Is the Big Bang defined as before or after Inflation? Is the Big Bang defined as before or after Inflation? Seems like a simple enough question to answer right? And if just yesterday I were to encounter this, I'd have given a definite answer. But I've been doing some reading while writing up my thesis and I'm finding conflicting definitions of the Big Bang.
Everyone agrees that in standard Big Bang cosmology, the Big Bang is defined as the singularity; the moment in time when the scale factor goes to zero. Okay, but when you include the theory of inflation, it gets a bit murky.
So here's what I mean by conflicting definitions. As an example, in The Primordial Density Perturbation by Lythe and Liddle, they define the Big Bang as the beginning of the era of attractive gravity after inflation. However, Modern Cosmology by Dodelson defines the Big Bang as coming before inflation; it effectively uses the old definition that the Big Bang is the moment when the scale factor approaches zero.
This contradiction is evident in multiple places. When doing a google search for it, one can find many persuasive explanations for both definitions. All definitions agree that we cannot any longer define it as the singularity where $a=0$. But every one makes sense in its own way and so, I become more and more confused about which is right the more of them I read.
The argument for the Big Bang coming after is that inflationary theory diverges from the standard Big Bang cosmology around $10^{-30}s$ before we'd expect to run into the singularity, when inflation ended, and that we have no evidence to anything coming before that, thus the big bang is now defined as the initial conditions for the hot, expanding universe that are set up by and at the end of inflation.
The argument for the Big Bang coming before seems to be that inflation is still a period where the scale factor grows and as such, the Big Bang can be defined as the closest value to zero (which is before inflation), or rather, the earliest time as the scale factor approaches zero. This essentially seems to be based on saying "well, we defined it as the moment when the scale factor was smallest before inflation was added. Why would we not continue to have that as the definition after inflation is added?"
The former argument has merit because it defines the start of the epoch where the universe is describable (practically) by the standard Big Bang cosmology. But the latter argument has merit because of its simplicity and that it uses the spirit of the original definition; the smallest scale factor and the moment when the expansion of the universe seems to begin.
Thus, my root question: Which definition is correct? Do we say the Big Bang came before or after inflation?
P.S. I realize that asking this here only serves to add one or more additional persuasive arguments to an already crowded debate. However, this is Physics.SE, so I figure whatever we decide here can be definitive. Even if we can't find a truly correct answer, this can set the record straight, or at least, firmly crooked.
 A: In my  opinion it all hinges on whether one includes quantization of gravity or not. 
The classical Big Bang just uses General Relativity and solutions of its equations.  A singularity   has a well defined meaning in the classical approach.
As physicists are convinced that the underlying framework of nature is quantum mechanical it is expected that gravity will also be quantized . The Inflationary epoch  appears in the model where effective quantization of gravity is assumed. Quantization carries the Heisenberg Uncertainty principle  which in  effect eliminates singularities that appear in the classical theories, for example the 1/r electric potential does not create a singularity in the quantum mechanical formulation.
The classical theories generally  are emergent from  the underlying  quantum mechanical framework. This  does not invalidate the theoretical models that describe so successfully the classical data. It just limits their region of validity. The Big Bang model is successful in its region of validity and it is modeled as if there is a singularity at the very beginning because that is what the data say.
When the limits of the validity in energy density, space and time of classical GR are reached, then QM has to be invoked and the data modeled with it, which is what the inflationary model does. That does not invalidate the classical BB. It will still emerge and apply at its region of validity . The various definitions you state are due to this confusion of trying to keep both the classical and QM frame at once . 
When a solid model of quantized gravity in a Theory Of Everything appears these confusions will be resolved. My view is that the classical Big Bang is no longer validated for the small dimensions, and thus its singularity, as the inflationary period is validated by the cosmic microwave background. Nevertheless, the emergent behavior at large dimensions is as if there exists a singularity at the beginning, so in large dimensions the Big Bang is valid. It is similar to the difference between the hydrogen atom quantum mechanically, and two charged balls attracting each other macroscopically. There is no singularity in the hydrogen atom, the two charged balls modeled as points in their center of mass have singularities there, and their macroscopic   behavior fits the model with singularities. 
A: So I've done some further research into this question and the result I found is quite surprising. There truly is no set definition. Some cosmologists will tell you (as John Rennie mentioned) to avoid using the term "Big Bang" unless you absolutely have to. However, that is a luxury not afforded to all cosmologists.
The more surprising thing is that among the cosmologists who can't avoid using the term, there is still no consensus and nobody ever talks about how there are two almost equally used definitions. I've confronted several faculty cosmologists about this and every single one says the same thing: you sort of just pick a definition you like best and go with that. Nobody mentions their specific definition in papers or that there are mutually exclusive definitions, it is just ignored as a problem. But I'm ranting, so let me formally answer the question.
There are two valid definitions and they are, for the most part, as I defined in the question. For those of us who can't avoid using "the Big Bang" in our work, the choice of definition seems to rely in part on the area of research of the individual. Most of the time, inflationary cosmologists will choose the definition of the Big Bang as the curvature singularity before the onset of inflation (except for those studying eternal inflation, for whom the period before the onset of inflation is ill-defined). Observational cosmologists and those studying the universe after inflation tend to choose the definition of the Big Bang as the end of inflation and the onset of the hot, radiation-dominated expansion era. Sometimes, I have found that this time is referred to as the "Hot Big Bang", which seems to be quite a useful way of distinguishing this definition from that of the curvature singularity (which I suppose would then be called the "Cold Big Bang").
So the answer to my question of what is the correct definition of the Big Bang is that there are two correct definitions with no canonical way of distinguishing them. It is equally valid to claim the Big Bang happened before inflation as it is to claim the Big Bang happened at the end of inflation. While it can be aggravatingly confusing that no one seems to acknowledge this dichotomy, the definition used in a given circumstance can often be inferred from context; researchers tend to use the definition that marks the beginning of the era(s) they are researching.
A: To my mind, the Big Bang doesn’t refer to a distinct event but to a cosmogonic theory as a whole, that “predicts” (
should we say “retrodicts”?) many different events of the deep past. For example, there is such established term as “Big Bang nucleosynthesis” that describes an epoch several seconds past the Beginning of Time. The Beginning of Time in the narrow sense must be rigorously referred to as “cosmological singularity”. But even professional physicists do not always distinguish the terms consistently.
Back to the original question. Inflation certainly occurs after the cosmological singularity, but before several important events of the Big Bang. In other words, the inflation is an optional epoch that can occur within the Big Bang era and theory.
A: We have no idea how long inflation lasted. You need 50-70 $e$-folds (with the $e$-fold time being vaguely in the neighborhood of $10^{-30}\text{ s}$) to match the data, but that's only a lower bound. This paper describes a model that just barely attains the minimum of 50 $e$-folds, while this paper describes a model in which the expected number of $e$-folds may be up to $10^{10^{10}}$, which is roughly $10^{10^{10}}$ seconds, or $10^{10^{10}}$ years, or $10^{10^{10}}$ times what we normally call the "age of the universe".
So putting the big bang before inflation, however well motivated in principle, is useless in practice, since you can't use it as a reference point for anything other than the start of inflation. When someone asserts that recombination happened 300,000 years after the Big Bang, or when the universe was 300,000 years old, they must be measuring from the end of inflation or thereabouts, because if they were measuring from the start of inflation then they'd just be wrong.

Modern Cosmology by Dodelson defines the Big Bang as coming before inflation; it effectively uses the old definition that the Big Bang is the moment when the scale factor approaches zero.

Does it? It says that inflation is the proposal "that the universe expanded exponentially fast when it was but $10^{-35}$ sec old." I don't think that's meant to suggest that the duration of inflation was $10^{-35}\text{ s}$, or that the beginning of inflation was $10^{-35}\text{ s}$ after the beginning of time. It seems to be the time corresponding to a temperature of $10^{15}\text{ GeV}$ in a model that's radiation-dominated back to $t=0$, or, as you put it, it's the time until we'd expect to run into the singularity in traditional big bang cosmology.
Dodelson never seems to talk about the Big Bang as a moment/event; he only uses the phrase to refer to Big Bang nucleosynthesis or to FLRW cosmology as a whole. I think that's a pretty common strategy: just don't bother to define a Big Bang time at all.
A: The Big Bang was originally defined as the zero time limit of the FLRW metric, so it's a mathematical construct and not primarily something physical. We have chosen to apply it to the zero time limit of the universe because we thought the FLRW metric was a good description of the universe, but then inflation gatecrashed the party and spoiled the fun.
So if you're going to use the phrase Big Bang in connection with the universe, as opposed to its purely mathematical meaning, then it's up to you to define what it means. As you've found, there is currently no consensus on its meaning.
Personally I would avoid using the term unless you're specifically referring to the FLRW metric.
A: The physicist Vilenkin (much involved in the development of the theory of field-based eternal inflation), in his popular science book "Many Worlds in One", describes the energy of the inflaton field as releasing itself in a hot bath of particles upon each of its potentially infinite number of endings in local (i.e., "bubble" or "pocket") universes, including the one we inhabit, where that release equates to "the" Big Bang.  He also deals with the timing issue mentioned by John Rennie, by sketching a difference between "cosmological" time and time within each of the LU's, which are generally expected to remain causally separated from each other, with no evidence to the contrary yet encountered.  All this is similar, if not identical, to the treatment given to the relation between the BB and inflation by Guth, in his well-indexed book titled "The Inflationary Universe".
However, regarding any beginning (in cosmological time) of phenomena as basic as radiation, the conclusion drawn by an earlier answerer is very far from certain. Vilenkin, Guth, and Borde collaborated on the BGV Theorem:  It's often construed as requiring a beginning for any inflation (asymptotically-exponential expansion) that might be eternal into the future.  However,in the last (2003) revision of their theorem, those 3 authors included a footnote, in their list of references at the end of that revision, which specifies that Aguirre and Gratton's "steady-state eternal inflation", described at https://arxiv.org/abs/astro-ph/0111191, is compatible with it, even though steady-state eternal inflation definitely does NOT require a beginning for the inflationary multiverse:  In that version of inflation, dual arrows of time extend in opposite directions from a Cauchy surface, and formally-identical versions of the BGV theorem apply separately in each of those temporal directions.         
A: I agree that there was a period of inflation. However, there was not yet anything to expand other than spacetime. Spacetime could not have expanded at light-speed because photons, carriers of light, did not yet exist. The photon epoch started after most leptons and anti-leptons were annihilated at the end of the lepton epoch, about 10 seconds after the Big Bang. That would be an eternity compared to the epoch of inflation.
Time Dilation would have been a major factor in the rate of spacetime expansion – Time slows as a thing speeds up and it stops at the speed of light. At time zero, Time would have been infinite and spacetime expansion would have been zero – nothing existed that could move.
We can’t know how long Time remained in that state because there were no laws of physics; quantum or otherwise. All quantum metrics are derived from speed of light and light did not yet exist.
The rate and duration of inflation depends on the frame of reference. Looking back from now to the epoch of inflation, it might appear that the rate of inflation exceeded light-speed. Looking forward from time zero, the expansion of spacetime during the epoch of inflation might have taken an eternity.
Another claim made by the standard big bang theory is that the temperature during the epoch of inflation was greater than one billion degrees centigrade. However, there is no explanation for the source of energy to create such an astronomical temperature. This leads me to believe that inflation began before theoretical time zero. There must have been an epoch beginning before time one (possibly before time zero, whose conclusion was the end of inflation. I think that would require some kind of eruption of matter/energy in a prior cosmic existence. Even this timeline is pure conjecture because it would require a metric for quantum Time, which did not yet exist.
A: 
Is the Big Bang defined as before or after Inflation?

The word "defined" makes this a question about opinion rather than fact. I always try to stick to the facts determined from empirical evidence. But hey ho, I'll do my best. 

Seems like a simple enough question to answer right? And if just yesterday I were to encounter this, I'd have given a definite answer. But I've been doing some reading while writing up my thesis and I'm finding conflicting definitions of the Big Bang.

That's because there's confusion in cosmology. 

Everyone agrees that in standard Big Bang cosmology, the Big Bang is defined as the singularity; the moment in time when the scale factor goes to zero.

Many cosmologists might agree about that, but IMHO it's going too far to say that everybody agrees about a singularity. A singularity sometimes denotes an issue, indicating that a theory has gone awry. I think it's possible to shed some light on the early universe by thinking about gravity and relativity and black holes, and appreciating that there is no central point-singularity in a black hole.     

Okay, but when you include the theory of inflation, it gets a bit murky.

The theory of inflation is definitely "a bit murky". Not the expanding universe, inflation. I like to think that one reaches a position wherein inflation is a solution to a problem that does not exist. 

So here's what I mean by conflicting definitions. As an example, in The Primordial Density Perturbation by Lythe and Liddle, they define the Big Bang as the beginning of the era of attractive gravity after inflation.

That's not a definition I'd use. By the way, note that gravity is always attractive, and that it alters the motion of light and matter through space, but it doesn't make space fall down. The big crunch didn't happen when the universe was small and dense, and it isn't ever going to happen. 

However, Modern Cosmology by Dodelson defines the Big Bang as coming before inflation; it effectively uses the old definition that the Big Bang is the moment when the scale factor approaches zero.

I think that's better, but not quite right. It's kind of missing the trick. 

This contradiction is evident in multiple places. When doing a google search for it, one can find many persuasive explanations for both definitions. All definitions agree that we cannot any longer define it as the singularity where a=0. But every one makes sense in its own way and so, I become more and more confused about which is right the more of them I read.

None of them are exactly right. 

The argument for the Big Bang coming after is that inflationary theory diverges from the standard Big Bang cosmology around 10−30s.

Forget that. There was no inflation. 

before we'd expect to run into the singularity

Forget that too. There was no singularity. 

when inflation ended, and that we have no evidence to anything coming before that, thus the big bang is now defined as the initial conditions for the hot, expanding universe that are set up by and at the end of inflation.

There's no evidence for inflation. But there is confusion in cosmology.

The argument for the Big Bang coming before seems to be that inflation is still a period where the scale factor grows and as such, the Big Bang can be defined as the closest value to zero (which is before inflation), or rather, the earliest time as the scale factor approaches zero. This essentially seems to be based on saying "well, we defined it as the moment when the scale factor was smallest before inflation was added. Why would we not continue to have that as the definition after inflation is added?"

Big bangs are associated with things getting bigger. IMHO it makes no sense to define the big bang as something that occurs after the universe got a whole lot bigger fast.  

The former argument has merit because it defines the start of the epoch where the universe is describable (practically) by the standard Big Bang cosmology. But the latter argument has merit because of its simplicity and that it uses the spirit of the original definition; the smallest scale factor and the moment when the expansion of the universe seems to begin.

If there was no inflation, things get a whole lot simpler, don't they? 

Thus, my root question: Which definition is correct? Do we say the Big Bang came before or after inflation?

Does not compute. There was no inflation. On that basis the second definition is more correct, but not wholly correct. PS: I noticed your comment about the documentary. If you'd like to consult me, please don't hesitate to email me on myname at btconnect dot com. LOL! 
