ЯДЕРНАЯ ФИЗИКА, 2012, том 75, № 9, с. 1155-1175
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
SUPERNOVA BANGS AS A TOOL TO STUDY BIG BANG
© 2012 S. I. Blinnikov*
Institute for Theoretical and Experimental Physics, Moscow, Russia Received November 15, 2011
Supernovae and gamma-ray bursts are the most powerful explosions in observed Universe. This educational review tells about supernovae and their applications in cosmology. It is explained how to understand the production of light in the most luminous events with minimum required energy of explosion. These most luminous phenomena can serve as primary cosmological distance indicators. Comparing the observed distance dependence on redshift with theoretical models one can extract information on evolution of the Universe from Big Bang until our epoch.
1. INTRODUCTION: EXPLODING STARS
A Nova is a sudden increase of stellar luminosity reaching L ~ 105Lo, where Lo = 3.839 x 1033 erg/s is the solar luminosity, i.e., the power of photon emission. "Stella Nova" means a "new star" in Latin. Nova bursts are hydrogen explosions on the surface of an old star, a white dwarf, in a binary system. Only a small fraction of the star mass is ejected <10"4Mo at the Nova outburst.
A Supernova (SN) is an enormous explosion that marks the final stage of evolution of a star. The burst of photons reaches L ~ 1010Lo (and higher) and the star may be virtually destroyed. Supernovae (SNe), the plural of Supernova, are extremely important for understanding our Galaxy. They heat up the interstellar medium, produce and distribute heavy elements throughout the Galaxy, and accelerate cosmic rays.
SNe are among the most luminous phenomena in the Universe: a standard event produces around 1049 erg in photons during the first months after explosion. Recalling that solar luminosity is Lo & 4 x x 1033 erg/s we see that it would take ~108 yr for our Sun to emit this amount of light. Even larger energy is stored in the kinetic energy of ejecta, around 1051 erg for a typical SN. Sometime, the energy emitted in photons is much higher than the average 1049 erg. Discovery of extremely luminous events like SN 2006gy has demonstrated that some of SNe produce a factor of 10 to 100 more visible photons than other, already powerful explosions.
In this work I explain how to understand the production of light in the most luminous events with minimum required energy of explosion.
SNe can serve as good cosmological distance indicators. In some cases one can use a "standard
E-mail: Sergei.Blinnikov@itep.ru
candle" method. For variable events one takes as the standard candle the peak luminosity (this epoch is called "maximum light"). A special class, so-called Type Ia supernovae (SNe Ia) are currently the most favored for this method. I will give a definition of different SN types (Ia, Ib/c, IIP, IIn). Although SNe Ia are not the same at maximum light, i.e., they are not standard candles directly, they can be "standardized". The standardization is based on statistical correlations found for nearby events [1—3]. Thus, they are secondary distance indicators, because one needs to find distance to their host galaxies from other indicators, see reviews [4, 5].
In the last part of the paper I explain old and novel approaches to measuring distances for cosmology using these most luminous phenomena in the Universe. They can serve as primary cosmological distance indicators. Comparing the observed distance dependence on redshift d(z) with theoretical models one can extract information on evolution of Universe from Big Bang until our epoch.
2. COSMOLOGY IN THE SIMPLEST MODELS
2.1. Uniformity and Isotropy
In the simplest models, the Universe is approximated as having uniform density of matter everywhere, where the density would be the average taken over some large volume, say (10 Mpc)3 or even (100 Mpc)3. This assumption of uniformity — throughout all space, and isotropy in every direction — is the cosmological principle. This is approximate, but reasonable, as supported by cosmic microwave background (CMB) observations.
The first general relativistic (GR) cosmologi-cal models for the Universe with matter, homogeneous and isotropic in space, but not static in time,
were constructed by Alexander Friedmann in 19221924 [6,7].
It is convenient to take the metric in FriedmannRobertson-Walker (FRW) [8, 9] form:
ds2 — dt2 -
(1)
- a2 (t)
dr2
1 — kr2
+ r2(d02 + sin2 0dp2)
S - Sg + Sm -
c3 f „ ,—1 167TG I C
J R^/^gd4x + - J Cm^f—gdAx.
The factor 1/c appears if we write dx° = cdt.
This action joins properties of matter with spacetime geometry (paraphrasing Wheeler [11]: "Matter bends space, and space gives matter its marching orders").
"We will use latin i, j, k for 4-components and greek a, 3, y for 3-components.
The variations of Sm with respect to 5gik produce the energy—momentum tensor Tik:
1 5Sm
Tik = 2c
V^g Sg-
ik
This expression, i.e. the functional derivative one should understand as 1
öS„
with a(t) — scale factor, and k = ±1 for curved Friedmann worlds, and k = 0 for the flat 3-space (Einstein—de Sitter model).
Formula (1) is a simple transformation of original formulae from [6, 7] (this is the form used, e.g., by Landau and Lifshitz [10]). Sometimes, it is written in literature on cosmology that Robertson and Walker did not know about Friedmann. This is not true: Walker [9] cites Robertson [8], while the latter extensively uses Friedmann's works [6, 7] with proper references.
There is no center at r = 0: in all cases the point r = 0 can be chosen quite arbitrary, in any place of uniform space, like a pole on a uniform 2D sphere. We consider clusters of galaxies as points with coordinates r, d, p fixed, i.e. they are comoving coordinates, and the expansion of the Universe is determined by the scale factor a(t). Friedmann has shown that Einstein's equations can be satisfied by his metric. To check this one has only to find the scale factor a(t) as function of t.
All standard textbooks on cosmology, e.g. Landau and Lifshitz [10], have derivation of Friedmann solution. Normally, those solutions are derived from Einstein equations, which connect Ricci tensor and scalar with the energy—momentum tensor Tik !).Here I give a simple pedagogical derivation directly from Hilbert action.
The Hilbert action S is a sum of a gravity term containg R and a term with the Lagrangian density of all kinds of matter Lm (although Hilbert himself has written Lm for a particular case only):
(2)
= bj T^V^gd'x
(see any book on GR), but we do not need this again!
I will show that the main equation for cosmology can be derived for simple cases directly from S, not using Tik.
2.2. Friedmann Equation from Variational Principle
We take the simplest case of a perfect fluid and the Lagrangian density Lm must be related to some form of the energy per unit volume E. Let us pull out the contribution of the restmass density p, and a contribution of pressure:
E = pc2 + pn. We will consider only adiabatic perturbations,
d
This gives that
- P
dp 7
0.
dp dE 7 = P + £'
and dn — Pdp/p2,
p
dP
n = -£ +
pp
0
where the last term is just enthalpy. If we put
Lm = pc2 + pn = E,
then we get from 5Sm/5gik an expression for Tmn for a perfect fluid:
Tmn — (P + E Pgm
(3)
where E and P are the energy density and pressure (respectively) as measured in the rest frame, and um is the four-velocity of the fluid which is
um = (1,0,0,0)
in the comoving coordinates (with r, d, p fixed). Thus, it appears that the Lagrangian density for a perfect fluid Lm is just the energy density E.
For our derivation we need only to know the Ricci scalar curvature R = gikRik (where Rik = Rrmnk is the Ricci tensor) expressed through the metric. From
the metric one can mechanically compute the connection coefficients r%mn and the curvature tensor looking in a textbook or using a symbolic manipulation software. Be careful with signs of Riemann and Ricci tensors — they are very different in various textbooks!
Denoting a = da/dt, we get the Ricci scalar (with the sign convention from [10]):
6
R = —:r(ad + a2 + k), a2
and the action with the Lagrangian
„3
C
c
g
l6nG
Now
a3r2 sin 0 yj\ — kr2
Then from Ricci's R and E, multiplied with a3 from <7, we have a Lagrangian for finding a(t):
c3 1
L[a(t)] = — ■ 6 (a2d + ad2 + ka) + -Ea3. 16nG c
We can get rid of a doing this term by parts in /Ldt:
a2adt = / a2da = a2a
— / dda =
to
In my oversimplified approach I just omit those inconvenient terms outside the integral. This leads to a correct Lagrangian (in units c = 1)
L
3
(aa2 — ka) + E a3.
8nG
We can take any barotropic equation of state to relate E and p. All we need to assume now that E depends only on a (via p) not on a. Then we have a "Hamiltonian"
tt dl ■ t 3
H = —a — L = ——
da 8nG
(aa2 + ka) — Ea3
There is no explicit dependence on time t, and H must
be constant, that is 3
-(aa2 + ka) — Ea3 = const,
8nG
or
• 2 , 8nG „ 3 .
aa + ka =-Ea + const.
3
Now if we take the simplest case of zero pressure matter ("dust" made of galaxies),
E
-a0\3
we see that the constant can be absorbed into the definition of po^o and we get the Friedmann equation:
= const — 2 J a2adt.
I have written down the limits of integration only for a term outside the integral. It should be understood that the same limits apply to integrals.
There are nontrivial questions on the exclusion of second derivatives from Sg (this is needed for obtaining correct equations of motion) and on taking variations at the boundaries. Although these questions have been intensively discussed in special literature, one cannot say that they are finally solved. They are touched upon only in a couple of textbooks [12, 13], see, e.g., Sections 4.1.3—4.1.5 in [13] (cf. also a discussion in Section 1.9 of a recent book [14]). It appears that a suitable form of action contains terms with extrinsic curvature of the boundary. Perhaps the first to suggest this action was York [1
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