Does dark matter really exist or is just a made-up theory in order to cover up the loopholes in the Big Bang theory, and also to explain why universe is expanding at an accelerated rate?
The question seems to be based on some misconceptions, so I'll try to clear them up.
Dark matter is not the same as dark energy.
Dark matter and dark energy are two completely separate and unrelated concepts. It's unfortunate that many articles trying to entice readers with tales of the unexplored frontiers of astronomy tend to use the phrase "dark matter and dark energy" quite a lot, typically with the buzzwords "mysterious," "unknown," and even "spooky."
Obviously, the best introduction here is to read the Wikipedia articles on dark matter and dark energy. You can also see the dark-matter and dark-energy tags (and the questions that use them) for more information. Physics also has some good questions about these topics.
I'll give you the important points:
Dark matter . . .
- . . . interacts via gravity
- . . . has no electric charge (and so does not interact with light outside of possibly gravitational lensing, which does not involve electric charge)
- . . . may be composed of one (or more) of several hypothetical (and some not-so-hypothetical) particles, such as axions or WIMPS (one of the more popular choices)
- . . . is responsible for some stars moving faster than they should
Dark energy . . .
- . . . is responsible for the acceleration of the expansion of the universe, though not the expansion of the universe
- . . . can be represented by the cosmological constant
The one thing that the two share? We know very little about them.
Dark energy does not solve as many problems in the Big Bang theory as you may think.
In fact, it causes more problems than it solves.
You can come up with many different universes that begin with a Big Bang but don't have an accelerated expansion (not counting inflation, though you don't have to have that in a universe under some circumstances). They don't somehow suffer from any problems that universes with dark energy don't have. In fact, they're better because you don't have to somehow explain dark energy.
Courtesy of Rob Jeffries, here's one tidbit about the one fairly important role of dark energy in the Big Bang:
Our universe is close to flat (see the many papers by the WMAP team), and quite a lot of measurements have confirmed that we're not living on some wacky-shaped universe. The issue is that the measurements indicate that the universe isn't quite flat. Again, see the WMAP data. You can attribute part of the percentage offset from flatness to experimental error, or you can attribute it to the idea that perhaps the universe isn't so flat after all.
Dark energy provides an explanation for this inherent not-quite-flat-ness that characterizes our universe. In those other toy universes (without dark matter), you need to explain this near-flatness some other way. Not in our universe, it seems.
Dark matter and dark energy aren't just theories.
Well, sort of. Coming from someone like me (who is firmly on the "theorist" side of the scientific spectrum), that's quite a whopper. But from a certain point of view, it's completely accurate.
Dark matter is actually a theory. It's an explanation for some screwed-up galactic rotation curves, where some stars near the outer edges of a galaxy move much faster than they should.
So it's actually a theory to explain a phenomenon.
Dark energy, on the other hand, is an experimental phenomenon. We don't have a good theory to explain it. But it's a phenomenon responsible for many different problems. There's quite a lot of evidence that says, "Something odd is going on."
What I mean by all this is that there's a heck of a lot of evidence for them. Our theories of them were created to explain evidence, and not vice versa (evidence gathered to support or refute a theory).
I advise, once again, that you look at the Wikipedia articles on both concepts to start learning more (especially the sections on evidence). Physics and Astronomy have plenty of good information. arXiv has many pre-prints that may be helpful. And there are lots of books, magazines, and (reputable) web sites that can give you even more detailed information.
The modern dark matter theory assumes that a spiral galaxy is a system of orbiting objects in the same way that a solar system is a system of orbiting objects.
That is not the case. The stars in an elliptical galaxy are not in orbit around the galactic center. They revolve around it, but they are not in an equilibrium state where they would be both falling into and coasting away from the central object at essentially the same speed. All the stars are either falling in towards the center -- in a long spiral journey -- or are drifting out towards the intergalactic void in a long spiral journey. You could intuitively draw a circle on a photo of a spiral galaxy demarcating the border separating the ingoing from the outgoing stars.
It has to be remembered that our galaxy, for instance, is roughly a billion times larger in diameter than what might be determined to be the diameter of our solar system. The motion of the stars, radially and otherwise, is relatively much, much slower. Profoundly slower.
In our solar system, all the mass of the primordial rotating disk has long since either become part of the sun or has coasted outward. The system is "cleared out" -- all except for a few planets whose very existence is close to being a miracle . A few lucky lumps and orbs, have, in the chaos of motion, somehow fallen into a highly improbable equilibrium state. The sun has 98.5 per cent of all the mass in the solar system, and most of the rest is Jupiter, so the rest is really just debris.
In contrast, it takes a stupendous amount of time for a galaxy to clear out. The stars would all have gone through their life cycle and would no longer be producing light. We are watching matter inexorably falling in, and matter inexorably exiting (with maybe just a few stars ultimately ending up in an unlikely equilibrium state).
The stars in a galaxy initially got their roughly equivalent momentum from interactive gravitational forces. But the point is, when we observe a spiral galaxy, we are always looking at something that is just at the beginning of a process that has already run its course in a solar system. There is no need for 'dark matter' at the periphery of galaxy, because there is nothing being held in orbit.