Limitations of cosmography in extended theories of gravity

Transcription

1 DSU 2015 Yukawa Institute for Theoretical Physics, Kyoto Dec Limitations of cosmography in extended theories of gravity Álvaro de la Cruz-Dombriz Astronomy, Cosmology and Gravitation Centre, ACGC - UNIVERSITY of CAPE TOWN Collaborators: V. C. Bus/, P. K. S. Dunsby, O. Luongo, L. Reverberi and D. Sáez- Gómez Department of Theoretical Physics, COMPLUTENSE UNIVERSITY of MADRID A.L.MAROTO Collaborators: J.A.R.CEMBRANOS, A.DOBADO, V.GAMMALDI, R.LINEROS and

2 Outline 2 I. Some rudiments in Cosmography II. Extended theories of gravity: state-of-the art III. Limitations of cosmographic approach in extended theories - - Biased results. Spotting ΛCDM? - - Ruling out and reconstructing higher-order theories IV. Prospects and Conclusions

3 Outline 3 I. Some rudiments in Cosmography II. Extended theories of gravity: state-of-the art III. Limitations of cosmographic approach in extended theories - - Biased results. Spotting ΛCDM - - Ruling out and reconstructing higher-order theories IV. Conclusions and Prospects

4 ü Cosmography rudiments (I) In order to test GR and the Copernican Principle, a useful tool is to use frameworks able to encompass a large class of models/theories Such model independent methods instead of a case-by-case approach have been used to infer the Dark Energy EoS and reconstruct classes of DE theories Cosmography approach just relies on the Copernican principle and the expression of the FLRW scale factor in terms of an auxiliary variable, such as redshift(s), time, etc. H(z) =ȧ a = H 0 + H 0z (z z 0 )+ H 0zz 2 (z z 0) S. Weinberg, Gravita(on and Cosmology: Principles and Applica(ons of the General Theory of Rela(vity E. R. Harrison, Nature, 260, 591 (1976).

5 ü Cosmography rudiments (I) In order to test GR and the Copernican Principle, a useful tool is to use frameworks able to encompass a large class of models/theories Such model independent methods - instead of a case-by-case approach have been used to infer the Dark Energy EoS and reconstruct classes of DE theories Cosmography approach just relies on the Copernican principle and the expression of the FLRW scale factor in terms of an auxiliary variable, such as redshift(s), time, etc. H(z) =ȧ a = H 0 + H 0z (z z 0 )+ H 0zz 2 (z z 0)

6 ü Cosmography rudiments (I) In order to test GR and the Copernican Principle, a useful tool is to use frameworks able to encompass a large class of models/theories Such model independent methods - instead of a case-by-case approach have been used to infer the Dark Energy EoS and reconstruct classes of DE theories Cosmography approach just relies on the Copernican principle and the expression of the FLRW scale factor in terms of an auxiliary variable, such as redshift(s), time, etc. H(z) =ȧ a = H 0 + H 0z (z z 0 )+ H 0zz 2 (z z 0) or as alternative independent variable [or Padé polynomials, etc.]

7 ü Cosmography rudiments (and II) How well this expansion is in comparison with exact models and other parameterisations?

8 ü Cosmography rudiments (and II) How well this expansion is in comparison with exact models and other parameterisations? ΛCDM model S. Capozziello, R. Lazkoz and V. Salzano, Phys. Rev. D (2011), arxiv: [atro- ph CO]

9 ü Cosmography rudiments (and II) How well this expansion is in comparison with exact models and other parameterisations? ΛCDM model CPL model S. Capozziello, R. Lazkoz and V. Salzano, Phys. Rev. D (2011), arxiv: [atro- ph CO]

10 Outline 10 I. Motivation and some cosmography rudiments II. Extended theories of gravity: state-of-the art III. Limitations of Cosmographic approach in extended theories - - Biased results. Spotting ΛCDM - - Ruling out and reconstructing theories with higher orders IV. Conclusions and Prospects

11 Extended theories of gravity: a motivation The search for a fully consistent Quantum Gravity theory is a very active field of research. GR is consistent if treated in the frame of quantum effective field theories but it breaks down at Planck scale J.F. Donoghue and T. Torma, gr- qc/

12 Extended theories of gravity: a motivation The search for a fully consistent Quantum Gravity theory is a very active field of research. GR is consistent if treated in the frame of quantum effective field theories but it breaks down at Planck scale J.F. Donoghue and T. Torma, gr- qc/ ü Extended theories of gravity ü Proposals must Emulate certain gravitational aspects of General Relativity must Explain the cosmological evolution in different eras are motivated by the cosmological constant (Λ) problem, dark energy, dark matter, singularities Scalar/Vector-Tensor gravity: Brans-Dicke theories, f(r) theories, Horndeski Extra dimensions theories: Brane-world theory, String theory Massive gravity Born-Infeld inspired gravity

13 The degeneracy problem ü Def.: several extended gravity theories lead to identical results with either General Relativity (GR) or the Concordance (ΛCDM) Model

14 The degeneracy problem ü Def.: several extended gravity theories lead to identical results with either General Relativity (GR) or the Concordance (ΛCDM) Model ü Therefore, the only use of these degenerate results cannot distinguish between GR and the alternative suggested theory(ies) ü Consistency tests

15 The degeneracy problem ü Def.: several extended gravity theories lead to identical results with either General Relativity (GR) or the Concordance (ΛCDM) Model E.g. AdlCD and A. Dobado, Phys. Rev. D74: , 2006 f(r) model with Robertson-Walker solution the same as ΛCDM solution for dust + Λ.

16 The degeneracy problem ü Def.: several extended gravity theories lead to identical results with either General Relativity (GR) or the Concordance (ΛCDM) Model ü Therefore, the only use of these degenerate results cannot distinguish between GR and the alternative suggested theory(ies) ü Consistency tests - Evolution of geodesics and Raychaudhuri equation - Importance of averaging and backreaction mechanism - Evolution of scalar perturbations - Black holes properties and thermodynamics - - Dark matter: astrophysical fluxes and (in)direct detection experiments [UCT Cosmology group] [Madrid UCM-Th] - Model/theory independent tests fit with data catalogues

17 Outline 17 I. Motivation and some cosmography rudiments II. Extended theories of gravity: state-of-the art III. Limitations of Cosmographic approach in extended theories - - Biased results. Spotting ΛCDM? - - Ruling out and reconstructing theories with higher orders

18 ü Differences between auxiliary variables: y vs. z Mock data generated from a fiducial flat ΛCDM model with redshift distribution Union2.1 catalogue and Two sets of parameters and 100 simulations How frequent the true cosmographic values fall in 1, 2, 3σ confidence regions

19 ü Differences between auxiliary variables: y vs. z Mock data generated from a fiducial flat ΛCDM model with redshift distribution Union2.1 catalogue and Two sets of parameters and 100 simulations How frequent the true cosmographic values fall in 1, 2, 3σ confidence regions V. Bus/, AdlCD, P. Dunsby, D. Sáez- Gómez arxiv: [astro- ph.co], PRD (2015)

20 ü Differences between auxiliary variables: y vs. z Mock data generated from a fiducial flat ΛCDM model with redshift distribution Union2.1 catalogue and Two sets of parameters and 100 simulations How frequent the true cosmographic values fall in 1, 2, 3σ confidence regions Overes/ma/on of errors Biased results Well behaved V. Bus/, AdlCD, P. Dunsby, D. Sáez- Gómez arxiv: [astro- ph.co], PRD (2015) DSU Kyoto Decenber 2015

21 ü Differences between auxiliary variables: y vs. z y- paramet)ization z- paramet)ization One simula/on, ü For ΛCDM j 0 =1 V. Bus/, AdlCD, P. Dunsby, D. Sáez- Gómez arxiv: [astro- ph.co], PRD (2015)

22 ü Is Cosmography able to spot the correct XCDM model? Mock realizations of data for a flat XCDM Constraints for θ 1 (fourth order), θ 2 (fifth order) and direct constraint of parameters V. Bus/, AdlCD, P. Dunsby, D. Sáez- Gómez arxiv: [astro- ph.co] PRD (2015) Fitting to the model spots deviations from ΛCDM with less effort Some evidence of when considering θ 1, but dissapears assuming θ 2!!!

23 Outline 23 I. Motivation and some cosmography rudiments II. Extended theories of gravity: state-of-the art III. Limitations of cosmographic approach in extended theories - - Biased results. Spotting ΛCDM? - - Ruling out and reconstructing higher-order theories

24 Cosmography as a tool to reconstruct DE models Capozziello et al., Bamba et al. Astrophys. Space Sci. 342, 155 (2012) Nonetheless in theories with higher derivatives, the appearance of extra parameters apart from the cosmographic ones, imposes some limitations in the method E.g. 1: K-essence when scalar potential and kinetic term expanded around z=0 and expressed in terms of the cosmographic parameters 1σ ü It requires assumption on the model today ü Thus, a one-to-one correspondence between cosmographic parameters and the model emerges 2σ V. Bus/, AdlCD, P. Dunsby, D. Sáez- Gómez arxiv: [astro- ph.co], PRD (2015)

25 E.g. 1: K-essence ü Generic realization of ΛCDM ü Gaussian processes 1σ 1σ 2σ 2σ Fourth-order cosmographic expansion No assump/on of the model today V. Bus/, AdlCD, P. Dunsby, D. Sáez- Gómez arxiv: [astro- ph.co], PRD (2015) R. Nair, S. Jhingan and D. Jain, JCAP 01 (2014) 005 [arxiv: ]

26 E.g. 1: K-essence PRELIMINARY (Calcula/ons by L. Reverberi, ACGC Cape Town) SNIa Union 2.1 data and/or H(z) data O. Farooq and B. Ratra, Astroph. J. Leg. 776:L7 (2013) arxiv:

27 E.g. 2: f(r) theories Higher- order deriva/ves theories Two extra parameters

28 E.g. 2: f(r) theories Higher- order deriva/ves theories Two extra parameters ü f(r)-derivatives cosmographic parameters one-to-one correspondence must be abandoned ü Sensible priors for α and β are required A. Aviles et al., Phys. Rev. D 87, no. 4, (2013) Phys. Rev. D 87, no. 6, (2013) However, whenever these values are assumed, an instability occurs L. Pogosian and A. Silvestri, Phys. Rev. D 77, (2008) Phys. Rev. D 81, (2010) ü Cosmological values and may still produce viable cosmological models

29 E.g. 2: f(r) theories ü Mock data generated from the given f(r) model [exact background evolution] ü Simulations with three different hypotheses: True values of {α, β }, { α = 1, β = 0}, broad marginalization (α N (1, 0.05) and β N (0.07, 0.05)) V. Bus/, AdlCD, P. Dunsby, D. Sáez- Gómez arxiv: [astro- ph.co], PRD (2015) o o The probability of f 0 is highly dependent on the choice of {α, β} which may even lead to rule out the true values The errors are very large for every case leading to a completely degenerated fit, such that a wide range of completely different and viable f(r) models - e.g., W. Hu & I. Sawicki PRD lie in the 1σ region

30 E.g. 2: f(r) theories SNIa Union 2.1 data + H(z) data O. Farooq and B. Ratra, Astroph. J. Leg. 776:L7 (2013) arxiv: PRELIMINARY (Calcula/ons by L. Reverberi, ACGC Cape Town)

31 Outline 31 I. Motivation and some cosmography rudiments II. Extended theories of gravity: state-of-the art III. Limitations of Cosmographic approach in extended theories - - Biased results. Spotting ΛCDM - - Ruling out and reconstructing theories with higher orders IV. Conclusions and Prospects

32 Conclusions on Cosmographic Approach Cosmography results do depend upon both the chosen auxiliary variable (redshifts z or y) and the expansion order. Parameter y is highly disfavoured Reliability of cosmography to spot ΛCDM around close-enough XCDM competitors remains very limited with results once again depending upon the expansion order For extended theories of gravity, the method provides a sort of clear picture for theories with no higher-order derivatives, although not competitive - larger errors when compared to other mehods For extended theories with higher derivatives in either geometrical - like f(r) - or matter sector - like Galileons -, there are extra free parameters requiring priors and marginalisation. Large errors emerge Other neglected limitations: spatial cuvature (Clarkson 2011), lensings effects (Wald 1998, Bacon 2014) and local gravitational redshift (Wojtak 2015) may lead to extra scatter in Hubble diagrams

33 Is there any hope for Cosmography? 1. Clear definition of - auxiliary variables (redshifts, Padé polynomials, etc.) - testing against mock data 2. Establish a trade-off between number of data points, number of cosmological parameters and Bayesian evidence, so criteria can be provided 3. Motivated priors over extra parameters : stability conditions, absence of ghosts, behaviour of perturbations vs. blind tests

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35 Backup slides

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37 Cosmography Hence we can construct observable magnitudes by starting from such expansions, and then compare with the observational data. Luminosity distance H 0 d L (z) =z + 1 q 0 z j 0 + q 0 +3q 2 0 z [2 + 5j 0(1 + 2q 0 ) q 0 (2 + 15q 0 (1 + q 0 )) + s 0 ] z j0 2 l 0 j 0 [27 + 5q 0 ( q 0 )] + 3q 0 [2 + q 0 ( q 0 (11 + 7q 0 )) 5s 0 ] 11s 0 } z m 0 + l 0 ( q 0 ) 10j 2 0(19 +28q 0 ) 3q 0 [8 + q 0 ( q 0 ( q 0 (19 + 9q 0 )) 70s 0 ) 95s 0 ] +j 0 [ q 0 ( q 0 ( q 0 )) 35s 0 ] + 8(3 + 13s 0 )} z 6 + O(z 7 ) LCDM model and cosmographic parameters H 2 H 2 0 = m (1 + z) 3 +(1 m) j 0 =1

38 Fourth-order cosmographic expansion E.g. 1: K-essence Fifth-order cosmographic expansion

39 E.g. 2: f(r) theories Higher- order deriva/ves theories Two extra parameters E.g. 3: Galileons theories

40 =f ; V ( ) = Rf f (R). (20) R that, as shown in figures R Note 1-4, viable modified gravities (2.10) produce someoccurs. oscilin Ref. [27]. In general, we have shown that the transition to the phantom epoch lations along cosmological evolution, a fact out before for this behavior kind of models Moreover, thethe election of phantom conditions at pointed z = 0 gives an anomalous of the For the model (14), the scalar field and its potential in in Ref. [27]. parameters In general, for we both have models shown that to the phantom epoch occurs. cosmological whenthe z >transition 0. terms of the Ricci scalar become Moreover, the election of phantom conditions at z = 0 gives an anomalous behavior of the parameters fornboth > 0. 4cosmological Scalar-tensor representation ofwhen f (R)z gravity 2 1 models Are viable modified gravities viable? = 1 bn (R/cH0 ) (21), 2 n 2 is equivalent It is Scalar-tensor well known that f (R) gravity a kind of Brans-Dicke theory with a null 4 problem? representation of f (R)togravity [1 + d(r/ch Where is the 0) ] [6, 23] and references therein), kinetic term, and a scalar potential, (see for example, 2 2 n 2 n bch (R/cH ) 1 n + d(r/ch ) It is well known that0 f (R) gravity is equivalent theory with a null 0 0 to a kind of Brans-Dicke do contain a singularity what makes the cosmological evolution singular.2 V (They (R)) =. 4 2 = d2 xn (seeg for R V ( ) [6, + 223]Land kinetic term, and a scalarspotential, example, m.references therein), (4.1) [1 + d(r/ch ) ] 0 By varying the action with respect field between both theories S= d4 xto the g scalar R V ( ), + 2the2 Lrelation (4.1) Scalar-tensor picture: m. In general is it obtained, is not possible to get the explicit expression and f (R) action(2.1) is recovered, of the scalar potential terms the scalar =, the relation between both theories By varying the in action withofrespect to the field scalar Vfield R = V ( ) = (R), f (R) (4.2) V ( ) sinceisthe first expression in (22) isisrecovered, not invertible = (R)R V ( (R)). obtained, and f (R) action(2.1) analyticallywhile for athe general n. Nevertheless, this possible scalar field and the potential areisrelated with the particular f (R) action by, (4.2) An exact case: n=1 R = V ( ) = (R), f (R) = (R)R V ( (R)). for the case n = 1, such that the scalar potential yields = fr (R), V ( (R)) = fr (R) R f (R). (4.3) While the scalar field and the potential are related with the particular f (R) action by, V ( ) = ch02 b + (1 ±fr (R) b(1, ) = d ) V ( (R)) =(22) fr (R) R. f (R). 6 (4.3) genera param Raych Condi H/V ( )!1 1, V bch02. V!Φ"#H02 9 Note that in this case the potential is not uniquely defined, as depicted in Fig. 2. It is straightforward to 9 check!1 R that the sudden singularity, where, occurs for (23) Provid the SE may b contrib tion o lows t directi 3 sudden This is the case Asince H singularity!! V ( ) and the first derivative!10!8!6!4!2 0 is neg Φ of the potential V ( 1), so consequently the assum tion in sudden singularity occurs. Hence, in order tos.construct A. Appleby, R. A. Battye and A. A. Starobinsky, JCAP 1006, 005 (2010), arxiv: Figure 2: Evolution of the scalar potential (22) for n = 1. DSG, Class. Quant.The Grav.singular (2013), arxiv: behaviour lies at = 1. The lower branch is lowing like g