Model-independent Way to Determine the Hubble Constant and the Curvature from the Phase Shift of Gravitational Waves with DECIGO

This is a note on paper: https://doi.org/10.3847/2041-8213/ad3553

Introduction

• Hubble Tension:

CMB v.s. Supernova

• the spatial curvature of the Universe:

  1. pure CMB data: closed Universe ${\mathrm{\Omega }}_k=-{0.044}_{-0.015}^{+0.018}$
  2. combination of Planck lensing data and low redshift baryon acoustic oscillations (BAOs): flat Universe ${\mathrm{\Omega }}_k=0.0007\pm 0.0019$

Methodology

1. traditionally:

   model -> Friedman equation

   $$E(z)\equiv {\left(\frac{H(z)}{H_0}\right)}^2={\mathrm{\Omega }}_m(1+z{)}^3+{\mathrm{\Omega }}_k(1+z{)}^2+{\mathrm{\Omega }}_{\mathrm{\Lambda }}$$

   -> theoretical angular diameter distance $D_A^{\text{th}}(z;\mathit{p})$

   $$D_A^{\text{th}}(z;\mathit{p})=\{\begin{array}{ll}{\displaystyle \frac{D_H}{(1+z)\sqrt{|{\mathrm{\Omega }}_k}|}{\sinh }^{-1}\left(\sqrt{|{\mathrm{\Omega }}_k|}{\int }_0^z\frac{\mathrm{d}{z}^{\prime }}{E(z^{\prime }{)}^{1/2}}\right)}& \text{for}{\mathrm{\Omega }}_k>0\\ {\displaystyle \frac{D_H}{(1+z)}\int \frac{\mathrm{d}{z}^{\prime }}{E(z^{\prime }{)}^{1/2}}}& \text{for}{\mathrm{\Omega }}_k=0\\ {\displaystyle \frac{D_H}{(1+z)\sqrt{|{\mathrm{\Omega }}_k}|}{\sin }^{-1}\left(\sqrt{|{\mathrm{\Omega }}_k|}{\int }_0^z\frac{\mathrm{d}{z}^{\prime }}{E(z^{\prime }{)}^{1/2}}\right)}& \text{for}{\mathrm{\Omega }}_k<0\end{array}$$

   here $\mathit{p}=({\mathrm{\Omega }}_m,{\mathrm{\Omega }}_k,{\mathrm{\Omega }}_{\mathrm{\Lambda }})$ -> get observed distance $D_A^{\text{obs}}(z)$ -> obtain cosmological parameters $\mathit{p}$ by chi-square-minimized (Markov Chain Monte Carlo method)

2. this work:

   gravitational wave (GW)'s phase shift -> acceleration parameter $X(z)$($H(z)$)

   $$X(z)\equiv \frac{H_0}{2}(1-\frac{H(z)}{(1+z)H_0})$$

   -> using artificial neural network (ANN) to get function ($z\to X(z)$) -> theoretical angular diameter distance $D_A^{\text{th}}(z;\mathit{p})$

   $$D_A^{\text{th}}(z;\mathit{p})=\{\begin{array}{ll}{\displaystyle \frac{D_H}{(1+z)\sqrt{|{\mathrm{\Omega }}_k}|}{\sinh }^{-1}\left(\sqrt{|{\mathrm{\Omega }}_k|}\frac{H_0}{1+z}{\int }_0^z\frac{\mathrm{d}{z}^{\prime }}{H_0-2X(z^{\prime })}\right)}& \text{for}{\mathrm{\Omega }}_k>0\\ {\displaystyle \frac{c}{(1+z{)}^2}\int \frac{\mathrm{d}{z}^{\prime }}{H_0-2X(z^{\prime })}}& \text{for}{\mathrm{\Omega }}_k=0\\ {\displaystyle \frac{D_H}{(1+z)\sqrt{|{\mathrm{\Omega }}_k}|}{\sin }^{-1}\left(\sqrt{|{\mathrm{\Omega }}_k|}\frac{H_0}{1+z}{\int }_0^z\frac{\mathrm{d}{z}^{\prime }}{H_0-2X(z^{\prime })}\right)}& \text{for}{\mathrm{\Omega }}_k>0\end{array}$$

   here $\mathit{p}=(H_0,{\mathrm{\Omega }}_k)$

GW

• The gravitational waveform without cosmic acceleration reads

  $$\tilde{h}(f)=\frac{\sqrt{3}}{2}\mathcal{A}{f}^{-7/6}{e}^{i\mathrm{\Psi }(f)}[\frac{5}{4}{A}_{\text{pol},\alpha }(t(f))]e^{-i({\phi }_{\text{pol},\alpha }+{\phi }_D)}$$

• gravitational waveform including the effects of the cosmic acceleration reads

  $$\tilde{h}(f{)}_{\text{acc}}=\tilde{h}(f)e^{i{\mathrm{\Psi }}_{\text{acc}}(f)}$$

  where

  $${\mathrm{\Psi }}_{\text{acc}}(f)=-{\mathrm{\Psi }}_N(f)\frac{25}{768}X(z){\mathcal{M}}_z{x}^{-4}$$

  The acceleration parameter $X(z)$ is defined as

  $$X(z)\equiv \frac{H_0}{2}(1-\frac{H(z)}{(1+z)H_0})$$$$\Rightarrow E(z{)}^{1/2}=\frac{H(z)}{H_0}=(H_0-2X(z))(1+z)$$

• the waveform $\tilde{h}(f{)}_{\text{acc}}$ depends on 11 parameters:

  $${\theta }^i=(\ln {\mathcal{M}}_z,\ln \eta ,\beta ,t_c,{\varphi }_c,{\overline{\theta }}_S,{\overline{\varphi }}_S,{\overline{\theta }}_L,{\overline{\varphi }}_L,D_L,X)$$

• set $m_1=m_2=1.4M_{\odot }$ and take $t_c={\varphi }_c=\beta =0$ for each fiducial redshift $z$, randomly generate ${10}^4$ sets of $({\overline{\theta }}_S,{\overline{\varphi }}_S,{\overline{\theta }}_L,{\overline{\varphi }}_L)$

  using a flat $\mathrm{\Lambda }$CDM model with the cosmological parameters derived by Planck 2018 measurements to generate datas

• divide sample into 50 bins,

  train the ANN on the simulated $X(z)$ data and predicte the $X(z)$ at other redshifts.

• ANN: input redshift $z$,

  output corresponding cosmic acceleration parameter $X(z)$ and its respective uncertainty ${\sigma }_X$ at that redshift.

QSOs

quasi-stellar objects (QSOs)

• the characteristic angular size of a distant radio quasar is

  $$\theta =\frac{2\sqrt{-\ln \mathrm{\Gamma }\ln 2}}{\pi B}$$

  $B$: the interferometer baseline

  $\mathrm{\Gamma }=S_c/S_t$: the ratio between the total and correlated flux densities

  The angular size of the compact structure in radio QSOs is

  $$\theta (z)=\frac{l_m}{D_A(z;\mathit{p})}$$

  $l_m$: the linear size scaling factor describing the apparent distribution of radio brightness within the core

• Taking $l_m=11.03\pm 0.25\text{pc}$ and following the redshift distribution of QSOs from Palanque-Delabrouille et al.(2016)

  simulate 1000 "angular size-redshift" data, assume the "measured" angular sizes follow a Gaussian distribution ${\theta }_{\text{means}}=N({\theta }_{\text{fid}},{\sigma }_{\theta })$, ${\theta }_{\text{fid}}$ is obtained from equation above under same model in GW.

Results and Discussion

1. the best-fit values of $H_0$ and ${\mathrm{\Omega }}_k$

• using the Markov Chain Monte Carlo method to minimize the ${\chi }^2$ objective function:

  $${\chi }^2=\sum _{i=1}^{1000}\frac{[D_{A,i}^{\text{th}}(z;\mathit{p})-D_{A,i}^{\text{obs}}(z){]}^2}{{\sigma }_{D_{A,i}^{\text{th}}}^2+{\sigma }_{D_{A,i}^{\text{obs}}}^2}$$

  here GW -> $X(z)$ -> $D_{A,i}^{\text{th}}(z;\mathit{p})$

  QSOs -> $D_{A,i}^{\text{obs}}(z)$

• result:

  $$H_0={67.19}_{-0.74}^{+0.77}\text{km}{\text{s}}^{-1}{\text{Mpc}}^{-1},{\mathrm{\Omega }}_k={0.022}_{-0.046}^{+0.051}$$

  at $68.3\mathrm{\%}$ confidence level.

  compare to used model $H_0=67.4\text{km}{\text{s}}^{-1}{\text{Mpc}}^{-1}$ and ${\mathrm{\Omega }}_k=0$

2. the uncertainties of the best-fit parameters as a function of QSO sample size $N$

3. the uncertainties related to GW signals and QSOs

Conclusion

1. a cosmological model-independent method to determine the Hubble constant and curvature parameter simultaneously based on GW from DECIGO and QSOs from VLBI

DECIGO -> GW -> phase shift -> $H(z)$ -> $D_{A,i}^{\text{th}}(z;\mathit{p})$

VLBI -> QSOs -> "angular size-distance" -> $D_{A,i}^{\text{obs}}(z)$

2. assume a fiducial cosmology to simulate GW and QSOs datas:

$N=1000$ QSOs -> ${\sigma }_{H_0}=0.75\text{km}{\text{s}}^{-1}{\text{Mpc}}^{-1}$ and ${\sigma }_{{\mathrm{\Omega }}_k}=0.048$ achieve $1\mathrm{\%}$ precision

the precision of $H_0$ with a fixed prior on ${\mathrm{\Omega }}_k$ can reach $0.5\mathrm{\%}$

the performance of the method on QSO samples of different sizes (from $N=50$ to $N=1000$): not a simple $1/\sqrt{N}$ but saturate at $N=500$ (unsolved)

3. potential ways to improve results:

higher angular resolution and lower statistical and systematic uncertainty from QSOs from VLBI

datas from other astronomical probes such SNe Ia and BAOs