1. ICGAC13, Seoul,July 6, 2017
Standard Sirens and Dark Sector with Gaussian Process
Rong-Gen Cai
Institute of Theoretical Physics
Chinese Academy of Sceinces

2. Outline：
1) Introduction
2) Dark sector interaction as example of Gaussian process
3) Reconstructing dark sector interaction with Lisa
4) Constraint ability of GW on cosmological parameters
with ET
5) Conclusion
Refs: arXiv: , ,

3. 1) Introduction
LIGO/Virgo: GW150914, GW , GW170104…
New era of cosmology and astronomy with GW
Fundamental Phys
Astrophysics
Cosmology
B. Schutz, Nature 323,
310 (1986)

4. 2) Dark sector interaction with Gaussian process
LCDM model looks fine with all observational data,
However, it suffers from two well-known problems:
fine tuning problem; 2) coincident problem
To alleviate the coincident problem,an interaction between
the dark energy and dark matter is introduced.
Usually an interacting form has to be assumed!
For a review: B. Wang et al, arXiv:
Rept. Prog. Phys. 79 (2016)

5. In arXiv: (PRD 81 (2010) ,
R.G. Cai and Q. Su investigated the possible interaction in
a way independent of specific interacting form by dividing
the whole range of redshift into a few bins and setting the
interacting term to be a constant in each redshift, it was
found that the interaction is likely to cross the non-interacting
line and has an oscillation behavior.
In arXiv: (PRL 113 (2014) ,
V. Salvatelli et al showed that the null interaction is excluded
at 99% confidence level,when they added the redshift-space
distortion data to the Planck data for the decaying vacuum
energy model by assuming an interaction form
and dividing the redshift into four bins

6. In arXiv: 1505. 04443, we present a nonparametric approach
to reconstruct the interaction term between dark energy and
dark matter directly from the observational data using the
Gaussian process.
The Gaussian process is a model independent method to
smooth the data. We set the nonparameterized interaction term Q(z) and reconstruct it from SNIa union 2.1 data sets.
We consider three cases:
1) decaying vacuum energy with w=-1
2) wCDM model with a constant w
3) CPL parametrization

7. The model:
In a flat universe with an interaction between dark energy and
dark matter,the Friedmann equation reads:
And the conservation equations are changed into:

8. Introducing dimensionless form:

9. One can reconstruct the interaction from the observed
distance-redshift relationship, once the equation of state
is given.
We will reconstruct D(z) and its derivatives using Gaussian
process.
T. Holsclaw et al, arXiv: ; ;
M. Seikel et al,
The GP allows one to reconstruct a function from data without assuming a parametrization for it.

10. At each point z, the reconstructed function f(z) is a Gaussian
distribution with a mean value and Gaussian error.
The function at different points z and \tilde z are related by a
covariant function k (z, \tilde z), which only depends on a set
of hyperparameters l and \sigma_f . Here l gives a measure of
the coherence length of the correlation in x-direction, while
\sigma_f denotes the overall amplitude of the correlation in
the y-direction. Both of them will be optimized by GP with
the observed data set.
The different choice of the covariant function may affect the
reconstruction to some extent.

11. Usually it is taken as
In arXiv: , Seikel and Clarkson considered various
cases and concluded that the Matern form can lead to more
reliable results in GP reconstruction.

12. To see the ability of the GP method to distinguish different
models and recover the correct behaviors of the models, we
create mock data sets of future SNIa according to DES for
two fiducial models: LCDM model and decaying vacuum
energy model ,
For the LCDM model, it is easy to calculate H/H_0, then
obtain the simulated data of D(z).
For the decaying vacuum energy:

13. Now it is to test how well the reconstructed q(z) agrees with
the fiducial models. If the GP method can recover the fiducial
models and has the ability to distinguish them,we can show
that the GP method is a valid one in the reconstruction for
our purpose.
The DES is expected to obtain high quality light curves for
about 4000 SNIa from z=0.05 to z=1.2 within 5years.

14. LCDM model with mock data:

15. Decaying vacuum energy model with mock data:

16. Decaying vacuum energy model with union 2.1:

17. wCDM model and CPL model:
If w lies between -0.9 and -1.1,
q=0 is captured within 95%
confidence level.

18. GWs as standard sirens:
Further works with GP method:
Null test of the cosmic curvature using H(z) and supernovae data
R.G. Cai, Z.K. Guo and T. Yang, PRD 93 (2016) , arXiv:
Dodging the cosmic curvature to probe the constancy of the speed of light
R. G. Cai,Z.K. Guo and T. Yang, JCAP 1608 (2016) 016, arXiv:
GWs as standard sirens:
Estimating cosmological parameters by the simulated data of
Gravitational waves from the Einstein Telescope
R.G. Cai and T. Yang, PRD 95 (2017) , arXiv:
Reconstructing the dark sector interaction with LISA
R.G. Cai,N. Tamanini and T. Yang, JCAP 1705 (2017) 031,
arXiv:

19. 3) Reconstructing dark sector interaction with Lisa
R. G. Cai , N. Tamanini and
T. Yang, JCAP 05 (2017) 031
【arXiv: 】
The aim of this work is to reconstruct the dark sector interaction
by using simulating MBHB standard siren events with Lisa.
The simulated datasets of MBHB standard sirens are taken from
the following work:
N. Tamanini et al, JCAP 04 (2016) 002 【arXiv: 】

20. How many MBHB standard sirens will be detected by Lisa
How many MBHB standard sirens will be detected by Lisa? In Tamanini et al paper, the rate and redshift distribution of MBHB merger events have been reproduced by using the results of semi-analytical simulations based on
arXiv: by E. Barausse.
Three different competing scenarios for the initial conditions for the massive BH population at high redshift have been considered:

21. The model: decaying vacuum energy
The equation of state w=-1
The aim is to reconstruct the function q(z)

22. The results:
Lisa MBHB standard sirens alone/with DES
with Lisa mission duration of 5 years and 10 years
LCDM model with as the fiducial model

23. The case with Lisa MBHB standard siren alone:

25. The case with Lisa and DES:

27. The comparison of the 95% CL errors on q(z):

28. Conclusions:
Using only Lisa data, the dark sector interaction can be
well reconstructed from z~1 to z~3 (for a 5years mission);
from z~1 to z~5 (for a 10 years mission), although the
reconstruction is inefficient at lower redshift.
With DES datasets, the interaction is well reconstructed in
the whole redshift range from z=0 to z~3 (5years mission);
from z=0 to z~5 (10 years mission).
The massive BH binary can be used to constrain the dark
sector interaction at redshift ranges not reachable by the usual
supernoave datasets which probe only the z <1.5 range.

29. From GW waveform of binary system, one can determine
4) Constraint ability of GW on cosmological parameters with ET
R.G. Cai and T. Yang, PRD 95 (2017) , arXiv:
From GW waveform of binary system, one can determine
the absolute distance of the source 【B. Schutz,1986】
One needs redshift information of the source. This can be
done with the companying EM signal.
Binary system consists of neutron star and black hole.
to see how accurately measure the cosmological parameters
such as the Hubble constant, dark matter energy density, and
the equation of state of dark energy with ET.

30. The pattern function of antenna:ET
The strain:
The pattern function of antenna:
The Fourier transform of
the time domain of waveform:
The Fourier amplitude:
W. Zhao et al, PRD 83 (2011) ; T.G.F. Li, 2015

31. The model:

32. To identify the redshift of the source, one takes the EM signal
To identify the redshift of the source, one takes the EM signal. The binary merger of a NS with a NS (BNS) or BH (BHNS) is hypothesized to be the progenitor of a short and intense burst of gamma rays (SGRB).
The expected rates of BNS and BHNS detections per year for the ET are about the order 10^3-10^7 (ET project). However, only a small fraction (~10^(-3)) is expected to have the observation of SGRB.
In this work we take the detection rate in the middle
range (10^5), thus 10^2 events with SGRB per year.

33. The NS mass distribution is chosen to be uniform in the
interval [1,2]M_sun, while for black hole, it is in [3,10]M_sun.
The ratio between BHNS and BNS events is taken to be 0.03.
The redshift distribution of the sources takes
B. Sathyaprakash et al, Class. Quan. Grav. 27, (2010);
W. Zhao et al, PRD 83, (2011)

34. We vary the observed number of sources from 100 up to 1000 to see that with how many events we can constrain the cosmological parameters as precisely as the current Planck results.
To constrain the Hubble constant and the dark matter density,
we set them free and fix other parameters as in the fiducial
model.

35. The results:
With about GW events,we can constrain the Hubble constant with an accuracy comparable to Planck result. However, for the dark matter density parameter, ….
With 1000 events,

36. The equation of state of dark energy with GP method:
700 GW events can give the same constraint
accuracy to w(z) as Planck in the low redshift
region.

37. 5) Conculsion
Cosmological parameters with Planck and GW:
1000 GW events with ET
30 GW events with Lisa

38. Cosmological parameters within 68% CL:

39. Thanks !

40. 4) Constraint ability of GW on cosmological parameters with ET
PRD 95 (2017) , arXiv: