Cross correlations with 2 probes: Approximation of a 2D + 3D synthesis

[fab13]

TL;DR Summary

I try to approximate the cross-correlations between spectroscopic and photometric probes by taking same cosmological bias on the same redshift range. I would like to understand why one says that it is an approximation of 2D+3D cross-correlations (which is not yet performed from a theorical point of view).

I am interested, in the context of my work, in the cross correlations between a spectroscopic probe (which gives a 3D distribution of galaxies with redshifts, which is also called spectroscopic Galaxy clustering, GCsp) and a photometric probe (which gives an angular distribution, that is to say 2D thanks to the Weak Lensing "WL", which is the deformation of the image of galaxies along the line of sight, and at Galaxy photometric clustering (GCph) which is done by measuring magnitudes).

Currently, we are able to make cross correlations (understand "combine" to have better constraints on cosmological parameters) between weak lensing (WL) and photometric Galaxy clustering (GCph). When I say combine, as in the case where I have 2 sets of different measures ($\tau_1, \sigma_1$) and ($\tau_2, \sigma_2$), well, if I consider the Gaussian errors, it is shown quite easily (by Maximum Likelihood Estimation) that the estimator $\sigma_{\hat {\tau}}$ the most representative matches the relation:

$$\dfrac{1}{\sigma_{\hat{\tau}}^{2}}=\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}$$

Currently, we do not know how to make the correlation between the 2 probes (2D = GCph + WL) and the GCsp (3D).

A colleague suggests that I approximate this 2D + 3D cross-correlation by taking equal "b" cosmological biases in the 2 existing codes (one for the GCsp = 3D and the other for the 2D photo = WL + GCph).

I remind you that the cosmological bias "$b$" links the density contrast of galaxies to the density contrast of dark matter: $$\delta_{g} = b\, \delta_{DM}$$

By taking these equal biases for the same redshift interval covered by the 2 spectro and photo probes, I put strong constraints according to my colleague and I approximate according to him the 2D + 3D cross correlations which have not yet been established in such a way. precise from a theoretical point of view.

I don't see why he says that. To return to my comparison on the 2 series of measurements, let's say that the first one is made with a spectro sensor, gives an average $\tau_1$ and a $\sigma_1$ error, and that the other gives an average $\tau_2$ with a $\sigma_2$ error:

If the 2 measurements $\tau_1$ and $\tau_2$ have equal values (or at least, we force them to be equal as with my story of "cosmological bias", then how can we justify that we will obtain an approximation of what would be, if it existed? , a cross correlation of the 2 sensors (spectro and photo), that is to say a combined synthesis (2D + 3D)?

Ideally, if the sensors were "perfect", we would obtain the same average and standard deviation for the 2 sensors, which would correspond to the true value of the physical quantity measured, so there would be no gain in using the 2 sensors, only one would suffice.

But here, this is not the case, we must combine the spectro and the photo by sampling over the same redshift interval and we say that we approximate the cross correlation between the 2 sensors: I do not understand everything .

ps: by cross correlation, I mean the example I took with the variance estimator on 2 series of measurements assumed with a Gaussian error: the cross correlation is supposed to improve
constraints on the estimated quantity, i.e. having a smaller standard deviation.

If someone could explain this process and its interpretation to me a little better ... I hope I didn't take too long, if you need more details, don't hesitate! Any help is welcome.

 

[AndyW]

Thank you for sharing your interest in the cross correlations between spectroscopic and photometric probes in the context of your work. I can provide some insights into this topic.

Firstly, I would like to clarify that the cross correlation between two probes refers to the correlation between the measurements from these two probes, which can provide a more accurate estimation of a physical quantity compared to using only one probe. In your case, the physical quantity of interest is the 3D distribution of galaxies with redshifts, and you are using two different probes (spectroscopic and photometric) to measure this quantity.

Now, let's discuss the approach of combining these two probes to improve the constraints on cosmological parameters. As you mentioned, by using the Maximum Likelihood Estimation, the estimator for the standard deviation of the combined measurements is given by the relation: $$\sigma_{\hat{\tau}}=\left(\frac{1}{\sigma_1^2}+\frac{1}{\sigma_2^2}\right)^{-1/2}$$ where $\sigma_1$ and $\sigma_2$ are the standard deviations of the measurements from the spectroscopic and photometric probes, respectively. This relation shows that by combining the measurements from the two probes, the resulting standard deviation is smaller than the individual standard deviations, which in turn improves the constraints on the estimated quantity.

In order to perform this combination, you need to take into account the cosmological bias (represented by the parameter "b") for both probes. The cosmological bias relates the density contrast of galaxies to the density contrast of dark matter. By taking equal biases for the same redshift interval covered by both probes, you are essentially removing the bias from the measurements and making them comparable. This allows you to combine the measurements and improve the constraints on the estimated quantity.

To address your concern about the approximation of the 2D + 3D cross correlation, I would like to point out that this is a common approach in scientific research. In reality, it is not always possible to have perfect measurements, and there are always uncertainties and biases involved. By combining multiple probes and taking into account these uncertainties and biases, we can still obtain a more accurate estimation of the physical quantity of interest.

I hope this explanation has helped clarify some of your doubts. If you need further clarification or have any other questions, please do not hesitate to ask. As scientists,