# Perspectives on Constraining a Cosmological Constant-Type Parameter with Pulsar Timing in the Galactic Center

## Abstract

**:**

## 1. Introduction

## 2. Calculating the Perturbation of the Orbital Component of the Time Shift Due to the Cosmological Constant

## 3. The Opportunity Offered by Hypotetical Pulsars in the Galactic Center

## 4. Summary and Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Notations and definitions

- $G:$ Newtonian constant of gravitation
- $c:$ speed of light in vacuum
- $\hslash :$ reduced Planck constant
- ${\ell}_{\mathrm{P}}\doteq \sqrt{\hslash G{c}^{-3}}:$ Planck length
- $\mathsf{\Lambda}:$ cosmological constant
- ${H}_{0}:$ Hubble parameter
- ${\varrho}_{\mathrm{crit}}\doteq \left(3/8\mathsf{\pi}\right){H}_{0}^{2}{G}^{-1}:$ critical density of the universe
- ${\varrho}_{\mathsf{\Lambda}}\doteq \left(1/8\mathsf{\pi}\right){c}^{2}\mathsf{\Lambda}{G}^{-1}:$ density due to the cosmological constant
- ${\Omega}_{\mathsf{\Lambda}}\doteq {\varrho}_{\mathsf{\Lambda}}{\varrho}_{\mathrm{crit}}^{-1}:$ normalized energy density of the cosmological constant
- ${m}_{\mathrm{p}}$: mass of the pulsar p
- ${m}_{\mathrm{c}}$: mass of the invisible companion c
- ${m}_{\mathrm{tot}}\doteq {m}_{\mathrm{p}}+{m}_{\mathrm{c}}$: total mass of the binary
- $\mu \doteq G{m}_{\mathrm{tot}}:$ gravitational parameter of the binary
- $a:$ semimajor axis of the binary’s relative orbit
- ${n}_{\mathrm{b}}\doteq \sqrt{\mu {a}^{-3}}:$ Keplerian mean motion
- ${P}_{\mathrm{b}}=2\mathsf{\pi}{n}_{\mathrm{b}}^{-1}:$ Keplerian orbital period
- ${a}_{\mathrm{p}}={m}_{\mathrm{c}}{m}_{\mathrm{tot}}^{-1}a:$ semimajor axis of the barycentric orbit of the pulsar p
- $e:$ eccentricity
- $I:$ inclination of the orbital plane
- $\omega :$ argument of pericenter
- ${t}_{p}:$ time of periastron passage
- ${t}_{0}:$ reference epoch
- $\mathcal{M}\doteq {n}_{\mathrm{b}}\left(t-{t}_{p}\right):$ mean anomaly
- $f:$ true anomaly
- $E:$ eccentric anomaly
- $u\doteq \omega +f:$ argument of latitude
- $\mathbf{r}:$ relative position vector of the binary’s orbit
- ${\mathrm{r}}_{z}:$ component of the position vector along the line of sight
- $r:$ magnitude of the binary’s relative position vector
- $\widehat{\mathit{\rho}}:$ radial unit vector
- $\widehat{\mathit{\nu}}:$ unit vector of the orbital angular momentum
- $\widehat{\mathit{\sigma}}\doteq \widehat{\mathit{\nu}}\times \widehat{\mathit{\rho}}:$ transverse unit vector
- ${\mathrm{r}}_{\rho}:$ radial component of the relative position vector of the binary’s orbit
- ${\mathrm{r}}_{\nu}:$ normal component of the relative position vector of the binary’s orbit
- ${\mathrm{r}}_{\sigma}:$ transverse component of the relative position vector of the binary’s orbit
- ${U}_{\mathsf{\Lambda}}:$ perturbing potential due to the cosmological constant
- ${\mathit{A}}_{\mathsf{\Lambda}}:$ perturbing acceleration due to the cosmological constant
- $\delta {\tau}_{\mathrm{p}}={\mathrm{r}}_{z}{c}^{-1}:$ periodic variation of the time of arrivals of the pulses from the pulsar p due to its barycentric orbital motion

#### Appendix A.2. Tables and Figures

**Table A1.**Relevant physical and orbital parameters of the S2 star and the SMBH at the GC along with their estimated uncertainties according to Table 3 of Gillessen et al. [73]; they are referred to the epoch $2000.0$. ${D}_{0}$ is the distance to ${\mathrm{Sgr}\phantom{\rule{3.33333pt}{0ex}}\mathrm{A}}^{\ast}$. The linear size of the semimajor axis of S2 is $a=1044\phantom{\rule{3.33333pt}{0ex}}\mathrm{au}$.

Estimated Parameter | Value |
---|---|

${M}_{\u2022}$ | $4.28\pm {\left.0.10\right|}_{\mathrm{stat}}\pm {\left.0.21\right|}_{\mathrm{sys}}\times {10}^{6}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{M}}_{\odot}$ |

${D}_{0}$ | $8.32\pm {\left.0.07\right|}_{\mathrm{stat}}\pm {\left.0.14\right|}_{\mathrm{sys}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$ |

${P}_{\mathrm{b}}$ | $16.00\pm 0.02\phantom{\rule{3.33333pt}{0ex}}\mathrm{yr}$ |

a | $0.1255\pm 0.0009\phantom{\rule{3.33333pt}{0ex}}\mathrm{arcsec}$ |

e | $0.8839\pm 0.0019$ |

I | $134.18\pm 0.40\phantom{\rule{3.33333pt}{0ex}}\mathrm{deg}$ |

$\Omega $ | $226.94\pm 0.60\phantom{\rule{3.33333pt}{0ex}}\mathrm{deg}$ |

$\omega $ | $65.51\pm 0.57\phantom{\rule{3.33333pt}{0ex}}\mathrm{deg}$ |

${t}_{p}$ | $2002.33\pm 0.01$ calendar year |

**Figure A1.**Average orbital time shift per orbit ${\overline{\Delta \delta \tau}}_{\mathrm{p}}^{\mathsf{\Lambda}}$, in $\mathsf{\mu}\mathrm{s}$, of a hypothetical pulsar in Sgr A${}^{\ast}$ obtained analytically from Equation (27) along with the value of Equation (1) for $\mathsf{\Lambda}$ as a function of the initial phase ${E}_{0}$. The orbital configuration of the S2 star, quoted in Table A1, was adopted. It can be noted that ${\overline{\Delta \delta \tau}}_{\mathrm{p}}^{\mathsf{\Lambda}}$ vanishes for two given values of ${E}_{0}$; the largest absolute value occurs for ${E}_{0}=342.08\phantom{\rule{3.33333pt}{0ex}}\mathrm{deg}$. By assuming a pulsar timing accuracy of ${\sigma}_{{\tau}_{\mathrm{p}}}=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}\mathrm{s}$, it translates to an upper bound on $\mathsf{\Lambda}$ of the order of $\left|\mathsf{\Lambda}\right|\le 9\times {10}^{-47}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-2}$ ($\lesssim {10}^{-}116$ in Planck units).

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Iorio, L.
Perspectives on Constraining a Cosmological Constant-Type Parameter with Pulsar Timing in the Galactic Center. *Universe* **2018**, *4*, 59.
https://doi.org/10.3390/universe4040059

**AMA Style**

Iorio L.
Perspectives on Constraining a Cosmological Constant-Type Parameter with Pulsar Timing in the Galactic Center. *Universe*. 2018; 4(4):59.
https://doi.org/10.3390/universe4040059

**Chicago/Turabian Style**

Iorio, Lorenzo.
2018. "Perspectives on Constraining a Cosmological Constant-Type Parameter with Pulsar Timing in the Galactic Center" *Universe* 4, no. 4: 59.
https://doi.org/10.3390/universe4040059