Perspectives on constraining a cosmological constant-type parameter with pulsar timing in the Galactic Center
Lorenzo Iorio
gr-qc astro-ph.GA physics.space-ph



ŝ
cos2i
sin2i
cos2ε
sin2ε
cosi
sini
cosε
sinε
√(1-e^2)
·
·
·
·
·
·
·
·
·
·
L̂
l̂
m̂
S^A
S^B
𝒮^A
𝒮^B
𝒮^p
𝒮^c
𝒮^A
𝒮^B
Ŝ^A
Ŝ^B
Ŝ^p/c
Ŝ^p
Ŝ^c
Ŝ_x^A
Ŝ_y^A
Ŝ_z^A
Ŝ_x^B
Ŝ_y^B
Ŝ_z^B
Ŝ_x
Ŝ_y
Ŝ_z
M_A
M_B
ρ̂
σ̂
ν̂
ê_x
ê_y
ê_z
n_b
n

 

r̂^̂'̂

.
.
sinf
cosf
cosu
sinu
L̂_x
L̂_y
L̂_z
P_b



Sgr A^∗
#1Equation (<ref>)
#1#2Equations (<ref>) to (<ref>)
#1Eq. (<ref>)
#1#2Eq. (<ref>)-eq. (<ref>)
#1#2∂#1∂#2
#1#2d#1/d#2
#1#1             

#1“#1"
.
.
#1#2#3[#1#2]#3

    Ω#1#2#1#2#1k̂ŵ_xŵ_yŵ_z#1#1
        R_ rm_ rI_ r 2I_ r 2I 2ω 2Ω_ r 2_ rΔΔ IΣ I#1#2cos^#1#2#1#2sin^#1#2sincossin 2cos 2sincossin 2cos 2cosωsinωcos 2ωsin 2ωcosΩsinΩcos 2sin 2cos_ rsin_ rcos 2_ rsin 2_ rcos Isin Icos 2Isin 2Icos I^'sin I^'cos 2I_ rsin 2I_ r
        1 - e^2√(1 - e^2)e^2#1#21 + #1#2e^2#1#21 - #1#2e^2
        #1(#1)#1[#1]#1{#1}#1⟨ #1⟩
        
        
        
        
        
        mn2e
        
        
        
        
        4pc
        4pc4pc1
        [1]
        
        L. IorioMinistero dell'Istruzione, dell'Università e della Ricerca
        (M.I.U.R.)-Istruzione
        
         Permanent address for correspondence: Viale Unità di Italia 68, 70125, Bari (BA),
        Italy
        Independent tests aiming to constrain the value of the cosmological constant Λ are usually difficult because of its extreme smallness (Λ≃ 1× 10^-52 m^-2, or 2.89× 10^-122 in Planck units). Bounds on it from Solar System orbital motions determined with spacecraft tracking are currently at the ≃ 10^-43-10^-44 m^-2 (5-1× 10^-113 in Planck units) level, but they may turn out to be somewhat optimistic since Λ has not yet been explicitly modeled in the planetary data reductions. Accurate (σ_τ_p≃ 1-10 s) timing of expected pulsars orbiting the Black Hole at the Galactic Center, preferably along highly eccentric and wide orbits,  might, at least in principle, improve the planetary constraints by several orders of magnitude.
        
        By looking at the average time shift per orbit Δδτ^Λ_p, a S2-like orbital configuration with e=0.8839, =16 yr would allow to obtain preliminarily an upper bound of the order of|Λ|≲ 9× 10^-47 m^-2 (≲ 2× 10^-116 in Planck units)if only σ_τ_p were to be considered.
        
        Our results can be easily extended to modified models of gravity using Λ-type parameters.
        
        
        
        keywords
        gravitation–pulsars: general–stars: black holes
        
        
        
        
        § INTRODUCTION
        intro
        The cosmological constant (CC) Λ<cit.> is the easiest way to explain certain large-scale features of the universe like the acceleration of its expansion  <cit.> and the growth of fluctuations by gravity <cit.> within General Relativity (GR) assumed as a fundamental ingredient of the standard ΛCDM model <cit.>; for a recent overview of the status and future challenges of the Einsteinian theory of gravitation, see, e.g., <cit.>. Interestingly, the CC was considered before Einstein for possible modification of the Poisson equation in the framework of the Newtonian gravity <cit.>.
        The CC can be expressed in terms of the Hubble parameter H_0 and the ratio Ω_Λ between the density due to the cosmological constant itself ϱ_Λ = 1/8 c^2 Λ G^-1 and the critical density ϱ_crit = 3/8 H_0^2 G^-1 as Λ = 3 H_0^2 Ω_Λ c^-2, where <cit.>H_0= 67.74± 0.46 km s^-1 Mpc^-1, Ω_Λ= 0.6911± 0.0062. As such, its most recent value inferable from the  measurements of the Cosmic Microwave Background (CMB) power spectra by the satellite Planck reads
        Λ = (1.11± 0.02)×10^-52 m^-2Lambda.
        
        In order to relate it to possible symmetry breaking in gravity <cit.>, the CC is sometimes written as a very tiny dimensionless parameter essentially by multiplying it by the square of the Planck length ℓ_P=√(ħ G c^-3)=1.61× 10^-35 m. Thus, one gets, in Planck units,
        Λ = 2.89× 10^-122.
        
        A CC-type parameterization occurs also in several classes of long range modified models of gravity aiming to explain in a unified way seemingly distinct features of the cosmic dynamics like inflation, late-time acceleration and even dark matter <cit.>.
        
        Ever since the time of Einstein, who, on the backdrop of what is mathematically feasible with the Poisson equation, included Λ in his GR field equations to obtain a non-expanding, static cosmological model <cit.>, the introduction of the CC has always been justified from an observational/experimental point of view by arguing that it would not be in contrast with any observed effects in local systems like, e.g., orbital motions in gravitationally bound binary systems because of its extreme smallness. As a consequence, there are not yet independent, non-cosmological tests of the CC itself for which only relatively loose constraints from planetary motions of the Solar System exist in the literature. So far, most of the investigations on the consequences  of the CC in local binary systems have focused on the anomalous pericenter precession induced by Λ<cit.> on the basis of a Hooke-type perturbing potential <cit.>
        U_Λ= -16Λ c^2 r^2Ulam
        arising in the framework of the Schwarzschild-de Sitter spacetime <cit.>.
        Equation <ref> yields the radial extra-acceleration <cit.> A_Λ = 13Λ c^2𝐫.Accel
        The latest upper limits on the absolute value of Λ, inferred within the framework of fT gravity from the anomalous perihelion precessions of some of the planets of the Solar System tightly constrained with the INPOP10a ephemerides <cit.>, are of the order of <cit.>|Λ| ≲ 2× 10^-43 m^-2bounda,
        
        corresponding to
        |Λ| ≲ 5× 10^-113
        in Planck units.
        
        The Earth-Saturn range residuals constructed from the telemetry of the Cassini spacecraft <cit.> yielded an upper limit of the order of <cit.>|Λ| ≲ 5× 10^-44 m^-2boundb,
        i.e.
        |Λ| ≲ 1× 10^-113
        in Planck units.
        <cit.> suggested that a challenging analysis of the telemetry of the New Horizons spacecraft might improve the limit of boundb by about one order of magnitude.
        On the other hand, the bounds of boundaboundb may be somehow optimistic since they were inferred without explicitly modeling Accel in the dynamical force models of the ephemerides. As such, its signature may have been removed from the post-fit residuals to a certain extent, being partially absorbed in the estimation of, e.g., the planets' initial state vectors.
        
        Such a possibility was investigated  by simulating observations of major bodies of Solar System in the case of some modified models of gravity <cit.>.
        
        Thus, more realistic constraints might yield larger values for the allowed upper bound on Λ.
        
        In this paper, we will show that the future, long waited discovery of pulsars revolving around the putative Supermassive Black Hole (SMBH) in the Galactic Center (GC) at Sgr A^∗<cit.> along sufficiently wide and eccentric orbits and their timing accurate to the σ_τ_p≃ 1-10 s level <cit.>, might allow, in principle, to substantially improve the planetary bounds of boundaboundb by several orders of magnitude, getting, perhaps, closer to the level of Lambda itself under certain fortunate conditions.
        
        The possibility that travelling gravitational waves can be used in a foreseeable future for local measurements of the CC through their impact on Pulsar Timing Arrays (PTA) is discussed in <cit.>.
        
        In Section <ref> we will analytically work out the perturbation Δδτ^Λ_p induced by Λ on the  the pulsar's timing periodic variation δτ_p due to its orbital motion around the SMBH; we will follow the approach put forth in <cit.> applying it to Accel. We will neglect the time shifts due to the CC on the propagation of the electromagnetic waves <cit.>. Despite it can be shown that, for certain values of the initial conditions, an extremely wide  orbital configuration like, say, that of the actually existing star S85 may yield values of the instantaneous changes Δδτ^Λ_pt as large as just ≃ 1-10 s, caution is in order because of, e.g., the very likely systematic bias induced on such an extended orbit by the poorly known mass background in the GC <cit.>. Moreover, also the accurate knowledge of the SMBH physical parameters like mass, angular momentum and quadrupole moment would be of crucial importance because of the competing pN orbital timing signatures Δδτ^pN_p which would superimpose to the CC effect. Finally, also the orbital parameters of the pulsar should be determined over a relatively short time interval Δ T with respect to its extremely long orbital period P_b. If, instead, a closer pulsar is considered, it makes sense to look at its net orbital time shift per orbit Δδτ_p^Λ. <cit.> recently investigated the possibility of constraining the SMBH's spin with such kind of rapidly orbiting pulsars. See also <cit.>. In Section <ref> it will be shown that a S2-type orbital geometry, summarized in Table <ref>, would allow, in principle, to improve the planetary bounds of boundaboundb by about 3-4 orders of magnitude. A strategy to overcome the potentially serious bias posed by the competing post-Newtonian (pN) orbital time delays driven by the SMBHS's mass, spin and quadrupole moment will be discussed as well. In Section <ref>, we summarize our findings and offer our conclusions.
        
        
        
        
        § CALCULATING THE PERTURBATION OF THE ORBITAL COMPONENT OF THE TIME SHIFT DUE TO THE COSMOLOGICAL CONSTANT
        calcolo
        Here, the analytical method devised in <cit.>, relying upon <cit.>, will be applied to the perturbing acceleration of Accel with some technical modifications. Indeed, since, in this case, the use of the eccentric anomaly E as fast variable of integration instead of the true anomaly f turns out to be computationally more convenient, Equations (30) to (31) of <cit.>, giving the radial and transverse components of the perturbation Δ𝐫 of the position vector 𝐫 and used in <cit.> as Equations (3) to (4), have to be replaced with Equations (36) to (37) of <cit.>, i.e.
        
            Δr_ρEDr   =rEa Δ aE -rEe + cos f1-e^2 Δ eE + rEesin f√(1-e^2) Δ EE, 
            
            
            Δr_σEDtau   = rEsin f 1-e^2 Δ eE + a √(1-e^2) Δ EE + rEcos I ΔΩE+ΔωE.
        
        Equation (32) of <cit.>, giving the out-of-plane component Δr_ν of the perturbation Δ𝐫 of the position vector 𝐫 and used in <cit.> as Equation (5), remains unchanged.
        Thus, the perturbation of the z component of the pulsar's position vector 𝐫 reads
        
            Δr_z    = rEasin Isin u Δ aE-rEsin Isinω+esin u1-e^2 Δ eE+rEcos Isin u Δ IE+ 
            
            
               + rEsin Icos u ΔωE + sin Ia1-e^2cos u +erEsin fsin u  √(1-e^2) Δ EEDz.
        
        From <cit.>, it is Δδτ_p = Δr_z c^-1 in a coordinate system whose reference z axis points towards the observer perpendicularly to the plane of the sky spanned by the reference x, y plane.
        In DrDz, the instantaneous shift Δ EE of the eccentric anomaly can be expressed, in turn, in terms of the perturbations ΔℳE, Δ eE of the mean anomaly and the eccentricity, respectively, according to Equation (A.5) of <cit.>, i.e.
        Δ EE = arEΔℳE+sin E Δ eE.casot
        The instantaneous shifts  of the osculating orbital elements are to be computed in terms of E as
        ΔκE =∫_E_0^EκttE^'dE^', κ=a, e, I, Ω, ω
        with the aid of the standard formulas of celestial mechanics
        
            sin f     = √(1-e^2) sin E1 - ecos Esinf, 
            
            
            cos f     = cos E -e1 - ecos E, 
            
            
            rE    = a1- ecos ErE, 
            
            
            tE    = 1 - ecos E
        
        applied to the usual Gauss equations for the variation of the elements yielding dκ/dt.
        The calculation of the perturbation ΔℳE of the mean anomaly has to be performed as shown in <cit.>, whose
        Equations (20) to (21) are to be calculated with E.
        The CC-induced instantaneous perturbations of the osculating orbital elements turn out to be
        
        
            Δ aE    = c^2Λ a ecos E-cos E_0-2 +ecos E+cos E_03 ^2 sma, 
            
            
            Δ eE    = c^2Λ1-e^2cos E-cos E_0-2 +ecos E+cos E_06 ^2 ecce, 
            
            
            Δ IE    = 0, 
            
            
            ΔΩE    = 0, 
            
            
            ΔωE    = c^2Λ√(1-e^2)12e^24 1 + e^2sin E_0 - e 6 E_0 - 6 E + sin 2 E_0 -.
            
               . -  4 1 + e^2sin E + e sin 2 Eomega, 
            
            
            ΔℳE    = c^2Λ72 e^2 12 e 7 + 6 e^2E_0 - E - 4 6 + 54 e^2 + 7 e^4sin E_0 +
            6 e sin 2 E_0 + .
            
            
               . + 3 8 + 72 e^2 + 7 e^4sin E + 2 e^3  9E-E_0 + e2sin E_0 - 9sin Ecos 2 E_0 +.
            
            
               +. 6 e^2  7 e sin E_0 + 12 E_0 - E + e sin Ecos E_0 -.
            
            
               -.  3 e 2 + 19 e^2sin 2 E + 7 e^4 sin 3 E.manom
        
        By inserting ecce and manom in casot, it is possible to explicitly infer the instantaneous perturbation of the eccentric anomaly
        
            Δ EE    = -c^2Λ72 e^21-ecos E12e7 + 6e^2E-E_0  +2sin E_0 - sin E + .
            
            
               . + e 6e36+5e^2sin E_0  -32+7e^2sin 2 E_0  -2e^3sin 3E_0  -3e71+8e^2sin E +..
            
            
               ..  +3 e E_0 - 3 e E + sin E + 2 e^2 sin E6ecos2E_0-24cos E_0  +..
            
            
               +.. 92 + 5e^2sin 2 E -e3+4e^2sin 3 E  .eccanom
        
        By inserting smaomega and eccanom in Dz and using sinfrE allows one to obtain the instantaneous perturbation Δδτ_p^ΛE of the orbital time shift of the pulsar p due to Λ. It is
        Δδτ_p^ΛE = cΛ asin I72^2ℒE; E_0, e, ωDTmega,
         where ℒE; E_0, e, ω is a function of  E and  the parameters E_0, e, ω definitely too cumbersome to be explicitly displayed. Thus, we show only the leading term of DTmega;
        
            Δδτ_p^ΛE   ≃cΛ asin I6^24 E_0 - EcosE + ω - sinE_0 - 2 E - ω - .
            
            
               . - 3 sinE_0 + ω + 2 sinE + ω +𝒪e^k, k≥ 1.Dtau
        
        It is important to note from DTmega that Δδτ_p^Λ is proportional to the fourth power of the semimajor axis a, which characterizes the size of the pulsar's orbit, and is inversely proportional to the mass of the SMBH.
        
        The net shift per orbit can be calculated from DTmega with E→ E_0+2: it turns out to be
        
            Δδτ_p^Λ   = - cΛ asin I12^211-ecos E_0√(1 - e^2)16 + 9 e^2cos E_0 + ..
            
            
               .. + 3 e 10 + 6 cos 2E_0 - e cos 3 E_0cosω - 16 sin E_0 sinω + .
            
            
               . + 6 e 2 -3 + e^2cos E_0 + e -6 + cos 2E_0sin E_0 sinω.DTave
        
        It can be noted that also DTave depends on the initial conditions through E_0.
        It is also important to stress that both DTmega and DTave were worked out without any a priori simplifying approximations about the pulsar's orbital configuration; they hold for all values of e. It is a key feature in view of the highly eccentric orbits revealed so far in the GC.
        
        
        
        
        
        
        
        
        § THE OPPORTUNITY OFFERED BY HYPOTETICAL PULSARS IN THE GALACTIC CENTER
        misuraLet us, now, move to the compact object located in Sgr A^∗. For an interesting multidisciplinary discussion about the possibility that it is, actually, a SMBH or something else, see the recent overview in <cit.>. However, our results will be unaffected by the alternative possibilities discussed there since their spacetimes are undistinguishable from that of a SMBH for the pulsars' orbital motions of interest here.
        
        In order to explore the opportunity offered by our results to effectively constrain the CC with pulsar timing in the GC, let us consider a putative pulsar whose orbital period  is short enough to allow to monitor at least one full revolution during a timing campaign. In this case, by suitably choosing the initial orbital phase E_0, it would be possible to profitably use DTave in order to maximize it; indeed, in principle, DTave can even vanish. To this aim, for the sake of concreteness, let us assume a S2-type orbital configuration characterized by =16 yr, e=0.8839<cit.>. It turns out that the maximum of the absolute value of DTave occurs for E_0=342.08 deg, which corresponds to almost an orbital period after the time of periastron passage, yielding an upper bound on the CC as little as
        |Λ|≲ 9× 10^-47 m^-2 (≲ 2× 10^-116 in Planck units)boundS2
        for a timing accuracy of σ_τ_p≃ 1  s. It should be noted that boundS2 is 3-4 orders magnitude better than the (likely optimistic) planetary bounds of boundaboundb. Fig. <ref> depicts the plot of DTave as a function of E_0. If we modify some of the parameters of the pulsar's orbital configuration by adopting, say, =30 yr, e=0.987, I=90 deg, it is possible to improve the bound on the CC to the level
        |Λ|≲ 4× 10^-48 m^-2 (≲ 1× 10^-117 in Planck units)boundX
        for E_0=354.04 deg.
        About the figures in boundS2boundX, inferred by considering only σ_τ_p as source of observational error, it must be stressed that they should be regarded with caution as preliminary and just indicative of the potential of the approach proposed. If not explicitly modeled and simultaneously estimated in actual pulsar timing data reductions, the CC-induced signature may be partially removed from the resulting residual. As such,  the resulting bounds may be weaker than those in boundS2boundX. Further dedicated analyses should be made by simulating observations and fitting a full orbital model to them in order to assess how good the input values are recovered. 
        A possible source of systematic uncertainty is represented by the mismodelled part of the competing averaged orbital time shifts induced by the standard post-Newtonian (pN) effects due to the current experimental errors in the SMBH's parameters entering their formulas. For example, according to Equation (35) of <cit.>, the amplitude of the 1pN gravitoelectric average time shift Δδτ_p^GE is proportional to μ_∙ c^-3=22 s, while the mass of the SMBH is currently known at a ≃ 7% level of accuracy <cit.>. Analogous considerations hold for the Lense-Thirring (Equation (51) of <cit.>) and quadrupole (Equation (83) of <cit.>) average shifts. In principle, such an issue could be circumvented if N pulsars j with different orbital configurations will be discovered. Indeed, in this case, it could be possible to write down for each of them an analytical expression
        Δδτ^exp_j = Δδτ^GE_j+Δδτ^LT_j+
        Δδτ^Q_2_j+Δδτ^Λ_j, j=1,2,… Nlinear
        for their measured average orbital time shift Δδτ^exp_j as a sum of the pN terms plus the CC one by treating μ_∙, S_∙, Q_2^∙, Λ, which enter each term of linear as multiplicative scaling parameters, as unknowns of the resulting linear system of algebraic equations. Solving for them, it would be possible to obtain, among other things,  an expression for Λ independent, by construction, of the mismodeled SMBH's physical parameters. Such an approach could be extended also to other dynamical effects impacting the pulsar's average orbital time shift like, e.g., third-body perturbations.
        
        
        Recently, the upper bound
        |ω̇_S2|≲ 1.6× 10^-3 yr^-1=9.2 deg cty^-1hees
        on the periastron precession of the real star S2 was inferred in <cit.>. By combining hees with the well known analytical expression for the Λ-induced  pericenter precession (see the references cited in Section <ref>)
        ω̇_Λ=12Λ c^2√(1-e^2),
        
        it is possible to infer a tentative upper limit on the CC of the order of
        |Λ|≲ 3× 10^-35 m^2 (≲ 8× 10^-105 in Planck units).
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        For much more distant pulsars, major sources of systematic uncertainty would be given by the still poorly mass background and the difficulty of effectively constraining the  parameters of extremely wide orbits <cit.> and of the Black Hole itself over a relatively short observational time interval Δ T with respect to the expected extremely long orbital period P_b of the neutron star.
        
        
        
        
        
        
        
        
        § SUMMARY AND CONCLUSIONS
        fine
        In this paper, we analytically calculated the perturbation Δδτ_p^Λ induced by the CC Λ on the orbital part of the time variation δτ_p of a hypothetical pulsar p orbiting the SMBH in Sgr A^∗. We did not restrict to any particular orbital configuration, and our results are, thus, exact with respect to the eccentricity e; it is an important feature since most of the main sequence stars discovered so far in the GC move along highly eccentric orbits. We obtained both the instantaneous change Δδτ_p^ΛE and the net shift per orbit Δδτ_p^Λ: they are proportional to cΛ a^4sin Iμ_∙^-1. A distinctive feature of both of them is their explicit dependence on the initial value E_0 of the orbital phase. Our results hold also for a wide class of long-range modified models of gravity generating an extra-potential quadratic in the distance r.
        
        We applied our results to some putative scenarios by adopting, for the sake of definiteness, the orbital configurations of one actually existing main sequence star orbiting Sgr A^∗. By considering a S2-type orbit with = 16 yr, it is meaningful to look at the averaged time shift Δδτ_p^Λ. It turns out that, for a careful choice of the initial orbital phase E_0, it would be possible, in principle, to infer an upper bound |Λ|≲ 9× 10^-47 m^-2, corresponding to ≲ 2× 10^-116 in Planck units, by assuming a pulsar timing accuracy of σ_τ_p≃ 1 s. It would be 3-4 orders of magnitude better than the current, likely optimistic, constraints from Solar System's planetary orbital motions.
        On the other hand, it should be stressed that the very same aforementioned bound on Λ, derived by accounting for only σ_τ_p, may be optimistic in view of possible partial removal of the sought signature if not explicitly modeled and solved for in actual data reductions. As a suggestion for further dedicated investigations, simulating the observations and fitting a complete dynamical orbital model to them  would be needed in order to assess how accurately the input values can be recovered.
        The bias due to the errors in the physical parameters of the SMBH entering the competing pN net shifts per orbit could be eliminated by setting up  suitably designed linear combinations of the time delays measured for several pulsars. In the case of much more distant pulsars, using the orbital averaged time shift Δδτ_p^Λ is unfeasible; only instantaneous values Δδτ_p^ΛE could be, in principle, measured.
        
        On the other hand, too wide and slow orbits may be impacted by the still poorly known mass background in the GC, and it would be difficult to effectively constrain the pulsar's orbital parameters over a relatively short time interval with respect to its extremely long orbital period.
        
        
        § ACKNOWLEDGEMENTS
        
        I would like to thank two attentive referee for their precious critical remarks.
        
        
        
        § NOTATIONS AND DEFINITIONS
        appen
        Some basic notations and definitions used in the text are listed below <cit.>. In the case treated in this  paper, the unseen companion c of the pulsar p is the SMBH of mass M_∙, so that m_c=M_∙≫ m_p and a_p≃ a.
        
          G: Newtonian constant of gravitation
        
          c: speed of light in vacuum
        
          ħ: reduced Planck constant
        
          ℓ_P≐√(ħ G c^-3): Planck length
        
          Λ: cosmological constant
        
          H_0: Hubble parameter
        
          ϱ_crit≐3/8H_0^2 G^-1: critical density of the universe
        
          ϱ_Λ≐1/8c^2Λ G^-1: density due to the cosmological constant
        
          Ω_Λ≐ϱ_Λϱ_crit^-1: normalized energy density of the cosmological constant
        
          m_p: mass of the pulsar p
        
          m_c: mass of the invisible companion c
        
          m_tot≐ m_p + m_c: total mass of the binary
        
          μ≐ Gm_tot: gravitational parameter of the binary
        
          a:  semimajor axis of the binary's relative orbit
        
          ≐√(μ a^-3):   Keplerian mean motion
        
          = 2^-1: Keplerian orbital period
        
          a_p=m_c m^-1_tot a: semimajor axis of the barycentric orbit of the pulsar p
        
          e:  eccentricity
        
        
          I:  inclination of the orbital plane
        
        
          ω:  argument of pericenter
        
        
          t_p: time of periastron passage
        
          t_0: reference epoch
        
          ℳ≐t - t_p: mean anomaly
        
          f:  true anomaly
        
        
          E: eccentric anomaly
        
          u≐ω + f:  argument of latitude
        
        
          𝐫: relative position vector of the binary's orbit
        
          r_z: component of the position vector along the line of sight
        
        
          r: magnitude of the binary's relative position vector
        
          : radial unit vector
        
          : unit vector of the orbital angular momentum
        
          ≐×: transverse unit vector
        
          r_ρ: radial component of the relative position vector of the binary's orbit
        
          r_ν: normal component of the relative position vector of the binary's orbit
        
          r_σ: transverse component of the relative position vector of the binary's orbit
        
          U_Λ: perturbing potential due to the cosmological constant
        
          A_Λ: perturbing acceleration due to the cosmological constant
        
          δτ_p=r_z c^-1: periodic variation of the time of arrivals of the pulses from the pulsar p due to its barycentric orbital motion
        
        
        § TABLES AND FIGURES
