Theory space and stability analysis of General Relativistic cosmological solutions in modified gravity

Abstract Some aspects of two General Relativistic cosmological solutions, an exact $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM-like cosmological solution $$j=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ( j is cosmographic jerk parameter), and a specifically designed toy cosmological solution $$j=1+3\varepsilon (q-1/2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mi>ε</mml:mi> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ( q is cosmographic deceleration parameter, $$0&lt;|\varepsilon |&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mo>|</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>|</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ) that is capable of accommodating a phantom crossing scenario as suggested by DESI DR2, are studied within the context of f ( R ) gravity, by portraying them as a flow in the 2-dimensional theory space spanned by the quantities $$r=\frac{R f'}{f}, m=\frac{R f''}{f'}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>f</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:mi>f</mml:mi> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>R</mml:mi> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>′</mml:mo> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mfrac> </mml:mrow> </mml:math> . For the f ( R ) theories exactly reproducing a background $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM-like expansion history $$j=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , it is shown by means of a cosmographic reconstruction approach that the curvature degree of freedom need not necessarily behave like an effective cosmological constant, and that cosmologies under different possible such theories lead to different possible values of $$\Omega _{m0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> . With the theory space analysis, it is also shown that $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM-mimicking f ( R ) cosmologies that asymptote to General Relativistic $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM in the limit $$q\rightarrow 1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , are prone to instability under small homogeneous and isotropic perturbation, casting a doubt on achieving an exact <jats:al