Cosmological Constant Effects on the Properties of Mass Twin Compact Stars
CCosmological Constant Effects on the Properties of MassTwin Compact Stars
Noshad
Khosravi Largani , ∗ and David Edwin
Álvarez-Castillo , ∗∗ Department of Physics, Alzahra University, Tehran, 1993893973, Iran Bogoliubov Laboratory of Theoretical Physics, JINR Dubna, 141980 Dubna, Russia
Abstract.
We present a systematic investigation of the cosmological constant e ff ects incompact stars interiors in the framework of Einstein’s gravity. Considerationof a cosmological constant Λ in compact stars is motivated by the mechanismof acceleration of the observable universe, where Λ is usually related to thedark energy. In particular, we consider compact star mass twins, hybrid neutronstars that populate both the second and third branch of the mass-radius diagram.For those models, the need of consideration of excluded volume e ff ects in theequation of state, resulting from the finite size volume of nucleons, leads to asti ff ening of matter causing compact stars to acquire higher mass and radiusvalues. We demonstrate that certain values of the cosmological constant canalso modify the compact star properties but in an opposite way. In addition, wefind that the inclusion of Λ can have a similar e ff ect to the existence of pastaphases at the hadron-quark interface. The physics of neutron stars is currently a very active topic of investigation, covering manyaspects of the contemporary physics. These include, but not only, astrophysical processes likeenergetic emissions, stellar evolution, studying the dense matter properties in compact star in-teriors, and element nucleosynthesis. The recent detection of gravitational radiation from theevent GW170817 [1] helped to clarify many of the questions related to the aforementionedphenomena, like corroborating that the only possibility of creation of certain heavy elementsis through the fusion of neutron stars and its association with kilonovae, as well as determina-tion of masses of the binary system. On the other hand, Cosmological studies are dedicatedto the understanding of the dynamics of universe as a whole as well as considering the roleof the fundamental interactions within it. In this respect, Λ is one of the necessary ingredi-ents for the description of the observable universe, providing a mechanism for its acceleratedexpansion, somehow related to the so called dark energy.In this work, we take neutron stars as a probe of the possible e ff ects of the cosmologicalconstant as a free parameter, and allow it to vary inside the star, where matter e ff ects mightchange its strength, in contrast to the conditions existing in vacuum. We consider two up-to-date realistic equations of state (EoS) and look at the resulting macroscopic changes in ∗ e-mail: [email protected] ∗∗ e-mail: [email protected] a r X i v : . [ a s t r o - ph . H E ] N ov he stars, namely mass and radius. Of these two chosen EoS, the second features two masstwin stars located in the second and third branch in the mass-radius diagram and brings thepossibility of probing a critical point in the QCD phase diagram [2]. The approach in thispaper is indeed motivated by the fact that whenever one faces a new phenomenon that cannotbe explained with the already existing theories, two evident possibilities exist: a) assumingthat there is a completely new theory which has not been developed yet, or b) trying to modifythe current valid paradigm in order to broaden the domain of exploration. Our Λ parameterin this study might account for vacuum energy e ff ects [3] as well as to correspond to somealternative gravity theories parameter under certain limits, thus providing a useful benchmark. Although it seems that Einstein’s General Relativity is working perfectly well in our solarsystem, some problems including a reason for accelerated expansion of the universe and thelack of precise data beyond this scale has made some physicists to think out the box and toconsider alternative theories of gravity such as Scalar-Tensor theories, f ( R ) models, to namea few. Moreover, many approaches have started to modify many of the di ff erent aspects ofthe cosmological standard model, the so called Λ CDM. Nevertheless, the recent GW170817event detection of simultaneous electromagnetic and gravitational signals was able to rule outmany of these possibilities [4].
The standard way of modelling neutron stars starts by considering a static, spherical and notrotating stellar object under a given theory of gravity which will require interior and externalsolutions matching properly at the boundaries. In the case of general relativity, this treatmentleads to the Tolman-Volkov-Oppenheimer equations (TOV) [5, 6] (here written in the naturalunits c = G = dmdr = π r (cid:15) (1) dPdr = − ( (cid:15) + P )( m + π r P ) r ( r − m ) (2)where integration of the first equations will determine the total mass M of the star, where asthe second one will allow for determination of the pressure profile as a function of radius aswell as the total stellar radius R when P ( r = R ) =
0. This equations are to be solved by choos-ing a central density (cid:15) c for a given star and increasing it to obtain another more massive upto the so called maximum mass M Max , the last configuration stable against radial oscillationsproduced when gravity takes over. Most of alternative theories of gravity attempts of de-scription of compact stars end up with modifications to these equations. Those modificationswill modify the mass-radius diagram by increasing or decreasing either values, like chang-ing the maximum mass or minimal radius. For the most general free variation of parametersthat can be included the TOV equations see [7], where a classification of the correspondingmodifications for most popular alternative gravity theories can be found.
In order to test the e ff ect of the cosmological constant parameter we fix the equation of state.We utilise two realistic EoS that fulfil modern constraints from terrestrial laboratory exper-iments as well as astrophysical observations. In one hand, we select the DD2 relativisticensity functional model EoS [8] which is used to describe a classical neutron star whoseinterior contains only protons and neutrons and leptons like electrons and possibly muons aswell. Its great quality relies on the fact that it has been calibrated to nuclear physics mea-surements of properties like parameter values at saturation as well as those of the symmetryenergy.In addition, we consider in our study a multi-polytrope (MP) EoS whose parameters werefit to realistic EoS in the lower density regions and features a strong first order phase transitionat a high density value which can represent the transition from hadron to quark matter insidecompact stars. This EoS model has been introduced in [9] where the authors discuss the highmass twins phenomenon (HMTs) where stars with very similar masses are located in di ff erentbranches of the mass-radius diagram therefore having considerable di ff erent radii ( ≈ − ff erence). This case is of two-fold relevance since it serves to solve several microscopicissues like the reconfinement, masquerades, and the hyperon puzzle (see [10] for an extendeddiscussion) as well as providing mass and radius predictions potentially observable by nowa-days facilities like the NICER X-ray detector [11] and gravitational wave observatories thatcan estimate tidal deformabilities that strongly related to those stellar properties [12–14]. In this section we present results of our investigation on the e ff ects of the cosmological con-stant inside compact stars. We start by considering the same general relativity TOV of equa-tions with the inclusion of a Λ parameter [15, 16], where the mass equation (1) remains thesame but the one for the pressure (2) is modified as: dPdr = − ( (cid:15) + P )( m + π r P + Λ r / r ( r − m + Λ r / . (3)It is then clear that the term Λ r / ff ectivelymodify the internal pressure profiles of stars. These new equations are solved under the sameinitial and boundary conditions as in the zero cosmological constant case. The resulting mass-radius diagrams for both equations of state considered here are presented in figure 1, wherethe left panel corresponds to the pure hadronic DD2 EoS and the right panel to the MP-HMTsEoS.Within our figures, we also present shaded regions that correspond to mass estimatesfor several objects. Such values are often derived from measurements of other quantitiestherefore in order to extract the corresponding mass values, general relativity is applied tothe modelling of the involved physical process. Examples include binary systems dynamics,gravitational redshifts, gravitational wave emission and tidal deformabilities, among others.With that being said, it is necessary to include the modifications to gravity into the mass es-timation from the measured quantities. Nevertheless, we include the standard mass measure-ments in our mass-radius plots as a reference. We summarise our findings in the followingsection. In this paper we have investigated changes that can be possibly occur in the compact starmorphology by considering the variation of a parameter that we associate with the cosmolog-ical constant Λ . The main motivation is to explore possible e ff ects of Λ with matter insideneutron stars, where the quantum vacuum might behave di ff erently than in the cosmic voids.
10 11 12 13 14 15 16R [km]0.60.811.21.41.61.822.2 M [ M O . ] PSR J0437-4715PSR J1614-2230PSR J0348+0432M M L =0.0 m -2 L =2.5·10 -10 m -2 L =5.0·10 -10 m -2 L =7.5·10 -10 m -2 L =1.0·10 -9 m -2 L =1.5·10 -9 m -2 L =2.5·10 -9 m -2 L =5.0·10 -9 m -2 L =7.5·10 -9 m -2 L =1.0·10 -8 m -2 M [ M O . ] PSR J0437-4715 PSR J1614-2230PSR J0348+0432M M L =0.0 m -2 L =1.0·10 -10 m -2 L =2.5·10 -10 m -2 L =5.0·10 -10 m -2 L =7.5·10 -10 m -2 L =1.0·10 -9 m -2 L =2.5·10 -9 m -2 L =5.0·10 -10 m -2 L =7.5·10 -9 m -2 L =1.0·10 -8 m -2 Figure 1.
Mass-Radius for compact stars under the influence of a cosmological constant Λ . Theshaded colourful areas correspond to measurements of neutron stars, see [14] for details. Left column:DD2 EoS. Cosmological parameter values can shrink mass and radius values, eventually changing thetopology from a standard hadronic EoS (black lines) into a quark EoS (red lines) in the framework ofgeneral relativity with zero cosmological constant. Right column: High mass twins (MP-HMTs) EoS.The strength of the Λ parameter can wipe out the mass twins (black lines) leaving a quark-like EoS (redlines) with a characteristic kink due to the strong first order phase transition in the HMTs EoS. Thisphenomenon is similar to an internal softening of the EoS when a mixed phase featuring geometricalstructures appear [25, 26], however no kink is present in that case. In the other hand, alternative theories of gravity often modify the compact star structure aswell. For instance, the inflaton model presented in [17] features a scalar field that modifiesthe internal stellar pressure. Other examples include chameleons fields [18] and f ( R ) theo-ries [19], as well as more involved approaches like Palatini gravity [20]. In our endeavour forexploring related e ff ects we have chosen two state-of-the-art equations of state and solved thegeneral relativistic equations with cosmological constant as presented in [16, 21]. We havefound that the typical values for Λ to have an observable e ff ect in compact stars are of theorder of Λ = − − − m − . Our first observation is that the Λ values we obtain di ff erin up to five orders of magnitud from those of [16], apparently due to the sti ff ness of ourEoS sets in contrast to the soft EoS presented there. The sti ff ness in our models is indeedan important requirement to support massive compact stars, like the most massive measuredone of 2M (cid:12) [22]. Moreover, we have found that the strength of Λ can change the topologyof mass-radius curve, see figure 1. The general tendency is that higher values of cosmolog-ical constant tend to lower the maximum mass and shrink the radius. This e ff ect is quitethe opposite to the rise of pressure from the microscopic description of the equation of statewhen for instance an excluded volume of the nucleons produced by the Pauli blocking oftheir internal quarks is considered [23, 24]. As Λ is increased pure hadronic stars changetheir appearance from what is the standard hadronic mass-radius curve to the typical one ofa quark star, an object composed of a plasma of quarks bound only by gravity. Furthermore,this e ff ect is also present in the HMTs stars, with the Λ highest values causing the instabilitiesin the mass-radius diagram to disappear thus frustrating the existence of mass twins. A simi-lar mass-radius behaviour has been observed when the EoS is modified by considering pastaphases at the hadron-quark phase transition [25, 26]. However, for the cosmological param-ter case, there remains a sharp kink in the mass-radius relation that evidences the nature ofits EoS and the strong first order phase transition in it. The authors would like to express their gratitude to the organisers of the AYSS 2018 con-ference for their hospitality and support. D.E.A-C. acknowledges support from the Ter-Antonyan-Smorodinsky program for collaboration between Armenian Institutions and JINR.
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