Primordial gravitational waves

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Primordial gravitational waves

• Sabino Matarrese
• Dipartimento di Fisica Galileo Galilei
• Universita degli Studi di Padova
• INFN, Sezione di Padova
• email matarrese_at_pd.infn.it

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Sources of GWs
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Primordial gravitational waves (I)

• GWs are tensor perturbations of the metric.
  Restricting ourselves to a
• flat FRW background (and disregarding scalar and
  vector modes)
• ds2a2(?)d?2 - (dij
  hij(x,t)) dxi dxj
• where hij are tensor modes which have the
  following properties
• hij hji
  (symmetric)
• hii 0 (traceless)
• hiji 0 (transverse)
• and satisfy the equation of motion

Possible source term. It vanishes in linear
theory and for a perfect fluid.
d/dt
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Primordial gravitational waves (II)

• GWs have only (9?6-1-3) two independent degrees
  of freedom,
• corresponding to the two polarization states of
  the graviton
• behaviour
• k aH (outside the horizon) ? const
  decaying mode
• k aH (inside the horizon) ? ? eikt/a
  (gravitational wave it freely
• 
  streams, experiencing redshift and
  
• 
  dilution, like a free photon)

polarization tensor
free massless, minimally coupled scalar field
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Classify Inflationary Models

• The shape of the inflaton potential V (f)
  determines the observables.
• It is standard to use three parameters to
  characterize the shape
• e slope of the potential ? (V/V)2
  1
• ? curvature of the potential ?V/V 1
• ? jerk of the potential ?(V/V)(V/V)
  e2

slow-roll conditions
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Slow-roll parameters and observables
Scalar (comoving curvature) perturbation
power-spectrum
Tensor (gravity-wave) perturbation power-spectrum
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Generic predictions of single field slow-roll
models vs. WMAP 3yr data
from Spergel et al. 2006
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Gravity waves the smoking gun of inflation

• The spectra PR(k) and PT(k) provide the contact
  between theory and observations. The WMAP (SDSS)
  dataset allows to extract an upper bound, rlt0.28
  (95 CL) (Spergel 2006), or elt0.017. This limit
  provides un upper bound on the energy scale of
  inflation
• V1/4 lt 2.6 x 1016 GeV
• A positive detection of the B-mode in CMB
  polarization, and therefore an indirect evidence
  of gravitational waves from inflation, once
  foregrounds due to gravitational lensing from
  local sources has been properly treated, requires
  e lt 10-5, corresponding to
• V1/4 gt 3.5 x 1015 GeV

Note r V1/4
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The search for primordial gravitational waves
Battye Shellard 1996
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B-modes detectability of IGB
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Tensor-to-scalar ratio
Cooray 2004
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Super-Inflation getting a blue spectrum (nTgt0)
of tensor modes
Baldi, Finelli Matarrese 2005
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CMB weak-gravitational lensing a foreground for
the (indirect) detection of primordial
gravitational waves via B-mode polarization
Carbone, Springel, Baccigalupi, Bartelmann
Matarrese 2007
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CMB weak-gravitational lensing a foreground for
the (indirect) detection of primordial
gravitational waves via B-mode polarization
All-sky map of the lensing potential obtained
from the Millennium Simulation (Carbone et al.
2007) ? see also Fosalba et al. 2007 Das Bode
2007
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Second-order tensor modes
Second-order metric
Second-order tensor modes
tensor projector
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Secondary effects vs. IGB
The B-mode polarization produced by primordial
gravitational waves can be hidden by
gravitational lensing and/or by second- -order
vector and tensor modes, unless the inflation
energy scale is larger than 1015 GeV. Our ability
of delensing polarization maps is the crucial
issue in this problem (e.g. Hirata Seljak 2004
Sigurdson Cooray 2005).
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GW from non-linear cosmological perturbations
tensor-mode projection operator

• Tensor (and vector!) metric modes
• are generated by scalar (e.g.
• density) perturbations as soon as
• the latter become non-linear. As
• a result GW are produced during
• the later stages of cosmological
• structure formation with typical
• period of the order of the Hubble
• time.

Carbone Matarrese 2005
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GW from the collapse of DM halos

• During the formation of non-
• spherical DM halos low-frequency
• GW are emitted, thus forming a
• stochastic background with
• amplitude comparable with the
• primordial one for an inflation
• energy scale few x 1015 GeV.
• The typical GW strain amplitude for
• a non-spherical halo of mass M and
• size L at a distance D is
• This also produces CMB temperature
• anisotropy and polarization by secondary
• effects

few events contributing to the stochastic
background
Carbone, Baccigalupi Matarrese 2005
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Inflation vs. Cyclic-Ekpyrotic Universe in terms
of GW
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Secondary tensors
Matarrese, Mollerach Bruni 1998 Mollerach,
Harari Matarrese 2004 Ananda, Clarkson Wands
2007 Baumann, Steinhardt, Takahashi Ichiki
2007 computed the GW background produced at
second-order by scalar modes in various epochs.
According to Baumann et al. these second-order
modes may dominate the primary background on
intermediate scales. For cyclic/ekpyrotic models
they always dominate.
Baumann et al. 2007
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Detectability of second-order tensor modes
Baumann et al. 2007
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Neutrino free-streaming and GW

• The effect of neutrino free-streaming (via the
  tensor mode contribution of the anisotropic
  stress) was recognized by Weinberg (). Cosmic
  neutrinos induce a damping term in the linear
  evolution of GW, having a large effect for those
  waves which crossed the Hubble radius during
  radiation dominance
• At second-order cosmic neutrinos have two effects
  (Mangilli, Bartolo, Matarrese Riotto 2008)
• they produce a damping term (analogous to the
  first order one)
• they act as an extra source of GW
• Similar effects should be expected for any
  particle species freedom undergoing
  free-streaming (on suitable scales)

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Curvaton and GWs
In the curvaton scenario for the generation of
perturbation, the production of primary tensor
modes is suppressed by the requirement that
inflaton Perturbations have negligible
amplitude. Bartolo, Matarrese, Riotto
Väihkönen (2007) have shown that second-order
tensor modes can have a non-negligible amplitude,
being proportional to the non-Gaussianity
strength fNL For fNL 100 one can easily
attain values as large as WGW 10-15 in the
frequency range relevant for BBO or DECIGO
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Conclusions future prospects

• Inflation provides a causal mechanism for the
  generation of ultra-low frequency gravitational
  waves
• Their direct (or indirect via CMB polarization)
  detection with the specific features predicted by
  inflation would provide strong independent
  support to the model
• However, gravitational (weak) lensing of the CMB
  photons by the intervening DM distribution
  produces a foreground to GW detection via
  conversion of E into B modes. High-precision
  cleaning of this effect is crucial for GW
  detection down to interesting inflation energy
  scales
• (Non-linear) scalar modes unavoidably generate
  tensor modes by mode-mode coupling, which in some
  models may even overcome the primordial
  (inflationary) background