Understanding
Cosmological perturbation
theory for Large-scale Structure
(in 1D)
Matthew McQuinn
(University of Washington)
in collaboration with Martin White (Berkeley)
arxiv: 1502.07389
``[One] grows stale if he works all the time on insoluble problems, and
a trip to the beautiful world of one dimension will refresh his
imagination better than a dose of LSD.''
Freeman Dyson
We have this amazing
model for the CMB
Planck CMB
measurements
Green curve is linear
theory calculation
things eventually go nonlinear
(and perturbation theory gets more complicated)
credit: Bolshoi collaboration
linear perturbation theory still explains
galaxy clustering on large scales
BOSS
Anderson et al ’12:
for given (standard) cosmology, clustering can be
calculated at percent accuracy using linear theory
not perturbative
It is not clear whether we have a
successful perturbative theory
G
ti m
e-R
large-N theory
1l
oo
pS
PT
th
eo
ry
Carlson et al. 2009
matter power
spectrum
divided by
some smooth
function
PT
G
R
linear theory
simulation
(models
trying to
explain this)
Lagrangian Resum.
Many theories predict infinite correlation function for LCDM and the Lagrangian theories
--what do the worst for P(k)-- do excellently in the correlation function
Standard Cosmological
perturbation theory
This can be solved perturbatively in powers of
n
the initial matter overdensity, δL .
Assumptions:
• the universe is a invicid, pressureless fluid
• still converges at large scales despite RMS
field value being >1 on small scales
• the flow is irrotational (irrelevant for 1D)
perturbative solution to these equations
(these are the 1D solutions but 3D are also known)
Makino et al ’92; Jain & Bertshchinger ’94
Don’t worry, you don’t need to memorize this!
Lagrangian Perturbation
theory
𝛙 is the displacement and q Lagrangian coordinate, ɸ
is gravitational potential, H is Hubble function
Theory initially introduced by Zeldovich (1970) at
linear order, who showed that it works better at
capturing structures.
Expansion essentially proceeds by expanding RHS in
initial overdensity.
After standard Eulerian and Lagrangian
theories, all developments circa 2010 try
to resum terms in these expansions to
accelerate the convergence
(In ~2010 there was a really interesting
development that I will get to.)
We specialize to case of 1D dynamics to understand PTs
1. same equations as in 3D and, hence, same assumptions
2. 1st order Lagrangian perturbation theory is exact prior to
shell crossing
3. Can compute P(k) in standard Eulerian perturbation
3D is
theory to any order with 1D integral: infinite order
Lagrangian theory solution (McQuinn & White 2015)
Z
Infinite
⌘
⇣
R
ikq
k2 01 dkk 2 PL (k)(1 cos[kq])/⇡
order P (k) = dqe
e
1
solution:
linear power spectrum
advantages (continued)
4) Can calculate nonlinear power with ~107 particle
simulation to percent level for 1000 band powers
(need ~1010 to capture BAO scales in 3D)
This allows us to test many 1D models!
5) Striking number of similarities/analogies with 3D
(MM & White 2015)
what 1D looks like
Testing CDM-like 1D cosmology
z=0
order n
PSPT
Z 1
2
=
0
Z
n
2
2
X
[
k
(q)
]
ikq
= dqe
2n n!
1
dk 2PL
(1 cos[kq])
2
⇡ k
Standard perturbation theories do not describe nonlinear
evolution on any scale accurately.
Testing CDM-like 1D cosmology
z=0
order n
PSPT
Z 1
2
=
0
Z
n
2
2
X
[
k
(q)
]
ikq
= dqe
2n n!
1
dk 2PL
(1 cos[kq])
2
⇡ k
Standard perturbation theories do not describe nonlinear
evolution on any scale accurately.
Testing CDM-like 1D cosmology
z=0
order n
PSPT
Z 1
2
=
0
Z
n
2
2
X
[
k
(q)
]
ikq
= dqe
2n n!
1
dk 2PL
(1 cos[kq])
2
⇡ k
Standard perturbation theories do not describe nonlinear
evolution on any scale accurately.
Testing CDM-like 1D cosmology
z=0
order n
PSPT
Z 1
2
=
0
Z
n
2
2
X
[
k
(q)
]
ikq
= dqe
2n n!
1
dk 2PL
(1 cos[kq])
2
⇡ k
Standard perturbation theories do not describe nonlinear
evolution on any scale accurately.
Testing CDM-like 1D cosmology
z=0
order n
PSPT
Z 1
2
=
0
Z
n
2
2
X
[
k
(q)
]
ikq
= dqe
2n n!
1
dk 2PL
(1 cos[kq])
2
⇡ k
Standard perturbation theories do not describe nonlinear
evolution on any scale accurately.
effective theories
•
Can write dynamical equations for fields that have
been smoothed so do not depend on high
wavenumbers (Baumann et al. 2012, Carrasco et al
2012). At lowest beyond-linear order this looks like
•
New term
compared
to SPT
A huge success: Adding these terms gets rid of
the sensitivity to modes with k’ >> k and hence
infinities that appear in some cosmologies in SPT
(e.g. Pajer & Zaldarriaga 2014).
Comparison of effective theories
in CDM-like 1D cosmology
CDM-like 1D cosmology
z=1
z=0
(similar improvement
found in 3D; Carrasco
et al 2012)
Effective theory adds a term that scales as k2P(k), which is not
forbidden by symmetries, that standard theories do not.
(Baumann et al 2009; Carrasco et al 2012; Mercolli & Pajer 2014)
1D Power-law cosmologies
SPT is convergent
SPT depends on cutoff
Power-law cosmologies, P(k)~kn, have self-similar solutions.
PT at 1-loop is P(k)~kn + #k2n+1. EFT adds kn+2 and “k4”
terms with free coefficients … these really help!
Lowest nontrivial contributions in 1D
linear correlation function
RMS displacment
(15% difference between standard and effective theories)
RMS displacement is almost
non-perturbative at BAO.
Lagrangian theory does not
expand in this term and this is
why it is more successful.
1D BAO
Plotted cosmology has the same dimensionless power spectrum as LCDM.
conclusions
• standard perturbation theories do not work at
any mildly nonlinear scale in all the 1D
cosmologies we consider (even at infinite order)
• effective theories add exactly the term that
standard theories are missing
• many analogies between 1D and 3D