The observation that type Ia supernovae (SNe Ia) are fainter than expected given their redshifts has led to the conclusion that the expansion of the universe is accelerating. The widely accepted hypothesis is that this acceleration is caused by a cosmological constant or some dark energy field that pervades the universe. We explore what the supernovae data tell us about this hypothesis by answering the question: Can these data be explained with a model in which the strength of gravity varies on a cosmic timescale? We conclude that they can and find that the supernovae data alone are insufficient to distinguish between a model with a cosmological constant and one in which G varies. However, the varying-G models are not viable when other data are taken into account. The topic is an ideal one for undergraduate physics majors.

1.
See, for example,
S.
Weinberg
,
Cosmology
(
Oxford U. P.
,
New York
,
2008
), pp.
1
100
.
2.
J. B.
Hartle
, “
General relativity in the undergraduate physics curriculum
,”
Am. J. Phys.
74
,
14
21
(
2006
).
3.
See, for example,
S.
Colafrancesco
, “
Dark matter in modern cosmology
,”
AIP Conf. Proc.
1206
,
5
24
(
2010
).
4.
See, for example,
P. J. E.
Peebles
and
B.
Ratra
, “
The cosmological constant and dark energy
,”
Rev. Mod. Phys.
75
,
559
606
(
2003
).
5.
E. V.
Linder
, “
Resource Letter DEAU-1: Dark energy and the accelerating universe
,”
Am. J. Phys.
76
,
197
204
(
2008
).
6.
A. G.
Riess
 et al (
The High-Z Team
), “
Observational evidence from supernovae for an accelerating universe and a cosmological constant
,”
Astron. J.
116
,
1009
1038
(
1998
).
7.
S.
Perlmutter
 et al (
The Supernova Cosmology Project
), “
Measurements of omega and lambda from 42 high-redshift supernovae
,”
Astrophys. J.
517
,
565
586
(
1999
).
8.
The units of ρ are mass per unit volume [ML3], where M is the mass unit and L the unit of length. However, it is common to choose units so that c=1, thereby eliminating the distinction between mass and energy. For clarity, we keep the symbol c in all expressions. Note that [Λ]=[K]=L2.
9.
J. D.
Barrow
, “
Varying constants
,”
Philos. Trans. R. Soc. London, Ser. A
363
,
2139
2153
(
2005
).
10.
C. J. A. P.
Martins
, “
Cosmology with varying constants
,”
Philos. Trans. R. Soc. London, Ser. A
360
,
2681
2695
(
2002
).
11.
M.
Kowalski
 et al, “
Improved cosmological constraints from new, old, and combined supernova data sets
,”
Astrophys. J.
686
,
749
778
(
2008
).
12.
R. R.
Caldwell
,
M.
Kamionkowski
, and
N. N.
Weinberg
, “
Phantom energy and cosmic doomsday
,”
Phys. Rev. Lett.
91
,
071301
-1–4 (
2003
).
13.
J. C.
Wheeler
and
R. P.
Harkness
, “
Type I supernovae
,”
Rep. Prog. Phys.
53
,
1467
1557
(
1990
).
14.
S.
Chandrasekhar
,
An Introduction to the Study of Stellar Structure
(
Dover
,
New York
,
1967
).
15.
P.
Höflich
, “
Physics of type Ia supernovae
,”
Nucl. Phys. A
777
,
579
600
(
2006
).
16.
S.
Perlmutter
, “
Supernovae, dark energy, and the accelerating universe
,”
Phys. Today
56
(
4
),
53
60
(
2003
).
17.
We have described the bolometric magnitude, that is, the magnitude of a star assuming we can measure the flux across all wavelengths. In practice, fluxes are measured in wavelength bands defined by standard filters, such as the B-band filters (Ref. 18). An observed supernova spectrum is redshifted with respect to the spectrum in the rest frame of the supernova. Therefore, a filter will transmit a flux that differs from the one that would be measured in the supernova’s rest frame. Astronomers use K corrections to map the measured flux back to its value in the object’s rest frame. Given a model of the object’s spectrum, it is possible, in principle, to infer the bolometric flux and hence the bolometric magnitude of the object. The distance modulus data compiled in Ref. 11 are derived from B-band magnitude data.
18.
H.
Johnson
and
W.
Morgan
, “
Fundamental stellar photometry for standards of spectral type on the revised system of the Yerkes spectral atlas
,”
Astrophys. J.
117
,
313
352
(
1953
).
19.
E.
Gaztañaga
,
E.
García-Berro
,
J.
Isern
,
E.
Bravo
, and
I.
Domínguez
, “
Bounds on the possible evolution of the gravitational constant from cosmological type-Ia supernovae
,”
Phys. Rev. D
65
,
023506
-1–9 (
2001
);
E.
García-Berro
,
Y.
Kubyshin
,
P.
Loren-Aguilar
, and
J.
Isern
, “
The variation of the gravitational constant inferred from the Hubble diagram of type Ia supernovae
,”
Int. J. Mod. Phys. D
15
,
1163
1174
(
2006
).
20.
E.
Komatsu
 et al, “
Five-year Wilkinson microwave anisotropy probe observations: Cosmological interpretation
,”
Astrophys. J., Suppl. Ser.
180
,
330
376
(
2009
).
21.
The absolute luminosity of a supernova cannot be determined independently of the Hubble constant H0. Consequently, in the fit of the modulus function to the data, it is only the shape of the function that contains useful information about the cosmology. The offset Q depends both on H0 as well as on the flux corrections.
22.
If the interval [a,b] is divided into N intervals of width h=(ba)/N, the midpoint rule is abf(x)dxhi=1Nf(a+(i0.5)h).
23.
R.
Brun
 et al, “
ROOT data analysis package
,” ⟨root.cern.ch⟩.
24.
The lifetimes can be computed given values for the parameters b and H0. However, because we cannot extract a value of H0 from the fits, we compute the lifetimes using the nominal value 70kms1Mpc1 for the Hubble constant. We write all lifetimes in terms of the parameter H0 to make clear how the numerical values will change if H0 differs from the nominal value.
25.
If the reported modulus uncertainties are Gaussian distributed, we expect χ2 to be sampled from a probability density with mean NPND, where N=307 is the number of data points and P2 is the number of adjustable parameters. ND is the number of degrees of freedom. Therefore, for a fit that neither overfits nor underfits, we expect χ2/ND1. The quantity P would be exactly equal to 2 if the constraints that define the parameter estimates were linear in the parameters.
26.
See, for example,
J. D.
Barrow
and
P.
Parsons
, “
The behavior of cosmological models with varying-G
,”
Phys. Rev. D
55
,
1906
1936
(
1997
);
E.
García-Berro
,
J.
Isern
, and
Y. A.
Kubyshin
, “
Astronomical measurements and constraints on the variability of fundamental constants
,”
Astron. Astrophys. Rev.
14
,
113
170
(
2007
).
27.
E. V.
Linder
(private communication,
2010
).
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