Modified Einstein's gravity as a possible missing link between sub- and super-Chandrasekhar type Ia supernovae
aa r X i v : . [ a s t r o - ph . S R ] M a y Prepared for submission to JCAP
Modified Einstein’s gravity as apossible missing link between sub- andsuper-Chandrasekhar type Iasupernovae
Upasana Das a and Banibrata Mukhopadhyay ∗ a a Department of Physics, Indian Institute of Science, Bangalore 560012, IndiaE-mail: [email protected], [email protected]
Abstract.
We explore the effect of modification to Einstein’s gravity in white dwarfs for thefirst time in the literature, to the best of our knowledge. This leads to significantly sub-and super-Chandrasekhar limiting masses of white dwarfs, determined by a single modelparameter. On the other hand, type Ia supernovae (SNeIa), a key to unravel the evolutionaryhistory of the universe, are believed to be triggered in white dwarfs having mass close to theChandrasekhar limit. However, observations of several peculiar, under- and over-luminousSNeIa argue for exploding masses widely different from this limit. We argue that explosions ofthe modified gravity induced sub- and super-Chandrasekhar limiting mass white dwarfs resultin under- and over-luminous SNeIa respectively, thus unifying these two apparently disjointsub-classes and, hence, serving as a missing link. Our discovery raises two fundamentalquestions. Is the Chandrasekhar limit unique? Is Einstein’s gravity the ultimate theory forunderstanding astronomical phenomena? Both the answers appear to be no!
Keywords: modified gravity, supernova type Ia - standard candles, white and brown dwarfs
ArXiv ePrint: ∗ Corresponding Author. ontents α = 0 84.2 Cases with α < α > Modification to Einstein’s theory of gravity, although has been applied to neutron stars, hasnever been explored for white dwarfs. Perhaps the reason for not doing so, until the presentwork, is the larger size and lower density of white dwarfs, which apparently argue for theirmuch weaker gravitational field compared to neutron stars. In this work, we aim at exploringthe effect of modification to Einstein’s gravity in white dwarfs. This shows that modifiedgravity effect is quite non-negligible in high density white dwarfs and could be significantdepending on the value of a model parameter. We furthermore argue that our result mayhave far reaching astrophysical implications.It has been understood that our universe exhibits accelerated expansion, which hasbeen firmly established by the observations of extremely luminous stellar explosions, knownas type Ia supernovae (SNeIa). SNIa is one of the most widely studied astronomical events.These SNeIa are believed to result from the violent thermonuclear explosion of a carbon-oxygen white dwarf, when its mass approaches the famous Chandrasekhar limit of 1 . M ⊙ ,where M ⊙ is the solar mass. The characteristic nature of the variation of luminosity withtime of SNeIa is believed to be powered by the decay of Ni to Co and, finally, to Fe.This feature, along with the consistent mass of the exploding white dwarf, allows SNeIa tobe used as a ‘standard’ for measuring far away distances (standard candle) and, hence, inunderstanding the expansion history of the universe [1].However, the discovery of several peculiar SNeIa provokes us to rethink the commonlyaccepted scenario. Some of these SNeIa are highly over-luminous, e.g. SN 2003fg, SN 2006gz,SN 2007if, SN 2009dc [2, 3], and some others are highly under-luminous, e.g. SN 1991bg, SN1997cn, SN 1998de, SN 1999by, SN 2005bl [4–10] (see also [11]). The luminosity of the formergroup of SNeIa (super-SNeIa) implies a huge Ni-mass (often itself super-Chandrasekhar),invoking highly super-Chandrasekhar white dwarfs, having mass 2 . − . M ⊙ , as their mostplausible progenitors [2, 3, 12–15]. On the other hand, the latter group (sub-SNeIa) producesas low as ∼ . M ⊙ of Ni [16]. Attempted models to explain sub-SNeIa, often based on– 1 –umerical simulations, include explosion due to the merging of two sub-Chandrasekhar whitedwarfs [17], explosion of a single sub-Chandrasekhar white dwarf triggered externally due toaccretion of a helium layer (sub-Chandrasekhar mass model) [18]. However, they entailcaveats, such as, the simulated light-curve in the merger scenario fades slower than thatsuggested by observations [17], along with other spectroscopic discrepancies in individualmodels [19]. The models, in order to explain super-SNeIa progenitor mass, include rapidly(and differentially) rotating white dwarfs [20], binary evolution of accreting, differentiallyrotating white dwarfs [21], highly magnetized white dwarfs [22, 23]. However, they alsoharbor several doubts such as, existence of supermassive ( > . M ⊙ ), stable, highly rotatingwhite dwarfs [24], stability of highly magnetized white dwarfs [25]. Nevertheless, the issuesrelated to highly magnetized white dwarfs have been addressed, e.g., by considering varyingmagnetic fields within them [26, 27].Even if we keep aside all the aforementioned caveats, a major concern arises that such alarge array of models is required to explain apparently the same phenomena, i.e., triggeringof thermonuclear explosions in white dwarfs. It is unlikely that nature would seek mutuallyantagonistic scenarios to exhibit sub- and super-SNeIa, which are sub-classes of the sameSNeIa. This is where the current work steps in. Our work attempts to unify the phenomeno-logically disjoint sub-classes of SNeIa by effectively a single underlying theory, which, hence,serves as a missing link. This is achieved by invoking a modification to Einstein’s theory ofgravity or general relativity in white dwarfs.The validity of general relativity has been tested mainly in the weak field regime, forexample, through laboratory experiments and solar system tests. The expanding universe,the region close to a black hole and neutron stars are the regimes of strong gravity. Thequestion is, whether general relativity is the ultimate theory of gravitation, or it requiresmodification in the strong gravity regime. It is important to note that a modified theory ofgravity, which explains observations that general relativity cannot (as we will demonstratehere), should also be able to reproduce observations in the regime where general relativity isadequate (which we will establish below as well). Indeed, it was shown long back that suchmodified gravity theories reveal significant deviations to the general relativistic solutions ofneutron stars [28]. As neutron stars are much more compact than white dwarfs, so far,modified gravity theories have been applied only to them in order to test the validity of suchtheories in the strong field regime. The current venture with white dwarfs is a first in theliterature to the best of our knowledge.In the next section, we briefly recall the basic equations of general relativity and thenmove on to describe the modification of Einstein’s theory which we invoke. Subsequently,we discuss the perturbative solution procedure employed in section 3 and the final resultsobtained in section 4. Finally, we end with conclusions in section 5. We mostly use geometrized units while deriving various equations, which is, c = G = 1, unlessotherwise mentioned, where c is the speed of light and G Newton’s gravitation constant. Wealso use the metric signature ( − , + , + , +).The standard method to arrive at an equation of motion in any field theory is applying avariational principle. In general relativistic field theory, one starts with the Einstein-Hilbert– 2 –ction in 4 dimensions given by [29] S = Z ( L G + L M ) √− g d x = Z (cid:20) π R + L M (cid:21) √− g d x, (2.1)where g is the determinant of the metric tensor g µν (which describes the nature of theunderlying curvature of spacetime), L G the Lagrangian density of the gravitational field and L M the Lagrangian density of the matter field. L G in general relativity is simply R/ π ,where the Ricci scalar R is defined by R = g µν R µν , when R µν is the Ricci tensor, which isdefined as R µν = ∂ λ Γ λ µν − ∂ ν Γ λ λν + Γ λ λσ Γ σ µν − Γ λ σν Γ σ λµ , (2.2)where Γ λ µν = (1 / g λσ ( ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν ), is known as the Christoffel symbol [29]. Ifthe above action, equation (2.1), is varied with respect to g µν and then extremized such that δS = 0, then one obtains the famous Einstein’s field equations G µν = R µν − R g µν = 8 πT µν , (2.3)where G µν is the Einstein tensor and T µν the energy-momentum tensor of the matter field.Now, in a modified gravity theory, the left hand side of equation (2.3) is modified leavingthe right hand side unchanged. One such a very popular class of modified gravity theory isknown as the f ( R ) theory, in which L G is replaced by f ( R ) / π , where f is an arbitraryfunction of R . The action for f ( R ) gravity is thus [30, 31] S = Z (cid:20) π f ( R ) + L M (cid:21) √− g d x, (2.4)varying which with respect to g µν , one arrives at the following modified field equation F ( R ) G µν + 12 g µν ( F ( R ) R − f ( R )) − ( ∇ µ ∇ ν − g µν (cid:3) ) F ( R ) = 8 πT µν , (2.5)where F ( R ) ≡ ∂f ( R ) /∂R , the covariant derivative ∇ µ acting on a vector A ν is defined as ∇ µ A ν = ∂ µ A ν − Γ σ µν A σ and the d’Alembertian operator (cid:3) ≡ ∂ µ ( √− gg µν ∂ ν ) / √− g . Thusan f ( R ) theory reduces to general relativity for f ( R ) = R and, hence, F ( R ) = 1.For the present purpose, we choose the Starobinsky model [32] or the R -squared modelof modified gravity defined as f ( R ) = R + αR , (2.6)where α is a constant having the dimension of length-squared. Henceforth, in the rest of thearticle, by modified gravity effects we would mean the effects of Starobinsky model. Thismodel/theory can be conformally related to a scalar-tensor theory [33], and this connectionhas also been explored very recently in the context of quark stars [34]. These theories havebeen furthermore tested against binary pulsar observations and solar system measurements.From such observations, one obtains the bound on the linear coupling constant of the scalar-tensor theory, α , to be few factor times 10 − (e.g. [28, 35, 36]). In the Starobinsky model, | αR | may be thought of as playing the role equivalent to α , and we will show below that thedimensionless model parameter | αR | in the present work has been indeed restricted to few– 3 –actor times 10 − . Moreover, the strictest astrophysical upper bound on the value of α itselfhas been set by the Gravity Probe B experiment, namely | α | . × cm [37]. Hence,the value of α may not be chosen arbitrarily. In this context, we furthermore mention thatbounds on α have been obtained from other systems as well. For example, the E¨ot-Washlaboratory experiment sets the limit as α . − cm , while the precession of the pulsar B inthe double-pulsar binary system PSR J0737-3039 gives α . . × cm [37]. This apparenthuge difference in the bounds may be explained invoking an underlying “chameleon” effect[38]. This in turn would cause α to vary depending on the characteristic length scale ordensity of the system under consideration.However, similar effects, as of Starobinsky f ( R )-model, could also be obtained in othermodified gravity theories, e.g. Born-Infeld gravity (e.g. [39]). Although, the Starobinskymodel was originally proposed to explain inflation in the early universe, later, it has alsobeen applied to describe neutron stars [40, 41]. Also, modified Starobinsky models, forexample, with logarithmic and cubic corrections, have been used to obtain viable neutronstar solutions [42]. For the Starobinsky f ( R )-model, the modified field equation is of the form G µν + α (cid:20) RG µν + 12 R g µν − ∇ µ ∇ ν − g µν (cid:3) ) R (cid:21) = 8 πT µν . (2.7) Now that we have obtained the modified field equation, equation (2.7), our next step wouldbe to derive from it the corresponding modified
Tolman-Oppenheimer-Volkoff (TOV) equa-tions for this model and subsequently solve them to obtain the structure of the sphericallysymmetric white dwarf. Recall that the TOV equation in general relativity is obtained fromequation (2.3).
Obtaining the modified TOV equations exactly from equation (2.7) is quite complicated andlaborious. Here we adopt a simpler and more intuitive way to deal with the problem, namelythe perturbative method, which has been extensively applied in neutron stars [40–45]. So farwe have not commented about the magnitude of α in equation (2.6), except its astrophysicalconstraint. In the perturbative approach, α is considered to be a small parameter, such that αR ≪
1. Thus the αR term in the Starobinsky model can be considered as a first ordercorrection to general relativity, neglecting higher order corrections. Note that for α = 0,equation (2.7) reduces to equation (2.3), giving back the zeroth-order results correspondingto general relativity.Now, as a first step towards constructing the modified TOV equations in the perturba-tive approach, let us consider the spherically symmetric metric describing the interior of thestar ds = g µν dx µ dx ν = − e φ α dt + e λ α dr + r ( dθ + sin θdφ ) , (3.1)where φ α and λ α are functions of the radial coordinate r . Note that perturbative constraintimplies that g µν also has to be expanded in terms of α as g µν = g (0) µν + αg (1) µν + O ( α ), where– 4 – (0) µν is the metric in general relativity. We consider the matter source to be a perfect fluiddescribed by T µν = ( ρ α + P α ) u µ u ν + P α g µν , (3.2)where ρ α is the density, P α the pressure and u µ the 4-velocity of the fluid. Again, for aperturbative solution we have ρ α = ρ (0) + αρ (1) + O ( α ) and P α = P (0) + αP (1) + O ( α ).All the zeroth order quantities (e.g., φ (0) , λ (0) , ρ (0) and P (0) ) correspond to the solution ofEinstein’s equations in general relativity. Taking all these into account and neglecting termsof O ( α ) and higher, the temporal component ( µ = ν = t ) of equation (2.7) yields − πρ α = − r − + e − λ α (1 − rλ ′ α ) r − + α (cid:20) R (0) ( − r − + e − λ (0) (1 − rλ (0) ′ ) r − )+ 12 R (0)2 + 2 e − λ (0) ( R (0) ′ r − (2 − rλ (0) ′ ) + R (0) ′′ ) (cid:21) , (3.3)while the radial component ( µ = ν = r ) yields8 πP α = − r − + e − λ α (1 + 2 rφ ′ α ) r − + α (cid:20) R (0) ( − r − + e − λ (0) (1 + 2 rφ (0) ′ ) r − )+ 12 R (0)2 + 2 e − λ (0) R (0) ′ r − (2 + rφ (0) ′ ) (cid:21) , (3.4)where prime ( ′ ) denotes single derivative with respect to r and double prime ( ′′ ) denotesdouble derivative with respect to r . Note that we are seeking perturbative solutions onlyup to order α and, hence, for the terms already multiplied by α , we invoke the zeroth orderquantities λ (0) , φ (0) and R (0) . The zeroth order Ricci scalar is defined as R (0) = 8 π ( ρ (0) − P (0) ) , (3.5)which can be obtained by taking the trace of equation (2.3), when note that R = R (0) forequation (2.3). One can furthermore simplify equations (3.3) and (3.4) by using the temporaland radial components of equation (2.3), given by, − πρ (0) = − r − + e − λ (0) (1 − rλ (0) ′ ) r − and 8 πP (0) = − r − + e − λ (0) (1 + 2 rφ (0) ′ ) r − , respectively.Note that the exterior solution of the star is simply the vacuum solution of Einstein’sequations which yields the Schwarzschild metric. Keeping that in mind we assume e − λ α = 1 − M α r , (3.6)where M α = M (0) + αM (1) + O ( α ), is the mass of the star and M (0) = 4 π R ρ (0) r dr , is thezeroth order mass (in general relativity), which corresponds to e − λ (0) = 1 − M (0) r . Usingequation (3.6) and its derivative in equation (3.3), followed by some algebra, one obtains themass equation dM α dr = 4 πr ρ α − α (cid:20) πr ρ (0) R (0) − c G R (0)2 r + R (0) ′ (cid:18) πr ρ (0) + 3 M (0) − c G r (cid:19) − c G R (0) ′′ r − GM (0) c r !(cid:21) , (3.7)– 5 –here we have plugged back c and G to make the equation dimensionful.Next, from equation (3.4) we obtain the following dimensionful equation for the gravi-tational potential φ α ( r ) dφ α dr = G ( πr P α c + M α ) r (1 − GM α c r ) − α (1 − GM α c r ) (cid:20) πrR (0) P (0) Gc − c rR (0)2 + R (0) ′ c − GM (0) r + 4 πP (0) r Gc !(cid:21) , (3.8)which can be replaced in the equation of relativistic hydrostatic equilibrium dP α dr = − (cid:18) ρ α + P α c (cid:19) dφ α dr , (3.9)which is obtained from the conservation of the energy-momentum tensor, g νr ∇ µ T µν = 0.Thus equations (3.7) and (3.9) together form the modified set of TOV equations, whichreduce to the usual TOV equations in general relativity for α = 0. In this context, wemention that in the non-perturbative and exact approach, one can no longer invoke solutionsof Einstein’s equations as zeroth order terms, which is what makes that approach moredifficult to handle numerically. Furthermore, if no perturbative constraints are imposed,then α can be arbitrarily large, which, however, might not respect astrophysically determinedconstraints obtained from perturbative calculations [35–37]. In order to solve the modified TOV equations, one must also supply an equation of state(EoS) relating the pressure and density within the star. In the current work, we use the EoSobtained by Chandrasekhar [46] for non-magnetized, non-rotating white dwarfs, which areconstituted of electron degenerate matter. The pressure and density of such a system arerespectively given by [46] P α = πm e c h [ x (2 x − p x + 1 + 3 sinh − x ] (3.10)and ρ α = 8 πµ e m H ( m e c ) h x , (3.11)where x = p F / ( m e c ), p F is the Fermi momentum, m e the mass of electron, h Planck’sconstant, µ e the mean molecular weight per electron (we choose µ e = 2 for our work) and m H the mass of hydrogen atom. Eliminating x from equations (3.10) and (3.11) yields theEoS for the white dwarf.Finally, the modified TOV equations, accompanied by the above EoS, can now be solvednumerically, subjected to the boundary conditions M α ( r = 0) = 0 and ρ α ( r = 0) = ρ c , where ρ c is the central density of the white dwarf. Note that a particular ρ c , supplied from theEoS, yields a particular mass M ∗ and radius R ∗ for a white dwarf. Hence, by varying ρ c , onecan construct the mass-radius relation for a given EoS. In the current work, we vary ρ c from2 × g/cm to a maximum of 10 g/cm .– 6 – .3 Relativistic Lane-Emden equations for modified gravity In Newtonian stellar structure theory, in order to capture a better physical insight, the hydro-static equilibrium condition combined with Poisson’s equation is recast into a dimensionlessform for a polytropic fluid. This helps in obtaining scaling relations for M ∗ and R ∗ with ρ c ofthe corresponding polytrope, known as the Lane-Emden formalism. With a similar aim, weapply the relativistic Lane-Emden formalism [47] to the modified TOV equations and obtainanalytical expressions for M ∗ and R ∗ (also see [48]). Here the hydrodynamic quantities aredefined in terms of new dimensionless density and mass, θ and η , respectively. The zerothorder density in general relativity transforms as ρ (0) = ρ c θ (0) n , (3.12)where n is the polytropic index. For the polytropic EoS P (0) = Kρ (0)Γ = Kρ /n ) c θ (0)( n +1) , (3.13)where K is a constant and Γ = 1 + (1 /n ). The radial coordinate transforms as r = aξ, (3.14)where a has the dimension of length and is defined as a = ( n + 1) Kρ (1 − n ) /nc πG ! / . (3.15)The zeroth order mass in general relativity transforms as M (0) = 4 πρ c a η (0) . (3.16)Similarly, all the hydrodynamic quantities in our chosen modified gravity theory also trans-form as follows: ρ α = ρ c θ nα , (3.17) P α = Kρ Γ α = Kρ /n ) c θ ( n +1) α , (3.18)and M α = 4 πρ c a η α . (3.19)Thus, in terms of the new dimensionless variables, the two modified TOV equations can becast into the modified relativistic Lane-Emden form. Equation (3.7) becomes dη α dξ = I m = ξ θ nα − αR (0) (cid:20) ξ θ (0) n − ξ θ (0) n − σθ (0) ) − R (0) dR (0) dξ (cid:18) ξ ( n + 1) σ − η (0) − ξ θ (0) n (cid:19) − ξ ( n + 1) σ R (0) d R (0) dξ − η (0) ( n + 1) σξ !(cid:21) , (3.20)while equation (3.9) becomes dθ α dξ = − (1 + σθ α )( η α + ξ σθ ( n +1) α ) ξ (1 − η α ( n +1) σξ ) + αR (0) (1 + σθ α )(1 − η α ( n +1) σξ ) (cid:20) ξσθ (0)( n +1) − ξθ (0) n (1 − σθ (0) )2+ 1 R (0) dR (0) dξ σ ( n + 1) − η (0) ξ + ξ σθ (0)( n +1) !(cid:21) , (3.21)– 7 –here σ = P c / ( ρ c c ), P c being the central pressure of the star. The boundary conditionsrequired to solve these equations are θ α ( ξ = 0) = 1 and η α ( ξ = 0) = 0. Note that αR (0) is a dimensionless quantity and R (0) = πGc ρ c θ (0) n (1 − σθ (0) ). Furthermore, note that for α = 0, the above equations reduce to the relativistic Lane-Emden equations corresponding tothe TOV equations in general relativity, whereas for α = σ = 0 we obtain the Lane-Emdenequations for a Newtonian system.The radius R ∗ of the star is given by R ∗ = aξ = (cid:18) ( n + 1) K πG (cid:19) / ρ (1 − n ) / nc ξ , (3.22)where ξ corresponds to the first zero of the function θ α ( ξ ). The mass M ∗ of the star is givenby M ∗ = 4 π (cid:18) ( n + 1) K πG (cid:19) / ρ (3 − n ) / nc η α ( ξ ) , (3.23)where η α ( ξ ) = R ξ I m dξ (see equation 3.20).Now, for high density ( ρ c & × g/cm ), relativistic white dwarfs, the EoS associatedwith equations (3.10) and (3.11) can be simply described by a n = 3 polytropic EoS with K = (1 / /π ) / hc/ ( µ e m H ) / . The mass and radius for such white dwarfs hence become M ∗ = 4 π (cid:18) KπG (cid:19) / η α ( ξ ) (3.24)and R ∗ = (cid:18) KπG (cid:19) / ρ − / c ξ . (3.25)Note that in the corresponding Newtonian case, M ∗ is completely independent of ρ c ,giving rise to the limiting mass. However, in both general relativity and modified gravity, M ∗ implicitly depends on ρ c through the parameter σ , which determines η α ( ξ ). Finally, we move on to describe the results obtained from our calculations, which are illus-trated in Figure 1, for α = 0 (confirming known results) and in Figure 2, for different valuesof α . The choice of different α -s and corresponding different results imply the variation of α with space (and/or space-time). A more physical model should automatically account forthe variation of α or equivalent parameter(s) with the density of white dwarfs. The presentwork just argues for different results at different α -s corresponding to the different regimes ofdensity. In future, we should repeat the present work using a viable f ( R ) model (e.g. [38, 49])consistent with solar system constraints, which reveals the density dependent modificationto the gravity effects, by exploiting, e.g., the so-called chameleon effect. α = 0In Figure 1, we recall the well known results where we compare the Newtonian solutionswith those in general relativity, i.e., the α = 0 case. Figures 1(a) and (b) confirm that in theNewtonian case, with the increase of ρ c , R ∗ decreases and M ∗ increases, until it saturates to amaximum mass M max ∼ . M ⊙ , which is the famous Chandrasekhar limit. We furthermore– 8 – R * M * ρ c (b)(a) Figure 1 . Comparison of solutions in the Newtonian case (solid lines) and the general relativistic or α = 0 case (dotted lines). (a) Mass-radius relations. (b) Variation of ρ c with M ∗ . ρ c , M ∗ and R ∗ arein units of 10 g/cm , M ⊙ and 1000 km, respectively. confirm that in the general relativistic case, with the increase of ρ c , R ∗ decreases but M ∗ increases until it reaches M max = 1 . M ⊙ at ρ c = 3 . × g/cm . A further increase in ρ c results in a slight decrease in M ∗ , indicating the onset of an unstable branch, which isabsent in the Newtonian case [46]. For low density white dwarfs having ρ c < g/cm , theNewtonian and general relativistic M ∗ − ρ c curves are identical. However, for ρ c & g/cm ,general relativistic effects become important, resulting in a slightly smaller M ∗ compared tothe Newtonian case and eventually leading to a smaller M max . Once this M max is approached,by further gaining mass, white dwarfs contract, causing an increase in the core temperatureand, finally, leading to runaway thermonuclear reactions, which result in SNeIa. α < α < α much less than the strictest astrophysicalupper bound. Figure 2(b) shows that for ρ c < g/cm , all the three M ∗ − ρ c curves areindistinguishable from the α = 0 case (note that the α = 0 curves in Figure 2 are identicalto the dotted lines in Figure 1). As ρ c increases beyond 10 g/cm , the curves deviate moreand more due to modified gravity effects. This feature beautifully establishes the necessaryconstraint that a modified gravity theory should replicate general relativistic results in theappropriate regime, which for white dwarfs is the low density regime. This furthermore, very– 9 – R * M * ρ c (a) −3.50028100100 (b) Figure 2 . Unification diagram for SNeIa. (a) Mass-radius relations. (b) Variation of ρ c with M ∗ .The numbers adjacent to the various lines denote α/ (10 cm ). ρ c , M ∗ and R ∗ are in units of 10 g/cm , M ⊙ and 1000 km, respectively. importantly, reveals that modified gravity has a tremendous impact on white dwarfs which sofar was completely overlooked, whereas general relativistic effect itself is non-negligible. Notethat the values of M max for all the three cases correspond to ρ c = 10 g/cm , an upper limitchosen to avoid possible neutronization. Interestingly, all values of M max are highly super-Chandrasekhar, ranging from 1 . − . M ⊙ . The corresponding values of ρ c are large enoughto initiate thermonuclear reactions, e.g. they are larger than ρ c corresponding to M max of α = 0 case, whereas the core temperatures of the respective limiting mass white dwarfsare expected to be similar.This could explain the entire range of the observed super-SNeIamentioned above [2, 3, 12–15]. While the general relativistic effect is very small, modifiedgravity effect could, according to the perturbative f ( R )-model, lead to ∼ α -equivalent parameter were shown to reveal large deviations in their mass[28]. The modified gravity effect particularly is pronounced at the high density regime, evenif α is very small. We also find that unlike the α = 0 case, the M ∗ − R ∗ relations for whitedwarfs having α < ρ c increases, M ∗ always increasesas seen in Figure 2(b). The results of the Lane-Emden solutions for several α < ρ c = 10 g/cm , which yields M ∗ = M max . We observe that M max (andthe corresponding R ∗ ) for the three α < able 1 . Maximum mass from relativistic Lane-Emden solutions for modified gravity with α < n = 3 and σ = 2 . × − , where α = α/ (10 cm ). α ξ η α ( ξ ) M max ( M ⊙ ) R ∗ (1000 km) | αR (0) | max | − g (0) tt /g tt | max | − g (0) rr /g rr | max -1 5.7832 2.49083 1.772 0.6027 0.00184 0.0016 0.0052-2 5.1032 3.00800 2.139 0.5318 0.00369 0.0031 0.0108-2.5 4.8507 3.26938 2.325 0.5055 0.00462 0.0038 0.0138-3 4.6375 3.53262 2.513 0.4833 0.00554 0.0045 0.0168-3.5 4.4547 3.79753 2.701 0.4643 0.00646 0.0052 0.0199 solving the modified TOV equations) agree with the values in Table 1.The last three columns of Table 1 list three extra parameters that give a measure forensuring the perturbative validity of the solutions for a chosen α , which we hereby define.Recall that we solve the modified TOV equations only up to O ( α ) and since the product αR is first order in α , we replace R in it by R (0) , given by equation (3.5). The maximumvalue of | αR (0) | max occurs at the center of the white dwarf, and for the perturbative validityof the entire solution, | αR (0) | max ≪ g (0) tt /g tt and g (0) rr /g rr , which should be close to 1 for the validity of perturbative approach [50]. Weconsider the maximum deviation of these quantities from 1, such that | − g (0) tt /g tt | max ≪ | − g (0) rr /g rr | max ≪ − α -s are perfectlyin accordance with the observational constraints [35–37]. Note, furthermore, that there isno universal quantity which gives an absolute measure of the allowed deviation from generalrelativity in the perturbative approach and, hence, we discuss above at least three suchpossible quantities. An additional estimate of the error may be obtained from a quantitydefined in [40], which we denote here as, δ P = ( dP α /dr ) / ( dP (0) /dr ) −
1. For perturbativevalidity of the solution | δ P | .
1, a condition satisfied for the cases listed in Table 1. α > α > M ∗ − ρ c curves overlap withthe α = 0 curve in the low density region. However, with the increase in the magnitude of α ,the region of overlap recedes to a lower ρ c . Modified gravity effects set in at ρ c & , × and 2 × g/cm , for α = 2 × cm , 8 × cm and 10 cm respectively. For agiven α , with the increase of ρ c , M ∗ first increases, reaches a maximum ( M max ) and thendecreases, like the α = 0 case. With the increase of α , M max decreases and, interestingly, for α = 10 cm , it is highly sub-Chandrasekhar (0 . M ⊙ ). In fact, M max for all the chosen α > . − . M ⊙ . This is a remarkable finding since itestablishes that even if the values of ρ c for these sub-Chandrasekhar maximum/limiting masswhite dwarfs are lower than the conventional value at which SNeIa are usually triggered, anattempt to increase the mass beyond M max , for a given α , will lead to a gravitational in-stability. This presumably will be followed by a runaway thermonuclear reaction, providedthe core temperature increases sufficiently due to collapse. One might wonder if such lowdensity sub-Chandrasekhar limiting mass white dwarfs can attain conditions suitable to ini-tiate a detonation, which would give way to a SNIa. Interestingly, the occurrence of such adetonation has already been demonstrated in white dwarfs having densities as low as ∼ – 11 –/cm , provided certain background conditions are satisfied [51]. Thus, once the maximummass is approached, a SNIa is expected to trigger just like in the α = 0 case. The explosionsof these sub-Chandrasekhar white dwarfs could explain the sub-SNeIa [4–9], like SN 1991bgmentioned above, because a small progenitor mass will consequently yield a small Ni massleading to an under-luminous event. Note that, as evident from Figure 2(b), the M ∗ − ρ c curves for the α > ρ c s unlike the α < ρ c exceeds a certain value for a given positive α , the numerical/mathematical solutionsreveal a region of negative mass within the white dwarf with an overall positive M ∗ . With afurther increase in ρ c , the entire M ∗ becomes negative. These are unphysical scenarios and,hence, in Figure 2, we present the α > ρ c for which the mass ispositive throughout the white dwarf.We now check the validity of the perturbative approach for the α > M max . For α = 2 × cm , | αR (0) | max = 7 . × − , | − g (0) tt /g tt | max =6 . × − and | − g (0) rr /g rr | max = 2 × − ; for α = 8 × cm , | αR (0) | max = 7 . × − , | − g (0) tt /g tt | max = 6 . × − and | − g (0) rr /g rr | max = 2 . × − ; and for α = 10 cm , | αR (0) | max = 7 . × − , | − g (0) tt /g tt | max = 6 . × − and | − g (0) rr /g rr | max = 2 × − . Thisensures that the solutions are within the perturbative regime and are perfectly in accordancewith the observational constraints [35–37]. Also, | δ P | < α > α (perturbative limit) lead to a decrease in themass of neutron stars, while larger positive values of α (non-perturbative regime) lead toan increase in mass, with respect to that in the general relativistic case. Our preliminarycalculation, which is beyond the scope of the current paper, shows a similar trend in whitedwarfs. In future, we plan to report the non-perturbative results for white dwarfs based onestablished numerical techniques [53, 54]. Based on a simple f ( R )-model, we show, for the first time in the literature to the best ofour knowledge, that modified gravity effects are significant in high density white dwarfs.Consideration of such effects in white dwarfs appears to be indispensable, since it appearsto be remarkably explaining and unifying a wide range of observations for which generalrelativity may be insufficient. Importantly, we are also able to show that the f ( R )-modelchosen in our work successfully reproduces the low density white dwarfs and their basicproperties, which are already explained in the paradigm of general relativity (and Newtonianframework).We note here that the perturbative method is adequate for the present study, as thenwe have a handle on α characterizing our model, which cannot be arbitrarily large, allowingit to be constrained directly by astrophysical observations. In our work, for the super-Chandrasekhar limiting mass white dwarfs, α ranges from − to − . × cm , whilefor the sub-Chandrasekhar limiting mass white dwarfs, the range is 2 × to 10 cm .Hence, the range of α chosen in our work is well within the astrophysical bound set by theGravity Probe B experiment, namely | α | . × cm [37].Furthermore, even though α is assumed to be constant within individual white dwarfshere, there is indeed an implicit dependence of α on the central density, particularly of thelimiting mass white dwarfs presumably leading to SNeIa, as is evident from Figure 2(b).– 12 –his indicates the existence of a chameleon-like effect in observed SNIa progenitors. A moresophisticated calculation, which invokes an (effective) α that varies explicitly with density, islikely to yield results similar to those we have already obtained in this work.Depending on the magnitude and sign of α , we are not only able to obtain both highlysuper-Chandrasekhar (for α <
0) and highly sub-Chandrasekhar (for α >
0) limiting masswhite dwarfs, but we can also establish them as progenitors of the peculiar, super-SNeIaand sub-SNeIa, respectively. Thus, an effectively single underlying theory, inspired by theneed to modify Einstein’s theory of general relativity, appears to be able to unify the twoapparently disjoint sub-classes of SNeIa, and, hence, serves as a missing link, which have sofar hugely puzzled astronomers. The significance of the current work lies in the fact that itnot only questions the uniqueness of the Chandrasekhar mass-limit for white dwarfs, but italso argues for the need of a modified theory of gravity to explain astrophysical observations.
Acknowledgments
B.M. acknowledges partial support through research Grant No. ISRO/RES/2/367/10-11.U.D. thanks CSIR, India for financial support. The authors would like to thank K. Y. Ek¸sifor useful discussion. Thanks are also due to the anonymous referee for the suggestions toimprove the presentation of the paper.
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