruffini R., 2014, Phys. Rev. C, 89, 015801 hydrostatic balance equations) to obtain the structures of WDs at different temperatures.Due to the large size of observed WDs in our data sample, we can study them within the Newtonian framework.The hydrostatic balance equations are given by d() d = −   () ()  is the Newton gravitational constant and  () is the mass of the WD within a radius .Since we consider the effect of temperature, one, in principle, needs to simultaneously solve the radiative energy transport and the energy conservation equations, which are respectively given by d () d = − 3 ()() () 64 2  () 3 (10) and d () d = 4 2 () (), (

Figure 1 .
Figure 1.The green scattered points are masses and radii of WDs with their respective error bars obtained from the Gaia-EDR3 survey, which are within a radius of 50 pc.Different coloured lines depict WDs' theoretical mass-radius curves for different temperatures assuming  e = 2.

Figure 2 .
Figure 2. Variation of uncertainties of fundamental constants as a function of temperature along with their corresponding 3- error bars.Note that the sizes of error bars might look different as the vertical axis is plotted in logarithmic scale.