Page 1

DES-2024-0831

FERMILAB-PUB-24-0293-PPD

MNRAS 000, 1–13 (2024)

Preprint 10 June 2024

Compiled using MNRAS LATEX style file v3.0

The Dark Energy Survey Supernova Program: Slow supernovae show

cosmological time dilation out to 𝑧 ∼ 1.

Ryan M. T. White1★ , Tamara M. Davis1 , Geraint F. Lewis2 , Christopher Lidman3,4 , Paul Shah5 ,

T. M. C. Abbott6, M. Aguena7, S. Allam8 , F. Andrade-Oliveira9, J. Asorey10, D. Bacon11, S. Bocquet12 ,

D. Brooks5 , D. Brout13, E. Buckley-Geer14,8 , D. L. Burke15,16, A. Carnero Rosell17,7 , D. Carollo18,

J. Carretero19 , L. N. da Costa7, M. E. S. Pereira20, J. De Vicente21 , S. Desai22 , H. T. Diehl8 ,

S. Everett23, I. Ferrero24, B. Flaugher8 , J. Frieman8,25 , J. Garc�a-Bellido26 , E. Gaztanaga27,11,28 ,

G. Giannini19,25 , K. Glazebrook29, R. A. Gruendl30,31, S. R. Hinton1, D. L. Hollowood32,

K. Honscheid33,34 , D. J. James13 , R. Kessler14,25 , K. Kuehn35,36 , O. Lahav5 , J. Lee37 , S. Lee23,

M. Lima38,7, J. L. Marshall39 , J. Mena-Fern�ndez40 , R. Miquel41,19 , J. Myles42, A. M�ller29,

R. C. Nichol11, R. L. C. Ogando43 , A. Palmese44 , A. Pieres7,43 , A. A. Plazas Malag�n15,16 ,

A. K. Romer45 , M. Sako37, E. Sanchez21 , D. Sanchez Cid21 , M. Schubnell9 , M. Smith46 ,

E. Suchyta47 , M. Sullivan46 , B. O. S�nchez48,49 , G. Tarle9 , B. E. Tucker4, A. R. Walker6 ,

N. Weaverdyck50,51, and P. Wiseman46,

(DES Collaboration)

Affiliations are listed at the end of the paper.

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

We present a precise measurement of cosmological time dilation using the light curves of 1504 type Ia supernovae from the

Dark Energy Survey spanning a redshift range 0.1 ≲ 𝑧 ≲ 1.2. We find that the width of supernova light curves is proportional

to (1 + 𝑧), as expected for time dilation due to the expansion of the Universe. Assuming type Ia supernovae light curves are

emitted with a consistent duration Δ𝑡em, and parameterising the observed duration as Δ𝑡obs = Δ𝑡em(1 + 𝑧)𝑏, we fit for the form of

time dilation using two methods. Firstly, we find that a power of 𝑏 ≈ 1 minimises the flux scatter in stacked subsamples of light

curves across different redshifts. Secondly, we fit each target supernova to a stacked light curve (stacking all supernovae with

observed bandpasses matching that of the target light curve) and find 𝑏 = 1.003 � 0.005 (stat) � 0.010 (sys). Thanks to the large

number of supernovae and large redshift-range of the sample, this analysis gives the most precise measurement of cosmological

time dilation to date, ruling out any non-time-dilating cosmological models at very high significance.

Key words: supernovae: general – cosmology: observations

1 INTRODUCTION

Time dilation is a fundamental implication of Einstein’s theory of

relativity in an expanding Universe — the observed duration of an

event, Δ𝑡obs, should be longer than the intrinsic emitted (or rest-

frame) duration, Δ𝑡em, by a factor of one plus the observed redshift,

𝑧,

Δ𝑡obs = Δ𝑡em(1 + 𝑧).

(1)

The idea of using time dilation to test the hypothesis that the Universe

is expanding dates back as far as Wilson (1939) and was revisited

★ E-mail:ryan.white@uq.edu.au

by Rust (1974). One of the first observational hints of time dilation

was the observation by Piran (1992) and Norris et al. (1994) that the

duration of gamma-ray bursts (GRBs) was inversely proportional to

their brightness – they used this to argue that at least some GRBs

must be cosmological. The first measurements of cosmological time-

dilation using supernovae were made by Leibundgut et al. (1996) for

a single Type Ia supernova (SN Ia) at 𝑧 = 0.479 and Goldhaber et al.

(1997) for seven supernovae at 0.3 <𝑧< 0.5. Most relevant to this

work, we take Goldhaber et al. (2001) as the current state of the art

in identifying cosmological time dilation in SN Ia photometry. They

used 35 supernovae in the redshift range 0.30 ≤ 𝑧 ≤ 0.70, to test a

model with a factor (1+𝑧)𝑏 time dilation and found 𝑏 ∼ 1.07�0.06.

To avoid degeneracy between the natural variation of light-curve

� 2024 The Authors

arXiv:2406.05050v1 [astro-ph.CO] 7 Jun 2024

2

White et al.

width and time dilation, Foley et al. (2005) and Blondin et al. (2008)

observed time dilation in the evolution of spectral features of high-𝑧

Type Ia supernovae (SNe Ia). The former found inconsistency with

no time dilation at the 99% confidence level and the latter finding

𝑏 = 0.97 � 0.10. Most recently, Lewis & Brewer (2023) inferred

𝑏 = 1.28

+0.28

−0.29

using the variability of 190 quasars out to 𝑧 ∼ 4.

Despite these successes, there remains continued discussion of hybrid

or static-universe models such as Tired Light (Zwicky 1929; Gupta

2023) that do not predict expansion-induced time dilation.

In this study, we measure cosmological time dilation using SNe Ia

from the full 5-year sample released by the Dark Energy Survey

(DES) (DES Collaboration et al. 2024), which contains ∼ 1500 SN Ia

spanning the redshift range 0.1 ≲ 𝑧 ≲ 1.2 — significantly larger and

higher-redshift than any sample of supernovae previously used for a

time-dilation measurement. Such a large sample of SNe is important

in reducing statistical uncertainty and such a high-redshift sample

is the ideal regime to robustly identify time dilation. Over half the

DES-SN5YR sample are at 𝑧 > 0.5, compared to only 12% in the

previous gold-standard Pantheon+ sample (Brout 2019) and 37% in

the Goldhaber analysis (Goldhaber et al. 2001). Therefore the DES

sample should have observed light-curve durations more than 1.5

times longer than their rest frame durations (up to 2.2 times longer

for those at 𝑧 ∼ 1.2). This means their time dilation signal should be

significantly larger than the intrinsic width variation expected due to

SNe Ia diversity in their subtypes.

We test the model that time dilation occurs according to,

Δ𝑡obs = Δ𝑡em(1 + 𝑧)𝑏

.

(2)

If standard time dilation is true we should find 𝑏 = 1. If no time

dilation occurs we should find 𝑏 = 0.

For this work we aim to keep supernova modelling assumptions to

a minimum to avoid circularity in our arguments (because most mod-

els of supernova light curves are generated assuming time-dilation

occurs). We therefore take two data-driven approaches to measuring

time dilation:

(i) Firstly, we simply take all the light curves, divide their time

axis by (1 + 𝑧)𝑏 relative to the time at peak brightness, and find the

value of 𝑏 that minimises the flux scatter.

(ii) Secondly, we ‘de-redshift’ all the supernova light curves and

stack them to define a data-driven SN Ia ‘reference light curve’. Then

for each individual SN Ia we measure the observed light curve width

(𝑤) relative to the appropriate reference, Δ𝑡obs = 𝑤Δ𝑡reference. This

allows us to see if the time-dilation occurs smoothly with redshift

and find the best-fit value of 𝑏, where the expected result would be

𝑤 = (1 + 𝑧) corresponding to 𝑏 = 1.

The first method is entirely data-driven and has no time-dilation

assumptions. In the second method for ease of computation we create

the stacked reference by dividing the time axis of the light curves by

(1+𝑧). This method therefore includes an assumption of time-dilation

in the generation of the reference. Even though this assumption is

justified by the result of the first method, the second method should

strictly be considered a consistency check. We note it is possible

to remove any circularity by keeping the reference light curves in

their rest frame and fitting to 𝑤 = ((1 + 𝑧target)/(1 + 𝑧reference))𝑏,

which requires multiple different reference light curves per target

supernova; mathematically this is very similar to our approach but

requires even more data. To further check that this method rules out

no time dilation we re-test method two without de-redshifting the

reference light curves; it dramatically fails the consistency check,

see Appendix C.

This paper is arranged as follows. In Section 2 we discuss the use

of type Ia supernovae as standard clocks, and the challenges that

need to be taken into account when comparing SNe Ia light curves

observed in different bands across different redshifts. In Section 3, we

present the data used in this study, while Section 4 we describe our

approach of defining a reference light curve and the determination of

the redshift dependence of the time dilation signal. We discuss our

results in Section 5 and conclude in Section 6 that the null hypothesis

of no time dilation is inconsistent with the data.

2 TYPE IA SUPERNOVAE AS STANDARD CLOCKS

Any investigation into the physics at large length scales in the uni-

verse relies on known quantities, be they standard candles, rulers,

sirens, or clocks. SNe Ia have long fit the bill of a standardisable can-

dle on the basis of their extreme brightness and consistency (Tripp

1998; M�ller-Bravo et al. 2022; Scolnic et al. 2023), allowing their

observation over cosmic distances with only little uncertainty in their

intrinsic properties. As SNe Ia are the explosions of a white dwarf

approaching the Chandrasekhar limit (Hoyle & Fowler 1960; Ruiter

2019), their properties are reasonably uniform across their popula-

tion compared to other SN types; not only are they standardisable in

brightness, but also in time (Phillips 1993; Leibundgut et al. 1996).

Hence, the observed duration of SN Ia explosions are well suited

to investigating time dilation as a result of an expanding universe

(Wilson 1939; Rust 1974).

The presence of a time dilation signal in SNe Ia data tests the gen-

eral relativistic prediction of an expanding universe having a factor

of (1 + 𝑧) time dilation (Wilson 1939; Blondin et al. 2008). This

signal needs to be corrected for in supernova cosmology analyses

(Leibundgut & Sullivan 2018; Carr et al. 2022) and so conclusively

quantifying the effect of time dilation is foundational to our cos-

mological model, especially considering the continued discussion of

hybrid or static-universe models such as Tired Light (Zwicky 1929;

Gupta 2023) that do not predict expansion-induced time dilation.

2.1 The importance of colour

SNe Ia are known to spectrally evolve over the duration of their

∼ 70 day bright period. The early-time spectrum is relatively blue

with spectral features dominated by transitions from intermediate

mass elements. The spectrum then reddens on the order of days from

heavier element emission lines and the cooling of the continuum

(Filippenko 1997). Previous papers have described the redward evo-

lution of SNe Ia spectra over time (Takanashi et al. 2008; Blondin

et al. 2012; Branch & Wheeler 2017), while photometric evidence

of this phenomenon is seen in the light curve peaking later in redder

bandpasses than in bluer ones (as in Figure 1) for a light curve in

the rest-frame optical. As such, the photometric behaviour of a light

curve is dependent on the rest-frame wavelength range observed.

This means that, to a good approximation, a typical high-𝑧 SN Ia

observed in a redder band should have the same photometric and

spectral characteristics as a medium-𝑧 SN Ia observed in a bluer band

(Figure 2). Since our photometric bands are fixed, they sample dif-

ferent rest-frame wavelength ranges as the supernovae are redshifted.

Therefore, it is critical to design a method that ensures time dilation

measurements compare light curves measured at similar rest-frame

wavelengths.1

1 A note on language: The phrase ‘rest-frame’ wavelengths arises from the

usual assumption that redshifts are due to recession velocities. The fact red-

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Time Dilation with DES SNe Ia

3

-20

0

20

40

60

Time Since Peak Brightness, t - tpeak (days)

0.0

0.2

0.4

0.6

0.8

1.0

Normalised

Flux

r-band data

i-band data

z-band data

Figure 1. The normalised (in flux) light curve of a SN Ia at 𝑧 = 0.4754

shows intrinsically broader light curves at redder wavelengths that tend to

peak later (at least in the optical regime observed across our dataset). The

𝑥-axis represents time in the observer frame, and SALT3 model fits (solid

lines) are overlaid onto the data in each band.

400

500

600

700

800

Wavelength (nm)

0.0

0.2

0.4

0.6

0.8

1.0

Relativ

e

T

ransmittance/In

tensity

DES g

DES r

DES i

z = z1

z = z2

Figure 2. For a supernova at redshift 𝑧1 observed in a given filter, there exists

a higher redshift 𝑧2 > 𝑧1 such that another supernova at 𝑧2 observed in a

redder filter will have a similar rest frame effective wavelength as the nearer

supernova. We show here the spectrum of SN2001V (Matheson et al. 2008)

at 𝑧1 ∼ 0.01 and again artificially redshifted to 𝑧2 ∼ 0.35. Here we can

clearly see the Si II absorption line (615nm) redshifted from the 𝑟 band to

roughly the same position along the bandwidth in the 𝑖 band. Overlaid are the

transmission curves of the DES filters from Abbott et al. (2018) Fig. 1.

2.2 Stretch-Luminosity relation

Our fitting methods are independent of individual supernova light

curve models and are based only on the assumption that supernova

shifts occur is not in question here (so it is fine to use (1 + 𝑧) to calculate

matching rest-frame wavelengths, and this contains no time-dilation assump-

tion). The question is whether that redshift arises due to a recession velocity,

which would also cause time-dilation.

light curves are all similar. One dissimilarity between SNe Ia is the

‘stretch’ in their light curve – an intrinsic width variation of up to

∼ 20% that is separate from time dilation and strongly correlated

with the peak brightness (Phillips et al. 1999). With this amount of

data at such high redshifts we can simply treat this intrinsic variation

as noise. The stretch variation between SNe Ia essentially acts as

random (but intrinsic) scatter in the obtained widths of light curves.

This does not affect the overall trend of light curve width against

redshift. We therefore make no correction for the stretch-luminosity

relation in this work, to maintain maximal model-independence.

As long as there is a representative sample of the entire popula-

tion of SNe Ia at every redshift, this simple analysis should measure

time-dilation without bias. However, Malmquist bias can influence

the result since brighter supernovae have wider light curves. If faint

supernovae are under-represented at high-redshifts one might ex-

pect a slight bias toward a higher inferred time dilation at high-

𝑧. Thankfully, the DES data are well-sampled to such high-𝑧 that

Malmquist bias has minimal impact on our results. M�ller et al.

(2022) showed that the full stretch distribution is well represented in

the DES-SN5YR sample out to 𝑧 ∼ 1.1 (see also Fig. A2), meaning

that the stretch-luminosity relation should have a negligible effect on

our results.

Previous studies (e.g. Nicolas et al. 2021) have also found that

the stretch distribution of the SN Ia population drifts slightly with

redshift (getting wider by ∼ 3% between 0.0 <𝑧< 1.4). Even though

we do not see this drift in the DES-SN5YR sample (see Fig. A2),

we quantify the possible impact of this effect on our time dilation

measurement in Appendix A and find it to be small.

3 DATA

We exclusively use the data of the 1635 type Ia supernovae measured

by the Dark Energy Survey Supernova Program (DES Collaboration

et al. 2024). The DECam instrument on the 4m Blanco telescope at

the Cerro Tololo Inter-American Observatory (Flaugher et al. 2015)

observed most of the photometrically classified SN Ia candidates in

the 𝑔, 𝑟, 𝑖, and 𝑧 bands according to the criteria in Smith et al. (2020).

The flux is determined by difference imaging (Kessler et al. 2015).

High-redshift SNe Ia typically show negligible flux in the ultraviolet

wavelength region, so 𝑔 and 𝑟 band light curves are only useful for

SNe at 𝑧 ≲ 0.4 and 𝑧 ≲ 0.85 respectively (see DES Collaboration

et al. 2024, Fig. 2). The SALT3 (Kenworthy et al. 2021) template fits

with a cadence of 2-day time sampling in each band were available for

each SN candidate. We use these fits to estimate the peak flux of each

supernova so we can normalize the observed flux values; the time

relative to the 𝐵-band maximum in the SALT3 fit for each light curve

is also used to define the time since peak brightness measurement in

our analysis. We otherwise discard the SALT supernova information.

We performed an initial quality cut on the sample of 1635 SNe Ia,

requiring the probability of being a type Ia PROBIa > 0.5 as classified

with SuperNNova (M�ller & de Boissi�re 2020; M�ller et al. 2022;

Vincenzi et al. 2024). This kept the sample of usable light curves

high while removing possible type II supernova contaminants. We

removed individual data points from each light curve that had an

error in their flux (FLUXCALERR2 for the flux value FLUXCAL) greater

than 20; this was done to restrict our fitting to the highest quality ob-

servations, particularly cutting those with very low signal-to-noise at

2 FLUXCALERR is the Poisson error on FLUXCAL, which is the variable used

for flux in SNANA corresponding to mag = 27.5 − 2.5 log10 (FLUXCAL).

MNRAS 000, 1–13 (2024)

4

White et al.

2−1

2−2

2−3

2−4

2−5

Width Factor, δ

0.2

0.4

0.6

0.8

1.0

T

arget

SNe

Redshift

z

102

103

104

Reference

P

opulation

Figure 3. For each of the SNe Ia in our sample, we constructed a reference

light curve with a 𝛿 = 2−𝑥 parameter according to Equation 3, with Δ𝜆𝑓

band FWHM. We counted how many data points populated the reference

curves (i.e. the number of points in Fig. 4 for example) changing 𝛿 in integer

steps of powers of 2. This plot shows the reference populations for a target

SN measured in the observer frame 𝑖 band, and is largely similar for the

different bands differing mainly by a linear shift on the vertical axis. That is,

an analogous plot in a bluer band would see the colour distribution shifted

downwards in target redshift and a redder band upwards.

high redshift (whose observations had comparatively low FLUXCAL).

In the analysis, we did not attempt to fit SNe light curve widths if

their light curve had fewer than 5 data points and if their reference

curve had fewer than 100 data points (discussed in Section 4). This

was done on a per-band basis; we estimated the width of each SN

light curve in each band where it satisfied these criteria. Individual

light curves were also omitted from the analysis if the 𝜒2 width fitting

did not converge. All together, after these quality cuts we were left

with width measurements of 1504 unique SN Ia across the dataset.

4 FITTING SUPERNOVA PHOTOMETRY TO A

REFERENCE LIGHT CURVE

The photometric analysis in Goldhaber et al. (2001) relied funda-

mentally on template fitting to measure the expansion-induced time

dilation signal in SNe Ia. With the wealth of data now available we

can in principle obtain the same dilation signal using the data alone,

independent of a light-curve template. Herein we describe such a

template-independent method involving scaling the time axis of each

of the 1504 individual light curves to fit their own unique reference

curve composed of all the matching photometry in the DES dataset.

The time dilation signal in each light curve is the multiplicative in-

verse of said scaling factor, and this can be found for each of the 1504

SN Ia light curves in the data. This differs from a traditional template-

fitting method in that we do not assume the shape of a light curve

but instead let the data from other SNe compose the template-like

reference curve.

4.1 Reference curve construction

The main functionality of this method is to use the photometric data

of many SNe to automatically create a reference light curve unique

to any one target SN. The target SN is the SN whose width we are

trying to measure.

Since the shape of a SN Ia light curve is dependent on the rest-

frame effective wavelength at which it is observed (Fig. 1; see also

Takanashi et al. 2008; Blondin et al. 2012), the reference photometry

must be composed only of light curves that have the same (or very

similar) intrinsic shape as the target SN. Hence, we must choose ref-

erence photometry that samples the same rest-frame effective wave-

length as the target light curve. This effect is shown in Fig. 2, where,

for example, we might compare a low-𝑧 supernova in some band

to a higher-𝑧 supernova observed in a redder band with the same

(or similar) rest frame effective wavelength. We can compare the

photometry between the two events provided that their rest-frame

effective wavelength (and hence their light curve shape/evolution) is

alike.

If we chose instead to fit each target light curve in some band

against all of the photometry from that band, we would expect a non-

linear change in slope as a function of redshift on a width-vs-redshift

plot. The explanation for this lies in the fact that SNe Ia spectra get

redder over time; the light curves measured in a redder band are

intrinsically wider than those measured in a bluer band as shown

in Fig. 1. Hence, with this hypothetical method (comparing to all

photometry), we would observe a bluer than average rest-frame curve

for a high redshift SNe which would bias the obtained width to an

intrinsically thinner value. Conversely, we would be biased towards

a wider (redder) than average width for low redshift supernovae. To

avoid this bias, we use the aforementioned method of only using

reference photometry with a similar rest-frame wavelength as our

target light curve.

To find relevant light curves to populate the reference curve, we

pick all light curves out of a calculated redshift range. To fit a single

(target) SN light curve at redshift 𝑧 imaged in a band of central

wavelength 𝜆 𝑓 , we can populate the reference curve with SNe within

the redshift range

𝜆𝑟 (1 + 𝑧)

𝜆 𝑓

− 𝛿

Δ𝜆 𝑓

𝜆 𝑓

≤ 1 + 𝑧𝑟 ≤

𝜆𝑟 (1 + 𝑧)

𝜆 𝑓

+ 𝛿

Δ𝜆 𝑓

𝜆 𝑓

(3)

whose photometry is measured in a band of central wavelength 𝜆𝑟 .

Here 𝛿 is a free parameter which, together with the band full width at

half maximum (FWHM) Δ𝜆 𝑓 , describe the acceptable wiggle room

in the relative band overlap. A derivation of this formula is given

in Appendix B and a graphical representation of this construction is

shown on the left-side plots in Fig. 6.

We show in Fig. 3 the number of points in the reference curves

(hereafter referred to as reference population) for all of the DES SNe

with a variable 𝛿 parameter. Ideally, this 𝛿 parameter should be as

small as practical to ensure that the reference curve is consistent in

shape (i.e. the spread of rest frame effective wavelengths is small). In

practice, we find a value of 𝛿 = 2−4 is the minimal value that provides

a large enough reference population for high/low redshift target SNe

(on the order of ∼ 102 needed to satisfy the Section 3 criteria at

𝑧 ∼ 1.1). For medium range target SNe redshifts (in the context of

the DES-SN sample), we note that the reference population is large

(∼ 103) even for 𝛿 ≲ 2−4.

After populating the reference curve with data points we then

normalise the photometry in flux; as the curve is populated with

the data of several SNe at different redshifts, the curve must be

homogeneous in flux. To do this, we utilised the peak flux in the

SALT3 model light curves provided for each SNe. The data in each

constituent curve is normalised by this value before being added to

the reference. For convenience we also use the time of peak brightness

given by SALT3 as the reference point about which to stretch the light

MNRAS 000, 1–13 (2024)

Time Dilation with DES SNe Ia

5

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

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Normalised

Flux

(Pre-Correction)

−20

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40

60

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0.0

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1.0

1.2

1.4

Normalised

Flux

(Corrected)

z ∼ 0.21 (r band)

z ∼ 0.48 (i band)

z ∼ 0.74 (z band)

Figure 4. For a target SN Ia at 𝑧 ≃ 0.48, the 𝑖-band reference curve consists of

data from the 𝑟, 𝑖, and 𝑧 bands. These data are chosen at redshifts according

to equation (3) (and visually shown in the left plots of Fig. 6). Top: the data

in different bands are not in phase (in the observer frame). It is visually

obvious that light curves appear wider in time at higher redshift. Bottom:

after a (1 + 𝑧) correction to the SN Ia light curves, we see a consistent trend

across all bandpasses and time. This alone is evidence for some degree of

time-dilation.

curves (see equation 5). These are the only uses of SALT3 information

and we expect that the same time dilation signal would be obtained

in the data with any other consistent normalisation measures.

4.2 First measure of time dilation: minimising scatter in the

reference curve

After the flux of the reference curve is normalised, we see that the

different bandpass data in the curve are temporally stretched (see the

colour gradient of the top plot in Fig. 4). As the redder bandpasses

are sampled at higher redshift, this is an immediate indication of time

dilation. Without assuming our expected cosmological time dilation

of (1 + 𝑧), we can scale the data in all of the reference curves by a

factor of (1 + 𝑧𝑗)𝑏, where 𝑧𝑗 is the redshift of each constituent curve

in a reference and 𝑏 is a free parameter. We posit that minimising

the flux dispersion in the reference curve is analogous to finding the

optimal temporal scaling, simultaneously minimising the dispersion

in time. Hence, finding the value of 𝑏 that minimises the flux scatter

gives us our correction factor.

To investigate this, we generated reference curves for each of the

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

b

0.130

0.135

0.140

0.145

0.150

Reference

Flux

Disp

ersion

Figure 5. By scaling the reference photometry in time according to (1+𝑧)𝑏 for

some free parameter 𝑏, we find 𝑏 ∼ 1 minimises the reference flux dispersion

across the entire SNe sample. The reference flux dispersion represents the

median dispersion of flux across the entire sample of normalised reference

light curves in each band (here averaged for the𝑟𝑖𝑧 bands), where the errorbars

indicate one standard deviation in these values. We note that this figure yields

a signal of (1 + 𝑧) time dilation in the DES dataset, independent of the rest

of the analysis.

target SNe per observing band and scaled the data in time according

to the aforementioned relation in terms of 𝑏 (Fig. 4 shows this scaling

for 𝑏 = 1 as an example). Then, we binned the timeseries data into

30 equal-width time bins and found the standard deviation of the

flux within each bin. We calculated the median of these 30 standard

deviations as a representative estimate of the total flux scatter for that

reference curve with that tested 𝑏 value. We then took the median of

those results across all the SN reference curves as our estimate of the

dispersion for that 𝑏, which is shown in Fig. 5. That is, our reference

flux dispersion is

𝜎rf(𝑏) = Med ({{Med(𝜎𝑖 𝑗 (𝑏))|∀𝑗 ∈ (1, ..., 30)}|∀𝑖 ∈ (1, ..., 𝑁sn)})

(4)

for 𝑁sn supernova light curves in that band, and 𝜎𝑖 𝑗 (𝑏) being the

flux standard deviation of the 𝑖th light curve in the 𝑗th timeseries

bin. This process was repeated for each of the 𝑟𝑖𝑧 observing bands,

omitting the 𝑔 band due to the smaller number of SNe. We crudely

estimate the error (for each 𝑏) in this method as being the standard

deviation of the median dispersions across all light curves. We find

an optimal scaling corresponding to 𝑏 ∼ 1 (Fig. 5) across the entire

dataset, which is the expected dilation factor of ∼ (1 + 𝑧).

If there was no time dilation we would expect the minimum dis-

persion in the reference curve to be at 𝑏 ∼ 0 (i.e. no time scaling) in

Fig. 5. The fact that we find 𝑏 ≈ 1 is evidence for time dilation of

the expected form. This a rough but completely model independent

measure of time dilation and it is the paper’s first main result.

4.3 Second measure of time dilation: Finding each light curve

width

After constructing the reference curves for a target SN, we are ready

to fit for the width, 𝑤, of each individual target light curve and look

for a trend with redshift. This method enables a more precise measure

of 𝑏.

MNRAS 000, 1–13 (2024)

6

White et al.

0.00

0.25

0.50

0.75

1.00

Target SNe z

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Reference

z

g Band

r Band

i Band

z Band

-10

0

10

20

30

40

Time Since Peak Brightness (days)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Normalised

Flux

Reference Photometry

Target Photometry (Obs. Frame)

Target Photometry (Scaled)

0.00

0.25

0.50

0.75

1.00

Target SNe z

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Reference

z

g Band

r Band

i Band

z Band

-20

-10

0

10

20

30

40

Time Since Peak Brightness (days)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Normalised

Flux

Reference Photometry

Target Photometry (Obs. Frame)

Target Photometry (Scaled)

0.00

0.25

0.50

0.75

1.00

Target SNe z

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Reference

z

g Band

r Band

i Band

z Band

-30

-20

-10

0

10

20

30

40

Time Since Peak Brightness (days)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Normalised

Flux

Reference Photometry

Target Photometry (Obs. Frame)

Target Photometry (Scaled)

Figure 6. We show the reference curve construction and subsequent target SN fit for 3 SNe at redshifts 𝑧 ≃ 0.22, 𝑧 ≃ 0.43, and 𝑧 ≃ 1.02 and in fitting bands 𝑟,

𝑖, and 𝑧 respectively (in descending order). The left plots show the allowed ranges for reference curve SN sampling given the target redshift (and 𝛿 = 2

−4). The

vertical line of dots is plotted at the target SN redshift, with each dot representing the redshift of a DES supernova (vertical axis). The dots that fall in the narrow

coloured bands are the SNe that make up the reference population, as those data all share approximately the same rest-frame wavelength in their respective

bands. The right plots show the constructed (1 + 𝑧) time-scaled reference curve (small coloured points) with respect to the target SN photometry (blue points)

and subsequent target photometry scaled on the time axis to fit the reference (best-fit widths of 1.42, 1.49, and 2.17 respectively). Due to the statistics associated

with such large reference curve populations, the contribution of any individual reference point uncertainty to the overall reference curve uncertainty is negligible

and not plotted; the uncertainty in the target data has a much higher contribution to the uncertainty in the fitting.

MNRAS 000, 1–13 (2024)

Time Dilation with DES SNe Ia

7

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ligh

tcurv

e

width

w

g

1 + z

(1 + z)1.021�0.027

Data

Binned Data

г

(1 + z)1.004�0.008

1.0

1.2

1.4

1.6

1.8

2.0

1 + z

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ligh

tcurv

e

width

w

i

(1 + z)0.988�0.008

1.0

1.2

1.4

1.6

1.8

2.0

1 + z

ζ

(1 + z)0.997�0.008

Figure 7. Using the reference-scaling method described in Section 4.3, we plot the fitted SNe widths of light curves observed in the 𝑔, 𝑟, 𝑖, and 𝑧 bands (left to

right, top down respectively). The lines of best fit (blue dashed) are in excellent agreement with the expected (1 + 𝑧) time dilation (black dotted). The binned

data are purely to visualise rough trends in 50 data point bins. 361 SNe in the 𝑔 band passed the quality cuts described in Section 3, while the 𝑟 band has 1380

SNe, the 𝑖 band 1465, and the 𝑧 band 1381. The reduced chi-square values, 𝜒2

𝜈, of each fit (left to right, top down) are 0.537, 0.729, 0.788 and 0.896 respectively.

We first normalise the target data to the peak flux using the SALT3

fit (as with the reference curves). The free parameter in the fit is the

scaling parameter 1/𝑤, whereby changing this value would stretch

and squash the data relative to 𝑡peak (the time since peak flux) until

the 𝜒2 is minimised. That is, we assume the SN Ia light curve of the

𝑖th supernova is of a mathematical form similar to that described in

Goldhaber et al. (2001),

𝐹𝑖 (𝑡) ≃ 𝑓𝑖

(𝑡 − 𝑡peak

𝑤

)

(5)

and change 𝑤 until the data most closely matches the reference. Here,

𝑓𝑖 (𝑡) corresponds to the 𝑖th target light curve; 𝐹𝑖 (𝑡) corresponds to

the 𝑖th reference curve where each point is now scaled in time by

(1 + 𝑧) relative to 𝑡peak as per the results of Section 4.2.

To fit the target light curve width using its reference curve, we

minimised the 𝜒2 value of the differences in the target flux compared

to the median reference flux in a narrow bin around time values of the

target photometry. That is, for each target light curve we minimised

𝜒2

𝑖 =

𝑁𝑝

∑

𝑗

( 𝑓𝑖 𝑗 − Med {𝐹𝑖 (𝑡)|∀𝑡 ∈ [𝑡𝑖 𝑗/𝑤 − 𝜏, 𝑡𝑖 𝑗/𝑤 + 𝜏]})

2

𝜎2

𝑖 𝑗

(6)

for 𝑁𝑝 number of points in the 𝑖th target SN light curve ( 𝑓𝑖). The

points in the reference curve (𝐹𝑖) bin that are averaged and compared

to each target SN flux value ( 𝑓𝑖 𝑗 – with error 𝜎𝑖 𝑗) are selected within

the time range [𝑡𝑖 𝑗/𝑤 − 𝜏, 𝑡𝑖 𝑗/𝑤 + 𝜏]; here 𝑡𝑖 𝑗 is the time since peak

brightness of each target data point scaled by the fitted width 𝑤, and

𝜏 is the half bin width either side of the central time value 𝑡𝑖 𝑗.

During the fitting process, the bounds of this narrow bin around

each time value changes as the target data is scaled in time but remains

the same width. We chose a bin width, 2𝜏, of 4 rest-frame days (i.e.

�𝜏 = �2 of a central value); ideally this would be as low as practical

to maximise intrinsic similarity between the target data point position

and the reference curve slice, but needs to be large enough to provide

a sufficiently populated sample of the reference to compare to. We

find that a width of 4 days (just under the width of a minor tick span in

Fig. 4) is low enough that the reference curve does not significantly

change in flux but still contains enough points even for high/low

redshift target SNe with small reference populations. With this 𝜏 = 2

value we find ≳ 50 data points per time slice at the highest and lowest

redshifts, where a 𝜏 = 1 yields a prohibitively small ≲ 20 data points

per slice even in the most well sampled photometric band (𝑖-band).

In fitting the data, we did not include any target SN data points that

extended past the maximum time value in the reference curve; the

late-time light curves of SNe dwindle slowly and are less constraining

for width-measurements than those near the peak. We also omitted

any points that had observation times prior to the first reference curve

point from the fitting procedure.

MNRAS 000, 1–13 (2024)

8

White et al.

1.0

1.2

1.4

1.6

1.8

2.0

1 + z

0.5

1.0

1.5

2.0

2.5

3.0

Ligh

tcurv

e

width

w

1 + z

(1 + z)1.003�0.005

Data

Binned Data

1

2

3

4

Num

b

er

of

Av

eraged

Bands

Figure 8. We show here the width value for each SNe averaged across all available bands. Since cosmological time dilation is independent of the observed band

of any SN, we are able to average the widths over observed bands to form a more robust estimate of the light curve width. This relationship of (1 + 𝑧)1.003�0.005

time dilation (reduced chi-square 𝜒2

𝜈 ≃ 1.441) is comprised of the 1504 unique SNe across the 4 bandpasses, where the error bars here are the Gaussian

propagation of the errors in each band. Points are coloured according to how many bandpasses were used in computing the averaged width. A linear model fit to

the data recovers 𝑤 = (0.988 � 0.016)(1 + 𝑧)+(0.020 � 0.024) (with the same 𝜒2

𝜈 to 4 significant figures), consistent with our power model fit above.

We note that this method of fitting is not fundamentally limited to

target SN data with pre-peak brightness observations in each band.

Given enough target data (on the order of several well space points in

time), the mapping of this data to the corresponding reference curve

phases is unique regardless of whether pre-peak data is available.

The uncertainty in each estimated width was found via Monte

Carlo uncertainty propagation, where the target data was resampled

200 times according to its Gaussian error; for each iteration we fit

the width and the final error is the standard deviation in these widths.

To provide some measure of the error in the reference curve, we

imposed an error floor of 𝜎𝑚(1 + 𝑧) on the Monte Carlo uncertainty,

for 𝜎𝑚 representing the median (normalised) flux dispersion in the

reference curve. The 𝜎𝑚 dependence is from our rationale that the

fit width is only as good as the quality of the reference curve, and the

(1 + 𝑧) dependence arises from our scaling of the reference curve.

An example of how a reference curve is created is shown in Fig. 4.

We note the difference in the reference curve timescale before and

after (1 + 𝑧) correction, and the larger dispersion in flux between the

points at any one time prior to correction. Several examples of width

fitting and reference curve construction for low, medium, and high

redshift target SNe (in the context of the DES-SN sample) are shown

in Fig. 6.

We note that while the width fitting for the whole dataset was calcu-

lated in all four DECam bands, only the 𝑖 band data encompasses the

entire redshift range of the DES-SN sample. Due to the spectral shift-

ing inherent in redshifted data, the 𝑔 and 𝑟 filters are unable to detect

SNe at sufficiently high redshift (𝑧 ≳ 0.4 and 𝑧 ≳ 0.85 respectively)

as the observed wavelengths shift to lower emitted wavelengths (see

Fig. 2 of DES Collaboration et al. 2024) and become fainter as a re-

sult. Fig. 6 shows that fitting 𝑧 ≲ 0.2 SNe in the 𝑧-band would require

negative redshifted SNe in the other bands to populate the reference;

hence there is an inherent redshift floor for 𝑧-band fits leaving the

𝑖-band as the only suitable bandpass for the entire redshift range.

The widths obtained in all four bands separately are shown in

Fig. 7. We see the truncated 𝑔, 𝑟 and 𝑧 band data, and fit widths

consistent with the expected (1 + 𝑧) relationship across all bands.

The averaged widths of all the bands are shown in Fig. 8, again

showing excellent agreement with the (1 + 𝑧) expected theory.

As mentioned in the introduction, this method has an element of

circularity because we de-time-dilated the observed light curves to

generate each reference light curve. As a cross-check to ensure that

we are not just getting the answer we put in we repeated the analysis

without de-redshifting the data. This effectively makes the reference

light curves more noisy and wider (like the top plot of Fig. 4). If

time dilation is absent we should get a consistent 𝑏 = 0 fit in this

case. However, if (1 + 𝑧) time dilation is present we should find a

slope inconsistent with 𝑏 = 0 and an intercept offset from 𝑤 = 1.0

(because the reference light curve will itself be time-dilated). We

find, as expected, that the 𝑏 = 0 result is excluded strongly by this

test (see Appendix C and Fig. C1).

5 DISCUSSION

As we see in Fig. 7, there is a clear and significant non-zero time

dilation signature in the DES SN Ia dataset, conclusively ruling out

any static universe models. Our method described in Section 4 detects

a time dilation signature in all of the 𝑔, 𝑟, 𝑖, and 𝑧 DECam bandpasses

MNRAS 000, 1–13 (2024)

Time Dilation with DES SNe Ia

9

1.0

1.5

2.0

2.5

3.0

g

Band

Width

Raw Widths

Binned Widths

1.0

1.5

2.0

2.5

3.0

r

Band

Width

1.0

1.5

2.0

2.5

3.0

i Band Width

1.0

1.5

2.0

2.5

3.0

z

Band

Width

Figure 9. Since most SNe were observed in multiple bands, the fit widths in

each band for each SNe should be intrinsically correlated as they arise from

the same event. Hence, the widths for the same target SN should show some

agreement between bands. We plot their agreement relative to the 𝑖 band

which had the most SNe pass the quality cuts. Each of the data points here

corresponds to a width in Fig. 7 of bands 𝑔, 𝑟, or 𝑧 against the 𝑖 band widths.

A 1:1 dashed line is shown to represent perfect agreement and binned points

are plotted to represent the trends in the agreement.

as expected. The power-law fits to the data in each bandpass are all

consistent with the expected (1 + 𝑧) law to within 2𝜎.

Since there is a well documented stretch-luminosity relationship in

Ia light curves (Phillips 1993; Phillips et al. 1999; Kasen & Woosley

2007), it is possible that Malmquist bias could skew the data to larger

widths at high redshift where we may not see the less-luminous

SNe. Regardless, this does not greatly influence the quality of our

fits since the DES SN data extend to such high redshifts that the

intrinsic dispersion in widths is significantly smaller than the time

dilation signal. In fact, we find that the standard deviation in the

width residuals (i.e. of 𝑤𝑖 − (1 + 𝑧𝑖)1.003 for all SNe) within Fig. 8

is only ∼ 0.15 (which includes observational uncertainty as well as

intrinsic stretch dispersion). At 𝑧 ∼ 1 we would expect a time dilation

factor of ∼ 2 and so the contribution of time dilation far outweighs

the intrinsic light curve stretch in the supernova population.

Nicolas et al. (2021) (see also Howell et al. 2007) showed that

there is a 5𝜎 ‘redshift drift’ in the stretch of their unbiased multi-

survey sample of SNe Ia; that is, higher redshift SNe Ia tend to be

intrinsically wider. Although we do not see this trend in the DES-

SN5YR sample, we nevertheless quantify how such a drift could

affect a time dilation measurement (Appendix A). Given that the

drift is so small, we find its impact would be minimal even if it is

hidden in the data (|Δ𝑏| ≲ 0.02).

As a test of the robustness of our method, we reran all the width-

fitting code with a requirement of pre-peak observations in each light

curve. At most, this changed the power law fit by Δ𝑏 = −0.004 for

the 𝑔 band. The calculated 𝑏 values in the other bands were increased

by one or two thousandths (including the averaged fit of Fig. 8), or

not at all. Interestingly, including this pre-peak restriction reduced

our number of unique SNe widths by only 24 in total. This reduction

does not let us conclusively say if this method is robust at fitting

light curves without pre-peak data, and future analyses may look

at purposefully degrading the dataset (e.g. by manually removing

pre-peak data) to investigate this.

Our method creates a unique reference light curve for each SN as

a function of bandpass and redshift, and so we are able to infer a time

dilation signature no matter the photometric band. This is in contrast

to Goldhaber et al. (2001) who showed time dilation in the 𝐵-band

and suggested it would hold in the other bands (see also Wang et al.

2003). Fig. 9 compares the calculated widths in each band relative

to the 𝑖-band sample (which has the most SNe of any DECam band).

The apparent discrepancies toward high widths in the 𝑖 band might

be explained by dust effects or noise domination in the observation of

high redshift supernovae (M�ller et al. 2022, 2024). With that said,

we see generally broad agreement between the widths in the different

bands as expected of our source wavelength-flexible model.

To avoid de-redshifting reference light curves we devised another

method, similar in concept to method two above, that would yield a

posterior distribution of 𝑏. This entailed generating a reference curve

without first scaling it in time (as in the second method), and fitting

target data with a free 𝑏 (as opposed to 𝑤) to the reference data of

each constituent band. That is,

Δ𝑡target = Δ𝑡reference

( 1 + 𝑧target

1 + 𝑧reference

)𝑏

(7)

and we attempted this with the same 𝜒2 minimisation procedure as

in equation (6). With this method we tried 1) separating the reference

data into the constituent bands, fitting the 𝜒2 to each band reference

and minimising the weighted sum of these 𝜒2. The sum was weighted

according to how many points made up each constituent band ref-

erence curve, and this method is preferred as it assesses the fit to

all bands fairly. We also tried 2) fitting 𝑏 using the 𝜒2 on the entire

scaled reference (i.e. all bands together), but this is not preferred as

the reference is not necessarily composed of points equally sampled

from all bands and so can prefer fitting 𝑏 towards a single bands ref-

erence (i.e. a particular redshift). This method was abandoned overall

as we did not have enough data (even with the DES dataset) to ac-

curately fit 𝜒2 values to each band. We expect that this procedure

would be viable in the future with an even larger SNe Ia dataset of

a comparable redshift distribution, or by using a Bayesian approach

MNRAS 000, 1–13 (2024)

10

White et al.

(which will be the topic of future work). We instead performed the

analysis with flux-scatter-minimisation and width-fitting methods as

the former does not rely on fitting target SN data and the latter fits to

a unified (in phase) reference curve composed of all available bands.

Finally, in the spirit of the results of Goldhaber et al. (2001),

we similarly state that our method with the DES dataset would dis-

favour the null hypothesis of no cosmological time dilation to a

1.003/0.005 ≃ 200𝜎 significance. (If 𝜎 values were still meaningful

at that extreme!) Our uncertainty estimate is a statistical uncertainty

only; in Appendix A we look at a possible systematic effect due to

evolution of the stretch of supernova light curves as a function of red-

shift. We find it to be small, with a likely upper bound of 𝜎

sys

𝑏

≃ 0.01.

Even with an upper limit to the uncertainty of 𝜎

sys

𝑏 +𝜎stat

𝑏

≃ 0.015this

remains the most precise constraint on cosmological time dilation.

6 CONCLUSIONS

Using two distinct methods, we have conclusively identified (1 + 𝑧)

cosmological time dilation using the multi-band photometry of 1504

SNe Ia from the Dark Energy Survey that met our quality cuts. We

make this detection with the most model-independent methods yet in

the literature and with the largest survey of high redshift supernovae.

For both methods, we create a ‘reference curve’ unique to each

supernova (and each bandpass) which describes the expected light

curve shape without accounting for the stretch variation associated

with SN Ia subtypes. Doing this relied on the scale of the available

DES data (the number of SNe, the frequency of imaging, and the

redshift range) and would not be possible with a significantly smaller

survey. Creating this reference curve only relies on the assumption

that SNe Ia should be standard candles/clocks.

Using this reference curve we show an inherent preference of

∼ (1 + 𝑧)1 time dilation in the data, first by minimising the flux

scatter in the data via a redshift-dependent temporal scaling, and

then with a more traditional light curve width estimation. The latter

allows for numerical estimates with uncertainty with which we obtain

a factor of (1+𝑧)1.003�0.005 time dilation signature – the most precise

constraint on cosmological time dilation yet.

We discuss factors and choices that affect our fits and notably see

no indication that Malmquist bias or light-curve stretch significantly

impacts our results. Our results infer a cosmological time dilation

signature aligning strongly with the expected theory, corroborating

past findings (Leibundgut et al. 1996; Goldhaber et al. 2001; Blondin

et al. 2008; Lewis & Brewer 2023) with more SNe and at a higher

redshift than ever before.

ACKNOWLEDGEMENTS

RMTW, TMD, RCa, SH, acknowledge the support of an Australian

Research Council Australian Laureate Fellowship (FL180100168)

funded by the Australian Government. AM is supported by the ARC

Discovery Early Career Researcher Award (DECRA) project number

DE230100055.

Funding for the DES Projects has been provided by the U.S. De-

partment of Energy, the U.S. National Science Foundation, the Min-

istry of Science and Education of Spain, the Science and Technology

Facilities Council of the United Kingdom, the Higher Education

Funding Council for England, the National Center for Supercomput-

ing Applications at the University of Illinois at Urbana-Champaign,

the Kavli Institute of Cosmological Physics at the University of

Chicago, the Center for Cosmology and Astro-Particle Physics at

the Ohio State University, the Mitchell Institute for Fundamental

Physics and Astronomy at Texas A&M University, Financiadora de

Estudos e Projetos, Funda��o Carlos Chagas Filho de Amparo �

Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desen-

volvimento Cient�fico e Tecnol�gico and the Minist�rio da Ci�ncia,

Tecnologia e Inova��o, the Deutsche Forschungsgemeinschaft and

the Collaborating Institutions in the Dark Energy Survey.

The Collaborating Institutions are Argonne National Laboratory,

the University of California at Santa Cruz, the University of Cam-

bridge, Centro de Investigaciones Energ�ticas, Medioambientales y

Tecnol�gicas-Madrid, the University of Chicago, University Col-

lege London, the DES-Brazil Consortium, the University of Edin-

burgh, the Eidgen�ssische Technische Hochschule (ETH) Z�rich,

Fermi National Accelerator Laboratory, the University of Illinois at

Urbana-Champaign, the Institut de Ci�ncies de l’Espai (IEEC/CSIC),

the Institut de F�sica d’Altes Energies, Lawrence Berkeley National

Laboratory, the Ludwig-Maximilians Universit�t M�nchen and the

associated Excellence Cluster Universe, the University of Michigan,

NSF’s NOIRLab, the University of Nottingham, The Ohio State Uni-

versity, the University of Pennsylvania, the University of Portsmouth,

SLAC National Accelerator Laboratory, Stanford University, the Uni-

versity of Sussex, Texas A&M University, and the OzDES Member-

ship Consortium.

Based in part on observations at Cerro Tololo Inter-American

Observatory at NSF’s NOIRLab (NOIRLab Prop. ID 2012B-0001;

PI: J. Frieman), which is managed by the Association of Universities

for Research in Astronomy (AURA) under a cooperative agreement

with the National Science Foundation.

The DES data management system is supported by the Na-

tional Science Foundation under Grant Numbers AST-1138766

and AST-1536171. The DES participants from Spanish institutions

are partially supported by MICINN under grants ESP2017-89838,

PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-2016-

0597, and MDM-2015-0509, some of which include ERDF funds

from the European Union. IFAE is partially funded by the CERCA

program of the Generalitat de Catalunya. Research leading to these re-

sults has received funding from the European Research Council under

the European Union’s Seventh Framework Program (FP7/2007-2013)

including ERC grant agreements 240672, 291329, and 306478. We

acknowledge support from the Brazilian Instituto Nacional de Ci�n-

cia e Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-

2).

This manuscript has been authored by Fermi Research Alliance,

LLC under Contract No. DE-AC02-07CH11359 with the U.S. De-

partment of Energy, Office of Science, Office of High Energy Physics.

DATA AVAILABILITY

The data are available on Zenodo and GitHub as described in the

DES supernova cosmology paper (DES Collaboration et al. 2024)

and DES-SN5YR data release paper (Sanchez et al. 2024). The gen-

erated width fits and associated uncertainties for all 1504 SNe are

included in the analysis GitHub (see Code Availability section), as

are supplementary plots not included in the paper.

CODE AVAILABILITY

Our code in all analysis and plotting relied on the open source Python

packages NumPy (Harris et al. 2020), Matplotlib (Hunter 2007),

Pandas (pandas development team 2023), and SciPy (Virtanen et al.

MNRAS 000, 1–13 (2024)

Time Dilation with DES SNe Ia

11

2020) – specifically the Nelder-Mead algorithm described in Nelder

& Mead (1965).

The code used to generate the width fits/reference curves and all

associated figures is available at github.com/ryanwhite1/DES-Time-

Dilation.

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8364959

APPENDIX A: STRETCH DRIFT WITH REDSHIFT

There is evidence that the stretch distribution of SNe evolves with

redshift, as the fraction of older and younger progenitors evolves.

Nicolas et al. (2021) give the following relation for the evolution of

the SN stretch distribution,

𝑃(𝑥1) = 𝛿(𝑧)N(𝜇1, 𝜎2

1

)+(1−𝛿(𝑧))

(

𝑎N(𝜇1, 𝜎2

1

)+(1 − 𝑎)N(𝜇2, 𝜎2

2

)

)

,

(A1)

where N(𝜇, 𝜎2) is a normal distribution with mean 𝜇 and variance

𝜎2 and the values of the parameters were (𝑎, 𝜇1, 𝜇2, 𝜎1, 𝜎2, 𝐾) =

(0.51, 0.37, −1.22, 0.61, 0.56, 0.87); and the fraction of young su-

pernovae in the population is given by,

𝛿(𝑧) =

(

𝐾−1(1 + 𝑧)−2.8 + 1

)−1

.

(A2)

The distribution given by equation (A1) is shown in the upper panel

of Fig. A1 for several redshifts, where the vertical dashed lines show

the resulting change in the mean 𝑥1. The relationship between 𝑥1 and

the stretch of the supernova is given by (Guy et al. 2007),

𝑠 = 0.98 + 0.091𝑥1 + 0.003𝑥2

1

− 0.00075𝑥3

1

(A3)

and this is shown in the lower panel of Fig. A1.

Since light curve width is directly proportional to stretch, that

means that light curves at redshift 𝑧 = 1 should be approximately

3% wider than those at 𝑧 = 0. This is therefore substantially sub-

dominant to the factor of 2 widening expected from time-dilation

over the same range.

Compounding this effect, supernovae with wider light curves tend

to be brighter, so selection effects might also cause you to find a shift

to wider light curves at higher redshifts.

Nevertheless, in the DES-SN5YR data there is no indication that

the mean stretch parameter 𝑥1 changes with redshift, see Fig. A2.

Despite the consistency of 𝑥1 in the DES sample we want to

quantify how large the potential drift in the light curve widths could

be if equation (A1) holds. Thankfully this over-estimation can be

readily quantified. When you make a mock data set with this intrinsic

widening included you find you would actually get a line of Δ𝑡 ≈

1.03(1 + 𝑧) − 0.05 (see Fig. A3). In other words, it would change the

slope by ∼3%. This is in contrast to the recovered linear model fit

in the Fig. 8 caption, hence indicating that this redshift-dependent

stretch is not evident in the DES-SN5YR data.

The impact of high-redshift supernovae tending to have a few per-

cent wider stretch than their low-redshift counterparts would cause us

to slightly overestimate 𝑏. The magnitude of the impact on 𝑏 depends

on your redshift distribution, we estimate a shift of |Δ𝑏| ≲ 0.01 for

the DES data, and we consider this a likely upper limit to the system-

atic uncertainty on our result. Since our aim in this paper is to fit the

light curves with the minimal modelling assumptions (and since we

do not see an 𝑥1 trend in our light curve fits) we have chosen not to

correct for this trend. Instead we note that any potential effect would

only be a small deviation around the slope of 𝑤/(1 + 𝑧) ∼ 1 that we

see.

MNRAS 000, 1–13 (2024)

12

White et al.

-3

-2

-1

0

1

2

3

x1

0.0

0.2

0.4

0.6

0.8

1.0

x1

distribution

main

secondary

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Redshift,

z

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Redshift, z

0.98

0.99

1.00

Mean

Stretch

Figure A1. Upper panel: Distribution of 𝑥1 values predicted by Nicolas et al.

(2021). The grey and black dashed Gaussians show the two components of

the supernova population. The coloured lines show the total distribution for

several different redshifts. The vertical dashed lines show the mean of the

redshift distribution (in the same colours as the legend). One can see that the

mean drifts from low to high 𝑥1 as redshift increases. Lower panel: The black

line shows the evolution of the mean stretch (𝑠) of the supernova population

with redshift, where colours match the redshifts in the upper legend. The

intrinsic light curve width is proportional to 𝑠, and therefore light curves are

expected to be about 3% wider at 𝑧 = 1 than at 𝑧 = 0. This is much less than

the factor of two widening due to time dilation.

0.2

0.4

0.6

0.8

1.0

Redshift, z

0.8

0.9

1.0

1.1

1.2

Stretch

Best fit: s = 0.0030z + 0.96

−3

−2

−1

0

1

2

-0.2

-0.1

0.0

0.1

0.2

SN

Colour

Figure A2. The distribution of stretch in the DES-SN5YR data as a function

of redshift (calculated from the SALT3 fitted 𝑥1 values using equation A2),

with 𝑥1 shown on the right axis. Fitting a straight line to this distribution

shows no significant trend in the stretch with redshift.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1 + z

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Ligh

t

Curv

e

Width,

w

Time dilation only: ∆t = (1 + z)

with stretch drift: ∆t = 1.03(1 + z) - 0.05

no time dilation: ∆t = 0

Figure A3. The effect of adding the predicted stretch evolution of SNe Ia vs

redshift is to cause the width vs redshift plot to be slightly steeper. If this

result is present we therefore expect to slightly overestimate 𝑏, as we will

attribute that widening to time dilation.

APPENDIX B: REFERENCE CURVE SELECTION

DERIVATION

We begin with the definition of redshift,

1 + 𝑧 =

𝜆𝑜

𝜆𝑒

(B1)

where 𝑧 is the source redshift, 𝜆𝑜 is the observed wavelength of

light and 𝜆𝑒 is the original emission wavelength of light. If we are

sampling from a band with central wavelength 𝜆𝑟 during reference

curve construction, we need to find a central redshift 𝑧𝑟 for our

reference SNe selection, given that we are fitting the target light

curve in a band of central wavelength 𝜆 𝑓 . The idea is to match

the original emitted wavelengths and so we can divide by another

instance of equation (B1),

1 + 𝑧

1 + 𝑧𝑟

=

𝜆 𝑓 /𝜆𝑒

𝜆𝑟 /𝜆𝑒

=

𝜆 𝑓

𝜆𝑟

(B2)

Then, we can rearrange to find an expression for our target redshift

𝑧𝑟 ,

𝜆𝑟 (1 + 𝑧) = 𝜆 𝑓 (1 + 𝑧𝑟 )

(B3)

𝑧𝑟 =

𝜆𝑟 (1 + 𝑧)

𝜆 𝑓

− 1

(B4)

We can then append a term �Δ𝑧 on equation( B4) to give us a range of

applicable redshift values as in Section 4.1. Finally, it is useful in the

broader context of the paper (and Fig. 3) to show this redshift range

in terms of some fraction of the band FWHM of the band that the

target SN was observed in, 𝛿Δ𝜆 𝑓 . To do this we set Δ𝑧 = 𝛿Δ𝜆 𝑓 /𝜆 𝑓

and shift the term into the fraction within equation (B4),

𝑧𝑟 =

𝜆𝑟 (1 + 𝑧) � 𝛿Δ𝜆 𝑓

𝜆 𝑓

− 1

(B5)

which yields the redshift sampling range of equation (3) that we use

in the analysis.

MNRAS 000, 1–13 (2024)

Time Dilation with DES SNe Ia

13

APPENDIX C: NULL TEST — NO DE-REDSHIFTING OF

REFERENCE LIGHT CURVES

To confirm that our method is able to rule out no time dilation we

repeated the analysis without de-redshifting the reference curves.

That means that the reference curves would look like the top panel of

Fig. 4. If the data were not time dilated, then we should fit consistently

𝑏 = 0 in this case. Fig. C1 clearly shows that this null test fails. Be-

cause time-dilation is present in the data, it means that the reference

curves are now wider than they should be — making the measured

light curve widths narrower. Despite this, the trend for higher-redshift

light curves to be wider persists, in very strong contradiction to the

no-time-dilation hypothesis.

AFFILIATIONS

1 School of Mathematics and Physics, The University of Queensland,

QLD 4072, Australia

2 Sydney Institute for Astronomy, School of Physics, A28, The Uni-

versity of Sydney, NSW 2006, Australia

3 Centre for Gravitational Astrophysics, College of Science, The

Australian National University, ACT 2601, Australia

4 The Research School of Astronomy and Astrophysics, Australian

National University, ACT 2601, Australia

5 Department of Physics & Astronomy, University College London,

Gower Street, London, WC1E 6BT, UK

6 Cerro Tololo Inter-American Observatory, NSF’s National Optical-

Infrared Astronomy Research Laboratory, Casilla 603, La Serena,

Chile

7 Laborat�rio Interinstitucional de e-Astronomia - LIneA, Rua Gal.

Jos� Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil

8 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL

60510, USA

9 Department of Physics, University of Michigan, Ann Arbor, MI

48109, USA

10 Departamento de F�sica Te�rica and Instituto de F�sica de Part�cu-

las y del Cosmos (IPARCOS-UCM), Universidad Complutense de

Madrid, 28040 Madrid, Spain

11 Institute of Cosmology and Gravitation, University of Portsmouth,

Portsmouth, PO1 3FX, UK

12 University Observatory, Faculty of Physics, Ludwig-Maximilians-

Universit�t, Scheinerstr. 1, 81679 Munich, Germany

13 Center for Astrophysics | Harvard & Smithsonian, 60 Garden

Street, Cambridge, MA 02138, USA

14 Department of Astronomy and Astrophysics, University of

Chicago, Chicago, IL 60637, USA

15 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box

2450, Stanford University, Stanford, CA 94305, USA

16 SLAC National Accelerator Laboratory, Menlo Park, CA 94025,

USA

17 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife,

Spain

18 INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11,

I-34143 Trieste, Italy

19 Institut de F�sica d’Altes Energies (IFAE), The Barcelona Insti-

tute of Science and Technology, Campus UAB, 08193 Bellaterra

(Barcelona) Spain

20 Hamburger Sternwarte, Universit�t Hamburg, Gojenbergsweg

112, 21029 Hamburg, Germany

21 Centro de Investigaciones Energ�ticas, Medioambientales y Tec-

nol�gicas (CIEMAT), Madrid, Spain

22 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285,

India

23 Jet Propulsion Laboratory, California Institute of Technology,

4800 Oak Grove Dr., Pasadena, CA 91109, USA

24 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box

1029 Blindern, NO-0315 Oslo, Norway

25 Kavli Institute for Cosmological Physics, University of Chicago,

Chicago, IL 60637, USA

26 Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de

Madrid, 28049 Madrid, Spain

27 Institut d’Estudis Espacials de Catalunya (IEEC), 08034

Barcelona, Spain

28 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de

Can Magrans, s/n, 08193 Barcelona, Spain

29 Centre for Astrophysics & Supercomputing, Swinburne Univer-

sity of Technology, Victoria 3122, Australia

30 Center for Astrophysical Surveys, National Center for Supercom-

puting Applications, 1205 West Clark St., Urbana, IL 61801, USA

31 Department of Astronomy, University of Illinois at Urbana-

Champaign, 1002 W. Green Street, Urbana, IL 61801, USA

32 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064,

USA

33 Center for Cosmology and Astro-Particle Physics, The Ohio State

University, Columbus, OH 43210, USA

34 Department of Physics, The Ohio State University, Columbus, OH

43210, USA

35 Australian Astronomical Optics, Macquarie University, North

Ryde, NSW 2113, Australia

36 Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001,

USA

37 Department of Physics and Astronomy, University of Pennsylva-

nia, Philadelphia, PA 19104, USA

38 Departamento de F�sica Matem�tica, Instituto de F�sica, Univer-

sidade de S�o Paulo, CP 66318, S�o Paulo, SP, 05314-970, Brazil

39 George P. and Cynthia Woods Mitchell Institute for Fundamental

Physics and Astronomy, and Department of Physics and Astronomy,

Texas A&M University, College Station, TX 77843, USA

40 LPSC Grenoble - 53, Avenue des Martyrs 38026 Grenoble, France

41 Instituci� Catalana de Recerca i Estudis Avan�ats, E-08010

Barcelona, Spain

42 Department of Astrophysical Sciences, Princeton University, Pey-

ton Hall, Princeton, NJ 08544, USA

43 Observat�rio Nacional, Rua Gal. Jos� Cristino 77, Rio de Janeiro,

RJ - 20921-400, Brazil

44 Department of Physics, Carnegie Mellon University, Pittsburgh,

Pennsylvania 15312, USA

45 Department of Physics and Astronomy, Pevensey Building, Uni-

versity of Sussex, Brighton, BN1 9QH, UK

46 School of Physics and Astronomy, University of Southampton,

Southampton, SO17 1BJ, UK

47 Computer Science and Mathematics Division, Oak Ridge National

Laboratory, Oak Ridge, TN 37831

48 Department of Physics, Duke University Durham, NC 27708, USA

49 Universit� Grenoble Alpes, CNRS, LPSC-IN2P3, 38000 Greno-

ble, France

50 Department of Astronomy, University of California, Berkeley, 501

Campbell Hall, Berkeley, CA 94720, USA

51 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berke-

ley, CA 94720, USA

This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 000, 1–13 (2024)

14

White et al.

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Ligh

tcurv

e

width

w

g

1 + z

(1 + z)−0.346�0.039

No Time Dil

Data

Binned Data

г

(1 + z)−0.016�0.013

1.0

1.2

1.4

1.6

1.8

2.0

1 + z

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Ligh

tcurv

e

width

w

i

(1 + z)0.087�0.012

1.0

1.2

1.4

1.6

1.8

2.0

1 + z

ζ

(1 + z)0.221�0.011

Figure C1. Light curve widths measured with respect to a reference curve that has not been de-time-dilated. We nevertheless still see a persistent trend of

increasing light curve width with redshift. The vertical offset from the (1 + 𝑧) line arises because the non-de-time-dilated reference curves are wider than

rest-frame light curves, i.e. this offset is yet another indication of time dilation. The black horizontal dashed line indicates no time dilation and the blue dashed

lines are (poor) (1 + 𝑧)𝑏 model fits to the data. If there was no time dilation, these fits would be horizontal lines with 𝑏 = 0.

MNRAS 000, 1–13 (2024)