Lets assume that in the delayed eraser in Kim experiment https://en.wikipedia.org/wiki/Delayed-choice_quantum_eraser#The_experiment_of_Kim_et_al._(1999) we can split the original photon into 2 photons, namely signal and idler photon, but we only have D0 detector and no other detectors, no beam splitters , no mirrors. We will let the idler photons travel freely in the universe without being detected or analysed by any device. As we have no "which path" information about the signal photons shall we receive interference pattern on D0?
No, an interference pattern will not be detected.
The universe does not care whether the which-way information is "detected" in any way ─ the only thing that matters is that the which-way information is available even in principle, and if it is (as is the case here) then interference will not be present.
The only thing that can 'restore' the interference pattern is if the which-way information is actively erased, as performed via detectors 1 and 2 in the standard Wikipedia diagram, and particularly if you post-select out of the $D_0$ measurements: that is, the total set of the $D_0$ measurements will always look like a formless blob, and it is only once you look at the hits which were observed in coincidence with $D_1$ and $D_2$ clicks that the data can be decomposed into the two corresponding complementary interference patterns.
As we have no "which path" information about the signal photons shall we receive interference pattern on D0?
"we" might not have detected the which-way information, but that doesn't mean that the information does not exist. Unless and until it has been coherently erased ─ a step that always involves post-selecting out a subset of the data in the prospective interference pattern ─ it must always be assumed to exist.
In the original setup, a laser generates individual photons, 351.1nm, that passes through a double slit.
After the slits, there is Spontenaeous parametric downconversion of the photon into two photons of 702.2nm. It prepares an entangled two photons state. This converts the photon into two identical, orthogonally polarized entangled photons with half the frequency.
Spontaneous parametric down-conversion (also known as SPDC, parametric fluorescence or parametric scattering) is a nonlinear instant optical process that converts one photon of higher energy (namely, a pump photon), into a pair of photons (namely, a signal photon, and an idler photon) of lower energy, in accordance with the law of conservation of energy and law of conservation of momentum. It is an important process in quantum optics, for the generation of entangled photon pairs, and of single photons.
The signal photon goes to D0 detector.
During an experiment, detector D0 is scanned along its x axis, its motions controlled by a step motor. A plot of "signal" photon counts detected by D0 versus x can be examined to discover whether the cumulative signal forms an interference pattern.
Now in your case, the idler photon is sent far away into space, and you are assuming no interaction with this photon, no detection. Since this experiment needs to be repeated many times (only single photons are shot at a time), you are assuming (all these idler photons are shot far into space) that none of these idler photons are detected or interacted with.
Now it is very important to understand that there is a difference between signal and idler photons. Signal photons beams (from slit A and slit B) are recombined before the detector D0. The idler photons beams (from slit A and slit B) are not recombined. This means, that the idler photons carry information (because they are entangled with the signal photons) about the signal photons' which way.
In your case, the answer is you want to see an interference pattern at D0, you need to meet certain requirements, and one of them is to repeat the experiment many times (as only a single photon is shot at a time) at the detector D0. The other requirement is that you need to coherently erase which way information (that the idler photons have). Another requirement is that you do post selection of the D0 photons.
It is very important to understand that the signal photons do not carry which way information, because there is a lens, that recombines the two beams coming from slit A and slit B (only the case if the signal photons are projectively measured on the position basis on the focal plane of said lens).
Recombining the beams results in interference phenomena at detection screens positioned just beyond each exit port.
So basically as long as you recombine the two beams (signal photons) coming from slit A and slit B before detector D0, and you coherently erase which way information (that the idler photons have), and you do post selection of the D0 photons, you will see an interference pattern at D0.