The paper linked in the OP by David Crawford examines a critical consequence of the standard ( consensus, big bang ) cosmology -- time dilation.
Redshift and time dilation
Time dilation is a natural consequence of expanding universe cosmology in Einstein's general relativity (GR). Time dilation makes events in the distant (early) universe appear to happen slower at Earth than their intrinsic duration in the distant past. For example, an event with an intrinsic duration of 1 day occurring in the distant universe might be observed to last for 1.2 days, or 1.5,days, or 2 days or longer if it occurred further and further away.
In standard cosmologies, time dilation is associated with redshift, which is typically denoted as "z". An event with an intrinsic duration t_0 will have an apparent duration t(z) = t_0 ( 1 + z ).
Redshift, as the name suggests, is related to distant objects appearing "redder" that is there observed wavelengths are shifted to longer values than the intrinsic emitted values. Intrinsically "blue" light might appear green if slightly redshifted, red if shifted more, or even be shifted into the near or far infrared.
Like for time dilation, there are simple mathematical formulations to describe the wavelength. If the intrinsic emitted wavelength is labeled "a" the observed wavelength b = ( 1 + z ) a. Redshift of distant objects can be measured by identifying specific lines from specific elements and comparing the observed values. For example, the most prominent line of the most common element in the universe is Balmer series line at 656 nm (n = 3 -> 2 transition). If this line is observed at 787 nm, the object has a redshift of 0.2. [787 = ( 1 + 0.2 ) * 656 ] The same line from an object at z = 0.5 would be observed at b = 984 nm, etc.
In standard GR, expanding cosmologies redshift corresponds to distance as more distant parts of the universe has expanded more than more nearby parts.
It is this connection between redshift, expansion, and time dilation that Crawford's paper attempts to test. In non-standard cosmologies, such as static, or non-expanding, universes redshifts arise not from expansion, but some other effect ("tired light" is often used term, whatever that means) and there is no time dilation correlated with the redshift. For example, using the previously noted Balmer line of hydrogen and event with 1 day duration. If the same event was observed at redshift of 0.5 (the Balmer line observed at 984 nm) it would seem to last 1.5 days in a standard cosmology and only 1 day in the static, non-expanding cosmology.
So what is needed to test time dilation is a type of event with a well defined duration that can be observed in distant universe so that observed durations can be compared to their intrinsic durations. Crawford proposes that supernovae of type Ia are appropriate test objects. So, what are they?
Type Ia supernovae and cosmology
Type Ia supernovae are a specific sub-class of exploding stars that are all quite similar to each other. The escape of light from the expanding ejecta of the explosion leaves characteristic spectral features (relative brightnesses at different wavelengths in the same object at the same time) and light curves. A light curve is the pattern of brightness of an object with time. Pulsating variable stars have light curves as they expand and contract over and over. Tumbling asteroids have light curves as different faces are pointed to an observer. Quasars have light curves as the engine ingests material.
Supernova light curves rise from faint (usually unseen) to brightnesses similar to an entire galaxy in a few weeks and then fade on similar time scales (or a bit slower) often with a long "tail". In Type Ia supernovae (the only ones I'll write about from here on) the rise and fall of the light curve is almost, but not identical among all objects. The differences in their light curves has be shown to be related to the peak brightness with the brightest objects taking the most time to rise and the most time to fall in brightness. Or, to state otherwise, the duration, or width, of the light curve is related to the brightness. Thus astronomers can use a local sample of supernovae with well established distances to calibrate the relationship between peak brightness and light curve width.
With the brightness/width correlation established the intrinsic brightness of a cosmologically distant supernova can be determined by measuring the light curve width. Once the intrinsic brightness is known, the ratio of the intrinsic brightness to the brightness measured at Earth can be used to determine the geometry of the Universe. The relationship between distance and redshift is a fundamental property of any cosmology.
Now there is a further complication to this story, namely that the width of the light curve depends on the wavelength observed due to properties of the supernova ejecta, such that if you observe the same supernova in a blue filter and a red filter it will evolve slower in the red filter (wider light curve).
To use Type Ia supernovae as a test of cosmological time dilation requires that we be able to separate these effects: intrinsic width due to differences in intrinsic brightness, intrinsic width due to intrinsic wavelength effects, and time dilation. Which brings us to the Crawford paper...
Analysis of the Crawford paper
I'll skip the introduction and go straight to Figure 1 where he shows the dependence of the light curve width on wavelength, but before proceeding we must discuss the two primary ways that astronomers observe objects: spectroscopy and photometry.
A spectrum the brightness of an object at every wavelength. If we took a narrow beam of sunlight through a slit into a dark room and passed it through a prism we would see rainbow-like pattern projected onto the wall (this is what Newton did). This is the spectrum of the Sun. If we were to look closely we could see narrow dark likes where a very narrow color seemed to be mostly or completely gone. These come from absorption lines of specific atoms in the solar atmosphere. One of those lines would be the 656 nm hydrogen Balmer line discussed above. Each element has a specific pattern of lines and if we used a proper spectroscope to scan carefully the solar spectrum we would see thousands of lines where the brightness dipped a little or a lot. The wavelength dependent pattern of emission will be designated as S(a). (The (a) indicating that the spectrum is a function of the intrinsic wavelength a.)
The best observation of the supernova would be a series of spectra [ S(b,t) ] detailing its evolution, but spectra require lots telescope time (or large telescopes) than photometry, so photometry is often used. Photometry is about taking the starlight and measuring the total light detected after it passes through a filter. These filters are not dissimilar to the filters used in color photography to separate visible light into Red, Green, and Blue. Astronomical filters also exist for shorter wavelengths (ultraviolet) and longer wavelengths. The filter brightness, F, is a convolution of the filter response function, g(b), and the observed spectrum , S(b). (In pseudo LaTeX) the measured light curve for the filter would be
F(t) = \int S(b,t) g(b) db
where "\int" is the way an integral is written in the LaTeX typesetting mark-up language.
Ok, now to figure 1 of Crawford's paper...
Crawford took the template supernova spectra (found here:
start [SALT]) and measure the width at every available wavelength. The data in the "salt2_template_0.dat" file used are a series of template spectra at fixed points in the time history of a supernova. At every wavelength the history of the brightness was examine, the peak brightness found, and the width determined by finding the time when the brightness at that wavelength reaches 1/2 the peak before and after peak and measuring the time between those two half-peak points as the width. The width "W" is plotted in figure 1 for all available wavelengths after dividing by a reference width of 22 days. So far so good.
Since the width, W, is a function of wavelength, a, we can ask whether a simple curve can be fit. Crawford fits a power law ( a function in the form W(a) = C * a^e ) and gets C is about 1.9 and e = 1.2 when expressing wavelength in micrometers (or microns). The fit is pretty good for short wavelengths (a = 0.2 to 0.5 micron) passing through the central locus of the points. (Power laws don't have "wiggles" so it shouldn't be expected to match every feature, just the overall trend.) At longer wavelengths (a > 0.5 micron or 500 nm) the fit passes reasonably through the points above the green line, but does a poor job with those below. The green line separates the points Crawford used in his fit (above) from those he excluded. Excluding points from a fit isn't necessarily a bad practice as there are many reasonable causes to do so (large error bars, data collection problems, existing explanations for the deviation that are being set aside to look at something else, etc.), but his choice here represents a significant comprehension error in understanding the data set. We'll get to that shortly.
Leaving out the scaling constants, the width is proportional to a simple power law function of wavelength:
W(a) ~ a^(1.2)
So how is this impacted by redshift (and redshift alone)?
If we call the width of a measured (at Earth) supernova as Q(b), how does this behave?
The first thing we have to remember is that the observed wavelength, b, does not uniquely describe an intrinsic wavelength, a, since b = a (1+z) so the observed width is a function of b and z, Q(b,z). Observing W(a) at Earth from a supernova at redshift z will have a width Q(b,z) = a^(1.2), where we substitute a = b/(1+z) into the expression to get:
Q(b,z) = X(z) * b^1.2/(1+z)^1.2
Wait! What's X(z)?
I've inserted a time dilation factor X(z) into the expression. In a standard, BB/GR, expanding cosmology X(z) = 1+z in a non-expanding static cosmology it might be X(z) = 1. The whole point of this paper is to measure X(z), even if it's not written this specific way in the paper.
In an expanding BB cosmology with X(z) = 1+z, Q(b,z) "simplifies" to
Q(b,z) = b^1.2 * (1+z)^(-0.2)
(note that 1/(x^2) can be written as x^(-2).)
So it has the expected dependence on observed wavelength that comes from the intrinsically slower evolution at longer wavelengths when observing the same supernova with different filters, but only a weak dependence on redshift. (1+z)^(-0.2) is 1 at z = 0 and 0.87 at z = 1.
The two effects "red light curves are intrinsically slower" and "time dilation" roughly cancel for a sequence of supernovae at different redshifts at a specific observed wavelength, b. How does this work?
First we need to remember that redshift is causing the light observed from distant objects to shift in wavelength. For example, the light observed at 500 nm at Earth is 500 nm if observed for a local supernova (z = 1), but was emitted at 400 nm if the supernova has z = 0.25 and was emitted at 250 nm if the supernova was at z = 1. In the latter two cases the measured width, Q, corresponds to intrinsic light curve widths of W(400 nm) and W(250 nm), respectively.
The same applies to filters. If we have a filter that measures all light received between 400 and 500 nm ( b = [400,500] ) the intrinsic range of emitted wavelengths of the supernova light measured in that filter would be a = [400,500] for the local z = 0 supernova (no shift, b=a), a=[320,400] for the z=0.25 supernova, and a=[200,250] for the distant z = 1 supernova. So at higher and higher redshifts, the light observed in a specific filter was emitted at shorter and shorter intrinsic wavelengths, where we have already seen, the light curve changes faster.
The shifting of the light in the filters is true no matter what the origin of the redshift is as it is merely a property of the shift of the wavelength and not the origin of the shift. But, in a static, non-dilating cosmology the light curves should get significantly faster with redshift in a fixed, ~(1+z)^(-1.2), since the filter measures light from bluer, and therefore faster changing, parts of the supernova emission without the compensating effect of time dilation. [
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Trigger warning: If you think Crawford's paper will demonstrate there is no time dilation, brace yourself, because it's time for...
Crawford's blunder:
Let's go back to the "green line" of exclusion in figure 1. Quoting from the opening paragraph of section 3.2:
"The green line ... shows the lower[sic] limit of where the fitted widths are valid. After all if this variation in widths is of cosmological origin it must be a smooth function of wavelength."
Oh, boy! What can we say here. It isn't of cosmological origin. The source of the data for the sequence of template spectra used to make figure 1 explicitly says that these are models of the sequence intrinsic spectral changes in a standard type Ia supernova.
So what are spectral templates? They are smoothed and idealized versions of carefully calibrated, well observed normal type Ia supernovae from nearby galaxies where the tiny redshifts are not cosmological, but originate from relative motions of galaxies. Observations from several supernovae have been combined and then smoothed fits are made to represent their "average" at several fixed epochs. In essence, what the authors of this paper:
http://adsabs.harvard.edu/abs/2005A%26A...443..781G
are saying about their templates is that for "normal" supernovae this is what they really look like if you have a perfect instrument placed nearby. (Don't get to close, it will burn you!) Things go downhill from here...
Crawford then stumbles through his assumptions to the statement "the raw light curves would have widths that would be proportional to (1+z)^(-0.199)." where I think by "raw" he means the reported, observed values in standard astronomical filters. But the conclusion of the section seems to betray his deep misunderstanding: "If the standard time dilation is present then the rest frame spectrum must have a dependence of [wavelength]^1.199 which seems rather large should be rather obvious in the widths between different filters for low redshift supernovae." This is a really odd thing to say since he just showed that (at least the blue part) of the spectrum does have an intrinsic wavelength dependence of a^(1.2).
His "counterpoint" in section 3.3 (and Table 2) is in part that the z = 0 starting points of the widths (which he calls V_0 in Table 2) should have a strong a^(1.2) dependence, but the actual list of filters used in the observations include many which are on the red half of range in Figure 1, and whether he wants to exclude them or not, include individual wavelengths with low W(a). I don't know exactly how those would affect the variations in intrinsic width for various filter bands (how supernova light curves are actually observed and reported) but it does not surprise me that the redder filters do not give drastically slower light curves given the low values of W(a) in the red for some wavelengths.
Aside: It occurs to me that the methodology in figure 1 was not the best. If you want to understand how filters respond, why not make some idealized "artificial" filters. A boxcar or Gaussian of some width evaluated at all possible wavelengths on the templates and then compute a fitting function for the central wavelength of a filter. I suspect it would be similar to what he has in the blue, but the red may not have fit the trend.
Figure 2
For his next analysis, Crawford takes a large database of observed supernovae (I didn't track it down) and measures the width of the observed light curves using a simple template (presumably like the one used in the template spectrum analysis of figure 1) and fits a width. After dividing by the V_0 (Table 2) for that filter, which seems reasonable, he plots them in Figure 2 relative to their redshift and then fits a line to the collective set. His line indicates that V(z) (what he uses instead of Q(b,z)) as the wavelength independent (or normalized) light curve width is consistent with being a constant (V(z) = 1). This is also not inconsistent with the Q(F,z) = Q(F) * z^(-0.2) I showed above for an expanding cosmology. Crawford somehow believes that the expanding universe model should have a positive slope for his V(z) (apparently V(z) = 1 + z, which would be the time dilation you'd expect if redshift itself had no impact on observed light curve widths) and so plots his "expanding cosmology" line on the plot to show how bad it "fails".
Wrapping up...
There are a few other things I could write, but this has gone on long enough. I'm not sure why the author made such large errors, but they should have been caught before publication. So, shame on the editors of "Open Astronomy" for not having it properly reviewed. Someone who really knew the topic could do a much more complete job than I did.
