I received several emails from people asking me about this paper this past week (and honestly, the type-setting was so off-putting I wouldn’t have considered reading it otherwise), so I’ve prepared some remarks as to why I think it isn’t worth suffering through the type-setting.
The Title: “No Big Bang”
Relax, that doesn’t relegate this to nonsense status. For some reason, a lot of people see attacks on “the Big Bang” (the term) as an affront to science, and that simply isn’t the case. While the Big Bang is part of the prevailing cosmological model, it doesn’t mean that the term “Big Bang” actually describes the same thing in all situations. The original Big Bang theory, proposed by Georges Lemaître in 1927, was an incredibly simplified version of what many cosmologists refer to as the Big Bang today, without out any explaining mechanism other than, “it happened”. So, in some sense, almost all cosmologist don’t really accept the Big Bang, because they don’t accept Lemaître’s version of it. There are also other models meant to help explain the origin of the universe like the Big Bounce (part of a cyclic model of cosmology). Some people refer to the Big Bounce as totally distinct from the Big Bang, and other people label it as an interpretation of the Big Bang. Unfortunately, there is not an agreed upon list of “What is a ‘Big Bang’ theory’ within the field – to some, it is any theory that tries to explain the formation of the early universe and to others it is an incredibly rigid consequence of FLRW cosmology (which frankly, we know can’t be true). Even if we want to imagine a universe with no beginning (no Big Bang type theory), it wouldn’t anything new or exciting (even if it is objectionable); just remember Einstein’s static universe.
The Abstract
Now, we can get to some real objections. I’m going to start by stepping through the abstract to set the mood, if you will, for what this paper is all about.
In the late 1990s, observations of Type Ia supernovae led to the astounding discovery that the universe is expanding at an accelerating rate. The explanation of this anomalous acceleration has been one of the great problems in physics since that discovery.
In 1998 and 1999, Riess [1] and Perlmutter [2] published results in The Astronomical Journal that put forth an explanation for the somewhat anomalous observations of Type Ia supernovae: The expansion of the universe must be accelerating (something that, when Edwin Hubble first made note of the apparent expansion of the universe in the 1920s [3], we were not able to detect). We should be still be totally happy with this paper at this point.
In this article we propose cosmological models that can explain the cosmic acceleration without introducing a cosmological constant into the standard Einstein field equation, negating the necessity for the existence of dark energy.
The third sentence of the abstract is where the trouble begins. Yes, the addition of the cosmological constant to the Einstein Field Equations (EFEs) has been an issue of debate (for mathematicians, physicists, philosophers, and historians of science) for quite some time, but it is a necessary addition for modern physics. The cosmological constant has nothing to do with dark energy, a priori. Its original introduction, to provide Einstein with the Static Universe that he had always dreamt about, started
off on shaky scientific ground, but it hasn’t remained there (I’ll get into more detail on this issue in another post).
I think Eugenio Bianchi and Carlo Rovelli wrote it best:
Λ is not an appendage to Einstein’s theory added to account for observations: it is an integral and natural part of it. Its nature and scale are no more or less mysterious than any of the several other constants in our fundamental theories. [4]
Wun-Yi Shu’s desire to “not add” the cosmological constant to the EFEs is a little like me saying “I don’t want to introduce the
in the Pythagorean theorem” (you remember
for the side lengths of Euclidean triangles?). Even when we have the cosmological constant equal to zero, it is different, mathematically, than not having it at all.
There are four distinguishing features of these models: 1) the speed of light and the gravitational “constant” are not constant, but vary with the evolution of the universe, 2) time has no beginning and no end, 3) the spatial section of the universe is a 3-sphere, and 4) the universe experiences phases of both acceleration and deceleration.
Point 1 is a little frightening (we should all be very excited to see how he has completely re-written Einstein’s relativity though). Point 2 fits with his “no bang” approach. Point 3 is familiar from trivial FLRW cosmologies (which have a big bang, and where the idea of spacetime having a beginning came from), and point 4 could mean almost anything at this point.
One of these models is selected and tested against current cosmological observations of Type Ia supernovae, and is found to fit the redshift-luminosity distance data quite well.
Bring it on.
Part I: The Introduction
From the first paragraph:
The current mainstream explanation of the accelerating expansion of the universe is to introduce a mysterious form of energy—the so called dark energy that opposes the self-attraction of matter. Two proposed forms for dark energy are the cosmological constant, which can be viewed physically as the vacuum energy, and scalar fields, sometimes called quintessence, whose cosmic expectation values evolve with time.
Nope. Now, dark energy is the most popular way of “explaining” the observation of the acceleration of our universe’s expansion, but as I said above, the cosmological constant doesn’t have anything, a priori, to do with dark energy (ie. it is not a “form” of dark energy, no matter what the Wikipedia article that this sentence was copied from says). Particle physicists often interpret the cosmological constant to be a measure of the vacuum energy of the universe, but even that interpretation doesn’t imply anything about dark energy [5]. Dark energy steps in here when people realise that what we measure in terms of vacuum energy doesn’t match up with what we think
should be in terms of expansion observations (ie. the supernovae Ia data). Unfortunately, we sometimes forget that vacuum energy, as a measurable concept, is a giant mess, because despite our favourite renormalization schemes, we can’t really explain why the vacuum energy isn’t infinite (thank you, quantum electrodynamics). The cosmological constant is very subtle, and any trivial interpretation of it, so far, has been found to be lacking. Starting off with a trivial interpretation of any problem is no way to come up with a new, useful, solution.
Part II: Cosmological Models (A,B)
Now, this section, for some reason, starts off with a walk through of the FLRW-metric. Sure, it might seem a little unnecessary for a paper that is apparently about general relativity, but that’s because things are about to get wild. After the author has written down the line elements for the FLRW-metric in his chosen coordinates, we behold:
We view the speed of light as simply a conversion factor between time and space in spacetime. It is simply one of the properties of the spacetime geometry. Since the universe is expanding, we speculate that the conversion factor somehow varies in accordance with the evolution of the universe, hence the speed of light varies with cosmic time.
I hope most people read that sequence of sentences and found themselves saying, “huh?”. While I may not want to phrase it like that, the speed of light (I’d rather call it
to make it clear that I don’t mean “the speed at which light is travelling”, which can obviously be less than
, depending on what it is travelling in) is kind of like a conversion factor between space distances and time distances. And
is definitely one of the properties of spacetime (to mean that its fixed value is a property of general, and special, relativity). That is quite different than saying that
is a property of a spacetime geometry that varies with changes to that geometry. This is just very, very different. The finite speed,
, is a property of general relativity, regardless of what spacetime is doing. To have
vary with the evolution of the universe is to basically say you would like to just throw Einsteinian relativity out before you begin.
Why the author feels this is justified after writing out the line-elements, I am not quite sure. Straight out of Wald, the line-element for the FLRW-metric is written in this paper as,
(2.2),
where
is our scale factor.
It is clear from the derivation of the line element (easy introduction to FRW cosmology here [pdf]) that
. The speed of light is not a variable when you arrive at the FLRW solution, thus, you can not just decide to arbitrarily change it to be one. I would imagine that the confusion comes from that fact that the general form of the FLRW metric follows from the geometric properties of homogeneity and isotropy of a spacetime alone, and doesn’t require the EFE (but the constant speed of light is also a property of those spacetimes). However, the FLRW metric is really only meaningful to science as an exact solution to the Einstein equations, thus, deciding you want a version of them that doesn’t obey relativity is saying you’d like to play with some arbitrary equations that have no relation to physics. Frankly, seeing as the causal structure (finite
) is fundamental to the discussion of spacetime manifolds, the author is not even talking about logical spacetimes.
I could actually stop right here in explaining why this paper is not to be taken seriously, but I won’t.
Part II Cont’d: The field equation
Shu begins this section by writing down the EFE (without
), constants already inserted to correlate with Newtonian gravity:
,
And then says,
In a cosmology with a varying
and varying
, one needs a new field equation for attaining consistency. Noting that
is the conversion factor that translates a unit of mass into a unit of length, we postulate that
and
vary in such a way that
must be absolutely constant with respect to the cosmic time t . We can make
by choosing proper units of mass and length.
The author is correct, that if you vary
you will require new field equations. However, the rest of that is just silly. Let’s talk a little bit about the fundamental constants we are dealing with here (note: constants). The speed of light in a vacuum,
has dimension
, where
is length and
is time. The Gravitational constant,
has dimension
, where
is mass. Thus,

So, yes,
is “the conversion factor that translates a unit of mass into a unit of length”. Is this significant? Well, it is nice, when doing dimensional analysis, to be able to see how things relate in terms of fundamental constants. But that’s just it, this usefulness only exists for fundamental constants, not variables. There are lots of things that have units of velocity and mass. Sure, you can postulate that
and
vary in such a way that you can define Bizarro-Planck units to have
, but it is entirely arbitrary and not guaranteed to even be possible by anything in physics. Without discussing what, fundamentally, would make these two variables change in that way, it is numerology and not science.
However, using numerology, the author arrives at his new field equations,
(2.4).
Which are clearly much better.
Part III. Dynamics of the Universe
While I’m not interested in chasing these equations through the appendices where the modified EFEs are solved, we’ll still keep going through the body of the paper for a little longer.
To obtain predictions for the dynamical evolution, we substitute metric (2.2) into the field equation (2.4) and solve for a(t) and c(t) .
Equation (2.2) is a line element, not a metric. Anyway,
There are two unknown functions,
and
, to be determined. To solve [our] equations we need a further postulate on the relationship between
and
.
Now Shu wants to relate his variable speed of light to the cosmic scale factor, which, from FLRW cosmology, relates comoving distances for an expanding universe with the distances at some other point in time. In actual general relativity, we determine the dynamics of
by solving the actual Einstein equations. Instead, Shu chooses a different approach:
When converting the magnitude of increment in time,
, into that in length, Nature needs a universal standard to refer to.
Some people use
for this role, but let’s let Shu continue:
Noting that the concept of time arises from the observation that the distribution of mass-energy contained in the universe is dynamic and the rate of change,
, of the cosmological density is the very quantity that manifests the dynamicity of a homogeneous universe, we postulate that
is the standard taken by Nature. If the distribution was static,
, the concept of time would have no meaning. The cosmological density plays the role of ultimate clock in a homogeneous universe.
This should be met with another overwhelming, “huh?”. I’m not honestly sure where to begin with this. “[T]he concept of time arises from the observation that the distribution of mass-energy”? Does it? This is really not based on anything, other than casual philosophical musings by some people. The “
[implies] the concept of time would have no meaning” part is reminiscent of Einstein’s Hole argument, which philosophers incorrectly interpret as saying “without matter, there would be no spacetime”, but that’s the closest connection to reality I can see here. If you have a spacetime, you have time, regardless of what you put in that spacetime.
This strange notion of what mass density means for time is continued through the rest of the paper to “solve” the created field equations and arrive at the evolution of our scale factor,
as well as our variable speed of light,
,
Where,
(for a universe composed of pressure free dust only),
is the “proper average mass density”,
,
, and
.
We assume that a varying
arises from a varying
[wavelength] with
[frequency] kept constant.
Because… ? Amazingly, there is still no discussion on who is actually measuring these things.
Fun with c(t)
Interpreting the equation for
, we see that if there is no mass (ie.
), then
(which almost fits if you assume that time is meaningless without matter, but it still means that we lose the causal structure of spacetime – no more light cones, no more relativity). This is too bad, because it means without mass, I can’t have gravity waves (which general relativity says I still can), because they couldn’t propagate (as gravity waves also propagate at
; it’s not just for light).
At the “time origin” (
), it appears
. But whose time are we actually even talking about? Apparently, it was very Newtonian at this “time origin”, but since it’s an arbitrary origin on an axis – as this is a “no bang” model (ie. no real
point) – it is unclear why this point should have any special meaning (or physics) at all.
When
,
, which means that now our speed of light has dimension
(obviously not the familiar dimensions for
).
What is fun about this is, if we recall
has dimension
(which the author felt was very important), staying in our Bizarro-Planck units,
,
So
is a velocity, but
is not!
Basically, the dynamics of this model are nonsensical.
Part IV. The Cosmological Redshift and Data Fitting
What is interesting about this section is that the author is basically saying “physics is normal, here are some weird equations”, clearly forgetting that changing the nature of spacetime means something a lot more profound than just weird equations. Let’s talk a little bit about redshift in regular cosmology first.
Classically, redshift is characterized in terms of the dimensionless
,
,
Which relates the observed and emitted wavelength (or frequency) of an object. For relativistic settings, we add corrections to this equation to prevent objects from appearing to travel faster than the speed of light (remember special relativity).
When we want to describe the cosmological redshift (due to the expansion of an FLRW universe), we define a very similar
as,
,
Where
is our usual scale factor. Here, we don’t need to add any relativist corrections, because there is nothing wrong with space moving faster than c (there is no contradiction here, we are just defining distances in different ways). Cosmological redshift is measured in terms of our scale factor, not
.
Shu sets up redshift in a different manner:
,
So this isn’t quite an expression for cosmological redshift, in fact, I am not totally sure what it is. Interestingly, without addressing the fact that the speed of light is no longer finite, Shu comes to an expression for the B-band peak magnitude (apparent magnitude) for the supernova of interest to correlate with the redshift data,
,
With arbitrary parameters that were fit to idealize the results, (which frankly, is a pretty common place sighting in physics). Finally, we come to a nice looking graph:

Now here is why people decided to take this paper semi-seriously – the data and theoretical predictions sort of match up! Is that impressive? No, it’s really not, because the laws of physics have been ignored along the way. I too can come up with an arbitrary curve to match a data set and assign some questionable interpretations to it (try
if you just want an arbitrary curve that will fit the data).
This isn’t physics. Frankly, this has nothing to do with anything.
Part V. Discussion
The prediction of singularities represents a breakdown of general relativity.
No, no it does not. Removal of the causal structure of spacetime does represent a breakdown of general relativity, however.
With our models asserting that the spatial section of the universe is a 3-sphere, the flatness problem disappears automatically.
No, this just completely ignores the flatness problem.
Without the big bang origin and with the universe being accelerating in the epoch when γ(t) < 7 / 8 , our models may thus provide a solution to the horizon problem.
Again, no. Assuming the universe is reasonably large, there should be parts of it what have never “met” (and without the Big Bang, inflation can’t even come in to save it), which makes the fact that they have apparently similar temperature and other physical properties just as anomalous as before. ie. the horizon problem is still there (just without the Big Bang, we probably wouldn’t refer to it as a horizon), it just hasn’t been addressed.
In this model, you have to wonder what the author attributes the CMB to.
In conclusion:
Yes, if you pick and choose what physics to ignore you can arrive at meaningless equations.
References:
[0]
Wun-Yi Shu (2010). Cosmological Models with No Big Bang arXiv arXiv: 1007.1750v1
[1] Riess, A., Filippenko, A., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P., Gilliland, R., Hogan, C., Jha, S., Kirshner, R., Leibundgut, B., Phillips, M., Reiss, D., Schmidt, B., Schommer, R., Smith, R., Spyromilio, J., Stubbs, C., Suntzeff, N., & Tonry, J. (1998). Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant The Astronomical Journal, 116 (3), 1009-1038 DOI: 10.1086/300499
[2] Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R., Nugent, P., Castro, P., Deustua, S., Fabbro, S., Goobar, A., Groom, D., Hook, I., Kim, A., Kim, M., Lee, J., Nunes, N., Pain, R., Pennypacker, C., Quimby, R., Lidman, C., Ellis, R., Irwin, M., McMahon, R., Ruiz‐Lapuente, P., Walton, N., Schaefer, B., Boyle, B., Filippenko, A., Matheson, T., Fruchter, A., Panagia, N., Newberg, H., Couch, W., & Project, T. (1999). Measurements of Ω and Λ from 42 High‐Redshift Supernovae The Astrophysical Journal, 517 (2), 565-586 DOI: 10.1086/307221
[3] Hubble, E. (1929). A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae Proceedings of the National Academy of Sciences, 15 (3), 168-173 DOI: 10.1073/pnas.15.3.168
[4] Bianchi, E., Rovelli, C., & Kolb, R. (2010). Cosmology forum: Is dark energy really a mystery? Nature, 466 (7304), 321-322 DOI: 10.1038/466321a
[5] Sean M. Carroll (2000). The Cosmological Constant LivingRev.Rel.4:1,2001 arXiv: astro-ph/0004075v2
Possible Reading of Interest
Note: There has been an unusual amount of anti-Big Bang hype this week (see “Big Bang? A Critical Review” by Ashwini Kumar Lal for some more nonsense).
Edit: I apologize for the date of publish appearing as July 27th (when I started it, as opposed to when I actually wrote it on August 1st) and messing some links up (was briefly at http://badphysics.wordpress.com/2010/08/01/nobang/).