Chris Clarkson - Dark Energy and Backreaction

backreaction in the concordance model

Chris Clarkson
Astrophysics, Cosmology & Gravitation Centre
University of Cape Town

Thursday, 26 January 12

key ingredients for cosmology

• universe homogeneous and isotropic on large scales (>100 Mpc)

• background dynamics determined by amount of matter + curvature present
(+ a theory of gravity)

• inﬂation lays down the seeds for structure formation from quantum
ﬂuctuations

• works ... if we include a cosmological constant or ‘dark energy’

• matter + curvature not enough


How does structure affect the background?


3 issues

•Averaging

•Coarse-graining of structure, such that small-scale effects are hidden
to reveal large scale geometry and dynamics.
•Backreaction

•Gravity gravitates, so local gravitational inhomogeneities may affect
the cosmological dynamics.
•Fitting

•How do we fit an idealized model to observations made from one
location in a lumpy universe, given that the ‘background’ does not in
fact exist? (Averaging observables is not same as spatial average)


scale of homogeneity of the dist
1 1 it seems 3natural to define the e
HD ⇤ ↵N ⇧ D = N ⇧Jd x , As pointed out in
matter flow.
3 3VD D
Averaging is difficult one introduces an ambiguity in
VD ⇤ Jd x is the Riemannian volume of D. The avereaging as m
3 expansions: the expansion oper
measured by observers comoving
•Define Riemannian averaging on ⌥. As mentionned before, the co
er the domain D: domain D
1 the matter distribution, so that
↵D = ↵↵ D ⇤ ↵(t, xi )Jd3 x ,the space-time metri
write down
VD D In the following we will then ret
scalar function ↵. One can then define the e⌥ective scale factor for
:
Riemannian volume ✏ element
⇣t aD where J ⇤ det(hij ) and VD ⇤
HD = . the Riemannian average over the
spatial average implies wrt
aD
some foliation of spacetime

that is well defined for any scala
how can average of a tensor preserve Lorentz invariance?
the function aD (t) obeying:


scale of homogeneity of the dist
1 1 it seems 3natural to define the e
HD ⇤ ↵N ⇧ D = N ⇧Jd x , As pointed out in
matter flow.
3 3VD D
Averaging is difficult one introduces an ambiguity in
VD ⇤ Jd x is the Riemannian volume of D. The avereaging as m
3 expansions: the expansion oper
measured by observers comoving
•Define Riemannian averaging on ⌥. As mentionned before, the co
er the domain D: domain D
1 the matter distribution, so that
↵D = ↵↵ D ⇤ ↵(t, xi )Jd3 x ,the space-time metri
write down
VD D In the following we will then ret
scalar function ↵. One can then define the e⌥ective scale factor for
: eg, to specify average
Riemannian volume ✏ element
energy density need ⇣t aD where J ⇤ det(hij ) and VD ⇤
full solution of the HD = . the Riemannian average over the
spatial average implies wrt
aD
field equations some foliation of spacetime

that is well defined for any scala
how can average of a tensor preserve Lorentz invariance?
the function aD (t) obeying:


Backreaction is generic

on small scales

but on large scales


Backreaction is generic

on small scales

but on large scales

backreaction terms
arise from averaging
& non-linear EFE


Buchert

Zalaletdinov


Buchert backreaction
from non-local
variance of
local expansion
rate

Zalaletdinov


Buchert backreaction
from non-local
variance of
local expansion
rate

Zalaletdinov

macroscopic Ricci
correlation tensor


Another view of the averaging problem

on l ity d ay
e nd ati ua to Hubble radius
i nﬂ eq

averaging gives corrections here



on l ity d ay
e nd ati ua to Hubble radius
i nﬂ eq

model = ﬂat FLRW + averaging gives corrections here
perturbations



on l ity d ay
nd ati
howfl do
e weua to
remove backreaction bits
Hubble radius
in eq
to get to ‘real’ background?
smoothed background today is not
same background as at end of inflation

?
model = flat FLRW + averaging gives corrections here
perturbations


Fitting isn’t obvious either

a single line of
sight gives D(z) {
averaged on the
Hubble rate along the
sky to give
line of sight
smooth ‘model’


Aren’t the corrections just ~10-5?



•No. Of course not. Λ is bs [Buchert, Kolb ...]




•Yes. Absolutely. Those guys are idiots. [Wald, Peebles ...]





•Well, maybe. I’ve no idea what’s going on. [everyone else ... ?]





•Well, maybe. I’ve no idea what’s going on. [everyone else ... ?]

•Corrections from averaging enter Friedmann and Raychaudhuri
equations

•is this degenerate with ‘dark energy’?

•can we separate the eﬀects [if there are any]?

•or ... is it dark energy? neat solution to the coincidence
problem


Could it be dark energy?

?
‘average’ expansion can
accelerate while local one
decelerates everywhere

regions with faster expansion dominate the volume over time
so, the average expansion will increase


Non-Newtonian in nature


Non-Newtonian in nature

curvature!


What to compute?

•general formalisms lack quantitative computability

•perturbative methods don’t give coherent picture

•but they do give predictions


Different aspects


Different aspects

•Non-linear perturbations [Newtonian vs non-Newtonian]


Different aspects


•Relativistic corrections

•linear and non-linear


Different aspects




•backreaction from averaging


Different aspects




•backreaction from averaging

•corrections to observables


Canonical Cosmology

•compute everything as power series in small parameter ε
gµ⇥ = gµ⇥ + ⇥
¯ (1)
gµ⇥ + ⇥
2 (2)
gµ⇥ + · · ·

‘real’ spacetime second-order
ﬁrst-order
‘background’ spacetime perturbation correction
FLRW

ﬁt to ‘background’ observables - SNIa etc


Canonical Cosmology

•compute everything as power series in small parameter ε
gµ⇥ = gµ⇥ + ⇥
¯ (1)
gµ⇥ + ⇥
2 (2)
gµ⇥ + · · ·

‘real’ spacetime second-order
first-order
‘background’ spacetime perturbation correction
FLRW
how do these

?
fit in?
fit to ‘background’ observables - SNIa etc


so, ....


so, ....

is it big or is it small ... ?


so, ....

is it big or is it small ... ?

(I don’t know either)

Simulations can’t see it

periodic BC’s and no
horizon give
Olbers paradox -
cancels backreaction


Perturbation theory
metric to second-order

Bardeen equation

no real backreaction from ﬁrst-order perturbations
- average of perturbations vanish by assumption


Perturbation theory
metric to second-order

second-order potentials induced by ﬁrst-order scalars

vectors and tensors give measure of relativistic corrections


Vectors
~1%


induced tensors
bigger than
primordial [today]


backreaction as correction to the background

•second-order modes give non-trivial ‘backreaction’

•Hubble rate depends on

•What is it?


backreaction as correction to the background

•second-order modes give non-trivial ‘backreaction’


•What is it?

determines amplitude
of backreaction


backreaction is concerned with the
homogeneous, average contributions

what does this depend on?


scaling behaviour d lity y
en ua da
to
eq Hubble radius

at ﬁrst-order

scale increasing

amplitude of second-order contributions


amplitude of second-order contributions

large equality scale suppresses
backreaction - but overcomes
factor(s) of Delta


backreaction - expansion rate

•second-order modes give non-trivial backreaction


•UV divergent terms don’t contribute on average [Newtonain]

•well deﬁned and well behaved backreaction

•but, this is only well behaved because of the long radiation era

•what would we do if the equality scale were smaller?


on we wish to probe, and some subtleties arise because we have to
hen we examineto the Hubble we areat second-order
Change averagedHubble as a function of redshift as a fractional in twothe backgroundthe rate.
the rate rate interested things:
ed Friedmann equation. In the Friedmann Friedmann equation, and theinterested
FIG. 1: The
˘
Hubble rate
the Hubble rate H calculated from the ensemble-averaged
D
equationchange are left shows the Hubble rat
we to Hubble

eld equations asBothresult of and EdS models are considered, and the di erentcomponents indicated
directly, H . a concordance averaging, which are three new averaging schemes,
D
if we put R = 0, H still doesn’t have a UV divergence.
of dark energy, it is common to think of these as e⇥ective fluid or
S D

hese two agree up to rate
Normalised Hubble perturbative order, when we take the ensemble
as function of redshift
e given by the generalised Friedmann equation. For a given domain
from averaging Friedmann
veraged Hubble rate we might expect to find. When we present HD
equation
⇧
˘ 2
HD ⌅ HD (98)

Friedmann equation and then taken the square-root. This does not
(55) directly (but note that if we square Eq (55), take the ensemble
equality scale domain
he Hubble rate as calculated directly from the Friedmann equation).
ce’ of the Hubble rate, which may be defined by
Clarkson, Ananda & Larena, 0907.3377
2
[HD ] = HD 2 HD . (99)
⌥⌃ ⌥⌃ ˘
FIG. 2: Plots for H
D with RD = 1/kequality , RS = 0, with the variance included, as a function of redshif

effective ﬂuid


effective fluid

•amplitudes apply to effective fluid approach [Baumann etal]
and Green and Wald [PPN]

•they claim backreaction is small from this


great!


great!

backreaction is small ...


backreaction - acceleration rate

•other quantities are much stranger
•time derivative of the Hubble rate represented in the
deceleration parameter

•UV divergent terms do not cancel out


backreaction - acceleration rate

•other quantities are much stranger
•time derivative of the Hubble rate represented in the
deceleration parameter

•UV divergent terms do not cancel out

dominates backreaction


divergent terms


oh no!


oh no!

backreaction is huge ...

but wait!

use observables - spatial averaging is meaningless!

•we fit our cosmology to an all-sky average of observed
quantities

•distance-redshift, number-count redshift, etc.

•if averaging/backreaction is significant then these should
fit to the wrong model (should use LTB metric!)

•can be computed using Kristian-Sachs approach


expectation value

observed deceleration governed by cut-oﬀ


UV cutoff


UV cutoff

•modes below equality scale dominate eﬀect - not physical


UV cutoff


•where should they be cut oﬀ?


UV cutoff



•inﬂation scale [last mode to leave the Hubble scale]?
backreaction >>>1


UV cutoff



backreaction >>>1

•dark matter free-streaming scale? [~ pc] backreaction >>1


UV cutoff



backreaction >>>1


•‘spherical scale’ [Kolb]


UV cutoff



backreaction >>>1


•‘spherical scale’ [Kolb]

•virial ‘scale’ ~ Mpc’s [Baumann etal] - gives O(1) backreaction


wtf?

amplitude of backreaction determined purely from cut-oﬀ

virial scales only sensible cutoﬀ O(1) backreaction
ignore them - maybe gauge or unphysical ?


fourth-order perturbation theory ... ?


Hubble rate at fourth-order


ok ... so,


ok ... so,

backreaction could be ...
anything ...


key questions

•convergence of perturbation theory function of equality
scale
•why are we so lucky?
•with less radiation, scales up to ‘virial scale’ must
contribute to backreaction - how would we compute this?
•model with no radiation era might be best model to
explore backreaction


What is the background?

Do observations measure the background?

What is ‘precision cosmology’?


Interfering with dark energy

•until we understand
backreaction precision
cosmology not secure
•what are we measuring?
•Zalaletdinov’s gravity
predicts decoupled
geometric and dynamical
curvature
•removes constraints on
dynamical DE
•evidence for acceleration
only provided by SNIa


a single line of
sight gives D(z) {
•We average this over the sky, and (re)construct model
- alternative aspect to ‘backreaction’

•average depends on z, so ‘looks’ spherically symmetric
and inhomogeneous
•would remove copernican constraints on void models!
•no need for LTB solution, kSZ constraints removed...


Conclusions Confusions

•Why are second-order perturbations so large?
•
•tells us that perturbation theory must be relativistic, not
Newtonian
•role of UV divergence must be understood to decide whether
backreaction is small - higher order or resummation methods
needed? must include tensors!
•do we need relativistic N-body replacement?
•what is the background? what is ‘precision cosmology’?
•‘void models’ may be mis-interpretation of backreaction ...

Curvature test for the Copernican Principle

• in FLRW we can combine Hubble rate and distance data to ﬁnd curvature

2
[H(z)D (z)] 1
k =
[H0 D(z)]2
⇥
dL = (1 + z)D = (1 + z) dA
2

• independent of all other cosmological parameters, including dark energy
model, and theory of gravity

• tests the Copernican principle and the basis of FLRW (‘on-lightcone’ test)
⇥
C (z) = 1 + H 2
DD D 2
+ HH DD = 0

Clarkson, Basset & Lu, PRL 100 191303


Using age data to reconstruct H(z)

Shaﬁeloo & Clarkson, PRD


Consistency Test for ΛCDM For flat ΛCDM mod-
d for all curvatures, where H(z) is given by the Friedmann equation,
els the slope of the distance data curve must satisfy
onsistency Test for ΛCDM For flat ΛCDM mod- lies outside the n-σ er
H(z) 2
H 2
m
1/ + Ωk (1 + z)3 + (1 − Ωm ). + Rearranging for
D=(z)of= (1 + z)3Ωm (1 +z)2 + ΩDE exp 3satisfy w(zdeviations from Λ, as
the A litmus test for flat ΛCDM
slope 0 Ωthe distance data curve must
z
1 )
dz ,
z) = 1/ wemhavez)3 + (1 − Ωm ). Rearranging for + z below for the general
Ωm Ω (1 + 0 1

Ωk The 2 If, on the other hand
we .have usual procedure is to postulate a several parameter form for w(z) and calculate
ernative method is to reconstruct w(z) by1 − D reconstructingfit for all suitable para
directly (z) the luminosity-distance
Ωm = w(z). Writing D(z) = (H ./c)(1 + z)−1d (z), we hav
1 − D (z) 2
[9? ] dL (z) Ωm = inverted to yield + z)
(5)
3 − 1]D (5) 20 that is good evidence
may be
[(1 + z)
[(1 .
3 − 1]D (z)2 (z) L

is that only one choic
2(1 + z)(1 + Ωk D )D − (1 + z) Ωk D + 2(1 + z)Ωk DD − 3(1 + Ωk D2 ) D
2 2 2
hin the flat ΛCDMflat ΛCDMwe measure D if we2 measure D to provide e
Within the paradigm, if paradigm, (z) at L (z) = 0 (z) at
2 [Ω + (1 + z)Ω ] D 2 − (1 + Ω D )} D
.
3 {(1 + z)
e z and calculate the rhs kof this equation, we should
m k
some z and calculate the rhs of this equation,parameterisation
ery we should
in the same answer independently of the for flat LCDM w(z) that ].could ne
this is constant redshift equation of state [? of
e-redshift curve D(z), we can reconstruct the dark energyof of the redshift Typicall
in
obtain the same answer independently
surement. Differentiating Eq. (5)ansatz used in [5],
ed ansatz for D(z), such as the Pade we then find that through various ensur
measurement. Differentiating Eq. (5) we then find that nated.
L (z) = ζD (z) + 3(1(1 + z)D α (1 − z) − 1 2 ] α
+ z)2 − (z)[1 + D (z)+
DL (z) = = ζD (z) √ 3(1 + z)2 D (z)[1 − D (z)2 ]the test To
L (z) flat ΛCDMγmodels. 2 − α − β −(6) .
+ +z+ Testing
= 0 for all β(1 + z) + 1 γ
strate that it will wo
have used then shorthandw(z) from Eq ΛCDM models.in Eq. (4) toisthe lum
e data, and the calculatesfor = 2[(1 + z)3 − 1]. a reconstruction method (6) g
= 0 ζ all flat (3). Such Note more
w(z) directly because we are fitting directly to the observable, and 3 can spot small The
this is completely independent of the value of Ωm . simulated data. vari
so
slate to dramatic a test for Λ as Zunckel & Clarkson, PRL, arXiv:0807.4304;Note err
mayWe this as variations in w(z). Unfortunately,param- + z) to larger and substit
use have used the shorthand ζ a this also leads − 1]. reported
follows: take = 2[(1 calculated
form this is directly. It is also Sahni L (z) which errors = 0 within .the e
see data. If etal 0807.3548 value of Ω
sedparameterising it completely not clear, however,= 0 theL (z)on our understand
by that for D(z) and fit to the independent of m
e taken most seriously. What is very nice about this method is that, if done in small re
f w(z) in a given useis independent of bins atΛ as z; this is not the case param-
We may bin this as a test for lower follows: take a for parameteri
grateeterised form for D(z)dand Bothto the data. If strong degeneracies
over redshift when calculating (z). fit methods suffer from L (z) = 0

A litmus test for flat ΛCDM

these are better fits to constitution data than LCDM

with Arman Shafieloo, PRD


A litmus test for ﬂat ΛCDM


should be zero!



A litmus testno dependence on Omega_m
for ﬂat ΛCDM


should be zero!



other tests

( )

Conclusions

• none.


Chris Clarkson - Dark Energy and Backreaction

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