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Title
Abstract
Figures
Introduction
CDM 2-POINT Statistics
Conclusions
References
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Physics
Classical Mechanics

A Lagrangian perturbation theory for modify gravity

Profile image of Alejandro Felipe AvilésAlejandro Felipe Avilés

2017, arXiv (Cornell University)

Abstract

We present a formalism to compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order. We emphasize the case of theories with scale dependent gravitational strengths, such as chameleons, but our formalism can be accommodated to other modified gravity theories. In the non-linear regime two qualitative features arise. One, as is well known, is that nonlinearities lead to a screening of the force mediated by the scalar field. The second is a consequence of the transformation of the Klein-Gordon equation from Eulerian to Lagrangian coordinates, producing frame-lagging terms that are important especially at large scales, and if not considered, the theory does not reduce to the ΛCDM model in that limit. We apply our formalism to compute the 1-loop power spectrum and the correlation function in f (R) gravity by using different resummation schemes. We further discuss the IR divergences of these formalisms.

Figures (17)
Figure 1: First order growth functions over ACDM growth function. We show the cases of the HS F4, F5 and F6 models and redshifts z = 0 and z = 0.5. At linear order the right hand side of Eq. (47) is set to zero, leading to the Zel’dovich solution
Figure 1: First order growth functions over ACDM growth function. We show the cases of the HS F4, F5 and F6 models and redshifts z = 0 and z = 0.5. At linear order the right hand side of Eq. (47) is set to zero, leading to the Zel’dovich solution
Figure 2: Second order normalized growth functions Dd, D? D® and DY, for the F4 model. Different triangular configu- rations are considered. See text for details.
Figure 2: Second order normalized growth functions Dd, D? D® and DY, for the F4 model. Different triangular configu- rations are considered. See text for details.
We shall use the above perturbations in the following section to find the third order Lagrangian displacement field. ky ~ ko. We can read from the equations for the growth functions that in the case of equilateral configuration we have DY = be ) The squeezed case corresponds to the limit of large scales where the theory reduces to ACDM, we can see from panel (c) in Fig. 2 that actually by considering the combination DY) — D® + D® ) (dotted red line in panel (c)) in this configuration we get D) = 0, as was shown analytically above. The frame- _— second order growth function for equilateral configuration is exactly zero and for the orthogonal is very similar in shape to the squeezed one, but reaching smaller values. In panel (d) we show the screening normalized growth functions for the three considered triangular configurations. We make note that the functions DY) and D® do not tend to 1 as k goes to zero, the EdS value, instead their limits are ~ 1.009, the Ip(to) value for the chosen cosmological parameters in the ACDM model.
We shall use the above perturbations in the following section to find the third order Lagrangian displacement field. ky ~ ko. We can read from the equations for the growth functions that in the case of equilateral configuration we have DY = be ) The squeezed case corresponds to the limit of large scales where the theory reduces to ACDM, we can see from panel (c) in Fig. 2 that actually by considering the combination DY) — D® + D® ) (dotted red line in panel (c)) in this configuration we get D) = 0, as was shown analytically above. The frame- _— second order growth function for equilateral configuration is exactly zero and for the orthogonal is very similar in shape to the squeezed one, but reaching smaller values. In panel (d) we show the screening normalized growth functions for the three considered triangular configurations. We make note that the functions DY) and D® do not tend to 1 as k goes to zero, the EdS value, instead their limits are ~ 1.009, the Ip(to) value for the chosen cosmological parameters in the ACDM model.
The growth functions depend on three wavevectors, but these are constrained to form a quadrilateral k = kj +k2+k3; thus, they depend only on 6 numbers for a given k. Furthermore, the relevant configurations for these quadrilaterals in 2-point statistics are the so-called double squeezed in which k = —k, and p = kg = —kg, which left us with only 3 degrees of freedom, highly simplifying the analysis. We can write the third order Lagrangian displacement kernel as
The growth functions depend on three wavevectors, but these are constrained to form a quadrilateral k = kj +k2+k3; thus, they depend only on 6 numbers for a given k. Furthermore, the relevant configurations for these quadrilaterals in 2-point statistics are the so-called double squeezed in which k = —k, and p = kg = —kg, which left us with only 3 degrees of freedom, highly simplifying the analysis. We can write the third order Lagrangian displacement kernel as
Figure 3: Functions Qi(k) and Q2(k) for F4 model: the dot-dashed red curves do not contain frame-lagging terms, the black curves have pure frame-lagging contributions (with the dashing showing the negative), and the solid blue is the full curve (with dashing for the negative in Q2), the dotted orange is the ACDM.
Figure 3: Functions Qi(k) and Q2(k) for F4 model: the dot-dashed red curves do not contain frame-lagging terms, the black curves have pure frame-lagging contributions (with the dashing showing the negative), and the solid blue is the full curve (with dashing for the negative in Q2), the dotted orange is the ACDM.
Figure 4: Left panel: Ri(k) function: the solid red curve does not contain frame-lagging terms (with the dashing showing its negative); the dot-dashed black curve has pure frame-lagging contributions; the solid blue is the full curve; and the dotted orange is the ACDM. Right panel: R2(k) function: the solid red curve does not contain frame-lagging terms (with the dashing showing its negative); dashed black is the negative of pure frame-lagging contributions; solid blue is the full Rz curve (with the dashing showing its negative); and solid orange is for ACDM. These squeezed configurations survey large scales and then should reduce to GR. We explicitly showed in Sect. IV that A- B= D?(p, —p)- DO (p, - p)+ D® (P,— p) = 0, thanks to the frame-lagging. By this reason the leading term that survives in Eq. (93) is of order O((k/p)*), yielding
Figure 4: Left panel: Ri(k) function: the solid red curve does not contain frame-lagging terms (with the dashing showing its negative); the dot-dashed black curve has pure frame-lagging contributions; the solid blue is the full curve; and the dotted orange is the ACDM. Right panel: R2(k) function: the solid red curve does not contain frame-lagging terms (with the dashing showing its negative); dashed black is the negative of pure frame-lagging contributions; solid blue is the full Rz curve (with the dashing showing its negative); and solid orange is for ACDM. These squeezed configurations survey large scales and then should reduce to GR. We explicitly showed in Sect. IV that A- B= D?(p, —p)- DO (p, - p)+ D® (P,— p) = 0, thanks to the frame-lagging. By this reason the leading term that survives in Eq. (93) is of order O((k/p)*), yielding
Figure 5: Power spectrum a(k) of divergences of Lagrangian displacements for the F4 model: The solid blue curve is the full power spectrum, the dot-dashed black contains only frame-lagging contributions; the red curve does not contain any frame- lagging, with the dashed showing its negative; the dotted brown is for the linear theory, that coincide with the linear matter power spectrum.
Figure 5: Power spectrum a(k) of divergences of Lagrangian displacements for the F4 model: The solid blue curve is the full power spectrum, the dot-dashed black contains only frame-lagging contributions; the red curve does not contain any frame- lagging, with the dashed showing its negative; the dotted brown is for the linear theory, that coincide with the linear matter power spectrum.
Figure 6: Ratios of power spectra to the linear corresponding models (F4 and F5) power spectrum with BAO removed at redshift z = 0. Long dashed black curves are for the linear theory; dashed blue are for CLPT; solid red are for SPT*; dot-dashed browr are for LRT; and dotted green are the linear ACDM. The vertical line shows the scale ky, = oz /2.
Figure 6: Ratios of power spectra to the linear corresponding models (F4 and F5) power spectrum with BAO removed at redshift z = 0. Long dashed black curves are for the linear theory; dashed blue are for CLPT; solid red are for SPT*; dot-dashed browr are for LRT; and dotted green are the linear ACDM. The vertical line shows the scale ky, = oz /2.
Figure 7: Ratios of power spectra to the linear corresponding models (F6 and ACDM) power spectrum with BAO removed at redshift z = 0. Long dashed black curves are for the linear theory; dashed blue are for CLPT; solid red are for SPT*; dot-dashed brown are for LRT. The F6 panel shows also the linear ACDM case, denoted with dotted green curve. The vertical line shows the scale kn, = o;/2.
Figure 7: Ratios of power spectra to the linear corresponding models (F6 and ACDM) power spectrum with BAO removed at redshift z = 0. Long dashed black curves are for the linear theory; dashed blue are for CLPT; solid red are for SPT*; dot-dashed brown are for LRT. The F6 panel shows also the linear ACDM case, denoted with dotted green curve. The vertical line shows the scale kn, = o;/2.
Figure 8: CLPT correlation function at redshift z = 0. Long dashed (black) is the linear ACDM model; dashed (red) is for ACDM in CLPT ; solid (blue) is for F4; dotted (orange) for F5; and dot-dashed (brown) is for F6. The lower panel shows the ratio with the Zel’dovich approximation in ACDM.
Figure 8: CLPT correlation function at redshift z = 0. Long dashed (black) is the linear ACDM model; dashed (red) is for ACDM in CLPT ; solid (blue) is for F4; dotted (orange) for F5; and dot-dashed (brown) is for F6. The lower panel shows the ratio with the Zel’dovich approximation in ACDM.
Figure 9: F4 and F5. SPT* power spectra with (solid curves) and without (dashed curves) screenings for models F'4 and F5 at redshifts z = 0 (black curves) and z = 0.5 (red curves). The lower panels shows the ratio with the SPT* power spectra fo: z =0. The simulated data were provided by [111].
Figure 9: F4 and F5. SPT* power spectra with (solid curves) and without (dashed curves) screenings for models F'4 and F5 at redshifts z = 0 (black curves) and z = 0.5 (red curves). The lower panels shows the ratio with the SPT* power spectra fo: z =0. The simulated data were provided by [111].
Figure 10: Same as Fig. 9 but for F6. [see Eq. (A19)], including the term
Figure 10: Same as Fig. 9 but for F6. [see Eq. (A19)], including the term
where A; = U;(q.) — Vi (q,) are the Lagrangian displacement differences, and q = qz — q, the Lagrangian coordinate difference. By using the cumulant expansion theorem we can recast the power spectrum as [46] This appendix briefly reviews the methods we use to compute the power spectrum and the correlation function. For completeness we write the relevant equations but we refer the reader to the original papers for detailed derivations. The matter power spectrum in LPT is given by [44] Appendix A: Lagrangian displacement polyspectra and the matter power spectrum
where A; = U;(q.) — Vi (q,) are the Lagrangian displacement differences, and q = qz — q, the Lagrangian coordinate difference. By using the cumulant expansion theorem we can recast the power spectrum as [46] This appendix briefly reviews the methods we use to compute the power spectrum and the correlation function. For completeness we write the relevant equations but we refer the reader to the original papers for detailed derivations. The matter power spectrum in LPT is given by [44] Appendix A: Lagrangian displacement polyspectra and the matter power spectrum
the integrals in Eqs.(A21), (A22) and (A23) reduce to one-dimensional Hankel transformations. In the CLPT scheme, the correlation function can be written compactly by Fourier transforming the power spectrum of Eq. (A19) and using Gaussian integrations, resulting in
the integrals in Eqs.(A21), (A22) and (A23) reduce to one-dimensional Hankel transformations. In the CLPT scheme, the correlation function can be written compactly by Fourier transforming the power spectrum of Eq. (A19) and using Gaussian integrations, resulting in

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